Lehrstuhl Statik und Dynamik

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A deterministic model combining NDT to estimate permissible bending loads on trees
(2023-09-25) Muench, Ingo; Loske, Simon
The stability of trees is the subject of numerous studies, as their loss poses a threat to life and limb. Bending loads are responsible for the two most common failure modes: uprooting and stem break. Uprooting is strongly related to soil conditions, water saturation and the root system itself. These parameters are difficult to measure in situ. Since both failure modes result in loss of vitality, arguments from evolutionary theory suggest that roots and stems should have similar load limits under normal conditions. Therefore, we propose a deterministic model to estimate the allowable bending load in the stem of European beech (Fagus sylvatica). The model assumes a non-linear stress distribution in the cross section. Destructive bending tests show that damage progresses in three stages. Furthermore, the local compressive and tensile strength in fibre direction are crucial parameters to determine the ultimate load. Since the tensile strength cannot be measured by NDT, the experimental data from the large-scale tests are used to relate this parameter to the compressive strength from the NDT data. This provides a method for determining the risk of a tree, for example in urban areas. It is also useful for estimating the allowable limit of engineering loads on trees.
Growth of green wood based on a phase field model
(2023-03-24) Wulf, Jan Bernd; Muench, Ingo
Tree engineering is a young discipline utilizing trees as structural elements, where the determination of limit loads in tree trunks is of great importance. Simple numerical models underestimate the load-bearing capacity of green wood in contrast to experimental bending tests. A well-known reason for this is the residual stress state of the living tree lowering compressive stress towards the trunks surface. This results in an overall stress state, which increases the load capacity, since the tensile strength of wood is commonly higher than its compressive strength. By determining the residual growth stress, a more accurate evaluation of the load-bearing capacity of a living tree is possible. The residual stress state is a non-linear and time dependent function in thickness direction of the trunk. In order to simulate growth and growth stress, a phase field model is employed. The morphology of a tree is the result of innumerable and often temporary environmental stimuli, which also change and interact with the genetically predisposed growth tropisms. Therefore, we use image processing to capture the individual tree morphology of an existing tree, which is based within the phase field model as predefined growth direction. This is the basis for primary growth in the model. Additionally the model simulates the secondary growth, which corresponds to the thickness of the trunk. Except in tropical areas, this growth is associated with growth rings, which we assign as an attribute to the modelled material. While in the branch structure several tropisms (e.g. gravitropism) are responsible for the off-centre accumulation of woody material, in the stem region we only follow the stress-induced growth. This mechanism can respond to either the principal tensile stress or the principal compressive stress in our model, as this difference is observed in hardwoods and softwoods. Since the wood matrix represents an anisotropic material with a distinct fiber direction, we approach it in our model by a transversely isotropic constitutive law, whose principal direction coincides with the growth direction.
A quadratic finite element for the relaxed micromorphic model
(2023-05-31) Sky, Adam; Muench, Ingo; Neff, Patrizio
In this work we discuss the relaxed micromorphic model and implementation details for a full three-dimensional formulation entailing a quadratic Lagrangian-Nédélec finite element and appropriate boundary conditions in the discrete setting. The relaxed micromorphic model is a generalized continuum theory with the capacity to capture more complex kinematical behaviour than in the classical Cauchy continua. Such behaviour is commonly found in materials with a pronounced micro-structure such as porous media and metamaterials. The theory introduces the microdistortion field, encompassing nine additional degrees of freedom for each material point in the continuum, effectively turning each material point into a deformable micro-body. The uncommon discrete formulation stems from the employment of the Curl operator in the energy functional of the relaxed micromorphic model, thus requiring H(curl)-conforming finite elements for well-posedness to be inherited in the discrete setting. The model further introduces the so called consistent coupling condition, which requires some technical considerations in order to be upheld correctly. This work demonstrates the finite element formulation, culminating with a numerical example.
On [H1]3×3 , [H(curl)]3 and H(sym Curl) finite elements for matrix-valued Curl problems
(2022-09-09) Sky, Adam; Muench, Ingo; Neff, Patrizio
In this work we test the numerical behaviour of matrix-valued fields approximated by finite element subspaces of [H1]3×3, [H(curl)]3 and H(symCurl) for a linear abstract variational problem connected to the relaxed micromorphic model. The formulation of the corresponding finite elements is introduced, followed by numerical benchmarks and our conclusions. The relaxed micromorphic continuum model reduces the continuity assumptions of the classical micromorphic model by replacing the full gradient of the microdistortion in the free energy functional with the Curl. This results in a larger solution space for the microdistortion, namely [H(curl)]3 in place of the classical [H1]3×3. The continuity conditions on the microdistortion can be further weakened by taking only the symmetric part of the Curl. As shown in recent works, the new appropriate space for the microdistortion is then H(symCurl). The newly introduced space gives rise to a new differential complex for the relaxed micromorphic continuum theory.
Higher order finite elements for relaxed micromorphic continua
(2022) Sky, Adam; Münch, Ingo; Neff, Patrizio
Structural optimisation of diffusion driven degradation processes
(2021-05-23) Waschinsky, Navina; Barthold, Franz-Joseph; Menzel, Andreas
In this article, we propose an optimisation framework that can contribute to the prevention of premature failure or damage to building structures and can thereby strengthen their longevity. We concentrate on structures that are contaminated by chemical substances and that have strong destructive effects on the material. The aim of this contribution is a mathematical algorithm that allows the optimisation of a structure exposed to chemical influences and thus the assurance of the static load capacity. To achieve this, we present a coupled mechanical-diffusion-degradation approach embedded in a finite element (FE) framework. Furthermore, we integrate an optimisation algorithm to reduce material degradation. In this paper, we establish shape optimisation of a structure with a gradient based optimisation algorithm.
A hybrid H1×H(curl) finite element formulation for a relaxed micromorphic continuum model of antiplane shear
(2021-05-17) Sky, Adam; Neunteufel, Michael; Münch, Ingo; Schöberl, Joachim; Neff, Patrizio
One approach for the simulation of metamaterials is to extend an associated continuum theory concerning its kinematic equations, and the relaxed micromorphic continuum represents such a model. It incorporates the Curl of the nonsymmetric microdistortion in the free energy function. This suggests the existence of solutions not belonging to H1, such that standard nodal H1-finite elements yield unsatisfactory convergence rates and might be incapable of finding the exact solution. Our approach is to use base functions stemming from both Hilbert spaces H1 and H(curl), demonstrating the central role of such combinations for this class of problems. For simplicity, a reduced two-dimensional relaxed micromorphic continuum describing antiplane shear is introduced, preserving the main computational traits of the three-dimensional version. This model is then used for the formulation and a multi step investigation of a viable finite element solution, encompassing examinations of existence and uniqueness of both standard and mixed formulations and their respective convergence rates.
Structural optimisation of diffusion-driven degradation processes
(2021) Waschinsky, Navina; Barthold, Franz-Joseph; Menzel, Andreas
In the field of structural engineering, structures are developed and calculated. The stresses and deformations resulting from mechanical loads are determined, and the structures are dimensioned to ensure load-bearing capacity, usability and durability in accordance with standards. The application of structural optimisation algorithms enables the development of more efficient and economical building structures, whereby maximum permissible stresses can be exhausted. However, standardised calculations take environmental influences, such as chemical impact, only via so-called exposure classes and resulting material properties into account. Detailed calculations on the influence of stresses and deformations of the structures, especially due to the long-term chemical influence and resulting material degradation, are often neglected. For example, specific stress constraints may be exceeded. Within the scope of the present work, a numerical programme is developed, enabling an efficient optimisation of mechanical structures that are additionally burdened by degradation processes due to diffusive concentrations. For this purpose, a mechanicalchemical- degradation coupled model is developed. Within the framework of classical structural mechanics, the developed material behaviour is presented, taking into account modified physical principles of continuum mechanics to describe a mechanical-chemicaldegradation coupled processes. With the help of the fundamentals of the Finite Element Method (FEM), the solution of the non-linear problem is outlined in detail. Furthermore, the developed structural analysis is embedded in a mathematical algorithm of gradient-based structural optimisation. The optimisation allows a deeper analysis and reduction of the harmful effects due to the influence of acting chemical concentrations. A variational approach to structural optimisation provides the simultaneous integration of analytically prepared sensitivity analysis with the structural analysis for embedding the continuum mechanical formulations. Thus, efficient structural optimisation of the introduced mechanical-chemical-degradation model is comprehensively presented. The mathematical model with the required derivations as well as discretisation is documented and implemented in a computer-based model.
Computational shape optimisation for a gradient-enhanced continuum damage model
(2020-01-28) Guhr, Fabian; Sprave, Leon; Barthold, Franz-Joseph; Menzel, Andreas
An isotropic gradient-enhanced damage model is applied to shape optimisation in order to establish a computational optimal design framework in view of optimal damage distributions. The model is derived from a free Helmholtz energy density enriched by the damage gradient contribution. The Karush–Kuhn–Tucker conditions are solved on a global finite element level by means of a Fischer–Burmeister function. This approach eliminates the necessity of introducing a local variable, leaving only the global set of equations to be iteratively solved. The necessary steps for the numerical implementation in the sense of the finite element method are established. The underlying theory as well as the algorithmic treatment of shape optimisation are derived in the context of a variational framework. Based on a particular finite deformation constitutive model, representative numerical examples are discussed with a focus on and application to damage optimised designs.
Variational sensitivity analysis of elastoplastic structures applied to optimal shape of specimens
(2020-05-19) Liedmann, Jan; Barthold, Franz-Joseph
The aim of this paper is to improve the shape of specimens for biaxial experiments with respect to optimal stress states, characterized by the stress triaxiality. Gradient-based optimization strategies are used to achieve this goal. Thus, it is crucial to know how the stress state changes if the geometric shape of the specimen is varied. The design sensitivity analysis (DSA) of the stress triaxiality is performed using a variational approach based on an enhanced kinematic concept that offers a rigorous separation of structural and physical quantities. In the present case of elastoplastic material behavior, the deformation history has to be taken into account for the structural analysis as well as for the determination of response sensitivities. The presented method is flexible in terms of the choice of design variables. In a first step, the approach is used to identify material parameters. Thus, material parameters are chosen as design variables. Subsequently, the design parameters are chosen as geometric quantities so as to optimize the specimen shape with the aim to obtain a preferably homogeneous stress triaxiality distribution in the relevant cross section of the specimen.
Numerical approach for a continuum theory with higher stress gradients
(2021-01-25) Ghasemi, Seyed Ali; Münch, Ingo; Liedmann, Jan; Barthold, Franz-Joseph
We use an extended balance of linear momentum derived from stress field analysis of higher order terms in power series expansion. Thus, the balance equation accounts for higher gradients of stress in the contiguity of continuum points. Interestingly, it does not coincide with the balance of linear momentum from strain gradient elasticity. As shown in [1], it exhibits an inverse sign for the extended term compared to strain gradient elasticity. We are interested in the mechanical interpretation of this inversed sign since it seems to inverse the stiffening effect of strain gradient elasticity. Therefore, we set up the weak form of our extended balance equation by means of Galerkin's approach. Then, we use the Finite Element Method to approximate the weak form with help of different shape functions. In this context we also use Isogeometric Analysis since it is very promising for a numerical model with higher gradients.
Topology evolution of composite structures based on a phase field model
(2021-01-25) Wulf, Jan Bernd; Münch, Ingo
he composition of fibers and matrix is of great importance in several fields of engineering, such as steel reinforcement in concrete for civil engineering or lightweight applications in the automotive and aviation industry, as it allows combining the advantages of both materials. If the bond between fibers and matrix is ideally strong enough, the mechanical deformation can be assumed to be equal in both materials. With this assumption we set up a phase field model evolving the topology of reinforcement. The phase field parameter represents regions of reinforcement in the sense of averaged increased stiffness since we do not intend to simulate single fibers. A similar model but for topology optimization based on equivalent stresses was introduced by Muench et al. [1]. In many matrix materials, viscoelastic behavior is observed. Therefore, we also consider viscoelasticity in our model for the matrix.
Two-scale shape optimisation based on numerical homogenisation techniques and variational sensitivity analysis
(2021-03-06) Kijanski, Wojciech; Barthold, Franz-Joseph
This contribution presents a theoretical and computational framework for two-scale shape optimisation of nonlinear elastic structures. Particularly, minimum compliance optimisation problems with composite (matrix-inclusion) microstructures subjected to static loads and volume-type design constraints are focused. A homogenisation-based FE2 scheme is extended by an enhanced formulation of variational (shape) sensitivity analysis based on Noll’s intrinsic, frame-free formulation of continuum mechanics. The obtained overall two-scale sensitivity information couples shape variations across micro- and macroscopic scales. A numerical example demonstrates the capabilities of the proposed variational sensitivity analysis and the (shape) optimisation framework. The investigations involve a mesh morphing scheme for the design parametrisation at both macro- and microscopic scales.
Shape optimization of the X0-specimen: theory, numerical simulation and experimental verification
(2020-09-20) Liedmann, Jan; Gerke, Steffen; Barthold, Franz-Joseph; Brünig, Michael
The paper deals with the gradient based shape optimization of the biaxial X0-specimen, which has been introduced and examined in various papers, under producibility restrictions and the related experimental verification. The original, engineering based design of the X0-specimen has been applied successfully to different loading conditions persisting the question if relevant stress states could be reached by optimizing the geometry. Specimens with the initial as well as with the two load case dependent optimized geometries have been fabricated of the aluminum alloy sheets (AlSi1MgMn; EN AW 6082-T6) and tested. The strain fields in critical regions of the specimens have been recorded by digital image correlation technique. In addition, scanning electron microscope analysis of the fracture surfaces clearly indicate the stress state dependent damage processes. Consequently, the presented gradient based optimization technique facilitated significant improvements to study the damage and fracture processes in a more purposeful way.
Nichtlokale Modellierung des Versagens- und Schädigungsverhaltens von elastisch-plastischen Materialien
(Universität Dortmund, 2004-08-10) Ricci, Sabine; Obrecht, Hans; Svendson, Bob
Die vorliegende Arbeit handelt von der numerischen Untersuchung großer elastisch-plastischer Deformationen und des Lokalisierungsverhaltens anisotrop geschädigter duktiler Materialien, die eine hydrostatische Druckabhängigkeit aufweisen.Das vorgestellte Modell basiert auf einer verallgemeinerten makroskopischen Theorie im Rahmen einer nichtlinearen Kontinuumsschädigungsmechanik unter Verwendung einer kinematischen Beschreibung des Schädigungsverhaltens.Um eine physikalisch korrekte Formulierung zu gewährleisten, wird die Nichtlokalität des Materialverhaltens sowohl in der Fließbedingung als auch in der Schädigungsbedingung berücksichtigt.Abschätzungen der Spannungs- und Verzerrungsentwicklungen werden aus einer direkten numerischen Integration der plastischen und schädigungsbezogenen Ratenbeziehungen ermittelt. Dieser zweiteilige Integrationsalgorithmus besteht aus einem inelastischen Prediktorschritt, dem ein elastischer Korrektorschritt folgt.Numerische Simulationen des elastisch-plastischen Deformationsverhaltens geschädigter Körper unterstreichen die Effizienz des vorgestellten Modells und verdeutlichen die Wirkung des Schädigungseinflusses auf das Lokalisierungs- und Deformationsverhalten.
Numerische Modellierung des druck- und ratenabhängigen elastisch-plastischen Verhaltens von metallischen Werkstoffen
(Universität Dortmund, 2004-07-20) Berger-Bickendorf, Simone; Obrecht, H.; Ungermann, D.
In der vorliegenden Arbeit wird ein nichtlineares Finite-Element-Modell zur Analyse des ratenunabhängigen und ratenabhängigen inelastischen Deformations- und Lokalisierungsverhaltens von Metallen entwickelt, die vom hydrostatischen Spannungszustand abhängig sind,ein unterschiedliches Verhalten im Zug- und Druckbereich sowie eine inelastische Volumendilatationaufweisen.Die kontinuumsmechanische Beschreibung der großen elastisch-plastischen und großenelastisch-viskoplastischen Deformationen basiert auf der multiplikativen Aufspaltung desMetrik-Transformationstensors.Einen Schwerpunkt dieser Arbeit bildet die Formulierung ratenunabhängiger und ratenabhängigerinelastischer Materialmodelle, mit denen das durch die genannten Phänomene gekennzeichnetebei Experimenten beobachtete inelastische Materialverhalten metallischer Werkstoffenumerisch simuliert werden kann. Dafür wird abweichend von der klassischenMetallplastizität eine von 1 I , 2 J und 3 J abhängige Fließbedingung bzw. Fließfunktion sowieein nicht-assoziiertes Fließgesetz verwendet.Die numerische Integration der konstitutiven Gleichungen erfolgt mittels eines explizitenAlgorithmus, der aus einem inelastischen Prädiktor und elastischen Korrektor besteht.Anhand ausführlicher numerischer Studien wird die Auswirkung des Materialmodells auf daselastisch-plastische und elastisch-viskoplastische Deformations- und Lokalisierungsverhaltenuntersucht. Dafür wird der Einfluss unterschiedlicher Materialparameter analysiert.Die Ergebnisse der numerischen Untersuchungen verdeutlichen, dass die Berücksichtigungder Abhängigkeit vom hydrostatischen Spannungszustand und der Ratenabhängigkeit desinelastischen Materialverhaltens einen signifikanten Einfluss auf das Deformations- undLokalisierungsverhalten haben kann. Das entwickelte Materialmodell für Metalle trägt somitzu einer deutlichen Verbesserung der Simulationsergebnisse bei, da das reale makroskopischeVerhalten genauer als bisher beschrieben werden kann.
Modellierung des reibungsbehafteten Rollkontakts elasto-plastischer metallischer Festkörper
(Universität Dortmund, 2000-05-24) Rauscher, Thomas
In der vorliegenden Arbeit wird ein stationärer Walzprozess eines elastisch-plastischen Körpers mit Materialverfestigung als unilateraler Rollkontakt unter Berücksichtigung trockener Festkörperreibung formuliert. Die mathematische Beschreibung des stationären Rollkontakts führt auf die Minimierung eines nichtlinearen Variationsfunktionals. Im Rahmen der Active-Set-Strategie wird ein zweidimensionaler Rollkontakt-Algorithmus entwickelt, in den ein Coulomb'sches Gleitreibungsmodell integriert ist. Unter Anwendung einer Penalty-Regularisierung der geometrischen Rollkontaktbedingungen wird das Problem mit Hilfe der Finite-Elemente-Methode näherungsweise gelöst. Die unbekannten Verteilungen von Normal- und Tangentialspannung in der Kontaktoberfläche werden aus der Transformation der Kontaktknotenkräfte mit Hilfe einer diagonalisierten Kontaktflächenmatrix berechnet. Anhand numerischer Simulationen wird das Verformungsverhalten des Walzguts sowie Kontaktspannungsverteilungen mit dem neu entwickelten Rollkontaktalgorithmus untersucht. Um den Einfluss der Reibung zu zeigen, sind auch die Haft- und Gleitzonen in der Kontaktfläche und die oberflächennahen Spannungsverteilungen im Walzgut dargestellt.

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