to those who believed in me

To those who believed in me

i

Zusammenfassung

In dieser Dissertation wird eine Mehrskalenmethode fur die Vorhersage von Großenef-fekten im inelastischen Bereich von Materialien mit einer charakteristischen innerenLange vorgestellt. Wenn die Große des betrachteten Bauteils mit der inneren Langevergleichbar ist, konnen diese Großeneffekte wegen der Sensitivitat bezuglich derBauteilgroße einen bedeutenden Einfluss auf die Materialantwort haben.Fur die Berechnung von großenabhangigen Materialien haben sich generalisierteKontinua bewahrt, da sie bereits eine innere Lange als Materialparameter beinhalten.Unter verschiedenen Formulierungen fur generalisierte Modelle garantiert der mikro-morphe Ansatz ein hohes Maß an Flexibilitat bezuglich der konstitutiven Annahmen.Die klassische mikromorphe Formulierung wird fur die Beschreibung von inelastischenPhanomenen wie Elastoplastizitat und duktiler Schadigung bei finiten Deformationenangepasst.Eine Mehrskalenmethode, in die die mikromorphen Modelle eingebettet sind, umdie Mesostruktur zu beschreiben, wird entwickelt. So kann der numerische Aufwandvon einskaligen Simulationen von mesoskopisch heterogenen Materialien vermiedenwerden. Die Mesostruktur wird durch Reprasentative Volumenelemente beschrieben,die als eigenstandige Randwertprobleme auf der Subskala innerhalb einer zweiskaligenFormulierung behandelt werden. Die Effizienz der resultierenden verschachteltenProzedur wird durch eine konsistente Linearisierung der mesoskopischen Mate-rialantwort gewahrleistet. Diese Linearisierung basiert auf Sensitivitatsanalysender Spannungsantwort bezuglich der aufgebrachten Randbedingungen, die aus dermakroskopischen Deformation berechnet werden.Da heterogene Mesostrukturen auch in Materialzwischenschichten zu finden sind,wird die numerische Homogenisierungsprozedur modifiziert. Ein makroskopischesInterface-Element, das speziell fur Deformationsmoden, die hauptsachlich in dunnenSchichten vorkommen, geeignet ist, wird entwickelt. Dies erlaubt die mehrskaligeBerechnung solcher Schichten, die auch die auftretenden Großeneffekte abbilden kann.Die Software-Werkzeuge AceGen und AceFEM werden benutzt, um effiziente Pro-grammcodes zu erzeugen und zu verwenden, da sie den symbolischen Ansatz derMathematica-Umgebung ausnutzen. Die auf automatischer Differentiation beruhendeFormulierung ist speziell fur die Sensitivitatsanalyse, die fur die konsistente Lin-earisierung benutzt wird, nutzlich.Die Leistungsfahigkeit der entwickelten Algorithmen wird anhand von numerischenBeispielen sowohl fur Ein- als auch fur Mehrskalenprobleme untersucht.

Schlagworte: Generalisierte Kontinua, Großeneffekte, Duktile Schadigung,Mehrskalenmethoden, FE2, Materialzwischenschichten

iii

Abstract

In this dissertation, a multiscale method to predict size-effects in the inelastic regimeof materials with an inherent characteristic length is presented. If the actual size of abody under investigation is comparable to the characteristic length, these size-effectscan influence the material response significantly due to the sensitivity with respect tothe size of the body.Generalized continua have proven useful for the computation of size-dependentmaterials as they naturally incorporate an intrinsic length as a material parameter.Among various formulations of generalized models, the micromorphic approachguarantees a high level of flexibility considering constitutive assumptions. Theclassical micromorphic formulation was adapted to describe inelastic phenomena suchas elasto-plasticity and ductile damage for finite deformations.A multiscale framework, in which the derived micromorphic-type models are embeddedfor the description of the mesostructure, is derived to circumvent computationallycostly single-scale simulations of mesoscopically heterogeneous materials. The me-sostructure is described by Representative Volume Elements which are treated asindependent subscale boundary value problems within a two-scale homogenizationframework. The efficiency of the resulting nested computational solution procedure isguaranteed by a consistent linearization of the mesoscopic response. This linearizationis based on sensitivity analyses of the stress response with respect to the imposedboundary conditions derived from the macroscopic deformation.Since heterogeneous mesostructures cannot only be found in bulk materials but alsoin thin material layers, the computational homogenization scheme is modified. Amacroscopic interface element especially suited for the deformation modes whichmainly occur in thin layers is developed. This enables multiscale computations of suchlayers also capturing the occurring size-effects.The software tools AceGen and AceFEM are used to generate and to use efficientcodes as the symbolic approach of the Mathematica software package is exploited. Theautomatic-differentiation-based formulation is especially beneficial for the sensitivityanalysis used for the consistent linearization.The computational performance of the developed algorithms is investigated in single-scale problems as well as in multiscale problems.

Keywords: Generalized Continua, Size-effects, Ductile Damage, Multiscale Methods,FE2, Material Interfaces

v

Acknowledgements

This thesis is the result of my research work during the past four years at the In-stitute of Continuum Mechanics (IKM) at the Gottfried Wilhelm Leibniz UniversitatHannover. The project was funded by the Gottfried Wilhelm Leibniz Universitat Han-nover through WIF II and by the German Research Foundation (Deutsche Forschungs-gemeinschaft DFG).My sincere thanks go to my advisor and principal referee Professor Peter Wriggers forhis support and encouragement during my time at his institute. He was always ableto provide sound advice whenever I needed it. Moreover, I would like to thank him forthe possibility of participating in many conferences and summer schools in Germanyand abroad, which were beneficial for my academic and personal development.I also want to thank my former supervisor Britta Hirschberger for giving me the chanceto broaden my knowledge in generalized continua and for guiding me through the firstperiod of my work.I also thank my second referee Professor Marc Geers and the chairman of the commit-tee Professor Hans Jurgen Maier.Without the knowledge and support of Professor Joze Korelc considering my countlessquestions about AceGen and AceFEM this work would never have been completed. Ireally enjoyed my stay in Ljubljana.Furthermore, I would like to thank my colleagues at the institute, especially DanielGottschalk, with whom I shared an office, for an enjoyable and productive atmosphereand for reminding me from time to time that engineering does not only consist of the-ory. Thanks also go to Daniel Nolte, with whom I experienced my first conference andwho shares my passion for football, Behrooz Safiei for the time during our Diplomatheses, Stefan Lohnert and Dana Muller-Hoeppe for their lectures in continuum me-chanics which brought me to the IKM, and Sebastian Zeller and Eva Lehmann for thediscussions on plasticity.I am grateful for the support of my family, in particular my mother Brigitte, my brotherAlexander, and my grandparents, without whom my life during my studies would havebeen a lot harder.Finally, I would like to thank Kathrin Menke for her love, her patience, and her abilityto cheer me up when things did not turn out the way I intended.

Hanover, May 2015 Heiko Clasen

Contents

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Computational solution of mechanical problems . . . . . . . . . 2

1.2.2 Fracture mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.3 Continuum damage formulations . . . . . . . . . . . . . . . . . 5

1.2.4 Macroscopic failure in multiscale frameworks . . . . . . . . . . . 7

1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Continuum mechanics 9

2.1 Standard continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.2 Balance equations . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.3 Constitutive models . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2 Generalized Continua . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.2.1 Higher Gradient Continua . . . . . . . . . . . . . . . . . . . . . 26

2.2.2 Micropolar continua . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.3 Micromorphic continua . . . . . . . . . . . . . . . . . . . . . . . 27

2.2.4 Micromorphic-type inelastic continua . . . . . . . . . . . . . . . 27

3 Finite Element Method 33

3.1 Variational principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 Principle of virtual work . . . . . . . . . . . . . . . . . . . . . . 34

3.1.2 Hu-Washizu principle for incompressibility . . . . . . . . . . . . 34

3.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Discretization in space . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.2 Discretization in time . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3 Numerical integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4 Symbolic-numeric solution scheme . . . . . . . . . . . . . . . . . . . . . 41

4 Single-scale benchmark tests 47

4.1 Prandtl-indentation test . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4.2 Strip with a hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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viii

5 Computational homogenization 57

5.1 Bulk homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.1.1 Meso-to-macro transition . . . . . . . . . . . . . . . . . . . . . . 605.1.2 Symbolic derivation of the macroscopic tangent . . . . . . . . . 655.1.3 Thermodynamic consistency . . . . . . . . . . . . . . . . . . . . 67

5.2 Interface homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . 695.2.1 Finite Element discretization of interface layers . . . . . . . . . 705.2.2 Meso-to-macro transition . . . . . . . . . . . . . . . . . . . . . . 75

6 Multiscale benchmark tests 79

6.1 Bulk homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 796.1.1 Size-effect by variation of the microstiffness . . . . . . . . . . . 796.1.2 Size-effect by variation of the RVE-size . . . . . . . . . . . . . . 836.1.3 Macroscopic convergence . . . . . . . . . . . . . . . . . . . . . . 85

6.2 Interface homogenization . . . . . . . . . . . . . . . . . . . . . . . . . . 856.2.1 Size-effect by variation of the microstiffness . . . . . . . . . . . 866.2.2 Size-effect by variation of the RVE-size . . . . . . . . . . . . . . 906.2.3 Macroscopic convergence . . . . . . . . . . . . . . . . . . . . . . 926.2.4 Investigation of a curved layer . . . . . . . . . . . . . . . . . . . 93

7 Concluding remarks 99

List of Figures 103

List of Tables 107

Bibliography 109

Curriculum vitae 121

Chapter 1

Introduction

1.1 Motivation

The correct prediction of failure is crucial for the estimation of the life-time of me-chanical components. A common definition of failure used in engineering is the excessof the maximum load bearing capacity of a structure. It is the responsibility of theengineer to decide what the maximum load bearing capacity of a mechanical compo-nent is. In materials that may deform elastoplastically, especially metals, persistentdeformations of a workpiece are no longer desired once the manufacturing process iscomplete. Thus, plastic yielding can be considered as failure in certain applications.During forming processes however, persistent deformations of a workpiece are desiredand not considered as failure. In such processes, the maximum load bearing capacityis determined by the maximum stress a material can endure before strain localizationoccurs. Lastly, a third criterion of failure is the complete loss of material integrity, i.e.the formation of a crack. In the present work, focus is set on the first two definitionsof failure: plastic yielding and strain localization.In a wide range of engineering applications, polycrystalline metals are used. In fact,only a few applications for components manufactured from single crystals (e.g. turbineblades) exist. On a macroscopic scale, polycrystalline materials are often describedusing phenomenological models as far as inelastic behavior and strain localization asa result of damage evolution are concerned. These deformation mechanisms are ofteninduced by microstructural processes. However, phenomenological models can pre-dict this behavior only within a certain range of accuracy, as they are derived frommacroscopic observations. Sophisticated models that account for microstructural ef-fects directly exist but are not applicable to many engineering problems as the size ofa component under investigation is often orders of magnitude larger than the size ofa microstructural unit cell. This would result in a computational effort which cannotbe handled reasonably by modern computers. For this reason, multiscale techniqueshave been developed in which a single computation of a microstructured componentis replaced by a macroscale computation governing the overall response. In such aprocedure, the macrolevel relies on information gained by microlevel computations de-scribing the phenomena of interest.

1

2 CHAPTER 1. INTRODUCTION

Heterogeneous, microstructured materials are also used to connect different workpieces.Often, they constitute interface layers, e.g. in soldered connections or in reinforced ad-hesive bondings, with a small thickness compared to the size of the component. Insuch interfaces, complex microstructural processes are also present. In problems wherethe main part of the deformation takes place in the material layer, they influence theoverall behavior significantly.The goal of this thesis is to present a method for modeling inelastic, size-dependentmaterial behavior within bulk and interface problems in a two-scale framework. Forthis, phenomenological models usually found in the macroscopic modeling of materialsare used to describe the mesostructure of a heterogeneous body. To account for nu-merical efficiency within the Finite Element Method, the resulting nonlinear equationsare solved by means of a Newton-Raphson scheme including a consistent linearization.Numerical simulations exhibit the performance of the developed tool.This can be regarded as the first step towards the integration of more sophisticatedmodels describing the true microstructural behavior, e.g. crystal plasticity for metals ordamaging mechanisms of rubbery polymers in a multiscale homogenization procedure.

1.2 State of the art

In this section, an overview of the computational solution methods of material failuremechanisms is given. For this, several numerical solution techniques for partial differ-ential equations in engineering problems are briefly introduced, including the FiniteElement Method, which will be used later on. Due to the scope of this thesis, thediscussion is restricted to methods applied to solid mechanics problems.Different approaches to the modeling of different types of discontinuities within a me-chanical problem, as they occur in fracture mechanics, are discussed. Since this thesisfocuses on problems where the failure path is known in advance, Cohesive Zone Modelsare discussed explicitly.Afterwards, the macroscopic modeling of continuum damage is introduced. The arisingchallenges within the Finite Element Method and relevant solution strategies (includinggeneralized continua, which are used in this work) are explained. Lastly, the multi-scale modeling of failure processes is reviewed, and macroscopic methods known fromsingle-scale fracture mechanics as well as multiscale cohesive interface elements arediscussed.

1.2.1 Computational solution of mechanical problems

Very often, the mathematical modeling of mechanical engineering problems leads topartial differential equations (PDEs) which, for realistic applications in general, cannotbe solved analytically. Consequently, numerical methods have been developed whichapproximate the solution to a certain accuracy. A convenient way to find solutions forthe PDEs is the discretization of the domain, after which the PDE including boundaryconditions is transformed into a set of algebraic equations that needs to be solved forthe degrees of freedom introduced by different discretization techniques.

1.2. STATE OF THE ART 3

An early approach of this class of solution algorithms is the Finite Difference Method,where the differential quotients which appear in the PDE are approximated by differ-ence quotients based on a Taylor-expansion. The resulting equation system can thenbe solved for the unknowns at the nodes where the differences are evaluated. With thismethod, an approximation of the solution field has to be interpolated over the domainas only discrete values at the nodes are computed.A different approach is the method of weighted residuals, which uses an approximationfunction for the solution with unknown parameters. This approximation function isweighted with a test function, which usually fulfills the homogeneous boundary con-ditions. The postulation of a vanishing weighted residual then leads to an equationsystem to determine the parameters of the approximation function. This way, themean weighted residual over the entire domain is zero. In contrast to the Finite Differ-ence Method, where the PDE is solved directly, the error resulting from the insertionof the approximation function is minimized. A special case of this method arises ifDirac-delta functions are used as test functions. In this case, the method is called col-location method and it fulfills the underlying differential equation exactly at the pointswhere the test functions are non-zero. However, the solution might not be accuratein between the collocation points. Further information can be found in [Collatz, 1951]and [Gross et al., 2011].The method of weighted residuals is the basis of the Finite Element Method, in whichspecial discretization techniques are used to describe the domain. Often, a Bubnov-Galerkin approach, in which the test function is approximated with the same shapefunctions as the approximation function, is chosen. As in the Finite Difference Method,an algebraic equation system has to be solved for the degrees of freedom, whereas theinterpolation functions to describe the solution over the whole domain are prescribeda priori. A detailed description of this method is given in chapter 3 of this thesis.For other computational methods, which are beyond the scope of this work, the readeris referred to the literature. The Boundary Element Method (BEM), where the gov-erning PDEs are transferred to boundary integral equations solved on a discretizedboundary, is described in [Sauter and Schwab, 2011]. In solid mechanics, the BEM isrestricted to small deformations and linear elasticity. However, this method can be ap-plied efficiently to linear elastic fracture mechanics as a priori knowledge of the near-tipstress and strain fields can be incorporated without much additional effort.For very large deformations, mesh-based methods require remeshing techniques to avoidlarge element distortions. To circumvent these problems, meshless methods have beendeveloped. In [Li and Liu, 2004], an overview of these methods is given. SmoothedParticle Hydrodynamics introduced by [Gingold and Monaghan, 1977] are based onmeshfree collocation methods. A different approach are the meshfree Galerkin meth-ods, first described in [Nayroles et al., 1992], where diffuse shape functions are used fora moving least squares interpolation.For systems containing a large amount of particles, which are commonly consideredas rigid as the deformation stems from particle movements rather than from particledistortions, discrete particle simulations as outlined in [Zhu et al., 2007] and in [Zhuet al., 2008] can be applied.


1.2.2 Fracture mechanics

Fracture mechanics is an important topic regarding the failure of mechanical compo-nents. Fracture occurs due to the macroscopic presence and propagation of cracksrepresenting a partial or complete rupture of material. Consequently, a crack repre-sents a strong discontinuity in the material as the crack faces do not interact in thecase of crack opening. In the following, several methods capturing fracture processesare listed.In the 1920s, an energy based crack propagation criterion for elastic media, which isstill used today, was developed in [Griffith, 1921]. An asymptotic analytical solution todescribe the stress field in the vicinity of a crack tip is given in [Irwin, 1957]. However,many engineering materials cannot be modeled as elastic continua, as a ductile frac-ture process zone in front of a crack tip can appear. Therefore, a model including thepresence of a ductile process zone near the crack tip was developed in [Dugdale, 1960].A detailed introduction to fracture mechanics and a summary of the models describedabove is given in [Gross and Seelig, 2011].From a computational point of view, some drawbacks occur within the computation offracture processes with the standard Finite Element Method. Standard elements arenot able to capture a sharp discontinuity, such as a crack, in the inside. Thus, crackshave to be aligned with element edges, which leads to a strong mesh dependency ofthe solution obtained. Although adaptive remeshing strategies can be used, the com-putation of the near-tip stress and strain fields remains an issue as the position of thecrack tip has to coincide with element nodes. Therefore it is difficult to obtain meshindependent solutions for crack propagation problems.A remedy to this problem is the formulation of Finite Elements with embedded dis-continuities as described in [Dvorkin et al., 1990], in [Simo et al., 1993], and in [Linderand Armero, 2007]. In this method, the strain field is enhanced in a way that FiniteElements are enabled to contain cracks. In fracture mechanical problems, a drawbackof this method is that cracks are not allowed to end inside an element which results ina loss of accuracy considering the near tip stress and strain fields.A method for incorporating the near tip fields explicitly is the Extended Finite Ele-ment Method (XFEM), going back to [Belytschko and Black, 1999] and [Moes et al.,1999], which is based on Partition of Unity Finite Element Methods (see [Melenk andBabuska, 1996]). In these methods, it is possible to include a priori knowledge ofthe solution in the element formulations. For fracture mechanics, this means that thecrack tip fields can be integrated into an Extended Finite Element via certain nodeenrichment techniques. An enrichment procedure with an additional effort to fulfill thepartition of unity is explained in detail in [Fries, 2008]. An application of the XFEMwith this particular way of node enrichment to finite deformation problems is providedby [Lohnert et al., 2011]. As the crack geometry is described via node enrichments andthe near tip fields are incorporated directly, the Extended Finite Element Method ren-ders a mesh independent solution of fracture problems. In [Fries and Belytschko, 2010],[Muller-Hoeppe, 2012] and in [Holl, 2014], several XFEM-approaches are summarizedand comprehensive literature reviews are given.Even earlier than the the work of Dugdale, the idea of a process zone in front of a


crack tip was explained in [Barenblatt, 1959], which was the beginning of the modelingof cohesive zones. In these Cohesive Zone Models (CZM), a constitutive traction-separation relation is assumed before complete rupture of the material occurs. Thetraction-separation laws can be used in cohesive interface elements as introduced in[Beer, 1985]. These elements are located in between the standard bulk elements to de-scribe ductile fracturing processes on element boundaries. For more information aboutinterface elements see section 5.2.1. The drawback of the CZM approach as far ascrack propagation is concerned is that many elements have to be used and the crackis automatically aligned with the mesh. Thus, the solution becomes mesh dependentjust as in fracture problems solved with the standard FEM. However, cohesive interfaceelements have proven to be well suited for problems where the failure path is known inadvance. Typical applications are matrix-inclusion-debonding in composite materials(see [Xu and Needleman, 1993] and [van den Bosch et al., 2006]), polymer coating ofmetal components (see [van den Bosch et al., 2007]) and delamination of layered struc-tures (see [Alfano and Crisfield, 2001]), only to name a few. A formulation of finitethickness cohesive zones is provided by [Paggi and Wriggers, 2011a] and [Paggi andWriggers, 2011b].In the cohesive band model proposed in [Remmers et al., 2013], a cohesive traction-separation formulation is used within a Partition of Unity Finite Element Method.This approach contains the modeling of cohesive zones with interface elements as aspecial case.

1.2.3 Continuum damage formulations

Rather than describing macroscopic cracks, continuum damage models deal with theeffective description of microstructural effects leading to the formation of microscopicvoids and cracks. These models introduce a set of continuous field variables accountingfor a decreasing load bearing capacity of the material for the general case of anisotropicmaterial behavior. In case of isotropy, a scalar damage variable is sufficient to com-pute the loss of material integrity. These phenomenological models with the purposeof failure prediction are based on the work of [Kachanov, 1958]. For a literature reviewthe reader is referred to section 2.1.3.A difficulty in damage formulations in a Finite Element framework can be seen in thecomputations carried out in [Lemaıtre, 1985]. There, damage evolution was computedfor a given Finite Element mesh of a notched specimen. It is obvious that the result-ing failure pattern strongly depends on the element size as a damaged element failscompletely without affecting the surrounding elements. This non-physical localizationpattern occurs in the softening regime as the governing PDE loses its ellipticity. This isa pathological problem and mesh refinements will not lead to a physically meaningfulsolution (see [Peerlings, 1999] and the references mentioned in chapter 4 of the presentthesis). Please note that this localization phenomenon is not restricted to damagewithin the Finite Element Method but occurs in other softening mechanisms and inother numerical tools as well.In the following, several frameworks for overcoming this problem are described. These


methods have in common that the damage evolution does no longer depend only onvariables of the material particle under consideration, but also take into account thesurrounding of a particle. This can be done either directly as in nonlocal damage mod-els or implicitly as in the other frameworks.A straightforward way to capture localization effects is the use of generalized continuaas they all incorporate gradients of kinematic fields which are not present in a stan-dard continuum model. Detailed descriptions of different kinds of generalized continuaincluding a literature review are presented in section 2.2.As already mentioned, quantities of material particles in the close surrounding of thematerial particle under consideration appear directly in a nonlocal formulation. In anonlocal damage model as in [Pijaudier-Cabot and Bazant, 1987] and in [Pijaudier-Cabot and Bode, 1992], weighted averages of the quantities of interest are used inthe constitutive relations instead of locally defined quantities. This ensures damageevolution in a band with a finite thickness which converges to a physically meaningfulsolution in the case of mesh refinements. However, this method requires sophisticateddata handling techniques to keep the computational effort reasonably low. Otherwise,the evaluation of a single integration point in a Finite Element becomes very time con-suming.Because of this issue, which contradicts the paradigm of simple implementation in theFinite Element Method, implicit gradient enhanced models have been developed. Inthese models, a PDE governs the behavior of an additional primary field variable cou-pled with a local quantity. This field variable can then be used instead of a nonlocalone. This approach offers the possibility of replacing nonlocal calculations with theintroduction of an additional degree of freedom for each node leading to a simple im-plementation. Such models, in which a Helmholtz-type PDE for the additional fieldand the standard balance of linear momentum are solved simultaneously, can be appliedin various frameworks. Gradient enhanced brittle damage is investigated in [Peerlingset al., 1996]. Gradient enhanced softening plasticity with a scalar additional field ispresented in [Engelen et al., 2003] and [Peerlings et al., 2012] for small strains andin [Geers, 2004] for finite strains. A gradient enhanced plasticity framework with anadditional tensorial field is provided by [Poh et al., 2011].Very similar to the implicit gradient approach is the phase-field description of fractureby [Kuhn and Muller, 2010], [Miehe et al., 2010], [Verhoosel and de Borst, 2013], and[Miehe and Schanzel, 2014] based on a variational approach to fracture by [Bourdinet al., 2008]. In these frameworks, the phase-field is an additional primary quantityalso governed by a Helmholtz-type PDE. A crack as a result of damage evolution isnot modeled as a sharp discontinuity as in the XFEM. Rather than that, a smearedrepresentation of the crack described by the phase-field is used to model fracture. Thisway, discontinuous element formulations are not necessary as the crack is representedby a continuous field. However, dense meshes are needed for a good approximation ofa sharp discontinuity.All these models share the presence of a characteristic inner length describing the sizeof the domain affected by gradient or nonlocal effects. Consequently, they are capableof predicting size-effects when the specimen size is varied for a constant characteristic


length. These effects also occur if the characteristic length is varied for a constantspecimen size. The theory of the modeling of size-effects in generalized continua isdescribed in section 2.2. Simulations can be found in chapters 4 and 6.

1.2.4 Macroscopic failure in multiscale frameworks

Computational multiscale methods can be applied to avoid dense meshes for the reso-lution of the microstructure of a material. Also, different continuum formulations andconstitutive models can be applied on different scales. This enables a great flexibil-ity to incorporate effects of interest on the scales on which they occur independently(with certain restrictions regarding thermodynamic consistency) of the models of otherscales.In many of these methods, the microstructural behavior is described by computations ofso-called Representative Volume Elements (RVEs). The results of these computationscan be used to describe an effective material behavior for different loading modes. Adifferent approach, often referred to as FE2-method, uses a nested solution scheme, inwhich the RVE computations are carried out instead of the evaluation of a constitutivelaw in the macroscopic problem. For further details the reader is referred to chapter 5,where different analytical and computational multiscale techniques are reviewed.In the conventional FE2-method, a problem arises for an overall softening behavior ofthe RVE which may be induced by softening elasto-plasticity or damage evolution oreven partial microscopic rupture. In this case, the microscopic volume element losesits representativeness as strain localization occurs on the microscopic scale. However,several numerical tools have been developed to enable the computation of macroscopicfailure induced by microstructural defects in a homogenization framework.A general introduction into this topic and a description of a model for multiscale ma-sonry cracking as proposed in [Massart, 2003], [Massart et al., 2007a] and [Massartet al., 2007b] are given in [Massart et al., 2011].In [Coenen, 2012], [Coenen et al., 2012a], and [Bosco et al., 2014], a two-scale methodusing Finite Elements with embedded discontinuities (as described in the previoussection) on the macrolevel to predict macroscopic strain localization is presented. Ac-cording to [Coenen et al., 2012b], the direction of the developing localization band iscomputed with the help of special boundary conditions accounting for the interactionof the macroscopic loading and orientation of the microstructure. In a similar compu-tational procedure, the Extended Finite Element Method was used to introduce macro-scopic discontinuities instead of embedded discontinuities ([Belytschko et al., 2008]).The evolution of cohesive macrocracks induced by microstructural processes with theXFEM in a two-scale setting is also presented in [Nguyen et al., 2011], [Nguyen, 2011],and [Nguyen et al., 2012]. In these works, adhesive cracks are investigated as well. Forthis, macroscopic multiscale interface elements, which gain the constitutive responsethrough microscopic analyses, are introduced. Macroscopic interface elements werealso used before in [Matous et al., 2008], [Hirschberger et al., 2008], and [Hirschbergeret al., 2009] to model thin, microstructured material layers and in [Vossen et al., 2014]for the prediction of polymer-metal-delamination.


Some of the computational approaches mentioned above are based on a microscopicmodel with an additional field equation. The methods presented in [Nguyen et al.,2012] and [Massart et al., 2007a] apply a gradient enhanced damage model as de-scribed by [Peerlings et al., 1996] to regularize the microscopic evolution of damage,whereas a micromorphic hyperlelastic model is used in [Hirschberger et al., 2008] topredict size-effects (see also section 2.2). This procedure with a generalized model forthe microstructure will also be investigated in this thesis.

1.3 Outline of the thesis

In chapter 2, the continuum mechanical foundations needed for the presented numericalmethod are described. This chapter is divided into two parts. Firstly, the nonlinearkinematics for standard continua are presented and the axiomatic balance equationsare explained before the constitutive models relevant for this work are formulated.Secondly, some forms of generalized continua are introduced. Focus is set on the de-scription of micromorphic-type inelastic continua suitable for the modeling of softeningelastic-plastic and damaging elastic-plastic materials.These continuum models are used within the Finite Element Method as shown in chap-ter 3. The general concept of this method is illustrated by the derivation of the weakforms of equilibrium and the introduction of discretization techniques in space andtime. For the solution procedure, a symbolic formulation well suited for the implemen-tation in automatic-differentiation-based frameworks is used.Examples demonstrating the performance of the derived Finite Elements in single-scalecomputations can be found in chapter 4. The indentation of a rigid punch into a ficti-tious elastic-plastic micromorphic-type material is computed to demonstrate the meshregularization capability of the model in the softening regime. The damaging elastic-plastic model is validated in a tension test of a strip with a hole made of copper (CW024 A).The application of the micromorphic-type continua in a two-scale homogenizationframework is explained in detail in chapter 5, where the general procedure is illus-trated for bulk materials before the modeling of thin material layers is investigated.A way to compute a consistent macroscopic tangent via sensitivity analyses, whichensures optimal macroscopic convergence rates within a nested solution scheme, is em-phasized.The capability of these models to predict size-effects within a multiscale method isshown in chapter 6 for the inhomogeneous deformation of a microstructured bulk ma-terial as well as for softening heterogeneous material interfaces.Concluding this thesis, a summary and an outlook on further extensions of this workare given in chapter 7.

Chapter 2

Continuum mechanics

In continuum mechanics, material properties (such as density, stiffness, etc.) are as-sumed to be distributed continuously within material bodies. The fields describingthe body (placement, velocity, stress, etc.) are considered to be continuous as well.Consequently, a material body is composed of a continuous set of material particlespossessing their respective properties.In this chapter, the basic governing equations for nonlinear mechanics are briefly sum-marized and explained. In section 2.1, kinematics, balance equations, and some ex-amples of constitutive equations for standard continua are investigated. Since notonly standard continua are used in this work, certain generalizations are described.Additional kinematic fields, balance equations, and constitutive models for so-calledgeneralized continua are discussed in section 2.2. This class of materials will again beextended to what can be called micromorphic-type inelastic media in section 2.2.4.Throughout this thesis, isothermal, quasi-static processes neglecting body forces areassumed. However, the equations are derived for the general case in order to simplifythem afterwards.

2.1 Standard continua

First, the kinematic relations for standard continua are explained. After the definitionof a motion, certain deformation measures, deformation rates, and transformation lawsbetween configurations are derived. Afterwards, the governing balance equations for aphysically meaningful framework are pointed out. Finally, some examples of elastic andinelastic material models, which are relevant for this thesis, are given and explained. Fora more detailed view on the derivation of the kinematic relations, balance equations andelastic constitutive equations, see e.g [Truesdell and Toupin, 1960], [Truesdell and Noll,2004], [Malvern, 1969], [Marsden and Hughes, 1983], [Ogden, 1984], and [Holzapfel,2000]. Readers especially interested in elastic-plastic deformations are referred to [Simoand Hughes, 1998], [Lubliner, 2008], [Bonet and Wood, 2008], and [de Souza Neto et al.,2008].The relations derived in this section are used as a basis for the extensions made in thesubsequent sections.

9

10 CHAPTER 2. CONTINUUM MECHANICS

2.1.1 Kinematics

Let a material body B, consisting of material particles, occupy the space B0 at thetime t0. Then, there exists a unique mapping χ0 onto the Euclidean vector space E3

such that the placement X ∈ B0 of each material particle P can be identified uniquely:

X = χ0(P, t0) . (2.1)

This mapping is called the reference configuration of the body B.This body occupies the space B at an arbitrary time t > t0. Analogously, there existsa mapping χ to determine the placement x ∈ B of each material particle P :

x = χ(P, t) . (2.2)

Then, χ is called the current configuration of the body B. The unique mapping ϕ thendescribes the motion of the body (see figure 2.1):

x = χ(χ−10 (X, t)) = ϕ(X, t) . (2.3)

This formulation of the motion of a body, depending on the initial position X of thematerial point P , is called the material or Lagrangian formulation. The observer keepstrack of the motion of the observed material particle.A motion ϕ is usually assumed to be bijective, so that the inverse mapping ϕ−1 isunique:

X = ϕ−1(x) . (2.4)

This is the spatial or Eulerian description of a motion where a fixed spatial point,occupied by different material points at different times, is observed.

x

X

u

B0

B

ϕ

χ

χ0 P

E

Figure 2.1: Motion of a body

2.1. STANDARD CONTINUA 11

Now that the placements X and x are defined, the displacement can be introduced asthe difference between them:

u = x−X . (2.5)

The first and second material time derivatives of the motion, known as velocity andacceleration respectively, can be expressed in terms of u:

v =dx

dt=

du

dt= x = u (2.6)

a =dx

dt=

du

dt= x = u . (2.7)

Being the material gradient of the placement x, the tensor F maps an infinitesimal lineelement dX from the reference configuration to the current configuration:

dx = F · dX (2.8)

F =∂x

∂X= 1+

∂u

∂X= 1 +H , (2.9)

where H = ∂u/∂X is the displacement gradient tensor. The tensor F describes both,stretching and rotation, of a line element. It can be uniquely decomposed into itsstretching and rotational parts by a polar decomposition

F = R ·U = V ·R , (2.10)

with the proper orthogonal rotation tensor R. The symmetric, positive definite tensorsU and V are called right stretch tensor and left stretch tensor respectively. They canbe expressed by their respective eigenvalues and eigenvectors:

U =3∑

i=1

λiNi ⊗Ni (2.11)

V =

3∑

i=1

λini ⊗ ni . (2.12)

In the above equations, λi are the principal stretches, Ni are the eigenvectors of U, andni are the eigenvectors of V. With these quantities at hand, R and F can be writtenas follows:

R =

3∑

i=1

ni ⊗Ni (2.13)

F =

3∑

i=1

λini ⊗Ni . (2.14)


With the help of these tensors, another pair of deformation measures used throughoutthis thesis can be defined:

C = FT · F = U ·U =

3∑

i=1

λ2iNi ⊗Ni (2.15)

b = F · FT = V ·V =3∑

i=1

λ2ini ⊗ ni . (2.16)

These tensors are called Cauchy-Green deformation tensors. According to the nomen-clature of U and V, C is called right Cauchy-Green tensor and b is called left Cauchy-Green tensor.With the help of these deformation tensors, strain measures for both configurations aredefined:

E =1

2(C− 1) (2.17)

e =1

2

(1− b−1

). (2.18)

Consequently, the Green-Lagrange strain tensor E is defined in the reference configu-ration and the Euler-Almansi strain tensor e is defined in the current configuration.The linearizations of both strain tensors around u = 0 lead to the definition of thesmall strain tensor ǫ:

ǫ =1

2

(∂u

∂X+

(∂u

∂X

)T)

. (2.19)

Although only quasi-static deformations will be assumed in this work, certain deforma-tion rates are used to derive physically meaningful constitutive models. The materialvelocity gradient is defined as follows:

F =∂v

∂X=

∂x

∂X. (2.20)

Its counterpart, the spatial velocity gradient, reads:

l =∂v

∂x=∂x

∂x= F · F−1 . (2.21)

The symmetrical part of this tensor is denoted as d:

d =1

2

(l+ lT

). (2.22)

To map infinitesimal volume elements from χ0 to χ, the determinant of the deformationgradient can be used:

J = detF (2.23)

dv = JdV , (2.24)


where dV and dv are the volume elements in the reference configuration and in the cur-rent configuration respectively. The change in area and orientation of an infinitesimalsurface element dA = NdA can then be calculated by Nanson’s formula:

nda = JF−T ·NdA , (2.25)

with the area element in the current configuration denoted as da and the outwardnormal unit vectors in the reference configuration and the current configuration denotedas N and n respectively.

2.1.2 Balance equations

The following axiomatic balance equations have been derived from physical observa-tions. The balance equations of mass, linear and angular momentum, and energy areinvestigated. Additionally, the entropy inequality (which is no balance equation in theoriginal sense) is given.

Balance of mass

The mass of a body in the reference configuration can be determined by the integralof the initial mass density ρ0 over the volume:

m =

∫

B0

ρ0 dV . (2.26)

If the body is considered to be a closed (i.e. no particles are added to it or separatedfrom it), non-relativistic system, the mass is constant during the deformation. Withthe mass density in the current configuration ρ, the mass reads:

m =

∫

B

ρ dv . (2.27)

This leads directly toρ0 = Jρ . (2.28)

If the mass does not change during the deformation (or equivalently in time), its rateis equal to 0:

m =d

dt

∫

B0

ρ0 dV =d

dt

∫

B0

Jρ dV =

∫

B0

(Jρ+ Jρ

)dV = 0 . (2.29)

With the rate of J computed by

J =∂ detF

∂F:dF

dt= J div v , (2.30)

the global and local forms of the mass balance equation can be formulated:∫

B0

(ρ+ ρ div v) dV = 0 (2.31)

ρ+ ρ div v = 0 . (2.32)


Balance of linear momentum

The linear momentum I of a body is defined as

I =

∫

B

ρv dv . (2.33)

It changes due to surface tractions t or body forces b, such that

I =d

dt

∫

B0

ρvJ dV =

∫

∂B

t da+

∫

B

ρb dv . (2.34)

With the assumption of mass conservation (2.32), this reduces to

∫

B

ρa dv =

∫

∂B

t da+

∫

B

ρb dv . (2.35)

Introducing the Cauchy stress tensor σ in such a way that the Cauchy theorem

σ · n = t (2.36)

holds (with n as outward normal unit vector), the above equation is rewritten as follows:

∫

B

ρa dv =

∫

∂B

σ · n da +

∫

B

ρb dv . (2.37)

The application of the divergence theorem then provides the global and local forms ofthe balance of linear momentum:

∫

B

(divσ + ρb− ρa) dv = 0 (2.38)

divσ + ρb− ρa = 0 . (2.39)

In case of quasi-static deformations neglecting body forces, the balance of linear mo-mentum reads:

divσ = 0 . (2.40)

To obtain a different stress measure, the integral over the boundary tractions is trans-ferred to the reference configuration via Nanson’s formula (2.25):

∫

∂B

σ · n da =

∫

∂B0

Jσ · F−T ·N dA =

∫

∂B0

P ·N dA . (2.41)

The quantityP = Jσ · F−T (2.42)

is called first Piola-Kirchhoff stress tensor and can be used to rewrite equation (2.40)in the reference configuration as follows:

DivP = 0 . (2.43)


Balance of angular momentum

The balance of angular momentum looks very similar to the balance of linear momen-tum, although, physically speaking, these equations are independent of each other. Letx0 be the placement of a momentarily fixed reference point in the current configuration.The angular momentum L0 with respect to x0 can by calculated by

L0 =

∫

B

(x− x0)× ρv dv . (2.44)

The change of angular momentum is then expressed by

L0 =d

dt

∫

B

(x− x0)× ρv dv =

∫

∂B

(x− x0)× t da+

∫

B

(x− x0)× ρb dv . (2.45)

Together with the balance of mass (2.32), Cauchy’s theorem (2.36), the balance of linearmomentum (2.39), and the divergence theorem, this equation leads to the symmetryof the Cauchy stress tensor for standard continua:

σ = σT . (2.46)

Introducing the Kirchhoff stress tensor τ with

τ = Jσ , (2.47)

it follows thatτ = τ T . (2.48)

Balance of energy

The energy E of a body is composed of its kinetic energy T and its internal energy U :

E = T + U (2.49)∫

B

ρe dv =1

2

∫

B

ρv · v dv +

∫

B

ρu dv . (2.50)

Here, e is the specific energy density and u is the specific internal energy density.The rate of energy E is equal to the sum of mechanical power and heat, P and Q,provided to the body:

E = T + U = P +Q . (2.51)

The mechanical power consists of the powers of the surface tractions and the bodyforces:

P =

∫

∂B

v · t da+∫

B

ρv · b dv , (2.52)

whereas the heat flux q through the surface and the distributed heat source r contributeto the heat:

Q = −∫

∂B

q · n da+

∫

B

ρr dv . (2.53)


This leads to

d

dt

(1

2

∫

B

ρv · v dv +

∫

B

ρu dv

)=

∫

∂B

v · t da+∫

B

ρv ·b dv+

∫

B

ρr dv−∫

∂B

q ·n da .

(2.54)Using the mass balance (2.32), Cauchy’s theorem (2.36), the balance of linear mo-mentum (2.39), the symmetry of the Cauchy stress tensor (2.46), and the divergencetheorem, this equation can be simplified to obtain the global and local forms of theenergy balance:

∫

B

ρu dv =

∫

B

(σ : d+ ρr − div q) dv (2.55)

ρu = σ : d+ ρr − div q . (2.56)

For an isothermal problem, the energy balance reduces to

ρu = σ : d =1

JP : F , (2.57)

where the stress power can also be expressed by P and F.

Entropy inequality

The entropy inequality describes the direction of a process. Processes with a decreasingentropy are physically not admissible. If (theoretically) the entropy is constant duringa process, this process is considered to be reversible.The entropy inequality states that the rate of entropy in a body is greater than or equalto the amount of entropy provided to the body by thermal loads:

dN

dt=

d

dt

∫

B

ρη dv ≥∫

B

ρr

Θdv −

∫

∂B

1

Θq · n da . (2.58)

In this equation, N is the entropy, η is the entropy density, and Θ is the absolutetemperature. Using the Helmholtz free energy density ψ defined as

ψ = u−Θη (2.59)

together with the mass balance (2.32), the balance of energy (2.56), and the divergencetheorem, this equation can be rewritten in a form known as Clausius-Duhem inequality:

∫

B

(σ : d− ρψ − ρηΘ− 1

ΘgradΘ · q

)dv ≥ 0 (2.60)

σ : d− ρψ − ρηΘ− 1

ΘgradΘ · q ≥ 0 . (2.61)

For an isothermal problem, this inequality renders the mechanical dissipation D, whichhas to be greater than or equal to zero:

D = σ : d− ρψ ≥ 0 . (2.62)


2.1.3 Constitutive models

Only considering the balance equations of the previous section does not suffice to de-scribe a thermomechanical problem, since they do not provide enough equations for the19 unknowns hidden in the mass density ρ, the displacement u, the stress tensor σ, theinternal energy u, the heat flux q, the entropy η and the temperature Θ. Thus, addi-tional equations are necessary to form a solvable system of equations. These additionalequations, called constitutive equations, describe the material behavior. They cannotbe chosen arbitrarily but should fulfill the following continuum mechanical principlesof axiomatic character to result in a material behavior which is in accordance withexperience.

Continuum mechanical principles

A more detailed description of the following principles can be found in the literaturealready mentioned at the beginning of this chapter. Additionally, a canonical overviewof these principles is given in [Altenbach, 2012].

Causality In a thermomechanical continuum, the motion ϕ and the temperature Θare treated as independent variables. They are the cause for the stress tensor, the heatflux, the energy, and the entropy, which are the response of the continuum, to change.In an isothermal problem, the only remaining independent variable is the motion ϕ.

Determinism The current state of a continuum is completely determined by itscurrent loading and its loading history.

Equipresence The set of independent variables contained in one constitutive equa-tion has to be contained in all other constitutive relations.

Consistency All constitutive relations have to be in accordance with the balanceequations and the entropy inequality.

Material objectivity The constitutive equations have to be independent of thechoice of a frame of reference. They have to be invariant with respect to the motion ofany observer.Let the motion ϕ be observed by observer O. Then a second observer O∗, resultingfrom an Euclidean transformation, observes the motion ϕ∗ in the following way:

x = ϕ(X) (2.63)

x∗ = ϕ∗(X) = Q · x+ c . (2.64)

Here, Q is a proper orthogonal tensor describing a rigid body rotation and c is a rigidbody translation. It follows that

F∗ = Q · F . (2.65)


Then, in order to be objective, a scalar a, a vector a, and a tensor of second order Ahave to transform in the following way:

a∗ = a (2.66)

a∗ = Q · a (2.67)

A∗ = Q ·A ·QT . (2.68)

This implies that the stress tensor σ as a functional of the motion ϕ has to transformin the following way:

σ = F (x) (2.69)

Q · F (x) ·QT = F (Q · x+ c) . (2.70)

Material symmetry The constitutive equations for solid bodies are invariant torotations applied to material coordinates before the deformation, if the rotation tensorQ is from the symmetry group of the material, i.e.

σ(F · Q) = σ(F) . (2.71)

Local action The state of a material point is only influenced by its close surroundingso that the constitutive equations depend on independent variables and their gradients.

Fading memory The recent history of the deformation and temperature has agreater influence on the current state than events that occurred in the distant past.

Isotropic Hyperelasticity

Hyperelasticity states that the work done by the stresses is independent of the loadingpath and can be described by a strain energy function Ψ :

∫ t2

t1

P : F dt = Ψ (F2)− Ψ (F1) . (2.72)

The function Ψ can be identified as the Helmholtz free energy from equation (2.59):

Ψ = ρ0ψ . (2.73)

In general, this function depends on the deformation gradient F. But since the principleof material symmetry is valid and isotropic material behavior is assumed, the rigid bodyrotation Q can be replaced by the rotation tensor R from the polar decomposition ofF (2.10) without further restrictions. Consequently, a strain energy formulated in F isequivalent to formulations in other deformation measures:

Ψ (F) = Ψ1(U) = Ψ2(C) = Ψ3(V) = Ψ4(b) . (2.74)


In the following, Ψ is assumed to depend on the left Cauchy-Green tensor b. Now, thestress tensor can be determined via the Coleman-Noll procedure introduced by [Cole-man and Noll, 1963], providing relations for the respective work conjugated quantities.Since elastic materials are dissipation-free, the dissipation inequality (2.62) is fulfilledwith equity:

D = σ : d− ρψ =1

Jτ : d− ρψ = τ : d− Ψ = τ : l− Ψ = 0 . (2.75)

Inserting the rate of the free energy function

Ψ =∂Ψ (b)

∂b: b (2.76)

and exploiting the symmetry of b, the equation above can be transformed to(τ − 2

∂Ψ (b)

∂b· b)

: l = 0 . (2.77)

This gives an expression for the Kirchhoff stress tensor τ :

τ = 2∂Ψ (b)

∂b· b . (2.78)

If the free energy is chosen to depend not on b directly but rather on its invariants Ib,IIb and IIIb, isotropic material behavior is obtained as the stress tensor is expressedby an isotropic tensor function. These invariants can be computed with b or with theprinciple stretches λi in the following way:

Ib = λ21 + λ22 + λ23 = trb (2.79)

IIb = λ21λ22 + λ22λ

23 + λ23λ

21 =

1

2

((trb)2 − tr(b · b)

)(2.80)

IIIb = λ21λ22λ

23 = detb = J2 . (2.81)

However, Ψ cannot be chosen arbitrarily in its arguments. According to [Ball, 1977],the strain energy function has to be polyconvex, i.e. convex in each of its argumentsF, adjF, and detF to ensure a well-posed mechanical problem.The strain energy function of interest in this work is the Neo-Hookean strain energydepending on Ib and J :

Ψ =µ

2(Ib − 3)− µ ln J +

Λ

4

(J2 − 2 lnJ − 1

). (2.82)

This function allows finite rotations as well as considerably large strains compared tosmall deformation theory. For small deformations (u ≈ 0), the material parameters µand Λ can be identified as the first and second Lame-constants. The parameter µ isalso known as shear modulus. Knowing Young’s modulus E and the Poisson ratio ν ofa material, they can be computed as follows:

µ =E

2(1 + ν)(2.83)

Λ =Eν

(1 + ν)(1− 2ν). (2.84)


The Kirchhoff stress tensor then reads

τ = µ (b− 1) +Λ

2(J2 − 1)1 . (2.85)

For some material models, it is convenient to uncouple the deviatoric and volumetricparts of the strain energy function. This split goes back to [Flory, 1961]. Let the

modified left Cauchy-Green tensor b be defined in the following way:

b = J− 23b . (2.86)

It follows thatdet b = 1 . (2.87)

With this definition, Ψ can be defined using b and J instead of b and J such that

Ψ = Ψ(b) + U(J) , (2.88)

with the deviatoric (or isochoric) part of the strain energy Ψ and the purely volumetricpart U . The following Neo-Hookean strain energy function will be used in this thesis:

Ψ =µ

2

(Ib− 3)+κ

4

(J2 − 2 lnJ − 1

). (2.89)

The bulk modulus κ is described by

κ =2

3µ+ Λ =

E

3(1− 2ν). (2.90)

From this, a different formula for τ in terms of the deviatoric part of b, called devb,and the determinant J emerges:

τ = µJ− 23

(b− 1

3Ib1

)+κ

2(J2 − 1)1 = µJ− 2

3 devb+κ

2(J2 − 1)1 . (2.91)

Please note that this strain energy function does not fulfill the condition of polycon-vexity for very large strains in the nearly incompressible regime (i.e. for large valuesof κ) [Hartmann and Neff, 2003]. Instead, a different formulation of U should be used.However, such large strains will not be investigated in this thesis so that the proposedstrain energy is acceptable.

Isotropic Hyperelasto-plasticity

Plastic (i.e. persistent) deformations in metals occur due to dislocation movements inthe crystal lattice of the material. The dislocations start to slip through the latticewhen a certain stress measure, i.e. the resolved shear stress on the respective slipsystem, exceeds a certain value. By these movements, the amount (and/or length) ofdislocations increases, which hinders the movements of other dislocations. This meansthat the critical stress for plastic slip is increasing. This results in an overall hard-ening behavior of metal components during plastic deformation processes [Bargel and


Schulze, 2008].This microstructural process can be captured phenomenologically by the J2-plasticitytheory on a macroscopic scale by introducing a yield criterion to determine the onset ofplastic flow, a flow rule governing the direction of plastic deformation, and a hardeningrule. For a more detailed description of this approach, the reader is again referred tothe following textbooks: [Simo and Hughes, 1998], [Lubliner, 2008], [Bonet and Wood,2008] and [de Souza Neto et al., 2008].A well-established assumption in the theory of finite inelastic deformations is the mul-tiplicative decomposition of the deformation gradient F into its elastic part Fe and itsplastic part Fp (figure 2.2):

F = Fe · Fp . (2.92)

dx

dxdX

B0 B

B

F

FeFp

Figure 2.2: Multiplicative split of the deformation gradient

This assumption, originating from [Kroner, 1960] and [Lee and Liu, 1967], postulates astress-free intermediate configuration for each material particle, which is reached afterthe application of Fp. In this configuration, the body occupies the space B and aline element is denoted dx. With this assumption at hand, the elastic part of the leftCauchy-Green tensor is defined as

be = Fe · FTe = F ·C−1

p · FT . (2.93)

In the above equation, Cp denotes the plastic part of the right Cauchy-Green tensor.With the help of these kinematics, a strain energy function in terms of the elasticdeformation tensor be and the determinant of Fe, which is called Je, is set up:

Ψ =µ

2

(Ibe

− 3)+κ

4

(J2e − 2 lnJe − 1

)+H(ξ) . (2.94)


This formulation is motivated by the assumption that the intermediate configurationis stress-free. Thus, the mapping Fp does not contribute to the strain energy. H(ξ) isthe hardening function responsible for isotropic hardening (i.e. an increase of the yieldstress during the deformation) and kinematic hardening, which models the Bauschingereffect that appears under cyclic loading conditions.For simplicity, the model used in this thesis is restricted to isotropic hardening, knowingthat the material response for cyclic loading lacks the Bauschinger effect. A quadraticterm depending on the equivalent plastic strain α is chosen:

H(ξ) = H(α) =K

2α2 . (2.95)

The material constant K is the isotropic hardening modulus. The thermodynamicdriving force for α is called h and is defined as follows:

h = −∂Ψ∂α

= −Kα . (2.96)

To decide whether plastic deformation takes place, the following von-Mises-type yieldcriterion, which compares the equivalent stress to the current yield stress, can be used:

Φ = τvM − (τy − h) ≤ 0 . (2.97)

Here, τy is the initial yield stress and τvM is the equivalent (or von-Mises) Kirchhoffstress:

τvM =

√3

2dev τ : dev τ . (2.98)

This particular choice of the equivalent stress is based on macroscopic observations inmetal plasticity which indicate that plastic deformations are isochoric. Consequently,the hydrostatic stress is excluded from the formulation. Additionally, the yield condi-tion (2.97) implies that the material behavior remains isotropic even for plastic defor-mations, as the yield function depends on an invariant of the stress deviator.The condition Φ = 0 is often called yield surface as it describes the bounds of the elas-tic domain in the stress space. A stress state fulfilling the yield condition (2.97) withequity (and thus lying on the yield surface) induces plastic flow causing the plastic de-formation gradient Fp to change. This change of plastic deformation is path dependentand requires evolution equations to determine the amount and the direction of plasticflow. Due to the increasing yield stress as a result of hardening, the elastic domainbounded by the yield surface is enlarged and the yield criterion changes during plasticdeformations as well.The evolution equations for the plastic deformation Fp and the internal variable αhave to be in accordance with the entropy inequality (2.62). Following the Coleman-Noll procedure once again, the following relations are derived (see previous section for


comparison):

τ = 2∂Ψ

∂be

· be (2.99)

Ψ = τ : l+∂Ψ

∂be

:(F · C−1

p · FT)+∂Ψ

∂αα (2.100)

D = τ :

(−1

2F · C−1

p · FT · b−1e

)+ hα > 0 . (2.101)

Please note that the stress tensor τ is defined in a similar manner compared to thehyperelastic case (2.78). This isomorphism justifies the kinematic assumption (2.92).In this work, only associative plasticity is considered. This means that the yield condi-tion Φ serves as a potential for the evolution of the plastic quantities. This postulatesthe evolution equations for the plastic variables as follows:

−1

2F · C−1

p · FT · b−1e = γ

∂Φ

∂τ=

√3

2

dev τ√dev τ : dev τ

γ (2.102)

α = γ∂Φ

∂h= γ . (2.103)

This formulation renders a rate independent model of elasto-plasticity as the evolutionof inelastic quantities is independent of any deformation velocity. In experiments,visco-plastic behavior in which the yield stress depends on the loading velocity canbe observed. However, since only quasi-static loading is considered in this thesis, theassumption of a rate independent model is justified.Equation (2.103) shows that the plastic multiplier γ can be interpreted as the rate ofthe equivalent plastic strain α. Equation (2.102) can be further reduced to

F · C−1p = −2γ

∂Φ

∂τ· F ·C−1

p . (2.104)

Since the yield criterion Φ depends on the stress deviator rather than on the stresstensor, plastic yielding takes place only in deviatoric directions. This means that, inaccordance with experimental observations, plastic deformations neither cause a changein volume nor cause a change of the volumetric material response. Knowing this, thedeterminant Je in equation (2.94) can be replaced by J , which gives

J = JeJp = Je . (2.105)

Isotropic Hyperelasto-plasticity including damage

During severe plastic deformations, not only sliding of dislocations happens. Due to themicrostructure of the material, deformations that may be considered homogeneous ona macroscopic scale may be recognized as heterogeneous when looking through a micro-scope. This leads to high localized strains causing ruptures in the material structure.As a consequence, these microstructural cracks and voids influence the macroscopic be-havior. In continuum mechanics, the material behavior that lies in between perfectly


intact and completely fractured material can be described by continuum damage me-chanics. This phenomenological approach is similar to the phenomenological plasticityapproach described earlier: Several internal variables are introduced to describe theinfluence of the deformation history. These damage variables adapt the stress tensorthat would act on a perfectly intact area element to the real, partly ruptured and thussmaller, remaining area element. Further details as well as an overview of differentexisting damage models are given in [Lemaıtre and Desmorat, 2005] and [Murakami,2012]. The specific model used in this work is a combination of the models describedin [Lemaıtre, 1984] and [Lemaıtre, 1985], and in [Steinmann et al., 1994].For isotropic material behavior, it is sufficient to introduce only one scalar damage vari-able D, which is equal to zero for intact material and equal to one for entirely damagedmaterial. In the present model, damage evolution is coupled with the evolution of theplastic deformation. Since metals are yielding plastically only in deviatoric directions(see previous section), the damage affects only the deviatoric part of the strain energyfunction (see [Steinmann et al., 1994]), which results in a decreased capability of thematerial to bear shear loads.

Ψ =µ

2(1−D)

(Ibe

− 3)+κ

4

(J2 − 2 lnJ − 1

)+K

2α2 . (2.106)

The stress tensor then reads

τ = (1−D)µJ− 23 devbe +

κ

2(J2 − 1)1 . (2.107)

This equation clearly shows that the material fails only in deviatoric directions, whichis acceptable for the deformations occuring during metal forming processes.The rate Ψ is then determined by

Ψ = τ : l+∂Ψ

∂be

:(F · C−1

p · FT)+∂Ψ

∂αα +

∂Ψ

∂DD . (2.108)

With the energy release rate Y defined as

Y = − ∂Ψ

∂D=µ

2

(Ibe

− 3)

, (2.109)

the dissipation inequality renders

D = τ :

(−1

2F · C−1

p · FT · b−1e

)+ hα + Y D > 0 . (2.110)

As the stresses computed in equation (2.107) are not the stresses acting on the remain-ing intact area element, but are rather a mean value over the whole area element, theyield criterion (2.97) is modified:

Φ =τvM1−D

− (τy − h) . (2.111)

This way, the stresses acting on the intact material are treated as a cause for plasticflow. Again, associative plasticity is considered:

F · C−1p = −2γ

∂Φ

∂τ· F ·C−1

p (2.112)

2.2. GENERALIZED CONTINUA 25

∂Φ

∂τ=

√3

2

1

1−D

dev τ√dev τ : dev τ

(2.113)

h = −Kα (2.114)

α = γ . (2.115)

It can be seen that there is a difference to the elastic-plastic model without damageas the damage parameter D is present in the evolution equations. Still, an evolutionequation for D itself has to be introduced. In this model, the energy release rate Y ischosen as the thermodynamic driving force for damage evolution. A quadratic potentialΦD, including the material parameter Y0, is defined to describe the evolution:

ΦD =Y0

2(1−D)

(Y

Y0

)2

(2.116)

D = γ∂ΦD

∂Y= γ

Y

(1−D)Y0. (2.117)

In this model, damage occurs not only under tensile loading but also under compressionas it depends on the deviatoric part of the elastic energy.

2.2 Generalized Continua

The standard continuum theories provide reliable results for problems where the size ofthe investigated body is large compared to the characteristic length of its microstruc-ture. However, if a body is about the same size as its microstructure, the sensitivityof the material response with respect to the size of the body will be significant. Thus,size-effects which cannot be captured within the standard continuum theories can bemeasured: In [Taylor, 1924] the manufacturing process of thin metal wires was investi-gated, and it was found that thinner wires had a greater tensile strength than thickerones, although both were made of the same material. Further research by [Brenner,1956] obtained qualitative results for this effect.Size-effects can be predicted by generalized continuum models, which share the presenceof additional kinematic variables contributing to the energy stored during a deformationprocess. Consequently, additional stress-like variables (often called couple stresses orhigher-order stresses), which need an additional balance equation, exist. An overviewcan be found in [Eringen, 1999], [Hirschberger, 2008], [Maugin and Metrikine, 2010],and [Altenbach et al., 2011]. A classification of various inelastic generalized theories isgiven in [Forest and Sievert, 2003] and [Hirschberger and Steinmann, 2009].In this section, a short introduction to several generalized frameworks is given beforethe focus is set on the micromorphic-type inelastic continuum proposed in [Forest,2009]. Extensions to inelastic finite deformation theory can be found in [Clasen andHirschberger, 2012].


2.2.1 Higher Gradient Continua

A first approach to a continuum description that includes size-effects is the incorpora-tion of higher gradients. The Taylor-expansion of the placement x yields the following:

∆x ≈ F ·∆X+1

2∆X · ∂F

∂X·∆X . (2.118)

With the abbreviation G = ∂F/∂X and omitting terms of even higher order this leadsto a second gradient theory. In case of hyperelasticity, the free energy (which does notonly depend on strains for generalized continua) takes the following form:

Ψ = Ψ (F,G) . (2.119)

For small deformations, a third gradient theory (including the second gradient theoryas a special case) was introduced by [Mindlin, 1965]. Over the decades, this model hasbeen extended to finite deformations and inelastic constitutive models. For an overviewof the literature, the reader is referred to [Bertram and Forest, 2013] and referencestherein.

2.2.2 Micropolar continua

A second approach, which is even older than the higher gradient continuum, is theso-called micropolar continuum described in [Cosserat and Cosserat, 1909]. In sucha continuum, every material particle has the standard three translational degrees offreedom and additionally three rotational degrees of freedom. It is assumed that thereis a microcontinuum attached to each material particle, occupying the space B0 in thereference configuration and B in the current configuration. The motion ϕ transformsone configuration into the other one (figure 2.3). The microplacement X is then affinelytransformed to the current configuration by the proper orthogonal tensor R:

x = R · X . (2.120)

The material gradient of R describes the change of the microrotation between thematerial particles:

G =∂R

∂X. (2.121)

In a hyperelastic micropolar continuum, the free energy depends on F, R, and itsgradient G:

Ψ = Ψ (F, R, G) . (2.122)

The formulation of inelastic micropolar media can be found in [de Borst, 1993] and[Steinmann, 1995].


x

X

u

B0

B

ϕ

χ

χ0 P

E

ϕ

B0

BX

x

Figure 2.3: Micropolar motion

2.2.3 Micromorphic continua

A more general framework, which is an extension of the micropolar continuum, is themicromorphic continuum. Contrary to the micropolar formulation, the microdefor-mation ϕ is not restricted to rigid-body-rotations. Thus, a full deformation gradienttensor F is introduced, transforming the microplacement X to the current configuration(figure 2.4):

x = F · X (2.123)

G =∂F

∂X. (2.124)

For micromorphic hyperelasticity the following free energy is postulated:

Ψ = Ψ (F, F, G) . (2.125)

This theory is based on the work of Eringen and Suhubi ([Eringen and Suhubi, 1964]and [Suhubi and Eringen, 1964]). Applications for finite deformation theory are givenin [Hirschberger et al., 2008] for elastic material layers and in [Regueiro, 2010] forelastic-plastic problems.

2.2.4 Micromorphic-type inelastic continua

By the introduction of a special kind of non-dissipative microdeformation ξ, which iscoupled with inelastic quantities ξ (e.g. the plastic deformation) as described in [Forest,2009], the micromorphic-type inelastic continuum is created. Let

ζ =∂ξ

∂X(2.126)


x

X

u

B0

B

ϕ

χ

χ0 P

E

ϕ

B0 B

Xx

Figure 2.4: Micromorphic motion

be the material gradient of the inelastic microdeformation. Then, the free energy issplit into two contributions Ψmeso and Ψ :

Ψ = Ψmeso(be, ξ) + Ψ (ξ, ζ, ξ) . (2.127)

Consequently, the dissipation D is enlarged in the following way:

D = τ : l + a · ˙ξ + b : ˙ζ − Ψ ≥ 0 . (2.128)

In this equation, a and b are stress-like quantities which are power-conjugated to ξ

and ζ respectively. Adapting the terminology from other generalized theories, thesequantities are referred to as couple stresses and double stresses respectively.Inserting the rate of the free energy

Ψ =∂Ψ

∂be

: be +∂Ψ

∂ξ· ξ +

∂Ψ

∂ξ· ˙ξ +

∂Ψ

∂ζ: ˙ζ , (2.129)

the following relations are deduced:

τ = 2∂Ψmeso

∂be

· be (2.130)

h = −∂Ψmeso

∂ξ− ∂Ψ

∂ξ(2.131)

a =∂Ψ

∂ξ(2.132)

b =∂Ψ

∂ζ. (2.133)


Although these results are very similar to the ones in section 2.1.3, it is obvious thatthe driving force for hardening h depends on the standard quantities as well as on themicrodeformation fields. With these abbreviations, the dissipation inequality reducesto:

D = τ :

(−1

2F · C−1

p · FT · b−1e

)+ h · ξ ≥ 0 . (2.134)

Since the potentials Ψmeso and Ψ are decoupled in the primary variables u and ξ, theDirichlet-principle is applied only to Ψ to render the governing balance equation forthe additional field ξ. This principle states that the variation of the stored energy hasto be equal to the variation of the energy provided to the system. Neglecting dynamicsand body forces and introducing the double stress vector t0, the following equationsare obtained:

∫

B0

δΨ dV =

∫

∂B0

t0 · δξ dA (2.135)

∫

B0

(∂Ψ

∂ξ· δξ +

∂Ψ

∂ζ: δζ

)dV =

∫

∂B0

t0 · δξ dA (2.136)

∫

B0

(a · δξ + b : δζ) dV =

∫

∂B0

t0 · δξ dA . (2.137)

Inserting the relation

Div(δξ · b

)= Div b · δξ + b : δζ , (2.138)

using the divergence theorem, and postulating Cauchy’s theorem for the double stressesas

b ·N = t0 (2.139)

leads to the balance of the microforces in global and local form:

∫

B0

(a− Div b

)dV = 0 (2.140)

a− Div b = 0 . (2.141)

For Ψmeso, a strain energy as already described in the previous sections can be assumed,whereas Ψ consists of a term coupling the fields ξ and ξ via the material parameter Hand a term including the gradient ζ and a second additional material parameter A. Aquadratic function in the form of

Ψ =H

2

(ξ − ξ

)·(ξ − ξ

)+A

2ζ : ζ (2.142)

is chosen. Consequently, a Helmholtz-type equation is obtained for the microdeforma-tion, where ∆X is the Laplace-operator with respect to the reference configuration:

ξ − A

H∆Xξ = ξ . (2.143)


This is the same kind of partial differential equation as in various gradient enhancedtheories (e.g. [Peerlings et al., 1996], [Engelen et al., 2003], and [Peerlings et al., 2012])derived from nonlocal continuum theories. However, equation (2.143) is only obtained,if the quadratic potential in equation (2.142) is chosen. Thus, the thermodynamicallyconsistent micromorphic-type framework is more flexible than the gradient enhancedtheory.In the gradient dependent models mentioned above, the homogeneous Neumann bound-ary condition

b ·N = t0 = 0 (2.144)

is assumed, which is adopted in this thesis. In the case of the quadratic potential(2.142), the relation

∫

B0

ξ dV =

∫

B0

ξ dV (2.145)

is obtained. This shows that the amount of inelastic distortion in homogeneous defor-mations is the same as in a standard inelastic continuum, which justifies assumption(2.144). This choice of boundary conditions will be crucial for the development of thehomogenization framework in chapter 5.

Micromorphic-type hyperelasto-plasticity

The first application of the theory mentioned above is the extension of the isotropichyperelastic-plastic model described in section 2.1.3. To avoid a large number of pri-mary variables, only a scalar microfield α is introduced, which is coupled with theequivalent plastic strain α. More sophisticated models arise if an additional tensorialfield is coupled with the inelastic deformation as described in [Poh et al., 2011]. How-ever, the present formulation is sufficient to predict size-effects in the inelastic regimewhile keeping computational costs low.The strain energy is chosen as follows:

Ψ = Ψmeso + Ψ (2.146)

Ψmeso =µ

2

(Ibe

− 3)+κ

4

(J2 − 2 lnJ − 1

)+K

2α2 (2.147)

Ψ =H

2(α− α)2 +

A

2β · β , (2.148)

with β being the material gradient of α:

β =∂α

∂X. (2.149)


Following the Coleman-Noll procedure once again, the thermodynamic driving forcesare derived:

τ = 2∂Ψmeso

∂be

· be = µJ− 23 devbe +

κ

2(J2 − 1)1 (2.150)

h = −∂Ψmeso

∂α− ∂Ψ

∂α= −Kα − H(α− α) (2.151)

a =∂Ψ

∂α= −H(α− α) (2.152)

b =∂Ψ

∂β= Aβ . (2.153)

The yield condition remains almost the same, with the difference that an additionalterm appears in the hardening h:

Φ = τvM − (τy − h) ≤ 0 . (2.154)

The evolution equations for the inelastic quantities Cp and α remain unchanged sincethe derivatives of Φ remain unchanged (see equations (2.102) and (2.103)). Again, thedissipation differs from the standard model as an additional term in h is present (seeequation (2.101)).This framework is the extension to finite deformations of the scalar microstrain gradientplasticity model provided by [Forest, 2009].

Micromorphic-type hyperelasto-plasticity including damage

As proposed in [Clasen and Hirschberger, 2012], the elastic-plastic continuum damagemodel from section 2.1.3 is extended to a micromorphic-type formulation by introducingthe additional scalar microdamage field D and its material gradient E:

E =∂D

∂X. (2.155)

Then, the following free energy emerges:

Ψ = Ψmeso + Ψ (2.156)

Ψmeso =µ

2(1−D)

(Ibe

− 3)+κ

4

(J2 − 2 lnJ − 1

)+K

2α2 (2.157)

Ψ =H

2

(D − D

)2+A

2E · E . (2.158)


The power-conjugate quantities then read:

τ = 2∂Ψmeso

∂be

· be = (1−D)µJ− 23 devbe +

κ

2(J2 − 1)1 (2.159)

h = −∂Ψmeso

∂α= −Kα (2.160)

Y = −∂Ψmeso

∂D− ∂Ψ

∂D=µ

2

(Ibe

− 3)− H(D − D) (2.161)

a =∂Ψ

∂D= −H(D − D) (2.162)

b =∂Ψ

∂E= AE . (2.163)

Again, the formulae for the evolution equations and the dissipation (equations (2.110) –(2.117)) remain unchanged except for the presence of an additional term in the energyrelease rate Y .

Chapter 3

Finite Element Method

Analytical solutions to the governing partial differential equations (PDEs) explained inthe previous chapter exist only for special cases of bodies and boundary conditions. Ingeneral mechanical problems, the balance equations are solved with the help of com-putational methods. Over the decades, accompanied by the development of more andmore powerful computers, the Finite Element Method (FEM), developed in the 1960s,evolved into a powerful, robust tool. The general procedure within this method is thediscretization of the domain into disjoint subdomains with finite sizes. These subdo-mains are called elements. By the discretization of the domain, the weak form of aPDE is transformed into a generally nonlinear equation system. This equation systemis then commonly solved by iterative solution procedures including linearizations of thenonlinear equation system.For an introduction into the method and its origins, the reader is referred to the text-books [Bathe, 1996], [Hughes, 2000], and [Zienkiewicz et al., 2005]. In [Wriggers, 2008],focus is set on nonlinear analyses. Applications of the FEM to elastic-plastic problemsare given in [Bonet and Wood, 2008] and [de Souza Neto et al., 2008]. The foundationsfor the variational principles used to derive the weak formulations are explained indetail in [Washizu, 1982].In this chapter, the Finite Element Method is decribed as it is necessary for the fol-lowing chapters of this thesis. Therefore, the variational foundations and discretizedequations are explained. The Newton-Raphson solution procedure for solving the ob-tained nonlinear equation system is discussed. Since the author follows the numeric-symbolic approach as described in [Korelc, 2009a], focus is set on the formulation ofa nonlinear Finite Element, in which the linearizations are carried out by automaticdifferentiation so that no analytical expressions for tangent matrices are required. Forthis the software package AceGen by Korelc ([Korelc, 2009b]) is used.

3.1 Variational principles

Since it is not possible to fulfill the balance equations at every material point (whichwould be an exact solution), and fulfilling the balance equations in particular pointsonly (which is called collocation method) might lead to spurious results, the PDEs are

33

34 CHAPTER 3. FINITE ELEMENT METHOD

solved in their weak forms. This is the so-called Galerkin approach of weighted resid-uals. For the derivation of the weak forms, variational principles are applied, reducingthe order of the PDE. Two of these principles are investigated here, namely the prin-ciple of virtual work and the Hu-Washizu-Principle, leading to standard displacementelements and constant pressure elements respectively.

3.1.1 Principle of virtual work

In chapter 2, the following balance equations were derived:

DivP = 0 (3.1)

Div b− a = 0 . (3.2)

Together with appropriate boundary conditions, a boundary value problem is set up:

u− upre = 0 on ∂Bu0 (3.3)

t0 − tpre0 = 0 on ∂Bt0

0 (3.4)


0 . (3.5)

Here, ∂Bu0 , ∂Bt0

0 , and ∂Bt00 are the respective boundaries for the described displacements

upre, tractions tpre0 and double tractions t

pre0 , which do not intersect. To obtain the

weak forms of the balance equations, they are multiplied by the variations of theirrespective primal variables, which fulfill the essential boundary condition (3.3). In anequilibrated state, the internal virtual work is equal to the external virtual work:∫

B0

DivP·δu dV+

∫

B0

(Div b−a)·δξ dV =

∫

∂Bt00

(t0−tpre0 )·δu dA+

∫

∂Bt00

(t0−tpre0 )·δξ dA .

(3.6)Integration by parts and application of Cauchy’s theorem (2.36) and (2.139), and thedivergence theorem lead to∫

B0

(P : δF+ a · δξ + b : δζ

)dV =

∫

∂Bt00

tpre0 · δu dA+

∫

∂Bt00

tpre0 · δξ dA . (3.7)

Assuming tpre0 = 0 (see (2.144)), the right-hand-side is further reduced:

∫

B0

(P : δF+ a · δξ + b : δζ

)dV =

∫

∂Bt00

tpre0 · δu dA . (3.8)

3.1.2 Hu-Washizu principle for incompressibility

Large elastic-plastic deformations are almost volume preserving since, according to theconstitutive models from section 2.1.3, the dominating plastic part of the deformationis purely isochoric. Within the Finite Element Method, incompressible (i.e. volumepreserving) material behavior might lead to spurious results due to volumetric lockingeffects. These effects occur especially when elements with a low interpolation order

3.1. VARIATIONAL PRINCIPLES 35

are used in a standard displacement formulation derived from the principle of virtualwork. If nevertheless linear shape functions are chosen, locking can be avoided by theconstruction of constant pressure elements (also called Q1P0-elements, see [Simo et al.,1985]) with the help of a special form of the Hu-Washizu principle. For this, the Cauchystress tensor is split into deviatoric and volumetric parts as follows:

σ = σ + p1 (3.9)

p =1

3trσ . (3.10)

Here, σ is the stress deviator and p is the hydrostatic pressure. Using relation (2.42),the first Piola-Kirchhoff stress tensor is split as well:

P = P+ pJF−T (3.11)

P = Jσ · F−T . (3.12)

Furthermore, an additional kinematic field θ representing the mean volumetric dilata-tion with

θ =1

V

∫

B0

J dV (3.13)

is introduced, replacing the determinant J in the strain energy function. The followingfree energy can be assumed:

Ψ = Ψmeso(be, ξ) + Umeso(θ) + Ψ(ξ, ζ, ξ) . (3.14)

With this assumption, the hydrostatic stress p is determined as

p =∂Umeso(θ)

∂θ. (3.15)

Then, the equations

DivP = 0 (3.16)

Div b− a = 0 (3.17)

θ − J = 0 (3.18)

determine the problem accompanied by the boundary conditions (3.3) – (3.5).The weak form is constructed in a similar way as the principle of virtual work. Here,an additional term coupling θ and J is present, with p acting as a Lagrange-multiplier:

∫

B0

DivP · δu dV +

∫

B0

(Div b− a) · δξ dV +

∫

B0

(θ − J)δp dV =

∫

∂Bt00

(t0 − tpre0 ) · δu dA+

∫

∂Bt00

(t0 − tpre0 ) · δξ dA . (3.19)


Again, Cauchy’s theorem ((2.36) and (2.139)) and the divergence theorem are appliedto give the following relation:∫

B0

((P+ pJF−T ) : δF+ a · ξ + b : δζ + (J − θ)δp

)dV =

∫

∂Bt00

tpre0 · δu dA .

(3.20)With this variational formulation, the volumetric and deviatoric parts are decoupledand it is possible to interpolate the variables F (and implicitly J) and θ with differentorders within a Finite Element, providing a method which improves the volumetric lock-ing behavior. For other numerical procedures that overcome locking (e.g. F-methods,enhanced strain elements, etc.), the reader is referred to [de Souza Neto et al., 2008]and to [Wriggers, 2008].

3.2 Discretization

Since the evolution equations are time dependent (or rather load step dependent, sincequasi-static deformations are considered), the problem has to be discretized not onlyin space but also in time. First, the spatial discretization is discussed. Afterwards, theevolution equations are approximated over a time (or load) step.

3.2.1 Discretization in space

The space occupied by the body is discretized with Finite Elements with convex shapeswhich share boundary areas and boundary edges but do not penetrate each other (seefigure 3.1). Points on the corners of elements are called nodes.

∂B0

∂Bh0

Element boundary

Ωe

Node

Figure 3.1: FE-discretization of a domain

The space B0 is approximated by the discretized space Bh0 composed of the Finite

Element domains Ωe:

B0 ≈ Bh0 =

ne⋃

e=1

Ωe , (3.21)

3.2. DISCRETIZATION 37

where e is the number of the respective element and ne is the overall number of elements.Within the isoparametric concept, the primary variables in each element u and ξ aswell as the placement X are approximated by the nodal values uI , ξI and XI and theshape functions NI :

u ≈ uh =

nn∑

I=1

NIuI (3.22)

ξ ≈ ξh=

nn∑

I=1

NI ξI (3.23)

X ≈ Xh =

nn∑

I=1

NIXI . (3.24)

Here nn is the number of nodes per element. To ensure the correct representation of arigid body movement the partition of unity has to be fulfilled by the shape functions.

nn∑

I=1

NI = 1 . (3.25)

Within the Bubnov-Galerkin approach, the variations of the primal variables are ap-proximated in a similar manner with the same shape functions:

δu ≈ δuh =nn∑

I=1

NIδuI (3.26)

δξ ≈ δξh=

nn∑

I=1

NIδξI . (3.27)

Since the spatial behavior of the field quantities is now described by the shape functions(the nodal values are independent of the coordinates), the derivatives of the primary

variables uh and ξhwith respect to X are carried out in the following way:

∂uh

∂X=

nn∑

I=1

uI ⊗∂NI

∂X(3.28)

∂ξh

∂X=

nn∑

I=1

ξI ⊗∂NI

∂X. (3.29)

For the formulation of the shape functions NI , a local coordinate system is introduced.Within this approach, the shape functions depend on the coordinates X of a referenceelement occupying a unit space with

NI = NI(X) (3.30)

−1 ≤ X ≤ 1 (3.31)

−1 ≤ Y ≤ 1 (3.32)

−1 ≤ Z ≤ 1 , (3.33)


I

I

II

II

IIIIII IVIV

V V I

V IIV III

XX YY

Z

(−1/− 1) (1/− 1)

(1/1)(−1/1)

(−1/− 1/− 1) (1/− 1/− 1)

(1/1/− 1)(−1/1/− 1)

(−1/− 1/1) (1/− 1/1)

(−1/1/1) (1/1/1)

Figure 3.2: Isoparametric elements

as shown in figure 3.2 for a two-dimensional bilinear and for a three-dimensional trilin-ear element.A line element of a reference element is then mapped to a line element of the actualFinite Element in the reference configuration via the Jacobian J defined as

J =∂Xh

∂X=

nn∑

I=1

XI ⊗∂NI

∂X. (3.34)

For the derivatives with respect to the placement X the chain rule is applied:

∂uh

∂X=

nn∑

I=1

uI ⊗(J−T · ∂NI

∂X

). (3.35)

The derivative of ξ can be computed likewise.The shape functions of Finite Elements of the Lagrangian type can be constructedby Lagrangian polynomials. For a one-dimensional element (only the coordinate Xexists), the shape functions have the following form:

NI =nn∏

J=1,n; J 6=I

XJ − X

XJ − XI

. (3.36)

Thus, a two-node linear element has the two shape functions

N1 =1

2(1− X) (3.37)

N2 =1

2(1 + X) . (3.38)

For the bilinear four-node quadrilateral (Q1) and the trilinear eight-node hexahedral(H1) element depicted in figure 3.2, the shape functions are obtained by multiplication

3.2. DISCRETIZATION 39

of the Lagrangian polynomials for the respective coordinates. For the Q1-element, theshape functions are

N1 =1

4(1− X)(1− Y ); N2 =

1

4(1 + X)(1− Y ) (3.39)

N3 =1

4(1 + X)(1 + Y ); N4 =

1

4(1− X)(1 + Y ) , (3.40)

whereas the H1-element contains the shape functions

N1 =1

8(1− X)(1− Y )(1− Z); N2 =

1

8(1 + X)(1− Y )(1− Z) (3.41)

N3 =1

8(1 + X)(1 + Y )(1− Z); N4 =

1

8(1− X)(1 + Y )(1− Z) (3.42)

N5 =1

8(1− X)(1− Y )(1 + Z); N6 =

1

8(1 + X)(1− Y )(1 + Z) (3.43)

N7 =1

8(1 + X)(1 + Y )(1 + Z); N8 =

1

8(1− X)(1 + Y )(1 + Z) . (3.44)

This way, the shape functions obtain the Kronecker-delta-property

NI(XJ) = δIJ . (3.45)

The Q1P0-element (or H1P0-element in three dimensions) arises if the Hu-Washizuprinciple is applied and the displacements (and in this case the micromorphic-typefield) are interpolated with linear shape functions and the variables related to thepressure, p and θ, are treated as constants over the element. Please note that theQ1P0-element derived in this way does not fulfill the Ladyzhenskaya-Babuska-Brezzicondition ([Ladyzhenskaya, 1969], [Babuska, 1973], [Brezzi, 1974], [Bathe, 2001]) fora given element patch and boundary conditions in general ([Bathe, 1996]) and thatnon-physical pressure modes (often referred to as checkerboard-modes) might occur. Aremedy to this problem via postprocessing methods is mentioned in [Wriggers, 2008].For the applications in this work however, the obtained solutions from the Q1P0-element without further post-processing steps display a physically meaningful behavior(see chapter 4).

3.2.2 Discretization in time

The evolution equations for the inelastic quantities C−1p , γ, and D are time (or load

step) dependent. To solve these evolution equations incrementally, they are discretizedin time and solved with a backward Euler update procedure. For this, the quantities ofthe current time step ∆t are denoted by a subscript n+ 1 and the quantities from theprevious time step (which are known as they are either initial conditions or calculatedin the previous time step) are denoted by the subscript n. For quantities which appearwithout a subscript indicating the time step, the subscript n+1 is skipped for brevity.


The rates of the scalar inelastic quantities α, γ, and D are approximated by

γ ≈ γn+1 − γntn+1 − tn

=∆γ

∆t(3.46)

α ≈ αn+1 − αn

∆t(3.47)

D ≈ Dn+1 −Dn

∆t. (3.48)

With this, the evolution equations (2.103) and (2.117) have the following form:

αn+1 − (αn +∆γ) = 0 (3.49)

Dn+1 −(Dn +∆γ

∂ΦD

∂Y

∣∣∣∣n+1

)= 0 . (3.50)

The more complex evolution equation for the plastic deformation (2.104) is discretizedwith the exponential mapping developed in [Simo, 1992], which leads to

F ·(C−1

p

)n+1

− exp

(−2∆γ

∂Φ

∂τ

∣∣∣∣n+1

)· F ·

(C−1

p

)n= 0 . (3.51)

3.3 Numerical integration

In the weak forms of equilibrium (3.8) and (3.20), integrals over the domain have to becomputed. Since the domain is composed of elements, the integrals over the domaincan be computed by integration of the elements:

∫

B0

(. . . ) dV ≈∫

Bh0

(. . . ) dV =ne⋃

e=1

∫

Ωe

(. . . ) dV . (3.52)

Within the isoparametric concept, the integral over a single element is then computedin the reference space Ω rather than in the actual element. With dV = dXdY dZ,one obtains ∫

Ωe

(. . . ) dV =

∫

Ω

(. . . ) detJ dV . (3.53)

In general, no analytical expression exists for this integral. Thus, a numerical integra-tion scheme has to be applied. In the Finite Element Method, the Gauss-integration isvery common. With this scheme, the integral of a function f(X) is approximated bythe values of the function at the Gauss-points Xg and the weighting factors wg:

∫

Ω

f(X) dV ≈ng∑

g=1

f(Xg)wg . (3.54)

3.4. SYMBOLIC-NUMERIC SOLUTION SCHEME 41

For the Q1-element with a 2x2 integration scheme (two Gauss-points in each direction),the Gauss-point coordinates are the following:

X1 =

(− 1√

3/− 1√

3

); X2 =

(1√3/− 1√

3

)(3.55)

X3 =

(− 1√

3/

1√3

); X4 =

(1√3/

1√3

). (3.56)

The weighting factors are

w1 = w2 = w3 = w4 = 1 . (3.57)

Commonly, the H1-element is integrated with a 2x2x2 scheme with the Gauss-pointcoordinates

X1 =

(− 1√

3/− 1√

3/− 1√

3

); X2 =

(1√3/− 1√

3/− 1√

3

)(3.58)

X3 =

(− 1√

3/

1√3/− 1√

3

); X4 =

(1√3/

1√3/− 1√

3

)(3.59)

X5 =

(− 1√

3/− 1√

3/

1√3

); X6 =

(1√3/− 1√

3/

1√3

)(3.60)

X7 =

(− 1√

3/

1√3/

1√3

); X8 =

(1√3/

1√3/

1√3

). (3.61)

Here again, the weighting factors are all equal to one:

w1 = w2 = w3 = w4 = w5 = w6 = w7 = w8 = 1 . (3.62)

For other Gauss-point distributions for the Q1- and H1-elements and for higher orderelements, the reader is referred to [Wriggers, 2008].In two-dimensional analyses, a constant sheet thickness s is assumed for the entireproblem. Consequently, a volume integral reduces to

∫

B0

(. . . ) dV = s

∫

B0

(. . . ) dA . (3.63)

The area integral is then evaluated by the numerical integration scheme as describedbefore, whereas the thickness s appears on both sides of every equation. Thus, itplays no vital role in the Finite Element solution. Knowing this, only volume integralswill be mentioned in the remainder of this thesis even though the integrals that willbe evaluated during the calculations of two-dimensional problems actually are areaintegrals.

3.4 Symbolic-numeric solution scheme

In this section, the solution procedure for the discretized problem with the symbolicapproach by [Korelc, 2009a] is pointed out. This procedure is especially well suited


for the use of automatic differentiation procedures which are used within the softwarepackage AceGen [Korelc, 2009b].First, the procedure is described for the weak form derived from the principle of virtualwork (3.8). For this, the weak form is rewritten to give an expression for the globalresidual vector G:

G · δp = (Rint −Rext) · δp = 0 (3.64)

G = Rint −Rext = 0 . (3.65)

In this equation, the primary variables u and ξ are assembled in the vector p. With

δF =∂F

∂p· δp (3.66)

δξ =∂ξ

∂p· δp (3.67)

δζ =∂ζ

∂p· δp , (3.68)

Rint and Rext emerge as

Rint =

∫

B0

(P :

∂F

∂p+ a · ∂ξ

∂p+ b :

∂ζ

∂p

)dV (3.69)

Rext =

∫

∂Bt00

tpre0 · ∂u

∂pdA . (3.70)

The discretized form of these quantities reads as follows:

G ≈ Gh = Rhint −Rh

ext = 0 (3.71)

Rhint =

ne⋃

e=1

(Rint)e =

ne⋃

e=1

∫

Ωe

(P :

∂F

∂pe

+ a · ∂ξ∂pe

+ b :∂ζ

∂pe

)dV (3.72)

Rhext =

nr⋃

r=1

∫

∂Ωt0r

tpre0 · ∂u

h

∂pe

dA , (3.73)

where r is the index of the elements forming the boundary of the discretized domain.

The vector pe contains all nodal values of the primary variables uh and ξh.

In the case of inelastic problems, a yield condition Φ = 0, depending on internalvariables C−1

p , α, γ, and, in case of damage, D, which are assembled in the vector h,has to be fulfilled at each Gauss point. For this reason, the algorithm is described ateach Gauss point with the inelastic variables hg and the contributions to the residual(Rint)g. Within the backward Euler update procedure, a load step is assumed to beelastic and is corrected if the yield condition is violated by the elastic trial stresses.Consequently, the trial yield function Φtr

g depends on the inelastic variables of the lastload step (hg)n:

Φtrg = Φg(Fg, ξg, (hg)n) . (3.74)


If Φtrg ≤ 0, the load step is elastic and the updated internal variables are

(hg)n+1 = (hg)n . (3.75)

Accordingly, the stress measures can be determined by

Pg =∂Ψg(Fg, ξg, ζg, (hg)n+1 , (hg)n)

∂Fg

(3.76)

ag =∂Ψg(Fg, ξg, ζg, (hg)n+1 , (hg)n)

∂ξg

(3.77)

bg =∂Ψg(Fg, ξg, ζg, (hg)n+1 , (hg)n)

∂ζg

. (3.78)

For an inelastic load step, the internal variables evolve according to their respectiveevolution equations. These, together with the yield condition Φg = 0, are taken in theirimplicit forms to assemble the nonlinear equation system

Qg((pe)n+1 , (hg)n+1 , (pe)n , (hg)n) = 0 . (3.79)

This generally nonlinear equation system is solved with a local Newton-Rapshon pro-cedure within the Finite Element routine. The linearization of this equation systemwith respect to (hg)n+1 leads to the following iteration rule:

((hg)n+1

)0= (hg)n (3.80)

(Ag)j =∂ (Qg)j

∂((hg)n+1

)j

(3.81)

∆ (hg)j = − (Ag)−1j

· (Qg)j (3.82)((hg)n+1

)j+1

=((hg)n+1

)j+∆(hg)j . (3.83)

After a desired tolerance (∆ (hg)j · ∆(hg)j < TOL) is reached, the updated inelasticvariables are obtained with an implicit dependency on (pe)n+1. To determine thisdependency, the total derivative of (3.79) with respect to (pe)n+1 is computed:

∂Qg

∂ (pe)n+1

+∂Qg

∂ (hg)n+1

·D (hg)n+1

D (pe)n+1

= 0 (3.84)

D (hg)n+1

D (pe)n+1

= −A−1g · ∂Qg

∂ (pe)n+1

. (3.85)

In an automatic differentiation framework, this dependency would lead to a wrongcomputation of the stress tensor. Thus, relation (3.76) is modified such that only onerule capturing both, elasticity and inelasticity, is obtained. Here, the symbol δ is thesymbol for the computational derivative used in automatic differentiation schemes (see[Korelc, 2009a]).

Pg =δΨg(Fg, ξg, ζg, (hg)n+1 , (hg)n)

δFg

∣∣∣∣D(hg)n+1D(pe)n+1

=0

. (3.86)


After the equation for the dependent residual Qg is solved, the inner contribution tothe element residual can be computed. The insertion of relations (3.86), (3.77), and(3.78) into (3.72) and the application of the chain rule lead to

(Rint)g =δΨg

δ (pe)n+1


=0

. (3.87)

The Gauss point integration then leads to the element residual (Rint)e:

(Rint)e =

ng∑

g=1

detJ(Xg) (Rint)g wg . (3.88)

With this relation at hand, the equation system (3.71) is completely determined. Dueto finite deformation kinematics and inelastic material behavior, this equation systemis highly nonlinear. Thus, the residual Gh is linearized with respect to the primaryvariables p, such that the Newton-Raphson procedure can be applied:

(pn+1)0 = pn (3.89)

Ki =DGh

i

D (pn+1)i(3.90)

∆pi = −K−1i ·Gh

i (3.91)

(pn+1)i+1 = (pn+1)i +∆pi . (3.92)

Similar to the local Newton-Raphson procedure, the iteration is stopped after ∆pi ·∆pi

is smaller than a certain value. The stiffness matrix K is obtained by assembling theelement stiffness matrices:

K =

ne⋃

e=1

D

D (pe)n+1

(Rint)e =

ne⋃

e=1

Ke . (3.93)

This equation is only valid if the external loads tpre0 are conservative as it is assumedin this thesis. Otherwise, the external contribution to the element residual (Rext)e hasto be linearized as well.Exploiting the chain rule, Ke is determined by

Ke =∂ (Rint)e∂ (pe)n+1

+

ni∑

g=1

∂ (Rint)e∂ (hg)n+1

·D (hg)n+1

D (pe)n+1

, (3.94)

where ni is the number of inelastic Gauss points in the element. Inserting relation(3.85) yields

Ke =∂ (Rint)e∂ (pe)n+1

−ni∑

g=1

∂ (Rint)e∂ (hg)n+1

·A−1g · ∂Qg

∂ (pe)n+1

. (3.95)

Consequently, the automatic differentiation rule for the contribution of a Gauss pointto the element stiffness matrix is

Kg =δ (Rint)g

δ (pe)n+1


=−A−1·∂Q

∂(pe)n+1

, (3.96)


such that

Ke =

ng∑

g=1

detJ(Xg)Kgwg . (3.97)

In the case that the weak form is derived from the Hu-Washizu principle (3.20), thesolution procedure is similar, but additional internal primary variables p and θ exist ineach element. Summarizing these quantities in the vector pe, they can be eliminatedfrom the equation system by static condensation procedures (see [Hughes, 2000] and[de Souza Neto et al., 2008]), as they are defined locally in each element. The globalstiffness matrix used in the iterative solution procedure (3.89) – (3.92) is now assembledfrom element stiffness matrices Ke obtained in a slightly different manner. On elementlevel, the linearization is carried out with respect to a vector which contains pe and pe.The element stiffness matrix then reads

Ke =

[(Ke)pepe

(Ke)pepe

(Ke)pepe(Ke)pepe

], (3.98)

such that

Ke ·[∆pe

∆pe

]= −

[(Rint)e(Rint)e

]. (3.99)

Solving the second equation of this equation system for ∆pe and inserting the resultinto the first equation leads to

(Kcond)e = (Ke)pepe− (Ke)pepe

·((Ke)pepe

)−1

· (Ke)pepe(3.100)

(Rcond)e = (Rint)e − (Ke)pepe·((Ke)pepe

)−1

· (Rint)e (3.101)

− (Rcond)e = (Kcond)e ·∆pe . (3.102)

The quantities (Rcond)e and (Kcond)e are used to assemble the global equation system.After each iteration step, the increments of the internal degrees of freedom can becomputed by

∆pe = −((Ke)pepe

)−1

·((Rint)e + (Ke)pepe

·∆pe

). (3.103)

Chapter 4

Single-scale benchmark tests

The constitutive models described in chapter 2 are implemented in the automatic-differentiation-based Finite Element Method from chapter 3. To prove the efficiencyand robustness of the resulting procedure, several relevant benchmark tests are carriedout. It will be shown that the micromorphic-type continuum is not only able to predictsize-effects but also provides mesh regularization in case of softening.For the constitutive models of choice, it is possible that material instabilities occur dueto a negative hardening modulus K or a large amount of damage. This causes troublein the standard FEM as the governing PDE divσ = 0 loses its ellipticity. This leads toa complete failure of the element in which the instability occurs without affecting thesupport of this element. This pathological localization behavior cannot be remedied bymesh refinements. However, in the continuum models used throughout this thesis, theadditional balance equation Div b = a provides the possibility of mesh regularizationas it includes the surrounding elements in the material behavior via the gradient ζ andthe microstiffness A.Investigations on the regularization characteristics of inelastic generalized continua goback to [de Borst, 1991] and can further be found in [Peerlings et al., 1996], [de Borstet al., 1999] and [Engelen et al., 2003]. Recent developments in localization analysisare shown in [Poh et al., 2011] [Clasen and Hirschberger, 2012] and [Peerlings et al.,2012].

4.1 Prandtl-indentation test

The goal of this example is to show the robustness of the Q1P0-element derived fromthe Hu-Washizu principle compared to the standard displacement Q1-element. There-fore, an indentation test into an elastic-plastic medium is performed and the resultsare compared to an analytical solution by Prandtl ([Prandtl, 1920]) for an ideally plas-tic body. For this test, a rigid punch (modeled by the imposed displacement v) isdriven into a fictitious elastic-plastic material modeled by two-dimensional elements ina plane-strain condition (see figure 4.1, measures in mm). The material parametersassumed for the material can be found in table 4.1. The hardening modulus is chosensignificantly smaller than Young’s modulus in a way that softening is induced (negative

47

48 CHAPTER 4. SINGLE-SCALE BENCHMARK TESTS

K), while the behavior still resembles perfect plasticity K = 0. The computations werecarried out for different mesh sizes for the Q1- and the Q1P0-elements. The coarsestmesh consists of 64 elements in X-direction and 32 elements in Y -direction (referredto as 64×32). The other meshes are 128×64, 256×128 and 512×256.

XY

v

3

2

0.75

(a) Boundary value problem (b) Analytical solution by [Prandtl, 1920]

Figure 4.1: Prandtl-indentation test

E/ Nmm2 ν τy/

Nmm2 K/ N

mm2 H/ Nmm2 A/N

200000 0.35 100 −100 1000 0.1

Table 4.1: Material data for Prandtl-indentation test

The computational potential of the constant pressure formulation is emphasized by acomparison of the force-displacement-diagrams, in which the overall reaction force ofthe nodes where v is imposed is plotted over their vertical displacements. The force-displacement-curves are plotted in figures 4.2 and 4.3. The improved volumetric lockingbehavior of the Q1P0-elements is obvious as the reaction forces are much lower, espe-cially for coarse meshes.In figure 4.4, the vertical displacement as well as the equivalent plastic strain α ob-tained from the Q1-element are plotted for the different meshes. The same quantitiesobtained from the Q1P0-element are given in figure 4.5 for comparison. Due to thepresence of the additional field equation for the plastic microstrain, both elements con-verge to a solution where the plastic deformation localizes in a narrow band, with theQ1P0-element converging much faster. This would not be the case for Finite Elementsolutions of standard continua as a clearly softening material behavior (K<0) is as-sumed.A similar solution was obtained by [Welschinger, 2011], where an incremental varia-tional framework for enhanced strain elements was used. In these elements, a localadditional strain field is introduced in each element to prevent volumetric locking. Theoriginal analytical solution computed by the theory of gliding lines as given in [Prandtl,

4.1. PRANDTL-INDENTATION TEST 49

1920] is shown in figure 4.1(b). A comparison of the solutions from the Q1P0-elementsand the solutions given by [Prandtl, 1920] and [Welschinger, 2011] justify the use ofthe constant pressure element in combination with a micromorphic-type inelastic con-stitutive model.

v/mm

Force/N

0 0.005 0.01

0

50

100

150

200

250

300

Q1 64×32

Q1 128×64

Q1 256×128

Q1 512×256

Figure 4.2: Force-displacement-diagrams for Q1-elements

v/mm

Force/N

0 0.005 0.01

0

50

100

150

200

250

300

Q1P0 64×32

Q1P0 128×64

Q1P0 256×128

Q1P0 512×256

Figure 4.3: Force-displacement-diagrams for Q1P0-elements


(a) Displacement 64×32 (b) Equivalent plastic strain 64×32

(c) Displacement 128×64 (d) Equivalent plastic strain 128×64

(e) Displacement 256×128 (f) Equivalent plastic strain 256×128

(g) Displacement 512×256 (h) Equivalent plastic strain 512×256

Figure 4.4: Displacement and plastic strain for Q1-elements

4.1. PRANDTL-INDENTATION TEST 51

(a) Displacement 64×32 (b) Equivalent plastic strain 64×32

(c) Displacement 128×64 (d) Equivalent plastic strain 128×64

(e) Displacement 256×128 (f) Equivalent plastic strain 256×128

(g) Displacement 512×256 (h) Equivalent plastic strain 512×256

Figure 4.5: Displacement and plastic strain for Q1P0-elements


4.2 Strip with a hole

To validate the damaging elastic-plastic model, the localization behavior of a stripwith a hole under tension is investigated. The test specimens were manufactured froma sheet of rolled copper (CW 024 A) and were taken out of the sheet in such a waythat the rolling direction is identical to the direction of tension in the test.The material parameters E, ν, τy, K, and Y0 were identified with a standard tensiontest with specimens according to DIN 50125–H 12.5×50 with a thickness of 1mm. Themicromorphic parameters H and A were identified in a similar test only that a holewith a diameter of 3mm was drilled in the center of each specimen.

E/ Nmm2 ν τy/

Nmm2 K/ N

mm2 Y0/N

mm2 H/ Nmm2 A/N

109000 0.35 250 900 0.35 10000 1

Table 4.2: Material data for copper (CW 024 A)

A comparison of the stress-strain-diagram of the tension test and the numerical simu-lation using the material data from table 4.2 is given in figure 4.6. The material data

E11/%

P11/

N

mm

2

0 5 10 15 20 25 30 35

0

50

100

150

200

250

300experiment simulation

Figure 4.6: Stress-strain-diagram for copper (CW 024 A)

were fitted in a way that the stresses and the loss of material stiffness (i.e. damageevolution) are predicted accurately for strains up to 10%. However, the damage evolu-tion is overestimated for strains larger than 15%. This results in an underestimation of

4.2. STRIP WITH A HOLE 53

the stresses and the stiffness. This deviation from the experiment could be remediedby the use of more sophisticated hardening and damage evolution laws. However, evenfor the simple assumptions made in this thesis, the material properties observed in theexperiment can be simulated with the presented framework.Unfortunately, it is not possible to determine the micromorphic material parametersH and A with a uniaxial tension test, as localization occurs only directly before thetotal failure of the specimen. Therefore, a strip with a hole was investigated. For theexperiment, a hole with 3mm in diameter was drilled in the center of specimens withthe same dimensions as in the uniaxial tension test. The displacement was measuredbetween the two measurement marks, which were 30mm apart from each other initially(figure 4.8(a)). This way, the zone in which strain localization occurs is determined apriori and is no longer governed by randomly distributed microstructural defects in thematerial.For the simulation, uniformly refined meshes of the type shown in figure 4.7(a), match-ing the geometry between the measurement marks, were used. The computations werecarried out with the H1P0-element. The convergence behavior of the element for sev-eral refinements (with ten elements in thickness-direction for each computation) canbe seen in figure 4.7(b). It becomes evident that the material response becomes meshindependent with ongoing mesh refinements and that mesh convergence is obtained.

(a) FE-mesh of astrip with a hole

u/mm

F/N

0 0.2 0.4 0.6 0.8 1.00

1000

2000

3000

2864 DOFs

10436 DOFs

36572 DOFs

153812 DOFs

342676 DOFs

(b) Force-displacement-diagrams

Figure 4.7: Mesh independence of localization behavior

For the computations mentioned in figure 4.7, the localization parameters from table4.2 were used. The comparison of the shapes and widths of the localization zones ob-served in the experiment and in the simulation (see figure 4.8) justifies the choice ofH and A. Furthermore, the force-displacement-diagrams are almost congruent despitethe overestimated loss of stiffness at the end of deformation (figure 4.9). The predictionof the geometry of the unloaded state is acceptable considering the relatively simpleconstitutive model: The largest diameter of the hole was measured as 3.98mm and


computed as 3.94mm and the smallest width of the specimen in the localization zonewas measured as 11.84mm and computed as 11.76mm. The change of geometry is thuspredicted with an error of approximately 1%.However, it is emphasized that the present numerical model does not account foranisotropic effects such as kinematic hardening, which plays an important role dur-ing the deformation of metals. The specimens in the experiments were all loaded inthe rolling direction. A different result can be expected for loading in other directions.Yet, the implementation of more sophisticated constitutive models is not within thescope of this thesis. Instead, the focus will be set on multiscale computations in thenext chapters.

(a) Experiment (b) Simulation (equivalent plastic strain)

Figure 4.8: Localization of plastic deformation

u/mm

Force/N

0 0.2 0.4 0.6 0.8 1.0

0

1000

2000

3000experiment simulation

Figure 4.9: Force-displacement-diagrams for a strip with a hole

The possibility of capturing size-effects with the micromorphic-type model is now inves-tigated by carrying out computations with the finest FE-mesh used in the computations

4.2. STRIP WITH A HOLE 55

before. The set of material parameters from table 4.2 is still valid, but the microstiff-ness A is varied. The resulting force-displacement-curves are given in figure 4.10.

u/mm

Force/N

0 0.2 0.4 0.6 0.8 1.00

1000

2000

3000

A = 100N

A = 10N

A = 1N

A = 0.1N

Figure 4.10: Size-effect due to a variation of the microstiffness

The results displayed in this picture clearly indicate the dependency of the entire in-elastic material response on the microstiffness A. A lower value of A leads to a strongerlocalization and consequently to a stronger damage evolution as the influence of ∇Ddecreases. Thus, a weaker material response is computed.

Chapter 5

Computational homogenization

Most engineering materials like fiber reinforced composites, particle reinforced adhe-sives, or concrete have heterogeneous structures. Even metals which appear to behomogeneous on a macroscopic scale have heterogeneously distributed material prop-erties if the inherent microstructure (e.g. grains in polycrystalline metals) is considered.In the present work, instead of resolving the inhomogeneities by the Finite Elementmesh on a single scale, a multiscale approach is followed. Analytical multiscale tech-niques go back to the works of Voigt [Voigt, 1889] and Reuß [Reuß, 1929], where theeffective behavior of a polycrystal was investigated. A summary of different approachesto multiscale-mechanics is given in [Hashin, 1983] and in [Nemat-Nasser and Hori, 1999].With increasing computational power, numerical methods have become popular as an-alytical methods are only valid for special cases of (elastic) microstructures, whereascomputational methods can be applied to arbitrarily shaped heterogeneities and con-stitutive models. Most multiscale methods are based on the concept of RepresentativeVolume Elements (RVEs). Typically, these volume elements are several orders of mag-nitude smaller than the macroscopic problem but nevertheless contain all informationabout the underlying microstructure. With this concept, it is possible to determine aneffective constitutive model for the macroscale by performing computations on RVE-level and averaging the material response for different RVE deformation modes. Thismethod is described in [Lehmann, 2013] where an effective material model for a poly-crystalline steel is developed. However, an a priori constitutive assumption on themacroscale is necessary in such homogenization schemes, which may be difficult to findfor complex microstructures.The FE2-method as described in [Miehe et al., 1999a], [Miehe et al., 1999b], [Feyel,1999], and in [Feyel and Chaboche, 2000] does not require a constitutive assumptionon the macrolevel. Instead, an RVE is attached to each integration point of the macro-scopic Finite Element problem. Boundary conditions computed from macroscopic dataare imposed on the edges of the RVEs and the stress responses as well as the algorith-mic tangent matrices are averaged over the RVEs and used on the macrolevel. Clearly,this solution scheme is more flexible than using an effective material model as no con-stitutive assumptions are made on the macrolevel. A drawback of this method is theincreased computational effort during the calculations. This can be remedied by theexploitation of efficient parallelization techniques where the different sub-problems are

57

58 CHAPTER 5. COMPUTATIONAL HOMOGENIZATION

solved on different cores. Also, fitting the material parameters on the macrolevel is nolonger required.Apart from the computation of polycrystalline metals in the already mentioned arti-cles, this computational procedure was successfully applied to model size-effects with astandard continuum on RVE-level and a higher gradient continuum on the macrolevelin [Kouznetsova et al., 2002] and in [Kouznetsova, 2002] and more recently in [Lesicaret al., 2014]. A different approach for the computation of material layers, where amicromorphic continuum was used on the RVE-level, can be found in [Hirschbergeret al., 2008]. This framework is suitable for the prediction of size-effects on the meso-scopic scale which is motivated from a physical point of view as these effects are inducedby mesostructural material behavior rather than by macroscopic effects.FE2-computations of heterogeneous material layers are carried out in [Hirschbergeret al., 2009]. A probabilistic analysis of size-effects in a two-scale fracture analysis ofquasi-brittle materials is provided by [Ibrahimbegovic and Matthies, 2012].

5.1 Bulk homogenization

According to the concept of [Hirschberger et al., 2008], the size-effects are to be cap-tured by a generalized continuum on RVE-level. Therefore, the micromorphic-typeinelastic materials described in section 2.2.4 are used to construct RVEs. To avoidmisinterpretations, the RVE-level is also called mesolevel and the expression microlevelonly refers to the micromorphic-type variables in the context of this thesis. In thefollowing, macroscopic quantities are indicated by a hat, while microquantities are stillwritten with a superscript bar. Mesoscopic (or RVE-level-) quantities appear in thestandard manner.In the multiscale procedure, a Q1-Element derived by the principle of virtual work,as described in section 3, but for a standard continuum, is used on the macroscale.Contrary to single-scale computations, a (usually heterogeneous) RVE responsible forthe material response is attached to each macroscopic Gauss-point (see figure 5.1, themesoscopic inclusion appears in gray).The governing equations on the macrolevel are similar to the ones in section 3.1.1:

Div P = 0 (5.1)

u− upre = 0 on ∂Bu0 (5.2)


0 . (5.3)

This leads to the principle of virtual work on the macroscopic scale:∫

B0

P : δF dV =

∫

∂Bt00

tpre0 · δu dA . (5.4)

The internal part of the discretized residual vector is consequently determined as

Rhint =

ne⋃

e=1

(Rint)e =ne⋃

e=1

∫

Ωe

P :∂F

∂pe

dV , (5.5)

5.1. BULK HOMOGENIZATION 59

Macrolevel node

Macrolevel Gauss-point

RVE-level Node

Macrolevel mesh

RVE-level mesh

Figure 5.1: FE2-discretization of a domain with RVE

where ne is the number of macroscale elements. The macroscopic stress tensor P,which has to be known at every macroscopic integration point, is obtained from therespective attached RVE by averaging techniques.To solve the nonlinear equation system on the macroscale, the internal part of themacroscopic element residual has to be linearized in order to use the Newton-Raphsonsolution procedure. For this, the homogenized algorithmic tangent A also obtainedfrom the microlevel is included to derive the tangent stiffness matrix Ke:

Ke =

ng∑

g=1

det J( Xg)Kgwg (5.6)

Kg =δ(Rint)g

δ (pe)n+1

∣∣∣∣DP

DF=A

. (5.7)

The resulting nested solution scheme is sketched in figure 5.2. The procedure is thefollowing: After the initialization of both scales (meshing, allocating memory, etc.),

a macroscopic load increment is applied and the deformation gradient F is computedat every macroscopic integration point. Then, for each integration point, a subscaleproblem is solved in several micro load steps (including sensitivity analyses) until the

macroscopic load factor is reached. Afterwards, the averaged stress P and the con-sistent tangent A are computed. These results are transferred back to the respectivemacroscopic integration point and the macroscopic step is solved iteratively. Thisimplies that the subscale computations have to be performed for every macroscopiciteration step in every macroscopic load step.


macro initialization meso initialization

macro load step

increase macro load factor

compute at Gauss points

meso step for each Gauss point

delete sensitivity history

meso load step

increase meso load factor

solve meso iteration step

compute sensitivities

final load step?

compute average stress and tangent

solve macro iteration step

final load step?no

yes

result

no

yes

yes

yes

no

no

Figure 5.2: Nested two-scale solution scheme

5.1.1 Meso-to-macro transition

Finite deformation kinematics on the macrolevel lead to a standard definition of themacroscopic deformation gradient F:

F =∂x

∂X. (5.8)

The macroscopic stress tensor P is considered to be the volume average of the meso-scopic stress tensor P:

P =1

V

∫

B0

P dV , (5.9)

where V is the volume of a RVE. This quantity can easily be determined by a numericalintegration as described in section 3.3.However, a physically meaningful transition between macroscale and mesoscale includessome kind of energy conservation criterion. In small deformation theory, the storedenergy can be conserved by the application of Hill’s theorem [Hill, 1972]. However,large deformations are more cumbersome so that only a criterion for the conservationof virtual work can be stated. The macroscopic internal virtual work is determinedby the stress P and the variation of the deformation gradient δF and has to be equalto the mesoscopic internal virtual work of the micromorphic-type inelastic continuum


derived in section 2.2.4:

P : δF =1

V

∫

B0

(P : δF+ a · δξ + b : δζ

)dV . (5.10)

This modified criterion was also used by [Hirschberger et al., 2008] for a micromorphichyperelastic mesoscale and by [Clasen et al., 2013] for micromorphic-type hyperelasto-plasticity.With the assumptions made earlier in this thesis, the right-hand-side of the aboveequation can be simplified. With

b : δζ = Div(δξ · b

)−Div b · δξ (5.11)

Div b = a (5.12)

b ·N = 0 (5.13)

and the application of the divergence theorem∫

B0

(a · δξ + b : δζ

)dV = 0 (5.14)

emerges. Thus, the final form of the criterion reads

P : δF =1

V

∫

B0

P : δF dV . (5.15)

This criterion can be fulfilled by the application of certain boundary conditions as shownfor infinitesimal deformations in [Nemat-Nasser and Hori, 1999] and in [Miehe andKoch, 2002], and for finite deformations in [Schroder, 2000]. These boundary conditionsare described in the following where focus is set on periodic boundary conditions, whichwill be used in this thesis.

Linear displacement boundary conditions

The first type of boundary conditions are the linear displacement boundary condi-tions, for which the deformation gradient tensor obtained from macrolevel kinematicsis applied in an affine manner (see figure 5.3(b)) to the RVE-boundary, such that

x = F ·X on ∂B0 . (5.16)

In general, the application of linear displacement boundary conditions leads to an over-estimation of the equivalent RVE-stiffness as it resembles the homogenization approachby Voigt [Voigt, 1889].

Uniform traction boundary conditions

A different approach is to apply uniform tractions with a constant value on each RVE-boundary. This method stands in analogy to the homogenization approach by Reuß[Reuß, 1929] and leads to an underestimation of the effective stiffness. Since no Dirichlet


boundary conditions are present for this case, the uniform traction boundary conditionsare not applicable to the present multiscale scheme, as it requires a sensitivity analysiswith respect to the imposed deformations on RVE-level for the computation of theexact algorithmic tangent. Consequently, further investigation of this type of boundaryconditions is omitted.

Periodic boundary conditions

A compromise between the aforementioned boundary conditions is the application ofperiodic displacement boundary conditions in combination with antiperiodic tractionson the boundaries. Contrary to the linear displacement approach, a mesoscopic fluc-tuation of the placement x is allowed:

x = F ·X+ x on ∂B0 . (5.17)

For convenience, we assume that the average fluctuation vanishes:

1

V

∫

B0

x dV = 0 . (5.18)

The boundary ∂B0 is split into two equally-sized, disjoint parts ∂B+0 and ∂B−

0 . Con-sidering figure 5.3(a), ∂B+

0 may contain the sides 1 and 2 and ∂B−0 may contain sides

3 and 4. We state the periodicity constraints of the fluctuation field and the boundarytractions as follows:

x+ = x− (5.19)

t+0 = −t−0 . (5.20)

These conditions are especially beneficial for periodic microstructures and the homoge-nized stiffness is always between the two bounds provided by the previous two methods.Due to these advantages, the periodic boundary conditions are used for the bulk ho-mogenization procedure in this thesis. The fact that these conditions fulfill equation(5.15) will be proved in the following.

1

2

3

4

I II

IIIIV

(a) Undeformed RVE

1

2

3

4

III

IIIIV

(b) Linear displacementboundary conditions

1

2

3

4

III

IIIIV

(c) Periodic boundaryconditions

Figure 5.3: RVE boundary conditions


Admissibility of periodic boundary conditions

First, the definition of the average stress tensor (5.9) is reformulated. For this therelation

Div (X⊗P) = PT +X⊗ DivP = PT , (5.21)

assuming DivP = 0, is needed. With the boundary conditions P ·N = t0, this renders∫

B0

PT dV =

∫

B0

Div (X⊗P) dV (5.22)

=

∫

∂B0

(X⊗P) ·N dA (5.23)

=

∫

∂B0

X⊗ t0 dA . (5.24)

Thus, the average stress is described by

P =1

V

∫

B0

P dV =1

V

∫

∂B0

t0 ⊗X dA . (5.25)

The right-hand-side of equation (5.15) is rewritten:

1

V

∫

B0

P : δF dV =1

V

∫

B0

(Div (δx ·P)− δx · DivP) dV . (5.26)

Exploiting equilibrium DivP = 0, boundary conditions P ·N = t0, and the divergencetheorem, this leads to the average work theorem:

1

V

∫

B0

P : δF dV =1

V

∫

∂B0

δx · t0 dA . (5.27)

The combination of the equations (5.15), (5.17), (5.25), and (5.27) leads to the follow-ing:

P : δF =1

V

∫

∂B0

t0 ·(δF ·X+ δx

)dA (5.28)

= δF :1

V

∫

∂B0

t0 ⊗X dA+1

V

∫

∂B+0

t+0 · δx+ dA +1

V

∫

∂B−

0

t−0 · δx− dA (5.29)

= P : δF+1

V

∫

∂B+0

t+0 · δx+ dA+1

V

∫

∂B−

0

t−0 · δx− dA . (5.30)

Using the periodicity constraints (5.19) and (5.20) renders the proof of the virtual workcriterion for periodic boundary conditions:

P : δF = P : δF+1

V

∫

∂B+0

t+0 · δx+ dA+1

V

∫

∂B+0

−t+0 · δx+ dA (5.31)

= P : δF . (5.32)

Thus, this kind of boundary condition is admissible for the present homogenizationframework.


Enforcement of periodicity constraints

In the present two-dimensional case, the periodicity constraints are enforced by theaugmented Lagrangian method (see [Wriggers, 2006] and references therein). For this,constraint elements for the RVE boundary nodes are introduced, which couple thedegrees of freedom on the opposite sides of the RVE. For example, let nodes 1 and 3be the nodes on the bottom edge, on which the displacement boundary conditions areimposed, and nodes 3 and 4 arbitrary adjacent nodes on the sides of the RVE (figure5.4). Then, the periodicity conditions can be written as follows:

(u3 − u1)− (u4 − u2) = gu = 0 (5.33)

(α3 − α1)− (α4 − α2) = gα = 0 . (5.34)

1

1

2

2

33

4

4

Figure 5.4: Node ordering for constraint elements on vertical and horizontal RVE edges

Introducing Lagrange multipliers for both constraints, λu and λα, and the penaltyparameter ρ, the following functional to ensure periodicity is constructed:

Π =

[λu

λα

]·[gu

gα

]+

1

2ρ(gu · gu + g2α) . (5.35)

To obtain the weak form used for the FE-framework the variation of this functional iscomputed:

δΠ =

[δλu

δλα

]·[gu

gα

]+ (λu + ρgu) · δgu + (λα + ρgα)δgα . (5.36)

From this weak form, which appears as an additional term in the overall weak form, therespective residual vectors and tangent matrices are obtained in a similar manner asdescribed in section 3.4. Using these elements on all nodes of the four edges of the RVEenforces the periodicity over the whole boundary. These elements are then assembledto the equation system together with the vectors and matrices of the solid elements.In the case of the elastic-plastic damaging model on RVE-level, the same procedure isused with D instead of α. Please note that for this case of constraint enforcement, theboundary nodes have to be placed in equidistant steps on opposite boundaries.


5.1.2 Symbolic derivation of the macroscopic tangent

In [Michaleris et al., 1994], an efficient, exact method for computing design sensitiv-ities for transient locally coupled problems and an application to elasto-plasticity ispresented. In [Korelc, 2009a] this method is used in an automatic-differentiation-basedFinite Element Method for finite inelastic deformations. Choosing the macroscopic de-formation gradient F as a sensitivity parameter for the stress tensor obtained from anRVE-simulation, the algorithmic tangent A for the macroscale can be determined bythis method as described in [Clasen et al., 2013]. This way, a consistent linearizationis obtained within a symbolic approach contrary to the work of [Miehe, 2002], wherethe linearization is achieved by analytical approaches, or the work of [Temizer andWriggers, 2008], which exploits boundary condition perturbation methods.In this section, this algorithm will be explained in detail as the computation of the cor-rect macroscopic tangent matrix is crucial for optimal macroscopic convergence rates(compare section 5.1 for the macroscopic FE-formulation). This computation is per-formed at the end of every RVE solution procedure which means that the sub-problemhas already been solved for the displacements u and the micromorphic type variablesα or D.The choice of boundary conditions allows to determine the exact homogenized algo-rithmic tangent A on each macroscopic integration point (thus for each RVE) which isdefined as follows:

A =DP

DF=

1

V

∫

B0

DP

DFdV . (5.37)

This integral is computed with the numerical integration scheme from section 3.3.Consequently, the stress tensorP is only computed at the integration points of the RVE.Due to the implicit dependencies of the stresses Pg at integration point g with respectto the nodal degrees of freedom pe of the respective element and the history variableshg of the respective integration point, the derivative DP/DF contains convective termsas follows:

DPg(F,pe(F),hg(F))

DF=∂Pg

∂F+∂Pg

∂pe

· Dpe

DF+∂Pg

∂hg

· Dhg

DF. (5.38)

Once the derivatives Dpe/DF and Dhg/DF are known, the sensitivity can be calcu-lated. Please note that this solution procedure requires initial design velocity fields forthe sensitivities with respect to F at the beginning of the analysis. The derivatives ofquantities from the previous time step are then stored as history variables. Thus, theyare known at the current time step.In a converged state, the RVE-problem fulfills the following equations for the indepen-dent (discretized) residual n+1Gh and the dependent residuals n+1Qg of each integrationpoint (i.e. the inelastic evolution equations in implicit form):

n+1Gh(n+1p, n+1hg(n+1p), np, nhg) = 0 (5.39)

n+1Qg(n+1pe,

n+1hg(n+1pe),

npe,nhg) = 0 . (5.40)


In the follwing, the superscript n+1 will be dropped for brevity. To determine thedependency Dpe/DF, the total derivative of the residual Gh with respect to each

component of F, namely Fi, is computed:

∂Gh

∂p· Dp

DFi

+

ngi∑

g

∂Gh

∂hg

· Dhg

DFi

+∂Gh

∂np· D

np

DFi

+

ngi∑

g

∂Gh

∂nhg

· Dnhg

DFi

+∂Gh

∂Fi

= 0 . (5.41)

The quantity ngi denotes the number of integration points (of the current element) atwhich the inelastic quantities evolve. In a similar manner, the total derivative of thedependent residual Qg is constructed:

∂Qg

∂p· Dp

DFi

+∂Qg

∂hg

· Dhg

DFi

+∂Qg

∂np· D

np

DFi

+∂Qg

∂nhg

· Dnhg

DFi

+∂Qg

∂Fi

= 0 . (5.42)

Rearranging this equation, one obtains the following relation:

Dhg

DFi

= −A−1g ·

(∂Qg

∂p· DpDFi

+∂Qg

∂np· D

np

DFi

+∂Qg

∂nhg

· Dnhg

DFi

+∂Qg

∂Fi

). (5.43)

The following variable is introduced to shorten the expressions:

Zg i = −A−1g ·

(∂Qg

∂np· D

np

DFi

+∂Qg

∂nhg

· Dnhg

DFi

+∂Qg

∂Fi

). (5.44)

This result is now inserted into equation (5.41) to give a formula for the computation

of the derivative Dp/DFi:

(∂Gh

∂p−

ngi∑

g

∂Gh

∂hg

·A−1g · ∂Qg

∂p

)· Dp

DFi

=−(

ngi∑

g

∂Gh

∂hg

· Zg i +∂Gh

∂np· D

np

DFi

+

ngi∑

g

∂Gh

∂nhg

· Dnhg

DFi

+∂Gh

∂Fi

). (5.45)

Comparing the left-hand-side of this equation to equation 3.95, it can be seen that theterm in brackets is the global stiffness matrix of the primal Finite Element problem,which is already known. Thus, the linear equation system

K · DpDFi

= −Ri (5.46)

can be easily solved for Dp/DFi and the result is inserted into equation (5.43), provid-

ing a result for the other derivative Dhg/DFi. A standard assembly routine is used tocompute the independent sensitivity pseudo load vector Ri from the element contribu-tions Re i:

Ri =ne⋃

e=1

Re i . (5.47)


This means that the derivatives with respect to p can be replaced by pe for eachelement, as the residual is computed elementwise.Please note that in the present case, due to the choice of boundary conditions, noNeumann boundary conditions are imposed on the RVE. This means that the externalpart of the residual Gh, Rh

ext, vanishes. As a result, the quantity Gh can be replacedby Rh

int, which further simplifies the solution procedure. However, if uniform tractionboundary conditions were used, this would not be the case.The application of this procedure for each component Fi makes it possible to computethe exact algorithmic tangent A = DP/DF, which leads to a correct linearization of themacroscopic residual. Thus, optimal convergence rates are obtained in the macroscopiciterative solution procedure.

5.1.3 Thermodynamic consistency

In other homogenization frameworks, in which additional field variables (e.g. the tem-perature field) are present on the mesoscopic scale (see e.g. [Ozdemir et al., 2008],[Temizer and Wriggers, 2011], [Sahraee et al., 2014], and references therein), a cor-responding additional field has to be present on the macroscopic scale. This impliesthat if mesoscopic heat conduction due to a change of the mesoscopic temperature Θis computed, also the heat conduction on the macroscopic scale due to the macroscopictemperature Θ has to be considered to ensure that the dissipation is transferred cor-rectly from one scale to the other.The criterion for thermodynamic consistency is the following. It is assumed that therate of the macroscopic free energy Ψ is the volume average of the rate of the mesoscopicfree energy:

˙Ψ =

1

V

∫

B0

Ψ dV = 〈Ψ〉 . (5.48)

Consequently, the homogenization is consistent if and only if the macroscopic dissipa-tion D equals the volume average of the mesoscopic dissipation:

D =1

V

∫

B0

D dV = 〈D〉 . (5.49)

In this section, the proof of thermodynamic consistency of the proposed frameworkwithout a macroscopic micromorphic-type degree of freedom will be discussed. For ananalogous argumentation in thermoinelasticity the reader is referred to [Sahraee et al.,2014].Consider a macroscopic material described by a free energy which depends on themacroscopic deformation and macroscopic internal variables:

Ψ = Ψ(F, ξ

). (5.50)

We assume that there are internal variables present on the macrolevel (just as in astandard inelastic framework as discussed earlier) although we are in no way interested


in computing them. Then the rate of the free energy follows as

˙Ψ =

∂Ψ

∂F:˙F+

∂Ψ

∂ξ· ˙ξ . (5.51)

With the definition of the thermodynamic conjugate force Ξ, the Coleman-Noll proce-dure can be applied to determine the macroscopic dissipation:

Ξ = −∂Ψ∂ξ

(5.52)

D = P :˙F− ˙

Ψ = Ξ · ˙ξ . (5.53)

Now, consider a mesoscopic free energy of a micromorphic-type continuum:

Ψ = Ψ(F, ξ, ξ,∇ξ

). (5.54)

It follows that

Ψ =∂Ψ

∂F: F+

∂Ψ

∂ξ· ξ +

∂Ψ

∂ξ· ˙ξ +

∂Ψ

∂∇ξ: ∇ ˙ξ (5.55)

〈Ψ〉 = 1

V

∫

B0

(∂Ψ

∂F: F+

∂Ψ

∂ξ· ξ +

∂Ψ

∂ξ· ˙ξ +

∂Ψ

∂∇ξ: ∇ ˙ξ

)dV . (5.56)

With the definitions of the couple stresses and double stresses a and b from equations(2.132) and (2.133), and relations (5.11) – (5.13), the last two terms in the integralcancel each other (as in equation (5.14)):

1

V

∫

B0

(a · ˙ξ + b : ∇ ˙ξ

)dV = 0 . (5.57)

Consequently, the rate of the averaged free energy reduces to

〈Ψ〉 = 1

V

∫

B0

(∂Ψ

∂F: F+

∂Ψ

∂ξ· ξ)

dV . (5.58)

With the averaging relations from section 5.1.1 and introducing the conjugate force Ξ,the following relation is obtained:

Ξ = −∂Ψ∂ξ

(5.59)

〈Ψ〉 = 1

V

∫

B0

P : F dV − 1

V

∫

B0

Ξ · ξ dV (5.60)

With the theorem of the conservation of virtual work from equation (5.15) applied tothe power and the derivation from the Coleman-Noll procedure on the mesoscale, itfollows that

〈Ψ〉 = P :˙F− 〈D〉 . (5.61)

5.2. INTERFACE HOMOGENIZATION 69

With this intermediate result at hand, the difference between the macroscopic andmesoscopic rates of the free energies can be computed:

˙Ψ− 〈Ψ〉 = P :

˙F− D − P :

˙F+ 〈D〉 = 〈D〉 − D . (5.62)

Using assumption (5.48), it follows that

D = 〈D〉 . (5.63)

This proves the thermodynamic consistency although the macroscopic counterpart ofthe micromorphic-type field on the mesoscale is not included in the framework. Thus,no additional variables or their corresponding stresses have to be passed from one scaleto the other to ensure the full functionality of the present homogenization procedure.Please note that this proof is also applicable to free energy functions depending only onthe elastic part of the deformation, rendering different expressions for the dissipation.For the sake of brevity, this case will not be discussed here and the reader is referredto section 2.1.3.

5.2 Interface homogenization

In many engineering problems, components contain thin interface layers of a differentmaterial with different material properties than the surrounding bulk material. Typicalapplications are adhesive bondings in the aerospace or the automotive industries. Butalso welded or soldered components contain such interfaces.From a computational point of view, many approaches to model interface layers havebeen developed over the decades. Nowadays, it is a common approach to use specialcontinuous Finite Elements which are formulated in the displacement jumps across theinterface rather than in the displacements themselves. Accordingly, a traction vectorinstead of a stress tensor is obtained as a constitutive response. In the most simplecase, a linear elastic constitutive model is chosen to couple the displacement jump andthe traction response ([Schellekens and de Borst, 1993]). More sophisticated modelsare the so-called Cohesive Zone Models capable of predicting the debonding of thesurrounding bulk (e.g. [Xu and Needleman, 1993] or [van den Bosch et al., 2007]).In [Hirschberger et al., 2008] and [Hirschberger et al., 2009], a macroscopic interfaceelement is used in an FE2-context to obtain the same versatile framework for heteroge-neous mesostructures without constitutive assumptions on the macrolevel as describedin section 5.1. In these works, a macroscopic deformation gradient is constructed fromthe displacement jump vector based on the formulations in [Larsson et al., 1998] and[Steinmann and Betsch, 2000]. This gradient is then used to impose boundary condi-tions on the subscale problems as described in the previous section.All the formulations mentioned above still lack deformation modes for stretching in theinterface direction, which causes serious problems especially in FE2 homogenizationprocedures. A remedy for this can be found in [McBride et al., 2012] and in [Javiliet al., 2014], where the macroscopic deformation gradient is modified so that in-planedeformations are predicted correctly.


In the next sections, a four-node interface element for two-dimensional problems, whichwill be used as a macroscopic element in the homogenization procedure, will be de-scribed. This element incorporates the large deformation theory from [Hirschbergeret al., 2008] and [Hirschberger et al., 2009] in a modified way to capture in-plane (orin-line in the two-dimensional case) deformations. Afterwards, boundary conditionswill be formulated that ensure the criterion of virtual work during the scale transition(see section 5.1.1).

5.2.1 Finite Element discretization of interface layers

In this section, the four-noded macroscopic Finite Element for the plane-strain analy-sis of thin layers will be described. The starting point is the formulation of interfacekinematics according to [Schellekens and de Borst, 1993]. In a two-dimensional bound-ary value problem, the interface is reduced to a line which is discretized as shown infigure 5.5. The deformation of the element is evaluated at the mid-line of the element,which is characterized by the tangential vector M and the normal vector N. Withinan orthonormal basis, these vectors can be computed as follows:

M =X23 − X14

‖X23 − X14‖=

[MX

MY

](5.64)

N =

[−MY

MX

]. (5.65)

1 2

34

-1 1

X

(a) Reference space

1

2

3

4MN

E

X14X23

θ

(b) Reference configuration

1

2

34

E

n m

x14x23

(c) Current configuration

Figure 5.5: Isoparametric four-node interface elements

Within the Finite Element approach, the placement of a material point on the mid-lineis determined with the shape functions (which only depend on X) and the materialpoints at both ends of the mid-line:

X = N1X14 +N2X23 (5.66)

N1 =1

2(1− X) (5.67)

N2 =1

2(1 + X) . (5.68)


In a large deformation analysis, according to equation (2.5), the nodal placements inthe current configuration xI can be determined by the nodal placements in the referenceconfiguration XI and the nodal displacements uI :

xI = XI + uI . (5.69)

Then, the tangential and normal vectors of the interface can also be calculated in thecurrent configuration:

m =x23 − x14

‖x23 − x14‖=

[mX

mY

](5.70)

n =

[−mY

mX

]. (5.71)

The angle of inclination θ can be expressed in terms of the tangential vector:

cos θ = MX

(5.72)

sin θ = MY . (5.73)

The vector uI of the nodal displacements of node I in the X-coordinate system canthen be transformed into the rotated θ-coordinate system as follows:

uIθ

= T · uIX

(5.74)[uIM

uIN

]=

[cos θ sin θ

− sin θ cos θ

]·[uIXuIY

]. (5.75)

From now on, every quantity with a basis in the θ-coordinate system is denoted bya subscript θ. Quantities without this subscript have a basis in the cartesian X-coordinate system.The assembly of all nodal vectors u

Iθis called u

θ:

uθ=[uT

1θ, uT

2θ, uT

3θ, uT

4θ

]T. (5.76)

With the displacements on the bottom side of the interface ub

θand on the top side of

the interface ut

θ, which are calculated as follows

ub

θ= N1u1θ +N2u2θ (5.77)

ut

θ= N1u4θ +N2u3θ , (5.78)

the displacement jump JuKθ= ut

θ− ub

θat any point of the mid-line X can be computed

with the help of a B-matrix:

JuKθ= B · u

θ(5.79)

B =

[−N1 0 −N2 0 N2 0 N1 00 −N1 0 −N2 0 N2 0 N1

]. (5.80)


With this quantity, a deformation gradient is constructed according to [Hirschbergeret al., 2008] and [Hirschberger et al., 2009], which includes the physical initial interfacethickness h0:

Fθ= 1+

1

h0JuK

θ⊗ N

θ. (5.81)

This tensor has to be transformed back into the global coordinate system:

F = TT · Fθ·T . (5.82)

This tensor describes the deformation of the interface for tensile and in-plane shearloading and is well suited for single-scale computations of thin material layers withdominant in-plane shear deformation modes. However, it has a major drawback as faras tensile loading modes are concerned. This drawback can be explained best by usinga simple example:Consider a horizontal interface element (θ = 0, N = [0, 1]T ) under pure tensile loading

(n = [0, 1]T , JuK = [0, JuNK]T ). In cartesian coordinates, the deformation gradient F

then reads:

F =

[1 00 1

]+

1

h0

[0

JuN

K

]⊗[01

]=

[1 00 1 + Ju

NK/h0

]. (5.83)

This means that there is no deformation in the tangential direction of the interface,which is clearly wrong if ν 6= 0 and lateral contraction is not hindered. This hasa significant influence in the two-scale homogenization framework as the boundaryconditions are imposed on the RVE by means of F. Consequently, the RVE cannot bestretched in lateral direction, which results in non-physical stress components in thisdirection, affecting the overall constitutive response.Moreover, a macroscopic stretching of the interface element in lateral direction wouldhave no effect on the behavior of the RVE at all as the corresponding entries in thedeformation gradient tensor are always constant.A remedy for this problem is the replacement of the unit tensor in equation (5.81) with atensor which is capable of describing lateral contractions. This was done by introducingcurvilinear coordinates for the description of surface kinematics by [McBride et al.,2012] based on [Steinmann, 2008]. More details about this approach can be foundin [Javili et al., 2014]. For details on differential geometry in curvilinear coordinatesystems the reader is referred to [Itskov, 2007].In the works mentioned above, the deformation gradient tensor is constructed by thesum of the dyadic products of the basis vectors of the different configurations:

F =3∑

i=1

gi ⊗Gi . (5.84)

In this context, lower indices are used for covariant quantities and upper indices forcontravariant indices.This definition is the basis for an approach to constructing a tensor

−→F to replace the


unity tensor as described by [McBride et al., 2012]. For this, we define a tensor Hθ

similar to the displacement gradient from equation 2.9:

Hθ= F

θ− 1 =

1

h0JuK

θ⊗ N

θ. (5.85)

Thus,H = TT · H

θ·T (5.86)

holds. Consequently, the modified deformation gradient can be written as

Fmod =−→F + H . (5.87)

In the present work, for a two-dimensional problem and one-dimensional material layers

discretized with linear interface elements,−→F can be expressed by the lateral stretch λ

and the tangential and normal interface vectors:

−→F = λm⊗ M+ n⊗ N (5.88)

λ =‖x23 − x14‖‖X23 − X14‖

. (5.89)

With this formulation, the tensor−→F introduces a deformation mode for stretching in

M-direction. The interface opening in N-direction is captured in H as in the originaldeformation as well as a shear opening in M-direction. As a result, the same elementcharacteristics as in the original interface formulation by [Hirschberger et al., 2009]are obtained while the drawback of artificial normal stresses in interface direction iseliminated. Thus, this element is suited as a macroscopic element in the homogenizationframework as the RVE response does no longer contain non-physical lateral stresses dueto boundary conditions derived from incorrect macroscopic kinematics.With the kinematics of the interface element at hand, knowing that the constitutivelaw is replaced by a RVE computation, the weak form of equilibrium is formulated in asimilar way as in section 5.1. Applying the principle of virtual work to the macroscopicequilibrium equation Div P = 0, once again, the following formula is obtained:

∫

B0

P : δF dV =

∫

∂Bt00

tpre0 · δu dA . (5.90)

Commonly, it is assumed that no external tractions act on the interface directly butare rather applied to the surrounding bulk material. Thus, the right-hand-side of thisequation is equal to zero. The internal part of the residual can then be computed inthe same way as in the bulk homogenization procedure:

(Rint)e =

∫

Ωe

P :∂F

∂pe

dV . (5.91)

The respective element contributions (Rint)e are then assembled to the global system asin standard Finite Element procedures as well as the element stiffness matrices, which


are computed in the same way as described in equations (5.6) and (5.7).As already mentioned in section 3.3, the volume integral reduces to an area integralin the two-dimensional case, as a constant material thickness is assumed. The inter-face element in a two-dimensional boundary value problem is formulated only in onedimension (the mid-line of the element) and is thus integrated with a one-dimensionalnumerical integration. The area integral can then be transformed to a line integral,incorporating the initial thickness of the interface h0:∫

Ωe

(. . . ) dV = s

∫

Ωe

(. . . ) dA = sh0

∫

Ωe

(. . . ) dL . (5.92)

It can be seen again that the thickness s has no influence on the result and can beneglected.For the one-dimensional numerical integration, it is possible to use the standard Gaus-sian quadrature. However, as shown in [Schellekens and de Borst, 1993], stress oscil-lations for high loads are less likely to occur when a two-point Lobatto integration isused instead. The formula (3.54) still holds, only the positions of the integration pointsare different (the weights are still w1 = w2 = 1):

X1 = −1; X2 = 1 . (5.93)

Still, the Jacobian J , which is scalar for one-dimensional problems has to be determined.For this, the length of a material line element dL is computed:

dL =

√dX2 + dY 2 . (5.94)

J is then defined as the derivative of dL with respect to the reference coordinate X:

J =dL

dX=

√√√√(dX

dX

)2

+

(dY

dX

)2

. (5.95)

With the Finite Element approach (5.66), this term can be identified as

J =1

2‖X23 − X14‖ . (5.96)

With this, the description of a macroscopic linear interface element suited for two-scalecomputations of material layers is complete.In actual computations, the macroscopic angle θ is used to rotate the generated meshfor the respective subscale problems such that the mesostructure is always aligned withthe macroscopic interface (see figure 5.6). In this figure, the macroscopic material layeris shaded in light gray. The mesostructural inclusion is shaded in dark gray. It can beseen that the RVEs are orientated differently at different integration points.This is especially useful for the computation of fiber reinforced material layers whenthe direction of the mesoscopic fiber coincides with the macroscopic tangential vectorof the interface M. The boundary conditions are then imposed on the top and bottomedges of the RVE as described in the next section. Consequently, the tensor F is nottransformed into the rotated coordinate system so that the macroscopic deformationis imposed on a rotated RVE.


Macrolevel node

Macrolevel integration point

RVE-level Node

Figure 5.6: RVE orientation for inclined interfaces

5.2.2 Meso-to-macro transition

In most practical applications for material layers, such as adhesive bondings, the stiff-ness of the layer material is significantly smaller than that of the surrounding bulkmaterial. Thus, the interface deformations (normal tension or in-plane shear) arelarger than the bulk deformations. This consideration gives rise to the applicationof boundary conditions different from the ones used in section 5.1.1, where periodicdisplacements and antiperiodic tractions are assumed in all directions. Instead, for themodeling of thin layers, hybrid boundary conditions as described in [Hirschberger et al.,2009] are applied to the subscale problems. The difference to the periodic conditionscan be seen in figure 5.7.

1

2

3

4

III

IIIIV

(a) Linear displacementboundary conditions

1

2

3

4

III

IIIIV

(b) Periodic boundaryconditions

1

2

3

4

III

IIIIV

(c) Hybrid boundary con-ditions

Figure 5.7: RVE boundary conditions for material layers

Linear displacement boundary conditions are imposed on the edges of the RVE whichare aligned with the material layer (edges 1 and 3) to model a perfect bonding of the

bulk and interface materials, as no displacement fluctuations are allowed in N-direction.Similar to periodic boundary conditions, periodic displacements and antiperiodic trac-tions are assumed in tangential direction to capture the in-plane behavior in a straight-


forward manner as the extension of the layer in M-direction is considerably larger thanthe thickness of the layer. This means that the following set of equations is valid:

x = F ·X+ x on ∂B0 (5.97)

x1 = x3 = 0 (5.98)

x2 = x4 (5.99)

t20 = −t40 . (5.100)

Here, the superscript numbers indicate the respective edges of the RVE boundary ∂B0

(see figure 5.7(c)).In the following, the admissibility of this type of boundary conditions will be provenby evaluating the virtual work criterion from equation (5.15). Similar to the bulk ho-mogenization procedure, the contribution of the micromorphic-type variables can beomitted.Exploiting the average work theorem (equation (5.27)) and the definition of the place-ment (5.97) renders

P : δF =1

V

∫

∂B0

t0 ·(δF ·X+ δx

)dA . (5.101)

This is the same equation as obtained in (5.28). Again, the integral is split to giveexpressions for the respective parts of the boundary (1–4):

P : δF =1

V

∫

1, 3

t1, 30 ·

(δF ·X+ δx1, 3

)dA+

1

V

∫

2, 4

t2, 40 ·

(δF ·X+ δx2, 4

)dA .

(5.102)According to the boundary conditions (5.98) – (5.100), this reduces to

P : δF =1

V

∫

1, 3

t1, 30 ·

(δF ·X

)dA+

1

V

∫

2, 4

t2, 40 ·

(δF ·X+ δx2, 4

)dA . (5.103)

Splitting the second integral again,

P : δF =1

V

∫

1, 3

t1, 30 ·

(δF ·X

)dA+

1

V

∫

2, 4

t2, 40 ·

(δF ·X

)dA+

1

V

∫

2, 4

t2, 40 · δx2, 4 dA

(5.104)is obtained. In this equation, the first two integrals are combined and rearranged andthe third integral is split such that

P : δF = δF :1

V

∫

∂B0

t0 ⊗X dA+1

V

∫

2

t20 · δx2 dA+1

V

∫

4

t40 · δx4 dA (5.105)

holds. The third integral is now rewritten by means of the boundary conditions (5.99)– (5.100) to give

P : δF = δF :1

V

∫

∂B0

t0 ⊗X dA+1

V

∫

2

t20 · δx2 dA− 1

V

∫

2

t20 · δx2 dA . (5.106)


It is obvious that the last two integrals cancel each other. The application of theaverage stress relation (5.25) yields

P : δF = δF :1

V

∫

B0

P dV = P : δF , (5.107)

proving the admissibility of the hybrid boundary conditions as they fulfill the virtualwork criterion.For the computation of the consistent tangent matrix A, the same rules as describedin section 5.1.2 are applied. The proof of thermodynamic consistency can be carriedout as already described in section 5.1.3.

Chapter 6

Multiscale benchmark tests

The applicability of the proposed homogenization procedure to bulk and interfaceproblems will be presented in the following. In section 6.1, the capability of themicromorphic-type elastic-plastic model to predict size-effects in the hardening regimeis investigated. In section 6.2, the interface homogenization framework is applied tomicromorphic-type elastic-plastic damaging microstructures. For this procedure, size-effects are obtained for a softening RVE response. A curved interface is investigated insection 6.2.4 for a softening elastic-plastic mesostructure.

6.1 Bulk homogenization

In this example, a sheet with a hole under plane strain conditions as can be seen infigure 6.1(a) (measures in mm) is investigated. The 336 macroscopic integration pointsare marked with red dots. The macroscopically homogeneous material consists of RVEsas depicted in figure 6.1(b) which are discretized by Q1P0-elements and contain 7266degrees of freedom. Consequently, approximately 2.4 · 106 degrees of freedom have tobe computed in this problem. For the computations, a cluster with 32 processors wasused, and the RVE computations were carried out in parallel.The RVE consists of a weak elastic-plastic matrix and four stiffer elastic inclusions.Computations for this boundary value problem are carried out in two different ways toshow different kinds of size-effects. The material parameter A, describing the charac-teristic length, is varied while all other parameters, including the size of the RVE, arekept constant in section 6.1.1. In section 6.1.2, A is kept constant and the response iscomputed for different RVE-sizes.The material data for the matrix and inclusion materials (shaded white and gray re-spectively) are given in table 6.1. For the enforcement of the periodicity constraints,an augmented Lagrange parameter of ρ = 100N/mm was used.

6.1.1 Size-effect by variation of the microstiffness

For the constant RVE-size of S = 5µm, the force-displacement curves for a final ver-tical displacement of v = 0.2mm for different values of A are plotted in figure 6.2. It

79

80 CHAPTER 6. MULTISCALE BENCHMARK TESTS

E/ Nmm2 ν τy/

Nmm2 K/ N

mm2 H/ Nmm2

Matrix 10000 0.3 20 1000 1000

Inclusion 50000 0.3 ∞ – 10000

Table 6.1: Material data for bulk homogenization

XY

v

2.5

1

1

Ø1

2.5

(a) Boundary value problem

S

S

(b) RVE

Figure 6.1: Boundary value problem for bulk homogenization

is obvious that the overall response shows a strong dependency on the microstiffness.For large values of A, the material response is stiffer as the evolution of plastic de-formation is limited by the strong influence of the surrounding material. For smallvalues of A, the micromorphic-type model resembles the local model and the plasticevolution is almost purely local and consequently not influenced by the surroundingmaterial. This effect can also be recognized in figures 6.3, 6.4, and 6.5, in which thestress field P22 is plotted for the macroscale and the equivalent plastic deformation αis plotted for selected RVEs located at (−0.519mm| − 0.044mm), (0.519mm|0.044mm)and (0.973mm|0.102mm) respectively.For smaller values of A, the stresses are lower and the plastic deformations are higherthan for larger values of A, which is in accordance with the force-displacement-diagrams.The forces measured for different values of A differ by approximately 30% and the maxi-mum equivalent plastic strains differ by 13%. Beyond the range of 10−1N > A > 10−5Nthe size-effect is faded out and the characteristic length is either too large or too smallto influence the behavior any further. Numerical experiments carried out in [Clasenet al., 2013] show the same behavior. This example proves that the behavior which wasalready observed in single-scale computations (see figure 4.10 in section 4.2) is reflectedin the multiscale procedure as well.


v/mm

Force/N

0 0.05 0.1 0.15 0.2

0

200

400

600

A = 10−5N

A = 10−3N

A = 10−1N


Figure 6.3: Macroscopic stress and mesoscopic equivalent plastic strain for A = 10−1N


6.1.2 Size-effect by variation of the RVE-size

In this investigation, the material data from table 6.1 are used once again. The aug-mented Lagrange penalty parameter ρ = 100N/mm remains unchanged. A constantmicrostiffness A = 10−3N is assumed.The influence of the RVE-size on the overall response is visualized in figure 6.6: Thebehavior is much stiffer for smaller RVE-sizes, which reflects the general rule of thumb“smaller is stronger” for size-dependent materials. An RVE-size of 1µm leads to anoverall force exceeding the force for an RVE-size of 50µm by approximately 30%. Thiseffect is in accordance with the experiments by [Taylor, 1924] and [Brenner, 1956]. Itcan be explained by the fact that the boundary layer of the inelastic deformation be-tween matrix and inclusion becomes larger with respect to the overall size of the RVE.Thus, the evolution of plastic distortion in space is limited by the gradient terms of themicromorphic-type formulation. Again, the size-effect fades out for smaller or largerRVEs.Figures 6.7, 6.8, and 6.9 show the macroscopic stress distributions and mesoscopicequivalent plastic distortions for the RVEs mentioned in section 6.1.1. Larger RVEslead to smaller stresses and larger inelastic deformations. In this particular exam-ple, the maximum equivalent plastic strains for S = 1µm and S = 50µm differ byapproximately 13%.

v/mm

Force/N

0 0.05 0.1 0.15 0.2

0

200

400

600

S = 50µm

S = 7.5µm

S = 1µm

Figure 6.6: Size-effect due to a variation of the RVE-size


Figure 6.7: Macroscopic stress and mesoscopic equivalent plastic strain for S = 1µm

Figure 6.8: Macroscopic stress and mesoscopic equivalent plastic strain for S = 7.5µm


Figure 6.9: Macroscopic stress and mesoscopic equivalent plastic strain for S = 50µm

6.1.3 Macroscopic convergence

In table 6.2, the macroscopic convergence behavior for the last two load steps of thecomputation from section 6.1.2 with S = 1µm is documented. An adaptive schemeto increase the prescribed displacement v was used, where the starting (and minimal)increment was 1µm and the maximum increment was 20µm.The deviation from the expected quadratic convergence rate in the last iterative stepof each load step is induced by the fact that the yield condition at the mesoscopicGauss-point level is only fulfilled with an accuracy of 10−12MPa.A very similar convergence behavior is obtained for the RVE-size of 50µm indicatingthe robustness of the consistently linearized multiscale framework.

6.2 Interface homogenization

The problem for the investigation of the interface homogenization framework consistsof an elastic macroscopic bulk material with a horizontal interface layer (shaded inlight gray). The bottom edge is fixed in both directions and the horizontal displace-ment u is imposed on the upper edge. A shear-dominant loading mode for the interfacewas chosen intentionally, as the assumed hybrid boundary conditions hinder the lat-eral contraction in M-direction. This effect is significant especially for heterogeneousmesostructures and leads to high, non-physical hydrostatic stress states if large dis-


load step v/mm ∆v/µm iteration rel. residual norm14 0.1845 20 2 3.35214 0.1845 20 3 1.106·10−1

14 0.1845 20 4 6.769·10−4

14 0.1845 20 5 8.764·10−8

14 0.1845 20 6 5.040·10−13

15 0.2 15.5 2 2.65115 0.2 15.5 3 1.076·10−1

15 0.2 15.5 4 6.863·10−4

15 0.2 15.5 5 6.456·10−8

15 0.2 15.5 6 4.139·10−13

Table 6.2: Macroscopic convergence for S = 1µm

load step v/mm ∆v/µm iteration rel. residual norm14 0.1821 20 2 4.13914 0.1821 20 3 3.594·10−1

14 0.1821 20 4 8.361·10−3

14 0.1821 20 5 8.361·10−6

14 0.1821 20 6 8.310·10−12

15 0.2 17.9 2 3.93515 0.2 17.9 3 4.501·10−1

15 0.2 17.9 4 1.804·10−2

15 0.2 17.9 5 4.725·10−5

15 0.2 17.9 6 3.202·10−10

Table 6.3: Macroscopic convergence for S = 50µm

placements in N-direction are present. However, this effect can be neglected in thecase of shear loading.The mesostructure consists of a micromorphic-type damaging elastic-plastic matrixmaterial and three stiffer elastic inclusions (shaded in dark gray). The boundary valueproblem (with measures in mm) and the RVE are depicted in figure 6.10. The ma-terial data are listed in table 6.4. For the discretization of the bulk on the macro-and mesolevel, Q1P0-elements were used. In a single discretized RVE, 4732 degreesof freedom have to be computed. On the macroscale, four interface elements, eachwith two integration points, are present, leading to an overall number of unknowns ofapproximately 38000. In the following subsections, size-effects induced by varying themicrostiffness A and the RVE-size S are presented.

6.2.1 Size-effect by variation of the microstiffness

In this section, the microstiffness A (which is equal for matrix and inclusions) is variedfor a constant RVE-size of S = 10µm and a constant interface thickness of h0 =


E/ Nmm2 ν τy/

Nmm2 K/ N

mm2 Y0/N

mm2 H/ Nmm2

Macroscopic bulk 210000 0.3 ∞ – – –

Mesoscopic matrix 10000 0.3 25 900 0.35 10000

Mesoscopic inclusion 50000 0.3 ∞ – – 10000

Table 6.4: Material data for interface homogenization

XY

1

1

2u

(a) Boundary value problem

S

S(b) RVE

Figure 6.10: Boundary value problem for interface homogenization

0.1mm. An augmented Lagrange stiffness of ρ = 1000N/mm is assumed. The force-displacement-diagrams for different values of A are plotted in figure 6.11. The hugeinfluence of the microstiffness is obvious. For a large A, the material response is muchstiffer as for a small value, which was already observed in previous examples. It canalso be seen that the model is capable of the prediction of softening after a peakload is reached. The maximum displacements which could be reached with a minimaldisplacement increment of 0.1µm were 0.0542mm for A = 5 · 10−5N and 0.0762mm forA = 10−4N. For A = 10−3N, larger displacements would have been possible but for acomparison with the other curves a final displacement of 0.1mm was sufficient.Due to the almost local behavior for small values of A, the thickness of the boundarylayer of the inelastic deformation between the matrix material and the inclusions is verysmall. This allows the damage variable D to evolve much faster than for larger valuesof A. Contour plots of the macroscopic horizontal displacements and the mesoscopicdamage variables for a selected RVE located at (1mm|1mm) are presented in figures6.12–6.14.


u/mm

Force/N

0 0.025 0.05 0.075 0.1

0

100

200

300

A = 5 · 10−5N

A = 10−4N

A = 10−3N


Figure 6.12: Macroscopic displacement and mesoscopic damage for A = 10−3N


Figure 6.13: Macroscopic displacement and mesoscopic damage for A = 10−4N

Figure 6.14: Macroscopic displacement and mesoscopic damage for A = 5 · 10−5N


6.2.2 Size-effect by variation of the RVE-size

In this section, the influence of the RVE-size S on the overall behavior is investigated.The microstiffness is kept constant at A = 10−4N for inclusion and matrix and S isvaried. The other parameters from table 6.4 as well as the stiffness ρ = 1000N/mmand the interface thickness h0 = 0.1mm are still valid.The force-displacement-diagrams from figure 6.15 show a qualitative behavior as al-ready seen in section 6.1.2: For smaller RVEs the response becomes stiffer. The curvefor S = 10µm is identical to the curve from the previous section for A = 10−4N andthe final displacement of 0.0726mm was reached with load increments of 0.1µm. Inthe simulation with S = 100µm, the final displacement of 0.02419mm was computedwith load increments of 0.01µm. It becomes evident that the prediction of the soften-ing regime becomes more and more cumbersome for more local behavior as only verysmall load increments lead to a converging behavior of the macroscopic iterative solu-tion procedure. For this reason, the computation for the RVE-size of S = 100µm wasstopped before the maximum damage value exceeded the one from the computationwith S = 10µm. However, the peak load was reached and a small part of the softeningresponse could be computed. The plots for the displacements and damage distributionscan be found in figures 6.16–6.18.

u/mm

Force/N

0 0.025 0.05 0.075 0.1

0

100

200

300

S = 100µm

S = 10µm

S = 1µm

Figure 6.15: Size-effect due to a variation of the RVE-size


Figure 6.16: Macroscopic displacement and mesoscopic damage for S = 1µm




6.2.3 Macroscopic convergence

Due to the significant sensitivity of the chosen elastic-plastic damaging model withrespect to the size of the load increments, only the stiffest responses from the compu-tations carried out in the previous sections are investigated. The macroscopic conver-gence behavior for the computation with S = 10µm and A = 10−3N is documented intable 6.5. Table 6.6 shows the behavior for S = 1µm and A = 10−4N. The convergencerate is close to the optimal one. Small deviations can again be explained with the errorin accuracy considering the yield criterion. Quadratic convergence behavior can also beobserved in the other computations, even in the softening regime if the load steps aresmall enough. This shows that the interface homogenization framework is linearizedconsistently as well.

load step v/mm ∆u/µm iteration rel. residual norm174 0.0999 0.1 2 1.352·10−2

174 0.0999 0.1 3 3.678·10−6

174 0.0999 0.1 4 3.343·10−10

175 0.1 0.1 2 1.410·10−2

175 0.1 0.1 3 3.794·10−6

175 0.1 0.1 4 3.504·10−10

Table 6.5: Macroscopic convergence for S = 10µm and A = 10−3N


load step v/mm ∆v/µm iteration rel. residual norm201 0.09966 0.5 2 1.516·10−1

201 0.09966 0.5 3 1.834·10−4

201 0.09966 0.5 4 8.404·10−9

202 0.1 0.34 2 1.114·10−1

202 0.1 0.34 3 1.016·10−4

202 0.1 0.34 4 4.504·10−9

Table 6.6: Macroscopic convergence for S = 1µm and A = 10−4N

6.2.4 Investigation of a curved layer

A circular material interface with a thickness of h0 = 1mm and a diameter of 50mm(shaded in light gray) connects a circular inclusion with its center at (0|0) with a squareframe. The macroscopic bulk material (frame and circular part) is assumed to be hy-perelastic. The circular part of the macroscopic material is rotated counterclockwise byimposing displacement boundary conditions on the part which is meshed with squareelements. The mesostructure consists of a micromorphic-type elastic-plastic matrixmaterial and three stiffer elastic inclusions (shaded in dark gray). The RVEs have thesame initial geometry as in the previous computations (see figure 6.10(b)). Each RVEwith a size of S = 0.1mm is rotated in a way that the macroscopic tangential directionof the interface coincides with the mesoscopic direction of the fiber-like inclusion. Theboundary value problem (with measures in mm) is shown in figure 6.19, and the mate-rial data can be found in table 6.7. Again, Q1P0-elements were used on both scales. 48macroscopic interface elements are present. Thus, approximately 450000 DOFs haveto be computed.Plots of the macroscopic vertical displacement v and the macroscopic shear stress σ12for a final angle of rotation of the circular part of ϑ = 0.004rad (approximately 0.23)are presented in figure 6.20. It can be seen that the weak interface decouples the de-formation of the circular part from the deformation of the square frame. The stressdistribution shows that the tractions are passed from one side of the interface to theother side in a consistent manner, although a slight disturbance is recognizable. Thiscan be explained by the fact that the reduced kinematics of the macroscopic interfaceelement are not as accurate as the standard bulk kinematics and the stress states insidethe interface may differ from the the stress state in the surrounding bulk. The stressoscillations at the center of the BVP are a post-processing issue. In the central area,all stresses are zero as a rigid body rotation is imposed via the boundary conditions.The macroscopic torque-angle-diagram, showing the transition from elastic behavior toinelastic behavior, is displayed in figure 6.21.The mesoscopic response is visualized in the subsequent figures 6.22–6.25 where theequivalent plastic strain α and the shear stress σ12 are plotted for the RVEs locatedat (50mm|0), (0|50mm), (−50mm|0), and (0| − 50mm) respectively. Due to the choiceof macroscopic boundary conditions, a heterogeneous macroscopic deformation occurs,which is reflected in the different responses of the RVEs at different integration points.The mesoscopic shear stresses are in a range which is in accordance with the macro-


scopic stress distribution.Although the material responds with a locally softening behavior for large parts ofthe RVEs (as a negative hardening modulus is assumed), the overall stiffness of themesostructure is sufficient to render macroscopic hardening. As already stated in theprevious example, overall softening of the material layer is difficult to predict with theproposed method as load steps have to be very small compared to the overall load.This is even more significant in the case of heterogeneous interface deformations asdescribed in the current example. A stabilization of the solution algorithm with anarc-length parameter to control the size of the load steps (see [Wriggers, 2008] and ref-erences therein) might enable larger macroscopic load steps while maintaining optimalconvergence rates as a result of the consistent linearization.

E/ Nmm2 ν τy/

Nmm2 K/ N

mm2 H/ Nmm2 A/N

Macroscopic bulk 210000 0.3 ∞ – – –

Mesoscopic matrix 10000 0.3 25 −100 1000 0.1

Mesoscopic inclusion 50000 0.3 ∞ – 10000 0.1

Table 6.7: Material data for interface homogenization (2)

X

Y

100

50

50

Ø50 100

Figure 6.19: Boundary value problem for interface homogenization (2)


(a) Vertical displacement (b) Shear stress

Figure 6.20: Macroscopic displacement and shear stress

ϑ/rad

Torque/Nm

0 0.001 0.002 0.003 0.004

0

100

200

300

400

500

Figure 6.21: Torque-angle-diagram


(a) Equivalent plastic strain (b) Shear stress

Figure 6.22: RVE-response at (50mm|0)


Figure 6.23: RVE-response at (0|50mm)



Figure 6.24: RVE-response at (−50mm|0)


Figure 6.25: RVE-response at (0| − 50mm)

Chapter 7

Concluding remarks

This work provides a consistent computational two-scale homogenization frameworkwhich accounts for size-effects in the inelastic regime induced by the mesoscopic ma-terial behavior. Micromorphic-type inelastic media as proposed in [Forest, 2009] con-stitute (heterogeneous) Representative Volume Elements describing the mesostructureof a body. These RVEs, which are attached to each quadrature point in a macroscopicboundary value problem discretized by Finite Elements, are treated as independentsubscale boundary value problems. Consequently, the described framework is a varia-tion of the FE2 solution scheme developed in [Miehe et al., 1999b] and [Feyel, 1999].The averaged stress response of an individual RVE to imposed boundary conditionsderived from a macroscopic deformation is computed as well as the stress sensitivitywith respect to the macroscopic deformation. These quantities replace a constitutiverelation on the macrolevel, providing a flexible framework which is solved by meansof a nested iterative procedure. The consistent linearization guarantees computationalefficiency as optimal convergence rates can be achieved on the macroscopic scale.Finite Elements for micromorphic-type models describing large strain hyperelasto-plasticity and damaging hyperelasto-plasticity are implemented and tested in single-scale computations. Volumetric locking due to the assumed plastic incompressibilityconstraint is avoided by using constant pressure elements. A variation of the inherentcharacteristic length of the micromorphic-type approach shows the possibility of pre-dicting size-effects. Additionally, due to the presence of gradient terms of the additionalkinematic field, mesh independence is guaranteed for softening material responses inthe case of damage evolution.The elastic-plastic micromorphic-type model is successfully used on the mesoscopicscale within the homogenization framework to predict size-effects either by variationof the characteristic length or by variation of the RVE-size. The material responsebecomes stiffer for a large characteristic length compared to the RVE-size. This effectis in accordance with observations of experiments involving heterogeneous materials.Periodic boundary conditions ensure the correct transition of the virtual work fromthe subscale to the macroscale. The thermodynamic consistency of the framework,although a macroscopic counterpart of the micromorphic-type variable is absent, isproven.The multiscale procedure is also applied to cohesive fracture problems where the macro-

99

100 CHAPTER 7. CONCLUDING REMARKS

scopic failure pattern is known a priori and can be discretized by cohesive interfaceelements. The simulations were carried out assuming a damaging elastic-plastic meso-structure. Hybrid boundary conditions especially suited for thin material layers fulfill-ing the virtual work criterion are introduced. In this framework, size-effects accompa-nying the variation of the characteristic length or the RVE-size can even be predictedin the softening regime.However, some enhancements of this homogenization procedure still have to be inves-tigated to obtain a tool for the two-scale computation of heterogeneous materials usedin practical applications. First of all, an adaptation of this method to three dimensionshas to be performed, as three-dimensional effects are especially important in microme-chanics and the assumption of plane strain conditions in an RVE is only relevant intheory.The chosen hybrid boundary conditions for the interface homogenization procedurelead to very stiff solutions in the case of loading in the direction normal to the inter-face. This is especially the case for heterogenous RVEs as the lateral contraction ofthe weaker material is not predicted correctly. This effect leads to high hydrostaticstress states rendering non-physical solutions. A remedy to this problem might be theapplication of the window-homogenization method as described in [Hain and Wrig-gers, 2008a], [Hain and Wriggers, 2008b], [Temizer et al., 2013], and references therein.Within this method, the boundary conditions are not imposed on the RVE directlybut rather on a surrounding “window” of average stiffness, weakening the constraintsinduced by the periodic boundary conditions.As of now, only phenomenological models normally used in macroscopic computationsare applied to the mesolevel to describe inelastic effects. In reality, these models areno longer applicable as they do not capture the actual microstructural processes. Inmetals, a crystal plasticity model (see [Steinmann and Stein, 1996], [Lehmann, 2013],and references therein) can be used to describe the inelastic deformation behavior onslip systems inside a single- or a polycrystalline RVE. To take size-effects into account,a generalized crystal plasticity model like the gradient dependent theories describedin [Gurtin, 2002], [Evers et al., 2004], or micromorphic-type theories as described in[Cordero et al., 2013] can be used. In [Gottschalk et al., 2014], grain boundary effectsarising in gradient crystal plasticity within polycrystals are investigated thoroughly.In rubbery polymers, a microstructural model accounting for crazing and fibrillationas described in [Basu et al., 2005] can be used on the small scale. For the predictionof size-effects, this model still has to be incorporated in a generalized framework.Still an unresolved issue within this framework is the computation of macroscopic de-formation localization induced by mesoscopic softening in bulk applications where thefailure path is not known in advance. Several different solution techniques for this kindof problem are already mentioned in section 1.2.4. These are for example the macro-scopic Finite Elements with embedded discontinuities by [Coenen et al., 2012a] and themacroscopic XFEM approach by [Belytschko et al., 2008] and [Nguyen et al., 2012].However, these methods require special Finite Element techniques on the macroscopicscale which are not straightforward from an implementational point of view. This isespecially the case if the benefits of a symbolic, automatic-differentiation-based ap-

101

proach are to be exploited. Within the present approach, the integration of a macro-scopic phase-field model as described in [Kuhn and Muller, 2010] and [Miehe et al.,2010] is preferred, as an additional macroscopic primary field with an associated freeenergy can be integrated easily without the necessity of complex macroscopic elementformulations. This approach is very similar to the micromorphic-type approach of themesoscopic models. Within the macroscopic phase-field approach in a homogenizationframework, the coupling of the phase-field and the mesoscopic failure mechanisms stillhas to be investigated. A simple approach might be the use of an averaged damagevariable as a source term in the governing phase-field equation.

102 CHAPTER 7. CONCLUDING REMARKS

List of Figures

2.1 Motion of a body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Multiplicative split of the deformation gradient . . . . . . . . . . . . . 212.3 Micropolar motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.4 Micromorphic motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.1 FE-discretization of a domain . . . . . . . . . . . . . . . . . . . . . . . 363.2 Isoparametric elements . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.1 Prandtl-indentation test . . . . . . . . . . . . . . . . . . . . . . . . . . 48(a) Boundary value problem . . . . . . . . . . . . . . . . . . . . . . . 48(b) Analytical solution by [Prandtl, 1920] . . . . . . . . . . . . . . . . 48

4.2 Force-displacement-diagrams for Q1-elements . . . . . . . . . . . . . . . 494.3 Force-displacement-diagrams for Q1P0-elements . . . . . . . . . . . . . 494.4 Displacement and plastic strain for Q1-elements . . . . . . . . . . . . . 50

(a) Displacement 64×32 . . . . . . . . . . . . . . . . . . . . . . . . . 50(b) Equivalent plastic strain 64×32 . . . . . . . . . . . . . . . . . . . 50(c) Displacement 128×64 . . . . . . . . . . . . . . . . . . . . . . . . 50(d) Equivalent plastic strain 128×64 . . . . . . . . . . . . . . . . . . 50(e) Displacement 256×128 . . . . . . . . . . . . . . . . . . . . . . . . 50(f) Equivalent plastic strain 256×128 . . . . . . . . . . . . . . . . . . 50(g) Displacement 512×256 . . . . . . . . . . . . . . . . . . . . . . . . 50(h) Equivalent plastic strain 512×256 . . . . . . . . . . . . . . . . . . 50

4.5 Displacement and plastic strain for Q1P0-elements . . . . . . . . . . . 51(a) Displacement 64×32 . . . . . . . . . . . . . . . . . . . . . . . . . 51(b) Equivalent plastic strain 64×32 . . . . . . . . . . . . . . . . . . . 51(c) Displacement 128×64 . . . . . . . . . . . . . . . . . . . . . . . . 51(d) Equivalent plastic strain 128×64 . . . . . . . . . . . . . . . . . . 51(e) Displacement 256×128 . . . . . . . . . . . . . . . . . . . . . . . . 51(f) Equivalent plastic strain 256×128 . . . . . . . . . . . . . . . . . . 51(g) Displacement 512×256 . . . . . . . . . . . . . . . . . . . . . . . . 51(h) Equivalent plastic strain 512×256 . . . . . . . . . . . . . . . . . . 51

4.6 Stress-strain-diagram for copper (CW 024 A) . . . . . . . . . . . . . . 524.7 Mesh independence of localization behavior . . . . . . . . . . . . . . . . 53

(a) FE-mesh of a strip with a hole . . . . . . . . . . . . . . . . . . . 53(b) Force-displacement-diagrams . . . . . . . . . . . . . . . . . . . . . 53

4.8 Localization of plastic deformation . . . . . . . . . . . . . . . . . . . . 54

103

104 LIST OF FIGURES

(a) Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54(b) Simulation (equivalent plastic strain) . . . . . . . . . . . . . . . . 54

4.9 Force-displacement-diagrams for a strip with a hole . . . . . . . . . . . 544.10 Size-effect due to a variation of the microstiffness . . . . . . . . . . . . 55

5.1 FE2-discretization of a domain with RVE . . . . . . . . . . . . . . . . . 595.2 Nested two-scale solution scheme . . . . . . . . . . . . . . . . . . . . . 605.3 RVE boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 62

(a) Undeformed RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . 62(b) Linear displacement boundary conditions . . . . . . . . . . . . . . 62(c) Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . 62

5.4 Node ordering for constraint elements on vertical and horizontal RVEedges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.5 Isoparametric four-node interface elements . . . . . . . . . . . . . . . . 70(a) Reference space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70(b) Reference configuration . . . . . . . . . . . . . . . . . . . . . . . 70(c) Current configuration . . . . . . . . . . . . . . . . . . . . . . . . 70

5.6 RVE orientation for inclined interfaces . . . . . . . . . . . . . . . . . . 755.7 RVE boundary conditions for material layers . . . . . . . . . . . . . . . 75

(a) Linear displacement boundary conditions . . . . . . . . . . . . . . 75(b) Periodic boundary conditions . . . . . . . . . . . . . . . . . . . . 75(c) Hybrid boundary conditions . . . . . . . . . . . . . . . . . . . . . 75

6.1 Boundary value problem for bulk homogenization . . . . . . . . . . . . 80(a) Boundary value problem . . . . . . . . . . . . . . . . . . . . . . . 80(b) RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

6.2 Size-effect due to a variation of the microstiffness . . . . . . . . . . . . 816.3 Macroscopic stress and mesoscopic equivalent plastic strain for A = 10−1N 816.4 Macroscopic stress and mesoscopic equivalent plastic strain for A = 10−3N 826.5 Macroscopic stress and mesoscopic equivalent plastic strain for A = 10−5N 826.6 Size-effect due to a variation of the RVE-size . . . . . . . . . . . . . . . 836.7 Macroscopic stress and mesoscopic equivalent plastic strain for S = 1µm 846.8 Macroscopic stress and mesoscopic equivalent plastic strain for S = 7.5µm 846.9 Macroscopic stress and mesoscopic equivalent plastic strain for S = 50µm 856.10 Boundary value problem for interface homogenization . . . . . . . . . . 87

(a) Boundary value problem . . . . . . . . . . . . . . . . . . . . . . . 87(b) RVE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.11 Size-effect due to a variation of the microstiffness . . . . . . . . . . . . 886.12 Macroscopic displacement and mesoscopic damage for A = 10−3N . . . 886.13 Macroscopic displacement and mesoscopic damage for A = 10−4N . . . 896.14 Macroscopic displacement and mesoscopic damage for A = 5 · 10−5N . . 896.15 Size-effect due to a variation of the RVE-size . . . . . . . . . . . . . . . 906.16 Macroscopic displacement and mesoscopic damage for S = 1µm . . . . 916.17 Macroscopic displacement and mesoscopic damage for S = 10µm . . . . 916.18 Macroscopic displacement and mesoscopic damage for S = 100µm . . . 92

LIST OF FIGURES 105

6.19 Boundary value problem for interface homogenization (2) . . . . . . . . 946.20 Macroscopic displacement and shear stress . . . . . . . . . . . . . . . . 95

(a) Vertical displacement . . . . . . . . . . . . . . . . . . . . . . . . . 95(b) Shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

6.21 Torque-angle-diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.22 RVE-response at (50mm|0) . . . . . . . . . . . . . . . . . . . . . . . . . 96

(a) Equivalent plastic strain . . . . . . . . . . . . . . . . . . . . . . . 96(b) Shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.23 RVE-response at (0|50mm) . . . . . . . . . . . . . . . . . . . . . . . . . 96(a) Equivalent plastic strain . . . . . . . . . . . . . . . . . . . . . . . 96(b) Shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.24 RVE-response at (−50mm|0) . . . . . . . . . . . . . . . . . . . . . . . . 97(a) Equivalent plastic strain . . . . . . . . . . . . . . . . . . . . . . . 97(b) Shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.25 RVE-response at (0| − 50mm) . . . . . . . . . . . . . . . . . . . . . . . 97(a) Equivalent plastic strain . . . . . . . . . . . . . . . . . . . . . . . 97(b) Shear stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

106 LIST OF FIGURES

List of Tables

4.1 Material data for Prandtl-indentation test . . . . . . . . . . . . . . . . 484.2 Material data for copper (CW 024 A) . . . . . . . . . . . . . . . . . . . 52

6.1 Material data for bulk homogenization . . . . . . . . . . . . . . . . . . 806.2 Macroscopic convergence for S = 1µm . . . . . . . . . . . . . . . . . . 866.3 Macroscopic convergence for S = 50µm . . . . . . . . . . . . . . . . . . 866.4 Material data for interface homogenization . . . . . . . . . . . . . . . . 876.5 Macroscopic convergence for S = 10µm and A = 10−3N . . . . . . . . . 926.6 Macroscopic convergence for S = 1µm and A = 10−4N . . . . . . . . . 936.7 Material data for interface homogenization (2) . . . . . . . . . . . . . . 94

107

108 LIST OF TABLES

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Curriculum vitae 121

Curriculum vitae

Name: Heiko Clasen

Date of Birth: January 7, 1986

Place of Birth: Lemgo, Germany

Nationality: German

Occupation

since 01.2011 Research assistant at the Institute of Continuum MechanicsGottfried Wilhelm Leibniz Universitat Hannover

University

10.2005 – 12.2010 Study of Mechanical EngineeringGottfried Wilhelm Leibniz Universitat Hannover

Degree: Dipl.-Ing. (with distinction)

11.2007 – 07.2009 Student assistant at the Institute of Machine Design andTribologyGottfried Wilhelm Leibniz Universitat Hannover

School

08.1996 – 06.2005 Secondary schoolEngelbert-Kaempfer-Gymnasium Lemgo

08.1992 – 07.1996 Primary schoolGrundschule Horstmar

Awards

2011 Award of the Dr. Jurgen and Irmgard Ulderup Foundation(Final Exam)

2011 Award of the Victor-Rizkallah-Foundation and theFoundation NiedersachsenMetall (Final Exam)

2007 Award of the Dr. Jurgen and Irmgard Ulderup Foundation(Intermediate Exam)

to those who believed in me

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