Received: 14 July 2020 Accepted: 10 November 2020
DOI: 10.1002/pamm.202000203
Shape optimization of the X0-specimen for biaxial experiments
Jan Liedmann1,∗, Steffen Gerke2, Franz-Joseph Barthold1, and Michael Brünig2
1 Lehrstuhl Baumechanik, Technische Universität Dortmund, August-Schmidt-Str. 8, 44227 Dortmund, Germany
2 Institut für Mechanik und Statik, Universität der Bundeswehr München, Werner-Heisenberg-Weg 39, 85579 Neubiberg
The mechanical damage and fracture behavior of ductile sheet metals strongly depend on the stress state and intensity. Thus,
for adequate characterization of the material behavior, it is crucial to have specimens that cover different and preferably
distinct stress states, especially in the inelastic domain. In this paper, the geometry of the X0-specimen is optimized to achieve
a distinct stress triaxiality distribution in the region of damage and fracture occurrence, depending on two different load cases
in a biaxial testing environment. The shape optimization is gradient based and the gradients of the objective and constraint
functions are computed analytically by means of variational principles. The resulting geometries show improvements in terms
of the intensity of the stress state numerically as well as experimentally.
© 2021 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH GmbH
1 Introduction
Ductile sheet metals play an important role in several engineering disciplines. The demand of improvements in terms of
weight, cost and energy consumption is growing. Appropriate characterization of the material behavior - especially during
inelastic deformations - is essential. This knowledge helps to prevent localization of irreversible strains in an early state of
deformation, which leads to material degradation. These effects strongly depend on the stress state and intensity. A high
tensile dominated state triggers nucleation and growth of micro-voids, whereas micro-cracks occur in a shear dominated stress
state. The stress triaxiality η and Lode parameter ω, defined by
σm I1 2 τ2 − τ1 − τ3
η = = √ and ω = − , with τ1 ≥ τ2 ≥ τ3, (1)σeq 3 3 J2 τ1 τ3
characterize the stress state. Here, I1 and J2 denote the first and second deviatoric invariant of the Kirchhoff stress tensor and
τi are the principle Kirchhoff stress components. New biaxial test specimens have been developed in [2] that cover a wide
range of stress triaxialities depending on the load case. Shape optimization of the presented specimens promises even more
homogeneous and distinct stress states and can be conducted efficiently within a finite element framework based on gradients
gained by a variational approach, see e.g. [1, 4].
2 Optimization tasks
The goal of the shape optimization is to improve the stress triaxiality intensity and homogeneity in the notched specimen area.
Two different load cases are considered, i.e. (ū1=1 / ū2 = 1) producing high stress triaxialities and (ū1=-1 / ū2 = 1) producing
low stress triaxialities. With the geometric design vector s, the optimization task for both load cases can be written as
min J(u(s), s), subject to cin ≤ 0, ceq = 0,
∈Rn (2)s
where J denotes the objective function and ceq and cin denote the vectors of equality and inequality constraints, respectively.
Depending on the load case, the objective function is chosen as
J(u(s), s) = ±‖ηcs‖, with a negative sign for (1/1) and a positive sign for (-1/1), (3)
where the vector ηcs collects the values of stress triaxiality in the cross section of the notched specimen area, obtained by the
solution of the mechanical analysis problem. One equality constraint is chosen to keep the cross section area constant and
three inequality constraints prevent destruction of the finite element mesh, viz.
 
Rt −Ri
ceq = Acs − 12 mm2 = 0, cin = R −R t o ≤ 0. (4)
D −Rt
The mechanical analysis problem is given by the elastoplastic initial boundary value problem based on a multiplicative kine-
matic assuming a plastic stress free configuration given by F −1p = Fe F, see [5], and is solved using the finite element method.
∗ Corresponding author: e-mail jan.liedmann@tu-dortmund.de, phone +49 231 755 7244, fax +49 231 755 7244, ORCID iD 0000-0001-8759-6940
This is an open access article under the terms of the Creative Commons Attribution License, which permits use,
distribution and reproduction in any medium, provided the original work is properly cited.
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2 of 3 Section 16: Optimization
ū2 prescr. displ.
symm. boundary cond.
R Rt
ū i1 Ro
D
Fig. 2: Design values. (a) Initial geometry, (b) Optimized for load case (1/1), (c)
Fig. 1: Initial finite element mesh Optimized for load case (-1/1)
For the gradient based solution of the inverse problem given in Sec. 2, the gradient of the objective function is computed based
on the variational approach explained in e.g. [1, 4]. The total variation of the stress triaxiality reads
( )
1 1 − 3 η ∂τ ∂τ ∂τδη = I τ dev : δτ , with δτ = δu + δs + δh , (5)
σeq
n
3 2σeq ∂u ∂s ∂hn
where the total variation of the Kirchhoff stress tensor δτ is obtained considering the fact that the weak equilibrium condition
(R = 0) has to hold for any geometric design change. The discrete version of the total variation of the weak equilibrium reads
!
δR = K δu + P δs + H δhn = 0 and therefore, δu = −K−1 (P δs + H δhn) = S δs, (6)
which yields the sensitivity matrix S connecting the structural response with geometric design changes, cf. [4]. The values of
variations of history variables δh have to be saved and updated in each load increment within the finite element procedure.
3 Numerical investigations
The finite element mesh consisting of 5376 F̄ -elements is shown in Fig. 1. The design variables stored in the design vector
s are three radii and the penetration depth of the notch in thickness direction. In Fig. 2 the values of the four geometric
design variables are summarized for the initial and the optimized geometries depending on the specific load case. The stress
triaxiality distributions for the (1/1) and (-1/1) load case are illustrated in Fig. 3 and Fig. 4, respectively. For the (1/1) load
case, an improvement can be clearly observed, while for the (-1/1) load case, the stress triaxality distribution is very similar.
(a) (b) (c) (a) (b) (c)
Fig. 3: Stress triaxiality distribution for load case (1/1) Fig. 4: Stress triaxiality distribution for load case (-1/1)
4 Concluding remarks
For verification of the results obtained by the presented shape optimization procedure, the resulting optimized X0 geometries
have been fabricated and compared to the initial X0 geometry. The biaxial test series has been captured by digital image
correlation (DIC) and the fracture surfaces have been examined by scanning electron microscopy (SEM). It can be seen that
the fracture surface of the optimized geometry for the (1/1) load case exhibits significantly larger voids due to the higher stress
triaxiality. More details on the experimental investigations can be found in [3].
Acknowledgements Open access funding enabled and organized by Projekt DEAL.
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PAMM · Proc. Appl. Math. Mech. 20:1 (2020) 3 of 3
References
[1] F.-J. Barthold, Braunschweiger Schriften zur Mechanik 44, (2002).
[2] S. Gerke, P. Adulyasak and M. Brünig, International Journal of Solids and Structures 110, 209-218 (2017).
[3] S. Gerke et al., Damage and fracture of the optimized X0-specimen, PAMM 20, (2020).
[4] J. Liedmann and F.-J. Barthold, Structural and Multidisciplinary Optimization 61(6), 2237-2251 (2020).
[5] J. Simo, T. J. R. Hughes, Computational Inelasticity, (2000).
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