Parameter identification approaches with
application to different classes of materials
von der Fakultät Maschinenbau
der Technischen Universität Dortmund
zur Erlangung des akademischen Grades
Doktor-Ingenieur (Dr.-Ing.)
genehmigte Dissertation
von
Robin Schulte
aus Wickede (Ruhr)
Vorsitz: apl. Prof. PD Dr.-Ing.
Dipl.-Inform. A. Zabel
Hauptreferent: Prof. Dr.-Ing. habil. A. Menzel
Korreferenten: Prof. B. Kiefer, Ph.D.
Prof. Dr. M.G.R. Faes
Tag der Einreichung: 18.04.2023
Tag der mündlichen Prüfung: 27.10.2023
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Schriftenreihe des Instituts für Mechanik
Herausgeber: Institut für Mechanik
Fakultät Maschinenbau
Technische Universität Dortmund
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© by Robin Schulte 2023
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To my family

Acknowledgements
This work is related to the research I conducted during my time as a research assistant
at the Institute of Mechanics at TU Dortmund University and afterwards between 2017
and 2023. By this point, I would like to thank all the people who supported me during
this time in so many different ways.
At first, I want to thank in particular my doctoral advisor Prof. Andreas Menzel for
the opportunity of graduating in such an outstanding research team. It was a pleasure
to deal with all the different research challenges under his guidance and to benefit from
his inspiring scientific knowledge. He always had time for fruitful discussions in all the
years of our great collaboration.
In addition, I would like to thank Prof. Björn Kiefer for his support, enthusiasm in the
field of mechanics and for accepting to act as the co-referee of this thesis. Furthermore,
many thanks go to Prof. Matthias Faes and Prof. Andreas Zabel for serving as third
referee and chairman of the examination committee, respectively.
I will always remember the incredible time at the institute since it was so enjoyable to
be part of such a group with a remarkable and excellent working atmosphere. Therefore,
I thank Prof. Jörn Mosler for some great discussions and in particular Kerstin Walter
and Christina McDonagh for their incredible support in all organisational aspects and
the planning of our social events. Moreover, I am very grateful for the hardware and
software support provided by Matthias Weiss. Special thanks also go to Dr. Thorsten
Bartel and Prof. Richard Ostwald for their supervision and support in various projects.
In particular, I would like to deeply thank my office partners Dr. Karsten Buckmann
and Isabelle Noll for the incredible working atmosphere and fruitful discussions—not only
of scientific nature but also privately. Furthermore, I would like to thank Dr. Serhat
Aygün, Dr. Alexander Bartels, Dr. Rolf Berthelsen, Dr. Till Clausmeyer, Dr. Dinesh
Dusthakar, Volker Fohrmeister, Tim Furlan, Jan Gerlach, Gian-Luca Geuken, Dilek
Güzel, Fabian Guhr, Marius Harnisch, Dr. Tim Heitbreder, Dr. Raphael Holtermann,
Dr. Tobias Kaiser, Dr. Patrick Kurzeja, Henning Lammen, Alexander Niehüser, Felix
Rörentrop, Dr. Lars Rose, Markus Schewe, Dr. Alexander Schowtjak, Dr. Christian
Sievers, Leon Sprave, Dr. Tobias Waffenschmidt, Tillmann Wiegold, Hendrik Wilbuer
and Carina Witt. Thank you very much for the great time and moments at work and
afterwards.
Furthermore, my thanks go to all members of the research groups FOR 1509 and TRR
188 for the fruitful work and interdisciplinary discussions. In particular, I would like
to thank my S01 project-partners Dr. Alexander Schowtjak, Jan Gerlach, Dr. Richard
Ostwald and Dr. Till Clausmeyer.
i
Finally, I would like to sincerely thank my friends and especially my family—most of
all my parents—for their outstanding support over all the years, especially during the
phases of high workload. They are always there when I need them and make so many
moments in my life special and unforgettable.
Dortmund, October 2023 Robin Schulte
ii
Zusammenfassung
Die vorliegende Arbeit befasst sich mit verschiedenen Strategien der Parameteridentifi-
kation bezüglich der multi-objektiven Optimierung unter Berücksichtigung von integra-
len Größen und Feldgrößen, um effizient Parameter von komplexen Materialmodellen
zu identifizieren, wie beispielsweise Gradienten-erweiterte Schädigungsmodelle. Außer-
dem wird eine hybride Strategie entwickelt, um die Problematik der Bestimmung von
adequaten Startwerten zu überwinden. Zu diesem Zweck wird ein künstliches neuro-
nales Netz mit den simulierten Materialverhalten von diversen Parameterkombinatio-
nen trainiert. Anschließend wird die experimentell gemessene Materialantwort in das
Netz eingegeben um eine Vorhersage der Parameter zu erhalten, die im Anschluss als
qualitativ hochwertiger Startwert für eine multi-objektive Parameteridentifikation ver-
wendet wird. In dieser Arbeit werden die hybride und die weiteren Strategien unter-
sucht unter der Verwendung von verschiedener komplexer Materialmodelle und diver-
ser Gruppen von Materialien. Zusätzlich wird ein Gradienten-erweitertes, mit Visko-
elastizität unter finiter Dehnung gekoppeltes Schädigungsmodel entwickelt um effizient
Schädigungseffekte in ratenabhängigen Materialien abzudecken. Des Weiteren wird im
Kontext eines Laminat-basierenden Modells für ferroelektrische Materialien eine nume-
rische Untersuchung bezüglich numerisch effizienter Fischer-Burmeister Ansätze durch-
geführt, um die häufig auftretenden Karush-Kuhn-Tucker Konditionen zu lösen.
Abstract
This thesis deals with different parameter identification strategies regarding multi-objective
optimisations including integral and field data in order to efficiently identify parame-
ters of complex material models, e.g. gradient-enhanced damage models. In addition,
to overcome the difficulty of finding appropriate starting values, a hybrid strategy is
developed. For this purpose, an artificial neural network is trained with simulated ma-
terial behaviours of various parameter combinations. Subsequently, the experimentally
measured response is fed into the network to obtain a parameter prediction which is
afterwards employed as qualitative starting value for a subsequent multi-objective pa-
rameter identification. In this work, the hybrid and the other strategies are investigated
by employing different complex material models and various material groups. Moreover,
a gradient-enhanced damage model coupled to finite strain viscoelasticity is developed
to efficiently cover damage effects in rate-dependent material responses. Furthermore,
in the context of a laminate-based model for ferroelectric materials, a numerical inves-
iii
tigation is performed regarding numerically efficient Fischer-Burmeister approaches, to
solve the frequently occurring Karush-Kuhn-Tucker conditions.
iv

vi
Publications
Key parts of this thesis are based on peer-reviewed journal articles, which were either
published or submitted during the progress of this thesis.
1. T. Bartel, R. Schulte, A. Menzel, B. Kiefer, B. Svendsen: Investigations on enhanced
Fischer-Burmeister NCP functions: application to a rate-dependent model for ferro-
electrics, Archive of Applied Mechanics, 89:995-1010, 2019 [6].
2. R. Schulte, R. Ostwald, A. Menzel: Gradient-Enhanced Modelling of Damage for
Rate-Dependent Material Behaviour—A Parameter Identification Framework, Mate-
rials, 13(14),3156, 2020 [110].
3. A. Schowtjak, R. Schulte, T. Clausmeyer, R. Ostwald, A.E. Tekkaya, A. Menzel:
ADAPT — A Diversely Applicable Parameter Identification Tool: Overview and full-
field application examples, International Journal of Mechanical Sciences, 213:106840,
2022 [104].
4. R. Schulte, C. Karca, R. Ostwald, A. Menzel: Machine learning-assisted parameter
identification for constitutive models based on concatenated loading path sequences,
European Journal of Mechanics - A/Solids, 98:104854, 2023 [113].
For all of the listed articles, the author of this thesis contributed essential parts consid-
ering the underlying theories, the algorithmic formulations and the preparation of the
articles. Considering the peer-reviewed article [104], the author of this thesis together
with the co-author Alexander Schowtjak mainly elaborated the theory and the implemen-
tation of the parameter identification tool ADAPT, which was published open-source, as
well as the preparation of the paper. Specifically, the author of this thesis contributed to
general sections of the article, e.g. introduction, methods or conclusion, and was respon-
sible for the application of ADAPT to a gradient-enhanced damage formulation coupled
to finite plasticity (Section 3.4 of the paper) and to visco-elastic damage (Section 4.1 of
the paper). For the peer-reviewed journal articles [6, 110, 113], the author of this thesis
contributed essential aspects with regard to the outline of the theory, carried out all of
the numerical implementations and simulations, and prepared the articles.
vii
During and alongside the preparation of this work, the following publications were com-
pleted
1. D.J. Hartl, B. Kiefer, R. Schulte, A. Menzel: Computationally-efficient modeling
of inelastic single crystal responses via anisotropic yield surfaces: Applications to
shape memory alloys, International Journal of Solids and Structures, 136-137:38-
59, 2018 [30].
2. R. Schulte, A. Menzel, B. Svendsen: A Laminate-Based Material Model Applied to
Ferroelectrics, Proceedings of the 3rd Seminar on The Mechanics of Multifunctional
Materials, 18:105-108, 2018 [109].
3. R. Schulte, T. Bartel, A. Menzel, B. Kiefer, B. Svendsen: Investigations on dif-
ferent Fischer-Burmeister functions applied to the modelling of ferroelectrics, Pro-
ceedings in Applied Mathematics and Mechanics, 18:e201800331, 2018 [108].
4. R. Schulte, R. Ostwald, A. Menzel: A computational framework for gradient-
enhanced damage – implementation and applications, Proceedings in Applied Math-
ematics and Mechanics, 20:e202000215, 2021 [111].
5. A. Schowtjak, S. Wang, O. Hering, T. Clausmeyer, J. Lohmar, R. Schulte, R. Ost-
wald, G. Hirt, A.E. Tekkaya: Prediction and analysis of damage evolution during
caliber rolling and subsequent cold forward extrusion, Production Engineering: Re-
search and Development, 14:33-41, 2020 [103].
6. K. Langenfeld, A. Schowtjak, R. Schulte, O. Hering, K. Möhring, T. Clausmeyer,
R. Ostwald, F. Walther, A.E. Tekkaya, J. Mosler: Influence of anisotropic damage
evolution on cold forging, Production Engineering: Research and Development,
14:115-121, 2020 [59].
The contributions in the proceedings [108, 109, 111] are related to conference participa-
tions of the author of this thesis. Regarding the content of this thesis, the proceedings
[108, 109] are preliminary work which is included in the peer-reviewed article [6] and
the proceeding [111] is included in the peer-reviewed article [110]. The author of the
thesis contributed basic aspects regarding the outline of the theory, preparation of the
slides, carried out the talk, executed the corresponding numerical implementations as
well as the simulations and generated the contributions. With regard to the contribu-
tion [103], the author of this thesis only contributed to the preparation of the article
and considering the contribution [59], the author only assisted in the preparation and
thus, both works are not included in this thesis. Regarding the article [30], the author
contributed to the numerical implementations and preparation of the article, mainly to
the Sections 4.3, 4.4, Appendix B and Appendix C. However, this article is not included
in this thesis.
viii
Contents
Notation xiii
1 Introduction 1
1.1 Motivation and state of the art . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Ferroelectric materials and non-linear complementarity problems . 1
1.1.2 Gradient-enhanced continuum damage formulations . . . . . . . . 3
1.1.3 Parameter identification . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Objective of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Classic parameter identification based on homogeneous states of defor-
mation 13
2.1 Gradient-enhanced damage model coupled to plasticity . . . . . . . . . . 14
2.1.1 Large strain gradient-enhanced ductile damage formulation . . . . 14
2.1.2 Parameter ranges . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.3 Parameter influences . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Finite-strain viscoelasticity coupled with gradient-enhanced continuum
damage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2.1 Finite hyperelasticity with volumetric-isochoric decomposition . . 20
2.2.2 Extension of finite hyperelasticity to finite viscoelasticity . . . . . 21
2.2.3 Incorporation of damage . . . . . . . . . . . . . . . . . . . . . . . 22
2.2.4 Hyperelastic material model . . . . . . . . . . . . . . . . . . . . . 24
2.2.5 Damage function . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Laminate-model for ferroelectric materials . . . . . . . . . . . . . . . . . 26
2.3.1 Constitutive model . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3.2 Fischer-Burmeister-based algorithms . . . . . . . . . . . . . . . . 29
2.3.2.1 Original Fischer-Burmeister approach . . . . . . . . . . . 30
2.3.2.2 Extended Fischer-Burmeister-type NCP functions . . . . 34
2.3.3 Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.3.1 Comparison of different approaches at t= tI =0.8 s . . . . 36
2.3.3.2 Comparison of different approaches at t= tII =1.1 s . . . 38
2.3.3.3 Analyses with respect to f IV and fVFB FB . . . . . . . . . . . 40
2.3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.4 Classic non-linear optimisation algorithms . . . . . . . . . . . . . . . . . 43
2.4.1 Unconstrained optimisation algorithms . . . . . . . . . . . . . . . 45
ix
Contents
2.4.2 Constrained optimisation algorithms . . . . . . . . . . . . . . . . 48
2.5 Parameter identification for homogeneous states of deformation using
classic optimisation methods . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.5.1 Finite plasticity model coupled to damage . . . . . . . . . . . . . 49
2.5.2 Finite-strain viscoelasticity model coupled to damage . . . . . . . 54
2.5.2.1 Homogeneous deformation tests . . . . . . . . . . . . . . 55
2.5.2.2 Parameter identification . . . . . . . . . . . . . . . . . . 58
2.5.3 Laminate-based model for ferroelectric materials . . . . . . . . . . 61
3 Machine-learning assisted parameter identification based on homogeneous
states of deformation 67
3.1 Machine-learning assisted parameter identification introduced for a gradient-
enhanced damage formulation . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.1.1.1 Tension test . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.1.1.2 Compression test . . . . . . . . . . . . . . . . . . . . . . 73
3.1.1.3 In-plane torsion test . . . . . . . . . . . . . . . . . . . . 73
3.1.2 Neural network for parameter identification . . . . . . . . . . . . 74
3.1.2.1 Sampling of training data . . . . . . . . . . . . . . . . . 74
3.1.2.2 Random sampling . . . . . . . . . . . . . . . . . . . . . 75
3.1.2.3 Latin Hypercube Sampling . . . . . . . . . . . . . . . . 76
3.1.2.4 Structure of the neural network . . . . . . . . . . . . . . 77
3.1.3 Training of the neural network . . . . . . . . . . . . . . . . . . . . 77
3.1.3.1 Training and testing of the neural network . . . . . . . . 78
3.1.3.2 Optimisation of the hyperparameters . . . . . . . . . . . 81
3.1.4 Parameter identification for homogeneous states of deformation . 85
3.1.4.1 Network based on tensile data . . . . . . . . . . . . . . . 86
3.1.4.1.1 Steel DP800 . . . . . . . . . . . . . . . . . . . . 86
3.1.4.1.2 Steel 16MnCrS5 . . . . . . . . . . . . . . . . . 89
3.1.4.2 Two networks each based on two experiments . . . . . . 92
3.1.4.2.1 Steel DP800 . . . . . . . . . . . . . . . . . . . . 92
3.1.4.2.2 Steel 16MnCrS5 . . . . . . . . . . . . . . . . . 93
3.1.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.2 Machine-learning assisted parameter identification applied to a laminate-
based material model for ferroelectrics . . . . . . . . . . . . . . . . . . . 97
4 Parameter identification based on inhomogeneous states of deformation 105
4.1 Parameter identification tool for multi-objective optimisations . . . . . . 105
4.1.1 Optimisation process . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.1.2 Postprocessing and interpolation . . . . . . . . . . . . . . . . . . 108
4.1.3 Multi-objective optimisation . . . . . . . . . . . . . . . . . . . . . 110
4.1.4 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 112
x
Contents
4.1.5 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2 Application to self-diagnostic poly(dimethylsiloxane) elastomer . . . . . . 114
4.2.1 Inhomogeneous deformation tests . . . . . . . . . . . . . . . . . . 114
4.2.2 Parameter identification based on inhomogeneous states of defor-
mation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.3 Application to the sheet metal DP800 . . . . . . . . . . . . . . . . . . . . 120
4.3.1 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3.2 FE-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.3.3 Parameter identification of a plasticity model . . . . . . . . . . . 124
4.3.4 Parameter identification of gradient-enhanced material models . . 127
4.3.5 Starting value generated via machine-learning assisted parameter
identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5 Concluding remarks 141
A Numerical time integrations and algorithmic pseudo-code of the finite-
strain viscoelasticity 147
A.1 Numerical time-integration of the local constitutive relations . . . . . . . 147
A.2 Algorithmic pseudo-code . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
B Derivation of the smoothed Fischer-Burmeister nonlinear complementarity
problem function 153
Bibliography 155
xi

Notation
Within this work, a consistent notation is employed which is explained by the context.
Nevertheless, some of the basic relations are summarised in the following for reasons of
clarity and comprehensibility.
Tensors The tensors of first, second and fourth order can be expressed in terms of
their coefficients by employing a representation of the Cartesian basis vectors {ei}, with
i = 1, 2, 3 spanning the three-dimensional Euclidean space R3, namely
v = vi ei ,
T = Tij ei ⊗ ej ,
S = Sijkl ei ⊗ ej ⊗ ek ⊗ el .
In this work, Einstein’s summation convention is applied to obtain a clear notation and
typically non-bold letters are employed for scalars, bold-face lower-case italic letters for
first-order tensors (i.e. vectors), bold-face upper-case italic letters for second-order ten-
sors and bold-face upper-case sans-serif letters for fourth-order tensors.
Inner tensor products The contractions of inner tensor products are depicted by
dots and the number of dots specifies the number of contractions, namely
v · u = vi ui ,
T · v = Tij vj ei ,
T ·K = TijKjk ei ⊗ ek ,
T : K = TijKij ,
S : T = Sijkl Tkl ei ⊗ ej .
If the number of contractions between two tensors is of the same level as the order of
these tensors, a scalar is always obtained as result.
xiii
Notation
Outer tensor products Within this work, not only the standard outer tensor or
dyadic products, respectively, indicated with the symbol ⊗ are employed, but also the
non-standard outer tensor products ⊗ and ⊗, specifically
v ⊗ u = vi uj ei ⊗ ej ,
T ⊗K = TijKkl ei ⊗ ej ⊗ ek ⊗ el ,
T ⊗K = TikKjl ei ⊗ ej ⊗ ek ⊗ el ,
T ⊗K = TilKjk ei ⊗ ej ⊗ ek ⊗ el .
In summary, while the result of the dyadic product of two first-order tensors is a second-
order tensor, a forth-order tensor is obtained by the dyadic products of two second-order
tensors. Furthermore, the non-standard outer tensor products can be depicted in an
alternative formulation as
[T ⊗K ] : L = T ·Lt ·Kt ,
[T ⊗K ] : L = T ·L ·Kt ,
[T ⊗v ] · u = T · u⊗ v ,
with T t denoting the transpose of T . Moreover, the symmetric and skew-symmetric
part of a second order tensor T are defined as
T sym
1 1
= [T + T t ] = [Tij + Tji ] ei ⊗ ej ,
2 2
T skw
1 1
= [T − T t ] = [Tij − Tji ] ei ⊗ ej .
2 2
Identity tensors The second-order identity tensor I is defined in terms of the Kro-
necker delta δij as {
I = δij ei ⊗
1, if i = j
ej , with δij = .
0, if i 6= j
The fourth-order identity tensor I and the symmetric part of the fourth-order identity
tensor Isym are denoted as
I = I⊗I ,
sym 1I = [I ⊗ I + I ⊗ I] .
2
xiv
Operators The employed gradient and divergence operations are denoted in index
notation following the right-gradient and right-divergence definition—here exemplarily
given for second-order tensors—as
∇ ∂ ∂TijXT = [Tij ei ⊗ ej]⊗ ek = ei ⊗ ej ⊗ ek ,
∂Xk [ ∂Xk ]
∇ · ∇ ∂Tij ⊗ ⊗ ⊗ ∂TilX T = XT : I = ei ej ek : [ el el ] = ei ,
∂Xk ∂Xl
where the subscript of the Nabla operator ∇• typically relates to placements referred to
the corresponding configuration.
xv

1 Introduction
The thesis at hand is based on the research which was conducted at the Institute of
Mechanics during the work in two research projects. One part of the research was car-
ried out within project P6 Microstructural Interactions and Switching in Ferroelectrics
of the research group FOR 1509 Ferroic Functional Materials: Multi-scale Modeling
and Experimental Characterization. The second part was performed within project
S01 Scientific Service Project - Model integration of the transregional research group
CRC/Transregio 188 Damage Controlled Forming Processes. In both projects, a col-
laboration between the scientific research areas of process technologies, characterisation
and modelling was essential. The first project dealt with the microstructural evolution
of the domain switching in ferroelectrics, the second project covered the damage evolu-
tion during metal forming processes. The research in this work was initiated within the
projects and further extended to this final thesis.
1.1 Motivation and state of the art
While the computational capabilities have increased massively over the recent decades,
the requirements of the manufactured products have simultaneously equally increased,
especially with respect to energy efficiency in all directions, e.g. in the manufacturing
process as well as regarding the whole life cycle. Thus, the need for different materi-
als and more complex material models covering more material mechanisms rises which
results in a larger amount of material parameters. Furthermore, the simulations have
to predict the material behaviour realistically in order to optimise the products in the
preferred directions. To this purpose, it is important that the model parameters are
accurately calibrated. Even though parameter identification (PI) has been known for
decades, and as a common tool, there is still ongoing improvement—especially with
regard to machine learning.
1.1.1 Ferroelectric materials and non-linear complementarity
problems
During the last decades smart materials have gained a high relevance in many engineer-
ing applications, especially in the area of sensing and actuating. These materials pos-
1
1 Introduction
(a) The unit cell of (BaTiO3) in the paraelectric (b) Two-dimensional view of a crystallographic aggre-
phase—due to the symmetry—on the left and in the gate which was cooled below the Curie temperature.
ferroelectric phase on the right. The orientation of Domains as subregions of equal spontaneous polarisa-
Pspon, the spontaneous polarisation vector, is defined tion are visible, such that macroscopically an isotropic
by the displaced titanium ion. Reprinted from M. state without any polarisation occurs. Reprinted from
Kamlah, Continuum Mech. Thermodyn. 13 (2001), M. Kamlah, Continuum Mech. Thermodyn. 13 (2001),
219-268,[46], with permission from Springer Nature. 219-268, [46], with permission from Springer Nature.
Figure 1.1: The piezoelectric property of the materials is based on the crystallographic microstructure.
The crystallographic effects are briefly indicated in both two figures. For further details, please be
referred to Kamlah [46], Lines and Glass [68], Tagantsev et al. [125].
sess the special property that a mechanical loading is transferred into a non-mechanical
material response and vice versa. Piezoelectric materials are a subgroup of this class
of material and are applied if a fast electro-mechanical material response is required,
e.g. in sonar, acoustics and ultrasound and precision actuators, see e.g. [37, 68]. Their
material behaviour is based on the polarisation of the crystal structure. In addition,
ferroelectric materials lose their piezoelectric properties above a certain temperature,
the Curie-temperature. Below this temperature, a spontaneous polarisation takes place
based on the unit cell structure where the centres of the positive and negative charges no
longer coincide, see Figure 1.1a. Among the most frequently used piezoelectric materials
are barium titanate (BaTiO3) and lead zirconate titanate (PZT), which mostly occur
as polycrystals consisting of different grains with differently oriented crystals. Within
these microstructures different domains with uniform spontaneous polarisation direc-
tions exist. Considering a tetragonal crystal structure below the Curie-temperature, six
different spontaneous polarisation directions are possible. The composition or, say, the
mixture of these different phases of the material governs the averaged polarisation and,
as a consequence, the macroscopic material behaviour, see Figure 1.1b. For more infor-
mation on this topic, the reader is referred to, e.g., [40, 46, 68, 125].
Such a material behaviour can be predicted by using mathematically sound material
models that fulfil the thermodynamic equations. These models can be divided into three
2
1.1 Motivation and state of the art
different groups. On the one hand, phenomenological models with continuous evolution
equations can be formulated to characterise the macroscopic material response, see,
e.g. [46, 71, 80], while, on the other hand, micromechanically motivated models can be
applied which are based on unit cell switching, see, e.g. [36, 39, 65, 67, 115]. Further-
more, phase field or level set approaches can be applied if the microstructure needs to
be resolved in space and time, see, e.g. [45, 82, 105].
Laminate-based material models are dedicated to the micromechanical formulations, see,
e.g., the review provided in [66]. Single-crystalline ferroelectric materials offer high mag-
nitudes of actuation strains, the predictions of which can be achieved via this kind of
approach. In [65], energy minimising multi-rank laminates for ferroelectric single crystals
were introduced. Several different laminate approaches can be found in [129, 130, 136].
In [16, 17] a laminate-based model was developed, which is capable of simulating rate-
and external stress-dependent domain evolution via suitable Gibbs energy and dissipa-
tion functions. Due to the fact that the underlying phase fractions are subjected to phys-
ically motivated constraints, the overall framework includes several Karush-Kuhn-Tucker
(KKT) conditions. In this regard, nonlinear complementarity problem (NCP) functions
were shown to provide sophisticated schemes to fulfil KKT conditions as an alternative
to, e.g., active-set strategies. As one of the most frequently used NCP approaches, the
Fischer-Burmeister functions introduced in [19] have been applied to problems in solid
mechanics in [102] in the context of rate-independent crystal plasticity. Later, this ap-
proach was used in, e.g., [4, 5] in the context of material models for shape memory alloys
based on energy relaxation concepts which involves both KKT conditions due to rate-
independent evolution equations and due to mass constraints. In general, this approach
is also applicable to different problems including inequality constraints in the context of
contact mechanics.
However, it is well known that the original Fischer-Burmeister approach is prone to
numerical instabilities, in particular due to the fact that the related function is not
continuously differentiable everywhere. Therefore, several authors have suggested im-
provements of the original ansatz, e.g. in [12, 18, 47].
1.1.2 Gradient-enhanced continuum damage formulations
The adequate numerical prediction of material behaviour under complex loads and
boundary conditions is of high significance in the product development context, par-
ticularly in view of today’s trends towards more efficient and cost-effective production,
but also in view of component safety requirements and appropriate component lifetime
predictions. To this end, not only the material model formulation itself needs to be
physically sound and able to capture all relevant material phenomena, but the corre-
sponding model parameters also need to be identified properly. This process of model
calibration is typically carried out with the help of optimisation-based parameter iden-
tification schemes which is described in the next section. Since the accurate prediction
of many material mechanisms became more and more important in many engineering
3
1 Introduction
applications, effects like damage have to be simulated properly especially with respect
to metal forming processes. However, it is not only necessary to identify the point of
failure but more importantly to characterise the evolution of damage prior to failure in
order to e.g. control the damage evolution—and thus the material properties—during
metal forming processes.
The first development regarding damage mechanics was accomplished by Kachanov
[41]. In his work and in many following contributions in the field of classic damage me-
chanics, damage was considered as a local effect which was described by a single scalar
damage variable reducing the effective area which induces material degradation. Fur-
ther information regarding the classic damage mechanics can be found in [42, 55, 56, 64].
Nevertheless, the local character of these models does not consider the complex nature
of damage evolution, e.g. its anisotropic property. To overcome this difficulty, subse-
quent works incorporated the anisotropic effect by using a tensorial damage variable,
see e.g. [48, 61] and the references cited therein. Since the characterisation of damage is
important in metal forming processes, ductile damage models have a broad application
range. These models are either phenomenological damage models, e.g. the Lemaitre-
type damage models—initially developed by Lemaitre [62, 63] and since then extended
in several directions by many researchers—or damage models based on micromechanics,
e.g. the well known Gurson model [29] which was extended to the popular GTN-model
by Tvergaard and Needleman [131] and many further variants in the literature.
In addition, the local damage formulations have mesh dependency in common which
is a massive disadvantage, since the finer the mesh, the more the damage region shrinks
and even might vanish in the end. Alternatively, non-local damage formulations were
introduced which can be categorised based on the extension type of the formulation,
i.e. whether they have an integral or gradient shape. However, following Lasry and Be-
lytschko [60], Polizzotto et al. [90], gradient formulations offer various advantages such
that gradient-enhanced damage frameworks are more common in the literature. In [133],
Waffenschmidt et al. developed a gradient-enhanced damage model for fibre-reinforced
materials and large deformations and presented an implementation in Abaqus. The
scalar [1 − d]-type damage formulation with the local damage variable d only affects
the anisotropic elastic response and the local free energy is then enhanced by a gradi-
ent contribution—following the approach of Dimitrijevic and Hackl [15]—namely, the
gradient of the additional independent non-local damage variable. A penalty term is
additionally introduced to ensure the equivalence of the local and non-local damage
variable. This damage framework was further developed in Polindara et al. [88, 89] and
Kiefer et al. [51]. However, the underlying framework of these works is based on a user
element subroutine for the finite element program Abaqus. A user element formulation
in Abaqus has various disadvantages, i.e. a high effort in developing and testing, the
visualisation is not straight forward and requires some workarounds, additional element
features like contact require further difficult extensions and the need of different element
types requires a completely new implementation. These disadvantages were overcome
by Ostwald et al. [86] with a damage regularisation framework which employs the heat-
4
1.1 Motivation and state of the art
equation solution capabilities of Abaqus for the damage regularisation. Thus, the model
can be implemented as a user material subroutine avoiding the drawbacks of the pre-
viously mentioned approaches. This framework is in line with the works of Forest [20]
and Miehe [83]. Sprave and Menzel [119] further extended this framework and com-
puted complex boundary value problems based on a finite plasticity model coupled to a
gradient-enhanced damage formulation. For further, advanced concepts in the context
of gradient damage, see, e.g., the work by Liu and Jeffers [69] and the references cited
therein.
The modelling of damage effects is still an ongoing challenge not only with respect
to metals and its corresponding forming processes, but also with regard to many dif-
ferent material groups. It is for example highly relevant in biological applications and
Polindara et al. [88, 89], Waffenschmidt et al. [134] developed gradient-enhanced dam-
age formulations—on the basis of the already mentioned framework by Waffenschmidt
et al. [133]—in order to simulate damage effects and its evolution in biological appli-
cations, e.g. blood vessels. Not only biological applications have recently become more
important, but also alternative materials in many engineering applications have gained
popularity, e.g. in the automotive industry due to the need of lightweight components.
In this regard polymers are employed in many engineering applications. For example,
thermoplastic polymers are requested due to their good thermomechanical properties
and simple manufacturing processes, cf. [7]. Another interesting and promising material
is an elastomer with self-diagnostic capabilities. Self-diagnostic poly(dimethylsiloxane)
(PDMS) elastomers are produced by the addition of so-called mechanophore units which
generate a chemical response towards mechanical loads, as discussed by, e.g., Brighenti
and Artoni [11]. The specific PDMS elastomer containing a supramolecular detection
probe was presented and developed by Früh et al. [22]. In this case, the elastomer re-
acts with strain-dependent levels of fluorescence when illuminated with UV light, thus
enabling an in situ quantification of the material’s strain and, potentially, damage state.
Thus accurate constitutive models capturing various material mechanisms are required.
Different material models allow for the description of the mechanical behaviour of
rate-dependent—or rather viscoelastic—materials such as the previously mentioned elas-
tomer, mostly using either a convolution integral approach or a differential operator form
of the constitutive relation. To name a few, Reese and Govindjee [92], for example, dis-
cussed a non-linear viscoelastic model that applies a non-linear evolution law that is
valid for arbitrary deviations from thermodynamic equilibrium. Bergström and Boyce
[8] proposed a constitutive model for the time-dependent behaviour of elastomers, moti-
vated micromechanically by the relaxation of a single entangled chain in a polymer gel.
Simo and Hughes [117] extended the well-established finite hyperelasticity framework
with uncoupled volumetric-isochoric response towards viscous effects. In addition, Simo
[116] already developed a finite strain viscoelastic damage model which incorpoartes a
simple isotropic damage mechanism. Mergheim et al. [78] developed a constitutive model
containing the curing in thermosetting materials combined with an isotropic gradient-
enhanced damage model covering the process-induced damage evolution.
5
1 Introduction
1.1.3 Parameter identification
The manufacturing of modern engineering components requires the accurate prediction
of material properties, such as accumulation of damage and plasticity resulting from, e.g.,
metal forming processes applied to a workpiece. A full exploitation of the lightweight
potential of modern materials and components requires not only appropriate material
models, but also a precise calibration of such models. To this end, optimisation-based
parameter identification schemes have been established in the literature, with important
works on the subject including the contributions by, e.g., Kublik [58], Furukawa [23],
Gelin [25], Mahnken [72–74, 76], Hartmann [31–33], and Kleuter [52, 53]. The classic,
optimisation-based approaches include schemes for model calibration based on homoge-
neous and on inhomogeneous deformations, where the latter incorporate digital image
correlation (DIC) methods.
The work of Mahnken and Stein [73] was a pioneering contribution in this field
of research providing the theoretical and numerical aspects regarding the model cali-
bration of metals. Further application examples were given by Mahnken in [74]. In
Mahnken and Kuhl [75], a parameter identification framework for a gradient-enhanced
damage model was introduced. However, since at the beginning only integral data,
such as forces, was available for the consideration in the objective function, full field
data, e.g. displacement fields, could not be considered to capture anisotropic effects
or inhomogeneous deformation states. This became a common approach a few years
later after the concept of DIC was well known, e.g. by Scheday and Miehe [100] or
Kajberg and Lindkvist [43]. The incorporation of field data required a special treat-
ment, e.g. regarding the interpolation, which was investigated in detail in the works of
Kleuter et al. [53], Rieger [93], Scheday [101]. In [53], Kleuter et al. introduced a param-
eter identification scheme for finite viscoelastic material models where an error square
functional—comparing experimentally-obtained inhomogeneous displacement fields with
related finite element-based simulations—is minimised by using a gradient scheme. This
gradient-based calibration process for finite viscoelasticity was studied in more detail and
applied to different materials in [52]. The identification of material parameters for inelas-
tic materials, using experimental data representing non-uniform stress and displacement
distributions from the surface of the three-dimensional specimen, was established in the
work of Mahnken and Stein [76]. An application of a parameter identification scheme for
different constitutive models was presented in [72] where different identification meth-
ods, instabilities in least squares problems, and identification for small and non-uniform
finite deformations were discussed. Hartmann [31] estimated the constitutive constants
of hyperelasticity relations of the generalised polynomial-type and discussed the as-
pects related to the gradient and convexity behaviour in certain deformations. The
particular optimisation methods that are generally applicable were illustrated in, e.g.,
[2, 70, 85, 91, 123]. More recent contributions on the subject of parameter identification
include works that deal with full-field DIC including, e.g., sensitivity analyses [34] and
temperature phenomena [95, 97].
6
1.2 Objective of this work
Parameter identification methods based on full-field measurements can be categorised
into different classes. The most commonly used is the Finite Element Model Updating
(FEMU) method where a finite element simulation is performed within each iteration,
and where the difference between the simulated and experimentally measured data is
calculated in order to find the optimal solution, see e.g. [49, 72, 73]. Another method
is the Virtual Fields Method (VFM) based on appropriate virtual field sets and the
principle of virtual work, see e.g. [27, 98], though it is often difficult to gather the
deformation data over the whole volume. For further information, the reader is referred
to the overviews provided in, e.g., [3, 99] and references cited therein.
In order to improve the overall calibration procedure, the concept of artificial neural
networks (ANN), which recently became very popular in all kinds of research fields, was
integrated into the optimisation process. One of the first works in machine learning-
assisted parameter identification for constitutive models has been established by Huber
et al. [38]. Among the more recent works dealing with machine learning methods in the
context of parameter identification, Ktari et al. [57] carry out parameter identification
for anisotropic thin walled-tubes based on finite element analyses and supported by a
neural network scheme. The work of Yao et al. [135] deals with a rate-dependent phe-
nomenological model including coupled plasticity and damage. The authors investigate
the effect of different rates of deformation as well as temperature levels and then estab-
lish a hybrid approach for parameter identification, combining a neural network with
a subsequent genetic algorithm, where a single mode of deformation—namely a tensile
experiment—is considered.
Zhang et al. [141] do not invoke a neural network, but instead a hybrid strategy in-
volving a two-stage process in the context of parameter identification for thin-walled
aluminium tube models. They approximate a starting value for the optimisation algo-
rithm based on theoretical considerations, then carry out the optimisation-based solution
of the inverse problem. The work highlights that proper starting values for the optimi-
sation procedure are essential, but in some cases challenging to obtain.
The recent work of Guo et al. [28] presented a deep learning model, named CPINet,
which combines a convolutional neural network for denoising processing and strain fea-
ture extraction and a long short-term memory neural network for identifying path-
dependent constitutive model parameters based on strain field sequences, loads and ge-
ometry dimensions. As an example, the developed model is applied to an elastic-plastic
constitutive model with isotropic hardening.
1.2 Objective of this work
The main objective of this work is the investigation of different parameter identification
strategies which are modified, i.e. optimised, for the underlying material model in order
to obtain the optimal parameter set and to introduce a framework which overcomes
the general difficulty of finding appropriate starting values for the identification process
7
1 Introduction
in general. For this purpose, as application examples, three different complex material
models are chosen based on the variety of different captured material mechanisms as
well as on the broad distribution over various material groups. The first application
example is the gradient-enhanced damage model coupled to finite plasticity developed
by Sprave and Menzel [119] which provides many advantages regarding the simulation
of e.g. metal forming processes and which is based on the efficient gradient-enhanced
damage framework by Ostwald et al. [86]. Since the underlying damage framework by
Ostwald et al. provides numerous advantages over many other damage formulations, and
since damage effects are of high interest in different material groups, the framework in
this work was coupled to a finite strain viscoelasticity model. The basis of the consti-
tutive viscoelasticity model used in this work is a convolution integral approach. The
model is based on the well-established finite hyperelasticity framework with uncoupled
volumetric-isochoric response that was extended towards viscous effects by Simo and
Hughes [117]. The final example is completely different to the previous material models
and covers the group of ferroelectric materials. The laminate-based model was developed
by Dusthakar et al. [16, 17] and couples mechanical and electrical effects.
Besides the investigations regarding parameter identification and developing a finite-
strain viscoelasticity model coupled to gradient-enhanced damage, the work covers a
numerical study with respect to the treatment of KKT conditions by employing differ-
ent Fischer-Burmeister approaches for the NCP functions. This was done since KKT
conditions need to be solved for many material models, and Fischer-Burmeister func-
tions provide an efficient solution scheme. Thus, different approaches are compared in
terms of their effectiveness and efficiency.
Consequently, the work is structured as follows:
In Chapter 2, the classic parameter identification approaches based on homogeneous
states of deformation are performed and investigated. Therefore, three different material
models are introduced. The gradient-enhanced damage model coupled to finite plasticity,
developed by Sprave and Menzel [119], is briefly summarised in Section 2.1 accompanied
by investigations regarding the different parameter influences. In the following section,
the finite strain viscoelasticity model coupled to gradient-enhanced damage is developed
based on the damage framework by Ostwald et al. [86] and the viscoelasticity frame-
work by Simo and Hughes [117]. In Section 2.3, the laminate-model for ferroelectrics,
developed by Dusthakar et al. [16, 17], is briefly summarised and investigated towards
the numerical efficiency of solving the NCP functions by employing different Fischer-
Burmeister approaches. The presentation of the different underlying models is followed
by a brief summary of classic non-linear optimisation approaches from the literature.
Afterwards, in Section 2.5, all three material models are calibrated with respect to ex-
perimental data based on homogeneous deformation states. Three different strategies
are therefore investigated. At first, a Nelder-Mead simplex algorithm is applied in order
to identify all of the material parameters—except for those which are related to non-local
contributions—at once, i.e. elasticity-, plasticity- and damage-related parameters.
Secondly, an optimisation-based framework is presented that enables the identification
8
1.2 Objective of this work
of parameters for rate-dependent materials under large strains and subject to gradient-
enhanced damage by using an efficient two-step approach. In the first step, basic consti-
tutive parameters, in particular elastic parameters, are calibrated based on experiments
reflecting homogeneous states of deformation and are later fixed in the second step, pre-
sented in Section 4.2. The particular material considered in this case is the self-diagnostic
poly(dimethylsiloxane) (PDMS) elastomer. However, we restrict our elaborations to the
purely mechanical properties of the elastomer, where later solely the inhomogeneous
displacement field and the load-deflection curve are considered within the second step of
the calibration process. A possible future extension of the framework proposed in this
work could include an additional consideration of the fluorescence field emitted by the
self-diagnostic specimen within the optimisation-based parameter identification proce-
dure.
Thirdly, a very efficient strategy is chosen—which is however often not applicable. Since
most of the material parameters of ferroelectric material models are characteristic for
this class of material, they can be directly taken from the literature, in this case for
BaTiO3. Thus, only the two remaining model parameters had to be calibrated with
respect to the characteristic butterfly and dielectric hysteresis curves. The parameter
identification in this section was performed by Dusthakar et al. [17] and is only briefly
summarised.
Nevertheless, a particular challenge that all these different strategies have in common
is the identification of a proper starting value for the optimisation scheme. Profound
knowledge is required of the interplay between constitutive parameters, evolution equa-
tions, and the final constitutive response reflected by a given material model in order to
pre-determine a set of suitable starting values for the constitutive parameters involved.
This is a challenge that becomes all the more relevant with modern, sophisticated and
complex material models that might include dozens of parameters relating to effects
such as damage, plasticity, temperature-dependence, microstructural information, etc.—
which is applicable for all three application examples. Thus, in Chapter 3, we aim to
overcome this challenge by a machine learning-scheme which automatically generates a
set of high-quality starting values for the subsequent classic optimisation scheme. The
main contribution of this work is the overall hybrid framework where the starting point
for a full-field data based parameter identification is generated by an ANN which is
trained with concatenated loading path sequences and not by the identification of the
material parameters of a particular material. While the general structure of the approach
is presented at the beginning of the chapter, the detailed description and investigation
is given by directly employing the finite plasticity model coupled to gradient-enhanced
damage as an example. Some comments on the selected neural network including the
data sampling strategy and the network structure are presented in Section 3.1.2. The
training, testing, statistical error quantification and hyper parameter optimisation of the
neural network is then presented in Section 3.1.3, before the parameter identification re-
sults for two types of metals—a DP800 sheet metal and a 16MnCrS5 case hardening
steel—are presented in Section 3.1.4 based on experiments with homogeneous deforma-
9
1 Introduction
tions. The final prediction of the parameter set is used in Section 4.3.5 as the starting
value for a classic parameter identification based on inhomogeneous states of deforma-
tion to obtain an accurate optimal parameter set. Additionally, the hybrid strategy
is adapted to the laminate model for ferroelectrics in order to demonstrate that the
framework can be easily applied to different material models, finite-element solvers and
material groups with small adjustments. Even though most parameters were directly
obtained from the literature and only two model parameters remained for the parameter
identification, the identification of appropriate starting values was not straight forward.
Thus, the hybrid strategy can be advantageous even for seemingly simple problems.
In the beginning of Chapter 4, the multi-objective optimisation tool ADAPT—A
Diversely Applicable Parameter Identification Tool is briefly summarised. The optimi-
sation tool was published open-source and can deal with all different kinds of finite
element solvers and various kinds of experimental data—even incorporating different
experiments in the objective function at once while maintaining an efficient calibration
due to a parallelisation of the procedures. Thus, the tool is not only able to deal with
integral data, such as forces, but also to incorporate field data, such as displacement
or strain fields, and even microstructural data like pore evolution, e.g. regarding the
characterisation of damage. The presentation of the optimisation process is followed
by a description of the postprocessing and interpolation techniques, the multi-objective
ansatz, as well as the incorporation of boundary conditions and a short description of
characteristic implementation aspects.
Afterwards, the multi-objective optimisation technique is applied to the self-diagnostic
elastomer. While the elastic parameters were already identified previously in a first
step with respect to experimental data based on homogeneous deformation states, the
damage-related parameters are identified considering integral and field data regarding
inhomogeneous deformation states. For this purpose, the already identified elastic pa-
rameters are then kept fixed in view of the second step of model calibration. The inho-
mogeneous states of deformation activate gradient terms within the gradient-enhanced
damage formulation considered in this work, so that mesh-objective damage effects can
be captured within finite element-based simulations. With damage gradient terms acti-
vated due to inhomogeneous states of deformation, we proceed to the identification of
damage-related model parameters in the second step of the model calibration procedure.
Subsequently, ADAPT is applied to calibrate material models for the sheet metal DP800.
At first, a brief summary of the parameter identification for a simple plasticity model,
performed by Schowtjak et al. [104], is presented in combination with an investigation
towards the influence of different optimisation strategies. Afterwards, ADAPT is used
to directly identify all material parameters of the finite plasticity model coupled to
gradient-enhanced damage at once, including the non-local contributions since a multi-
objective optimisation with respect to inhomogeneous deformation states is performed.
Finally, the prediction of the material parameters by the ANN in Section 3.1.4 is used
as a starting value for a subsequent multi-objective parameter identification based on
inhomogeneous states of deformation. This procedure reduces the computational cost
10
1.2 Objective of this work
of a finite-element based multi-objective optimisation to a minimum since the starting
value already lies quite close to the optimal solution. This is especially advantageous if
the user is not familiar with the underlying material model and is unaware of any pa-
rameter range. The solution is compared to the result with an additional intermediate
optimisation step which is employed in order to improve the starting value with regard
to the data based on homogeneous deformation states.
11

2 Classic parameter identification
based on homogeneous states of
deformation
In this chapter, classic parameter identification strategies regarding one objective for
the error functional, e.g. forces taken from load-displacement curves, are applied to dif-
ferent classes of material models. As already mentioned in the introduction, calibration
processes are important in order to apply developed material models to specific mate-
rials, although they can be quite challenging. In general, the better the optimisation is
performed and the closer the found local minimum is to the global minimum, the more
reasonable and realistic the material behaviour can be simulated. Even the best and
most complex material model that captures all different kinds of material effects is not
capable of predicting the material behaviour sufficiently if the model is not calibrated
properly to the specific material. Thus, it is important to invest sufficient effort into the
parameter identification process. To this end, classic calibration strategies are presented
in the following considering at first only homogeneous states of deformation. This type
of information can be provided by standard experiments which are easily available and
already sufficient to identify many material parameters.
At first, three different classes of material models are presented. In Section 2.1, a
gradient-enhanced damage model coupled to finite plasticity developed by Sprave and
Menzel [119, 120] is briefly summarised and some of the model parameters are fitted
in Section 2.5.1. The material model was developed for different kinds of steel and the
focus in this work lies on the sheet metal DP800 and the case-hardened steel 16MnCrS5.
The formulation of this model is based on the gradient-enhanced damage framework
derived by Ostwald et al. in [86]. In Section 2.2, the damage framework was applied to
a different material class—polymers. Thus a finite strain viscoelasticity coupled with
gradient-enhanced continuum damage model was developed in Schulte et al. [110] and the
model formulation is given in Section 2.2. The corresponding parameter identification
regarding homogeneous deformation states is given in Section 2.5.2. The identification of
the damage-related model parameters is given in Chapter 4 where inhomogeneous states
of deformation are taken into account. While both previous material models cover differ-
13
2 Classic parameter identification based on homogeneous states of deformation
ent material classes, they are some way related with respect to their damage framework.
Thus, as another application case, a completely different material model is given in
Section 2.3. Here, a laminate-based electro-mechanically coupled material model for
ferroelectric materials developed by Dushtakar et al. in [16, 17] is summarised. Since
the underlying phase fractions are subjected to physically motivated strains, the consid-
eration of these constraints is quite important for the material model. To this purpose,
Fischer-Burmeister functions are incorporated into the model. In addition, considering
the impact of these functions on the numerical efficiency of the model, different ap-
proaches of the FB functions are analysed in the section. The corresponding parameter
identification following the classic parameter identification scheme was performed by
Dushtakar et al. in [17] and is summarised in Section 2.5.3.
In Chapter 3, an alternative approach to the classic optimisation strategy, employ-
ing machine learning as already described in the introduction, is presented in order to
improve the standard procedure in several directions.
Before the parameter identification is performed in Section 2.5, a brief overview of
the different classic optimisation algorithms is given in Section 2.4.
2.1 Gradient-enhanced damage model coupled to
plasticity
As we aim to apply different parameter identification procedures in the first application
example to different types of metal alloys, a sufficiently generalised constitutive model
needs to be employed. To this end, in this work we employ a constitutive model that is
able to capture large deformations, a full coupling between damage and plasticity, and
where a gradient-based damage regularisation scheme is included, thereby facilitating
the simulation of mesh-objective damage evolution in the context of the finite element-
based solution of inhomogeneous boundary value problems. The particular large-strain
gradient-enhanced ductile damage model utilised in this work is in line with the formu-
lation presented in [119] and is briefly reviewed in the following.
2.1.1 Large strain gradient-enhanced ductile damage formulation
In view of kinematics, let X ∈ B0 denote material placements in a domain B0 reflecting
the reference configuration of a body under consideration and let x = ϕ(X, t) ∈ Bt
denote spatial placements in the spatial domain Bt associated with the current config-
uration at time t. Moreover, F = ∇Xϕ represents the deformation gradient, mapping
infinitesimal referential line elements dX onto their spatial counterparts dx. Assuming
a multiplicative decomposition of the deformation gradient into elastic and plastic con-
tributions, F = F e · F p, the elastic Finger tensor be is expressible as be = F e · [F e]t, so
that spatial logarithmic elastic strains εe follow as εe = ln(be).
14
2.1 Gradient-enhanced damage model coupled to plasticity
The constitutive relations are modeled based on an overall Helmholtz energy poten-
tial Ψ being additively composed of local and non-local contributions, Ψ loc and Ψnl,
respectively, so that
Ψ(F , φ,∇Xφ, dφ,Ip) = Ψ loc(F , d ,Ipφ ) + Ψnl(φ,∇Xφ, dφ) , (2.1)
where dφ and φ denote the local damage variable and the global damage field, respec-
tively, and where Ip reflects plasticity-related internal variables, particularly F p and the
isotropic hardening variable, denoted as α in the following.
The local part of the Helmholtz energy takes into account an isochoric-volumetric
split, where the specific form is chosen as
Ψ loc(F , dφ,Ip) = Ψvol(εe, dφ) + Ψ iso(εe, dφ) + Ψp(α)
K
= fvol(dφ) [tr(ε
e)]2 +Gf iso(d ) εe,iso : εe,isoφ
2 (2.2)
h
+ αnp+1 ,
np + 1
with volumetric, isochoric and plasticity-related—or rather hardening-related—energy
contributions, Ψvol, Ψ iso, and Ψp, respectively, which are further specified in terms of
bulk modulus K, shear modulus G, and hardening-parameters h and np. For details on
the derivation of the isochoric parts of the logarithmic strains, εe,iso, based on a spectral
representation of εe, the reader is referred to, e.g., [119]. The non-local Helmholtz energy
contribution is specified as
nl ∇ cd ∇ · ∇ βdΨ (φ, Xφ, d ) = 2φ Xφ Xφ+ [φ− dφ] , (2.3)
2 2
with cd being a regularisation parameter and βd a penalty-type parameter, see related
works, such as [51, 86, 133]. This gradient-extended format is in line with the micromor-
phic approach discussed by Forest [20] and with the multi-field incremental variational
framework for gradient-extended standard dissipative solids established by Miehe [83].
As driving forces, we consider the Mandel stress tensor frequently employed in finite
plasticity applications, which, in the special case of isotropic elasticity, is symmetric and
can conveniently be represented in terms of m := 2 ∂beΨ · be. Moreover, we obtain an
isotropic hardening stress β := − ∂αΨ and a damage driving force q := − ∂d Ψ .φ
To complete the constitutive fram√ework, we specify the plastic potential Φp as
Φp
2
(meff , α) = ‖ dev(meff) ‖ − h [α0 + α]np , (2.4)
3
15
2 Classic parameter identification based on homogeneous states of deformation
with dev(•) = [•] − 1 [[•] : I]I being the deviator operator in spatial representation.
3
The damage potential Φd is specified as
Φd(q, dφ) = q
q nd
eff − qmin − qvar [1− f (dφ)] , (2.5)
where effective driving forces meff := m/fm(dφ) and qeff := q/fα(α) are introduced
and where qmin and qvar represent damage threshold parameters. The damage functions
f +• : R0 → ]0, 1] are chosen as f•(dφ) = exp(−η ξ• dφ) with η denoting a damage rate
factor, and with ξ• reflecting further damage-related constitutive coefficients. Suitable
parameter ranges and choices for this particular constitutive formulation are provided
in Section 2.1.2. Moreover, Equations (2.4) and (2.5) are combined with classic Karush-
Kuhn-Tucker conditions, together with associative flow directions, in order to model
plasticity and damage evolution, cf. [119].
2.1.2 Parameter ranges
In terms of classic PI approaches several different initial guesses for the parameter sets
need to be chosen. Thus, it is important to identify the possible ranges of each param-
eter. In addition, regarding the machine learning approach in Chapter 3, besides the
method of the sampling of the training data for the neural network shown in Section
3.1.3, the admissible interval of the parameters is also of importance. For some param-
eters the specific range is quite obvious, e.g. for Poisson’s ratio or Young’s modulus.
However, some model parameters do not possess an obvious physical interpretation and
thus the influence of those parameters on the model response needs to be analysed. In
Section 2.1.1, the applied material model is summarised and this influence of the cor-
responding parameters is shown in Figures 2.1 and 2.2 in Section 2.1.3 with respect to
a reference set of material parameters given in Table 2.1. In order to demonstrate the
characteristic behaviour, a simple uni-axial tension test with a homogeneous deforma-
tion state, consisting of a loading, unloading and another loading phase, is simulated.
Therefore, a total axial strain of 13.5 % is applied in loading direction in the first phase
and subsequently the force is reduced to zero. Thereafter, the strain is increased to 32 %.
It has to be mentioned that the shown influence of the parameters in general depends
on the fixed values of the other parameters. Nevertheless, the basic influence can be
demonstrated. With regard to the hybrid strategy of Chapter 3, in particular in view
of the sampling methods of the training data, a vast amount of the various parameter
combinations is taken into account for the neural network. Thus, the network is able to
provide a good prediction for the chosen class of material avoiding over-fitting.
2.1.3 Parameter influences
As already mentioned in the previous section, in order to choose initial values for the
parameter sets or later to define a training set for the neural network, it is important to
16
2.2 Finite-strain viscoelasticity coupled with gradient-enhanced continuum damage
Table 2.1: Values of the material parameters of the reference set which are fixed while one of the
parameters is analysed. For further details on the model formulation see Section 2.1.1. Reprinted from
Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113], with permission under
creative commons licence 4.0.
Symbol Description Reference set Unit
E Young’s modulus 200.0 GPa
ν Poisson’s ratio 0.3 −
σy0 initial yield stress 250.0 MPa
h hardening parameter 1300.0 MPa
np hardening exponent 0.25 −
qvar variable damage threshold 20.0 MPa
nd damage exponent 0.667 −
η damage rate factor 1.0 −
ξvol volumetric damage factor 1.0 −
ξiso isochoric damage factor 1.0 −
ξq threshold factor 10.0 −
ξm effective stress factor 1.0 −
ηα coupling factor 2.5 −
cd regularisation parameter 0.54 N
βd penalty parameter 0.0 MPa
qmin initial damage threshold 10.0 MPa
define the parameter ranges, as described in Section 2.1.2. The influence of the different
parameters on the material response therefore has to be analysed. The influence of the
parameters to be identified can be seen in Figures 2.1 and 2.2. Within the figures, each
parameter is varied with respect to a reference parameter combination shown in Table
2.1.
2.2 Finite-strain viscoelasticity coupled with
gradient-enhanced continuum damage
Viscoelastic material models can describe a large class of history- and rate-dependent
materials. These materials differ from ideally elastic materials by time-dependent effects
such as creep or relaxation. Polymeric rubber materials such as the elastomer considered
in this work are typical representatives of this class of materials. Constitutive models for
viscoelasticity either use a differential operator or convolution integral representation,
where the constitutive model considered in this work is based on the latter approach.
The convolution integral approach was described in detail in the works by Kaliske and
17
2 Classic parameter identification based on homogeneous states of deformation
(a) Influence of Young’s modulus E in MPa. (b) Influence of Poisson’s ratio ν.
(c) Influence of the initial yield stress σy0 in MPa. (d) Influence of the hardening parameter h in MPa.
(e) Influence of the hardening exponent np. (f) Influence of the variable damage threshold qvar in
MPa.
Figure 2.1: Impact of the different material parameters on the simulated load-displacement response
for an exemplary tension test—Part 1. Reprinted from Schulte et al., European Journal of Mechanics
- A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
18
2.2 Finite-strain viscoelasticity coupled with gradient-enhanced continuum damage
(a) Influence of the damage rate factor η. (b) Influence of the isochoric damage factor ξiso.
(c) Influence of the threshold factor ξq . (d) Influence of the effective stress factor ξm.
(e) Influence of the coupling factor ηα. (f) Influence of the constant damage threshold qmin
in MPa.
Figure 2.2: Impact of the different material parameters on the simulated load-displacement response
for an exemplary tension test—Part 2. Reprinted from Schulte et al., European Journal of Mechanics
- A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
19
2 Classic parameter identification based on homogeneous states of deformation
Rothert [44], Simo and Hughes [117], amongst others. The model formulation consid-
ered in this work is introduced as follows: Section 2.2.1 briefly summarises the classic
finite strain hyperelastic constitutive relations with volumetric-isochoric decomposition.
Section 2.2.2 proceeds with the generalisation of these relations to a finite strain vis-
coelastic model. An extension towards gradient-enhanced continuum damage is then
briefly covered in Section 2.2.3.
2.2.1 Finite hyperelasticity with volumetric-isochoric decomposition
This section briefly outlines the hyperelastic local constitutive setting considered in the
work at hand, which will then be extended towards finite viscoelasticity. The constitutive
framework at hand facilitates the accommodation of any arbitrary hyperelastic response
based on a local Helmholtz free energy function ψloc, enabling the use of higher order
constitutive relations such as the Yeoh model.
The isochoric, i.e., volume-preserving, part F̄ of the deformation gradient F is defined
as
1
F̄ := J− 3F , with J = det(F ) and det(F̄ ) = 1 . (2.6)
1
In this context, J 3 I is referred to as the volumetric part of the deformation gradient,
with I denoting the second-order identity tensor. The standard and isochoric right
Cauchy–Green deformation tensors are introduced via
C := F t · F and C̄ := F̄ t · F̄ = J− 23 C, (2.7)
respectively.
In view of the hyperelastic contribution to the constitutive response, a local Helmholtz
free energy function ψhyp of the form
ψhyp(C) = ψvol(J) + ψich(C̄) (2.8)
is considered, where an additive decomposition into volumetric and isochoric energy
contributions, ψvol and ψich, respectively, is taken into account.
Within this modelling framework, the hyperelastic part of the Piola–Kirchhoff stress
tensor follows as
Shyp := 2 ∂
−1
Cψhyp(C) = J ∂Jψvol(J)C + S̄hyp , (2.9)
where the isochoric contribution to the hyperelastic part of the Piola–Kirchhoff stress
tensor is abbreviat[ed as ]
2
S̄ −hyp := J 3 2 ∂C̄ψich(C̄)−
1
[2 ∂ ψich(C̄) : C]C
−1 (2.10)
3 C̄
20
2.2 Finite-strain viscoelasticity coupled with gradient-enhanced continuum damage
for notational simplicity. It is noted that the well-established relations ∂ J = 1 J C−1C 2
2
and ∂CC̄ = J
− [Isym3 − 1 C ⊗C−1] were employed here.
3
The corresponding hyperelastic part of the spatial Kirchhoff stresses, τ hyp = F ·Shyp ·
F t, then follows as ( )
τ hyp = J ∂Jψvol(J) I + dev 2 F̄ · ∂ ψ tC̄ ich(C̄) · F̄ . (2.11)
2.2.2 Extension of finite hyperelasticity to finite viscoelasticity
In view of a generalisation of the volumetrically-isochorically decoupled finite strain
hyperelastic modelling framework outlined in Section 2.2.1 to finite viscoelasticity, C(t)
is now assumed to be a function of time t. As a relation for the viscoelastic stress
response in terms of Piola–Kirchhoff stresses S at time t, the form
∑N
2
S(t) = Shyp(t)− J− 3 DEV(Qi(t)) (2.12)
i=1
is considered and is in line with classic works such as [116, 117], where DEV(•) := [•]−
1 [[•] : C]C−1. Here, Shyp(t) denotes the hyperelastic stress contribution defined in (2.9)3
and Qi(t), with i = 1, 2, . . . , N , represent stress-type tensor-valued internal variables.
This form of the stress extension leaves the volumetric part of the constitutive response
unaffected by viscous effects. Relation (2.12) reflects a finite strain generalisation of
a classic generalised relaxation model with N being the number of Maxwell elements,
each consisting of a (linear) spring and a (linear) dashpot. Conceptually speaking, in a
simplified and linear setting, each Maxwell element would be characterised by a given
spring stiffness, say Ei, and corresponding dashpot viscosity, say ηi; cf., e.g., [117].
For the evolution of the stress(-type internal v)ariables, rate equations of the type1 γi
Q̇i(t) + Qi(t) = DEV 2 ∂C̄ψich(C̄(t)) with lim Qi(t) = 0 , (2.13)τi τi t→−∞
are considered, cf. [116, 117], with •̇ denoting the material time derivative and where τi
are referred to as relaxation times. Parameters γi represent relative moduli subject to
the restriction
∑N
γi = 1− γ∞ , (2.14)
i=1
with γ∞ ∈ [0, 1) determining the quasi-static (hyperelastic) limiting case of the material
response. In a simplified and linear setting, as thoroughly introduced in [117], τi :=
ηi/Ei > 0 and γi = Ei/E0 ∈ [0, 1] would be the parameters defin∑ing the properties of
each Maxwell branch in the viscoelastic model. Here, E0 = E∞ + iEi is the stiffness
21
2 Classic parameter identification based on homogeneous states of deformation
obtained in the (hyperelastic) limiting case reflected by infinitely high strain rates. In
the context of the three-dimensional non-linear formulation at hand, this contribution
is represented by the general Helmholtz free energy potential ψhyp that facilitates the
incorporation of higher-order and even anisotropic constitutive relations.
The linearity of evolution equations (2.13) facilitates the standard closed-form con-
volution represen∫tation
γ ti ( )
Qi(t) = exp(−(t− s)/τi) DEV 2 ∂C̄ψich(C̄(s)) ds . (2.15)τi −∞
Combining (2.15) with (2.12) and integration by parts yields an explicit expression
for the stress response in terms of the∫Piola–Kirchhoff stress tensor of the formt
− − 2 d ( ( ))S(t) = J ∂ 1Jψvol(J)C (t)+J 3 (t) g(t−s) DEV 2 ∂C̄ψich(C̄(s)) ds , (2.16)
−∞ ds
where the relaxation function
∑N
g(t) = γ∞ + γi exp(−t/τi) (2.17)
i=1
is incorporated.
2
The spatial counterpart of (2.16) is obtained by using J− 3 F · [DEV(•)] = dev(F̄ ·
(•) · F̄ t), so that the convolu∫tion representation of the Kirchhoff stresses readst d ( ( ))
τ (t) = J ∂Jψvol(J) I + g(t− s) dev 2 F̄ · ∂C̄ψich(C̄(s)) · F̄ t ds . (2.18)
−∞ ds
For more detailed information on the derivation of well-established relations in the
context of (undamaged) finite strain viscoelasticity based on a volumetric-isochoric split
of the underlying Helmholtz free energy function, the reader is referred to classic mono-
graphs such as [10, 117], amongst other works.
The numerical solution of the convolution integrals introduced above is based on a
transformation towards a recurrence formula, enabling the computation of the constitu-
tive response based on the standard internal variable procedure, thereby bypassing the
need to store the entire strain history that is individual to each material point in view of
the later finite element application; see Appendix A for details on the numerical update
procedure.
2.2.3 Incorporation of damage
We considered a gradient-regularised damage formulation in view of a mesh-objective
solution of boundary value problems by using the finite element method, where the
22
2.2 Finite-strain viscoelasticity coupled with gradient-enhanced continuum damage
constitutive response can include damage-induced softening. To this end, an overall
Helmholtz free energy potential ψ is introduced, composed of local and non-local energy
contributions, ψloc and ψnloc, in terms of ψ = ψloc + ψnloc.
The local Helmholtz free energy contribution, in extension of (2.8), takes the form
ψloc(C, κ) = f (κ)ψ (J) + f
niso
d vol d (κ)ψich(C̄) , (2.19)
where the volumetric and isochoric damage functions, fd ∈ (0, 1] and fnisod ∈ (0, 1], are
evaluated based on a common internal damage variable κ, but allow for an adjustment of
the volumetric-isochoric damage distribution via the exponent niso ∈ R. The exponent
niso is a material parameter that facilitates the modelling of a damage process that affects
volumetric and isochoric contributions with different intensities.
The non-local energy contribution ψnloc enables the damage regularisation and con-
sists of two contributions, namely a non-local gradient term ψgradnloc (∇Xφ;F ) and a non-
local penalty term ψpltynloc(φ, κ), so that ψ
grad plty
nloc(F , φ,∇Xφ, κ) = ψnloc (∇Xφ;F ) +ψnloc(φ, κ),
with
ψgrad
cd cd
nloc (∇Xφ;F ) = ∇ φ ·C−1X · ∇Xφ = ∇xφ · ∇xφ , (2.20)2 2
ψplty
βd
nloc(φ, κ) = [φ− κ]2 . (2.21)2
Here, φ is the non-local damage variable that is coupled to the local damage variable κ
via the penalty parameter βd. Moreover, cd represents a length scale-type regularisation
parameter. With this particular choice of the non-local energy contributions, the Euler–
Lagrange equations for the non-local damage field variable in spatial format take the
form
cd∇x · [∇xφ]− βd [φ− κ] = 0 in Bt , (2.22)
∇xφ · n = 0 on ∂Byt = ∂Bt , (2.23)
with n denoting the surface normal unit vector; see, e.g., [86] for more details on the
associated relations, as well as on the numerical treatment within appropriately coupled
finite element formulations.
The incorporation of the damage functions in the local free energy potential (2.19)
induces Piola–Kirchhoff stresses of the form
Shyp := 2 ∂Cψloc(C, κ) = J fd(κ) ∂Jψvol(J)C
−1 + fnisod (κ) S̄hyp , (2.24)
corresponding to spatial Kirchhoff stre(sses given by )
τ = J f (κ) ∂ ψ (J) I + dev 2 fnisohyp d J vol d F̄ · ∂C̄ψ tich(C̄) · F̄ , (2.25)
23
2 Classic parameter identification based on homogeneous states of deformation
both of which are direct extensions of their purely hyperelastic counterparts introduced
in (2.9) and (2.11).
The damage function affecting the isochoric part of the free energy function further-
more affects the evolution of viscous s(tress-type inter)nal variables via1 γi
Q̇i(t) + Qi(t) = f
niso
d DEV 2 ∂C̄ψich(C̄(t)) , (2.26)τi τi
which basically reflects the damage-extended version of (2.13). This induces a convolu-
tion representatio∫n in terms of
γ ti ( )
Qi(t) = exp(−(t− s)/τ nisoi) fd DEV 2 ∂C̄ψich(C̄(s)) ds , (2.27)τi −∞
see (2.15) for the corresponding expression that is associated with an undamaged setting.
Finally, the Piola–Kirchhoff stresses that fully incorporate viscous and damage effects
take the form ∫ t
2 d ( ( ))
S(t) = Jfd ∂Jψ
−1
vol(J)C (t)+J
−
3 (t) g(t−s) fnisod DEV 2 ∂C̄ψich(C̄(s)) ds ,
−∞ ds
(2.28)
which allows for the accommodation of arbitrary free energy functions, in particular
energy functions of higher order such as the Yeoh model that appropriately captures the
Treloar-type behaviour of materials such as the particular elastomer considered in this
work.
The numerical solution of the above-introduced convolution integrals is based on a
transformation towards a recurrence formula, enabling the computation of the constitu-
tive response based on the standard internal variable procedure, thereby bypassing the
need to store the entire strain history that is individual to each material point in view of
the later finite element application; see Appendix A for details on the numerical update
procedure.
Since the general framework can be applied to all kinds of energy functions and
considering the type of material and its behaviour, the chosen hyperelastic material
model is briefly presented in Section 2.2.4. The applied damage function fd is specified
in Section 2.2.5.
2.2.4 Hyperelastic material model
Considering the nearly incompressible, non-linear elastic material behaviour (see for ex-
ample the stress-strain-response in Figure 2.24b), the Yeoh hyperelastic material model
was chosen [137, 138]. In [139], Yeoh and Flemming argued that Rivlin’s theory [94],
24
2.2 Finite-strain viscoelasticity coupled with gradient-enhanced continuum damage
where the strain energy function depends on the first two invariants of the right Cauchy–
Green tensor C, to be specific
∑∞
ψ = C [I − 3]i[I − 3]jRiv ij 1 2 , (2.29)
i,j=1
runs into difficulties with respect to the identification of the material parameters since
the contributions cannot be perfectly determined separately in the experiments. Here,
Cij are the corresponding material parameters, and I1 and I2 are the first and second
principal invariants related to the isochoric right Cauchy–Green tensor, respectively.
Yeoh neglected the contribution of the second invariant assuming that no serious error
would arise since the contribution would be sufficiently small with respect to the impact
of the first invariant. In this work, however, the Yeoh energy potential
∑3
ψich = Ci[tr(C̄)− 3]i , (2.30)
i=1
serves as the isochoric contribution to overall energy potential, where Ci denote the
underlying material parameters. This phenomenological material model was developed
for rubber elasticity and is applied in this work as a first choice with respect to the
mentioned material behaviour. The volumetric energy contribution ψvol is chosen in a
standard manner[as ]
1 1
ψvol = K [J
2 − 1]− ln(J) , (2.31)
2 2
where K denotes the bulk modulus of the material. The related energy contributions
ψich and ψvol can, however, be straightforwardly replaced within the general constitutive
framework elaborated in this work in order to account for different particular material
characteristics.
2.2.5 Damage function
The damage function mentioned in Section 2.2.3 was chosen to be of an exponential type,
following [86, 133]. Thus, a damage initiation threshold, as well as a damage saturation
rate can be used in the function
fd(κ) = 1− d = exp(−ηd〈κ− κd〉) , (2.32)
fulfilling the requirements for the damage function to be restricted to f : R+d → (0, 1].
Here, d denotes the classic damage variable with d=0 indicating zero damage and d=1
for 100% damage. The variable κ represents a related internal damage variable, while
κd > 0 is the damage threshold parameter, and ηd introduces the damage saturation
25
2 Classic parameter identification based on homogeneous states of deformation
parameter. In addition, 〈•〉 := max{0, •} represents the Macaulay brackets. Conse-
quently, damage is obtained in the material if the local damage variable exceeds the
damage threshold. The damage function for the isochoric energy contribution matches
this chosen function except for the exponent niso. Thus, a relation between both func-
tions is still given and only adjusted via this material parameter. The evolution of the
internal damage variable κ is based on the associated form
∂Φd(F , φ,∇Xφ, κ)
κ̇ = λ , (2.33)
∂q
where λ denotes a proper Lagrange multiplier, Φd(F , φ,∇Xφ, κ) = q(F , φ,∇Xφ, κ) −
κ ≤ 0 represents the damage condition, and q = − ∂[ψloc + ψnloc]/∂κ is the energy
release rate. Hence, apart from the balance of linear momentum, the Euler–Lagrange
equation governing the non-local damage variable needs to be solved simultaneously. For
further information regarding this regularised damage model and its implementation, see
Ostwald et al. [86]. The implementation was performed in a user material subroutine
(UMAT) in Abaqus following [86].
2.3 Laminate-model for ferroelectric materials
In this section, the laminate-model for ferroelectric materials is presented, starting with
a brief outline of the underlying material model in Section 2.3.1. Subsequently, the
different Fischer-Burmeister-type functions discussed in this section are introduced in
Section 2.3.2. Section 2.3.3 provides numerical results and a detailed discussion with
respect to the performances of the analysed schemes. This section is concluded with a
summary regarding the investigations of the different FB functions in Section 2.3.4.
2.3.1 Constitutive model
The constitutive model to simulate switching behaviour of ferroelectrics, in particular
BaTiO3, shall be described in this section. The model considers small strain kinematics
and is adopted from [17], to which the reader is referred to in view of a detailed discussion
of the constitutive model.
We consider tetragonal unit cells, as representative for e.g. BaTiO3 below the Curie
temperature, and shall denote the polarisation direction of the respective variants by pα,
with α = 1, . . . , nv = 6. The orthonormal base system {e1,2,3} is assumed to be aligned
with the crystallographic axes so that
p s1 = −p2 = p e1 , (2.34)
p3 = −p4 = ps e2 , (2.35)
p5 = −p s6 = p e3 , (2.36)
26
2.3 Laminate-model for ferroelectric materials
wherein ps denotes the polarisation magnitude. Similarly, the states of spontaneous
strains related to the respective variants are introduced as
ε1 = ε2 = [ ε
s s s
1 − ε2 ] e1 ⊗ e1 + ε2 I , (2.37)
ε3 = ε4 = [ ε
s
1 − εs2 ] e s2 ⊗ e2 + ε2 I , (2.38)
ε5 = ε6 = [ ε
s
1 − εs2 ] e3 ⊗ e3 + εs2 I , (2.39)
wherein εs and εs1 2 define the axial spontaneous strain magnitudes.
Moreover, let the directions of the p√olarisation vectors be represented by the unit
vectors aα = pα/‖pα‖, with ‖pα‖ = pα · pα, which define the structural tensors
mα = aα ⊗ aα. Based on this, the representation of the constitutive tensors – i.e. the
fourth-order elasticity tensors, the third-order piezoelectric tensors and the second-order
dielectric tensors – results in
Eα(aα) = λ
E I ⊗ I + 2µE Isym + αE1 [mα⊗ I + I ⊗ mα ]
+ 2αE2 mα ⊗mα + αE3 [mα ⊗ I + I ⊗mα ] , (2.40)
eα(aα) =− βe1 aα ⊗ I − βe2mα ⊗ aα − 1 βe3 [ I ⊗ aα + I ⊗aα ] , (2.41)2
 (a ) =− 2 γ I − 2 γα α 1 2mα . (2.42)
The contribution of each variant at the level of a material point within the continuum
considered is captured by adopting the concept of volume fractions embedded into the
theory of mixtures. To be specific, let these volume fractions be related to the respective
variants and be denoted by λα together with the constraints∑nv
λα ≥ 0 and λα = 1 . (2.43)
α=1
Based on this, volume averaged (at the level of∑a material point) quantities can be
introduced, which shall be denoted by [•]M(λ) = nvα=1 λα [•]α with λ = [λ1, . . . , λnv ].
Alternatively, laminate related fractions can be introduced, which directly refer to the
laminate microstructure of the material. The laminate fractions can be expressed in
terms of the volume fractions, ∑0 for j−1α= 1 λα = 1 ,
µj =  λj (2.44)
1−∑j−1 otherwise ,α= 1 λα
27
2 Classic parameter identification based on homogeneous states of deformation
with j = 1, . . . , nv − 1, and vice versa, i.e.
 α∏−1µα [ 1− µj ] for α = 1, . . . , nv − 1 ,
λα =  α∏
j=1
−1 (2.45)
[ 1− µj ] for α = nv .
j=1
cf. [9, 65, 136] and Figure 2.3. The laminate fractions are constrained by
0 ≤ µj ≤ 1 . (2.46)
Furthermore, volume averaging at materi∑al point level can be reformulated in terms of
laminate fractions, namely [•]M(λ(µ)) = nvα=1 λα(µ) [•] with µ = [µ1, . . . , µnv−1α ].
λ2 1− µ1
λ1 µ1
Figure 2.3: Illustration of the laminate microstructure with the definition of variant and laminate
volume fractions λα and µ
j for two different variants, see also, e.g., [65] for further details. The arrows
indicate the orientation of spontaneous polarisation vectors in the respective variant. Reprinted from
Bartel et al., Arch. Appl. Mech. 89 (2019), 995-1010, [6], with permission from Springer Nature.
With theses definitions at hand, we next introduce the electric Gibbs energy H —
denoted according to the definition provided in [118], for general discussions on the
nomenclature of thermodynamic potentials in the electro-mechanically coupled case see
also [81, 107] — based on which the stress tensor σ = ∂H/∂ε, the dielectric displace-
ments d = − ∂H/∂e, as well as quantities f j = − ∂H/∂µj energetically conjugate to
the laminate fractions can be derived. In this context, ε denotes the total strain tensor,
28
2.3 Laminate-model for ferroelectric materials
whereas e represents the electric field vector. We assume an additive decomposition of
H into a piezoelectric contribution and an additional remanent term of the form
H(εe, e,µ) = Hpiezo(εe, e,µ) +Hrem(µ) , (2.47)
Hpiezo = 1 ε[e : EM : εe − e · eM : εe − 1 e · M · e− pM · e , (2.48)2 2
1 1 ( ) ( )]
Hrem = psat ln 1− [ pN(µ) ]2 + ‖pM‖ arctanh pN(µ) , (2.49)
c 2
with εe = ε− εM, the additional material parameter c, and pN(µ) = ‖pM‖/psat.
The laminate fractions do not evolve for states within the admissible domain intro-
duced via { ∣ }
E := f j(εe, e,µ) ∣Φj(f j) := |f j| − gc,j(µ) ≤ 0 ; j = 1, . . . , nv − 1 , (2.50)
wherein the coercive field (ec90◦,180◦) related threshold quantities are defined as
gc,1(µ) = 2 ec s180◦ p µ
2 + ec s90◦ p [ 1− µ2 ] , (2.51)
gc,2(µ) = ec s90◦ p [ 1− µ1 ] , (2.52)
gc,3(µ) = [ 2 ec ps µ4 + ec ps 4180◦ 90◦ [ 1− µ ] ] [ 1− µ1 ] [ 1− µ2 ] , (2.53)
gc,4(µ) = ec ps [ 1− µ1 ] [ 1− µ2 ] [ 1− µ390◦ ] , (2.54)
gc,5(µ) = 2 ec ps [ 1− µ1 ] [ 1− µ2 ] [ 1− µ3180◦ ] [ 1− µ4 ] , (2.55)
cf. [136]. Rate dependent evolution equations, activated for non-admissible states, are
introduced as
1 〈 〉 j
µ̇j = |f j| − gc,j m f(µ) | | for j = 1, . . . , nv − 1 , (2.56)η f j ∑
wherein 〈[•]〉 = max{0, [•]}, which shows that the dissipation inequality nv−1 f jj=1 µ̇j ≥ 0
is satisfied.
2.3.2 Fischer-Burmeister-based algorithms
In this chapter, the original Fischer-Burmeister NCP function according to [19] is intro-
duced and some of its characteristics are discussed. In addition, some enhanced methods
based on this original approach are presented.
29
2 Classic parameter identification based on homogeneous states of deformation
2.3.2.1 Original Fischer-Burmeister approach
As pointed out in Section 2.3.1, the constitutive framework includes the consideration
of the physically-motivated restrictions
rjI := µ
j ≥ 0 (2.57)
rjII := 1− µj ≥ 0 , (2.58)
with j = 1, . . . , nv − 1. In view of the underlying energetic approach elaborated in
Section 2.3.1, an enhanced form of the electromechanical stored energy density is defined
as
n∑v−1 n∑v−1
H e e j j j jenh(ε , e,µ,ΓI ,ΓII) := H(ε , e,µ)− ΓI rI − ΓII rII , (2.59)
j=1 j=1
with Γ = [Γ 1, . . . , Γ nv−1I I I ] and Γ
1 nv−1
II = [ΓII , . . . , ΓII ] as Lagrange-type parameters. As
a consequence, the evolution equations (2.56) are rewritten as
j 1〈 〉 jµ̇ = |f j | − gc,j m fenhenh (µ) j , (2.60)η |fenh|
with f jenh =− ∂Henh/∂µj, which are subjected to the complementarity conditions
rji ≥ 0, Γ j j ji ≥ 0, ri Γi = 0, with i = I, II . (2.61)
However, it was shown for different kinds of constitutive models, e.g. crystal plasticity
models, that the consideration of such complementarity conditions can be challeng-
ing and that it has significant influence on the efficiency of the underlying algorithms,
see, e.g., [102]. Such complementarity conditions mainly occur in the context of rate-
independent evolution equations and physically-motivated restrictions of the values of
phase volume fractions or the like. A sophisticated way to circumvent the explicit con-
sideration of these equality and inequality constraints stems from [19] and has been
introduced to problems in continuum mechanics and material modelling by [102]. To be
more precise, the Fischer-Burmeister non-linear complementarity problem functions (or
short, Fischer-Burm√eister NCP functions), namely
ξj(rj, Γ ji i i ) := [r
j]2i + [Γ
j]2i − [rji + Γ ji ] = 0, (2.62)
may substitute all of the conditions in (2.61). Figures 2.4 and 2.5 reveal the main char-
acteristics of this kind of function, where the original Fischer-Burmeister NCP function
shall be labelled f 0FB here for later purposes. Roots of f
0
FB only exist in the case of r
j
i ≥ 0
and Γ ji = 0 or, vice versa, r
j
i = 0 and Γ
j
i ≥ 0. The projections f 0FB(0, Γ ji ) and f 0FB(rji , 0)
are not continuously differentiable at {rji , Γ ji } = {0, 0}.
30
2.3 Laminate-model for ferroelectric materials
1.0 1
0.5
0.0
2
0.5
-0.5
3
-1.0 2.5
-1.0 -0.5 0.0 0.5 1.0
r
Figure 2.4: Illustration of the Fischer-Burmeister NCP function as three dimensional plot (left) and
contour plot (right) depending on r and a generalised Lagrange-type parameter Γ . Reprinted from
Bartel et al., Arch. Appl. Mech. 89 (2019), 995-1010, [6], with permission from Springer Nature.
As shown in, e.g., [4, 5, 50], the Fischer-Burmeister NCP functions allow a sophisti-
cated implementation of complementarity conditions by solving the system of equations
[ ]t ·
R(V) = ζµ1 . . . ζ
1
µnv−1 ξI . . . ξ
nv−1 ξ1 nv−1I II . . . ξII = 0 (2.63)
without the[need of, e.g., cumbersome active set searc]h strategies. Here,
1 n −1 1 nv−1 1 nv−1 tV := µ . . . µ v ΓI . . . ΓI ΓII . . . ΓII (2.64)
denotes a generalised list of unknowns and the quantities
ζµj := µ
j − nµj − F j , (2.65)
with
∆t〈 〉 j
F j := ∆t µ̇j = |f j | − gc,j m f(µ) enh
η enh |f j
, (2.66)
enh|
reflect the time-discretised form of the evolution equations with nµj as the value of µj
at the former time step tn, whereas the index referring to the current time step tn+1 is
31
Γ
2 Classic parameter identification based on homogeneous states of deformation
f 0FB ∂ f
0
Γ FB
Γ
-1.0 -0.5 0.5 1.0
2.0
-0.5
1.5
r = 0
1.0 -1.0 r = 0.25
r = -0.25
0.5
-1.5
Γ
-1.0 -0.5 0.5 1.0 -2.0
Figure 2.5: Projections of the Fischer-Burmeister NCP function (left) and the corresponding derivatives
with respect to Γ (right) for different values of r. Reprinted from Bartel et al., Arch. Appl. Mech. 89
(2019), 995-1010, [6], with permission from Springer Nature.
skipped in order to simplify notation. A standard method to solve (2.63) is the Newton-
Raphson scheme with iteration index l, where the solution is obtained via incremental
updates of the form[ ]
l −1
l+1 l − dRV = V · lR . (2.67)
dV
The Jacobiancan in general berepresented by using the following submatrices viadζ j dζ j dζ j
µ µ µ
k
 dµ dΓ
k
I dΓ
k 
II
  dξj dξj dξjdR =  I I Ik  . (2.68)dV  dµk dΓI dΓ kIIdξj j j II dξII dξII
dµk dΓ k kI dΓII
To be more specific, these submatrices—all of which exhibit the dimension [nv − 1] ×
[nv − 1]—are given by
dζµj ∂F
j
= δjk − , (2.69)
dµk ∂µk
dζ jµj ∂F
= − , (2.70)
dΓ kI ∂Γ
k
I
dζ jµj ∂F
= − , (2.71)
dΓ kII ∂ΓkII 
dξjI jk √ rj= δ I − 1 , (2.72)
dµk
[rj ]2I + [Γ
j
I ]
2
32
2.3 Laminate-model for ferroelectric materials
 dξj jI = δjk √ ΓI − 1 , (2.73)
dΓ kI [rj ]2 + [Γ j]2I I
dξjI = 0jk ,   (2.74)dΓ kII
dξjII = −δjk √ rjII − 1 , (2.75)
dµk
[rj 2II ] + [Γ
j
II ]
2
dξjII = 0jk ,
dΓ kI  
(2.76)
dξjII = δjk √ Γ jII − 1 . (2.77)
dΓ kII [rj ]2 + [Γ j ]2II II
The terms in (2.72), (2.73), (2.75), and (2.77) reveal two general drawbacks of the
original Fischer-Burmeister approach in the context of its numerical implementation:
• Intuitive choices for the initial values of the underlying variables such as µj = 0 or
µj = 1 for some j = 1, . . . , nv−1 and, as a consequence, rjI = 0 or rjII = 0 together
with Γ ji = 0 lead to a singularity of the Jacobian introduced in (2.68).
• In contrast to, e.g., (2.69), these terms are likely to take values such as 1 and
−1 which are essentially independent from the chosen physical units. As a con-
sequence, the eigenvalues and the condition number of the Jacobian (2.68) also
depend on the chosen units. In other words, the robustness of the algorithm is
highly affected by the specific numerical values of, e.g., the underlying Young’s
moduli.
In order to overcome these general drawbacks, we propose the following:
1. Even if it is intuitively clear that rji = 0 is the correct solution for a specific time
step of the overall calculation (for some i, j), the initial guess for the underlying
variables should result in rji =6 0. In the context of the constitutive model used in
this section, we do not set µj = 0 or µj = 1 as initial values, but rather µj = δµ or
µj = 1− δµ, where δµ is a small number, e.g. δµ = 10−4.
2. In some cases, the initial guess of Γ ji = 0 may be far off the actual solution. It
turns out that an initial choice of Γ ji = Γ
0 > 0, e.g. Γ0 = 10
−4, significantly
increases the algorithm’s robustness, even if Γ ji = 0 are the correct solutions. The
values of Γ ji should be initialised in this manner in every load step.
3. Physical units should be used that lead to rather small values of the related quan-
tities, for instance GPa for Young’s moduli.
33
2 Classic parameter identification based on homogeneous states of deformation
Remark 1 In general, NCP functions of any kind, e.g. without further smoothing
techniques, exhibit the property of being non-differentiable, at least at certain “points”.
Therefore, it is generally concluded that Newton-type solution schemes are not applicable
for the underlying system of equations. In practice, however, such methods turn out to
be applicable and even suitable if—in the specific case of the Fischer-Burmeister NCP
functions—the recommendations 1.–3. made above are taken into account.
Remark 2 A significant drawback of the original Fischer-Burmeister approach is the
limitation in dealing with monotone complementarity problems since the underlying func-
tion is too flat in the positive orthant, cf. [12].
2.3.2.2 Extended Fischer-Burmeister-type NCP functions
As indicated above, the implementation of the original Fischer-Burmeister NCP function—
despite its beneficial features—may suffer from specific shortcomings. Therefore, several
approaches have been established in order to increase the efficiency of solution procedures
for systems of equations such as (2.63). In this section, we investigate the applicability
and performance of the following NCP functions based on the original Fischer-Burmeister
approach in t√he context of the constitutive model presented in Section 2.3.1:
f IFB := [[r√j 2 j 2i ] + [Γi ] + 2 δ2 − [rj ji + Γi ] ] √〈 〉 〈 〉 (2.78)
f II := λ [ [rj√ ]2 + [Γ
j]2 + 2 δ2FB i i − [rj + Γ j]]+ [1− λ] rj
2 j 2 2
i i √ i Γi + 2 δ (2.79)〈 〉4 〈 〉
f III
4 j 4 j 4 4 j j 4 j j 4 4
FB := √λ [ri ] + [Γi ] + 2 δ − [ri + Γi ] √+〈[1−〉 λ〈] 〉 ri Γi + 2 δ (2.80)
IV 2 2fFB := √[[rj√2i ] + [Γ
j 2
i ] + 2 δ
2 − [rj + Γ j]− α] rj Γ j 2i i i i + 2 δ (2.81)2 [〈 〉 〈 〉]
V 2fFB := [r
j]2 + [Γ j]2 + 2 δ2 − [rji i i + Γ ji ] + α rji Γ ji (2.82)
In these relations δ reflects a perturbation parameter, λ∈ ]0, 1] and α∈ [0,∞[ are scaling
parameters.
The NCP function given in (2.78) stems from [47] and is referred to as smoothed
Fischer-Burmeister NCP function. Figures 2.6 and 2.7 show the main characteristics of
these enhanced NCP functions: In comparison to the original approach, the smoothed
function is continuously differentiable everywhere. As shown in Appendix B, the roots
of the original NCP function are only calculated in an approximative sense, where the
accuracy depends on the specific choice for the perturbation parameter δ. More precisely
speaking, the equality constraint to be solved by using this approach is rj j 2i Γi = δ .
Following [18], Equation (2.79) reflects a smoothed penalised Fischer-Burmeister NCP
function on the basis of the NCP function introduced in [12], which is illustrated in
34
2.3 Laminate-model for ferroelectric materials
1.0 1
0.5
0.0
2.5 0.5
-0.5
-1.0 3 2
-1.0 -0.5 0.0 0.5 1.0
r
Figure 2.6: Illustration of the smoothed Fischer-Burmeister NCP function f IFB as three dimensional
plot (left) and contour plot (right) depending on r and a generalised Lagrange-type parameter Γ . Here,
the perturbation parameter δ = 0.05 has been chosen for illustration purposes. Reprinted from Bartel
et al., Arch. Appl. Mech. 89 (2019), 995-1010, [6], with permission from Springer Nature.
Figure 2.8 in terms of contour plots for different values of the scaling parameter λ. The
NCP-function provided in (2.80) is based on f IIFB with modifications inspired by [13, 14].
The fourth alternative investigated here is given by (2.81) which is a modification
of the regularised FB function as presented in [122] which is also referred to as scaled
penalised FB function therein. Exemplary visualisations of this function are given in
Figure 2.9 for different values of the scaling parameter α. It is also shown in the afore-
mentioned work that this specific FB-function may be regarded as a modified version of
the NCP function established by [12]. Furthermore, the smoothing approach introduced
in [47] has been applied, so that this function could be referred to as smoothed and
scaled regularised FB function. Finally, the last NCP function investigated is given in
(2.82) which is adopted from [121] with modifications according to [47].
2.3.3 Numerical examples
In this section the different NCP functions discussed in this contribution are applied
in the context of the constitutive model presented in Section 2.3.1 and compared to
each other in view of their convergence behaviour. Prior to the detailed discussion, the
general material response is visualised in Figures 2.10 and 2.11, cf. [17].
35
Γ
2 Classic parameter identification based on homogeneous states of deformation
f I IFB ∂Γ fFB
Γ
-1.0 -0.5 0.5 1.0
2.0
-0.5
1.5
r = 0
1.0 -1.0 r = 0.25
r = -0.25
0.5
-1.5
Γ
-1.0 -0.5 0.5 1.0 -2.0
Figure 2.7: Projections of the smoothed Fischer-Burmeister NCP function f IFB (left) and the corre-
sponding derivatives with respect to Γ (right) for different values of r. Here, the perturbation parameter
δ = 0.05 has been chosen for illustration purposes. Reprinted from Bartel et al., Arch. Appl. Mech. 89
(2019), 995-1010, [6], with permission from Springer Nature.
The results are obtained due to an electromechanical loading, implemented through a
constant compressive stress σ=−2.7 e3 ⊗ e3 MPa and a cyclically applied electric field
e= e3(t) e3 with e3(0) = 0, e3∈ [−1.25 kV/mm, 1.25 kV/mm], and the loading rate ė3 =
±1 kV/[mm s]. In the following, the behaviour of the different NCP functions is analysed
for two different time steps indicated in Figures 2.10 and 2.11, viz. tI = 0.8 s and tII =
1.1 s. These time steps have been selected due to the fact that the related convergence
studies turned out to be most meaningful. As can be seen in Figure 2.11(left), the first
time step tI marks a state shortly after λ6 becomes zero, and tII reflects a state shortly
before λ5 becomes one. In the unloaded state, all variants are energetically equally
favourable so that λi=1/6 with i=1, . . . , 6 are used as initital values.
2.3.3.1 Comparison of different approaches at t= tI =0.8 s
In this section, the influence of the weighting factor δ and scaling parameter λ are
discussed at t = tI = 0.8 s. The prescribed time increment is chosen as ∆t = 0.02 s so
that tI is reached in load step 40. Figures 2.12 and 2.13 show the values of the residual
depending on the number of iteration steps. Five methods are compared, namely the
smoothed Fischer-Burmeister approach based on f IFB, the smoothed penalized Fischer-
Burmeister NCP function using f IIFB for λ=0.1 and λ=0.9, and the modified version of
the latter using f IIIFB also with λ= 0.1 and λ= 0.9. The remaining NCP functions f
IV
FB
and fVFB are discussed separately in Section 2.3.3.3.
Neglecting the results for δ=10−2 at first, the algorithms based on f IIIFB show the same
(δ = 10−12, δ = 10−8) or at least a very similar (δ = 10−6) behaviour as the algorithm
based on f I IIFB. The convergence of the NCP function fFB with λ= 0.9 is similar to the
convergence of the previously mentioned NCP functions as well, for λ=0.1, however, the
algorithm based on f IIFB shows an instability due to the occurring jump before fulfilling
the chosen tolerance of 10−8. It can be concluded that for rather small values of λ, where
the stabilisation term within the function f IIFB is dominating, the convergence behaviour
worsens. The precise value of the perturbation parameter δ does not significantly affect
36
2.3 Laminate-model for ferroelectric materials
λ = 0.1 λ = 0.25
1.0 0.2 0.3 1.0 0.25 0.25
0.5
0.5 0.5
0.5
0.2
0.1 0.075
0.0 0.3 0.0
-0.5 -0.5 0.25
0.4 0.2
- -1.01.0 0.3 0.5
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
r r
λ = 0.5 λ = 1.0
1.0 0.75 0.25 1.0 0
1.5
0.5 0.001 0.5
0.0 0.0
0.25
0.25
2.5
-0.5 -0.5
1
-1.0 1.5 0.75 -1.0 3
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
r r
Figure 2.8: Illustration of the smoothed penalised Fischer-Burmeister NCP function f IIFB as contour
plots for different values of the scaling parameter λ. Here, the perturbation parameter δ = 0.01 has
been chosen for illustration purposes. Reprinted from Bartel et al., Arch. Appl. Mech. 89 (2019),
995-1010, [6], with permission from Springer Nature.
the convergence behaviour of the applied schemes. In general, all applied methods
exhibit a linear convergence even though a Newton-Raphson scheme is properly applied
which reflects a general drawback of algorithms based on Fischer-Burmeister functions,
cf. Remark 2. As can be seen in Figure 2.13(right), all analysed schemes converge
quadratically for δ = 10−2. However, such large values of δ may not be chosen due to
the fact that the underlying constraints may significantly be violated which becomes
obvious by the graphs in Figure 2.16, even though these refer to a different time step,
as well as by the derivation provided in Appendix B.
37
Γ Γ
Γ Γ
2 Classic parameter identification based on homogeneous states of deformation
α = 0. α = 1.
1.0 1 1.0 1 0
0.5 0.5
0.0 0.0 0
-0.5 -0.5
3 3 1
-1.0 2 -1.0 2
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
r r
α = 2. α = 4.
1.0 1 -0.5 1.0 1 0 2
0
0.5 0.5
1
-1
0.0 0.02 0 0
-0.5 -0.5
1 3 1
-1.0 3 -1.0 2
-1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0
r r
Figure 2.9: Illustration of the smoothed and scaled regularised Fischer-Burmeister NCP function f IVFB as
contour plots for different values of the scaling parameter α. Here, the perturbation parameter δ = 0.01
has been chosen for illustration purposes. Reprinted from Bartel et al., Arch. Appl. Mech. 89 (2019),
995-1010, [6], with permission from Springer Nature.
2.3.3.2 Comparison of different approaches at t= tII =1.1 s
The results shown in Figure 2.14 refer to the chosen value of δ=10−6 for the perturbation
parameter and the time step t= tII =1.1 s. The prescribed time increment is ∆t=0.02 s
and, as an alternative, ∆t= 0.002 s so that tII is reached in load step 55 and load step
550, respectively. In this time step, the algorithms based on both f IIFB and f
III
FB with a
dominating stabilisation term, i.e. λ = 0.1, show significant numerical instabilities for
∆t = 0.02 s. In contrast to the example above, the worst convergence behaviour is in
the case obtained with the NCP function f IIIFB, where changes with respect to the active
set occur due to the jumps in the value of the residual. For smaller time increments of
∆t=0.002 s, the convergence behaviour of the different approaches become more similar.
38
Γ Γ
Γ Γ
2.3 Laminate-model for ferroelectric materials
0.7 0.3
tII
0.6 tII
0.2
0.5
0.1
0.4 tI
0.3 0
0.2
-0.1
0.1
-0.2
0 tI
-0.1 -0.3
-1.4 -1 -0.6 -0.2 0.2 0.6 1 1.4 -1.4 -1 -0.6 -0.2 0.2 0.6 1 1.4
e3 e3
Figure 2.10: Material response predicted by the underlying constitutive framework according to [17] in
terms of ε33(e3) (left) and d3(e3) (right). The labels tI and tII indicate two representative time steps
used for the subsequent convergence studies. Reprinted from Bartel et al., Arch. Appl. Mech. 89 (2019),
995-1010, [6], with permission from Springer Nature.
1.2 1.2
λ µ
1 1 1 1
λ2 µ2
0.8 λ 0.83 µ3
0.6 λ4 0.6 µ4
λ5 µ5
0.4 0.4
λ6
0.2 0.2
0 0
tI tII tI tII
-0.2 -0.2
0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4
e3 e3
Figure 2.11: Material response predicted by the underlying constitutive framework according to [17]
in terms of λ ji(e3) (left) and µ (e3) (right) for monotonic loading. The labels tI and tII indicate two
representative time steps used for the subsequent convergence studies. Reprinted from Bartel et al.,
Arch. Appl. Mech. 89 (2019), 995-1010, [6], with permission from Springer Nature.
Even more significant discrepancies between the different approaches can be obtained
by setting the perturbation parameter to δ = 10−4. As illustrated in Figure 2.15, two
algorithms even diverge and do not provide a proper solution for the specified time step
t = tII = 1.1 s with ∆t = 0.02 s. Interestingly, the two algorithms which exhibit the
worst convergence behaviour in the above examples, namely f II and f IIIFB FB with dominant
stabilisation terms, i.e. λ = 0.1, perform rather well in this example. In contrast, the
smoothed Fischer-Burmeister approach f I as well as f IIIFB FB with λ = 0.9 oscillate and
do not converge. This is rather unexpected due to the fact that these two algorithms
performed best in the above examples. The algorithm based on the NCP function f IIFB
with λ= 0.9 also shows oscillations at the very beginning of the iteration process but
then first slightly deviates from the, say, failing algorithms based on f I IIIFB as well as fFB
and finally results in a proper solution. It is, however, noteworthy that all algorithms
39
λi ε33
µi d3
2 Classic parameter identification based on homogeneous states of deformation
0 0
-2 -2
-4 -4
f I f IFB FB
f II II-6 FB,λ=0.1 -6 fFB,λ=0.1
f II , IIFB λ=0.9 fFB,λ=0.9
-8 f IIIFB,λ=0.1
-8 f IIIFB,λ=0.1
f III , =0.9 f IIIFB λ FB,λ=0.9
-10 -10
5 10 15 20 5 10 15 20
local iterations local iterations
Figure 2.12: Evolution of the residua of the different FB functions plotted over the number of local
iterations at t= tI = 0.8 s with ∆t= 0.02 s and δ = 10
−12 (left) and δ = 10−8 (right). Reprinted from
Bartel et al., Arch. Appl. Mech. 89 (2019), 995-1010, [6], with permission from Springer Nature.
0 0
f IFB
-2 -2 f IIFB,λ=0.1
f IIFB,λ=0.9
-4 -4 III
I fFB,λ=0.1fFB
f IIIII FB,λ=0.9-6 fFB,λ=0.1 -6
f IIFB,λ=0.9
-8 f IIIFB,λ=0.1
-8
f IIIFB,λ=0.9
-10 -10
5 10 15 20 5 10 15 20
local iterations local iterations
Figure 2.13: Evolution of the residua of the different FB functions plotted over the number of local
iterations at t= tI = 0.8 s with ∆t= 0.02 s and δ = 10
−6 (left) and δ = 10−2 (right). Reprinted from
Bartel et al., Arch. Appl. Mech. 89 (2019), 995-1010, [6], with permission from Springer Nature.
except the one based on f IIFB with λ=0.1 will diverge in this or a subsequent time step
so that the results shown in Figure 2.10 and 2.11 cannot be obtained. Smaller values of
the time increment such as ∆t=0.002 s provide similar results for all analysed schemes.
The rather large value of δ=10−4 results in, e.g., λ5 =1.001 which might be considered
inaccurate. Even higher values of δ such as 10−2 further increase the stability of the
respective algorithm but definitely violate the underlying constraints in an unacceptable
manner, cf. Figure 2.16.
2.3.3.3 Analyses with respect to f IV and fVFB FB
In this section, the focus is placed on the performances of the algorithms based on the
NCP functions f IVFB and f
V
FB. According to our investigations, those algorithms always
behave similarly to the algorithm using f IFB in the context of the present constitutive
40
Residua in log10 Residua in log10
Residua in log10 Residua in log10
2.3 Laminate-model for ferroelectric materials
0 f I 0FB
f I
f IIFB,λ=0.1
FB
-2 -2 II
f II
f
, =0.9 FB
,λ=0.1
FB λ II
III fFB,λ=0.9
-4 fFB,λ=0.1 -4 III
III fFB,λ=0.1fFB,λ=0.9
f IIIFB,λ=0.9-6 -6
-8 -8
-10 -10
5 10 15 20 5 10 15 20
local iterations local iterations
Figure 2.14: Evolution of the residua of the different FB functions plotted over the number of local
iterations at t= tII = 1.1 s for δ= 10
−6 with ∆t= 0.02 s (left) and ∆t= 0.002 s (right). Reprinted from
Bartel et al., Arch. Appl. Mech. 89 (2019), 995-1010, [6], with permission from Springer Nature.
0 0
f IFB
-2 -2 f IIFB,λ=0.1
f IIFB,λ=0.9
-4 -4
f IIIf I FB,λ=0.1FB
III
f II ,λ=0.1 fFB,λ=0.9-6 FB -6
f IIFB,λ=0.9
-8 f IIIFB,λ=0.1 -8
f IIIFB,λ=0.9
-10 -10
5 10 15 20 5 10 15 20
local iterations local iterations
Figure 2.15: Evolution of the residua of the different FB functions plotted over the number of local
iterations at t= tII = 1.1 s for δ= 10
−4 with ∆t= 0.02 s (left) and ∆t= 0.002 s (right). Reprinted from
Bartel et al., Arch. Appl. Mech. 89 (2019), 995-1010, [6], with permission from Springer Nature.
framework independent of the precise values of α, cf. Figure 2.17. This also holds for
examples where f IFB turns out to yield inaccurate results or even diverging iterations.
2.3.4 Summary
In this section, the numerical stability and efficiency of five different modified Fischer-
Burmeister NCP functions is investigated. For this purpose, these are applied to the
laminate-based material model for ferroelectrics taken from [17] which is also summarised
in Section 2.3.1.
The analysis of the convergence behaviour of the different approaches based on a
Newton-Raphson scheme reveals that unambiguous comparative statements regarding
the effectiveness and efficiency of the respective algorithms are hardly possible. For
rather small time increments, all applied schemes perform similarly and lead to robust
41
Residua in log10 Residua in log10
Residua in log10 Residua in log10
2 Classic parameter identification based on homogeneous states of deformation
1.2 1 1.2 1
r r
I II
1 1
1 Γ 1 ΓI II
2 2
r r
I II
0.8 2Γ 0.8 2Γ
I II
3 3
0.6 r 0.6 rI II
3 3
Γ Γ
I II
0.4 4 0.4 4r r
I II
4 4
Γ Γ
0.2 I 0.2 II
5 5
r r
I II
0 5 0 5Γ Γ
I II
5 5 5 5
r Γ r Γ
-0.2 I I -0.2 II II
0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70
n n
Figure 2.16: First (left) and second (right) set of constraints and corresponding Lagrange multipliers
plotted over the number of global iterations at t= t =1.1 s, while using the f IIIII FB function with δ=10
−2,
λ=0.1 and ∆t=0.02 s. Reprinted from Bartel et al., Arch. Appl. Mech. 89 (2019), 995-1010, [6], with
permission from Springer Nature.
0 0
f I IFB fFB
-2 f IVFB,α=0.1
-2 fVFB,α=0.1
f IVFB,α=10.0 f
V
FB,α=10.0
-4 -4
f IVFB,α=100.0 f
V
FB,α=100.0
f IVFB,α=1000.0 f
V
FB,α=5000000.0-6 -6
-8 -8
-10 -10
5 10 15 20 5 10 15 20
local iterations local iterations
Figure 2.17: Convergence study of the impact of the scaling parameter α on the residua of the NCP
functions f IV (left) and fV (right) at t= t = 0.8 s, with ∆t= 0.02 s and δ = 10−6FB FB I . Reprinted from
Bartel et al., Arch. Appl. Mech. 89 (2019), 995-1010, [6], with permission from Springer Nature.
algorithms which allow the simulation of cyclic loading paths. However, smaller time
increments result in higher computational costs which necessitates approaches with im-
proved efficiency. When using larger time increments one needs to carefully adjust the
underlying parameters such as the weighting factor λ and the perturbation parameter δ.
Concerning the latter, higher values for δ may result in stable Newton schemes, however,
not necessarily. Moreover, higher values of δ result in more significant violations of the
underlying constraints. In this regard, one should rather use small values for the per-
turbation parameter δ, even δ= 10−12 worked out well according to our investigations.
In addition, the scaling parameter λ should be chosen to be large enough so that the
stabilising term does not dominate the actual Fischer-Burmeister NCP function. It is
noteworthy, though, that for sufficiently small values of δ and sufficiently large values of
λ the penalised Fischer-Burmeister function f IIFB and its modified version f
III
FB as well as
42
Residua in log10 i , i i ir Γ , r Γ
I I I I
Residua in log10 ir , i , i iΓ r Γ
II II II II
2.4 Classic non-linear optimisation algorithms
f IVFB and f
V
FB could not be shown to perform significantly better than the straight-forward
smoothed Fischer-Burmeister function f IFB. In future work it could be of interest to fur-
ther enhance the algorithms by introducing non-constant parameters δ, λ, and α similar
to updated Lagrange multiplier approaches.
2.4 Classic non-linear optimisation algorithms
In this section, a brief introduction and summary to the huge field of optimisation
problems is given following the works of Alt [2], Geiger and Kanzow [24], Luenberger
and Ye [70], Mahnken [72], Nocedal and Wright [85]. More detailed information can be
taken from these sources, the master thesis of [112], the PhD theses of Kleuter [52], Rose
[96], Scheday [101] or the habilitation thesis of Mahnken [74].
Optimisation problems occur in various application fields. In the field of material
modelling, parameter identification is generally known as the optimal fitting of the
parameters of an underlying material model with regard to experimental data under
consideration of necessary physical or mathematical constraints. In the perfect case, the
material parameters can be directly identified based on specific experiments where the
parameters impact the material behaviour uncoupled from each other, e.g. the Young’s
modulus E and the Poisson’s ratio ν in a one-dimensional tension test within the elastic
region. Unfortunately, this is generally not possible for many material models depicting a
non-linear, inelastic, anisotropic or coupled—e.g. a thermo-mechanical model—material
behaviour. Nevertheless, it is possible in many cases to identify a certain subset of the
parameters based on some specific experiments which improves the overall efficiency and
performance of the optimisation process.
Furthermore, it is very important to be aware of the following possible sources of non-
conformances to be able to analyse the deviations between the measured and predicted
material behaviour, even if the ”optimal” parameter set is used:
• Modelling errors: The complex real material behaviour cannot be captured com-
pletely by developed material models and consequently the parameter identifica-
tion is always restricted to the precision of the applied material model.
• Measuring errors: Since specimens are not completely identical and because even
standardised experiments, e.g. following ISO-standard, cannot eliminate measuring
errors, several experiments have to be performed under the same conditions and
statistically evaluated to achieve reliable experimental data.
• Simulation errors: Numerical errors occur during the optimisation process as well
as during the simulations with the developed material model and thus have to be
taken into account.
In the end, simulations are performed in order to predict the real material behaviour
based on a given material model, the corresponding parameters, loading history and
43
2 Classic parameter identification based on homogeneous states of deformation
its initial conditions. This is denoted as the direct problem. However, to identify the
optimal set of material parameters for a specific material, the inverse or indirect problem
has to be solved. For this purpose, the simulated material response is compared to
the experimentally measured response for a given material model, loading history and
initial conditions. Since the amount of experimental material data is generally larger
than the total set of model parameters, the inverse problem is over-determined and thus
a unique solution is usually not achievable. Nevertheless, the optimal parameter set
can be obtained by employing a least-square minimisation. The inverse problem can be
formulated as
1∥ ∥2
f(κ) := ∥w · [Mf(κ)− f exp]∥ → min , (2.83)
2 ∀κ∈K
where w is a set of weighting factors which can be used to increase or decrease, respec-
tively, the impact of specific data points on the optimisation, e.g. if the experimental
data is not equally reliable over the whole range or if the simulated response has to be
more accurate at certain points. Since usually the simulated and experimentally mea-
sured data points do not coincide,M symbolises an abstract projection operator which
is in the simplest case a linear interpolation in order to compare both sets. Further-
more, f(κ) denotes the simulated force vector with the current parameter set κ which
is compared to the experimentally measured force vector f exp if the displacements are
prescribed. In the case of a homogeneous deformation where only scalar information is
taken into account, the force vector contains the integral force at discrete time steps of
the loading path. The whole set of admissible material parameters K is defined as
K = {κ |h(κ) = 0, g(κ) ≤ 0} , (2.84)
with h(κ) and g(κ) being the sets of equality and inequality constraints, respectively.
These constraints are necessary for many parameters, e.g. even for the two most known
parameters of an isotropic, linear-elastic material model, the Young’s modulus E > 0
and the Poisson’s ratio −1 ≤ ν ≤ 1/2.
In all scientific and engineering fields optimisation is one of the basic tasks. Follow-
ing Nocedal and Wright [85], optimisation problems can be categorised based on the
characteristic of the objective function and its corresponding constraints—e.g. linear,
non-linear, convex—on the amount of variables or many other categories. One of the
basic distinctions is between zero-order methods where only the objective function eval-
uations are taken into account and between first-order methods where, additionally to
the function evaluations itself, their corresponding gradients are required. The different
algorithms are otherwise also classified into being either deterministic or stochastic. The
most known example of the latter group is the Monte-Carlo method. Algorithms of this
group choose the parameter set of the next iteration step purely random-based without
any consideration of the achieved information of previous steps. These algorithms are
very simple to implement but computationally also very inefficient. The efficiency of
44
2.4 Classic non-linear optimisation algorithms
a purely stochastic method can be improved by employing an evolution-strategy based
algorithm. In contrast, the deterministic algorithms are generally more efficient. Com-
mon examples are the gradient-based Gauss-Newton and the gradient-free Nelder-Mead
simplex method. In the literature, the algorithms are mainly classified into constrained
and unconstrained methods.
2.4.1 Unconstrained optimisation algorithms
In the case of unconstrained optimisation algorithms, even if some variables may possess a
form of natural constraint, the constraints do not affect the solution at all. Nevertheless,
unconstrained algorithms can consider bounds of the parameters if a penalty method is
incorporated in the implementation.
Gradient-based optimisation algorithms are very efficient since the information of the
gradient is considered in the objective function as well. In this group of algorithms, the
parameter set of the next iteration κi+1 is calculated based on the previous set κi and
a specific search direction si which incorporates the information of the gradient, i.e.
κi+1 = κi + αi si , (2.85)
where αi is denoted as the step size parameter. While the step size parameter has to
fulfil αi > 0, the following condition for the search direction has to be ensured
∇tκf(κi) · si(κi) < 0 . (2.86)
Furthermore, the following one-dimensional minimisation problem
f(κi+1(αi)) = f(κi + αi si)→ min (2.87)
can be used to calculate the step size parameter αi and the optimal step size can be
determined by the roots of the necessary condition
dα f(αi) = 0 , (2.88)i
which can be solved in closed form for some special cases or numerically by using regula
falsi or the Newton-Raphsom scheme. Considering the definition of the search direction,
many different approaches have been developed. A straightforward approach is the
method of steepest decent, where the search direction is defined as si := −∇κf(κi) such
that it is tangential to the corresponding isoline. Another possibility is the conjugate-
gradient method which is very convenient for large non-linear optimisation problems
since the Hessian matrix H(κi) := ∇2κκf(κi) is not required, the step computation is
not based on matrix operations and it does not require a large vector storage. In contrast,
the Newton-Raphson method—with a search direction of s = −H−1i (κi) · ∇κf(κi)—
enables a quadratic convergence. However, a locally positive definite Hessian matrix
45
2 Classic parameter identification based on homogeneous states of deformation
is required and its inversion is computationally rather expensive. The computational
cost can be reduced by employing Quasi-Newton methods where the inverse Hession is
replaced by different updating relations, e.g. Davidson, Fletcher and Powell (DFP) or
the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.
Since the derivation of the gradient of the objective function is not always straightfor-
ward and can imply a high effort, it can be overall more efficient to employ simpler but
numerically less efficient algorithms, or some libraries in e.g. Python provide the oppor-
tunity to determine the gradient numerically. While the latter option might be helpful,
it can be even more efficient to use gradient-free strategies. Thus, a zero-order method,
the Nelder-Mead simplex algorithm is employed in this work for many application cases
since it is provided in several optimisation packages, e.g. the scipy optimisation package
in Python, and is a simple, robust, reliable and quite efficient algorithm. The method
was developed by Nelder and Mead [84] and is only briefly summarised here. If an ob-
jective function contains n different parameters, the method spans a simplex with n+ 1
corner points, denoted as P0, P1, . . . , Pn. The final simplex, containing the final mini-
mum, is obtained once the vertex with the highest function value Pmax was replaced with
a new point within each iteration until a specified tolerance is reached. For this purpose,
the method comprises th√ree∑main operations—reflection, expansion and contraction—toreplace the vertex with the highest function value such that the simplex is shrunken to
the final simplex where i[yi − yc]2/n is below the user-defined tolerance. Here, yi
denotes the function value of the corner points Pi and yc denotes the function value of
the centroid of the simplex Pc. The operations of this method are schematically visu-
alised in Figure 2.18. In each iteration, at first a reflection is performed for Pmax such
that the reflected point P ? is obtained via
P ? = [1 + α]Pc − αPmax , (2.89)
where α denotes the reflection coefficient, which is defined positive. Thus, the reflected
point is on the line intersecting Pc and Pmax and placed opposing to Pmax. Quantity
y? is the corresponding function value of P ?. If y? ∈ (ymin, ymax), the vertex Pmax is
replaced by the reflected point P ? and the optimisation process continues with the new
simplex. However, if y? ≤ ymin, a new minimum is found and the expansion operation is
performed to expand the simplex further in the new minimum direction. The expanded
point P ?? is achieved by
P ?? = γ P ? + [1− γ]Pc , (2.90)
where γ > 1 is used as expansion coefficient. If the function value y?? of P ?? is lower
than the previous function minimum of the simplex, Pmax is replaced with the expanded
point P ??. However, if y?? > ymin, Pmax is only replaced by the reflected point P
? and the
optimisation process continues with the new simplex. Nevertheless, if y? is still higher
than any other function value of the other vertices, the contraction operation has to be
46
2.4 Classic non-linear optimisation algorithms
performed. If in addition y? < y , the reflected point P ?max is set as Pmax. The contracted
point is defined as
P ?? = β Pmax + [1− β]Pc , (2.91)
with β ∈ (0, 1) denoting the contraction coefficient, which controls the magnitude of the
contraction, such as α and γ for the other operations. However, if y?? is still higher than
ymax and y
?, i.e. the contraction was not successful, each of the vertices Pi has to be
reduced to [Pi + Pmin]/2 in order to shrink the overall simplex.
P ⋆⋆
P ⋆min P
Pmin P ⋆
Pmax Pi Pmax Pi
(a) Reflection operation. (b) Expansion operation.
P ⋆ ⋆min P Pmin P
P ⋆⋆
P ⋆⋆
Pmax Pi Pmax Pi
(c) Contraction operation version 1. (d) Contraction operation version 2.
Pmin
Pmax Pi
(e) Shrinking operation.
Figure 2.18: Schematic visualisation of the different Nelder-Mead simplex operations following [84].
47
2 Classic parameter identification based on homogeneous states of deformation
A well laid out flowchart and more detailed information is given in [84].
Alternatively, even though a stochastic method causes a high computational effort,
it can be employed for the inverse problem since it is a rather simple and robust strat-
egy which can overcome local minima based on the random selection in each iteration.
However, as already mentioned before, the efficiency can be improved by using evolu-
tion strategy-based algorithms which adopt the biological evolution process and which
are driven by the principles of mutation and selection. At the beginning, a set of start
vectors is selected, defined as parent generation, and subsequently, following a random
number generation providing a Gaussian distribution around the parent generation, a
corresponding number of offspring vectors is generated. This symbolises the principle
of mutation in evolution processes. In the subsequent selection step, a set of different
parameter combinations providing the lowest objective function values is selected as the
parent generation of the next iteration step following the principle of survival of the
fittest. Since the standard, single-branch algorithm, where only one parameter set is
chosen for the parent generation, is not very efficient, a multi-branch evolution strategy-
based algorithm can improve the overall efficiency where many different sets are selected
simultaneously.
The two latter mentioned algorithms are implemented in the optimisation tool—
developed in [104]—applied within this work and summarised in Section 4.1.
2.4.2 Constrained optimisation algorithms
In the case of constrained optimisation algorithms, constraints of the parameters are cru-
cial with regard to the solution of the objective function. The corresponding constraints
can be either simple bounds of the parameter, e.g. for the Poisson’s ratio−1 ≤ ν ≤ 1/2, a
common linear constraint, for example if the sum of the parameters should be less equal
1, or even non-linear inequality constraints. Following this the optimisation problem
takes the form
f(κ)→ min , s. t. h(κ) = 0 and g(κ) ≤ 0 , (2.92)
where h(κ) and g(κ) denote the sets of equality and inequality constraints, respectively.
However, the solutions to this problem can be found by reformulating the optimisation
problem into the Lagrange functional
L(κ,µ,λ) := f(κ) + µt h(κ) + λt g(κ)→ stat , (2.93)
which is required to be stationary and where µ := [µ1, . . . , µ
t
n ] and λ := [λ1, . . . , λ
t
h ng ]
are denoting the vectors of the corresponding Lagrange multipliers. Integers nh and ng
are the numbers of equality and inequality constraints, respectively. If the following
Karush-Kuhn-Tucker (KKT) conditions are satisfied, the necessary stationary require-
48
2.5 Parameter identification for homogeneous states of deformation using classic
optimisation methods
ment of the Lagrange functional is fulfilled and a local constrained minimum exists so
that
∇ !κL(κ?,µ?,λ?) = ∇κf(κ?) +∇κh(κ?) · µ? +∇ ? ?κg(κ ) · λ = 0 ,
∇µL(κ?) = h(κ?) = 0 ,
∇ ? ?λL(κ ) = g(κ ) ≤ 0 , (2.94)
λ? ≥ 0 ,
λ? · g(κ?) = 0 ,
where i and j are the running indices of the equality and inequality constraints. Fur-
thermore, it is essential that the evaluation of the Hessian of the Lagrange function
at the optimal point has to be positive definite. If this is applied to convex problems,
it provides the advantage that these KKT conditions are already the sufficient condi-
tions for a global minimum. The solution of the KKT conditions can be obtained by
employing different kinds of algorithms, e.g. the active-set strategy, penalty method,
complementarity function approaches by Fischer-Burmeister, see Section 2.3.2, or stag-
gered solution schemes. Three of the algorithms are presented and applied in Kiefer
et al. [51].
2.5 Parameter identification for homogeneous states of
deformation using classic optimisation methods
In this section, several parameter identification procedures are performed considering
homogeneous states of deformation. Thus, only specific loading path sequences of char-
acteristic standard experimental tests, e.g. the forces during a loading phase of a tensile
test, are taken into account for the error functional of the inverse identification. In the
following, the three previously introduced material models are calibrated with respect
to selected materials by using classic parameter identification methods. While for the
first two application cases a Nelder-Mead simplex algorithm was chosen, a constrained
Sequential Quadratic Programming (SQP) method was employed for the last application
example.
2.5.1 Finite plasticity model coupled to damage
The first application example is the gradient-enhanced damage model coupled to finite
plasticity presented in Section 2.1. However, since only homogeneous states of defor-
mation are considered in this chapter, the parameters associated with the local parts
of the model cannot be identified and are deactivated by setting and fixing the penalty
parameter βd at a value of 0.0, see Tables 2.2–2.4. Thus, the local damage variable dφ
49
2 Classic parameter identification based on homogeneous states of deformation
is no longer related to the global damage variable φ. The employed experiments are
presented in Section 3.1.1.
Table 2.2: Finally identified parameter sets for the sheet metal DP800 and the case-hardened steel
16MnCrS5 based on the first initial parameter set 1. The fixed parameters are not calibrated within
this section.
Symbol Description Initial guess 1 DP800 16MnCrS5 Unit
f objective function value 100 44.81 7.62 %
E Young’s modulus 23 23 36.106 GPa
ν Poisson’s ratio 0.035 0.035 1.81 · 10−5 −
σy0 initial yield stress 99.99 102.49 174.46 MPa
h hardening parameter 499.94 512.434 463.361 MPa
np hardening exponent 0.1 0.1 0.1 −
qvar variable damage threshold 5 5 8.213 MPa
nd damage exponent fixed at 0.667 −
η damage rate factor 1 1 1.101 −
ξvol volumetric damage factor fixed at 1.0 −
ξiso isochoric damage factor 0.3 0.3 0.264 −
ξq threshold factor 2 2 1.581 −
ξm effective stress factor 0.15 0.15 0.184 −
ηα coupling factor 1 1 0.15 −
cd regularisation parameter fixed at 0.54 N
βd penalty parameter fixed at 0.0 MPa
qmin initial damage threshold 4.999 4.999 2.081 MPa
For this application, the Nelder-Mead simplex algorithm turned out to be a proper
choice. Since the simplex algorithm tends to get stuck in local minima, several parameter
identification procedures with different starting sets had to be performed to find the
best local minimum. For this purpose, three different starting values for the parameters
were chosen as shown in Tables 2.2–2.4. The different sets were selected based on the
parameter ranges mentioned in Section 2.1.2 in order to have one set located at the
lower bounds of the parameters, one located in the middle and another one located
at the upper bounds. Nevertheless, not all of the parameters could be changed with
regard to this scheme since the underlying material model is quite sensitive with respect
to some material parameter combinations. This demonstrates the difficulty of finding
manually appropriate starting values especially if the user is not familiar with the model
and the parameter influences. To emphasise this issue some of the parameters are fixed
for simplicity reasons since it would be too time consuming for the user while finding
parameter sets related to successful simulations close to the chosen direction.
50
2.5 Parameter identification for homogeneous states of deformation using classic
optimisation methods
Table 2.3: Finally identified parameter sets for the sheet metal DP800 and the case-hardened steel
16MnCrS5 based on the first initial parameter set 2. The fixed parameters are not calibrated within
this section.
Symbol Description Initial guess 2 DP800 16MnCrS5 Unit
f objective function value 36.96 1.13 1.24 %
E Young’s modulus 114.998 182.348 131.317 GPa
ν Poisson’s ratio 0.175 0.162 0.217 −
σy0 initial yield stress 99.99 106.159 11.39 MPa
h hardening parameter 499.94 1243.474 760.481 MPa
np hardening exponent 0.1 0.157 0.163 −
qvar variable damage threshold 25 0.005 19.763 MPa
nd damage exponent fixed at 0.667 −
η damage rate factor 5 3.918 6.143 −
ξvol volumetric damage factor fixed at 1.0 −
ξiso isochoric damage factor 1.5 1.342 1.968 −
ξq threshold factor 10 15.194 12.022 −
ξm effective stress factor 0.75 0.528 2.97 · 10−7 −
ηα coupling factor 4.998 0.404 3.001 −
cd regularisation parameter fixed at 0.54 N
βd penalty parameter fixed at 0.0 MPa
qmin initial damage threshold 24.996 40.783 30.694 MPa
Analysing the results with the first initial guess with regard to the sheet metal DP800
in Table 2.2 and the Figures 2.19 and 2.20a, it can be concluded that the Nelder-Mead
simplex algorithm gets stuck in a local minima quite close to the initial guess. The al-
gorithm could not find a proper direction to further reduce the objective function value.
The final objective function value and especially the comparison of the corresponding
simulated and experimentally measured tensile curve demonstrate that this local minima
is far away from an appropriate solution. In comparison, the second initial guess already
provides an error function value lower than the obtained local minimum of the first cal-
ibration procedure. Furthermore, the found local minimum of the second identification
process fits the experimentally measured tensile behaviour quite well with a deviation of
only 1.13%, cf. Table 2.3. If the stress-strain curves in Figures 2.19 and 2.20b are taken
into account, the very accurate match of the experimental data is visible, even though
the stress-strain behaviour of the second initial guess was also not close to the realistic
material behaviour, see Figure 2.20b. Starting with the third initial guess a local mini-
mum with an objective function value of 1.2% is obtained which is very close to the final
function value of the second calibration procedure. However, comparing all of the final
parameter values, a huge deviation in most of the parameters can be observed, cf. Tables
51
2 Classic parameter identification based on homogeneous states of deformation
2.3 and 2.4. In contrast, both of finally obtained stress-strain curves fit the experimen-
tally measured curve very accurately, see Figures 2.19 and 2.20. A difference is mainly
visible at the beginning of plastification. Considering that the last two local minima
differ vastly in the obtained parameters but that both accurately fit the experimental
data, it indicates that the material model is very complex and that the combination
of the different material properties allows multiple parameter combinations to show a
nearly equal material behaviour—at least with respect to a rather simple tensile test if
all material parameters are simultaneously identified.
Table 2.4: Finally identified parameter sets for the sheet metal DP800 and the case-hardened steel
16MnCrS5 based on the first initial parameter set 3. The fixed parameters are not calibrated within
this section.
Symbol Description Initial guess 3 DP800 16MnCrS5 Unit
f objective function value 100 1.2 0.87 %
E Young’s modulus 206.999 226.05 132.959 GPa
ν Poisson’s ratio 0.315 0.277 0.296 −
σy0 initial yield stress 99.99 158.54 56.36 MPa
h hardening parameter 499.94 1174.807 774.599 MPa
np hardening exponent 0.1 0.159 0.217 −
qvar variable damage threshold 44.999 39.645 47.463 MPa
nd damage exponent fixed at 0.667 −
η damage rate factor 9 7.564 9.863 −
ξvol volumetric damage factor fixed at 1.0 −
ξiso isochoric damage factor 2.7 0.222 0.53 −
ξq threshold factor 18 15.194 16.183 −
ξm effective stress factor 1.35 6.19 · 10−5 1.319 −
ηα coupling factor 8.997 2.03 7.978 −
cd regularisation parameter fixed at 0.54 N
βd penalty parameter fixed at 0.0 MPa
qmin initial damage threshold 44.993 19.923 39.97 MPa
The chosen three initial guesses were also used in the next step for the calibration
of the model with regard to the cased-hardened steel 16MnCrS5. As it can be seen in
Tables 2.2-2.4, in summary all of the obtained local minima have a quite low objective
function value which is quite reasonable since all of the initial guesses are already rel-
atively close to the experimentally measured stress-strain curve, cf. Figure 2.22. While
the first obtained local minimum exhibits the highest function value and the identi-
fied material parameters are very different to the other results, the other two objective
function values are relatively close to each other. Furthermore, some of their material
parameters are only slightly different, e.g. Young’s modulus or the hardening parameter,
52
2.5 Parameter identification for homogeneous states of deformation using classic
optimisation methods
(a) Stress-strain curves of the tension test, with strain (b) Shear stress-shear strain curves of the in-plane tor-
defined as ε=∆l/l0. sion [test, with t]he shear strain defined as the relatedshear component of the Green-Lagrange strain tensor
E= F t · F − I /2 with the factor two.
Figure 2.19: For the DP800 sheet metal, the simulation results with the optimal parameter sets, ob-
tained from the parameter identification with the three different initial guesses, cf. Tables 2.2–2.4, are
compared to the experimental curves. The calibration was based only on tensile data, i.e. only the
displacements of the two loading phases and the forces of the unloading phase of the tensile test were
considered within the objective function. The shear deformation was not included in the error functional
and is used for validation purpose.
even though most of them still extremely differ. Comparing the stress-strain curves of
the results, as expected, the first solution shows the least agreement with the experi-
mental data while the other two solutions are quite accurate with regard to the tensile
behaviour. However, the third solution suits the plastification and damage evolution
more accurately, cf. Figures 2.21a and 2.22. The difference is clearly more recognisable
in the validation with the compression test, see Figure 2.21b. While the simulation with
the second local minimum is aborted, the simulation with the third identified parameter
set finishes successfully. The visible decrease in the compressive stress of the last result
is huge, and thus also the deviation from the experimentally measured curve. This is
based on the tension-compression asymmetry which is not included in the underlying
version of the material model. Finally, the third parameter identification provided the
most accurate result regarding elasticity, plasticity and damage evolution, and the only
deviations from the experimental data are due to the fact that the Lüders extension and
the tension-compression asymmetry are not incorporated in the employed version of the
material model.
In Section 3.1, an alternative approach to this parameter identification is presented
in the form of a hybrid strategy. For this purpose, a neural network is trained with the
material model in order to provide a good starting point for a subsequent multi-objective
parameter identification, shown in Section 4.3.5. This strategy makes it possible to
overcome the difficulty of finding appropriate initial guesses while additionally reducing
53
2 Classic parameter identification based on homogeneous states of deformation
(a) Stress-strain curves of the tension test based on (b) Stress-strain curves of the tension test based on the
the first initial guess, with strain defined as ε=∆l/l0. second initial guess, with strain defined as ε=∆l/l0.
(c) Stress-strain curves of the tension test based on
the third initial guess, with strain defined as ε=∆l/l0.
Figure 2.20: For the DP800 sheet metal, the calibration performance based on the different initial pa-
rameter sets is visualised regarding the simulation results with the initial guess and the finally identified
parameter sets. However, the initial guesses of set 1 and 3 result in incomplete simulations and only the
4th and 11th iteration, respectively, provided a parameter combination with a successful simulation.
Thus, the simulation results of those sets are visualised in the figure instead of the corresponding initial
guess.
the effort for the user. In addition, a classic multi-objective parameter identification
using the optimisation tool ADAPT, see [104], is given in Section 4.3.4.
2.5.2 Finite-strain viscoelasticity model coupled to damage
For the next application example, the finite-strain viscoelasticity model coupled to dam-
age, presented in Section 2.2, is calibrated to a special polymer—a self-diagnostic PDMS
elastomer. In this case, once again the Nelder-Mead simplex algorithm is used for the
identification of the material parameters. However, since the gradient-enhanced dam-
age formulation is based on the same framework, in a first step, only the viscoelastic
54
2.5 Parameter identification for homogeneous states of deformation using classic
optimisation methods
(a) Stress-strain curves of the tension test, with strain (b) Stress-strain curves of the compression test, with
defined as ε=∆l/l0. strain defined as ε=∆l/l0.
Figure 2.21: For the 16MnCrS5 case-hardened steel, the simulation results with the optimal parameter
sets, obtained from the parameter identification with the three different initial guesses, cf. Tables 2.2–
2.4, are compared to the experimental curves. The calibration was based only on tensile data, i.e. only
the displacements of the two loading phases and the forces of the unloading phase of the tensile test were
considered within the objective function. The compression test was not included in the error functional
and is used for validation purpose.
material parameters are identified with respect to three characteristic tests based on
homogeneous deformation states. At first, the corresponding experiments are presented,
followed by the description of the applied parameter identification strategy and the final
results.
2.5.2.1 Homogeneous deformation tests
In the following, all accomplished types of experiments for the self-diagnostic PDMS
elastomer are briefly presented in order to provide an overview of the experimental data
required for the subsequent parameter identification of the implemented material model.
For most of the experiments, a micro-testing machine by Kammrath & Weiss GmbH
(Dortmund, Germany) was used; cf. Figure 2.23a. The machine can be combined with a
CCD camera system from Chemnitzer Werkstoffmechanik GmbH (Chemnitz, Germany),
which is of importance considering the need to track the displacement field for the inho-
mogeneous deformation tests in Section 4.2.1. The inhomogeneous deformation tests are
required for the full parameter identification of the damage-related material parameters
in Section 4.2.2. Figure 2.23b presents a screenshot of the software VEDDAC strain. The
software was linked with the CCD camera and provided additional information on the
displacement field during the experiment. The CCD camera additionally took a photo
during each time step and delivered the changes in the displacement field directly to the
DDS3 software of the testing machine. In the post-processing, the software VEDDAC
could be used to extract all required displacement information from the photos taken
during the experiment. Since the material was a transparent soft polymer, a stochastic
55
2 Classic parameter identification based on homogeneous states of deformation
(a) Stress-strain curves of the tension test based on (b) Stress-strain curves of the tension test based on the
the first initial guess, with strain defined as ε=∆l/l0. second initial guess, with strain defined as ε=∆l/l0.
(c) Stress-strain curves of the tension test based on
the third initial guess, with strain defined as ε=∆l/l0.
Figure 2.22: For the 16MnCrS5 case-hardened steel, the calibration performance based on the different
initial parameter sets is visualised regarding the simulation results with the initial guess and the finally
identified parameter sets. However, the initial guesses of set 1 and 3 result in incomplete simulations
and only the 4th and 11th iteration, respectively, provided a parameter combination with a success-
ful simulation. Thus, the simulation results of those sets are visualised in the figure instead of the
corresponding initial guess.
pattern was sprayed onto the specimens with graphite spray. Consequently, the software
was able to track all specific positions of the specimen up to the end of the experiment.
The following experiments were conducted with virgin material samples, thereby avoid-
ing the influences of Mullin’s effect or the strain history on the material response. The
environmental conditions were room conditions, including room temperature. As a first
step, experiments with homogeneous deformation states were conducted. In order to
receive sufficient experimental data for the calibration of the viscoelastic material pa-
rameters, three different types of experiments were carried out. Apart from uni-axial
tensile tests (cf. Figure 2.24), relaxation tests (see Figure 2.25), as well as creep tests (cf.
Figure 2.26) provided the experimental data basis for the following parameter identifi-
56
2.5 Parameter identification for homogeneous states of deformation using classic
optimisation methods
cation; see Section 2.5.2.2. Tensile tests with different strain rates were not considered
here, as we instead focused on capturing the viscosity-related parameters of the ma-
terial via the explicit incorporation of creep and relaxation characteristics within the
error square functional. In the case of all these experiments, a homogeneous rectangular
specimen with an initial length of 25 mm, thickness of 2.2 mm, and width of 10 mm was
used. An example of the specimen is shown in Figure 2.23b at the beginning of the
corresponding loading path.
(a) Micro-testing machine by Kammrath & (b) Rectangular specimen in the micro-testing machine un-
Weiss GmbH. der tension visible via the VEDDAC strain software.
Figure 2.23: Experimental setting—the micro-testing machine by Kammrath & Weiss GmbH in com-
bination with the CCD camera system by Chemnitzer Werkstoffmechanik GmbH providing load-
displacement data, as well as the displacement field on the surface. Reprinted from Schulte et al.,
Materials 13(14):3156 (2021), [110], with permission under creative commons 4.0.
1.5 0.45
1.45 0.4
1.4 0.35
1.35
0.3
1.3
0.25
1.25
0.2
1.2
0.15
1.15
1.1 0.1
1.05 0.05
1 0
0 5 10 15 20 25 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5
Time [min] Stretch [-]
(a) Strain-time curve. (b) Nominal stress (P11)-stretch (λ) curve.
Figure 2.24: Tensile test results of the micro-testing machine, where the strain path shown in (a) is
applied which results in the nominal stress shown in (b). Reprinted from Schulte et al., Materials
13(14):3156 (2021), [110], with permission under creative commons 4.0.
57
Stretch [-]
Piola stress P [MPa]
2 Classic parameter identification based on homogeneous states of deformation
3 0.72
0.7
2.5
0.68
2 0.66
0.64
1.5
0.62
0.6
1
0.58
0.56
0 15 30 45 60 75 90 105 120 0 15 30 45 60 75 90 105 120
Time [min] Time [min]
(a) Strain-time curve. (b) Nominal stress (P11)-time curve.
Figure 2.25: Relaxation test results of the micro-testing machine, where the stretch level shown in (a)
is fixed which results in the nominal stress decrease over time shown in (b). Reprinted from Schulte et
al., Materials 13(14):3156 (2021), [110], with permission under creative commons 4.0.
2.5.2.2 Parameter identification
In the context of parameter identification procedures, it is advantageous to compute
the Piola stress tensor as a quantity, facilitating the comparison of experimental and
simulated material response; cf. Section 2.5.2.1. The Piola stress tensor P is related to
the Piola–Kirchhoff stress tensor S and the Kirchhoff stress tensor τ via
P = τ · F−t = F · S , (2.95)
where the sensitivity of the Piola stress with respect to the deformation gradient is
denoted as A, in particular,
∂P ∂2ψloc
A = = ⊗ , (2.96)∂F ∂F ∂F
or, expressed in index notation,
∂2ψloc
AaAbB = . (2.97)
∂FaA ∂FbB
Moreover, A relates to the spatial elasticity tensor e via
A = F−1[e −1aBcD Bb abcd + τacδbd]FDd . (2.98)
It is noted that the tangent operator, as typically required within implicit finite ele-
ment formulations, respectively so-called constitutive drivers, is based on A = dP /dF .
58
Stretch [-]
Piola stress P [MPa]
2.5 Parameter identification for homogeneous states of deformation using classic
optimisation methods
8
0.5
0.45 7
0.4 6
0.35
5
0.3
4
0.25
0.2 3
0.15
2
0.1
1
0.05
0 0
0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 160
Time [min] Time [min]
(a) Nominal stress (P11)-time curve. (b) Displacement (u)-time curve.
Figure 2.26: Creep test results of the micro-testing machine, where the stress shown in (a) is fixed over
time which results in the displacement change shown in (b). Reprinted from Schulte et al., Materials
13(14):3156 (2021), [110], with permission under creative commons 4.0.
The Yeoh-type hyperelastic energy function was combined with the constitutive vis-
coelasticity model, presented in Section 2.2, including two Maxwell elements. Apart
from the Yeoh material parameters C1, C2 and C3, the viscoelastic parameters γ1 and
γ2, denoting the relation of the stiffness of each Maxwell element with respect to the
pure elastic Young’s modulus, and τ1, τ2, representing the ratio of the viscosity to the
stiffness in each Maxwell element, need to be identified. Regarding the parameters, it
has to be mentioned that the material was assumed to be nearly incompressible, and
thus, Poisson’s ratio was fixed to ν= 0.49. Thus, seven material parameters needed to
be identified.
Since these seven material parameters could be identified via experiments of homoge-
neous deformation states, strain- and stress-driven constitutive drivers were implemented
in MATLAB instead of full finite element (FE) simulations in order to reduce the com-
putational cost within each iteration of the parameter identification. The pseudo-codes
of both types of constitutive drivers are depicted in Algorithms 1 and 2 in the Appendix
A.2. The fminsearch-algorithm in MATLAB was used for the parameter identification,
minimising the goal function with respect to the difference in the simulated and experi-
mental reactio∑n force∣over all load steps,∣i.e.
f =wT
∣
wT ∣ 2 ∑ ∣ ∣2PT,exp − PT,sim ∣ R R ∣t t∣ t (κ)∣ ∣+ w wt ∣P
R,exp R,sim ∣
∑ t
− Pt (κ)∣
t ∣ ∣ t2
+ wC wC C,exp C,simt ∣ut − ut (κ)∣ , (2.99)
t
59
Piola stress P [MPa]
Displacement [mm]
2 Classic parameter identification based on homogeneous states of deformation
where wT, wR, and wC denote the weighting factors for the tensile, creep, and relaxation
test, respectively. The additional weighting factors wTt , w
R C
t , and wt were introduced to
emphasise specific time steps t of each experiment and to appropriately scale stress and
displacement contributions.
In the following, the viscoelastic material parameters are identified via the three tests
based on homogeneous deformation states, presented in Section 2.5.2.1.
If the parameters of the Yeoh-model were solely fitted with respect to the tensile
test, the simulated material response of the identified parameters matched the experi-
mental data perfectly. Next, the relaxation parameters were obtained with respect to
the relaxation and creep test, though the material behaviour highly depended on the
previously identified Yeoh parameters. Thus, the experimental relaxation and creep re-
sponse could not be sufficiently matched via optimisation of the relaxation parameters
only. Consequently, all of the material parameters were identified simultaneously for all
three different experiments in order to obtain the best parameter set reflecting the com-
plete material behaviour. The comparison of the simulated and experimental material
behaviour is shown in Figure 2.27 for the optimised parameter set. The corresponding
parameter set is presented in Table 2.5.
In the graphs shown in Figure 2.27, the experimental results of the relaxation and
creep tests are nearly perfectly matched, and only slight differences in the diagrams can
be seen at certain points. The simulated material response of the tensile test overes-
timated the experimental curve, but captured the general tendency. The tensile test
could be captured better by increasing the associated weight within the objective func-
tion (2.99) at the cost of the accuracy with which creep and relaxation tests were cap-
tured. The focus here was set on the rate-dependent constitutive characteristics that
were captured with high accuracy. Though beyond the scope of this work, a precise
capturing of tensile, creep, and relaxation test at the same time could be achieved by
introducing higher order energy functions and a larger number of independent Maxwell
branches within the constitutive framework provided. The accompanying increase in pa-
rameter identification complexity due to a significantly increased number of constitutive
parameters could then be dealt with by using parameter correlation matrices.
Considering the curve of the tensile test, especially the values and the signs of the Yeoh
parameters are important, since the first parameter C1 weights the linear behaviour of
the stress strain-response, whereas C2 scales the quadratic term of the isochoric energy
contribution resulting in the decreasing slope. The third parameter C3 corresponds
to the third-order polynomial contribution of the energy function and thus yields the
increasing slope at the end of the loading path. Thus, different values for the Yeoh
parameters would probably fit the tensile response way better; the dependence on the
relaxation parameters, however, would result in a worse fit of the relaxation and creep
behaviour. Higher order contributions to the Helmholtz free energy, in particular the
isochoric contribution, would therefore provide a potential in order to capture the tensile
test response better.
60
2.5 Parameter identification for homogeneous states of deformation using classic
optimisation methods
The identified viscoelastic material parameters were used as fixed values for the sub-
sequent optimisation of the damage-related parameters in Section 4.2.
Table 2.5: Optimised set of material parameters for the Yeoh material model, as well as the relaxation
parameters for the two Maxwell elements. Reprinted from Schulte et al., Materials 13(14):3156 (2021),
[110], with permission under creative commons 4.0.
Symbol Description Final set Unit
C1 0.19550588 MPa
C2 Yeoh parameters 0.11198637 MPa
C3 0.00685930 MPa
γ1 0.12862148 -Relative moduli
γ2 0.36026686 -
τ1 1879.5892 sRelaxation times
τ2 68.729741 s
2.5.3 Laminate-based model for ferroelectric materials
This section gives a short summary of the parameter identification process of the laminate-
based model for ferroelectric materials performed in Dusthakar et al. [17]. Most of the
required parameters have already been identified in experiments and taken directly from
the literature. These parameters define the linear piezoelectric response behaviour and
the coercive electric field constants. The remaining two model, respectively fitting, pa-
rameters need to be calibrated with respect to characteristic experimental data. In the
case of piezoelectric materials, butterfly and dielectric hysteresis curves contain the re-
quired data. In the following, the calibration is performed with regard to single crystal
BaTiO3 samples. The Sequential Quadratic Programming (SQP) method was used for
the parameter identification process minimising the least square error of the objective
function.
As it can be seen in Table 2.6, the piezoelectric coupling coefficients as well as the
permittivity parameters were taken from Zgonik et al. [140]. The computed values by
Schröder and Keip [106] were selected for the transversely-isotropic elasticity compo-
nents. From Yen et al. [136] the spontaneous polarisation and strain coefficients were
taken in addition to the coercive electric field limits for 90◦ and 180◦ domain switching.
All of the parameters were identified for single crystal tetragonal BaTiO3. Furthermore,
the magnitude of the saturation polarisation was assumed to be equal to the spontaneous
polarisation value with regard to a perfect single crystal without any impurities.
Nevertheless, the developed model by Dusthakar et al. [17] contains further model
parameters with an impact on the non-linear hysteric behaviour. These remaining two
fitting parameters were calibrated in [17] with respect to the characteristic experimen-
tally measured butterfly and dielectric hysteresis curves taken from Shieh et al. [114], see
61
2 Classic parameter identification based on homogeneous states of deformation
Experiment Simulation 0.8
0.4 0.7
0.6
0.3
0.5
0.4
0.2
0.3
0.1 0.2
0.1
Experiment Simulation
0 0
1 1.1 1.2 1.3 1.4 1.5 0 20 40 60 80 100 120
Stretch [-] Time [min]
(a) Tensile test—comparison of experimental and simu- (b) Relaxation test—comparison of experimental and sim-
lated material response. ulated material response.
7
6
5
4
3
2
1
Experiment Simulation
0
0 20 40 60 80 100 120
Time [min]
(c) Creep test—comparison of experimental and simulated
material response.
Figure 2.27: Comparison of experimental and simulation results of the (a) tensile, (b) relaxation, and
(c) creep tests for uni-axial stress states, using the optimised set of material parameters; cf. Table 2.5.
Reprinted from Schulte et al., Materials 13(14):3156 (2021), [110], with permission under creative
commons 4.0.
Figure 2.28. Shieh et al. based their studies on unpoled [001]-oriented BaTiO3 tetrag-
onal single crystal samples with a length of 5 mm, a width of 5 mm and a depth of
2 mm. A combined electromechanical loading path was applied to the samples consist-
ing of a constant compressive stress of six different levels in the range of 0 MPa and
2.7 MPa and a cyclic electric field of Emax3 = ±1.25 kV at a frequency of 0.2 Hz. In the
following numerical simulations, the strains ε33 and dielectric displacements D3 were
62
Piola stress P [MPa]
Displacement [mm]
Piola stress P [MPa]
2.5 Parameter identification for homogeneous states of deformation using classic
optimisation methods
Table 2.6: Material parameters of the laminate-based material model developed by Dusthakar et al. [17].
The values in the table of the parameters from [17] were taken from the literature [106, 136, 140].
Reprinted from Dusthakar et al., Mechanics of Materials 117:235-254 (2018), [17], with permission from
Springer Nature.
Symbol Description Value Unit Reference
e311 lateral piezoelectric coefficient −0.7 C/m2 [140]
e333 axial piezoelectric coefficient 6.7 C/m
2 [140]
e131 piezoelectric shearing coefficient 34.2 C/m
2 [140]
0 vacuum permittivity 8.854× 10−12 F/m [140]
11 dielectric permittivity 2200 0 F/m [140]
33 dielectric permittivity 56 0 F/m [140]
E1111 elastic stiffness 237.4× 103 MPa [106]
E1122 elastic stiffness 92.6× 103 MPa [106]
E 31133 elastic stiffness 111× 10 MPa [106]
E3333 elastic stiffness 151× 103 MPa [106]
E1313 elastic stiffness 61× 103 MPa [106]
P s spontaneous polarisation 0.26 C/m2 [136]
ηs1 spontaneous strain coefficient 0.67 % [136]
ηs2 spontaneous strain coefficient −0.42 % [136]
Ec90◦ coercive electric field for 90
◦ switching 0.26 kV/mm [136]
Ec180◦ coercive electric field for 180
◦ switching 0.23 kV/mm [136]
P sat saturation polarisation 0.26 C/m2 -
ηp viscosity-type parameter 0.01 m2/Cs -
calculated for the applied boundary conditions by using a constitutive driver routine,
following. [50, 79, 80].
In [17], the viscosity-type parameter ηp was fixed at a constant value during the
parameter identification process due to the fixed electrical loading frequency in the
experiments. The parameter ηp was set to ηp = 0.01 m2/Cs. The remaining two fitting
parameters were calibrated by minimising a least-square function f(κ), where κ denotes
the remaining parameter vector κ = {m, c}. Scalarm is the viscous-like parameter which
was introduced in the rate-dependent dissipation equation and c is the hysteresis shape
exponent which influences the remnant energy contribution. Both parameters have to be
greater than 0. Thus, Dusthakar et al. have chosen the following non-linear constrained
optimisation problem
κ = arg min{f(κ)} such that κ = {m > 0, c > 0} , (2.100)
κ
63
2 Classic parameter identification based on homogeneous states of deformation
(a) Experimentally measured butterfly curves. (b) Experimentally measured dielectric hysteresis
loops.
Figure 2.28: The Figure of the experimentally measured butterfly curves and dielectric hysteresis loops
for a [001]-oriented BaTiO3 tetragonal single crystal is taken from Dusthakar et al. [17]. They extracted
the data from Shieh et al. [114]. The single crystal was subjected to six different levels of a constant
compressive stress while a cyclic electric field of Emax3 = ±1.25 kV at a frequency of 0.2 Hz. Reprinted
from Dusthakar et al., Mechanics of Materials 117:235-254 (2018), [17], with permission from Springer
Nature.
where the least-square functional is depicted as∑ kT1
f(κ) = ∑N1 ∣wki ∣∣ ∣2εki,sim(κ, Eki,exp)− εki,exp ki,exp ∣(E )∣ (2.101)
2 1 33 3 33 3
k=1 i=1
∑ Nk2 ∣∣ ∣2
+ wkj ∣Dkj,sim(κ, Ekj,exp)−Dkj,exp ∣2 3 3 3 (Ekj,exp3 )∣  . (2.102)
j=1
While index T = 6 denotes the amount of different curves for the six applied stress
magnitudes, cf. Figure 2.28, the indices Nk1 and N
k
2 depict the number of data points that
were extracted from both characteristic curves with respect to each kth-set of externally
applied stress magnitude. In the objective function, for each stress level, the difference of
the simulated strain εki,sim ki,exp kj,sim kj,exp33 (κ, E3 ) and the dielectric displacements D3 (κ, E3 )
to the experimentally measured ones, εki,exp(Eki,exp) and Dkj,exp(Ekj,exp33 3 3 3 ), are taken into
account. Therefore, an interpolation between the simulated results and the experimental
data is performed. wki1 and w
kj
2 represent the non-negative, scalar weighting factors which
are defined as
wki1 := ∆t
ki,exp/∆tki,sim and wkj := ∆tkj,exp/∆tkj,sim2 , (2.103)
64
2.5 Parameter identification for homogeneous states of deformation using classic
optimisation methods
Figure 2.29: The corresponding loading curve for the simulations performed within each iteration of the
calibration process. The applied potential difference is plotted over time. Reprinted from Dusthakar et
al., Mechanics of Materials 117:235-254 (2018), [17], with permission from Springer Nature.
with ∆t denoting the time step size between any two time steps of the underlying loading
curve. In Shieh et al. [114], the experimental curves were presented for one complete
triangular loading cycle, though no information on the loading path preceding the shown
curves was provided. However, based on the obtained data from [114], Dusthakar et al.
[17] started with the remnant configuration of the samples, cf. Figure 2.28. Therefore,
the poled sample is loaded and unloaded once at a frequency of 0.2 Hz with respect
to the simulations in order to reach the remnant state of the specimens, marked by
the shaded region OA in Figure 2.29, as shown in the experiments. The part of the
simulated material response which is considered for the comparison to the experimental
data starts at point A, see Figure 2.29.
First, Dusthakar et al. [17] used the ga solver (generic algorithm) of the Optimization
Toolbox in MATLAB in order to find proper initial values for both remaining model
parameters. This initial process was performed independently for each of the different
stress magnitudes. A suitable starting point was chosen for each parameter from the
obtained list of parameter values. Secondly, the chosen initial values were used in a
following parameter identification by using the fmincon solver, performing a constrained
non-linear minimisation, in combination with the SQP-based algorithm in MATLAB.
This time, all of the different curves are simultaneously taken into account. Finally,
after considering all data of the applied six different stress magnitudes, the optimal
viscous-type and hysteresis shape parameters are identified as
m = 1.494 and c = 3.648 . (2.104)
Considering the large effort which was necessary in [17] to identify these two remaining
parameters, an alternative approach for the generation of qualitative starting points is
presented in Section 3.2.
65

3 Machine-learning assisted
parameter identification based on
homogeneous states of deformation
In general, regarding the calibration of all different kinds of material models, three
basic approaches are established in order to identify an initial guess for a parameter
identification procedure:
• Guess of a random starting point.
• Definition of possible intervals for each parameter and selection of random param-
eter values within the respective intervals in order to generate starting values.
• Stepwise parameter identification, e.g. in case of elastoplasticity: Identification of
elastic parameters in a first step. Thereafter, these elastic parameters are fixed
and the calibration of plastic parameters is performed. This procedure can be
continued for, respectively applied to, every parameter class. As a result, one
starting point is generated.
The first strategy can be considered the most challenging, inefficient and costly approach.
The second method reduces the search space, though it might not be possible to identify
a proper interval for every parameter. The third strategy might lead to a very good
starting point but requires special algorithmic treatment. Another application specific
strategy was performed in Dusthakar et al. [17] and summarised in Section 2.5.3 which
improved the quality of the starting point by investing a high effort. In conclusion, all
approaches require knowledge of the corresponding material model or success in guessing
a proper starting point.
The work in this chapter seeks to contribute to this discussion by proposing a ma-
chine learning-based two-stage hybrid optimisation approach. In the first stage a model-
specific neural network predicts certain starting values, where the latter are then directly
transferred to a classic optimisation algorithm in order to obtain a mathematically op-
timal solution in a second stage. In addition, an intermediate stage of a subsequent
67
3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
68
Experimental data hom. def.
Trained ANN
Prediction of model parameters 𝜿𝜿pred
Optimisation hom. def.
Optimal parameter set 𝜿𝜿opthom
Optimisation with DIC data
Experimental data with DIC
Optimal parameter set 𝜿𝜿optDIC
Optimal parameter set 𝜿𝜿predDIC
Figure 3.1: Overall machine learning-assisted parameter identification scheme proposed in this work. In the first step, experimental data
based on homogeneous deformation states is fed into a neural network, cf. Figure 3.2, yielding a high-quality prediction of suitable parameters
for the underlying constitutive model. The predicted parameters κpred are used as starting values for a classic optimisation scheme, yielding
the optimal solution κopthom in a fully automated manner. The obtained solution might then again be considered as a starting value for
the DIC-based calibration of non-local model parameters, which is necessary for the chosen material model. Alternatively, the predicted
parameters κpred can be directly chosen as starting values for the DIC-based calibration. Reprinted from Schulte et al., European Journal
of Mechanics - A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
x times
Generate randomised
parameter sets
Run simulations
Associated parameters Stress/strain curves
80 % 20 %
Training Validation
New set of hyperparameters
ANN: estimation of
model parameters
based on provided
ANN input curve
Predicted parameters 𝜿𝜿pred
Update weight Model: compute
and bias stress/strain curves
based on provided
Backpropagation parameters
algorithm
Predicted curves
ANN performance
evaluation
Original curves
Averaged mean error
No
Minimum 
found?
Yes
Trained ANN
Figure 3.2: Visualisation of the training and validation procedure applied to the artificial neural net-
work (ANN). A superordinate scheme for the optimisation of the network’s hyperparameters is already
incorporated here. The ANN is used to predict a set of parameters κpred, which is then fed into a
subsequent classic optimisation procedure, cf. Figures 3.1 and 3.3. Reprinted from Schulte et al., Euro-
pean Journal of Mechanics - A/Solids 98:104854 (2023), [113], with permission under creative commons
licence 4.0.
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
Predicted model
parameters 𝜿𝜿pred as 
initial guess
New parameter set
No Constraints
fullfilled?
Yes
Simulation
Obj. function value
No Convergence
?
Yes
Optimal parameter set 𝜿𝜿opt
Figure 3.3: Flowchart depicting a classic deterministic optimisation algorithm, such as a basic Nelder-
Mead simplex algorithm, which we feed with the parameter set κpred identified by the neural network,
cf. Figures 3.1 and 3.2. Reprinted from Schulte et al., European Journal of Mechanics - A/Solids
98:104854 (2023), [113], with permission under creative commons licence 4.0.
calibration with respect to the homogeneous states of deformation is included in order
to evaluate as to whether the neural network already provides a sufficient solution. While
the first and the intermediate stages perform the simulations by using a FE-model in
terms of one-element with corresponding boundary conditions (by analogy with a consti-
tutive driver) enabling us to compute uni-axial stress states, the second stage is based on
simulations of an inhomogeneous boundary value problem in order to evaluate full-field
data. This strategy provides the advantage that the network only has to be trained once
per model and can subsequently be applied to different materials without further effort.
Furthermore, the already trained network can be provided along the corresponding ma-
terial model such that the user immediately obtains the optimal starting value for the
classic parameter identification. The scheme we propose makes it easier to take into ac-
count any combination of experimental loading path sequences for the model calibration,
where loading path sequences refer to both homogeneous modes of deformation—such
70
3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
as tension, compression, shear—as well as to modes of loading, particularly loading-
unloading sequences occurring in, e.g., experimental step tests. Furthermore, different
types of loading path sequences such as temperature levels can be considered. In line
with the loading path sequences occurring in the considered experimental tests, a model-
specific neural network is trained by repeated evaluation of the model response for the
pre-defined loading path sequences at random parameter sets.
The particular experimental data to which the model is to be fitted is then delib-
erately decomposed into the same loading path sequences, which are then normalised,
concatenated, and used as input data for the neural network, enabling the algorithm
to predict proper model-related constitutive parameters. These parameters are then
transferred as starting values to a classic optimisation procedure, see Figure 3.1, which
is an inverse parameter identification using full-field data in combination with finite el-
ement simulations. In this work, the FEMU method, cf. Section 1.1.3, is applied for an
application example of the second stage in Section 4.3.5. Besides the hybrid strategy,
the FEMU method is employed for all multi-objective optimisation schemes in Chapter
4.
In Section 3.1, the general procedure of the developed hybrid strategy is directly ex-
plained, presented and analysed by employing a specific application example. For this
purpose, the finite plasticity model coupled to gradient-enhanced damage, cf. Section
2.1, is considered as the underlying material model since it takes many material char-
acteristics into account and thus is a very complex material model. Hence, it is quite
difficult to calibrate the model to specific materials, especially if the user is not pro-
foundly familiar with the model and its parameters. Furthermore, a second application
example is given in Section 3.2, where a neural network is trained for the laminate-based
model for ferroelectrics, presented in Section 2.3, in order to improve the complex start-
ing point generation of the model parameters, cf. Section 2.5.3, and to demonstrate the
capability of the hybrid strategy to cope with all kinds of loading path sequences.
3.1 Machine-learning assisted parameter identification
introduced for a gradient-enhanced damage
formulation
In this section, the hybrid strategy is presented alongside a first application example of a
finite plasticity model coupled to a (gradient-enhanced) damage framework, cf. Section
2.1. In order to demonstrate some advantages of the model, the model is calibrated to
two different materials, a DP800 sheet metal and a case-hardened steel 16MnCrS5, and
different loading path sequences are taken into account. First, the required experiments
are shown in Section 3.1.1, followed by the presentation of the structure of the networks
in Section 3.1.2 and the corresponding training process in Section 3.1.3. In Section 3.1.4,
the quality of the predictions of the neural networks is compared to the results of a sub-
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
sequent classic parameter identification with respect to the experimental data presented
in the next section. This parameter identification process based on the homogeneous
states of deformation is closed with a short discussion of important aspects of the overall
process. The final second step of the hybrid strategy, where the parameter prediction
of the artificial neural network is employed as a starting set for a subsequent classic
multi-objective parameter identification, is demonstrated in Section 4.3.5.
3.1.1 Experiments
The goal of this work is not only to train a neural network with a constitutive model, to
be specific a large strain gradient-enhanced ductile damage model, but also to calibrate
the model to experimentally measured material behaviour. Two exemplary materials
for the application in metal forming processes are selected, i.e. a sheet metal DP800 and
a case-hardened steel 16MnCrS5. Since both materials apply to different production
process chains, distinct fundamental experiments were conducted for the materials. The
experiments were conducted by Sprave and Menzel [119] and Hering [35]. All experi-
ments were carried out under room conditions.
3.1.1.1 Tension test
For both materials, the common tension test at sufficiently low loading rates is the first
characteristic fundamental test. Since the key goal of this work is to generate a neural
network which autonomously predicts a high quality starting set of parameters for a
subsequent classic parameter identification for efficiency reasons, the simulations are
restricted to homogeneous states of deformation resulting in a prediction of sufficient
accuracy. Thus, it would be ideal to have (quasi) homogeneous states of deformation in
the experiments as well. In the case of the DP800, Sprave and Menzel [119] extracted the
homogeneous states of deformation of the tensile test following the procedure described
in the paper, namely by relating the total tensile force to changes in length of a certain
segment of the test specimen. The tensile test was conducted on a micro-tensile machine
by Kammrath & Weiß with a 10 kN load cell. The corresponding displacement response
is measured by an optical extensometer with the digital image correlation (DIC) system
Veddac 7. This leads to the DP800 tensile reaction force-strain curve in Figure 3.4.
The initial cross-sectional area is 5.775 mm2. Since it is not straightforward to extract a
comparable flat sample for the case-hardened steel 16MnCrS5 due to its availability as
rod material, respectively specimen, a cylindrical tension sample following DIN 50125-B
8x40 with a cross section diameter of 8 mm and an initial length for the tactile macro-
extensometer of 40 mm is employed. This tension test was conducted on the universal
testing machine Zwick Roell Z250 and the tension test followed the DIN EN ISO 6892-1
(2017) norm. For more information please be referred to [35].
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3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
Figure 3.4: Tension test data for both considered materials, i.e. DP800 and 16MnCrS5, with strain
defined as ε= ∆l/l0. Both experiments include an unloading phase at ε≈ 0.135, but refer to different
sizes of test specimen, cf. [35, 119]. Reprinted from Schulte et al., European Journal of Mechanics -
A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
In the first step, a strain of ε= 0.135 is applied on the samples for both materials.
Subsequently, an unloading phase is conducted where the force is reduced to zero followed
by a reloading until a total strain of 0.32 is reached, cf. Figure 3.4.
3.1.1.2 Compression test
Since the case-hardened steel 16MnCrS5 is typically available as a rod material, a com-
pression test is natural to identify characteristic behaviour of the material. The geometry
of the sample is a cylinder with a diameter of 10.2 mm and a height of 15 mm. For the
compression test, an initial load of 20 N, a rate of the compressive stress of 30 MPa/s
and a data sampling rate of 0.0067 s−1 were chosen. The measured compressive reaction
force-strain curve is shown in Figure 3.5.
3.1.1.3 In-plane torsion test
In the case of the sheet metal DP800, an in-plane torsion test, detecting simple shear-
type loading, is more convenient to be carried out than a compression test. The results
depicted in Figure 3.6 were obtained and published by Kolpak et al. [54]. Further
information on the theory of in-plane torsion tests can be found in, e.g., Traphöner et
al. [128]. For this test, a circular specimen with a diameter of 80 mm was used including
circumferential grooves. In the groove root, the thickness of the specimen reduces to the
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
Figure 3.5: Compression test of the case-hardened steel 16MnCrS5 with strain defined as ε=∆l/l0, cf.
[35]. Reprinted from Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113],
with permission under creative commons licence 4.0.
half of the initial thickness t0 =1.5 mm, and the groove radius is 4 mm while the radius
distance is 12.4 mm. The Piola shear stress is calculated via
M
τ(r) = , (3.1)
2π t(r) r2
where M is the measured torque, t the local initial sheet thickness and r the radial
position. The shear strain is measured via DIC. More detailed information regarding
the experiment can be found in [54].
3.1.2 Neural network for parameter identification
In the following, the general structure of the employed artificial neural networks (ANN) is
discussed, starting with the generation of the required training data and then discussing
the influence of the various parameters to be identified. The structure of the networks,
in particular the hyperparameters, is subsequently addressed.
3.1.2.1 Sampling of training data
In order to provide a feasible amount of training data for the neural network, randomised
parameter combinations are simulated by the constitutive model to generate a dataset
of ”curves” enabling supervised learning approaches. Considering the goal of this work,
i.e. to generate a parameter set as a starting point for a subsequent classic parameter
74
3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
Figure 3.6: In-plane torsion test of the sheet metal DP800 with the shear strain, here s[caled by fa]ctor
two, defined as the related shear component of the Green-Lagrange strain tensor E= F t · F − I /2,
cf. [54]. Reprinted from Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023),
[113], with permission under creative commons licence 4.0.
identification, only a sufficient approximation is required and, thus, only homogeneous
states of deformation are taken into account. This results in lower computational costs
for each simulation with a different parameter set than compared to the simulation
of more complex boundary value problems. Consequently, a larger amount of various
parameter combinations can be simulated within reasonable computational time. To
be specific, 6× 104 simulations were conducted for each of the exemplary experiments.
However, not all of the different variations result in a reasonable combination of the
model parameters and in a successful simulation. For example only 22521 of the 6 ×
104 combinations were completed successfully in the case of the tensile test simulation.
This demonstrates the sensitivity of the model with respect to the possible parameter
combinations.
Since not only the amount of data but also the distribution of the different parameter
combinations is important for the generation of a qualitative training set, the sampling
of the data has an impact on the prediction capabilities of the neural network. Hence,
two basic sampling methods are compared in the following.
3.1.2.2 Random sampling
For the first method, the Random Sampling (RS), a classic randomising function is
used to generate 6 × 104 values for each parameter within the admissible parameter
range. Thus, the required amount of random values within the interval is given, although
an even distribution of all 6 × 104 values in the interval is not guaranteed, albeit the
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
probability for a broad distribution being high. This random sampling method can be
mathematically described as
Rk ∈ [0, 1] , with k = 1, . . . , 6× 104 ,
κ ∈ [κmin, κmaxi i i ] , (3.2)
κk = κmin max mini i +Rk [κi − κi ] ,
where Rk denotes the random number, κ
k
i the i-th material parameter in its interval of
the corresponding minimum and maximum value.
3.1.2.3 Latin Hypercube Sampling
An alternative basic method to random sampling is Latin Hypercube Sampling (LHS).
For this method, the interval is divided into subintervals and a random value is gen-
erated within each of these subintervals. This leads to an even distribution over the
total parameter range. In addition, the position of the values for each subinterval is
random within the generated array for each parameter. Thus, well mixed parameter
combinations over the whole training data set are ensured.
An LHS function returns a Latin Hypercube sample matrix X of size n× p. For each
column of X, the n values are randomly arranged with one value from each subinterval
(0, 1/n), (1/n, 2/n), ... , (1− 1/n, 1), and randomly permuted, where n is the specified
amount of samples and where p is the amount of variables, i.e. material parameters.
Basically, a random perturbation different for each column, respectively parameter, is
applied to the midpoint in each interval, e.g. for the first interval 0.5/n. Furthermore,
the distribution of the intervals is random within each column.
An example for four parameters with tensample points is
0.1893 0.2569 0.0147 0.55830.8038 0.1089 0.9378 0.19500.5995 0.6818 0.3649 0.3097 0.3225 0.8736 0.4487 0.8055 0.9183 0.9854 0.1598 0.2509X =   . (3.3)0.0131 0.3864 0.5924 0.7511 0.7916 0.7131 0.2760 0.6662 0.6600 0.5420 0.6877 0.91000.2740 0.0450 0.7816 0.0631
0.4200 0.4855 0.8760 0.4889
To be specific, each column of X contains one random number in each of the 10
intervals (0, 0.1), (0.1, 0.2), (0.2, 0.3), (0.3, 0.4), (0.4, 0.5), (0.5, 0.6), (0.6, 0.7), (0.7, 0.8),
(0.8, 0.9), and (0.9, 1). In summary, the LHS provides a more even distribution of the
parameter ranges and consequently a better training data set. Nevertheless, a large
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3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
amount of samples in combination with a small parameter range and a low sensitivity of
the material response with respect to parameter changes reduces the advantage of LHS
over the simple random sampling.
3.1.2.4 Structure of the neural network
Considering the application in this work, a feedforward neural network is selected. To be
specific, a multilayer perceptron regressor (MLPregressor) is used here. The activation
function is set to the rectified linear unit function f(x) = max{0, x}. As a solver, we
employ a stochastic gradient-based optimiser suggested by Kingma, Diederik, Ba, see
[87]. The learning rate is initially set to 0.0001, while the shuffling of samples within
each iteration is turned off. Furthermore, the random number generation for weights
and bias initialisation is fixed to 1. The definition of the hidden layers is discussed in
the following paragraph. All other settings maintain the default values as described in
the documentation of the scikit-learn 1.0.2 package.
In order to find the optimal depth of the neural network, six different integer pa-
rameters are introduced to generate a variety of networks while maintaining a reduced
amount of parameters. Therefore, three of the parameters a, b, and c are used to generate
a polynomial p of the form
p(x) = a x2 + b x + c . (3.4)
The polynomial is evaluated at six positions, i.e. x= 0, 1, 2, 3, 4, 5, to obtain the corre-
sponding amount of neurons for the different layers. The amount of layers is defined by
the last three parameters. They describe the frequency of the layers with a specified
amount of neurons.
To give an example, the parameter set pNN =[1, 9, 2, 6, 60, 170] generates the polyno-
mial p(x) = 6x2 + 60x + 6 and the corresponding network with the fully connected hid-
den layers [170, 236, 314, 314, 314, 314, 314, 314, 314, 314, 314, 404, 404, 404, 404, 404, 404,
404, 404, 404, 506, 506, 620, 620]. The first two numbers of neurons each occur only in
one layer. The next two numbers each appear in 9 layers and the last two numbers each
in two layers. Even though the use of only six integers with this method restricts the
solution space for the possible hyperparameters, it enables a sufficiently large subset of
different combinations and an efficient structure to find a sufficient set of hyperparame-
ters.
3.1.3 Training of the neural network
This section presents the training and validation process of the neural networks based
on different sampling techniques. Subsequently, the influence of a hyperparameter opti-
misation is demonstrated.
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
Figure 3.7: Random selection of 100 curves of the training data set already arranged in the input format
for the neural network. The input data consists of three concatenated normalised modes, where the
modes correspond to the structure of the desired experimental data shown in Figure 3.4. Reprinted
from Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113], with permission
under creative commons licence 4.0.
3.1.3.1 Training and testing of the neural network
As input data for the neural network, the load-displacement curves are not directly used.
Since the displacement is prescribed in the loading phases and the force is reduced to
zero during the unloading phases, the relevant information for the network are the forces,
respectively stresses, for the loading phases and displacements, respectively strains, for
the unloading phases. Since the stresses and strains have different units, the values
are normalised based on the maximum occurring value over all simulations for each
component. For reasons of simple implementation, the input data is arranged as follows:
normalised stresses of the first loading phase, normalised stresses of the second loading
phase followed by the normalised strain of the unloading phase. A random selection
of 100 curves of the input data for the tensile test is shown in Figure 3.7. In order to
demonstrate the distribution of the training data, the frequency is shown in a heatmap
for both loading phases in Figure 3.8. Some exemplary curve shapes are shown in Figure
3.9.
In Section 3.1.2, the neural network was introduced, mentioning the structure of the
network as well as the different sampling techniques and the parameter ranges. In the
following, the mentioned structure with the initial hyperparameters is used in order to
compare the performance of the network based on the different sampling methods. For
this purpose, three distinct criteria are chosen:
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3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
Figure 3.8: Heatmap of all curves of the training data set already arranged in the input format for
the neural network demonstrating the frequency distribution and thus the modeling range. The input
data consists of three concatenated normalised modes, where the modes correspond to the structure of
the desired experimental data shown in Figure 3.4. Reprinted from Schulte et al., European Journal of
Mechanics - A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
Figure 3.9: Random selection of 5 curves of the training data set already arranged in the input format
for the neural network showing a view of the various curve shapes. The input data consists of three con-
catenated normalised modes, where the modes correspond to the structure of the desired experimental
data shown in Figure 3.4. Reprinted from Schulte et al., European Journal of Mechanics - A/Solids
98:104854 (2023), [113], with permission under creative commons licence 4.0.
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
• The first crite∑rion is the mean absolute errorntd δj
δ = AEMAE , (3.5)
n
j=1 td
with ntd being the number of test data sets. Moreover, δ
j
AE denotes the absolute
error
∑ ∣∣∣ ∫ ∫ ∣∣ ∣jinp pred ∣ ∣n ∣s ∣{∣∣ ∫•i dt−∣∣ ∣∣ ∫•i dt∣ ∣δjAE = ∣∣}∣∣ , (3.6)
i=1 max ∣∣ •inpdt∣∣, ∣∣ •predi i dt∣∣ ∣∣
with j denoting the number of the corresponding test data set, ns being the num-
ber of input sections, respectively modes, e.g. ns = 3, considering the deviation
between the original curve of the test data set, used as input for the network, and
the simulated curve based on the corresponding predicted parameter set. The de-
viation is summarised for all three curve sections and subsequently averaged over
the whole test data set to determine the mean absolute error.
• The percentage of test data sets which have a curve deviation of less than 10 %.
The 10 %-criterion was chosen in order to obtain the amount of test data sets with
a sufficient accuracy for the application in this work.
• The average deviation of the original and predicted material parameters of the test
data set.
The network performance based on the random sampling method results in a mean
absolute error of δMAE =10.68 %. In addition, 65.85 % of the test data expose an absolute
error of less than 10 %.
In contrast, the performance based on the Latin Hypercube Sampling technique leads
to a mean absolute error of δMAE =7.90 %. Furthermore, 76.2 % of the test data feature
an absolute error of less than 10 %.
The performance of both networks is visualised in Figures 3.10 and 3.11, respectively.
While the deviation of each curve of the test data set from the median curve is plotted
in horizontal direction, the absolute error, calculated via equation (3.6), is assigned
in vertical direction. The heatmap shows the frequency of test data sets regarding
both properties. Hence, a high frequency at the bottom of the figure reflects a high
performance of the neural network. As it can be seen in Figure 3.10, the heatmap shows
a clustering of the data sets over the whole bottom region. Thus, most of the curves
are predicted quite accurately and independent of the deviation to the median curve.
Nevertheless, the figure presents some data sets with a less accurate prediction and the
sets have a larger frequency with a higher deviation to the median curve. Comparing
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3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
these results with those of Figure 3.11 which are based on the LHS, the average error
of the prediction decreases even further, and especially those curves lying closer to the
median curve have a nearly perfect match between the original curve and its prediction.
Furthermore, the amount of data sets with a higher deviation also decreases.
When only taking the third criteria into account, i.e. the parameter deviation shown
in Table 3.1, a summarising statement is not directly obvious since in some parameters
the RS method performs better than the LHS. Interestingly, the performance of the
elastic (E, ν) and plastic parameters (σy0 , h, np) is slightly better with the RS and
slightly worse for the damage related parameters.
In conclusion, the performance of the network based on the Latin Hypercube Sampling
is clearly better than the random sampling. Consequently, the LHS trained network is
used in the following.
While the curve deviation shows the prediction capabilities of the neural network, the
deviation of each parameter of the original and predicted values is not necessarily based
on the network performance. Since the network identifies the features of the inputs—
the curves—and since the network predicts a set of parameters producing a low curve
deviation, the ANN provides a good result. The possible larger parameter deviation
is probably based on modelling effects, e.g. several parameter combinations can lead to
comparable results, i.e. different local minima exist. Considering the results in Table 3.1,
the prediction of the first five parameters—elastic and plastic—is quite good, whereas
the damage related material parameters show a larger deviation. Hence, either these
parameters depend on each other leading to local minima or the experimental setting
is not sufficient to identify these parameters. Probably both aspects apply since one
unloading phase between two loading phases might be not enough in order to identify
the relevant local minima.
3.1.3.2 Optimisation of the hyperparameters
As already mentioned in Section 3.1.2.4, the hyperparameters need to be optimised to
find the network with the best performance. Thus, a random search algorithm was
chosen to find the best possible parameter set. Since the result of the previous section
was that the LHS method is superior over the RS approach, the hyperparameters were
only optimised for the LHS technique.
While the initially chosen set of hyperparameters already showed quite good results,
the overall net performance could be further improved with this set of hyperparameters,
i.e. poptNN =[3, 1, 3, 27, 40, 260]. This leads to the network structure [260, 260, 260, 327, 327,
327, 448, 623, 852, 852, 852, 1135, 1135, 1135] following the scheme described in Section
3.1.2.4. The overall runtime of the complete training process increased by approximately
60 % on a standard computer. Nevertheless, considering that the training of a specific
model only has to be performed once and subsequently can be applied to different
experimental inputs, an increase in the network performance is worth a possible increase
in computational time.
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
Figure 3.10: Heatmap of the network performance for the initial hyperparameters based on the RS
showing the frequency of the 2k test data sets with respect to their curve deviation from the median
curve and curve deviation of predicted and original curve. A clustering in the bottom of the figure
reflects a high network performance. Reprinted from Schulte et al., European Journal of Mechanics -
A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
Figure 3.11: Heatmap of the network performance for the initial hyperparameters based on the LHS
showing the frequency of the 2k test data sets with respect to their curve deviation from the median
curve and curve deviation of predicted and original curve. A clustering in the bottom of the figure
reflects a high network performance. Reprinted from Schulte et al., European Journal of Mechanics -
A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
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3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
Table 3.1: Average deviation of the original and predicted parameters of the complete test data set
for the initial hyperparameter set based on the RS and LHS. The fixed parameters are not calibrated
within this work. Reprinted from Schulte et al., European Journal of Mechanics - A/Solids 98:104854
(2023), [113], with permission under creative commons licence 4.0.
Symbol Description Deviation w/ RS Deviation w/ LHS
E Young’s modulus 11.70 % 11.72 %
ν Poisson’s ratio 10.52 % 10.92 %
σy0 initial yield stress 9.18 % 9.51 %
h hardening parameter 8.77 % 9.19 %
np hardening exponent 12.53 % 12.86 %
qvar variable damage threshold 21.19 % 21.58 %
nd damage exponent fixed fixed
η damage rate factor 20.19 % 19.97 %
ξvol volumetric damage factor fixed fixed
ξiso isochoric damage factor 20.06 % 19.31 %
ξq threshold factor 24.89 % 24.89 %
ξm effective stress factor 21.11 % 20.67 %
ηα coupling factor 22.28 % 21.74 %
cd regularisation parameter fixed fixed
βd penalty parameter fixed fixed
qmin initial damage threshold 21.69 % 21.57 %
The mean absolute error of the curve deviation, calculated for each test data set
via equation (3.5), decreased to δMAE = 6.5 %. Furthermore, the amount of test data
sets with a curve deviation of less than 10 % increased to 88.65 %. Hence, the network
performance improved noticeably with respect to the initial guess. The performance is
visualised in Figure 3.12 and can be compared to Figure 3.11 showing that the overall
prediction accuracy of the network for nearly all data sets increases considerably. The
dispersion of the absolute error decreases over the whole region of the test data. Thus, the
heatmap reflects the good performance of the network with optimised hyperparameters.
In Figure 3.13, the corresponding loss curve shows the evolution of the squared error
over the training epochs.
Comparing the values of the parameter deviations in Table 3.2 and 3.1, it can be seen
that the prediction accuracy increases for all of the different material parameters, except
for the first three parameters which increase negligibly.
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
Figure 3.12: Heatmap of the network performance for the optimal hyperparameters showing the fre-
quency of the 2k test data sets with respect to their curve deviation from the median curve and curve
deviation of predicted and original curve. A clustering in the bottom of the figure reflects a high net-
work performance. Reprinted from Schulte et al., European Journal of Mechanics - A/Solids 98:104854
(2023), [113], with permission under creative commons licence 4.0.
Figure 3.13: Loss curve of the network performance for the optimal hyperparameters demonstrating the
evolution of the squared error over the training epochs. Reprinted from Schulte et al., European Journal
of Mechanics - A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
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3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
Table 3.2: Average deviation of the origninal and predicted parameters of the complete test data set for
the optimal hyperparameter set. The fixed parameters are not calibrated within this work. Reprinted
from Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113], with permission
under creative commons licence 4.0.
Symbol Description Parameter deviation
E Young’s modulus 11.82 %
ν Poisson’s ratio 10.96 %
σy0 initial yield stress 9.28 %
h hardening parameter 7.02 %
np hardening exponent 9.64 %
qvar variable damage threshold 21.32 %
nd damage exponent fixed
η damage rate factor 19.38 %
ξvol volumetric damage factor fixed
ξiso isochoric damage factor 18.97 %
ξq threshold factor 24.64 %
ξm effective stress factor 20.46 %
ηα coupling factor 21.07 %
cd regularisation parameter fixed
βd penalty parameter fixed
qmin initial damage threshold 21.05 %
3.1.4 Parameter identification for homogeneous states of
deformation
After the testing of the neural network, experimental data is used to find a prediction
of the material parameter set for a specific material. Therefore, the ANN predicts the
model parameters of the experimental curve based on the knowledge obtained from the
training process. Th∫is can be m∫athematically expressed via∑ns
δEP = {•∫
exp
i dt− ∫•predi dt } , (3.7)
i=1 max •expi dt, •predi dt
where the • can be either referred to stresses or to strains depending on the segment of
the curve, i.e. loading or unloading phase. Depending on the prediction capabilities of
the trained networks, the deviation of the experimental and predicted curve can already
be sufficiently small enough. Since the main objective of the approach developed in this
work is to find an optimal starting value for a subsequent optimisation-based parameter
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3 Machine-learning assisted parameter identification based on homogeneous states of
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identification, it is advantageous, though not necessary, that the prediction is already
sufficiently accurate. Thus, in the final step, the optimisation tool minimises the error
functional f(κ) in order to obtain the optimal parameter set κopt, i.e.
 ∑ ∫ ∫ {•∫expns i dt− ∫•sim  i
(κ)dt 
minf(κ) = }→ κopt . (3.8)i=1 max •expi dt, •simi (κ)dt
By doing so, the deviation of the experimental and simulated curves based on the current
material parameter set κ is minimised until the optimal solution κopt is found. In this
case, the classic Nelder-Mead simplex algorithm [84] was a proper approach for the
calibration of the material parameters. A more detailed description of the method and
the underlying tool can be found in [104].
The optimal material parameters for two different materials shall be identified. The
first material is a sheet metal DP800 and the second one is a case-hardened steel 16Mn-
CrS5. Both are commonly used in metal forming processes and, consequently, possible
damage effects within the material need to be taken into account in order to achieve an
accurate prediction of the material properties occurring after the corresponding process
routes. The accurate prediction of damage evolution during, e.g., forming processes al-
lows the full utilisation of the lightweight potential of modern engineering components,
cf. [126, 127]. As mentioned before, different types of experiments are available for both
materials. Hence, if only one network is trained to obtain a prediction for the parameters
of both materials, only the tension test can be used for the input of the neural network.
Otherwise, two networks have to be trained based on a tension and an in-plane torsion
test for the first metal and on a tension and compression test for the second material.
Both methods shall be compared with respect to their prediction capabilities.
3.1.4.1 Network based on tensile data
In a first step, one network only based on tensile data was trained. The corresponding
network performance was shown in Section 3.1.3.2. The following results are obtained
on the basis of LHS and the optimal hyperparameters.
3.1.4.1.1 Steel DP800
To begin with the experimental tensile data of the DP800 from Figure 3.4 was trans-
formed into the proper input format for the network. The network immediately presented
a predicted parameter set for the material, see Table 3.3. The comparison of the corre-
sponding simulated curve based on the predicted parameters with the experimental curve
are shown in Figure 3.14. The predicted curve already matches the experimentally mea-
sured curve quite well, except that the curve lies slightly below the experimental curve
over all three load phases. The deviation between the curves is only δEP =3.47 % which
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3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
is already a very good prediction by the network. To demonstrate the influence of the
hyperparameters, the deviation of the prediction increases to δEP =19.28 % if the initial
values for the hyperparameters, shown in Section 3.1.2.4, are used.
Table 3.3: Predicted material parameters of the DP800 based on LHS and using the optimal set
of hyperparameters. The predicted parameters are compared to the optimised parameter set of the
subsequent Nelder-Mead-simplex algorithm. The fixed parameters are not calibrated within this work.
Reprinted from Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113], with
permission under creative commons licence 4.0.
Symbol Description NN-prediction Nelder-Mead Unit
f objective function value 3.47 1.02 %
E Young’s modulus 141.77 228.415 GPa
ν Poisson’s ratio 0.274 0.346 −
σy0 initial yield stress 100.043 150.398 MPa
h hardening parameter 1226.002 1205.834 MPa
np hardening exponent 0.166 0.168 −
qvar variable damage threshold 26.187 30.312 MPa
nd damage exponent fixed at 0.667 −
η damage rate factor 2.478 1.359 −
ξvol volumetric damage factor fixed at 1.0 −
ξiso isochoric damage factor 0.973 0.747 −
ξq threshold factor 10.307 7.352 −
ξm effective stress factor 0.802 0.001 −
ηα coupling factor 3.673 2.701 −
cd regularisation parameter fixed at 0.54 N
βd penalty parameter fixed at 0.0 MPa
qmin initial damage threshold 25.623 28.392 MPa
Even if the prediction of the neural network is already very good, a subsequent Nelder-
Mead simplex algorithm can increase the match of the experimental results even further.
Figure 3.14d shows that nearly a perfect match of the experimental material response
can be achieved by the material model with the optimised parameter set κopt, though,
the stress response is slightly overestimated at the beginning of plastification. The
difference based on equation (3.8) decreases to f(κopt)=1.02 %. The comparison of the
corresponding stress-strain curves, see Figure 3.14d, reflects the mentioned behaviour.
Even though the improvement of the curve deviation is marginal, the difference in some
of the material parameters is quite distinct, e.g. for the Young’s modulus, see Table
3.3. This demonstrates that a different combination of the parameters can lead to a
comparable material response shown in the stress-strain curves. Hence, it is still very
important to pre-identify as many material parameters as possible in order to find the
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
(a) Stress curves of the first loading phase of the tension (b) Strain curves of the unloading phase of the tension test,
test. with strain defined as ε=∆l/l0.
(c) Stress curves of the second loading phase of the tension (d) Stress-strain curves of the tension test, with strain de-
test. fined as ε=∆l/l0.
Figure 3.14: For the DP800 sheet metal, the simulation results with the predicted parameter set of the
neural networks, based on the training data of the tension test, are compared to the curves simulated
with the optimised parameter set of the Nelder-Mead simplex algorithm and the experimental curves.
Here the LHS generated the corresponding data basis. See Section 3.1.3 with regard to the separation
and normalisation of the phases. Reprinted from Schulte et al., European Journal of Mechanics -
A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
88
3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
Figure 3.15: For the in-pla[ne torsion ]test of the DP800 sheet metal, the curve of the Piola shearstress over the shear strain, with the shear strain defined as the related shear component of the Green-
Lagrange strain tensor E= F t · F − I /2 with the factor two, is compared to the shear stress-shear
strain curve simulated with the optimised parameter set of the Nelder-Mead simplex algorithm. Here
the LHS generated the corresponding data basis. Reprinted from Schulte et al., European Journal of
Mechanics - A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
most suitable parameter set. Nevertheless, by using the ANN for the generation of a
good starting point there is at least the possibility of using such a large amount of
parameters.
Since the constitutive model was so far only calibrated with respect to tensile data in
the case of the artificial neural network as well as the classic optimisation approach, the
final optimised set of parameters almost perfectly matches the tensile material behaviour.
However, it will be important to find out how the model is capable of predicting the other
characteristic behaviour patterns of the material with the optimised set of parameters.
Therefore, the in-plane torsion test is simulated, once again assuming a homogeneous
state under simple shear, cf. [54, 128], and compared to the experimentally measured
material response, cf. Section 3.1.1. The results are shown in Figure 3.15. With a
curve deviation of δEP =5.62 % the prediction of the shear loading case is quite accurate,
though the stress level is overestimated. Since the Piola stress level was slightly higher
for the simulated tension test with the optimised parameter set, the deviation from
the experimental curve increases to f(κopt) = 7.21 %. Nevertheless, with both curves a
sufficient match of the material behaviour is achieved. In Section 3.1.4.2, these results
are compared to the solution if the simple shear behaviour is included in the training
data of the neural network.
3.1.4.1.2 Steel 16MnCrS5
Next, the experimental data of the tensile test of the case-hardened steel 16MnCrS5,
cf. Figure 3.4, is entered into the network. Considering the results in Figure 3.16, the
predicted curve of the ANN matches the experimental curve quite well. The stress level is
only slightly underestimated in the middle of the first loading phase and slightly overes-
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
timated in the second loading phase. The overall deviation of the curves is δEP =6.10 %.
The corresponding predicted material parameters are shown in Table 3.4. Nevertheless,
the simulated material behaviour can never match this experimental curve perfectly
since the experimentally observed Lüders extension is not included in the model.
Table 3.4: Predicted material parameters of the 16MnCrS5 based on LHS and using the optimal set
of hyperparameters. The predicted parameters are compared to the optimised parameter set of the
subsequent Nelder-Mead-simplex algorithm. The fixed parameters are not calibrated within this work.
Reprinted from Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113], with
permission under creative commons licence 4.0.
Symbol Description NN-prediction Nelder-Mead Unit
f objective function value 6.1 2.65 %
E Young’s modulus 173.751 134.759 GPa
ν Poisson’s ratio 0.277 0.311 −
σy0 initial yield stress 107.568 104.813 MPa
h hardening parameter 868.199 789.801 MPa
np hardening exponent 0.321 0.271 −
qvar variable damage threshold 29.463 49.998 MPa
nd damage exponent fixed at 0.667 −
η damage rate factor 5.4 5.14 −
ξvol volumetric damage factor fixed at 1.0 −
ξiso isochoric damage factor 1.138 1.439 −
ξq threshold factor 9.327 6.166 −
ξm effective stress factor 1.029 0.833 −
ηα coupling factor 4.899 4.916 −
cd regularisation parameter fixed at 0.54 N
βd penalty parameter fixed at 0.0 MPa
qmin initial damage threshold 13.769 12.364 MPa
After the classic optimisation, the simulated material response almost perfectly matches
the experimental curve of the 16MnCrS5, except for the Lüders extension which is not in-
cluded in the model, cf. 3.16. The deviation following equation (3.8) is f(κopt)=2.65 %.
After the final optimised parameter set was found, the prediction accuracy with re-
spect to the compressive material behaviour is informative. Though, in contrast to the
DP800 where the shear behaviour was well predicted, the prediction of the compression
test shows only a nearly perfect result until a certain strain level is reached, cf. Fig-
ure 3.17. Afterwards, the stress decreases significantly. This is due to the fact that a
tension-compression asymmetry is not included in this version of the material model and
that this material behaviour is not included in the training data of the neural network.
90
3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
(a) Stress curves of the first loading phase of the tension (b) Strain curves of the unloading phase of the tension test,
test. with strain defined as ε=∆l/l0.
(c) Stress curves of the second loading phase of the tension (d) Stress-strain curves of the tension test, with strain de-
test. fined as ε=∆l/l0.
Figure 3.16: For the 16MnCrS5 case-hardened steel, the simulation results with the predicted param-
eter set of the neural networks, based on the training data of the tension test, are compared to the
curves simulated with the optimised parameter set of the Nelder-Mead simplex algorithm and the ex-
perimental curves. Here the LHS generated the corresponding data basis. See Section 3.1.3 with regard
to the separation and normalisation of the phases. Reprinted from Schulte et al., European Journal of
Mechanics - A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
Figure 3.17: For the compression test of the 16MnCrS5 case-hardened steel, the curve of the compression
Piola stress over the strain, with strain defined as ε=∆l/l0, is compared to the compression stress-strain
curve simulated with the optimised parameter set of the Nelder-Mead simplex algorithm. Here the LHS
generated the corresponding data basis. Reprinted from Schulte et al., European Journal of Mechanics
- A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
The curve deviation reaches f(κopt) = 9.69 % which is quite high compared to the very
good fit of the tensile test.
3.1.4.2 Two networks each based on two experiments
In order to evaluate whether the prediction accuracy can be further improved for both
loading paths, the training data contains simulations of both types of experiments in the
following. However, as mentioned before, only different types of experiments are available
for both materials. Hence, two different networks have to be trained based on the
combination of tensile data with in-plane torsion data and tensile data with compression
data, respectively. It is important to mention that, for the previous networks based only
on tensile data, 22521 simulations were completed successfully. In comparison, the
total data set for the following networks is clearly lower since only the intersection of
material parameter combinations which were successfully finished for both types can
be considered. In the case of tensile and compression data 7168 different parameter
combinations and in the case of tensile and in-plane torsion data 9155 parameter sets
can be used.
3.1.4.2.1 Steel DP800
In a first step, the training of the neural network based on tensile and in-plane torsion
data was conducted with the optimised hyperparameter set of Section 3.1.3.2, denoted
in the following as network A. Since the hyperparameter set was not fitted to the new
network, it was only to be expected that the prediction would probably be less accurate
than when using only one type of training data. This expected less accuracy is visible in
Figure 3.18, though the difference is quite marginal. The corresponding mean absolute
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3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
error is δMAE =12.49 % and only 64.0 % of the test data expose an absolute error of less
than 10 %. Afterwards, the hyperparameters were calibrated for the network by using
both types of data for the training process, denoted as network B, and the network per-
formance could be improved to a mean absolute error of δMAE = 11.86 %. Furthermore,
65.6 % of the test data feature an absolute error of less than 10 %. Not only the net-
work performance seemed to improve marginally, but the prediction accuracy also only
slightly improved, cf. Figure 3.18. While the stress-strain behaviour is overestimated
with network A, it is marginally underestimated with network B. The curve deviation
can be further improved with the subsequent Nelder-Mead simplex algorithm. Never-
theless, even after the optimisation, the accuracy of the in-plane torsion data is still
overestimated while the tensile curve matches nearly perfectly, showing almost the same
result as the network which was only trained with tensile data. However, the stress-
strain curve of the tensile test fits more accurately since the stress level is no longer
overestimated at the beginning of plastification.
3.1.4.2.2 Steel 16MnCrS5
Analogously to the sheet metal, the first network, where the training process is based
on the combination of tensile and compression data, was trained with the set of hyper-
parameters from Section 3.1.3.2. For this setting, denoted as network A, the prediction
accuracy improved markedly for both types of loading data compared to the prediction
of the network based only on tensile data, see Figure 3.20. Especially the damage evo-
lution under compression is avoided, i.e. q̇ = 0 for tr(εe) < 0, so that a very good fit
of the compressive material response is achieved. Though, since the impact of damage
evolution under tensile loading is minor, damage does not occur at all. Consequently, a
tension-compression asymmetry needs to be included into the material model in order
to accurately simulate the correct material behaviour. Even though the prediction accu-
racy of network A is quite good, the corresponding network performance shows a mean
absolute error of δMAE =12.59 %. In addition, 60.3 % of the test data expose an absolute
error of less than 10 %. Next, the hyperparameters were optimised for the network based
on both types of simulation data, denoted as network B. The network performance could
be improved markedly to a mean absolute error of δMAE = 9.84 %. Furthermore, 70.8 %
of the test data feature an absolute error of less than 10 %, the prediction accuracy of
the experimental curve decreased massively, not lying close to the experimental curve
at all, cf. Figure 3.20. Even with the subsequent Nelder-Mead simplex algorithm it was
not possible to find a minimum lying close enough to the experimental curve. However,
by using the predicted parameter set of network A with the inferior performance, the
Nelder-Mead simplex algorithm was capable of finding a nearly perfect solution, where
not only the loading and unloading states of the tensile test but also the loading phase
of the compression test nearly perfectly matched.
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
(a) Stress curves of the first loading phase of the tension (b) Strain curves of the unloading phase of the tension test,
test. with strain defined as ε=∆l/l0.
(c) Stress curves of the second loading phase of the tension (d) Shear stress curves of the in-plane torsion test.
test.
Figure 3.18: For the DP800 sheet metal, the simulation results with the predicted parameter sets of the
neural networks A and B, based on the combined training data of the tension and in-plane torsion test,
are compared to the curves simulated with the optimised parameter set of the Nelder-Mead simplex
algorithm and the experimental curves. Here the LHS generated the corresponding data basis. See
Section 3.1.3 with regard to the separation and normalisation of the phases. Reprinted from Schulte et
al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113], with permission under creative
commons licence 4.0.
94
3.1 Machine-learning assisted parameter identification introduced for a gradient-enhanced
damage formulation
(a) Stress-strain curves of the tension test, with strain de- (b) Shear stress-shear strain curves of the in-plane tor-
fined as ε=∆l/l0. [sion test, w] ith the shear strain defined as the relatedshear component of the Green-Lagrange strain tensor E=
F t · F − I /2 with the factor two.
Figure 3.19: For the DP800 sheet metal, the simulation results with the predicted parameter sets
of the neural networks A and B, based on the combined training data of the tension and in-plane
torsion test, are compared to the curves simulated with the optimised parameter set of the Nelder-
Mead simplex algorithm and the experimental curves. Here the LHS generated the corresponding data
basis. Reprinted from Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113],
with permission under creative commons licence 4.0.
3.1.4.3 Discussion
The previously presented results highlight the challenges and the benefits of using arti-
ficial neural networks for the prediction of a starting point for a subsequent parameter
identification for constitutive material models. First of all, the distinct smaller set of
training data—less than the half amount of the tensile data—might be one reason for
the overall, sometimes worse network performance of the previous two sections. In ad-
dition, the robustness is not quantifiable yet since a larger data set would be required,
especially experimental data. Furthermore, the experimental curve can be an outlier of
the training data of network B where the network is not capable of giving an accurate
prediction. Nevertheless, it could be advantageous to set up a preceding feature extrac-
tion, e.g. an autoencoder, in order to identify the influence of each parameter in more
detail. This becomes even more important for more complex material models since it
was already shown in Ktari et al. [57] that the so-called hybrid strategy works perfectly
for simple types of material models with a low amount of corresponding parameters
to be identified. Furthermore, a preceding feature extraction can reduce the network
complexity and thus reduce the computational costs of the training process.
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
(a) Stress curves of the first loading phase of the tension (b) Strain curves of the unloading phase of the tension test,
test. with strain defined as ε=∆l/l0.
(c) Stress curves of the second loading phase of the tension (d) Stress curves of the compression test.
test.
Figure 3.20: For the 16MnCrS5 case-hardened steel, the simulation results with the predicted parameter
sets of the neural networks A and B, based on the combined training data of the tension and compression
test, are compared to the curves simulated with the optimised parameter set of the Nelder-Mead simplex
algorithm and the experimental curves. Here the LHS generated the corresponding data basis. See
Section 3.1.3 with regard to the separation and normalisation of the phases. Reprinted from Schulte et
al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113], with permission under creative
commons licence 4.0.
96
3.2 Machine-learning assisted parameter identification applied to a laminate-based material
model for ferroelectrics
(a) Stress-strain curves of the tension test, with strain de- (b) Stress-strain curves of the compression test, with strain
fined as ε=∆l/l0. defined as ε=∆l/l0.
Figure 3.21: For the 16MnCrS5 case-hardened steel, the simulation results with the predicted parameter
sets of the neural networks A and B, based on the combined training data of the tension and compres-
sion test, are compared to the curves simulated with the optimised parameter set of the Nelder-Mead
simplex algorithm and the experimental curves. Here the LHS generated the corresponding data basis.
Reprinted from Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113], with
permission under creative commons licence 4.0.
3.2 Machine-learning assisted parameter identification
applied to a laminate-based material model for
ferroelectrics
After the general structure of the hybrid approach was explained and demonstrated pre-
viously with regard to a finite plasticity model coupled to (gradient-enhanced) damage,
another application example is presented in this section. For this purpose, a different
material model is chosen—an electro-mechanically coupled laminate-based model for
ferroelectric materials, see Section 2.3. This example provides several advantages em-
phasising the benefits and capabilities of the hybrid approach. The chosen model is
completely different to the previous example and describes another class of materials
exhibiting an electro-mechanical coupling and, thus, the characteristic loading path se-
quences which have to be considered for the parameter identification are totally different.
Furthermore, in Dusthakar et al. [17], the authors had to develop a sophisticated strategy
to obtain proper starting values for a subsequent PI of the two remaining model parame-
ters, cf. Section 2.5.3. The general experimental setting is depicted in that section. The
authors employed a generic algorithm in order to find initial guesses for the material
parameters. To this end, they performed the calibration procedures independently for
each of the different applied stress magnitudes to receive a list of parameter values, and
a suitable starting point was subsequently chosen for each parameter. This starting set
was used for the calibration of the parameters where the experimental data of all stress
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3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
magnitudes was considered simultaneously. The overall effort of the calibration process
of the parameters can be reduced by employing the presented new hybrid strategy. The
implemented tool is able to deal with different kinds of simulation solvers and various
kinds of concatenated loading path sequences by incorporating only small adjustments
in the code. To give another example of a machine learning framework, the Keras API
of TensorFlow is chosen to generate the ANN for this material model.
For the generation of the training data, the range has to be defined of both fitting
parameters κ = {m, c}, with m denoting the viscous-like parameter and c the hysteresis
shape exponent, influencing the remnant energy contribution, cf. Section 2.3 and 2.5.3.
While the first parameter is restricted to a range of m ∈ [1, 5], for the second parameter
c ∈ [1, 10] turned out to be an admissible range. In total a set of 10000 parameter
combinations was generated by using the previously mentioned Latin Hypercube Sam-
pling. However, not all of these parameter combinations lead to successful simulations
of the defined loading path, cf. Section 2.5.3, such that only approximately half of the
combinations where successful and thus usable for the training data set.
The characteristic information of this material class is shown by the butterfly and
dielectric hysteresis curves depicted in Section 2.5.3. Thus, two characteristic curves are
generated for each loading setting. In Shieh et al. [114], a total of six different compres-
sive stress levels were applied between 0 MPa and 2.7 MPa, such that 12 different curves
have to be considered in the parameter identification process. For further details please
be referred to Section 2.5.3. Since the cyclic electric field E3 is prescribed, the strains
ε33 and dielectric displacements D3 for the different stress magnitudes are normalised
and concatenated in order to generate the input of the ANN. Since in [17] most of the
parameters for BaTiO3 were already taken from the literature, only the two remaining
parameters m and c constitute the corresponding output of the ANN.
As in the previous example, 20% of the overall simulated data set was separated as
test data and not part of the network training. After the ANN was trained, the test data
set is employed to analyse the deviation of the predicted to the original parameters of
the test data set. As it can be seen in Table 3.5, the average deviation of the parameters
is less than 1% for both parameters in the case of the optimal set of hyperparameters.
Even the occurring maximum deviation of the first parameter for one specific set is only
1.14% and for the second parameter only 6.8%. Thus, the trained ANN is capable of
predicting the laminate-based material model very accurately.
Table 3.5: Average deviation of the original and predicted parameters of the complete test data set for
the initial hyperparameter set based on the LHS.
Symbol Description Parameter deviation
m viscous-like parameter 0.23%
c hysteresis shape exponent 0.91%
However, the prediction of the parameters by the network for the experimental data
of BaTiO3 varies from the optimal parameter set identified with the SQP algorithm in
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3.2 Machine-learning assisted parameter identification applied to a laminate-based material
model for ferroelectrics
[17]. While the viscous-like parameter is at least near the optimal solution, the hys-
teresis shape exponent is higher than the identified result, cf. Table 3.6. Nevertheless,
comparing the butterfly and hysteresis curves on the left sides of Figures 3.22-3.25, it
can be seen that the deviation of both simulated responses—with the predicted and the
calibrated parameter set—deviate markedly from the experimentally measured curves.
While the deviation of the hysteresis curves is on a comparable level for the six differ-
ent applied constant compressive stress magnitudes and the calibrated solution is always
slightly better than the predicted solution, the deviation of the butterfly curves increases
distinctly with a decreasing applied compressive stress level. While the butterfly curves
generally fit better with the calibrated solution, the predicted solution is capable of
reflecting some sections of the curves better, e.g. the connection point of the butterfly
wings for some stress levels. Considering the predicted and calibrated solution, the pre-
dicted parameters can already be used as proper starting values for a subsequent classic
parameter identification leading to the same minimum. Nevertheless, taking the param-
eter deviations of the test data set and the curves in the figures into account, it is clear
that the simulated material response of the model is not capable of perfectly matching
the experimental data. Two important notable differences are e.g. the asymmetry and
the variation of the butterfly curves under different applied compressive stress levels.
Thus, the experimental data of the butterfly curves was symmetrised with respect to
the y-axis—at E3 = 0 kV/mm by mirroring the wing of the positive side—in order to
investigate if the prediction quality by the network increases. It is shown in Table 3.6
and on the right sides of Figures 3.22-3.25 that the prediction is closer to the opti-
mal solution with the symmetrised experimental data as input for the ANN. Hence, the
ANN is capable of perfectly predicting the simulated material response of the model, but
since the experimental data deviates markedly from the simulated material behaviour,
the ANN has to extrapolate for a prediction which decreases its overall accuracy with
respect to experimental data.
Table 3.6: Predicted two remaining material parameters of the laminate-based model based on LHS
and using the optimal set of hyperparameters. Comparison of the predictions based on the original ex-
perimental data and on the symmetrised experimental data to the optimal set of the SQP optimisation.
Symbol Description NN-prediction NN-prediction sym. SQP Unit
f objective function value 76.69 69.42 54.77 -
m viscous-like parameter 1.664 1.615 1.494 -
c hysteresis shape exponent 9.766 7.323 3.648 -
This application demonstrates the capabilities of the hybrid strategy by employing
ANNs. Even if all material parameters can be uniquely identified in the testing phase,
indicating that the parameters are independent from each other, the predicted solution
using experimental data might clearly deviate. Nevertheless, for the hybrid strategy,
the prediction is still an appropriate starting value for a subsequent classic parameter
identification.
99
3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
(a) Butterfly curves based on the original experimental (b) Butterfly curves based on the symmetrised experimen-
data under a constant compressive stress of 2.7 MPa. tal data under a constant compressive stress of 2.7 MPa.
(c) Butterfly curves based on the original experimental (d) Butterfly curves based on the symmetrised experimen-
data under a constant compressive stress of 1.7 MPa. tal data under a constant compressive stress of 1.7 MPa.
(e) Butterfly curves based on the original experimental (f) Butterfly curves based on the symmetrised experimental
data under a constant compressive stress of 1.2 MPa. data under a constant compressive stress of 1.2 MPa.
Figure 3.22: Simulated and experimentally measured butterfly curves under various constant com-
pressive stress magnitudes part I—the trained ANN predicts the parameters based on the original
experimental data on the left side and on the symmetrised experimental data of the butterfly curves on
the right side. Symmetrisation of the butterfly curves is performed due to model properties.
100
3.2 Machine-learning assisted parameter identification applied to a laminate-based material
model for ferroelectrics
(a) Butterfly curves based on the original experimental (b) Butterfly curves based on the symmetrised experimen-
data under a constant compressive stress of 0.9 MPa. tal data under a constant compressive stress of 0.9 MPa.
(c) Butterfly curves based on the original experimental (d) Butterfly curves based on the symmetrised experimen-
data under a constant compressive stress of 0.5 MPa. tal data under a constant compressive stress of 0.5 MPa.
(e) Butterfly curves based on the original experimental (f) Butterfly curves based on the symmetrised experimental
data under a constant compressive stress of 0 MPa. data under a constant compressive stress of 0 MPa.
Figure 3.23: Simulated and experimentally measured butterfly curves under various constant com-
pressive stress magnitudes part II—the trained ANN predicts the parameters based on the original
experimental data on the left side and on the symmetrised experimental data of the butterfly curves on
the right side. Symmetrisation of the butterfly curves is performed due to model properties.
101
3 Machine-learning assisted parameter identification based on homogeneous states of
deformation
(a) Hysteresis curves based on the original experimental (b) Hysteresis curves based on the symmetrised experimen-
data under a constant compressive stress of 2.7 MPa. tal data under a constant compressive stress of 2.7 MPa.
(c) Hysteresis curves based on the original experimental (d) Hysteresis curves based on the symmetrised experimen-
data under a constant compressive stress of 1.7 MPa. tal data under a constant compressive stress of 1.7 MPa.
(e) Hysteresis curves based on the original experimental (f) Hysteresis curves based on the symmetrised experimen-
data under a constant compressive stress of 1.2 MPa. tal data under a constant compressive stress of 1.2 MPa.
Figure 3.24: Simulated and experimentally measured hysteresis curves under various constant com-
pressive stress magnitudes part I—the trained ANN predicts the parameters based on the original
experimental data on the left side and on the symmetrised experimental data of the butterfly curves on
the right side. Symmetrisation of the butterfly curves is performed due to model properties.
102
3.2 Machine-learning assisted parameter identification applied to a laminate-based material
model for ferroelectrics
(a) Hysteresis curves based on the original experimental (b) Hysteresis curves based on the symmetrised experimen-
data under a constant compressive stress of 0.9 MPa. tal data under a constant compressive stress of 0.9 MPa.
(c) Hysteresis curves based on the original experimental (d) Hysteresis curves based on the symmetrised experimen-
data under a constant compressive stress of 0.5 MPa. tal data under a constant compressive stress of 0.5 MPa.
(e) Hysteresis curves based on the original experimental (f) Hysteresis curves based on the symmetrised experimen-
data under a constant compressive stress of 0 MPa. tal data under a constant compressive stress of 0 MPa.
Figure 3.25: Simulated and experimentally measured hysteresis curves under various constant com-
pressive stress magnitudes part II—the trained ANN predicts the parameters based on the original
experimental data on the left side and on the symmetrised experimental data of the butterfly curves on
the right side. Symmetrisation of the butterfly curves is performed due to model properties.
103

4 Parameter identification based on
inhomogeneous states of
deformation
Considering several different kinds of material models, at least some of the material
parameters can be directly identified based on standard experiments incorporating only
homogeneous states of deformation, e.g. the Young’s modulus or the yield limit. How-
ever, the more complex the material models, the more model parameters exist which
cannot be directly identified based on the previously mentioned experiments since these
experiments do not provide sufficient information. Hence, especially for non-local ma-
terial models, experiments with inhomogeneous states of deformation are required. In
order to take the different deformation states into account, field data—such as strains
or displacements—has to be considered in the objective function of the parameter iden-
tification process. Thus, a multi-objective optimisation is necessary where not only the
load-displacement data contributes to the objective function but also the field data. A
parameter identification framework was already developed in a basic form by Schulte
[112]. In this chapter, a complex multi-objective parameter identification process is
presented. In this regard, a parameter identification framework was implemented in
Schowtjak et al. [104] which is summarised in Section 4.1 and subsequently applied to
different materials and corresponding material models in Sections 4.2 and 4.3. However,
since the difficulty of generating appropriate starting values for the parameter identi-
fication remains, the second step of the hybrid strategy is presented in Section 4.3.5
where the predicted parameter set of the ANN for the DP800 sheet metal is used as a
starting set for the multi-objective optimisation of the finite plasticity model coupled to
gradient-enhanced damage.
4.1 Parameter identification tool for multi-objective
optimisations
In order to be able to consider all kinds of data for the objective function, e.g. strains,
displacements or microstructural data, the parameter identification tool ADAPT—A
105
4 Parameter identification based on inhomogeneous states of deformation
Diversely Applicable Parameter Identification Tool was implemented, cf. [104]. ADAPT
provides material parameters for constitutive models for the application to continuum
mechanical analyses for finite deformations. The tool is implemented in Python since it
is a free programming language including a library with a remarkable amount of different
packages. The free, open source code is available on GitHub [26].
4.1.1 Optimisation process
In Figure 4.1, the flowchart of ADAPT is depicted. Subsequent to the selection of an
initial guess for the parameter set, the FE-model is simulated. Next, the results are
compared to the experimental data while iteratively updating the material parameters
until convergence is reached. ADAPT is implemented in a way in order to ensure the
efficiency, robustness and flexibility of the underlying optimisation scheme.
Figure 4.1: The Flowchart shows the parameter identification process as implemented in ADAPT. The
iterative optimisation process is controlled by a Python framework. Reprinted from Schowtjak et al.,
International Journal of Mechanical Sciences 213:106840 (2022), [104], with permission under creative
commons licence 4.0.
106
4.1 Parameter identification tool for multi-objective optimisations
As in the previous sections, the parameter set is defined as κ. For the corresponding
inverse parameter identification, the following optimisation problem
min(f(κ)),∀κ ∈ K, with K = {κ |h(κ) = 0, g(κ) ≤ 0} (4.1)
has to be solved, cf. Equation (2.84) and Section 2.4, with f(κ) representing the scalar
objective function which describes the error between experiment and simulation. Con-
tributions h(κ) = 0 and g(κ) ≤ 0 denote the equality and inequality constraints,
respectively. The error between the data from the experiment •expi and the simulation
•simi for each o√f the sample points can be calculated by using the root sum of squares√√√ ∑nsp1 2f(κ) = w [•sim(κ)− •expi i i ] , (4.2)nsp i=1
which is generally used as the objective function in a common formulation. Integer
nsp represents the number of sample points in space and time and wi are weighting
coefficients, providing the possibility to introduce different weights to different sample
points. In the equation, • can denote any quantity of interest, such as reaction forces
or local displacements and strains. In the following, only purely mechanical problems
are examined. However, the framework generally also allows for the inclusion of thermal
effects, as discussed in, e.g., [95, 97]. Furthermore, electro-mechanically coupled material
models can be calibrated as well to the corresponding characteristic experimental data,
as it was performed by Dusthakar et al. in [17], cf. the summary in Section 2.5.3. The root
sum of squares-function provides the advantage that statistical outliers are particularly
penalised due to its quadratic form.
In contrast to strains which only account for displacements arising in deformations,
displacements—experimentally measured with DIC—usually contain rigid body mo-
tions. The rigid body motions have to be excluded from the experimental data in order
to compare the experimentally measured displacements to the simulation results. Rela-
tive displacements are an alternative approach which can be considered in the objective
function following Rose and Menzel in [97], or the strains can be directly calculated from
the DIC-data.
In general, in ADAPT the optimisation algorithms of the SciPy package optimise are
incorporated in order to solve the minimisation problem, see [132]. However, the user is
able to implement additional algorithms if necessary due to the open-source publication
of the tool. The default optimisation algorithm in ADAPT is the Nelder-Mead-simplex
algorithm as a gradient-free method since it turned out to be efficient in solving complex
problems with a larger number of model parameters (>5) [84]. The chosen default algo-
rithm provides a robust process since the derivative is not required, but in general many
iterations are needed. For many optimisation algorithms, the efficiency highly depends
on the chosen initial guess for the parameter set. If a good starting point is selected, the
107
4 Parameter identification based on inhomogeneous states of deformation
calibration process can be very efficient. However, if a disadvantageous initial guess was
chosen, it can be difficult to efficiently achieve proper results. Stochastic algorithms can
be used for these cases since they scan large ranges of the parameter space, cf. Section
2.4. Hence, such algorithms are helpful for specific problems, but in general require
a large number of objective function evaluations to obtain an optimal parameter set.
Furthermore, the algorithms permit a higher probability of finding the global minimum
or at least a local minimum with a lower functional value. As an alternative to the
Nelder-Mead-simplex algorithm, the differential evolution algorithm is implemented in
ADAPT. Since not all of the possible optimisation algorithms allow the incorporation
of constraints for the parameter space, the penalty method is implemented in order to
consider the violation of parameter boundaries in the calculation of the objective func-
tion. While the incorporation of this method reduces the efficiency of the calibration,
it ensures the consideration of important constraints for the model parameters, e.g. the
Poisson’s ratio.
4.1.2 Postprocessing and interpolation
In addition, the framework of ADAPT contains a postprocessing subroutine in Python
which automatically evaluates the output databases (odb-file) for Abaqus by extracting
all the relevant data. However, all node and element sets have to be correctly specified in
the input-file and config-file, since the user is otherwise required to modify the routine.
In general, the experimentally measured data points and the FE node points do not co-
incide with respect to their locations in space and time. Thus, an interpolation in space
and in time is required. The interpolation is divided implementation-wise into integral
data, e.g. force-displacement data and field data, e.g. displacement or strain fields. In
general, forces are measured at different time steps or at displacements associated with
the current loading of the specimen. Regarding the computational efficiency, the data
set with the higher resolution is interpolated onto the one containing a lower amount of
sample points. In the case of field data, the data is generally measured in the section
on the surface of the specimen by using DIC over the time. Besides displacements and
strains, field data sets can be basically any kind of data which can be experimentally
measured and numerically predicted, e.g. void fractions, texture or hardness. In contrast
to the one-dimensional interpolation, Python offers no package containing an interpola-
tion algorithm for irregular—with respect to the spatial distribution—two-dimensional
data sets. Therefore, the interpolation framework
• = [1− a− b] •1 + a •2 + b •3 , (4.3)
108
4.1 Parameter identification tool for multi-objective optimisations
was implemented, cf., e.g., Scheday[101] and Kleuter [52], with
[y3 − y1] [x− x1]− [x3 − x1] [y − y1]
a = − and (4.4)[y3 y1] [x2 − x1]− [x3 − x1] [y2 − y1]
[y1 − y2] [x− x1] + [x2 − x1] [y − y1]
b = − − − − − . (4.5)[y3 y1] [x2 x1] [x3 x1] [y2 y1]
In this scheme, any considered quantity • is interpolated from the three support points
•i with its corresponding coordinates xi and yi to the particular one • at location x, y.
Usually the scheme is applied to displacement and strain fields. While in the case of
strains, the integration point positions are used, the node point coordinates are chosen
in the case of displacements. It has to be mentioned that, while the measured strain
field is located on the surface of the specimen, the simulated strains are calculated at
the integration points which may be located on or close to the surface. If the integration
point lies within the finite element, the distance to the surface of the specimen depends
on the discretisation of the FE mesh. However, it was demonstrated in a sensitivity
study that the simulated strains of the integration points closest to the surface of the
sample can be considered if a sufficiently large amount of hexahedral elements is given
across the thickness of sheets. A sketch of the interpolation scheme is given in Figure 4.2.
DIC-data FE-mesh
Figure 4.2: Schematic overview of the spatial two-dimensional interpolation. The experimentally mea-
sured displacements uexpi at the locations xi, yi are interpolated onto the FE nodes. Reprinted from
Schowtjak et al., International Journal of Mechanical Sciences 213:106840 (2022), [104], with permission
under creative commons licence 4.0.
It is important to ensure that the interpolation error is reduced to a minimum. There-
fore, the three support points have to be chosen in a way that they span a triangle around
the query point. Furthermore, the vertices should have the minimum distance to the
query point, as shown in Figure 4.3a. Thus, a nearest neighbour search algorithm was
chosen from the Python library to obtain the support points closest to the query point.
Furthermore, the ray casting algorithm [124], as shown in Figure 4.3b, was used to ensure
that the query point lies within the triangle spanned by the three vertices.
109
4 Parameter identification based on inhomogeneous states of deformation
(a) Interpolation sketch (b) Ray casting algorithm scheme
Figure 4.3: The corresponding support points are selected based on the comprising of a triangle sur-
rounding the query point x with the least distance of the of this query point to each support point xi
with coordinates x and y, schematically depicted in (a). In (b), the ray casting algorithm is illustrated
which is employed to ensure that the query point lies within the triangle spanned by the three support
points. The query point lies inside the triangle if a straight line has an odd number of intersections,
cf. p1, or if lying outside if the intersection number is even, see p2. Reprinted from Schowtjak et al.,
International Journal of Mechanical Sciences 213:106840 (2022), [104], with permission under creative
commons licence 4.0.
Furthermore, if the time spacing in the experiment differs from the simulation time,
an additional linear interpolation of the quantities in time is required and implemented
following
•exp t2 − ti= •exp ti − t1i − 1 +
exp
− •2 , (4.6)t2 t1 t2 t1
with ti representing the step times of the simulation. Since the resolution of the experi-
mental time steps is generally higher than within the simulations, the experimental data
is interpolated onto the simulation time steps, where t1 and t2 are the experimental time
steps before and after ti, respectively.
As mentioned before, the overall interpolation process is automatically processed in
ADAPT. Furthermore, considering the computational cost, the user can select whether
the interpolation should be performed only once prior to the optimisation process or
within each iteration, where the latter is not required in most application cases.
4.1.3 Multi-objective optimisation
In many optimisation problems more than one objective is usually required for the
optimisation process. Thus, multiple objective functions have to be defined and simul-
taneously optimised. The straight forward solution of the multi-objective optimisation
is to reformulate the set of n objective functions
f(κ) = [f1(κ), f1(κ), ..., f
t
n(κ)] (4.7)
110
4.1 Parameter identification tool for multi-objective optimisations
into one single objective function where the normalised sum of all components is con-
sidered ∑n fi(κ)
f(κ) = , (4.8)
i=1 f̂i
with f̂i denoting a scalar-valued quantity larger than zero which is of the same unit as
the associated objective function fi(κ). The single objective function values of the first
iteration turned out to be a proper choice for the normalisation. Though, as a conse-
quence, the solution depends on the initial parameter set. In general, a multi-objective
optimisation problem does not possess a unique solution, since the different objective
functions might lead to opposite directions. The design of a lightweight component is
a perfect example for this case where a sufficient stiffness has to be achieved while si-
multaneously reducing the weight of the component as much as possible. In these cases
with no unique optimal solution, a Pareto-optimal solution is chosen instead. A Pareto-
	 

		 
	 
		 
 f1
Figure 4.4: Visualisation of the Pareto method when a minimisation problem of two objectives is
required. The ideal point depicts the perfect minimum if both objectives could be minimised indepen-
dently to their minimal objective function value. However, since both objectives cannot be optimised
independently, the Pareto front depicts all possible points from which the overall optimum cannot be
further improved without increasing the function value of one objective. Reprinted from Schowtjak
et al., International Journal of Mechanical Sciences 213:106840 (2022), [104], with permission under
creative commons licence 4.0.
optimal solution is found if an improvement cannot be achieved without worsening the
other attributes, as shown in Figure 4.4. If the solution of each objective is combined to
111
		 
 f2
4 Parameter identification based on inhomogeneous states of deformation
a vector, the ideal vector f∗ is defined. By using the global criterion method its distance
√√√√∑n [ ∗ − ]2fi fi(κ)f(κ) =
f ∗
(4.9)
i=1 i
to the vector function f(κ) is minimised and thus can be handled as a single-objective
minimisation problem. If the correlation of the particular parameters are to be analysed,
the impact of each parameter on the objective function can be investigated. Fossum [21]
proposed an approach in this regard. Thereby, the derivative qi = df(κ)/dκi has to be
evaluated and it has to be investigated whether single parameters κi impact the objective
function similarly. Hence, unit vectors
q
q∗ ii = || || for ||qi|| =6 0 (4.10)qi
associated to two different material parameters, i.e. κi and κj with i 6= j, are co-linear
if θij = q
∗
i · q∗j = 1 such that the parameters correlate. In the ideal case, where the
underlying model is formulated optimally and the data set is appropriate, the matrix
coefficients are θij = 0 if i 6= j and θij = 1 if i = j.
4.1.4 Boundary conditions
Depending on the complexity of the corresponding material model, the computational
cost of each iteration is important for the overall computation time. Thus, the reduc-
tion of the model using e.g. symmetry properties results in saving computational time.
The consideration of an eighth of the sample is often sufficient for the simulations, as
in the case of an appropriately symmetric flat test specimen made of isotropic material
or of anisotropic material oriented in such manner that the material symmetry planes
coincide with the geometric symmetry planes. Moreover, it should be guaranteed that
the experimental data reflects symmetry conditions as well. The displacement bound-
ary conditions are directly taken from the measured DIC-data, see Figure 4.5 and [110].
Using the measured displacements for the displacement boundary conditions of the sim-
ulations is required if only a section of the specimen is simulated. This is explained in
more detail in Section 4.2.
4.1.5 Implementation details
In order to enable the employment of other functionalities, an implementation of a mod-
ular manner was chosen for ADAPT. In the provided implementation, the optimisation
process can be directly started using Abaqus as the FE-solver. However, considering the
modular structure, a different FE-software can be employed if a script for the extraction
112
4.1 Parameter identification tool for multi-objective optimisations
Figure 4.5: The displacement boundary conditions for the simulations of a section of the specimen
are obtained from the experimentally measured DIC-data, see [110]. Reprinted from Schowtjak et al.,
International Journal of Mechanical Sciences 213:106840 (2022), [104], with permission under creative
commons licence 4.0.
of relevant output data is implemented. Even though a graphical user interface (GUI)
was not generated for ADAPT due to its large amount of modification features, the
framework can be easily applied to different material models and boundary value prob-
lems if specific node and element sets are defined. In order to check the performance of
the overall optimisation process the results can be visualised such that all kinds of diffi-
culties which may occur are identifiable. In order to prevent the loss of any data, e.g. the
current solution state, in the case of unforeseen aborts or computer shutdowns, log-files
containing all important information such as warnings are automatically generated. Fur-
thermore, an iter -file containing all data related to the parameter identification process
is created to save the optimisation history. In general, the overall parameter identifica-
tion process using FE-simulations is numerically expensive and especially increases with
the complexity of the underlying material model or the boundary value problem. There-
fore, ADAPT contains a restart option for the Nelder-Mead-simplex algorithm where
the history of the iteration results in the iter -file is used. In contrast, in the case of
gradient-based optimisation methods, the optimisation can be simply restarted by using
the parameter set of the last saved iteration. Furthermore, several different experiments
can be considered for the parameter identification process simultaneously in ADAPT.
In this case, the numerical efficiency is improved by paralysing the computation of these
FE-models.
113
4 Parameter identification based on inhomogeneous states of deformation
4.2 Application to self-diagnostic poly(dimethylsiloxane)
elastomer
A first application example of a multi-objective optimisation process is given for the vis-
coelastic material model coupled to gradient-enhanced damage in a finite strain setting,
presented in Section 2.2. While the viscoelastic material parameters were already iden-
tified in a first step with regard to homogeneous states of deformation in Section 2.5.2,
the damage-related parameters are calibrated in this section based on inhomogeneous
states of deformation.
4.2.1 Inhomogeneous deformation tests
Apart from the experiments with homogeneous deformation states, an additional ex-
periment with an inhomogeneous deformation state is carried out in order to identify
the damage related material parameters. The general experimental setting, e.g. test-
ing machines, has already been presented in Section 2.5.2.1. Considering the regu-
larised gradient-enhanced damage framework, specific information on localisation effects
is needed for the identification. Hence, with regard to the possible specimen dimensions,
a notched rectangular shape was chosen, see Figure 4.6a. A constant displacement rate of
20.0µm/s was applied until the sample with an initial length of 10 mm, width of 10 mm,
radius of 2 mm and a thickness of 2.2 mm tore apart. In Figures 4.6a-4.6d, prominent
states of the experiment are visualised. Moreover, Figure 4.6e shows the corresponding
load-displacement-curve.
4.2.2 Parameter identification based on inhomogeneous states of
deformation
In the following, the previously identified Yeoh and relaxation parameters, see Section
2.5.2, are used to identify the damage-related material parameters ηd, κd, niso, and cd
via the tensile test with inhomogeneous deformation states presented in Section 4.2.1. In
contrast to the identification of the viscoelastic parameters, the constitutive driver is not
sufficient enough to capture the material behaviour of the gradient-enhanced damage
model. Thus, a finite element (FE) formulation is required. Therefore, the material
model was implemented into a UMAT in Abaqus, as already mentioned in Section 2.2.5.
In addition, a parameter identification tool was implemented in Python, using the
Nelder–Mead-Simplex algorithm of the scipy-package. In contrast to the calibration of
the Yeoh and relaxation parameters, it did not suffice to solely consider the difference
in the Piola stress P11 or the (clamping) displacement u1, depending on the experiment,
for the goal function f of the optimisation. Since the gradient terms—capturing the
mesh-objectivity of damage—of the regularised damage framework are activated by in-
homogeneous deformation states, the difference in the displacement field between the
114
4.2 Application to self-diagnostic poly(dimethylsiloxane) elastomer
(a) Specimen before loading. (b) Specimen at t = 554.64 s.
(c) Specimen at t = 1141.0 s, just before tearing starts. (d) Specimen at t = 1407.0 s, after tearing.
55
50
45
40
35
30
25
20
15
10
5
0
0 3 6 9 12 15 18 21 24 27
Displacement [mm]
(e) Load-displacement-curve.
Figure 4.6: Experimental results of the tensile test of a notched specimen presenting photos of the
sample at characteristic time steps as well as the load-displacement-curve. Reprinted from Schulte et
al., Materials 13(14):3156 (2021), [110], with permission under creative commons 4.0.
simulated and experimental material response needs to be added to the goal function
apart from the difference in the reaction force
∑∑nnp ∥ ∥ ∑2 ∣ ∣2
f = w ∥u uexp − usimti ti (κ)∥ + w ∣F expF t − F simt (κ)∣ , (4.11)
t i=1 t
115
Reaction force [N]
4 Parameter identification based on inhomogeneous states of deformation
where nnp denotes the number of node points considered, wu the weighting factor for the
displacement contribution, and wF the weighting factor for the impact of the reaction
force. The number of node points was the number of element nodes on the surface of
the specimen in the FE simulations. The experimental data was interpolated onto those
node points via a 2D-interpolation scheme prior to the parameter identification following
Kleuter [52], see Section 4.1.2.
100
Experiment Initial Final #1 Final #2 Final #3
90
80
70
60
50
40
30
20
10
0
0 3 6 9 12
Displacement [mm]
(a) Sketch of the simulated part of the specimen (b) Load-displacement-curves, comparing the experimental to the
using symmetry properties of the sample. simulated material response for the initial guess and the final param-
eter set including two variations. The shown displacement is taken
from the bottom node of the right boundary in the symmetry plane.
Figure 4.7: (a) Sketch of the chosen segment of the specimen with the applied boundary conditions.
(b) Comparison of the load-displacement curves of the experiment with the results of the initial param-
eter set, the optimised Parameter Sets #1 (niso = 1.0), #2 (niso = 0.9), and #3 (niso = 1.1). Reprinted
from Schulte et al., Materials 13(14):3156 (2021), [110], with permission under creative commons 4.0.
For the purpose of reducing the computational cost of the FE simulation within
each iteration of the parameter identification, the symmetry properties of the sample
were used; cf. Figure 4.7a. Furthermore, considering the material properties of the soft
polymer, only the segment shown in the figure was used to exclude boundary effects
by the clamping jaws. In order to still use the experimental boundary conditions in
the simulations, the experimentally measured displacements, taken via the CCD camera
system, were applied to the right boundary of the specified segment. As can be seen
from the photos of the specimen during the experiment (cf. Figure 4.6) and from the
sketch of the segment, the displacements were not uniform over the boundary of the
chosen segment of the sample.
The initial guess for the damage-related model parameters was ηd = 0.002, κd = 0.4,
niso = 1.0, and cd = 0.04 and resulted in no damage initiation in the material; cf. Fig-
ure 4.8a. In Figure 4.7b, the comparison of the load-displacement curves is presented.
Since the stress-strain path of the homogeneous tensile test already overestimated the
experimental curve, the simulated response for the tensile test with inhomogeneous de-
116
Reaction force [N]
4.2 Application to self-diagnostic poly(dimethylsiloxane) elastomer
formation states lay above the experimental result as well. Furthermore, considering
the large total stretch values present in combination with the identified Yeoh material
parameters, the increasing response for the reaction force was triggered by parameters
C2 and C3 weighting the hyperelastic energy contributions of second and third order.
(a) Damage values for initial guess. (b) Damage values for Final #1.
Figure 4.8: Comparison of the contour plot of the damage value d at the final load step for (a) the
initial parameter set and (b) the optimised Parameter Set #1. Reprinted from Schulte et al., Materials
13(14):3156 (2021), [110], with permission under creative commons 4.0.
(a) von Mises stress for the initial guess. (b) von Mises stress for Final #1.
Figure 4.9: Comparison of the contour plot of the von Mises stress at the final load step for (a) the
initial parameter set and (b) the optimised Parameter Set #1. Reprinted from Schulte et al., Materials
13(14):3156 (2021), [110], with permission under creative commons 4.0.
Since parameter niso strongly influences the simulated deformation behaviour of the
specimen in each iteration, niso was fixed to 1.0 in a first step, thereby neglecting a
different damage contribution of the volumetric and isochoric part. The corresponding
optimised parameter set, denoted as Final #1, ηd = 0.202531, κd = 0.15185, and cd =
0.52593, was rounded to five decimal digits. Considering the load-displacement curves,
the response of the final set lay closer to the experimental behaviour than the initial
guess; see Figure 4.7b. The stress distribution of the initial guess and the final set was
comparable; the magnitude of the final set, however, was less than half of the stress of the
initial set; cf. Figure 4.9. The von Mises stress distribution within the specimen—with
the maximum stress value obtained at the upper surface, corresponding to the region
dominated by the initially circular notch—was in line with observations made by, e.g.,
Kleuter [52].
117
4 Parameter identification based on inhomogeneous states of deformation
(a) von Mises stress for Final #2. (b) von Mises stress for Final #3.
Figure 4.10: Comparison of the contour plot of the von Mises stress at the final load step for (a) the
optimised Parameter Set #2 with niso = 0.9 and (b) the optimised Parameter Set #3 with niso = 1.1.
Reprinted from Schulte et al., Materials 13(14):3156 (2021), [110], with permission under creative
commons 4.0.
The contour plot of the damage value d in Figure 4.8 compares the damage value
obtained for the initial parameter set with the damage value obtained for the Final #1
set. Initially, no damage was initiated, and with the Final #1 set a damage value of
nearly 75% occurred in the symmetry plane of the loading direction.
Figures 4.7b and 4.10 show the influence of the different damage functions for the vol-
umetric and isochoric contributions. The set Final #2 included niso =0.9, and Final #3
considered niso =1.1. A higher value for niso flattens the load-displacement curve, while
a lower value provides an increased path. Another important impact of the parameter
is indicated in Figure 4.10, where apart from the difference in the stress magnitude, the
deformation of the sample was already different for slight changes in the parameter with
regard to the necking of the sample at the symmetry plane of the loading direction.
The advantage of the regularised damage framework is the mesh objectivity of the
simulated material response. To demonstrate this feature, the boundary value problem
was calculated with three different meshes for the optimised parameter set. In addition
to the mesh containing 1548 elements, used for the results in Figures 4.7–4.10, a coarser
mesh with 1092 and a finer mesh with 2613 elements were used in order to analyse the
mesh sensitivity of the results. As can be seen in Figure 4.11, only marginal differences
in the contour plots of the damage function values and the von Mises stress, as well as the
load-displacement curves are visible. Thus, the gradient-enhanced damage model works
properly. In contrast, it is noted that the local damage model diverged at different load
steps, depending on the mesh discretisation. In the case of the fine mesh, the solution
diverged at a displacement of 3.2 mm, for the basis mesh at 3.375 mm and for the coarse
mesh at 3.415 mm. The displacement was taken from the bottom node of the right
boundary in the symmetry plane.
118
4.2 Application to self-diagnostic poly(dimethylsiloxane) elastomer
(a) Damage values for a coarser mesh with 1092 ele- (b) von Mises stress for a coarser mesh with
ments. 1092 elements.
(c) Damage values for the basis mesh with 1548 ele- (d) von Mises stress for the basis mesh mesh with 1548
ments. elements.
(e) Damage values for a finer mesh with 2613 elements. (f) von Mises stress for a finer mesh with
2613 elements.
40
30
20
10
0 1092 elements 1548 elements 2613 elements
0 3 6 9 12
Displacement [mm]
(g) Load-displacement curves for different discretisations.
The displacement shown was taken from the bottom node of
the right boundary in the symmetry plane.
Figure 4.11: Contour plots of the damage function values and the von Mises stress, as well as load-
displacement curves regarding the parameter set Final #1 for different discretisations. Reprinted from
Schulte et al., Materials 13(14):3156 (2021), [110], with permission under creative commons 4.0.
119
Reaction force [N]
4 Parameter identification based on inhomogeneous states of deformation
4.3 Application to the sheet metal DP800
After an application example of a multi-objective optimisation was given for a polymer,
two different material models are calibrated for a DP800-graded steel to demonstrate
the application of ADAPT, summarising the work of Schowtjak et al. [104]. In the first
case, a simple von Mises plasticity model is calibrated while analysing the impact of
different objectives for the error functional, such as forces and strains or displacements.
Subsequently, the more complex material model coupling gradient-enhanced damage to
finite plasticity, cf. Section 2.1, is used in a parameter identification process. In this
case, the model is calibrated based on inhomogeneous states of deformation employing
a multi-objective optimisation.
In order to obtain the appropriate local minimum with the lowest objective function
value, several different starting sets of the parameters were used for all calibrations,
since otherwise no proper minimum is guaranteed. In order to overcome unsuitable
local minima, where the Nelder-Mead simplex algorithm may get stuck, an evolutionary
algorithm was additionally applied to find the appropriate minimum.
4.3.1 Experiments
Regarding the required experimental data, a short summary of the work of Schowtjak
[104] is given, where several different tests were performed at the Institute of Metal Form-
ing and Lightweight Components (IUL), TU Dortmund University, on a Zwick / Roell
Z250 tensile testing machine under quasi-static conditions. The length of the specimen
is 80 mm and the crosshead velocity was prescribed at 2 mm/min. In order to achieve
statistical solid results, the experiments were carried out several times. Nevertheless,
the different experimental results depicted no significant deviation. In the following, the
experimental data with the load-displacement curve lying closest to the average response
has been chosen for the investigations.
An isotropic material behaviour considering elasticity and plasticity was obtained
from the analysis of standardised tensile tests within different loading directions with
respect to the rolling direction. Following DIN EN ISO 6892-1, the Young’s modulus
and the Poisson’s ratio were identified as E = 202 GPa and ν = 0.3, respectively.
The notched specimen, shown in Figure 4.12, was used in the tensile test for the
subsequent inverse parameter identification processes. The shape of the specimen was
chosen in order to specify the region where necking occurs and to initiate several differ-
ent states of deformation which can be measured by the DIC-system. Since quasi-static
loading conditions were required, the local spatial logarithmic strain rate within loading
direction was measured during the experiment and remained under or equal to 0.028/s.
During the tensile test, a GOM Aramis DIC-system captured the displacement and
(spatial) logarithmic total strain fields. In addition, the elongation of the specimen
was measured by the DIC-system by employing a virtual extensometer placed outside
the notch—assuming a homogeneous strain field—and had an initial length of 30 mm.
120
4.3 Application to the sheet metal DP800
300
R15
t=1.5
Figure 4.12: Technical drawing of the notched specimen for the DP800-graded steel. Reprinted from
Schowtjak et al., International Journal of Mechanical Sciences 213:106840 (2022), [104], with permission
under creative commons licence 4.0.
The corresponding measured load-displacement curve is shown in Figure 4.13. The
experimental data was captured by the load cell as well as by the DIC-system at equal
macroscopic displacement increments, i.e. in total at 60 frames. In Figure 4.14, the
evolution of the equivalent strain εeq in loading direction is depicted for the three rep-
resentative fr√am∑es marked in Figure 4.13. Assuming incompressibility, the equivalent
strain ε neq = i=1(2/3 λi)
2 is calculated from the optically measured strain, with λi
the principal stretches, and εeq is used as measure for the magnitude of deformation.
Considering the calibration process of the models, only the data frames prior to the
sudden fracture—as a result of the immediate localisation—are taken into account since
the fracture behaviour is not covered by the models. The different components of the
strain and displacement fields are shown in Figures 4.15 and 4.16, respectively, at the
load step right before fracture. Further information regarding the experiments can be
found in [104].
4.3.2 FE-model
Since the simulation of the FE-model has to be performed within each iteration of the
parameter identification process, the overall efficiency of the calibration can be tremen-
dously reduced if the computational cost of the simulation is reduced. Thus, considering
the sample geometry, see Figure 4.12, using the symmetry properties is very convenient.
As a consequence, only the top right eighth of the specimen depicted in Figures 4.14,
4.15 and 4.16 is modelled and symmetry boundary conditions are applied as can be
seen in Figure 4.17a. However, this is only valid for symmetric material response as
observed in the experiments. The displacements of the experiment are applied to the
top surface of the specimen. In Abaqus 960 C3D8R and C3D8T elements for Sections
121
15
20
4 Parameter identification based on inhomogeneous states of deformation
25
1 2 3
20
15
10
5
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Extensometer displacement in mm
Figure 4.13: Load-displacement curve of the notched specimen, see Figure 4.12, of the tensile test
under quasi-static loading speed. The displacement was measured by the DIC-system by employing a
virtual extensometer with an initial length of 30 mm in the homogeneous region of the sample. The
red marked points show the load steps where the strain fields depicted in Figure 4.14 were measured.
Reprinted from Schowtjak et al., International Journal of Mechanical Sciences 213:106840 (2022), [104],
with permission under creative commons licence 4.0.
Figure 4.14: Evolution of the equivalent strain εeq at the three characteristic time steps, shown in
Figure 4.13. The left image is associated to point 1 at a loading state before the maximum force is
reached, the middle one to point 2 at maximum force and the right image to point 3 right before fracture.
Reprinted from Schowtjak et al., International Journal of Mechanical Sciences 213:106840 (2022), [104],
with permission under creative commons licence 4.0.
122
Force in kN
4.3 Application to the sheet metal DP800
Figure 4.15: DIC-measured logarithmic strain fields in x, y, z-directions within the notch of the specimen
at point 3, cf. Figure 4.13, right before fracture. The corresponding coordinate system is visible in Figure
4.17b. Reprinted from Schowtjak et al., International Journal of Mechanical Sciences 213:106840 (2022),
[104], with permission under creative commons licence 4.0.
Figure 4.16: DIC-measured displacement fields in x, y, z-directions within the notch of the specimen at
point 3, cf. Figure 4.13, right before fracture. The rigid body motions are computationally eliminated
using the Aramis software. Alternatively, the procedures described in Sections 4.1.1 and 4.3.5 can be
applied. The corresponding coordinate system is visible in Figure 4.17b. Reprinted from Schowtjak
et al., International Journal of Mechanical Sciences 213:106840 (2022), [104], with permission under
creative commons licence 4.0.
4.3.3 and 4.3.4, respectively, were chosen. The global coordinate system is positioned in
the middle of the specimen, see Figure 4.17b.
123
4 Parameter identification based on inhomogeneous states of deformation
(a) Boundary conditions of
the FE-model. (b) Mesh of the FE model.
Figure 4.17: Finite element model used in the simulations of the parameter identification. Symmetry
properties are taken into account in order to reduce the computational cost and only calculate the
top right eighth of the specimen. The boundary conditions are shown in (a)—symmetry boundary
conditions are applied in z-direction as well—while the mesh with the used global coordinate system
is presented in (b). Reprinted from Schowtjak et al., International Journal of Mechanical Sciences
213:106840 (2022), [104], with permission under creative commons licence 4.0.
4.3.3 Parameter identification of a plasticity model
In this section, a summary of the work by Schowtjak in the corresponding section in [104]
is given in order to investigate different optimisation strategies of the inverse parameter
identification process by employing a simple von Mises plasticity model. The model is
directly provided by Abaqus [1], formulated in incremental logarithmic strains and the
yield function is given as
Φ = σeq − σy = 0 , with σy = h [α + α]np0 , √ (4.12)
with σy as the yield stress following a Swift hardening law and σ
3
eq = dev(σ) : dev(σ)2
as the von Mises equivalent stress. While α is the hardening variable, α0, h and np remain
as the corresponding material parameters.
In the following, the multi-objective optimisation—following the Pareto-strategy where
the forces and strains are considered to obtain the Pareto-optimal solution—is compared
to strategies where only the forces, strains or displacements are considered in the ob-
jective function. Those are denoted as F -, ε- or u-strategy, respectively. The finally
identified optimal material parameters are presented in Table 4.1 and the strain fields
of the experiment and the simulations are exemplarily shown for the last time step in
Figure 4.18. Furthermore, the corresponding load-displacement curves are compared in
124
4.3 Application to the sheet metal DP800
0.35
0.18
(a) Experiment (b) F -strategy (c) ε-strategy (d) Pareto-strategy 0.0
Figure 4.18: Contour plots of the spatial logarithmic strains in loading direction εyy located within the
notch at the final loading step. Comparison of the experimentally measured strains to the simulations
based on the optimised parameter sets for all identification strategies in Table 4.1. For comparison
purposes, an interpolation of the experimental strain data onto the integration point-coordinates of the
FE-model has been performed. Reprinted from Schowtjak et al., International Journal of Mechanical
Sciences 213:106840 (2022), [104], with permission under creative commons licence 4.0.
Figures 4.19 and 4.20 and the simulated strains in loading direction are shown for the
last time step in Figure 4.21.
Table 4.1: Material parameters of the simple von Mises plasticity model—defining initial yield α0, hard-
ening coefficient h and hardening exponent np identified based on the different mentioned strategies and
their corresponding objectives. Reprinted from Schowtjak et al., International Journal of Mechanical
Sciences 213:106840 (2022), [104], with permission under creative commons licence 4.0.
Parameter F -strategy ε-strategy Pareto-strategy u-strategy Unit
a0 1.3 ·10−5 6.9 ·10−7 6.4·10−6 8.0 ·10−3 -
h 1271.4 5128.1 1241.6 7199.9 MPa
np 0.15 0.11 0.14 0.12 -
Since the simple von Mises plasticity model is not able to fully capture the experi-
mentally measured material behaviour—i.e. all localisation effects, especially the damage
related material softening effect—it is only employed to efficiently investigate various op-
timisation strategies.
As it can be seen in Table 4.2, the F -strategy shows the lowest deviation with regard
to the forces with an error of 908.3 N, which is taken as the global minimum with
respect to the forces f ∗F . However, the error regarding the strain fields is the highest
of all strategies since the strain fields were not considered in the objective function.
Accordingly, the lowest error with respect to the strains is achieved with the ε-strategy
as 1.9 · 10−5 which is assumed as global minimum f ∗ε . Nevertheless, the error regarding
the forces is more than 60000 times higher than the global minimum, and the comparison
of the load-displacement curves in Figure 4.19 demonstrate that the found solution of
this strategy is far away from a realistic mapping. An even worse result is obtained
from the u-strategy, see Table 4.2 and Figure 4.19. In contrast, the optimal solution
125
4 Parameter identification based on inhomogeneous states of deformation
Figure 4.19: Load-displacement-curve for all presented parameter identification strategies. In order to
compare both strategies close to the experimental response, a closer section of the figure is shown in
4.20. Reprinted from Schowtjak et al., International Journal of Mechanical Sciences 213:106840 (2022),
[104], with permission under creative commons licence 4.0.
Figure 4.20: Zoom of the load-displacement curves in Figure 4.19 for both calibration strategies close
to the experimental response. Reprinted from Schowtjak et al., International Journal of Mechanical
Sciences 213:106840 (2022), [104], with permission under creative commons licence 4.0.
considering the Pareto-strategy can be obtained by minimising the following objective
function √[ ]2 [ ]2
f ∗ − fF (κ) f ∗ − fε(κ)
f(κ) = F ε∗ + ∗ , (4.13)fF fε
which is normalised by the optima of the corresponding single objective problems. Even
though the single function values of the Pareto-optimal solution are 32% and 72% higher
than the chosen global minima of the forces and strains, the overall combined error is
minimised, cf. Table 4.2. It can be seen in Figures 4.19 and 4.18 that the Pareto
126
4.3 Application to the sheet metal DP800
(a) F -strategy (b) ε-strategy (c) Pareto-strategy
Figure 4.21: Contour plots of the strains in loading direction with regard to the FE-models at the
final loading step based on the corresponding optimised parameter sets for three exemplary calibration
strategies, cf. Table 4.1. Reprinted from Schowtjak et al., International Journal of Mechanical Sciences
213:106840 (2022), [104], with permission under creative commons licence 4.0.
Table 4.2: Final objective function values of the four different presented calibration strategies, i.e. based
solely on either forces F , strains ε or displacements u or on the combined solution regarding the Pareto
optimum. The error f• is calculated following the root sum of squares in Equation (4.2). Reprinted from
Schowtjak et al., International Journal of Mechanical Sciences 213:106840 (2022), [104], with permission
under creative commons licence 4.0.
Strategy Objective fF in kN fε
F -strategy F 0.9083 4.8 ·10−5
ε-strategy ε 57503.2425 1.9 ·10−5
Pareto-strategy F and ε 1.2026 3.2 ·10−5
u-strategy u 125005.4490 2.4 ·10−5
optimum depicts the closest solution regarding both quantities. Thus, in total, the
Pareto-strategy reflects the realistic material behaviour best, followed by the F -strategy
which is at least reasonable to some degree, and the other two strategies lead to quite
unreasonable results.
4.3.4 Parameter identification of gradient-enhanced material
models
As already mentioned in Section 4.3.3, the simple von Mises-based plasticity model was
not fully able to capture the softening behaviour of the material since damage was not
included in the model. However, in the gradient-enhanced model for coupled damage and
plasticity, see Section 2.1, not only damage is taken into account but also many aspects
of damage can be captured, such as localisation effects in the context of finite element
127
4 Parameter identification based on inhomogeneous states of deformation
applications. Consequently, the model is capable of matching the experimental material
response while additionally providing mesh independent simulation results. The fact
that the damage regularisation framework used provides mesh-independent results was
shown in [86]. The extensions to finite plasticity [119] and finite visco-elasticity [110]—cf.
Sections 2.2, 2.5.2 and 4.2—also provide mesh-independent simulation results.
Table 4.3: Material parameters of the initial guess and the optimised parameter set, calibrated with
regard to the load-displacement curve. For further details on the model formulation see 2.1. The
parameters indicated with ”fix” were not identified within the parameter identification procedure since
these were pre-defined in previous experiments, e.g. in the case of E and ν, see Section 4.3.1, or
these have been pre-defined manually. For example, the value for βd is set sufficiently high to ensure
the coupling between the non-local and local damage variables. Reprinted from Schowtjak et al.,
International Journal of Mechanical Sciences 213:106840 (2022), [104], with permission under creative
commons licence 4.0.
Symbol Description Initial guess Final set Unit
E Young’s modulus 202.0 fix GPa
ν Poisson’s ratio 0.3 fix −
α0 defining initial yield stress 1.4 · 10−5 2.6027 · 10−5 −
h hardening coefficient 1700.0 1386.9 MPa
np hardening exponent 0.25 0.1676 −
qmax max. damage threshold 1.0 0.8062 MPa
nd damage exponent 2/3 fix −
η damage rate factor 0.7 0.4751 −
ξvol volumetric damage factor 1.0 fix −
ξiso isochoric damage factor 0.4 0.5816 −
ξq threshold factor 1.0 0.5477 −
ξm effective stress factor 0.3 0.4296 −
ηα coupling factor 0.0 fix −
cd regularisation parameter 20.0 fix N
βd penalty parameter 10.0 fix MPa
qmin initial damage threshold 0.5 0.6025 MPa
In a first step, the model parameters were only calibrated with regard to the load-
displacement curve. The identification process was conducted with different starting
sets for the material parameters. The best fit for the parameters is presented in Table
4.3 with the corresponding starting values, and the comparison of the experimental and
simulated load-displacement curve is shown in Figure 4.22. The simulated distribution
of the damage value is depicted in Figure 4.23a. It can be seen that the damage initiation
and further evolution in terms of the damage variable d takes place in the necking region
of the notched specimen which is quite reasonable since the regime of highest tensile
load corresponds to the damage initiation regime. Nevertheless, Figure 4.23 perfectly
128
4.3 Application to the sheet metal DP800
25
20
15
10
5
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Extensometer displacement in mm
Figure 4.22: Comparison of the load-displacement curves of the simulations of the final parameter sets
with different calibration strategies with the experimental response. Reprinted from Schowtjak et al.,
International Journal of Mechanical Sciences 213:106840 (2022), [104], with permission under creative
commons licence 4.0.
d in %
72.4
36.2
0.0
(a) Parameter set 1 (b) Parameter set 2 (c) Parameter set 3
Figure 4.23: Contour plots of the damage variable d at the final load step for three different parameter
sets obtained via optimisations with different initial guesses. Parameter set 1 (a) represents the best
solution according to the force-response. These plots are associated with the load-displacement curves in
Figure 4.24. Reprinted from Schowtjak et al., International Journal of Mechanical Sciences 213:106840
(2022), [104], with permission under creative commons licence 4.0.
demonstrates the difficulties during the parameter identification process. First, the
importance of using many different starting sets for the model parameters in order to find
an appropriate minimum since, considering the amount of parameters, the appearance
of local minima substantially increases. Alternatively, an evolutionary algorithm should
129
Force in kN
4 Parameter identification based on inhomogeneous states of deformation
25
20
15 Experiment
Parameter set 1
Parameter set 2
10 Parameter set 3
5
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Extensometer displacement in mm
Figure 4.24: Load-displacement curves for three different parameter sets obtained via optimisations
with different initial guesses. These graphs are associated with the damage contour plots shown in
Figure 4.23. Reprinted from Schowtjak et al., International Journal of Mechanical Sciences 213:106840
(2022), [104], with permission under creative commons licence 4.0.
be used. Secondly, the specified load path of the experiment, in other words the data
available, is not sufficient in order to uniquely identify the damage related material
parameters, since, e.g., unloading steps are not considered. For example, the slope of
the unloading curve indicates the amount of damage induced in the material as related
to the reduction of the Young’s modulus, while the remaining strain is associated with
the amount of plastic strain stored in the material. Thus, the degradation effect can be
captured, see [119]. The results show that some starting sets can lead to local minima
where the simulated damage distribution does not predict the experimental material
behaviour sufficiently and where the damage amplitude in terms of the damage variable
d may show a significant difference.
Table 4.4: Result of the optimisation process of the combined solution associated with the Pareto
optimum for the calibration of the gradient-enhanced damage model calibrated with regard to the forces
and the displacement field. The error f• is given in the form of the root sum of squares, see Equation
(4.2). Reprinted from Schowtjak et al., International Journal of Mechanical Sciences 213:106840 (2022),
[104], with permission under creative commons licence 4.0.
Step Objective fF in N fu in mm
Initial F and u 36922.53 1.294 ·10−3
Final F and u 13.69 3.478 ·10−4
Thus, precise planning of the experimental setup prior to the parameter identifica-
tion is absolutely necessary. Depending on the capabilities of the material model, the
130
Force in kN
4.3 Application to the sheet metal DP800
experimental requirements need to be determined, e.g. considering different deformation
modes in terms of proportional and non-proportional loading paths, unloading sequences,
multi-cyclic responses and step tests. In addition, modelling errors have to be identified
in order to avoid or at least control them.
Table 4.5: Material parameters of the initial guess and the optimised parameter set, calibrated with
regard to the load-displacement curve and the displacement field, while Table 4.3 showed the calibration
results only based on the load-displacement data. Reprinted from Schowtjak et al., International Journal
of Mechanical Sciences 213:106840 (2022), [104], with permission under creative commons licence 4.0.
Symbol Description Initial guess Final set Unit
E Young’s modulus 202.0 fix GPa
ν Poisson’s ratio 0.3 fix −
α0 defining initial yield stress 1.4 · 10−5 1.3765 · 10−5 −
h hardening coefficient 1700.0 1404.4 MPa
np hardening exponent 0.25 0.1703 −
qmax max. damage threshold 1.0 1.611 MPa
nd damage exponent 2/3 fix −
η damage rate factor 0.7 0.9457 −
ξvol volumetric damage factor 1.0 fix −
ξiso isochoric damage factor 0.4 0.3752 −
ξq threshold factor 1.0 1.089 −
ξm effective stress factor 0.3 0.1123 −
ηα coupling factor 0.0 fix −
cd regularisation parameter 20.0 21.3755 N
βd penalty parameter 10.0 fix MPa
qmin initial damage threshold 0.5 0.5527 MPa
As mentioned in the previous section, by restricting the calibration process to load-
displacement data, especially field quantities—e.g. strains or displacements—are not
necessarily matched accurately. Therefore, the displacement field on the surface of the
specimen can be included into a multi-objective optimisation problem. Hence, specific
parameters such as the regularisation parameter, that control localising effects can be
identified more precisely. Furthermore, several types of deformation states—considering
experiments with inhomogeneous deformation states—are incorporated in the objective
function.
In order to improve the parameter identification results based on these experiments,
the experimentally measured displacement field is taken into account and compared to
the simulated response analogously to equation (4.13), though alternatively the displace-
ments are considered instead of the strains this time. The initial guess remains the same
as for the previous calibration restricted to load-displacement curve data. Nevertheless,
131
4 Parameter identification based on inhomogeneous states of deformation
this time, the regularisation parameter cd as well as the initial damage threshold qmin,
cf. Equation (2.5), are additionally calibrated, since the DIC-data enables their identifi-
cation. The Pareto-strategy is chosen since it was already shown in Section 4.3.3 to be
the most efficient one. The Pareto-optimum was chosen based on an equal weighting of
both the objective function contributions. The initial and the final objective function
values are shown in Table 4.4. The corresponding final parameter set is given in Table
4.5 and the comparison of the simulated and experimental material response is presented
with respect to the load-displacement curve in Figure 4.22 and to the displacements in
Figure 4.25. While the load-displacement curve and the displacement field in y-direction
are matched nearly perfectly, the displacement field in x-direction still shows slight dif-
ferences. For a moment it may appear confusing as to why the parameter identification
based on the Pareto-strategy results in a better agreement of the load-displacement
curve in contrast to the calibration based only on load-displacement curve data. How-
ever, with the Pareto-strategy, two more parameters were calibrated and especially the
initial damage threshold qmin influences the load-displacement behaviour of the material.
As a consequence, qmin can be already calibrated based on load-displacement data.
0.0
-0.28
-0.56
(a) Experiment ux (b) Simulation ux
1.2
0.6
0.0
(c) Experiment uy (d) Simulation uy
Figure 4.25: Contour plots of the displacements at the final load step with the optimised parameter
set, see Table 4.5 for (a) the experimental displacements ux, (b) the simulated displacements ux, (c) the
experimental displacements uy and (d) the simulated displacements uy. The measured displacement
data has been interpolated onto the FE nodes. Reprinted from Schowtjak et al., International Journal
of Mechanical Sciences 213:106840 (2022), [104], with permission under creative commons licence 4.0.
132
4.3 Application to the sheet metal DP800
After all, it was shown that the more complex gradient-enhanced damage model
captures the material behaviour of the DP800 way better than the simple model does.
By incorporating the displacement field into the objective function of the calibration
process, the material parameters are better fitted, although still no fully representative
set of parameters is obtained since the experiment lacks unloading steps.
4.3.5 Starting value generated via machine-learning assisted
parameter identification
In the previous section, all material parameters of the gradient-enhanced damage model
coupled to finite plasticity presented in Section 2.1 were identified by using classic pa-
rameter identification strategies. Therefore, the elastic parameters were pre-identified
in standard experiments and subsequently the plasticity- and damage-related material
parameters were calibrated. The overall identification process was quite time-consuming
considering that the parameter identification process was performed with different char-
acteristic initial guesses and different strategies in order to find an appropriate solution.
In order to improve the efficiency of the overall identification process—especially regard-
ing the difficulty in finding appropriate starting values for the calibration if the user is
not deeply familiar with the underlying material model—the advantage of the hybrid
strategy is analysed in the following. The hybrid approach was presented in Chapter 3
and applied to the mentioned finite plasticity model coupled to non-local damage.
Since the generation of qualitative starting values for the material parameters by
incorporating an ANN was based on homogeneous deformation states, some of the
damage-related material parameters could not be identified. However, in view of real-life
applications of coupled damage and plasticity formulations, the non-local regularisation-
related model parameters ensuring mesh independent results need to be calibrated as
well, see e.g. [119, 120]. Thus, a multi-objective parameter identification is carried out
with respect to inhomogeneous states of deformation. Therefore, not only the experi-
mentally measured load-displacement curve is considered but also the displacement field
on the corresponding surface measured via DIC. This enables a more accurate calibration
of the constitutive model to the material behaviour, since different deformation states
and modes are taken into account and as already mentioned in the previous sections.
Moreover, the mentioned regularisation-related parameters, such as the regularisation
parameter cd, can additionally be identified. The overall identification process is compa-
rable to the previous section, in the following however, the starting values were obtained
from the ANN and not guessed by the user. Furthermore, to increase the complexity
of the problem and showing the capability of this approach, no subsets of the mate-
rial parameters were pre-identified. In addition, unloading steps are considered in the
underlying experimental data.
To give an example of a subsequent calibration considering inhomogeneous states of
deformation, sheet metal DP800 is considered in the following. The experimental data
was generated by Sprave and Menzel, see [119]. The sketch of the specimen is depicted
133
4 Parameter identification based on inhomogeneous states of deformation
in Figure 4.27a, whereby the geometry of the specimen has a total length of 52 mm, a
maximum width of 10 mm, a minimum width of 3.85 mm and a radius of 39.4 mm. The
thickness of the specimen is 1.5 mm. The experiments were conducted with a micro-
tensile machine from Kammrath & Weiß with a load cell of 10 kN in combination with
the DIC-software Veddac 7. During the experiment, only a section of the domain with
a length in longitudinal direction of 10 mm in the centre of the specimen is taken into
account. Hence, the FE simulations are restricted to that section of the specimen,
visible in Figure 4.27a, where the experimentally measured data is directly used for the
applied Dirichlet boundary conditions. The section is discretised with 4000 8-noded
brick elements which were enhanced by the F-bar method. The nodes deal with three
displacement degrees of freedom and one non-local damage degree of freedom and both
fields are linearly interpolated. Further details can be taken from [119].
Table 4.6: Final material parameters of the DP800 after the multi-objective optimisation with respect
to the load-displacement curve and DIC-measured displacement data by a classic parameter identifica-
tion. The predicted parameter set κpred of the ANN and set κopthom after the subsequent Nelder-Mead
simplex optimisation are used as starting points for this calibration based on inhomogeneous states of
deformation. For the illustration of the evolution of the objective function value, the initial objective
function value with the predicted parameter set f (κpredDIC ) is set to 100 %. Thus, the initial objective
function value with κopthom is fDIC(κ
opt
hom)=111.91 %. Reprinted from Schulte et al., European Journal of
Mechanics - A/Solids 98:104854 (2023), [113], with permission under creative commons licence 4.0.
Symbol Description κpred κoptDIC DIC Unit
fDIC objective function value 61.67 62.72 %
fFDIC error in force 24.51 24.02 %
E Young’s modulus 171.455 173.188 GPa
ν Poisson’s ratio 0.151 0.304 −
σy0 initial yield stress 399.299 130.3 MPa
h hardening parameter 1030.137 1250.045 MPa
np hardening exponent 0.278 0.183 −
qvar variable damage threshold 1.35 3.636 MPa
nd damage exponent fixed at 0.667 −
η damage rate factor 2.203 2.609 −
ξvol volumetric damage factor fixed at 1.0 −
ξiso isochoric damage factor 0.184 1.617 −
ξq threshold factor 1.409 10.332 −
ξm effective stress factor 3.393 0.001 −
ηα coupling factor 0.246 2.715 −
cd regularisation parameter 266.111 30.591 N
βd penalty parameter fixed at 500.0 MPa
qmin initial damage threshold 256.837 29.748 MPa
134
4.3 Application to the sheet metal DP800
In the following, the DIC-based parameter identification procedure is carried out
with two different starting points. The first starting point is parameter set κpred as
predicted by the ANN and the second point is optimal parameter set κopthom including the
subsequent Nelder-Mead simplex optimisation step. Even though the simulations with
both parameter sets were able to match the homogeneous stress-strain behaviour quite
accurately and the deviation between the curves was only marginal, some of the identified
material parameters showed considerable differences, e.g. the Young’s modulus, see Table
3.3. In this section, the influence of the different starting points shall be analysed.
Even though the network was in this case trained with a tensile load path only, the
shear deformation was qualitatively predicted. Moreover, the final calibration performed
in this section considers full-field data including different deformation modes in order to
obtain a set of material parameters suitable for the prediction of the material response
in the context of more general loading paths.
As shown in Table 4.6, the final objective function value of both identifications is
nearly on the same level, although the deviation of the initial function values featured
a difference of ∆fDIC =11.91 %. Hence, two local minima were found due to the signif-
icantly differing starting values and final parameter sets. The objective function value
was calculated following Sprave and Menzel [119], i.e.
f = w∑fF + w f∆uDIC F DIC ∆u DIC with (4.14)
fF exp sim 2DIC = wt [F − F ]t (4.15)
∑t ∑nnp
f∆uDIC = wt/n ‖∆uexp −∆usim‖2np i i t , (4.16)
t i
where w• denote the weighting factors. In order to eliminate rigid body motions, the
difference of the displacement of two neighbouring measurement points is divided by their
distance. Thus, a simple strain-like quantity ∆u is obtained, see [97, 119]. To ensure a
comparable contribution of both terms—the force- and displacement-related terms—to
the objective function, both weighting factors are chosen such that the objective function
has the unit of stress squared. Thus, wF = [4/A0]
2 and w 2∆u = 0.7E , where factor 4
relates to the symmetry of the specimen and where 0.7 was determined empirically as a
good balance factor, see [119]. The first starting point leads to a minimum were the error
in displacement is lower compared to the result with the second starting point, though
the error in force is slightly higher. Comparing each of the parameters, some of them
lie close to each other, e.g. the Young’s modulus, even though this parameter was very
different in the initial points. Nevertheless, several other parameters, e.g. the Poisson’s
ratio, the initial yield stress or the initial damage threshold, show larger deviations.
In Figure 4.26, both simulated load-displacement curves are compared to the exper-
imentally measured one, both showing a quite accurate fit. Only two regions feature
larger deviations. Especially the force decrease at the end of the experiment due to
135
4 Parameter identification based on inhomogeneous states of deformation
material softening and damage evolution is not matched very accurately. With the sec-
ond parameter set κoptDIC, in the last steps, however, damage increases markedly and the
response approaches the values obtained in the experiment.
Figure 4.26: Load-displacement curve of the tension test of the DP800 considering inhomogeneous states
of deformation compared to the simulated curves with the final optimal parameter sets depending on the
starting point (SP), i.e. directly from the NN-prediction or after the subsequent Nelder-Mead simplex
calibration. Reprinted from Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023),
[113], with permission under creative commons licence 4.0.
The contour plots of the squared absolute error of the relative displacements depict
nearly the same deviations compared to the DIC-measured displacements, see Figure
4.27. This demonstrates that both solutions provide a comparable minimum for the
error in displacements and that the highest deviations cannot be captured by the model
itself due to, e.g., inhomogeneities in the specimen.
Examining the material parameters in Table 4.6 and the load-displacement curve in
Figure 4.26 of the first identified parameter set, it is indicated that no damage evolution
occurs. The contour plot of the isochoric damage function value after the final load step
demonstrates that damage does not evolve for the first set. In contrast, for the second
set, damage increases vastly in the final load steps in the necking region.
136
4.3 Application to the sheet metal DP800
(a) Sketch of the tensile specimen taken from Sprave and Menzel [119] marking the
plotted section of the specimen.
(b) Simulation with the optimal parameter set κpredDIC .
(c) Simulation with the optimal parameter set κoptDIC.
0 2 × 10−5 5 × 10−5 7 × 10−5
Figure 4.27: Contour plots of the squared absolute error of the relative displacements f∆uDIC, following
the calculation presented in Sprave and Menzel [119], at each node at time t=3520 s, at the beginning of
the last loading step, point P1 in Figure 4.26, for both optimal parameter sets. Reprinted from Schulte
et al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113], with permission under creative
commons licence 4.0.
These results demonstrate that the ANN already provides an adequate starting point
for a subsequent multi-objective parameter identification since the difference in the re-
sults is due to the fact that, for this complex material model and the experimental data
137
4 Parameter identification based on inhomogeneous states of deformation
used, many local minima exist which feature a comparable material behaviour. As a
consequence, this makes it not only difficult for the ANN to identify the features of the
model but also for the general parameter identification to find an appropriate minimum,
especially if the user is not familiar with the material model and not able to identify a
good starting point.
(a) Simulation with the optimal parameter set κpredDIC .
(b) Simulation with the optimal parameter set κoptDIC.
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Figure 4.28: Contour plots of the isochoric damage function value fiso(dφ) = exp(−η ξiso dφ) after the
final load step at t= 3730 s, point P2 in Figure 4.26, for both optimal parameter sets. Reprinted from
Schulte et al., European Journal of Mechanics - A/Solids 98:104854 (2023), [113], with permission under
creative commons licence 4.0.
Nevertheless, in future applications subsets of material parameters should be identified
based on subsets of the experimental data, as discussed in [77, 142]. To give an example,
parameters related to the purely elastic material response shall be identified based on
138
4.3 Application to the sheet metal DP800
data related to the elastic regime only, i.e. the regime where inelastic effects are not
activated, cf. Section 4.3.
139

5 Concluding remarks
This work dealt with the difficulty of identifying a proper set of material parameters
in order to be able to accurately predict a specific material behaviour. The effort and
difficulty to calibrate model parameters for the chosen material increases with the com-
plexity of a material model and the number of its material parameters, especially if some
of the parameters do not possess an obvious physical interpretation. Thus, the definition
of an appropriate starting point for a classic parameter identification algorithm is chal-
lenging, and the obtained optimum is often sensitive with respect to the starting point.
Therefore, three quite different and complex material models were chosen as application
examples. They differ not only with regard to the mechanisms which are simulated, but
also regarding the groups of materials.
For this purpose, in Chapter 2, a finite plasticity model coupled to gradient-enhanced
damage, developed by Sprave and Menzel [119], was described in Section 2.1 and was
later applied to metals which are used in metal forming processes. In addition, the un-
derlying damage framework, introduced by Ostwald et al. [86], was adapted to a finite
strain viscoelasticity model in Section 2.2 which was later on calibrated for polymers.
Furthermore, a laminate-based model for single- and polycrystalline ferroelectric ma-
terials, developed by Dusthakar et al. [17], was described and investigated regarding
the numerical treatment of the Karush-Kuhn-Tucker conditions in Section 2.3 and the
remaining two characteristic model parameters were identified later on.
Subsequently, the most challenging case in parameter identification is depicted re-
garding the first mentioned material model, where none of the parameters were pre-
identified via specific experiments and thus all of the parameters—except for the non-
local contributions—had to be identified at once based on homogeneous deformation
states. Therefore, a Nelder-Mead simplex algorithm was applied in combination with
three different starting values for the parameter sets for the sheet metal DP800 and the
case-hardened steel 16MnCrS5. With this calibration strategy, different local minima
were found which fit the experimental material behaviour properly, but the correspond-
ing identified material parameters clearly varied. This demonstrates that the combina-
tion of the different material mechanisms enables multiple parameter combinations to
depict a nearly equal material behaviour, if only a rather simple tensile test is considered
141
5 Concluding remarks
and too many material parameters are simultaneously identified. Thus, as many mate-
rial parameters as possible should be identified separately with respect to characteristic
experiments, e.g. the Young’s modulus and the Poisson’s ratio with regard to a tension
test in the elastic region.
As a consequence, a different strategy was applied for the next material model. In
order to identify the set of over 12 material parameters for the gradient-enhanced dam-
age model in rate-dependent materials under finite strains, an efficient scheme for the
process was carried out. At first, the basic constitutive model parameters, namely elastic
parameters, were calibrated with respect to experiments displaying homogeneous states
of deformation. Considering the deformation states, the computational cost of the iden-
tification was further reduced by using constitutive drivers for the simulations of all
three required types of experiments within each iteration of the calibration process. In
the next step, the already identified parameters were fixed during the identification of
the damage-related material parameters.
For the purpose of developing a general framework for the calibration of rate-dependent
materials coupled to damage, the constitutive viscoelasticity model by Simo and Hughes
[117] was coupled to the gradient-enhanced damage formulation. Thus, the advan-
tages of both models were combined, enabling different types of hyperelastic energy
formulations—considering the energy-independent viscoelastic derivations by Simo and
Hughes [117]—as well as mesh-independent results with respect to the damage evolution
due to the damage regularisation framework by Ostwald et al. [86].
In order to investigate whether the framework works even for very large deformations,
a self-diagnostic elastomer incorporating the mechanophore units within the polymeric
microstructure was chosen as an application. With regard to the material behaviour of
the soft polymer, the hyperelastic model by Yeoh and Fleming [139] was applied to the
viscoelastic damage model.
The results of the parameter identification in Section 2.5.2.2 showed that the general
framework of the implemented material model, as well as the parameter identification
works properly. Even though the chosen self-diagnostic elastomer was only a first at-
tempt at applying the framework, the experimental results already basically matched,
cf. Figures 2.27.
Nevertheless, further improvements could be achieved for example by using different
weighting factors for the contributions of the three different experiments with homoge-
neous deformation states to the goal function of the optimisation in order to improve
the calibration of the parameters with respect to the tensile test. However, after all, the
largest improvement could be achieved by a refinement of the experiments. The material
properties led to difficulties in the sample manufacturing—such as uniformly cutting the
shape—and in the clamping of the specimen, since the clamping jaws already gener-
ated deformations in the soft polymer. An improvement would be if the samples were
directly moulded or if clamping were conducted with a very accurate torque spanner.
Furthermore, experimental data based on different strain rates could be included in the
identification framework.
142
In the case of the third example, the laminate-based model for ferroelectric materials
was implemented incorporating many material parameters which are characteristic for
the class of ferroelectrics, e.g. the piezoelectric or the spontaneous strain coefficients,
cf. [17] or the summary in Section 2.3. Thus, regarding the calibration to BaTiO3,
Dusthakar et al. were able to take many parameters directly from the literature, see
[17] or from the summary of the identification process in Section 2.5.3, and only two
remaining model parameters had to be calibrated. However, considering that twelve
characteristic curves had to be considered for the optimisation process, Dusthakar et
al. separately applied a generic algorithm for the optimisation process of each of the
different stress levels in a first step Thus, they could select appropriate starting values
for each of the remaining model parameters for a subsequent optimisation considering
all experimental data simultaneously. Here, a SQP-based algorithm was chosen.
After all, it required luck to guess an appropriate initial guess or time and effort
to identify proper starting values for a parameter identification process. Thus, Chap-
ter 3 provided a combination of two approaches—artificial neural networks and classic
parameter identification—in order to overcome the severe difficulty in connection with
the application of complex material models to a specific material behaviour. By using
an ANN for the generation of a valid initial guess for a subsequent classic parameter
identification, as described in Chapter 3, the calibration process becomes more efficient.
Furthermore, if the trained ANN is provided along the corresponding constitutive ma-
terial model to the user, he is not required to be familiar with the model itself.
It was shown that it can therefore be advantageous to use an ANN ahead of a classic
PI. Nevertheless, it has to be ensured that a sufficient amount of training data is gen-
erated. Moreover, the prediction accuracy of the ANN is only as good as the material
model is able to capture the various features of the material behaviour. For the appli-
cations considered in this work, the results show that the Latin Hypercube Sampling
method in combination with a hyperparameter optimisation yielded the best network
performance. The ANN was able to provide a very good prediction of the experimen-
tal material behaviour which could be further improved to a nearly perfect fit with
the following classic PI. If different types of loading paths were included in the train-
ing process an overall better prediction of the different experiments could be obtained.
The results demonstrate that the ANN is capable of directly providing a quite accurate
solution, i.e. starting point, for a subsequent multi-objective parameter identification
even without the intermediate step of a subsequent optimisation based on homogeneous
deformation states. Nevertheless, in future work, the prediction accuracy of the frame-
work for complex material models can be further improved when subsets of material
parameters are calibrated with regard to specific loading path sequences.
Further research can be conducted in the ANN-architecture of the demonstrated sys-
tem. Autoencoders, which are an established type of ANN with the ability to learn
encodings and therefore extract features of given data, are a possible candidate. This
approach could reduce the ANN’s size and therefore increase the efficiency while main-
143
5 Concluding remarks
taining capabilities.
Nevertheless, since the user still has to choose a proper material model for the corre-
sponding material, one could provide another neural network prior to this hybrid iden-
tification approach or to every other classic parameter identification. This additional
network would be trained in order to classify the experimentally measured material be-
haviour, e.g. rubber or steel, and thereby provide an appropriate material model available
to the user.
Since the chosen material model is restricted to isotropic plasticity, the model is currently
extended to cover anisotropic effects to simulate the material behaviour more precisely.
In contrast to a classic parameter identification, the results of the ANN can show im-
mediately if the underlying dataset, e.g. regarding the loading paths, is chosen properly
in order to identify each of the parameters, thus providing sensitivity-like information.
However, if a parameter turns out to have no influence on the material response, a model
reduction should be performed, which also increases the efficiency of the model.
In addition, in order to explore the benefit of the hybrid strategy, it was applied
to a different material model, i.e. the laminate-based model for ferroelectric materials,
cf. Section 2.3. Since it was previously ascertained that it is very important to identify as
many material parameters as possible in corresponding specific experiments, most of the
parameters were taken from Table 2.6 following Dusthakar et al. [17] except for the two
remaining model parameters which remained to be identified. In [17], it was quite time
consuming to find proper starting values for the parameters. Thus, the hybrid approach
can still be advantageous even though most of the parameters are already fixed to a
specific material.
The results in Section 3.2 demonstrate that the trained ANN is able to perfectly
predict the simulated material response of the model and that the parameters are inde-
pendent from each other. However, it was shown that, if the experimentally measured
material response deviating massively from a possible simulated response, the ANN has
to extrapolate which decreases the prediction quality. Nevertheless, this prediction is
still sufficient for a subsequent classic calibration procedure.
However, experimental data with solely homogeneous deformation states is not always
sufficient for an accurate prediction, especially in terms of gradient-enhanced contribu-
tions, e.g. with regard to damage evolution. Thus, field data in combination with a
multi-objective optimisation are required. The parameter identification tool ADAPT,
summarised in Chapter 4, provides this feature and is available on GitHub [26] as
open source code. In this work, the tool is applied to the two mentioned damage frame-
works, see Sections 2.1 and 2.3.1, in order to identify the material parameters of the
self-diagnostic PDMS elastomer and the sheet metal DP800. The two step strategy
was employed for the polymer, where the elastic parameters were pre-identified, and for
the DP800 different strategies of the multi-objective optimisation as well as two various
complex material models were compared to each other.
144
In the case of the polymer, the results of the parameter identification based on the
inhomogeneous deformation states demonstrated once again that the framework of the
underlying model, as well as the calibration worked properly. Furthermore, since the
simulations for the experiments with inhomogeneous deformation states were conducted
before the sample started to tear in the necking region, the damage distribution with
the optimised parameter set—showing damage evolution of nearly 75% in the symmetry
plane of the loading direction, i.e., the necking region—reflected such a material be-
haviour.
In addition, in a future step, the load path for the inhomogeneous experiments could
be changed to cyclic loading, improving the damage characterisation, though increasing
the computational cost within each iteration of the parameter identification procedure.
Nevertheless, further experiments including unloading should be included in the calibra-
tion process to enhance the accuracy. Until now, only damage activated prior rupture
has been taken into account. As a long-term goal, the current framework should be com-
bined with a rupture model to improve the overall prediction of the material behaviour.
After all, the general parameter identification framework provides an efficient scheme
for the calibration of gradient-enhanced damage in rate-dependent material models un-
der finite strains. In the future, an interesting extension of the parameter identification
framework is the incorporation of the self-diagnostic properties of the elastomer in terms
of additional field data that is considered within the objective function.
In conclusion, it is generally beneficial to employ additional data for the overall cali-
bration process, especially if the weighting between different contributions is done prop-
erly, e.g. by using a Pareto-scheme to find the optimal solution with respect to all
contributions. Furthermore, the more material mechanisms a model can capture, the
more accurate the prediction of the realistic material behaviour becomes. However, si-
multaneously, all the more experimental data is required—e.g. unloading sequences or
field data—for an accurate parameter identification. Nevertheless, this results in higher
computational costs within each iteration, which becomes even more expensive if the
calibration process has to be performed with several different starting values. Thus, the
hybrid strategy provides an efficient alternative and the results in Section 4.3.5 demon-
strated that the ANN is capable of providing a quite accurate solution, i.e. starting
point, for a subsequent multi-objective parameter identification.
145

A Numerical time integrations and
algorithmic pseudo-code of the
finite-strain viscoelasticity
In this appendix, the numerical update procedure of the convolution integrals intro-
duced in Section 2.2.2 is shown in more details. Subsequently, the pseudo-codes of the
strain- and stress-driven constitutive drivers are presented which are employed for the
parameter identification regarding the viscoelastic material parameters.
A.1 Numerical time-integration of the local constitutive
relations
In order to transform the convolution integral expression obtained in Section 2.2.2 to a
standard recurre∫nce formula, we adopt the following internal algorithmic variables,t ( ( ))
H(i)
d
(t) := exp[−(t− s)/τi] fnisod (κ)DEV 2 ∂C̄ψich(C̄(s)) ds , (A.1)
−∞ ds
whereH(i)(t), i = 1, 2, . . . , N , i.e. we have one stress-type tensor-valued internal variable
H(i) for each Maxwell element considered in the generalised relaxation model, cf. Simo
and Hughes [117].
Using the semigroup property exp([t+∆t]/a) = exp(∆t/a) exp(t/a) and the property
of additivity of the integral over the interval of integration we obtain the recurrence
relation
H(i)(tn+1) =∫exp[−∆t /τ ]H(i)n i (tn)tn+1 ( ( )) (A.2)
+ exp[− d(tn+1 − s)/τ ] fnisoi d (κ)DEV 2 ∂C̄ψich(C̄(s)) ds .
t dsn
147
A Numerical time integrations and algorithmic pseudo-code of the finite-strain
viscoelasticity
Finally, integral (A.2) is approximated using the midpoin(t rule, so that(i) n+1 n )
Hn+1 = exp(−∆t/τ (i) nisoi)Hn + fd (κ) exp(−∆t/2 τi) S̃hyp − S̃hyp , (A.3)
using the abbreviations
n+1 n
S̃hyp := DEVn+1(2 ∂C̄ψich(C̄n+1)) and S̃hyp := DEVn(2 ∂C̄ψich(C̄n)) , (A.4)
where DEVn and DEVn+1 are computed with respect to Cn and Cn+1, respectively,
consistent with
1
DEVn+1(•) = [•]− [[•] : Cn+1]C−1
3 n+1
. (A.5)
The algorithmic approximation of the Piola-Kirchhoff stress tensor at time tn+1 then
takes the form
S = J f (κ ) ∂ ψ (J )C−1n+1 n+1 d n+1 J vol n+1 n+1
N
n n+1
∑ − 2
+ γ f iso 3
(i)
∞ d (κn+1) S̃hyp + γi Jn+1 DEVn+1(Hn+1) (A.6)
i=1
n+1
with S̃hyp representing relation (A.4) evaluated at the current time tn+1 and with κn+1 as
the (updated) internal variable representing the evolution of damage within the material.
For details on the straightforward time-integration of κ via a standard Euler backward
scheme, the reader may refer to, e.g., [86, 133]. The spatial counterpart of (A.6), i.e.
the current approximation of the Kirchhoff stress tensor is then computed by using a
standard push-forward operation in terms of
τ tn+1 = F n+1 · Sn+1 · F n+1 . (A.7)
By employing the fact that F ·C−1[ ] n+1F
t
n+1 = I and the relation
(i) t − 2
dev F̄ n+1H 3n+1F̄ n+1 = Jn+1 DEVn+1[2∂
◦
C̄W̄ (C̄n+1)] , (A.8)
the Kirchhoff stress tensor is finally com[ puted ]
◦′ · ◦ · tτ n+1 =∑Jn+1U (Θ[ n+1)I + γ∞dev 2F̄ n]+1 ∂C̄W̄ (C̄n+1) F̄ n+1N (A.9)
(i) t
+ γidev F̄ n+1 ·Hn+1 · F̄ n+1 ,
i=1
which is the spatial counterpart of (A.6).
148
A.2 Algorithmic pseudo-code
In order to evaluate the update formulas (A.3) the kinematic variables Cn+1 and
C̄n+1 are needed which are easily computed within the context of a strain-driven type
of algorithm as follows.
Let Sn = ϕn(B) be the current placement of the body at time tn, defined by the config-
uration ϕn : B → R3, which is assumed to be given. Let ∆tn be the time step size, and
assume that an incremental displacement field, denoted by
∆u : B → R3, (A.10)
is given. Then we compute the updated placement field Sn+1 = ϕn+1(B) simply by
setting
ϕn+1(X) = ϕn(X) + ∆u(X), (A.11)
which defines the confiuration ϕ 3n+1 : B → R at time tn+1 := tn + ∆tn. Then the
deformation gradient and Jacobian determinant are computed as
F n+1 = ∇Xϕn+1(X) and J = det(F n+1) . (A.12)
The right Cauchy-Green tensor and the volume-preserving part of the right Cauchy-
Green tensor follows as
2
Cn+1 = F
t
n+1 · F n+1 and C̄ −n+1 = J 3 Cn+1. (A.13)
A.2 Algorithmic pseudo-code
This section shows the pseudo-codes of the algorithms employed in Section 2.5.2.2 for
the identification of the viscoelastic material parameters. The strain-driven constitutive
driver is shown in Algorithm 1 and the stress-driven constitutive driver in Algorithm 2.
149
A Numerical time integrations and algorithmic pseudo-code of the finite-strain
viscoelasticity
n (i)
1 initialize internal variables S̃hyp, Hn ,∀ i = 1, . . . , N .
2 get material parameters C1, C2, C3, γ1, τ1, . . . , γi, τi.
3 initialize P n+1.
4 for every time step tn do
5 given: the stretch λ11.
6 while ‖P̂ n+1‖ < tol do
7 compute (k)F , (k)J , (k)n+1 n+1 C
(k) (k)
n+1, F̂ n+1, Ĉn+1.
8 compute initial elastic stress Kirchhoff tensor τ hypn+1.
n+1 (i)
9 compute algorithmic internal variables S̃hyp , Hn+1.
10 compute (k)Sn+1 according to (A.6).
11 compute Piola stress tensor (k)P n+1.
12 compute tangent operator (k)An+1.
13 partition stress tensor and the deformation gradient:
14 (k)P = (k)n+1 P
(k)
11 e11 ⊗ e11 + P̂ n+1
15 (k)F (k) (k)n+1 = F11 e11 ⊗ e11 + F̂ n+1.
16 partition the tangent operator:
(k)
17 (k)
dP̂ n+1
Ân+1 = .
(k)dF̂ n+1
18 update the transverse deformation gradient:
19 (k+1)F̂ ← (k)F̂ (k) (k)n+1 n+1 − Ân+1 : P̂ n+1.
20 end while
n+1 (i) n (i)
21 update internal variables {S̃hyp , Hn+1} ← {S̃hyp, Hn }.
22 end for
Algorithm 1: Strain-driven constitutive driver (relaxation)—uni-axial stress state.
150
A.2 Algorithmic pseudo-code
n (i)
1 initialize internal variables S̃hyp, Hn ,∀ i = 1, . . . , N .
2 get material parameters C1, C2, C3, γ1, τ1, . . . , γi, τi.
3 initialize F n+1.
4 for every time step tn do
5 given: Piola stress tensor P exp.
6 while ‖P − P exp‖ < tol do
7 compute (k)F , (k)J , (k)C , (k)F̂ , (k)n+1 n+1 n+1 n+1 Ĉn+1.
8 compute initial elastic Kirchhoff stress tensor τ hypn+1.
n+1 (i)
9 compute algorithmic internal variables S̃hyp ,Hn+1.
10 compute (k)Sn+1 according to (A.6).
11 compute Piola stress tensor (k)P n+1.
12 compute tangent operator (k)An+1.
13 compute (k)∆P n+1 =
(k)P (k)n+1 − P expn+1.
14 update the deformation gradient
15 (k+1)F ← (k)F (k) (k)n+1 n+1 − An+1 : ∆P n+1
16 end while
n+1 (i) n (i)
17 update internal variables {S̃hyp , Hn+1} ← {S̃hyp, Hn }
18 end for
Algorithm 2: Stress-driven constitutive driver (creep)—uni-axial stress state.
151

B Derivation of the smoothed
Fischer-Burmeister nonlinear
complementarity problem function
The roots of the original NCP function need to be solved in an approximative manner
for the smoothed Fischer-Burmeister NCP approach. Thus, the perturbation parameter
δ controls the accuracy and the equality constraint to be solved. The corresponding
derivation of the smoothed Fischer-Burmeister function provided that r ≥ 0 and Γ ≥ 0
reads as follows
r Γ = δ2 (B.1)
2 r Γ = 2 δ2 (B.2)
r2 + Γ 2 + 2 r Γ = r2 + Γ 2 + 2 δ2 (B.3)
[r + Γ ]2 = r2 + Γ 2 + 2 δ2 (B.4)
√
r + Γ = r2 + Γ 2 + 2 δ2 (B.5)
√
0 = r2 + Γ 2 + 2 δ2 − [r + Γ ] (B.6)
153

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