crystal plasticity and grain boundaries on small scales

Crystal plasticity and grain boundaries onsmall scales – modeling and numerical

implementation

Von der Fakultät für Bauingenieurwesender Rheinisch-Westfälischen Technischen Hochschule Aachen

zur Erlangung des akademischen Grades einer Doktorin der Ingenieurwissenschaftengenehmigte Dissertation

vorgelegt vonAtefeh Alipour Kiakalaee, M. Sc.

Berichter*innen: Prof. Dr.-Ing. Stephan WulfinghoffProf. Dr.-Ing. Stefanie ReeseProf. Dr. rer. nat. Bob Svendsen

Tag der mündlichen Prüfung: 20. August 2020

Diese Dissertation ist auf den Internetseiten der Universitätsbibliothek online verfügbar.

Dedicated to my beloved husband and mother

Acknowledgements

The present dissertation has been developed during my activities as a research assistant atAachen Institute for Advanced Study in Computational Engineering Science (AICES), theInstitute of Applied Mechanics (IFAM) at RWTH Aachen University and the Institute forMaterials Science at Kiel University. The financial support provided by the German ResearchFoundation (DFG) through Grant GSC 111 as well as WU 847/1-1 is gratefully acknowledged.At this point, I would like to express my gratitude to those who assisted me and contributed tothe successful completion of this dissertation.

First of all, I would like to express my sincere gratitude to my first supervisor Prof. Dr.-Ing.Stephan Wulfinghoff who has supported me during the development of this work. He gaveme the opportunity to make a new part in my life with my passion to learn. I thank himfor his great understanding, respectful and friendly behavior and his high level of trust in myresearch which led to a productive and creative atmosphere of work. I really appreciate manyinteresting technical discussions with him which have played an important role to get the bestout of my work and myself scientifically. In addition, I thank my second supervisor Prof.Dr.-Ing. Stefanie Reese for her support and precious comments during my studies. My specialthanks go to Prof. Dr. rer. nat. Bob Svendsen who has always been a great supporter andinspiration, and agreed to be the third reviewer of this dissertation. His guidance and helpthroughout my doctoral studies is greatly appreciated. I also thank Prof. Dr.-Ing. Sven Klinkelwho agreed to be the chairman of the dissertation committee.

Moreover, I would like to take the opportunity to thank my colleagues who contributed tosuch a friendly and respectful atmosphere in which we had fruitful scientific discussions. Aspecial thank goes toDr.-Ing. Hamidreza bayat, Dr.-Ing. ShahedRezaei, Dr.-Ing. TimBrepols,Dr.-Ing. Kavan Khaledi, Dr.-Ing. Julian Kochmann, Lukas Poggenpohl, Katharina Martin,Mohammad Khorrami, Thank Thank Nguyen and Marian Sielenkämper for their friendshipand the scientific discussions.

I also appreciate the support from AICES managing director, Dr. Nicole Faber and herprofessional team. Besides, I thank Eva-Maria Goertz, the secretary at IFAM,AxelMittendorf,IT administrator at IFAM, and Marina Pejić, the secretary at the Institute of Materials ScienceKiel university for their supports and kindness.

6

At this point, I would like to express my gratitude to my mother who has always supportedme with her endless kind words and taught me to strive for my goals. I also thank my siblings,Reza, Behzad and Arefeh as well as their families whom I can always count on. The greatestthanks go to my husband, Elias M. Zirdehi, for his affection, encouragement, understandingand patience, who always believes in me and stands by my side to give me full strength andendless support.

Bochum, August 2020 Atefeh Alipour

Summary

Metals are used for a wide range of applications in the industry due to their durability, strengthand ductility. Since dislocations are the fundamental reasons for plastic deformation in metals,the movement of dislocations, their interactions with each other as well as with the grainboundaries (GBs) have been investigated by numerous authors for some decades.

In this regard, the current dissertation represents a compilation of published articles of theauthor (and her coauthors). This thesis addresses geometrically nonlinear plastic deformationof face centered cubic (fcc) materials on the large and small scales using continuum ap-proaches. In large-scale applications, classical plasticity (size-independent) models are mostlyused which are in agreement with experimental data while inhomogeneous plastic deformationof materials on the microscale is investigated with strain-gradient theories by introducing aninternal length scale into the models.

In the study of large-scale crystalline materials, the main problem is to deal with the numer-ical issues when geometrically nonlinear plasticity approaches are implemented, e.g., lockingphenomenon. Moreover, single crystal simulations at room temperature often necessitate apower-law-type flow rule with high rate sensitivity exponent to capture the actual behavior ofthe material deformation, which leads to a complicated convergence of nonlinear equations. Tosolve such issues, a regularization method for the power law with high value of the sensitivityexponent in combination with a new concept for hybrid discontinuous Galerkin (DG) methods–control points– is presented in geometrically nonlinear crystal plasticity framework, leadingto a numerically efficient, robust and locking-free model (article 1).

On the micro-scale, the presence of grain boundaries results in pile-ups of dislocations andstrengthening of the material (Hall-Petch effect). Therefore, a grain boundary model in theconcept of geometrically nonlinear viscoplasticity is presented to improve single crystal models(article 2). This model is based on the dislocation density tensor and plastic surface deforma-tion which leads to a grain boundary yield criterion with isotropic and kinematic hardening.The grain boundary hardening effects are shown in cyclic shear deformation of bicrystals. Inthis model, the grain boundary strength is assumed to be a function of grain misorientation.

Subsequently, the grain boundary model in article 2 is extended by evaluating the grainboundary strength with regard to the grain misorientation using a geometrical transmissibil-ity parameter (article 3). To investigate the effect of mismatch between adjacent grains onthe grain boundary strength and dislocation transmission at the grain boundaries, randomlyoriented polycrystals are compared with textured polycrystals.

Zusammenfassung

Metalle werden aufgrund ihrer Haltbarkeit, Festigkeit und Duktilität für ein breites Spektrumvon Anwendungen in der Industrie genutzt. Da Versetzungen die grundlegenden Wirkme-chanismen für die plastische Verformung in Metallen darstellen werden die Bewegung vonVersetzungen, ihre Wechselwirkungen untereinander sowie mit den Korngrenzen seit einigenJahrzehnten von zahlreichen Autoren untersucht.

In diesem Zusammenhang stellt die aktuelle Dissertation eine Zusammenstellung der veröf-fentlichten Artikel der Autorin (und ihrer Koautoren) dar. Das Hauptziel ist die Untersuchungder geometrisch nichtlinearen plastischen Verformung von kubisch flächenzentrierten (kfz)Materialien auf der macro und micro Skala mit Kontinuummechanischen Ansätzen. In groß-skaligen Anwendungen werden meist klassische (größenunabhängige) Plastizitätsmodelle ver-wendet, die mit experimentellen Daten übereinstimmen, während die inhomogene plastischeVerformung von Materialien auf der Mikroskala mit Dehnungsgradiententheorien untersuchtwird, indem eine interne Längenskala in die Modelle eingeführt wird.

Bei der Untersuchung der großskaligen kristallinen Materialien besteht das Hauptproblemdarin, die numerischen Problemstellungen (z.B. das Phänomen des Lockings) zu behandeln,wenn geometrisch nichtlineare Plastizitätsansätze implementiert werden. Darüber hinaus er-fordern Einkristallsimulationen bei Raumtemperatur oft eine Fließregel vom Power-Law-Typmit einem Exponenten mit hoher Ratenempfindlichkeit, um das tatsächliche Verhalten derMaterialverformung zu erfassen, was zu einer komplizierten Konvergenz der nichtlinearenGleichungen führt. Um solche Probleme zu lösen, wird eine Regularisierungsmethode für dasLeistungsgesetz mit hohem Wert des Empfindlichkeitsexponenten in Kombination mit einemneuen Konzept für hybride diskontinuierliche Galerkin Methoden -Kontrollpunkte- in einemgeometrisch nichtlinearen Kristallplastizitätsrahmen vorgestellt, was zu einem numerisch effi-zienten, robusten und lockingfreien Modell führt (Artikel 1).

Auf der Mikroskala führt das Vorhandensein von Korngrenzen zu Ansammlungen vonVersetzungen und Verstärkung des Materials (Hall-Petch-Effekt). Deshalb wird ein Korngren-zenmodell im Konzept der geometrisch nichtlinearen Viskoplastizität vorgestellt, um Einkris-tallmodelle zu verbessern (Artikel 2). Dieses Modell basiert auf dem Versetzungsdichtetensorund der plastischen Oberflächenverformung, was zu einem Korngrenzenfließkriteriummit iso-troper und kinematischer Verfestigung führt. Die Korngrenzenverfestigungseffekte werden inder zyklischen Scherverformung von Bikristallen gezeigt. In diesem Modell wird die Korn-grenzenfestigkeit als ein Parameter angenommen, der nicht beeinflusst durch andere Faktorenwird.

10

Anschließend wird das Korngrenzenmodell in Artikel 2 erweitert, indem die Korngrenzen-festigkeit im Hinblick auf die unterschiedliche kristalline ausrichtungen der Körner mit Hilfeeines geometrischen Durchlässigkeitsparameters bewertet wird (Artikel 3). Um die Auswir-kung der unterschiedlichen Ausrichtung zwischen benachbarten Körnern auf die Korngren-zenfestigkeit und die Versetzungsübertragung an den Korngrenzen zu untersuchen, werdenzufällig orientierte Polykristalle mit texturierten Polykristallen verglichen.

Contents

1 Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 State-of-the-art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Dislocations in crystalline materials . . . . . . . . . . . . . . . . . . 31.2.2 Continuum study of dislocations . . . . . . . . . . . . . . . . . . . . 41.2.3 Grain boundary modeling in continuum theories . . . . . . . . . . . 5

1.3 Outline of the dissertation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Article 1:The concept of control points in hybrid discontinuous Galerkin meth-ods – application to geometrically nonlinear crystal plasticity 92.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Review of the hybrid DG formulation by Wulfinghoff et al. [2017] . . . . . . 12

2.3.1 Weak form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.3 Linearization of the residual . . . . . . . . . . . . . . . . . . . . . . 16

2.4 A symmetric discontinuousGalerkinmethod introducing the concept of controlpoints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4.2 Variational formulation and introduction of control points . . . . . . 172.4.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5 Review of single crystal viscoplasticity . . . . . . . . . . . . . . . . . . . . . 232.5.1 Free energy, dissipation and consistent flow rule . . . . . . . . . . . . 232.5.2 Time discretization by the midpoint rule . . . . . . . . . . . . . . . . 252.5.3 Regularization of the power law . . . . . . . . . . . . . . . . . . . . 252.5.4 Residuals and the linearization process . . . . . . . . . . . . . . . . 272.5.5 Algorithmic tangent operator . . . . . . . . . . . . . . . . . . . . . . 29

I

II Contents

2.6 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.6.1 Planar double slip single crystal under uniaxial load . . . . . . . . . . 302.6.2 Oligocrystal under uniaxial load . . . . . . . . . . . . . . . . . . . . 35

2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.8.1 Derivation of ∂uΓJuI in the linearization of the residual . . . . . . . . 382.8.2 The matrix form of dF T in Eq. (2.14) . . . . . . . . . . . . . . . . . 382.8.3 Evaluation of the dissipation inequality . . . . . . . . . . . . . . . . 382.8.4 Linearization of Eqns. (2.48) and (2.49) . . . . . . . . . . . . . . . . 392.8.5 Derivation of dSeS and dCC

etr in Eq. (2.64) . . . . . . . . . . . . . 412.8.6 Derivation of the algorithmic tangent operator . . . . . . . . . . . . . 41

3 Article 2:A grain boundary model for gradient-extended geometrically nonlinearcrystal plasticity: theory and numerics 433.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.3 Gradient-extended single crystal plasticiy . . . . . . . . . . . . . . . . . . . 46

3.3.1 Basics of single crystal elasto-viscoplasticity . . . . . . . . . . . . . 463.3.2 Continuum plastic surface deformation . . . . . . . . . . . . . . . . 473.3.3 Free energy, dissipation and consistent flow rule . . . . . . . . . . . . 48

3.4 Plastic deformation description on the grain boundaries . . . . . . . . . . . . 513.4.1 Dissipation, free energy and grain boundary yield criterion . . . . . 513.4.2 Radial return mapping algorithm . . . . . . . . . . . . . . . . . . . . 53

3.5 Computational implementation in material model . . . . . . . . . . . . . . . 533.5.1 Time discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.5.2 Modification of the Newton-scheme . . . . . . . . . . . . . . . . . . 543.5.3 Residuals and the linearization process . . . . . . . . . . . . . . . . 54

3.6 Finite element implementation . . . . . . . . . . . . . . . . . . . . . . . . . 573.6.1 Weak forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.6.2 Linearization of the variational forms . . . . . . . . . . . . . . . . . 58

3.7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.7.1 Example 1: plastic flow in a channel . . . . . . . . . . . . . . . . . . 613.7.2 Example 2: bicrystal under shear load with one slip system in each grain 623.7.3 Example 3: bicrystal under cyclic shear load . . . . . . . . . . . . . 67

Contents III

3.7.4 Example 4: bicrystal under shear load with two slip systems in eachgrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 713.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.9.1 Calculation of the stress power . . . . . . . . . . . . . . . . . . . . . 723.9.2 Evaluation of the dissipation inequality . . . . . . . . . . . . . . . . 733.9.3 Fourth-order box products in matrix notation . . . . . . . . . . . . . 743.9.4 Derivation of dF p in Eqns. (3.57) and (3.74) . . . . . . . . . . . . . 743.9.5 Derivation of ∂X#x in Eq. (3.60) . . . . . . . . . . . . . . . . . . . 753.9.6 Calculation of the weak form of Eq. (3.11) . . . . . . . . . . . . . . 763.9.7 Derivation of F etr

dSeF etrT in Eq. (3.66) . . . . . . . . . . . . . . . 76

3.9.8 Derivation of dSe . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.9.9 Derivation of dBmχ in Eq. (3.83) . . . . . . . . . . . . . . . . . . . 78

3.9.10 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4 Article 3:A grain boundary model considering the grain misorientation within ageometrically nonlinear gradient-extended crystal viscoplasticity theory 834.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3 Gradient-extended geometrically nonlinear crystal viscoplasticity . . . . . . . 86

4.3.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.3.2 Plastic surface deformation . . . . . . . . . . . . . . . . . . . . . . . 874.3.3 Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 884.3.4 Free energy density . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.3.5 Crystal plasticity flow rule . . . . . . . . . . . . . . . . . . . . . . . 914.3.6 Grain boundary yield criterion . . . . . . . . . . . . . . . . . . . . . 91

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.4.1 Comparison between textured polycrystals and random-oriented poly-

crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.4.2 Comparison between textured polycrystals with elongated grain shape

and random-oriented polycrystals . . . . . . . . . . . . . . . . . . . 1004.4.3 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

IV Contents

4.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1044.6.1 Proof ofm : B = M : B . . . . . . . . . . . . . . . . . . . . . . . 104

5 Conclusions and Outlook 105

List of Figures 109

List of Tables 113

Bibliography 115

1 Introduction

1.1 Motivation

Dislocations are the main reason of plasticity in the crystalline materials. They move throughthe grains, attract or repel each other and pile up at the grain boundaries or at other interfaces(e.g., with precipitates). The significant properties of crystalline materials (e.g., high strengthand processability) motivate many researchers to further study the behavior of metals (crystals)implementing material models.

Material models can be classified into discrete and continuum approaches. Discrete theoriesare often time-consuming due to the fact that individual dislocations and large numbers of theirinteractions are taken into account. Consequently, their applications are limited when theyare used for large-scale simulations. In contrast, continuum models, although less detailed,have the benefit of fast computation as a result of the average representation of dislocations(dislocation density).

Moreover, continuum plasticity theories in case of crystalline materials can be classifiedinto classical (size-independent) and gradient-extended (size-dependent) theories. Classicalapproaches are used on the large-scale where there is no size effect. The numerical issues areoften the most important problems when classical models for geometrically nonlinear plasticitymodels are implemented. For instance, a locking effect appears in finite element simulations,especially in geometrically nonlinear frameworks. Moreover, size-independent approaches arelimited when plastic deformation of the material is localized and strongly heterogeneous as aresult of micro-structured characteristics such as grain boundaries.

On the small-scale applications, the material strength increases by decreasing the grainsize. In contrast to classical theories, gradient-extended models are able to capture the sizeeffect phenomenon (Hall-Petch effect) by introducing an internal length scale into the models.Furthermore, heterogeneous plasticity on the small-scale, pile-ups of dislocations at the grainboundaries or other interfaces and transmission of dislocations through the grain boundariescan be modeled with gradient-extended approaches.

Since the presence of grain boundaries results in a modification of the mechanical response

1

2 1 Introduction

of the material, compared to single crystals, the influence of grain boundaries necessitates tobe investigated in modeling the crystalline materials on the small-scale. The grain boundaryacts as obstacle which to a certain degree impedes the movement of dislocations due to themismatch of the crystal lattices of adjacent grains. As a result, dislocations tend to pile up atthe grain boundary, increasing the back stress. Plastic deformation of the grain boundary be-comes increasingly activated when the back stress reaches a sufficiently high value and allowsdislocations to interact with the grain boundary. In computational works, the grain boundariesare mostly investigated by two extreme conditions of micro-hard (i.e., dislocations remain ineach grain and are not allowed to cross the grain boundary) and micro-free (i.e., dislocationstransmit freely the grain boundary neglecting the mismatch between adjacent grains).

In reality, the grain boundaries are influenced by different factors e.g., grain misorientation,grain boundary type, temperature, grain boundary orientation and structure of the atomic layersclose to the grain boundary. Consequently, the grain boundary strength can be changed duringthe deformation. Therefore, modeling the grain boundaries is one of the challenging issues onthe continuum micro-scale.

For many decades, researchers have been trying to propose gradient-extended models, forgrains and grain boundaries, to be able to capture accurate and correct results, matching withexperimental data qualitatively and quantitatively. Therefore, investigation of crystalline ma-terials on the small-scale is still an active field of study and needs more computational efforts.

The main goal of the dissertation is to investigate geometrically nonlinear viscoplasticityof fcc metals on the large- and small-scale. In the large-scale applications using classicalcontinuum plasticity theories, the numerical issues concerning the convergence problem dueto the power law flow rule as well as the locking effect are addressed by a combination ofa new hybrid discontinuous Galerkin approach with a regularization method for the powerlaw (Wulfinghoff and Böhlke [2013]). Implementation of a discontinuous Galerkin methodusing a geometrically nonlinear crystal plasticity framework leads to a locking-free and nu-merically efficient model. The performance of the regularized discontinuous Galerkin crystalviscoplasticity implementation is examined by a planar single crystal and a two-dimensionaloligocrystal.

Furthermore, in small-scale applications, the dissertation presents a new grain boundarymodel based on the dislocation density tensor and surface plastic deformation, leading to agrain boundary yield criterion with isotropic and kinematic hardening. In addition to micro-hard and micro-free grain boundaries, intermediate grain boundaries with specific strengthsare taken into account which are activated when the back stress resulting from dislocationpile-ups reaches a sufficiently high value. In this case, the initial grain boundary strength is

1.2 State-of-the-art 3

assumed as a constant and the effective factors on the grain boundary strength are neglected.The effect of isotropic and kinematic hardening, the grain boundary strength and the internallength scale on the material behavior are investigated in bicrystals under shear load.

The grain boundary model is then extended by evaluating the grain boundary strength withregard to the misorientation between neighboring grains. A transmissibility parameter fromexperimental data (Werner and Prantl [1990]) is used to evaluate dislocation transmission atthe grain boundaries. Therefore, each grain boundary takes a specific strength, regarding theorientation of crystalline lattices in adjacent grains. In order to study the effect of mismatchbetween adjacent grains on the grain boundary strength, randomly-oriented polycrystals arecompared with textured polycrystals under shear deformation.

1.2 State-of-the-art

1.2.1 Dislocations in crystalline materials

Dislocations were discovered by Orowan [1934], Polanyi [1934] and Taylor [1934] indepen-dently and almost at the same time while the first observation of dislocations by transmissionelectron microscopy was published in 1956 (Hirsch et al. [1956]). The story of the first dislo-cation picture is found in Hirsch et al. [2006] (see also Hirth [1985]).

Moreover, a vector field theory was developed by Burgers [1939] for the elastic dislocationfields which represents the magnitude and direction of the lattice distortion. Later, Nye [1953]developed a mathematical connection between the dislocation density tensor and the latticecurvature of a crystal (see also Bilby et al. [1955]). The early theories on dislocations are theones by Hirth and Lothe [1982] and Hull and Bacon [1984].

Generally, numerous atomistic and discrete simulation approaches have been developed toinvestigate the plasticity in crystals. In atomistic approaches, dislocations are modeled basedon atomic and interatomic potentials (e.g., Mishin et al. [1998] and Lee et al. [2003]). Discretedislocation dynamics model the interactions of dislocations by discretizing the dislocationline segments (e.g., Weygand et al. [2002] and Weygand et al. [2009]). Discrete theories aredetailed and contain much physical information often computationally expensive due to largenumbers of interactions between individual dislocations. Therefore, their implementations arelimited (or at least time-consuming) when they are used for large-scale problems. In contrast,continuum approaches model dislocations in an averaged sense. The mechanical behavior of adensity of dislocations is modeled, often in a phenomenological way. Crystal-plasticity modelson the continuum level require lower computational time, compared to discrete models, and

4 1 Introduction

are applicable to single- and poly-crystalline materials.

1.2.2 Continuum study of dislocations

Among the continuum approaches, size-independent plasticity models (e.g., vonMises [1913],Hill [1966], Peirce et al. [1982], Asaro [1983], Needleman et al. [1985], Needleman and Tver-gaard [1993], Steinmann and Stein [1996] and Ortiz and Stainier [1999]) are typically usedin case of large-scale applications on the millimeter- to meter-scale and show good agreementwith experimental data. When the material response is of single-crystalline type (Zaafaraniet al. [2006]) or polycrystals in which the individual grains are not predominant regarding thematerial response (Zhang et al. [2015]), macroscopic continuum approaches are valuable tobe used since the microstructure of the material does not need to be considered. Some recentworks on size-independent plasticity models are the ones by Klusemann et al. [2013], Zhanget al. [2015], Zecevic et al. [2016], Kweon and Raja [2017], Alipour et al. [2018] and Limet al. [2019]. However, the accuracy of such macroscopic plasticity models is limited whenstrongly inhomogeneous plastic deformations of materials on the microscale are investigated.They usually fail to model phenomena such as size effects due to their lack of an internal lengthscale (Hutchinson [2000]).

An early strain gradient plasticity theory was formulated by Aifantis [1987], in which sizeeffects in shear bands were modeled. Further gradient theories by Fleck and Hutchinson [1993]as well as Nix and Gao [1998] are based on the concept of geometrically necessary dislocations(see Ashby [1970]).

In addition, there are numerous gradient crystal plasticity theories which are thermodynami-cally based on the stored energy function. Ortiz and Repetto [1999] proposed a rank-one energy(see also Hurtado and Ortiz [2012], Hurtado and Ortiz [2013] and Wulfinghoff et al. [2015])and a logarithmic energy with similar properties was proposed later by Berdichevsky [2006].Moreover, Mesarovic [2010] represented a three-dimensional gradient crystal plasticity theoryin which a quadratic energy has been implemented. The different energy functions have beencompared by Forest and Guéninchault [2013] and Wulfinghoff et al. [2015].

Furthermore, gradient-extended crystal plasticity models can be classified into approacheswith geometrically linear deformations (e.g. Cermelli and Gurtin [2002], Cordero et al.[2013],Wulfinghoff et al. [2015],Mesarovic et al. [2015] andWulfinghoff [2017]) and extendedcrystal plasticity models for geometrically nonlinear crystal plasticity (Gurtin [2000], Liebeet al. [2003], Levkovitch and Svendsen [2006], Svendsen and Bargmann [2010], Bargmannet al. [2010], Gurtin [2010],Miehe [2014],Ha et al. [2017], Pouriayevali andXu [2017],Alipouret al. [2019] and Alipour et al. [2020]). For instance, in Gurtin [2000], a geometrically non-

1.2 State-of-the-art 5

linear theory is proposed using a defect energy chosen to be quadratic in the gradient of theplastic part of the deformation gradient. A viscoplastic flow rule is developed and the weakformulation of the nonlocal yield conditions is derived (see also Gurtin [2006]).

1.2.3 Grain boundary modeling in continuum theories

The early works regarding the presence of grain boundaries and their influence on dislocationmovement and the mechanical response of the material are the ones by Read and Shockley[1950] and Bilby et al. [1964]. It was found that the grain boundaries in microstructures canlead to heterogeneous plastic behavior as well as an increase of the strength of the material(smaller grain size results in stiffer material behavior Hall [1951] and Petch [1953]).

Therefore, the investigation of interface plasticity in gradient-extended models has beencarried out such that the grain boundaries are considered as obstacles to dislocation transmis-sion (see e.g. Aifantis and Willis [2005], Gurtin and Needleman [2005], Bayerschen et al.[2015] and Gottschalk et al. [2016]). Recently, a new grain boundary model was developed byWulfinghoff [2017] regarding geometrically linear plasticity based on the dislocation densitytensor and the results of discrete dislocation dynamics simulations. Discrete slip lines were in-vestigated on the continuum level, leading to a GB yield criterion with kinematic and isotropichardening.

Although grain boundaries are mostly assumed to be micro-hard (i.e., impenetrable to dislo-cations) or micro-free in some computational models (see e.g., Ekh et al. [2007], Wulfinghoffet al. [2013] and Bayerschen et al. [2015]), transmission of dislocations at the GBs depends onvarious factors, e.g., the grain boundary type, the dislocations reactions and the misorientationof adjacent grains (see e.g., Zaefferer et al. [2003], Gemperle et al. [2005], Bayerschen et al.[2016]). Therefore, numerous authors proposed various GB models to investigate disloca-tion transmission more precisely (e.g., Ashmawi and Zikry [2001], Ma et al. [2006], Gurtin[2008], Li et al. [2009], van Beers et al. [2013], McBride et al. [2016], Gottschalk et al. [2016]and Wulfinghoff [2017]). For example, Gurtin [2008] presented a gradient crystal plasticityformulation which accounts for both the misorientation in the crystal lattice between adjacentgrains and the orientation of the grain boundary relative to the crystal lattice of the adjacentgrains.McBride et al. [2016] extended the gradient crystal plasticity and grain-boundary theoryby Gurtin [2008] to the finite-deformation regime. They presented three alternative thermo-dynamically consistent plastic flow relations on the grain boundary and compared them usinga series of numerical experiments. Ma et al. [2006] introduced an additional activation energyinto the rate equation which accounts for the misorientation of the adjacent grains. Ekh et al.[2011] evaluated a transmission factor according to the degree of mismatch between the slip

6 1 Introduction

systems in adjacent grains. Moreover, van Beers et al. [2013] investigated the influence of graingeometry and multiple slip systems in the material behavior when the interface parameters arefixed in an intermediate situation between micro-free and micro-hard GB.

Regarding the misorientation of adjacent grains, many geometric slip transmission criteriahave been developed using experimental data (e.g., Lee et al. [1989], Luster andMorris [1995],Ravi Kumar [2010] and Bieler et al. [2014]) and have been implemented into computationalmethods (e.g., Shi and Zikry [2011], Spearot and Sangid [2014] and Hamid et al. [2017]). Forexample, Livingston and Chalmers [1957] investigated a geometric slip criterion, including theorientations of slip directions and the slip plane normals, to find the activated slip systems closethe location of dislocation pile-ups. Later, Werner and Prantl [1990] evaluated the mismatchbetween the slip systems in adjacent grains via a sum over all existing slip system combinations(see also Ravi Kumar [2010] and Beyerlein et al. [2012]). A review on slip transmission criteriais found in Bayerschen et al. [2016]. Recently, Hamid et al. [2017] applied a combination ofthe misorientation in adjacent grains and resolved shear stress on relative slip planes. Theyimplemented the criterion ofWerner and Prantl [1990] and assumed a linear function to predictthe grain boundary strength.

1.3 Outline of the dissertation

The present cumulative dissertation consists of three published journal papers and is organizedas follows.

Chapter 2 presents a new hybrid discontinuous Galerkin framework based on the conceptof control points on the interelement boundaries such that element shape functions can beformulated without nodes. Furthermore, the regularization of the power law proposed byWulfinghoff and Böhlke [2013] is extended for very high strain rate sensitivity exponents ina finite single crystal viscoplasticity framework. The regularization method in combinationwith the DG formulations lead to a numerically efficient, robust, and locking-free model.The performance of the regularized DG crystal viscoplasticity implementation is examinedon a planar single crystal and a 2D oligocrystal. Furthermore, the accuracy of the method istested for geometrically nonlinear deformations of crystals with very high strain rate sensitivityexponents.

In Chapter 3, a grain boundarymodel, motivated byWulfinghoff [2017], is presentedwithin ageometrically nonlinear crystal plasticity framework. The formulations are proposed regardingsurface related considerations which results in a grain boundary yield criterion with isotropicand kinematic hardening. Moreover, a micromorphic approach (Forest [2009]) is used which

1.3 Outline of the dissertation 7

makes it easy to differentiate between elastic and plastic regions in the body by introducinga field variable as additional degree of freedom associated with the internal variables presentin elasto-plasticity models of materials. The performance of the model and the influence ofdifferent GB hardening on the material behavior are investigated with some simple exampleswhen the effect of grain orientations on the grain boundary strength is neglected. Furthermore,the effects of grain boundary strength and the internal length scale on the material behaviorare discussed.

The main goal in Chapter 4 is to investigate the effect of mismatch between the adjacentgrains on the grain boundary strength in a geometrically nonlinear crystal viscoplasticityframework including the effect of the dislocation density tensor. Therefore, a transmissibilityparameter suggested byWerner andPrantl [1990] is implemented into the grain boundarymodelproposed in Chapter 3 to evaluate the dislocation transmission at the grain boundaries. Thegrain boundary strength is then evaluated based on the mismatch between neighboring grains.A comparison between randomly oriented polycrystals (with high average misorientation) andtextured polycrystals (with smaller average misorientation) shows the influence of mismatchbetween the slip systems in adjacent grains on the GB strength. Moreover, the effect ofmismatch in adjacent grains on the dislocation transmission at the grain boundaries and theHall-Petch slope is discussed in this Chapter.

Finally, Chapter 5 presents a summary of the results and a brief outlook.

2 Article 1:The concept of control points inhybrid discontinuous Galerkinmethods – application togeometrically nonlinear crystalplasticity

This article was published as:

Alipour A., Wulfinghoff, S., Bayat, H. R., Reese, S. and Svendsen, B. [2018], ‘The conceptof control points in hybrid discontinuous Galerkin methods–Application to geometricallynonlinear crystal plasticity’, International Journal for Numerical Methods in Engineering114, 1–23.

Disclosure of the individual authors’ contributions to the article:

S. Wulfinghoff developed the concept of control points in hybrid discontinuous Galerkinmethod and carried out the analytical linearization related to discontinuous Galerkin frame-work. He wrote part of the manuscript (section 2.4) and had the main idea to present hybriddiscontinuous Galerkin implementation of geometrically nonlinear plasticity, in the context ofsingle crystal plasticity. A. Alipour implemented the model into the academic finite elementsoftware FEAP (as an own element subroutine), set up and performed all simulations andinterpreted the results. Moreover, she wrote the main parts of the article. H. R. Bayat wrotesome parts of the introduction and provided A. Alipour with a very helpful discussion regard-ing the concept of discontinuous Galerkin method. S. Wulfinghoff, S. Reese and B. Svendsenread the article, contributed to the discussion of the results and gave valuable suggestions forimprovement. All authors approved the publication of the final version of the manuscript.

9

10 2 The concept of control points in hybrid discontinuous Galerkin methods . . .

2.1 Abstract

A new concept for hybrid discontinuous Galerkin (DG) methods is presented: control points.These are defined on the interelement boundaries. The concept makes it possible to formulateelement shape functions without nodes. Moreover, the theory is not restricted to certain ele-ment shapes. Furthermore, one can formulate the discrete model such that the displacementis either continuous or discontinuous at the control points. Classical continuous isoparametricelements are included as special case. As an additional new feature, a regularization techniquefor very high strain rate sensitivity exponents up to 1000 in finite single crystal viscoplasticityis presented and implemented into the new hybrid DG framework. In addition, the numericallinearization used in an earlier work is carried out analytically in this work. To the knowl-edge of the authors, this work presents the first hybrid DG implementation of geometricallynonlinear plasticity, here in the context of single crystal plasticity. The regularization methodin combination with the DG formulations facilitates a very simple implementation leading toa numerically efficient, robust, and locking-free model. Two examples are investigated: thedeformation of a planar double slip single crystal exhibiting localization in the form of shearbands and an oligocrystal under uniaxial load.

2.2 Introduction

The plastic deformation of single crystals results from the motion of dislocations and has beenextensively studied for many decades. There are many different continuum theories to studythe plastic deformation of single and polycrystals, which may be generally classified into clas-sical and strain-gradient theories. The classical continuum mechanical plasticity theories (vonMises [1913], Hill [1950]) are in good agreement with the experimental data for large-scaleviscoplasticity. However, strain-gradient theories are more accurate in small-scale plasticity(Gurtin et al. [2007], Fleck and Willis [2009], Wulfinghoff and Böhlke [2012], Cordero et al.[2013], Bayerschen et al. [2015], Wulfinghoff and Böhlke [2015]) since they introduce an in-ternal length scale into the theory, being in good agreement with experimental works (Ziemannet al. [2015]).

Nonuniform and localized deformation ofmacroscopic single crystals have been investigatedboth experimentally and theoretically, for rate-independent (Asaro and Rice [1977], Chang andAsaro [1981], Peirce et al. [1982]) and rate-dependent single crystals (Needleman et al. [1985],Rashid and Nemat-Nasser [1992], Izadbakhsh et al. [2011]). In a study on the rate sensitivityof crystals, Peirce et al. [1983] analyzed rate-dependent planar single crystals under tensile

2.2 Introduction 11

loading with an explicit numerical treatment and compared the results with a rate-independentstudy (Peirce et al. [1982]). Their results show that the rate sensitivity delays notably the shearband development. In addition, it allows the investigation of the effects of high latent hardeningratios on patchy slip. The analysis of localized deformations in rate-dependent single crystalswas reviewed later by Needleman et al. [1985], restricting the attention on the formulationand implementation of finite element methods for crystalline solids. The numerical simulationof shear band localization in ductile single crystal was investigated by Steinmann and Stein[1996] using a family of implicit constitutive algorithms.

Recent works on classical crystal plasticity models focused particularly on numerical issues.An explicit dynamic finite element scheme for structural analysis using rate-dependent crystalplasticity was studied by Dumoulin et al. [2009]. Further, Caminero et al. [2011] investigateda fully implicit algorithm for a large strain anisotropic elasto-plasticity model including elasticanisotropy. Moreover, a methodology was presented by Quey et al. [2011] to generate meshesfor large-scale Voronoi tessellations, including thousand of grains, to obtain meshes withhigh-quality elements and without regions that are overly or inadequately resolved. Klusemannet al. [2012] worked on the modeling and simulation of the deformation behavior of a bccthin sheet consisting of large rate-dependent grains of Fe-3%Si under the tension loading.More recently, Zeng et al. [2015] studied the smoothed finite element method for modelingrate-independent anisotropic crystalline plasticity, for both single crystal and polycrystallinesimulation.

Single crystal simulations at room temperature are significantly complicated by the fact thatthe actual behavior of the crystalline deformation necessitates power-law type flow rules with ahigh value of the rate sensitivity exponent. This issue makes the convergence of the nonlinearequations more complex using the Newton scheme. Therefore, the enhancement of the powerlaw material subroutine for crystals with high sensitivity exponents (nearly rate-independentmaterials) has been recently investigated in order to improve solving the nonlinear system ofequations (Wulfinghoff and Böhlke [2013], Wulfinghoff et al. [2013], Zecevic et al. [2016],Knezevic et al. [2016]) that will be extended in this work.

The first Discontinuous Galerkin (DG) method was introduced by Reed and Hill [1973] forthe solution of neutron transport problems. A penalty term on the element boundaries dueto the work by Nitsche [1971] was later added to guarantee a stabilized solution and to avoida possible non-uniqueness of the discrete solution caused by the discontinuities between thesubdomains. Later, DG methods were further developed in different variants. Among them,the interior penalty and the so-called local DG introduced by Arnold [1982] and Cockburn andShu [1998a], respectively.


Apart form early applications of DG methods in fluid mechanics (Bassi and Rebay [1997],Baumann and Oden [1999], Cockburn and Shu [1998b],Lomtev and Karniadakis [1997],Lomtev et al. [1998],Cockburn et al. [2002], Cockburn et al. [2004]), the field of solid me-chanics has benefited from DG methods as well. Mergheim et al. [2004] applied DG for thepre-failure regime in crack propagation problems. Engel et al. [2002] applied DG for thinbeams and plates and strain-gradient elasticity. An extension of the latter work in strain-gradient-dependent damage was carried out by Wells et al. [2004]. Alberty and Carstensen[2002] used a DG time discretization for elasto-plasticity and compared it with a backwardEuler and Crank-Nicholson scheme.

The first aim of the current work is to present the new hybrid discontinuous Galerkinframework based on the concept of control points. The second objective is to extend the reg-ularization of the power law proposed by Wulfinghoff and Böhlke [2013] for very high strainrate sensitivity exponents to geometrically nonlinear single crystal viscoplasticity. Finally, thethird objective is the analytical derivation of the element stiffness matrix for the applied hybridDG framework byWulfinghoff et al. [2017] which, up to now, had to be computed numerically.Notation. A direct tensor notation is preferred throughout the text. Vectors and 2nd-ordertensors are presented by small and capital bold letters, e. g. a orA, respectively. The transposeand inverse of a tensor is written as AT and A−1. The dyadic product and the inner productof two vectors are, respectively, denoted as a⊗ b = aibj ei ⊗ ej and a · b = aibi (usingsummation convention). In addition, the double contraction of two second order tensors isdenoted by A : B = tr(ATB) = AijBij and the double contraction of a fourth order tensorwith a second order tensor is represented by C : A = CklijAij . Furthermore, matrices (e.g.,the normalized Voigt notation) are marked by an underline, e.g. a.

2.3 Review of the hybrid DG formulation

by Wulfinghoff et al. [2017]

2.3.1 Weak form

A new hybrid discontinuous Galerkin quadrilateral element with a very simple formulationand implementation was introduced by Wulfinghoff et al. [2017]. Here, a brief review on thisframework is discussed. It constitutes the basis of the new hybrid DG method – introducingthe concept of control points – being presented in section 2.4. In order to understand thecontrol point based DG method, it is not necessary to understand all details of the methodof Wulfinghoff et al. [2017], but it is helpful to review some features of the method, being

2.3 Review of the hybrid DG formulation by Wulfinghoff et al. [2017] 13

therefore summarized in the sequel.Assume the deformation mapping x(X, t) to be given, in which X and x = u+X are

position vectors of a particle in the reference and current configuration at time t, respectively(u is the displacement field). The deformation gradient reads F = ∂x/∂X . The referenceconfiguration of the bodywithNe subdomainsΩe is considered as shown in Fig. 2.1. Moreover,the boundary conditions read

u = u on ∂Ωu, PN = t on ∂Ωt, (2.1)

in which u, P ,N and t are the displacement field, the first Piola-Kirchhoff stress tensor, theexternal normal and the traction vector in the reference configuration, respectively. Moreover,∂Ω = ∂Ωu ∪ ∂Ωt, ∂Ωu ∩ ∂Ωt = ∅ and prescribed quantities are denoted as (•). The modified

Figure 2.1: Left: Division of the body into subdomains. Right: Illustration with shrunksubdomains (Fig. taken from the work by Wulfinghoff et al. [2017]).

weak form of the quasi-static linear momentum balance Div (P ) = 0 reads (Wulfinghoff et al.[2017]): ∫

Ωe

P : δF dΩ−∫∂Ωe

(PN + θ(uΓ − u)

)· δu dS = 0 (2.2)

with a positive penalty constant θ (Nitsche [1971]). The displacement uΓ on the interface Γ

is kinematically decoupled from the displacement u within the subdomains Ωe. Accordingly,displacement jumps are allowed, i.e., the subdomains Ωe can detach from the interface Γ. Notethat the integration in Eq. (2.2) is carried out over the element Ωe. Furthermore, the modified


force balance on the union of all boundaries Γ (see Fig. 2.1) is given by∫Γ

(t+ + t− + θ(u+ − uΓ) + θ(u− − uΓ)

)· δuΓ dΓ = 0, (2.3)

whereu+ andu− are the displacements and t+ and t− represent the tractions at the subdomainboundaries which are assumed to be of quadrilateral shape as depicted in Fig. 2.1 and whichcan be computed from the stress in Ωe as PN . Note that the interface Γ and the subdomainsare a priori not kinematically coupled, i.e., displacements jumps are allowed.

2.3.2 Discretization

Assuming a constant deformation gradient within each subdomain Ωe, the displacement fieldin Ωe is interpolated by

u = u0 +H(X −X0) = u0 +H∆X, (2.4)

where u0 and H are the subdomain degrees of freedom. Here, u0 denotes the displacementof the element center (see the work by Wulfinghoff et al. [2017]), and H = F − I is thedisplacement gradient. The boundary integrals in Eqns. (2.2) and (2.3) are approximated by aquadrature scheme with quadrature points at the corners of the subdomains. Therefore, it canbe shown that (see the work by Wulfinghoff et al. [2017])

u0 =

Np∑I=1

wIuΓI , H =

Np∑I=1

wI(uΓI − u0)⊗∆XI , (2.5)

where uΓI and wI are the displacements on Γ and weights at the quadrature point I andX0 ischosen as

X0 =

Np∑I=1

wIXI . (2.6)

The vectors ∆XI are calculated by

∆XI = A−1

∆XI (2.7)

2.3 Review of the hybrid DG formulation by Wulfinghoff et al. [2017] 15

with ∆XI = XI −X0 and

A =

NΓ∑I=1

wI∆XI ⊗∆XI . (2.8)

Note that Eq. (2.5) represents an explicit solution of Eq. (2.2), i.e., only Eq. (2.3) has to besolved numerically. The related element internal force vector is given by (see Eq. (2.3) and thework by Wulfinghoff et al. [2017])

RI = |∂Ωe|

θwI(uΓI − uI)︸︷︷︸RθI

+PN I︸︷︷︸RpI

. (2.9)

Here, |∂Ωe| is the sum of element edge lengths and N I = w1IN

1I + w2

IN2I . Moreover, the

weights wiI and normals N iI (i=1,2) are shown in Fig. 2.2. Note that the weight wI is given

by wI = w1I + w2

I .

Figure 2.2: Tractions, normals and weights (Fig. taken from the work by Wulfinghoff et al.[2017]).

The nodal force contributionsRI (Eq. (2.9)) are treated in the same way as the element internalforces of conventional continuous finite elements, i.e., the theory can easily be implementedas element subroutine in existing finite element programs. The algorithm on element-levelis summarized in Box 1. The treatment of the boundary conditions does not require anyexceptional steps. See the work by Wulfinghoff et al. [2017] for more information regardingthe discontinuous Galerkin framework.


1. compute displacement in element center:

u0 =Np∑I=1

wIuΓI

2. get element displacement gradient and deformation gradient:

H =Np∑I=1

wI(uΓI − u0)⊗∆XI ; F = I +H

3. call material subroutine to obtain the first Piola-Kirchhoff stress P and ∂F TP

4. get corner displacements:uI = u0 +H∆XI

5. compute element internal force vector:RI =

(θwI(uΓI − uI) + P (w1

IN1I + w2

IN2I))|∂Ωe|

6. Use the analytically linearized element internal force vector (section 2.3.3) to obtain theelement stiffness matrix

Box 1: Element subroutine.

2.3.3 Linearization of the residual

In this section, the analytical linearization of the element internal force vector for the hybrid DGframework is presented for the first time. Up to now, it had to be carried out numerically (seethe work by Wulfinghoff et al. [2017]). The element internal forces in Eq. (2.9) are assembledin a usual manner in order to obtain a global nonlinear set of equations with unknowns uΓI .Therefore, its linearization is required to apply the Newton scheme and find the solution. Tothis end, one can obtain (see Eq. (2.9))

∂uΓJRθI = θwI (δIJI − ∂uΓJuI ) , (2.10)

in which (see 2.8.1)∂uΓJ

uI= (1 + aIJ)wJI (2.11)

with aIK = (∆XK · ∆XI). In addition, RpI in Eq. (2.9) can be written in matrix form

asRpI = N P , in which P = (P11, P12, P21, P22)T and

N =

(NI1 NI2 0 0

0 0 NI1 NI2

). (2.12)

2.4 A symmetric discontinuous Galerkin method introducing the concept of control points 17

Therefore, ∂uΓJRpI is obtained as follows

∂uΓJRpI = N

(∂F TP

) (∂uΓJ

F T). (2.13)

Here, using Eq. (2.5), dF T is calculated by

dF T = dHT =∑

J

∑K

wK∆XK ⊗ I (δJK − wJ) duΓJ . (2.14)

The matrix notation of dF T is given in 2.8.2.

2.4 A symmetric discontinuous Galerkin method

introducing the concept of control points

2.4.1 Motivation

The numerical treatment of the hybrid discontinuousGalerkinmethod proposed byWulfinghoffet al. [2017] and summarized in the previous sections leads to a discrete model with primarydegrees of freedom being defined on the skeleton Γ. These primary degrees of freedomof the discrete model are the displacements uΓI of points (the integration points on Γ),being detached from the displacement field of the subdomain interior Ωe. Furthermore, thedisplacement field u(X) within Ωe turns out to be a function of the displacements uΓI . Thisfunction is embodied by Eq. (2.5).

These observations motivate a new discontinuous Galerkin method. This new method getsrid of the intermediate step of introducing the displacement onΓ as a continuous function, whichis then reduced to points during the discretization process. Instead, the original, continuousformulation is enhanced by discrete points on Γ, which can – in principle – detach from thedisplacement field within Ωe. In case of the presence of a potential, the resulting methodis symmetric and – if applied to quadrilaterals – leads to similar results as the unsymmetricmethod of Wulfinghoff et al. [2017].

2.4.2 Variational formulation and introduction of control points

In hyperelasticity but also in the case of generalized standard materials (Halphen and Nguyen[1975]) the weak form of the boundary value problem corresponds to the stationary condition


of a potential Π. This potential is now enhanced by an extra term:

Π =

∫Ω

π(X,F ) dΩ−∫∂Ωt

t · u dS +Ne∑e=1

Ee(ueΓ). (2.15)

In hyperelasticity, π denotes the strain energy density function which is in general inhomo-geneous within the body with reference configuration Ω. In the case of generalized standardmaterials, π is interpreted as a time-discrete incremental potential (Ortiz and Stainier [1999],Miehe [2002]).

The new, rightmost extra-term in Eq. (2.15) is related to the control points, which are intro-duced on Γ which, again, denotes the union of the subdomain boundaries ∂Ωe (e = 1, ..., Ne).The displacements of the control points are collected in the vector ueΓ for each subdomain.Moreover, the functions Ee(ueΓ) will be further specified at a later stage. The stationaritycondition δΠ = 0 gives the weak form:

∫Ω

P : δF dΩ−∫∂Ωt

t · δu dS +Ne∑e=1

∂ueΓEe(ueΓ) · δueΓ = 0, (2.16)

whereP = ∂Fπ. It is noted that the newDGmethod may also be applied without the existenceof a potential π. In that case, Eq. (2.16) may be considered as starting point.The motion of the control points is assumed to be kinematically coupled to the body’s dis-placement field. This means that there exist explicit holonomic constraints which couple thedisplacements of the control points to the displacement field of the subdomains (details arediscussed below). An analogue constraint is assumed to hold for the variations.The non-discretized problem consists in finding a displacement field which is continuouslydifferentiable within the subdomains and continuous on Γ. The displacements of the controlpoints for this situation is discussed below.


2.4.3.1 Discretization within the subdomains

The displacement field within each subdomain Ωe is discretized by a conventional ansatz ofthe Ritz-Galerkin type (where the element index ’e’ is omitted for simplicity):

u(X) =N∑i=1

qiφi(X). (2.17)


Here, qi (i = 1, ..., N ) denote the unknown element internal degrees of freedom andφi(X) arevector-valued shape functions. The shapes of the subdomainsΩe may now in principle be morecomplex, i.e., there is no longer the restriction to quadrilaterals. The shape functions span asubspace of all continuously differentiable functions on Ωe. It is noted that Eq. (2.17) containsconventional isoparametric shape functions as special case. In addition, the deformationgradient for each element reads:

F = H + I =N∑i=1

qi∂Xφi(X) + I. (2.18)

Example: In analogy to the previous sections, quadrilateral elements are considered. It isthen a simple exercise that the ansatz (2.4) falls into the class of approximations defined byEq. (2.17):

u = u0 +H(X −X0) = u0 +H∆X, (2.19)

where the components of u0 andH represent the qi and N = 6, i.e., q = (u01, u02 , H11, H12,

H21, H22)T. The shape functions readφ1

= (1, 0)T, φ2

= (0, 1)T, φ3

= (X1 −X01, 0)T, φ4

=

(X2 −X02, 0)T, φ5

= (0, X1 −X01)T and φ6

= (0, X2 −X02)T.

2.4.3.2 Discretization of the control points

A total number of m control points is introduced on ∂Ωe at positions XΓC (C = 1, ...,m).The control points will ’carry’ the global degrees of freedom, while the internal degrees offreedom qi will be condensed out on element level (like in the theory of Wulfinghoff et al.[2017]). Again, the element index ’e’ is neglected. It is assumed that the control points’number of degrees of freedom of an element, given byM = md, satisfies the relationM ≥ N

(with the space dimension d ∈ 2, 3).In the simplest case, the control points are constrained to move with the element internal

displacement field, i.e., uΓC(XΓC) = u(XΓC). It is then easy to show that there exists a linearrelationship:

uΓ = Bq q (2.20)

with uΓ = (uTΓ1, ...,u

TΓm)

T and q = (q1, ..., qN)T (both vectors are defined on element level).Moreover,Bq can be expressed in terms of φi(XΓC). It is assumed that theN columns ofBq

are linearly independenti and thus span an N -dimensional subspace of RM .iIn fact, this requires a careful selection of the control points’ positions. A detailed discussion is beyond thescope of this work.


Example: The components of Bq, assuming 4 control points and 6 internal degrees of free-dom qi, can be written using Eq. (2.17) as follows (with φi = (φi1, φi2)T):

Bq =

φ11(XΓ1) φ21(XΓ1) ... φ61(XΓ1)

φ12(XΓ1) φ22(XΓ1) ... φ62(XΓ1)

φ11(XΓ2) φ21(XΓ2) ... φ61(XΓ2)

φ12(XΓ2) φ22(XΓ2) ... φ62(XΓ2)... ... ...

φ12(XΓ4) φ22(XΓ4) ... φ62(XΓ4)

. (2.21)

Amore interesting case is obtained if the control points are allowed to detach from the elements,i.e., uΓC(XΓC) 6= u(XΓC), in general. This is achieved by an ’enhancement’ of Eq. (2.20):

uΓ = Bq q +Bww =(Bq Bw

)(qw

)= BQQ. (2.22)

Here,Bw ∈ RM×P with P = M −N denotes a constant matrix and the vector w contains Pextra degrees of freedom. It is noted that forw = 0, the control points move with the elementboundary.

It is assumed that the columns ofBw are linearly independent and orthogonal to those ofBq,i.e.,BQ is invertible. This assumption implies that(

q

w

)=

(Aq

Aw

)uΓ = AQuΓ (2.23)

with AQ = B−1Q . It is noted that Aq is the pseudo-inverse of Bq, i.e., Aq = (BT

qBq)−1BT

q .Vice versa, Bq is the pseudo-inverse of Aq, i.e., Bq = AT

q (AqATq )−1. In analogy, Aw is the

pseudo-inverse of Bw. (and vice-versa). Moreover, it is a simple exercise to show that thelines ofAw are orthogonal to those ofAq.

Example: Obviously, Eq. (2.5) is of the form (2.23):

u0(uΓ) =

Np∑I=1

wIuΓI , H(uΓ) =

Np∑I=1

wI(uΓI − u0(uΓ))⊗∆XI , (2.24)

where the last two rows of the matrixAQ are left undefinedii.iiThis is free of consequences, since the actual values of w are not needed.


As a key observation, Eq. (2.23) implies that the element degrees of freedom q can be con-densed out on the element level: they are controlled by the displacements uΓ of the controlpoints. In fact, the values of the extra degrees of freedom w are not of interest, since they donot enter the weak form. With Eqns. (2.18) and (2.23) it follows that

δF = ∂qF δq =(∂qF

) (∂uΓq)δuΓ =

(∂qF

)AqδuΓ. (2.25)

Consequently, the first term of the weak form (2.16) is approximated as follows

∫Ω

P : δF dΩ ≈Ne∑e=1

δ(ueΓ)T∫

Ωe

(∂ueΓF

)TP dΩ

(2.23)=

Ne∑e=1

δ(ueΓ)T∫

Ωe

(Aeq)

T(∂qeF

)TP dΩ︸︷︷︸

RP

.

(2.26)Here, RP denotes an element internal force vector. Moreover, the linearization of the contri-butionRP is obtained as follows:

∫Ω

δF : ∂FP : ∆F dΩ ≈Ne∑e=1

δ(ueΓ)T

∫

Ωe

(Aeq)

T(∂qeF

)T(∂FP )

(∂qeF

)Aeq dΩ︸︷︷︸

kP

∆ueΓ.

(2.27)Here, kP is one part of the element stiffness matrix related to the contributionRP .

2.4.3.3 Specification of the penalty energy

The energy E(uΓ) (neglecting again the index ’e’) is assumed to be convex, zero for w = 0

and otherwise positive. These properties qualify E(uΓ) as a penalty energy. This means thatany displacements uΓ of the control points, which deviate from the displacement field withinthe elements (which depends on uΓ) is penalized. This choice is motivated from the θ-term ofother discontinuous Galerkin methods.

Example: In order to recover penalty terms similar to those of the previous sections, the penaltyenergy is assumed to take the following form

E(uΓ) = |∂Ωe|4∑

K=1

wKθ

2‖uΓK − uK(uΓ)‖2. (2.28)


The associated variation yields the contributionRθI to the element force vector:

δE(uΓ) =4∑I=1

δuTΓI

4∑K=1

wKθ(δIKI −

(∂uΓI

uK)T)

(uΓK − uK(uΓ))|∂Ωe|︸︷︷︸Rθ

I

. (2.29)

If the special displacement field (2.19) is used, the linearization ofRθI is identical to Eq. (2.10).

The sum ofRθI andR

PI constitutes the nodal internal force vector which has to be supplied on

the element level and which enters the global assembly process in the usual way.

2.4.3.4 Recovery of the continuous solution

The subsequent line of argument is a simplified explanation why the new discontinuousGalerkin method including control points may be expected to be consistent with the origi-nal, continuous problem. However, a rigorous mathematical proof remains to be developed.The non-discretized continuous solution can be expected to be recovered, if the ansatz func-tions φi(X) constitute a complete set of functions in the sense that any continuously dif-ferentiable displacement field u(X) can be represented within Ωe as the number N ofshape functions converges to infinity. Since M ≥ N , this implies also an infinite numberof control points at the boundary ∂Ωe. In this limit process, if M > N , one can always addansatz functions until M = N . In this case, the control points move with the elements, i.e.,uΓC(XΓC) = u(XΓC) and Eq. (2.20) holds. The displacement is then continuous at the con-trol points, since adjacent elements Ωe and Ωf are assumed to share the control points at thecommon boundary ∂Ωe ∩ ∂Ωf . The more shape functions and control points are added, themore ’continuous’ becomes the discrete solution. One may then expect the discrete, approxi-mate solution to converge to an exact (non-discrete) stationary point of the potential Π.For the case of an increasing mesh refinement, a linear displacement approximation within

the elements is in principle sufficient, if the exact solution is assumed continuously differen-tiable and the element size converges to zeroiii. In this case, the extra-degrees of freedom ware no longer needed. If the penalty energy is sufficiently large, w may be assumed to vanishin this situation also for M > N , such that the continuous solution is recovered even with alow-order ansatz. A more rigorous investigation is beyond the scope of this work. Instead, anumerical study will be carried out to support this assumption.

iiiThe exact solution may then be represented by a linear function within an infinitesimal region.

2.5 Review of single crystal viscoplasticity 23

2.5 Review of single crystal viscoplasticity

In this section a review on a continuum mechanical single crystal model is given. The defor-mation gradient F is assumed to be decomposed multiplicatively, i.e., F = F eF p into elasticand plastic parts (see the work by Lee [1969]). Moreover, Ce = F eTF e represents the elasticright Cauchy-Green tensor in the intermediate configuration. The elastic Green-Lagrangianstrain tensor is given by Ee = (Ce − I)/2 and the tensor Lp = F pF p−1 accordingly entersthe velocity gradient l by

l = F F−1 = F eF e−1 + F eLpF e−1. (2.30)

Based on the continuum model of crystal plasticity, one assumes Lp to be given as a superpo-sition of the contribution of the individual slip systems:

Lp =N∑α=1

γαMα, (2.31)

where α runs from 1 to N and N is the number of slip systems (e.g., N = 12 for fcc crystals).Here, γα are the slip rates and Mα = dα ⊗ nα represents the crystal geometry such thatdα and nα, respectively, indicate the slip direction and slip plane normal vectors. Moreover,the accumulated plastic strain is given by

γacc =∑α

t∫0

|γα| dt. (2.32)

Here and in the following, the sum runs over all slip systems.

2.5.1 Free energy, dissipation and consistent flow rule

The free energy per unit volume is assumed to take the following form

ψ = ψe(Ee) + ψh(γacc), (2.33)

inwhich the elastic part of the free energy is assumed to be given byψe(Ee) = (Ee : C : Ee)/2,

where C is the elastic stiffness tensor. For the second part ψh we choose (compare, e.g. the


work by Steinmann and Stein [1996])

ψh(γacc) =(τ∞ − τ0)2

h0 − h∞ln

(cosh

((h0 − h∞) γacc

τ∞ − τ0

))+

1

2h∞γ

2acc, (2.34)

where h0 and h∞ account for the initial and saturation hardening rate, τ0 is the initial criticalshear stress and τ∞ is an additional material parameter.Neglecting thermal effects, the dissipation per unit volume D may be obtained by subtractingthe time derivative of the free energy from the stress power, expressed here in terms of the firstPiola-Kirchhoff stress tensor P as

D = P : F − ψ ≥ 0. (2.35)

This equation gives the elastic second Piola-Kirchhoff stress tensor as follows (details are givenin 2.8.3)

Se = ∂Eeψe = C : Ee = F e−1PF pT. (2.36)

Therefore, Eq. (2.35) can be written in the following form (see Eq. (2.78))

D = Σe : Lp − qhγacc ≥ 0, (2.37)

with qh = ∂γaccψh and the Mandel stress tensor Σe = CeSe. Using Eq. (2.31) the dissipationcan be written as

D =∑α

dα · (Σenα)γα − qhγacc ≥ 0. (2.38)

It is noteworthy to mention that τα = dα · (Σenα) denotes the resolved shear stress associatedto slip plane α in the intermediate configuration. A thermodynamically consistent flow rule(satisfying Eq. (2.38)) is widely assumed as follows

γα = sgn(τα)γ0

⟨|τα| − qh − τ0

τD

⟩p, (2.39)

in which γ0 , τ0 , τD and p are, respectively, the reference shear rate, the initial yield stress, thedrag stress and the strain rate sensitivity. Moreover, sgn (•) denotes the sign of an expression.The shear flow stress is obtained by

τ cf (γacc) = τ0 + qh = τ0 + (τ∞ − τ0) tanh


τ∞ − τ0

)+ h∞γacc. (2.40)


2.5.2 Time discretization by the midpoint rule

In this work, the time discretization by Steinmann and Stein [1996] is adopted. By defin-ing hp = (tn+1 − tn)Lp = ∆tLp, the equation F p = LpF p is discretized in time by themidpoint rule:

F p − F pn =

1

2hp(F p + F p

n), (2.41)

where the index ′n+ 1′ is omitted, for simplicity. The midpoint rule is second-order accurate.In contrast to the implicit Euler scheme, it does not suffer from excessive violation of the plasticincompressibility condition det(F p) = 1. Moreover, by the definition of

fp = F pF pn−1 (2.41)

=

(I − 1

2hp

)−1(I +

1

2hp

), (2.42)

the elastic right Cauchy-Green tensor Ce is obtained by

Ce = F eTF e = fp−TCetr

fp−1 (2.43)

with the trial-value of Ce, given by Cetr

= (F etr

)TF etr and

F etr

= FF pn−1 = F eF pF p

n−1 = F efp. (2.44)

In addition, the flow rule (Eq. (2.39)) is discretized by the implicit Euler scheme:

∆γα = ∆t sgn(τα)γ0

⟨|τα| − qh − τ0

τD

⟩p= ∆γ0 sgn(τα)

⟨|τα| − τ c

τD

⟩p, (2.45)

in which ∆γ0 = ∆tγ0.

2.5.3 Regularization of the power law

In this work, the primary unknowns are chosen to be (Se, τ c), where τ c is the critical shearstress. In addition, an extended version of the regularization of the power law proposed byWulfinghoff and Böhlke [2013] is applied to improve the numerical solution of the nonlinearsystem of equations via the Newton scheme. As it is observed in Fig. 2.3, the approximatesolution x(k) is approaching the accurate answer x in each Newton iteration, starting with theinitial guess x0. Note that the blue region includes all technically resolvable numbers. Aftera certain number of iterations, because of the steep slope of the power law, the computationof ∆γα may be impossible due to the limited numerical range of the computer. Therefore, a


Figure 2.3: (a) The power law (with steep slope for large shear stresses) results in non-resolvableanswers. (b) Regularization of the power law.

non-resolvable answer is obtained (the red region in Fig. 2.3 (a)). In this work, this problemis solved by approximating the flow rule by a linear function with the slope m for large shearstresses, starting from τR, as it can be seen in Fig. 2.3 (b). As a result, the obtained answer x isin general different from the solution x, but it approaches x with increasing the slopem of thelinear part. The idea is now to first compute an approximate solution x, which is close to x andthen use x as new starting solution for the Newton scheme with an increased value ofm leadingto an improved approximate solution x. This procedure is repeated untilm is sufficiently high,such that all shear stresses are smaller than τR. This means that finally, the non-regularizedpower law solution is obtained. This adaptivity is a new feature in comparision with thework by Wulfinghoff and Böhlke [2013], where it was a fixed value for each time increment.Furthermore, the scheme byWulfinghoff andBöhlke [2013]was restricted to gradient-extendedmodels, where the value of m could be estimated due to the gradient-extension. This is nolonger necessary with the scheme presented in this work. Thus, the correct solution x can beobtained even for large strain increments.


Therefore, the flow rule (Eq. (2.45)) is modified as

∆γα =

0 |τα| ≤ τ c

∆γ0 sgn(τα)(|τα|−τc

τD

)pτ c < |τα| ≤ τR

sgn(τα)(m(|τα| − τR) + ∆γ0(∆τR

τD )p)

|τα| > τR

, (2.46)

where ∆τR = τR − τ c is calculated by the equality of slopes of the power law and the linearapproximation at the point τR by

∆τR = τD

(mτD

∆γ0 p

) 1p−1

. (2.47)

2.5.4 Residuals and the linearization process

By rearranging Eq. (2.36), the first residual is obtained:

rs = Ee − S : Se = 0 (2.48)

with S = C−1. Moreover, the second residual is given by (see Eq. (2.40))

rτ = τ cf − τ c = 0. (2.49)

Note thatEe(Se, τ c) and τ cf (Se, τ c) are dependent variables in Eqns. (2.48) and (2.49), as will

be explained in the sequel.As a result, the nonlinear system of Eqns. (2.48) and (2.49) is linearized as

rs + (∂Sers) : ∆Se + (∂τcrs)∆τ c = 0, (2.50)

rτ + (∂Serτ ) : ∆Se + (∂τcrτ )∆τ c = 0. (2.51)

Furthermore, the matrix form of Eqns. (2.50) and (2.51) is

A x = b, (2.52)

with x = [∆SeT,∆τ c]T and b = [−rsT,−rτ ]T. Here, the underline is used for normalized


Voigt notation. Moreover,A includes partial derivative components as follows

A =

(∂Sers ∂τcrs

∂Serτ ∂τcrτ

), (2.53)

in which∂Sers = −S−

∑α

(∂τα∆γα)Ms

α ⊗ M sα, (2.54)

∂τcrs = −∑α

(∂τc∆γα)Ms

α, (2.55)

∂Serτ = (∂γaccτcf )(∂Seγacc), (2.56)

∂τcrτ = (∂γaccτcf )(∂τcγacc)− 1, (2.57)

with ∆γα = ∆tγα (details are given in 2.8.4). Moreover, M s

α and M sα are defined as

Ms

α = sym(dα ⊗ nα) = sym

((Ce

(I − 1

2hp

)−1

dα

)⊗

((I +

1

2hp

)−Tnα

)),

(2.58)M s

α = ∂Seτα = (CeMα)s + 2S : (MαSe)s, (2.59)

hp =∑α

∆γαMα. (2.60)

Here, the notation (•)s denotes the symmetric part of a tensor. In addition, ∂Seγacc, ∂τc∆γα,and ∂τcγacc can be computed by

∂Seγacc =∑α

sgn(τα)(∂Se∆γα) =∑α

sgn(τα)(∂τα∆γα)M sα, (2.61)

∂τcγacc =∑α

sgn(τα)(∂τc∆γα) = −∑α

(∂τα∆γα), (2.62)

in which the derivative ∂τα∆γα, using Eq. (2.46), is given by

∂τα∆γα =

0 |τα| ≤ τ c

∆γ0 ( pτD )(|τα|−τc

τD

)p−1

τ c < |τα| ≤ τR

m |τα| > τR

. (2.63)

The unknowns (Se, τ c) are calculated by solving Eqns. (2.48) and (2.49) via the Newton-Scheme, which allows to compute the Kirchhoff stress τ = F eSeF eT.

2.6 Results and discussion 29

2.5.5 Algorithmic tangent operator

The algorithmic tangent operator in the global problem is calculated by the following equation

dCS = dSeS : dCetr Se : dCC

etr

= (F pn−1

s

F p−Tn ) : d

Cetr Se : (F p−Tn

s

F pn−1), (2.64)

inwhichS is the secondPiola-Kirchhoff stress,C = F TF is the rightCauchy-Green tensor andSe = fp−1Sefp−T is the second Piola-Kirchhoff stress with respect to the trial intermediateconfiguration, being obtained from the Kirchhoff stress tensor τ (refer to Eq. (2.44)):

τ = F eSeF eT = F etr

SeF etrT. (2.65)

In Eq. (2.64), dSeS and dCCetr are computed in detail in 2.8.5. In addition, the productA

s

AT is a fourth order tensor, being defined by(A

s

AT)

: B = A sym(B) AT for arbitrarysecond order tensorsA andB.Furthermore, d

Cetr Se is given in matrix notation by (see 2.8.6)

dCetr S

e=

((fp−1

s

fp−T, 0)− 2

∑α

(M ′

α

)s

( dx∆γα)T)

dCetrx, (2.66)

in which the tensor M ′sα is defined by

M′sα = d′α ⊗ n′α =

((I +

1

2hp

)−1

dα

)⊗

(fp−1Se

(I +

1

2hp

)−Tnα

)(2.67)

and (see 2.8.6)

dCetrx = −A−1

(12fp−T s

fp−1

0

). (2.68)

Note that dx∆γα in Eq. (2.66) is computed as in Eqns. (2.61) and (2.62) withx = (SeT, τ c)T.

2.6 Results and discussion

In this section, the performance of the regularized DG crystal viscoplasticity implementation isexamined by two examples, namely a planar single crystal and a 2D oligocrystal. Furthermore,the accuracy of the method is tested for geometrically nonlinear deformations of crystals withvery high strain rate sensitivity exponents. The adaptive time stepping algorithm of FiniteElement Analysis Program (http://www.ce.berkeley.edu/projects/feap) is used in this work.


2.6.1 Planar double slip single crystal under uniaxial load

A specimen with dimensions 20 × 60 mm is investigated with prescribed displacements atthe upper and lower side in longitudinal direction. One quadrant of the plane is simulatedusing symmetry conditions, and discretized into 40 rows of 20 quadrilateral elements each (seeFig. 2.4 (left)). In addition, the velocity of the top surface is considered to be 0.5 mm/s.

(a) (b) (c)

Figure 2.4: Left: Boundary conditions. Right: Distribution of the accumulated plastic strainat elongations (a) 2.5 mm (b) 5 mm (c) 5 mm via new DG method with controlpoints, where θ = 10−1|∂Ωe|−1 MPa and p = 200.

A planar double slip model is considered with initial slip directions being oriented about thelongitudinal direction by±30. Furthermore, the material is assumed to be elastically isotropicwith Lame parameters λ = 35104.88 MPa and µ = 23427.25 MPa, critical shear stressparameters τ0 = 0.84 MPa, τ∞ = 49.51 MPa, initial and saturation hardening moduli h0 =

541.48 MPa and h∞ = 1 MPa, the reference shear rate γ0 = 10−3 s−1 and drag stress τD =

60 MPa. In this example, the strength of one element at the lower left corner of the simulationdomain has been decreased to trigger localization (compare the work by Steinmann and Stein[1996]).

Fig. 2.4 (a) and (b) show the distribution of the accumulated plastic slip at elongations 2.5 mm

and 5 mm where θ = 10−1|∂Ωe|−1 MPa using the unsymmetric hybrid DG-method byWulfinghoff et al. [2017]. In addition, Fig. 2.4 (c) shows the results, as they were obtained bythe new, control point based DG method. Here, 6 internal degrees of freedom (N = 6) are


Figure 2.5: Load-displacement curves for different penalty parameters with p = 200 (20× 40elements in each quadrant).

introduced on each subdomain and 4 control points (m = 4) on the corners of the interfacesurrounding each subdomain. Applying this specific condition allows to compare the newDG framework with the one by Wulfinghoff et al. [2017]. Due to the fact that the resultsof the two DG methods (compare Figs. 2.4 (b) and (c)) are hardly differentiable, all othersimulations were carried out using the DG method of Wulfinghoff et al. [2017]. Note thatthe contours are obtained for a strain rate sensitivity exponent of p = 200. The results arein a reasonable agreement with the continuous nonlinear finite element method using Q1E4elements by Steinmann and Stein [1996] who used a slightly different constitutive model.Moreover, the computation of det(F p), regarding the midpoint rule (Eq. (2.41)), shows thatthe deviation of det(F p) from 1 is less than 10−10 in this simulation. It is noteworthy that theeffect of the penalty parameter θ is almost negligible for θ ≤ 1|∂Ωe|−1 MPa (see Fig. 2.5).In addition, the results for other small values of θ are almost identical to what is shown inFig. 2.4 (right), and they are therefore not depicted. The softening part of the deformationapproximately starts when the shear bands are fully developed. Here, the thickness of themodel is assumed to be 1 mm to obtain the reaction force in the load-displacement curves.

Global Newton iteration 1-2 3 > 3Strain rate sensitivity (p) 100 200 1000

Table 2.1: The gradual increase of p for the large value of p = 1000


Sensitivity exponent (p) 20 50 200 1000Number of time steps 14 34 195 337

Table 2.2: The number of time steps for different p at elongation 5 mm (≈ 16.67% averagetensile strain) where θ = 10−1|∂Ωe|−1 MPa (110× 220 elements).

Elements in each quadrant Number of time steps20× 40 21040× 80 25280× 160 293110× 220 337

Table 2.3: The number of time steps for different meshes at elongation 5 mm for θ =1|∂Ωe|−1 MPa and p = 1000.

The same example is investigated for 110 by 220 elements in each quadrant and p = 1000.Since the large value of p = 1000 is computationally very demanding, a modified globalNewton scheme, based on a gradual increase of p with increasing number of global Newtoniterations, is applied (see Table 2.1). Note that the final solution of the equations is obtainedfor p = 1000.

Fig. 2.6 confirms that, again, θ does not have much influence on the deformation even for avery high value of p, leading to nearly rate-independent material behavior. The shear bands areshown in Fig. 2.7 for two different elongations. As it is seen, secondary shear bands are formedin each quadrant as a result of refining the mesh and p = 1000. The contours of deformationfor other small values of θ look similar to what is shown in Fig. 2.7. Furthermore, the numberof time steps for different values of p is shown in Table 2.2. It is not surprising that the timeincrements required for convergence become smaller by increasing the value of p, due to theincreased nonlinearity of the power law.

Moreover, mesh convergence can be observed in Fig. 2.8 where the number of time steps islisted in Table 2.3 (the number of elements is considered for each quadrant). It is noted that thenew DGmethod was also compared to the one byWulfinghoff et al. [2017] for the hyperelastic’Cook’s membrane’-benchmark test discussed in that paper. The presentation of the details arebeyond the scope of this work, but it is noted that the new formulation converges to the correctsolution with a similar accuracy as the method by Wulfinghoff et al. [2017], when the mesh isgradually refined.


(a) (b)

Figure 2.6: Load-displacement curves for different penalty parameters with (a) p = 200 (b) p =1000 (110× 220 elements in each quadrant).

(a) (b)

Figure 2.7: Distribution of the accumulated plastic strain for p = 1000 at elongations (a) 3 mm(b) 5 mm, where θ = 10−1|∂Ωe|−1 MPa (110× 220 elements).


Figure 2.8: Mesh convergence for the planar single crystal for θ = 1|∂Ωe|−1 MPa and p =1000.

Figure 2.9: The geometry of oligocrystal.


(a) (b)

Figure 2.10: Load-displacement curves for the oligocrystal for (a) various penalty param-eters (p = 200) and (b) various strain rate sensitivity exponents (θ =103|∂Ωe|−1 MPa).

2.6.2 Oligocrystal under uniaxial load

In the following, the performance of the DG framework is examined on a 2D oligocrystalmade up of 20 randomly oriented single crystals discretized by 1185 elements (see Fig. 2.9).In this case, the bottom of the oligocrystal is constrained in y-direction and the upper side

is subjected to a prescribed displacement with velocity 0.5 mm/s. The material parametersread τ0 = 50.84 MPa, τ∞ = 549.51 MPa, h0 = 541.48 MPa, h∞ = 10 MPa, γ0 = 10−3 s−1

and τD = 60 MPa.Fig. 2.10 shows the effects of the strain rate sensitivity exponent p as well as the penalty

parameter θ on the deformation. As it is shown in Fig. 2.10 (a), the load-displacement curveis hardly distinguishable for different θ and a specific value of p = 200. This means that θdoes not have much influence on the deformation behavior of the oligocrystal. In addition,Fig. 2.10 (b) shows that the load-displacement curves for larger values of p are closer togetherin comparison with for the small values. It should be noticed that the number of load steps hasalmost no effect on the deformation behavior. This issue is considered for p = 20 in Fig. 2.10

(b).The distribution of γacc in the oligocrystal for two different strain rate sensitivity expo-

nents p = 20 and p = 1000 is shown in Fig. 2.11 for θ = 10−1|∂Ωe|−1 MPa. Since thedifference of the distribution is not pronounced, it can be deduced that the effects of p and θare less significant in the investigated oligocrystal deformation in comparison with the planar


(a) (b)

Figure 2.11: Distribution of accumulated plastic strain for (a) p = 20 (b) p = 1000, where theelongation is 7 mm and θ = 10−1|∂Ωe|−1 MPa.

single crystal under uniaxial load, where the deformation is dominated by localization.

2.7 Conclusion

A regularization method for the power law for crystals with high rate sensitivity exponents pin combination with a symmetric hybrid discontinuous Galerkin formulation – introducingthe concept of control points – has been presented, leading to a simple implementation intoexisting finite element codes. This new method has been motivated from the approach byWulfinghoff et al. [2017]. The new framework makes the application of general shape func-tions possible. For example, this allows one to achieve a reasonable mesh convergence by alowest order ansatz, which in case of a continuous framework is known to end up in severelocking effects. To the knowledge of the authors, the application of a hybrid DG frameworkto geometrically nonlinear plasticity has been investigated for the first time. In addition, theanalytic linearization of the internal force vector, which had to be computed numerically in the

2.7 Conclusion 37

work by Wulfinghoff et al. [2017], has been presented in this work. The performance of theframework has been examined by two examples: a planar single crystal and a 2D oligocrystalunder uniaxial load.

Moreover, it has been shown that the effect of the penalty parameter θ is almost negligiblein a large range of θ-values for the investigated examples. Moreover, secondary shear bandsare formed in each quadrant as a result of refining the mesh for high values of p, where thedeformation is still almost independent of θ. Furthermore, a good mesh convergence has beenfound.

As a second example, we have explored the performance of the DG framework on a 2Doligocrystal. A negligible influence of the penalty parameter θ on the deformation behaviorhas again been obtained for an even larger range of θ. This is most probably due to the lackof shear bands in the oligocrystal. Similarly to θ, it has been shown that the effects of thestrain rate sensitivity exponent p are less significant in the investigated oligocrystal deformationcompared with the planar single crystal. In future works, it is planned to extend the presentednumerical concepts to gradient-extended plasticity models (Wulfinghoff et al. [2015], Wulfin-ghoff [2017]).AcknowledgmentThe financial support of the project “Hybrid Discontinuous Galerkin Methods in Solid Me-chanics” within the priority program SPP 1748 “Reliable Simulation Techniques in SolidMechanics-Development of Non-standard Discretization Methods, Mechanical and Mathe-matical Analysis” by the German Science Foundation (DFG) is gratefully acknowledged byHamid Reza Bayat, Dr.-Ing. Stephan Wulfinghoff and Prof. Dr.-Ing. Stefanie Reese. Further-more, the authors Atefeh Alipour and Dr.-Ing. Stephan Wulfinghoff gratefully acknowledgethe financial support of Aachen Institute for Advanced Study in Computational EngineeringScience (AICES) by the DFG through Grant GSC 111. In addition, the funding related to theproject “Computational Methods for Microscale Plasticity StUpPD-180-15” is also greatfullyacknowledged.


2.8 Appendix

2.8.1 Derivation of ∂uΓJuIin the linearization of the residual

Using Eqns. (2.4) and (2.5), the differential of the displacement is obtained by

duI = du0 + dH∆XI

=∑

K

wK duΓK +∑

K

wK( duΓK −∑

J

wJ duΓJ) (∆XK ·∆XI)︸︷︷︸aIK

=∑

J

∑K

wK ((1 + aIK) δJK − wJaIK) I duΓJ

=∑

J

(1 + aIJ)wJ duΓJ ,

(2.69)

where the last rearrangement is a trivial exercise, exploiting Eq. (2.6).

2.8.2 The matrix form of dF T in Eq. (2.14)

The matrix form of ∂uΓJF T reads as follows

(∂uΓJF T)pm =

∑K

(∆X)Ki(p)δj(p)mwK (δKJ − wJ) , (2.70)

in which

i = (1 1 2 2) , j = (1 2 1 2) , p =

(1 2

3 4

). (2.71)

2.8.3 Evaluation of the dissipation inequality

Extend Eq. (2.35) asD = P : F − ∂Eeψe : Ee − qhγacc ≥ 0, (2.72)

with qh = ∂γaccψh and Ee = sym(F eT ˙

FF p−1). Therefore,

D = P : F −(F e∂EeψeF

p−T) : F − ∂Eeψe :

(F eTF

˙F p−1

)− qhγacc ≥ 0. (2.73)

The dissipation is assumed to vanish for virtually ′frozen′ plastic deformation:

D∣∣γα=0

=(P − F e∂EeψeF

p−T) : F = 0. (2.74)

2.8 Appendix 39

Since F is arbitrary, the first Piola-Kirchhoff stress can be computed as

P = F e∂EeψeFp−T, (2.75)

which must be equal to its definition as

P = τF−T = F eSeF eTF−T. (2.76)

Equality of Eqns. (2.75) and (2.76) leads to the elastic part of the second Piola-Kirchhoff stresstensor as (see Eq. (2.33))

Se = ∂Eeψe = C : Ee. (2.77)

Furthermore, Eq. (2.73) can be written as (with F p ˙F p−1 = −F pF p−1 = −Lp)

D = Se : (CeLp)− qhγacc ≥ 0, (2.78)

2.8.4 Linearization of Eqns. (2.48) and (2.49)

In this part, the analytical calculation of the components ofA in Eq. (2.52) is described.The inverse of Eq. (2.42) is

fp−1 =

(I +

1

2hp

)−1(I − 1

2hp

), (2.79)

while its differential reads

dfp−1 = −1

2

(I +

1

2hp

)−1

dhp

(I +

1

2hp

)−1(I − 1

2hp

)− 1

2

(I +

1

2hp

)−1

dhp

= −(I +

1

2hp

)−1

dhp

(I +

1

2hp

)−1

.

(2.80)From Ee = (Ce − I)/2 and Eq. (2.43), one obtains

dEe =1

2d(fp−TCetr

fp−1)

=1

2

(dfp−TfpTfp−TCetr

fp−1 + fp−TCetr

fp−1fp dfp−1)

=1

2

(dfp−TfpTCe +Cefp dfp−1

)= sym(Cefp dfp−1).

(2.81)


Furthermore (see Eqns. (2.42), (2.79) and (2.80)),

fp dfp−1 = −(I − 1

2hp

)−1(I +

1

2hp

)(I +

1

2hp

)−1

dhp

(I +

1

2hp

)−1

= −(I − 1

2hp

)−1

dhp

(I +

1

2hp

)−1

.

(2.82)

Consequently, Eq. (2.81) would be

dEe = −sym

(Ce

(I − 1

2hp

)−1

dhp

(I +

1

2hp

)−1), (2.83)

where hp is defined in Eq. (2.60) and its differential reads

dhp =∑α

Mα d∆γα, (2.84)

in whichd∆γα = (∂τα∆γα)M s

α : dSe + (∂τc∆γα) dτ c, (2.85)

with M sα = ∂Seτα which can be computed by the definition of τα as follows

τα = Σe : Mα = Se : (CeMα) = Ce : (MαSe) , (2.86)

and its differential is

dτα = (CeMα) : dSe + (MαSe) : dCe. (2.87)

From Eq. (2.36) and Ee = (Ce − I)/2, it follows that

Ce = 2S : Se + I, dCe = 2S : dSe. (2.88)

Substituting Eq. (2.88) in Eq. (2.87), we have

dτα =((CeMα)s + 2S : (MαS

e)s)︸︷︷︸

¯M s

α

: dSe, (2.89)

2.8 Appendix 41

where (•)s denotes the symmetric part. Thus, Eq. (2.83) changes as follows

dEe = −

(∑α

(∂τα∆γα)Ms

α ⊗ M sα

): dSe −

(∑α

(∂τc∆γα)Ms

α

)dτ c, (2.90)

in which M s

α and ∂τc∆γα are, respectively, defined in Eq. (2.58) and (2.62).

2.8.5 Derivation of dSeS and dCCetr

in Eq. (2.64)

The second Piola-Kirchhoff stress tensor and trial-value of the right Cauchy-Green tensor canbe written as

S = F−1τF−T(2.42)= F p

n−1fp−1Sefp−TF p−T

n = F pn−1SeF p−T

n , (2.91)

Cetr (2.43)= fpTCefp = F p−T

n F pTCeF pF pn−1 = F p−T

n F TFF pn−1 = F p−T

n CF pn−1. (2.92)

Therefore, their derivatives are given by

dSeS = F pn−1

s

F p−Tn , (2.93)

anddCC

etr

= F p−Tn

s

F pn−1. (2.94)

2.8.6 Derivation of the algorithmic tangent operator

The product rule leads to the decomposition of dSe = d(fp−1Sefp−T) into three parts:

dfp−1Sefp−T (2.80)= −

(I +

1

2hp

)−1

dhp

(I +

1

2hp

)−1

Sefp−T = −∑α

M′sα d∆γα,

(2.95)fp−1Se dfp−T =

(dfp−1Sefp−T

)T= −

∑α

(M′sα

)Td∆γα, (2.96)

fp−1 dSefp−T =((fp−1

s

fp−T, 0)

dCetrx

)dCetr (2.97)

where d∆γα = ( dx∆γα · dx) and M ′sα are defined in Eqns. (2.85) and (2.67), respectively.

The differential of the residuals (Eqns. (2.48) and (2.49)) according to the unknowns (Se, τ c)


as well as Cetr is required to obtain dCetrx. Thus, Eq. (2.48) can be rewritten as

rs = Ee − S : Se =1

2(Ce − I)− S : Se =

1

2

(fp−TCetr

fp−1 − I)− S : Se = 0. (2.98)

The differentials of Eqns. (2.49) and (2.98) read

drs =(∂Cetrrs

): dCetr

+ (∂Sers) : dSe + (∂τcrs) dτ c, (2.99)

drτ = (∂Serτ ) : dSe + (∂τcrτ ) dτ c. (2.100)

Thus, the matrix form of Eqns. (2.99) and (2.100), using Eq. (2.52), is obtained by(drs

drτ

)=

(∂Cetrrs

0

)dCetr

+A dx = 0 (2.101)

with∂Cetrrs =

1

2fp−T s

fp−1. (2.102)

Therefore, rearranging Eq. (2.101) leads to

dCetrx = −A−1

(12fp−T s

fp−1

0

). (2.103)

3 Article 2:A grain boundary model forgradient-extended geometricallynonlinear crystal plasticity: theoryand numerics


Alipour, A., Reese, S. and Wulfinghoff, S. [2019], ‘A grain boundary model for gradient-extended geometrically nonlinear crystal plasticity: theory and numerics’, International Jour-nal of Plasticity 118, 17–35.


S. Wulfinghoff developed the grain boundary model for gradient-extended geometrically non-linear crystal plasticity. A. Alipour implemented the model into the academic finite elementsoftware FEAP (as an own element subroutine), set up and performed all simulations, inter-preted the results and wrote the manuscript. S. Wulfinghoff and S. Reese read the article,contributed to the discussion of the results and gave valuable suggestions for improvement.All authors approved the publication of the final version of the manuscript.

43

44 3 A grain boundary model for gradient-extended geometrically nonlinear crystal plasticity . . .

3.1 Abstract

A grain boundary model is presented based on the dislocation density tensor within a geomet-rically nonlinear crystal plasticity framework. The formulations are derived using the linearmomentum balance equation and surface related considerations which lead to a grain bound-ary yield criterion with isotropic and kinematic hardening. To decrease the implementationeffort, a three-level solution algorithm is presented which allows to extend an already existinglocal single crystal material subroutine to account for gradient contributions. As an additionalfeature, the analytical linearization of the weak form regarding geometrically nonlinear crystalplasticity is presented. The effects of grain boundary strength and internal length scale on thematerial behavior as well as the role of grain boundaries as obstacles to dislocation transmis-sion are discussed in several examples. Further, the results show interesting grain boundaryhardening effects in cyclic loading.

3.2 Introduction

Single- and polycrystal plastic deformation has been extensively studied for many decadesvia different continuum approaches, which may be classified into classical and strain-gradienttheories. Although classical continuum plasticity approaches (e.g. von Mises [1913], Hill[1966], Peirce et al. [1982], Asaro [1983], Needleman et al. [1985], Quey et al. [2011], Zhanget al. [2015], Alipour et al. [2018]) are in good agreement with the experimental data in case oflarge-scale plasticity, they usually fail to model phenomena such as size effects due to their lackof an internal length scale. In contrast, strain-gradient theories (e.g., Aifantis [1984], Gurtinet al. [2007], Cordero et al. [2013], Bayerschen et al. [2015], Wulfinghoff and Böhlke [2015])are more accurate in small-scale plasticity since they introduce an internal length scale into thetheory, being in good agreement with experimental findings (e.g., Dimiduk et al. [2005], Yaoet al. [2014], Ziemann et al. [2015], Mu et al. [2016]).

In case of geometrically linear deformations, in contrast to the frameworks in which theplastic spin is neglected (e.g., Wulfinghoff and Böhlke [2012], Fleck et al. [2015]), the gradi-ent part of the full plastic distorsion Hp has been implemented in the models by numorousauthors (e.g., Gurtin [2002], Cordero et al. [2013], Wulfinghoff et al. [2015], Mesarovic et al.[2015], Wulfinghoff [2017]) based on the dislocation density tensor (Nye [1953]).

Moreover, there aremanyworks on extended crystal plasticity at large deformations (e.g.Gurtin[2002], Levkovitch and Svendsen [2006], Svendsen and Bargmann [2010]). For instance, inGurtin [2000], a geometrically nonlinear theory is proposed using a defect energy chosen to

3.2 Introduction 45

be quadratic in the gradient of the plastic part of the deformation gradient. A viscoplastic flowrule is developed and the weak (virtual power) formulation of the nonlocal yield conditions isderived (see also Gurtin [2006]). Since the presence of grain boundaries (GBs) in microstruc-tured specimens leads to strengthening of the material (i.e., specimens with smaller grainsizes are stiffer than the ones with larger grain sizes, Read and Shockley [1950], Hall [1951],Petch [1953], Bilby et al. [1964]), interface studies for gradient plasticity models have beencarried out, where the grain boundaries are considered as obstacles to dislocation transmission(see e.g., Aifantis and Willis [2005], Gurtin and Needleman [2005], Bayerschen et al. [2015],Gottschalk et al. [2016]). For example, Gurtin [2008] presented a gradient crystal plasticityformulation which accounts for both the misorientation in the crystal lattice between adjacentgrains and the orientation of the grain boundary relative to the crystal lattice of the adjacentgrains. McBride et al. [2016] extended the gradient crystal plasticity and grain-boundarytheory by Gurtin [2008] to the finite-deformation regime. They present three alternative ther-modynamically consistent plastic flow relations on the grain boundary and compare themusing a series of numerical experiments. Moreover, the resistance of grain boundaries againstplastic flow based on an interface yield condition is investigated by Wulfinghoff et al. [2013].Wulfinghoff [2017] later introduced a new grain boundary model based on the surface plasticdeformation pertaining to small deformations.

Although the grain boundaries are mostly assumed to be micro-hard (i.e., impenetrable todislocations) or micro-free (an infinite dislocation sink) in some computational models, variousmodels for the transmission of dislocations at GBs have been proposed by numerous authors(see e.g., Ashmawi and Zikry [2001], Bayerschen et al. [2016], Hamid et al. [2017]). Forinstance, Ma et al. [2006] introduced an additional activation energy into the rate equationwhich accounts for the misorientation of the adjacent grains. Ekh et al. [2011] evaluated atransmission factor according to the degree of mismatch between the slip systems in adjacentgrains. Moreover, van Beers et al. [2013] investigated the influence of grain geometry andmultiple slip systems in the material behavior when the interface parameters are fixed in anintermediate situation between micro-free and micro-hard GB.

In the current work, a grain boundary model, motivated by Wulfinghoff [2017], is studiedfor the first time on the concept of geometrically nonlinear crystal plasticity. We aim todevelop a grain boundary model in order to improve single crystal models. Therefore, theperformance of the model and the influence of different GB hardening on the material behaviorare examined with some simple examples when the effect of grain orientations on the grainboundary strength is neglected. Therefore, in section 3.3, the theory is introduced and thenumerical treatment is discussed in detail. Moreover, the extension of the model on the grain


boundaries is presented in section 3.4 which leads to both kinematic and isotropic hardeningon the GBs. The numerical computation is discussed in section 3.5 where a three-level solutionalgorithm is presented to reuse an already existing local crystal plasticity subroutine. The finiteelement implementation in the bulk and on the grain boundary is investigated in section 3.6.Finally, some examples on bicrystals in section 3.7 show the behavior of the current model.

A micromorphic approach (Forest [2009]) is used which makes it rather easy to differentiatebetween elastic and plastic regions in the body. In micromorphic approaches, a field variableis introduced as additional degree of freedom associated with the internal variables presentin elasto-plasticity models of materials. The micromorphic variable can be constrained to be(almost) equal to its counterpart (see Forest [2016]) if the coupling energy is used as a penaltyterm in the free energy. The model in the bulk is here similar to the models of Gurtin if thepenalty coefficient takes a very large value.Notation. A direct tensor notation is preferred throughout the text. Vectors and 2nd-order ten-sors are represented by small and capital bold letters, e.g. a orA, respectively. The transposeand inverse of a tensor is written asAT andA−1. Moreover, the dyadic product and the innerproduct of two vectors are, respectively, denoted asa⊗ b = aibj ei ⊗ ej anda·b = aibi (usingsummation convention). The double contraction of two second order tensors is indicated byA : B = tr(ATB) = AijBij and the double contraction of a fourth order tensor with a secondorder tensor is represented by C : A = CklijAij . Furthermore, matrices (e.g. the normalizedVoigt notation) are marked by an underline, e.g. a. Also, ∇x stands for the gradient operatorwith respect to the observed space. The norm of a vector (or a tensor) is written by ‖ a ‖ (or‖ A ‖) and sgn(•) represents the sign of an expression.

3.3 Gradient-extended single crystal plasticiy

3.3.1 Basics of single crystal elasto-viscoplasticity

In the continuum theory of plasticity at hand, the deformation gradient F is decomposedmultiplicatively, i.e., F = F eF p into elastic and plastic parts (Kröner [1959] and Lee [1969]).In addition, the elastic right Cauchy-Green tensor and the elastic Green-Lagrangian straintensorwith respect to the intermediate configuration are represented byCe = F eTF e andEe =

(Ce − I)/2, respectively. The velocity gradient l is given as follows

l = F F−1 = F eF e−1 + F eLpF e−1, (3.1)

3.3 Gradient-extended single crystal plasticiy 47

in which, based on the continuum model of crystal plasticity, the tensor Lp = F pF p−1 isassumed to be given as a superposition of the contribution of the individual slip systems α:

Lp =∑α

γαdα ⊗ nα, (3.2)

where α runs from 1 to N (e.g., N = 12 for fcc crystals). In addition, γα are the slip ratesandMα = dα⊗nα represents the crystal geometry in the sense that dα and nα, respectively,are the slip direction and slip plane normal vectors. The accumulated plastic slip is given by

γacc =∑α

∫t

|γα|dt. (3.3)

3.3.2 Continuum plastic surface deformation

Microscopic studies on the plastic deformation of crystal surfaces show discrete slip lines as aresult of moving dislocations crossing the surface. The amount of plastic surface deformationmay be captured along the line element dx (see Fig. 3.1, in which N is the surface normalvector). The plastic deformation of in-line elements (i.e., the line elements perpendicular to thesurface normal) corresponds to the plastic surface deformation due to crossing the dislocationsfrom the boundary. Therefore, it is sufficient to take the plastic deformation of in-line elementsinto account and neglect the deformation of out-line elements (i.e., which are parallel to thesurface normal) since they do not describe the the slip lines on the surface. Consequently, toaccount for the effects of such discrete slip lines on the continuum level, the plastic deformationof the surface is represented by (Wulfinghoff [2017]):

B = −Hpχ(N×), (3.4)

by projecting all line elements on the surface, in whichHpχ is defined byHpχ = F pχ − I .At this point, we constrain F pχ to be equal to F p (F pχ = F p). Later on, this constraint willbe enforced by a penalty method. The tensor (N×) projects any vector into the plane with thesurface normal vectorN and turns it about 90 in the clockwise direction. The projector (N×)

is calculated by(N×) = −εN = −εijkNkei ⊗ ej, (3.5)

where ε denotes the permutation tensor. It is noteworthy that B = 0 means that no plasticdeformation occurs on the boundary of a crystal (i.e., a micro-hard surface).


Figure 3.1: Volume V with external surface ∂V and in-plane line element dx (taken fromWulfinghoff [2017]).

3.3.3 Free energy, dissipation and consistent flow rule

Neglecting thermal effects, the dissipation D is obtained by

D = Pext −∫V0

ψ dV ≥ 0, (3.6)

in which Pext and ψ are the external power and the free energy density of the body, respectively.Moreover, V0 refers to the reference configuration. The power of external force is assumed totake the following form:

Pext =

∫∂V0

(t · u+m : B

)dA+

∫V0

ρ0b · u dV (3.7)

that includes the extra term m : B, where m is defined as generalized surface tractionbeing power conjugate to the surface plastic deformation B (see e.g., Gurtin [2000], Gurtinand Needleman [2005], Wulfinghoff [2017]). Here, ρ0, b and u denote the mass density inthe reference configuration, the body force and the displacement vector, respectively. Thetractions t and m are obtained by Cauchy’s lemma and its extension as (see Wulfinghoff[2017] for more details):

t = PN , m = −M (N×)(N×), (3.8)

where P and M are, respectively, the first Piola-Kirchhoff stress and a generalized stresstensor. The tractions t andm are assumed to be known on the Neumann boundaries ∂Vt and∂Vm, respectively. Moreover, the displacement u and the surface plastic deformation B areconsidered to be given on the Dirichlet boundary ∂Vu and ∂Vχ, respectively, where ∂V0 =

3.3 Gradient-extended single crystal plasticiy 49

∂Vt ∪ ∂Vu = ∂Vm ∪ ∂Vχ. The surface integral in Eq. (3.7) is transferred to a volume integralusing Eq. (3.8) together with Gauss’ theorem:∫

∂V0

m : B dA =

∫V0

(M : αχ −Xb : F

pχ)

dV, (3.9)

in whichαχ = CurlT(F pχ) = εijk∂XiF

pχlj el ⊗ ek (3.10)

is well-known as the dislocation density tensor andXb is defined by

Xb = CurlT(M ). (3.11)

Furthermore, the quasi-static linear momentum balance takes the form

Div(P ) + ρ0b = 0. (3.12)

It should be mentioned that Eqns. (3.11) and (3.12) will be used in the sequel to obtain theweak form. Therefore, the following identity is obtained by substituting Eqns. (3.8) and (3.9)in Eq. (3.7) (see 3.9.1):

Pext = Pint, Pint =

∫V0

(P : F −Xb : F

pχ+M : αχ

)dV. (3.13)

In addition, the free energy in Eq. (3.6) is assumed to be a function of the elastic Green-Lagrangian strain tensor Ee, the dislocation density tensor α and the accumulated plasticslip γacc. In this work, the free energy is extended by a penalty term with Hχ, in the sense ofthe micromorphic theory of Forest [2009]:

ψ = ψe (Ee) + ψg (αχ) + ψh (γacc) +1

2Hχ ‖ F pχ − F p ‖2︸︷︷︸

ψχ

. (3.14)

Here, Hχ is chosen as large as possible to guarantee that F pχ ≈ F p. The elastic part of thefree energy is given by ψe(E

e) = (Ee : C : Ee)/2, whereC is the fourth-order elastic stiffnesstensor. Moreover, ψg(αχ) = El2 ‖ αχ ‖2 /2 is assumed, where E is Young’s modulus of anisotropic polycrystal and l denotes an internal length scale parameter. Further, the hardeningpart in the free energy ψh(γacc) is assumed as follows (compare, e.g., Steinmann and Stein


[1996]):

ψh(γacc) =(τ∞ − τ c

0)2

h0 − h∞ln

(cosh


τ∞ − τ c0

))+

1

2h∞γ

2acc, (3.15)

where h0 and h∞ are the initial and saturation hardening moduli, τ c0 is the initial critical shear

stress and τ∞ is an additional material parameter.Therefore, substituting Eqns. (3.13) and (3.14) in Eq. (3.6) leads to the local dissipationinequality in the following form (see 3.9.2):

D = Σe : Lp −Xb : F p − qγacc ≥ 0. (3.16)

In Eq. (3.16), q = ∂γaccψh and Σe = CeSe is the Mandel stress tensor, where Se is the elastic

second Piola-Kirchhoff stress tensor. Moreover, it has been assumed that

Se = ∂Eeψe = C : Ee = F e−1PF pT. (3.17)

The generalized stress has been assumed purely energetic (i.e.,M = ∂αχψg) and

Xb = −∂F pχψχ = Hχ (F p − F pχ) . (3.18)

Note that ∂F pψχ = −∂F pχψχ. Consequently, using Eq. (3.2), the local dissipation is given by

D =∑α

(τα −X#

α

)γα − qγacc ≥ 0, (3.19)

where τα = dα · (Σenα) and X#α = dα ·

(X#nα

)represent, respectively, the resolved shear

stress and the back stress associated to the slip system α in the intermediate configuration,where X# = XbF pT. A thermodynamically consistent flow rule (satisfying Eq. (3.19)) isassumed as follows:

γα = sgn(τα −X#α )γ0

⟨|τα −X#

α | − (τ c0 + q)

τD

⟩p, (3.20)

in which γ0, τ c0 , τD and p are, respectively, the reference shear rate, the initial yield stress, the

drag stress and the strain rate sensitivity parameter. Moreover, the shear flow stress is obtainedby

τ cf (γacc) = τ c

0 + q = τ c0 + (τ∞ − τ c

0) tanh


τ∞ − τ c0

)+ h∞γacc. (3.21)

3.4 Plastic deformation description on the grain boundaries 51

3.4 Plastic deformation description on the grain

boundaries

3.4.1 Dissipation, free energy and grain boundary yield criterion

The displacement jump JuK on the grain boundaries is assumed to vanish since the grains’sizes are considered to be sufficiently large such that the influence of GB sliding (inverse Hall-Petch effect) can be neglected. A grain boundary yield criterion is motivated from the totaldissipation inequality (Eq. (3.6)), in which the grain boundary free energy ψΓ is implemented.In addition, the volume integral is assumed to be limited to a surface integral using the pillbox-argument (∂V0 → Γ) on the grain boundary (for details see, e.g., Wulfinghoff [2017]). Thedissipation on the GB takes the following form:

DΓ =

∫∂V0

(t · u+m : B

)dA−

∫Γ

ψΓ dΓ =

∫Γ

(JmK : B − ψΓ

)︸︷︷︸

DΓ

dΓ ≥ 0. (3.22)

It is important to mention that JBK ≈ 0 is assumed motivated by the data from three dimen-sional discrete dislocation dynamics (DDD) simulations (Bayerschen et al. [2015]), showingthe approximate continuity of plastic strain on the grain boundaries (see also Wulfinghoff[2017]). In fact, JBK, which represents the difference of plastic deformations on the adjacentgrains surfaces, leads to the storage of geometrically necessary dislocations (GNDs) at thegrain boundary and, therefore, the increase in the grain boundary energy. The data from DDDsimulation shows that the average plastic strain at the GBs seems to be continuous. By assum-ing a finite value of JBK, the GNDs store within the GBs and, therefore, the grain boundaryenergy becomes large for small values of JBK. In this case, the GB energy related to the storageof GNDs acts as a penalty energy which couple the surface plastic deformations of adjacentgrains. Moreover, the free energy density of the grain boundary is assumed as follows:

ψΓ = ψk(BΓ) + ψh(β) +1

2Hχ

Γ ‖ B −BΓ ‖2, (3.23)

where again a micromorphic approach is used for numerical reasons with a large numberHχΓ to

guarantee that B ≈ BΓ . Furthermore, β is a GB isotropic hardening variable. By assumingψk = Hk ‖ BΓ ‖2 /2 and ψh = Hhβ

2/2, in which Hk and Hh have constant values, the localdissipation is given by (see Eq. (3.22))

DΓ = (JmK− ∂BψΓ) : B − ∂BΓψΓ : BΓ − ∂βψhβ =

(mχ −XΓ

): B

Γ − qhβ ≥ 0, (3.24)


where it has been assumed that JmK = ∂BψΓ and the following abbreviations have been used:

mχ = ∂BψΓ, XΓ = ∂BΓψk, qh = ∂βψh. (3.25)

Motivated by Eq. (3.24), the following grain boundary yield criterion is assumed:

fΓ =‖mχ −XΓ ‖ − (mc + qh) ≤ 0, (3.26)

in which mc denotes the initial grain boundary yield strength. As it can be deduced fromthe GB yield criterion (Eq. (3.26)),XΓ corresponds to the back stress (kinematic hardening),whereas qh represents the increase of the grain boundary strength as an isotropic hardeningstress. Moreover, from the principle of maximum dissipation (see, e.g., Simo and Hughes[1998] and Eq. (3.24)) one obtains:

BΓ

= γΓ∂mχfΓ = γΓ mχ −XΓ

‖mχ −XΓ ‖, β = γΓ, (3.27)

where γΓ represents a consistency parameter satisfying the Kuhn-Tucker conditions:

γΓ ≥ 0, fΓ ≤ 0, γΓfΓ = 0. (3.28)

Accordingly, a time discretization of Eq. (4.24) reads:

∆BΓ = ∆γΓN , ∆β = ∆γΓ, (3.29)

in which ∆γΓ = ∆tγΓ and

N =mχ −XΓ

‖mχ −XΓ ‖, (3.30)

being evaluated at the end of the time step. It is noteworthy that the rate-dependent flow ruleinside the grains (Eq. (3.20)) is widely implemented due to the problems in capturing uniqueactive slip systems when a rate-independent flow rule is applied. However, since there is nosuch problem in the grain boundary model, a rate-independent GB yield criterion (Eq. (3.26))is assumed.

The implementation of the grain boundary model is carried out using the radial returnmapping algorithm. Implementing this algorithm, it is determined that the grain boundary isactivated (i.e., it allows the dislocations to transmit) or it acts as an obstacle against crossing thedislocations. This algorithm is given in section 3.4.2 in detail. It is noted that the linearizationpart in the algorithm is given in section 3.6.2.

3.5 Computational implementation in material model 53

3.4.2 Radial return mapping algorithm

In order to evaluate the grain boundary yield condition, a return mapping algorithm is imple-mented (compare, e.g., Simo and Hughes [1998]). It is assumed that all identities regardingthe previous time step are known (e.g., BΓ

n, βn, Hpχ

n ). The algorithm for each integrationpoint on the GB is given in Box 1. It is noted that N = N

tr, the proof of which is an easytask.

1. Get the surface plastic deformationB using Eq. (3.4).

2. Get the trial-values ofmχ,XΓ and qh (assuming the GB to be inactive):

mχtr= Hχ

Γ

(B −BΓ

n

), XΓtr

= HkBΓn, qtr

h = Hhβn.

3. Compute the trial-value of the grain boundary yield criterion f trΓ (Eq. (3.26)).

If f trΓ < 0 ⇒ the GB is not activated :

mχ = mχtr, BΓ = BΓ

n, XΓ = XΓtr

, β = βn.

Else ⇒ the GB is activated :

N = mχtr−XΓtr

‖mχtr−XΓtr‖, ∆γΓ =

f trΓ

Hk+Hh+HχΓ,

BΓ = BΓn + ∆γΓN , β = βn + ∆γΓ, mχ = Hχ

Γ

(B −BΓ

).

Box 1: Grain boundary return mapping algorithm.

3.5 Computational implementation in material model

3.5.1 Time discretization

In the current work, the time discretization for the relation F p = LpF p is adopted from thework by Steinmann and Stein [1996] in terms of the midpoint rule as follows:

F p − F pn =

1

2hp (F p + F p

n) , (3.31)

in which hp = (tn+1− tn)Lp = ∆tLp. The midpoint rule is second-order accurate and it doesnot suffer from excessive violation of the plastic incompressibility condition (see, e.g., Alipouret al. [2018]). It should be noted that the index ′n + 1′ is omitted, for simplicity. In addition,fp is defined by

fp = F pF pn−1 (3.31)

=

(I − 1

2hp

)−1(I +

1

2hp

), (3.32)


and the trial-value of Ce is given by

Cetr

= (F etr

)TF etr

= fpTCefp, (3.33)

in whichF etr

= FF pn−1 = F eF pF p

n−1 = F efp. (3.34)

Furthermore, the flow rule (Eq. (3.20)) is discretized using the implicit Euler scheme:

∆γα = ∆γ0 sgn(τα −X#α )

⟨|τα −X#

α | − τ c

τD

⟩p, (3.35)

where ∆γ0 = ∆tγ0 and τ c = τ c0 + q.

3.5.2 Modification of the Newton-scheme

In this work, a modification is applied in order to improve the numerical convergence of theNewton-scheme. For this purpose, an improved initial guess of the solution is computed. Theflow rule (Eq. (3.35)) is, in a first step, modified as follows by a regularized power law (fordetails see Wulfinghoff and Böhlke [2013] and Alipour et al. [2018]):

∆γα =

0 |τα −X#

α | ≤ τ c

∆γ0 sgn(τα −X#α )(|τα−X#

α |−τc

τD

)pτ c < |τα −X#

α | ≤ τR

sgn(τα −X#α )(m(|τα −X#

α | − τR) + ∆γ0(∆τR

τD )p)

|τα −X#α | > τR

,

(3.36)in which the flow rule is approximated by a linear function with the slope m for large valuesof the shear stress, starting from τR. Moreover, ∆τR = τR − τ c is computed by

∆τR = τD

(mτD

∆γ0 p

) 1p−1

. (3.37)

Once an improved starting solution is obtained, the value of m is increased to guarantee thatthe final solution corresponds to the non-regularized power law (Eq. (3.35)).

3.5.3 Residuals and the linearization process

In this section, the goal is to calculate the primary unknowns (Se, τ c,X#) by solving the resid-ual equations, described in the following, via a three-level Newton-scheme. Three residuals

3.5 Computational implementation in material model 55

are given by Eqns. (3.17) and (3.21) as well as the definitionX# = XbF pT as follows:

rs(Se, τ c,X#) = Ee(Se, τ c,X#)− S : Se = 0, (3.38)

rτ (Se, τ c,X#) = τ cf (γacc,S

e, τ cX#)− τ c = 0, (3.39)

r#(Se, τ c,X#) = X# −Hχ(F p(Se, τ c,X#)− F pχ

)F pT(Se, τ c,X#) = 0, (3.40)

in which S = C−1.In the current work, the main idea to calculate the primary unknowns is to reuse the local modelin Alipour et al. [2018]. The solution of Eqns. (3.38)-(3.40) is carried out in a partitionedmanner. First, we introduce the following abbreviations:

r =(

(rs)T, rτ)T, x =

((Se)T, τ c

)T. (3.41)

1. First r = 0 is solved for Se and τ c, withX# being fixed, via the Newton-scheme based onthe following linearization:

r + (∂xr)∆x = 0. (3.42)

This can be done using an already existing material subroutine for single crystals withoutgradient-extension. In the converged state (r = 0), x is linearized consistently w.r.t. X#:

dr = (∂xr) dx+ (∂X#r) dX# = 0 → ∂X#x = −(∂xr)−1(∂X#r). (3.43)

2. Next, the linearized equation r# = 0 is solved:

r# +[(∂xr

#)(∂X#x) + ∂X#r#]

∆X# = 0. (3.44)

Steps 1 and 2 are repeated until convergence in r and in r# is reached. This procedure permitsto reuse the local crystal plasticity implementation of Alipour et al. [2018], which needs tobe only slightly modified by adding the backstress X#, which is treated as a constant inputparameter. The aim is to reach quadratic convergence on all levels.The matrix ∂xr includes the following derivatives:

∂xr =

(∂Sers ∂τcrs

∂Serτ ∂τcrτ

), (3.45)


where (Alipour et al. [2018])

∂Sers = −S−∑α

(∂τα∆γα)Ms

α ⊗ M sα, (3.46)

∂τcrs = −∑α

(∂τc∆γα)Ms

α, (3.47)

∂Serτ = (∂γaccτcf )(∂Seγacc), (3.48)

∂τcrτ = (∂γaccτcf )(∂τcγacc)− 1, (3.49)

with ∆γα = ∆tγα. In addition, Ms

α and M sα are defined as

Ms

α = sym(dα ⊗ nα) = sym

((Ce

(I − 1

2hp

)−1

dα

)⊗

((I +

1

2hp

)−Tnα

)),

(3.50)M s

α = ∂Seτα = (CeMα)s + 2S : (MαSe)s, (3.51)

hp =∑α

∆γαMα. (3.52)

Here, the notation (•)s denotes the symmetric part of a tensor. Further, ∂Seγacc, ∂τc∆γα, ∂τcγacc

and ∂τα∆γα are obtained by

∂Seγacc =∑α

sgn(τα −X#α )(∂Se∆γα) =

∑α

sgn(τα −X#α )(∂τα∆γα)M s

α, (3.53)

∂τcγacc =∑α

sgn(τα −X#α )(∂τc∆γα) = −

∑α

(∂τα∆γα), (3.54)

∂τα∆γα(3.36)=

0 |τα −X#

α | ≤ τ c

∆γ0 ( pτD )(|τα−X#

α |−τc

τD

)p−1

τ c < |τα −X#α | ≤ τR

m |τα −X#α | > τR

. (3.55)

Moreover, Eq. (3.40) is linearized as follows:

r# +[(∂xr

#)(∂X#x) + ∂X#r#]︸︷︷︸

D

∆X# = 0. (3.56)

in whichD

(3.40)= I−Hχ

(I F pT +

((F p − F pχ)

T

I

))∂X#F p, (3.57)

3.6 Finite element implementation 57

where I is the fourth-order identity tensor. Moreover,(A B

)and (A

T

B) are fourth ordertensors, being defined by (see 3.9.3):

(A B

): C = ACB, (3.58)(

AT

B

): C = ACTB, (3.59)

where A, B and C are the arbitrary second order tensors. In addition, ∂X#F p in Eq. (3.57)is given in matrix notation by (see 3.9.4)

∂X#F p =

((I − 1

2hp

)−1

((I − 1

2hp

)−1

F pn

))((∑

α

Mα(∂x∆γα)T)∂X#x−

∑α

Mα(∂τα∆γα)MTα

) (3.60)

with (see 3.9.5)∂X#x = −(∂xr)−1 (∂X#r

), (3.61)

in which

∂X#rτ(3.39)= −

(∂γaccτ

cf

)(∑α

sgn(τα − X#α )(∂τα∆γα)MT

α

), (3.62)

∂X#rs =∑α

(∂τα∆γα)Ms

αMTα (3.63)

and M s

α is defined in Eq. (3.50).

3.6 Finite element implementation

3.6.1 Weak forms

The finite element implementation involves the computation of the global unknowns u,Hpχ(i.e., 6 degrees of freedom for each node in 2D) by the classical Newton algorithm. The weakforms of Eqns. (3.12) and (3.11) are given as follows (see also 3.9.6):

Gu =

∫V0

P : δF dV −∫∂Vt

t · δu dA−∫V0

ρ0b · δu dV = 0, (3.64)


Gχ =

∫V0

(M : δαχ −Xb : δF

pχ)

dV

︸︷︷︸GVχ

−∫∂Vm

m : δB dA+

∫Γ

JmK : δB dΓ

︸︷︷︸GΓχ

= 0. (3.65)

In the sequel, it will be assumed thatm = 0 on ∂Vm.

3.6.2 Linearization of the variational forms

Using P : δF = τ : lδ, the linearized form of Eq. (3.64) is obtained by

dGu = d

∫V0

(τ : lδ) dV =

∫V0

((ldτ l

Tδ ) : I + dδ :

(F etr

dSeF etrT

))dV, (3.66)

in which lδ = δF F−1, dδ = sym(lδ), ld = dF F−1, Se= fp−1Sefp−T is the second Piola-

Kirchhoff stress with respect to the trial intermediate configuration and τ = F etr

SeF etrT is

the Kirchhoff stress. In addition, F etr

dSeF etrT in Eq. (3.66) is computed by (see 3.9.7)

F etr

dSeF etrT

= cd : dd + cχ : dF pχ, (3.67)

in which dd = sym(ld) and cd and cχ are defined as follows:

cd =(F etr s

F etrT)

: ∂Eetr Se :

(F etrT s

F etr)

+(F etr s

F etrT)

: ∂X#Se : AX : Ad,

(3.68)

cχ =(F etr s

F etrT)

: ∂X#Se :

(AX : ∂F pχXb +Xb

T

I

). (3.69)

Note that the productAs

AT is a fourth order tensor, being defined by:(A

s

AT)

: B = A sym(B)AT (3.70)

for arbitrary second order tensorsA andB. In addition,AX,Ad and ∂F pχXb are obtained by

AX=

((I F pT) +

Xb

Hχ

T

I

), (3.71)

Ad =

(I− ∂X#F p :

(Hχ(I F pT

)+

(Xb

T

I

)))−1

: (HχaX) , (3.72)

3.6 Finite element implementation 59

∂F pχXb =−(I− ∂X#F p :

(Hχ(I F pT

)+

(Xb

T

I

)))−1

: Hχ

(I− ∂X#F p :

(Xb

T

I

)),

(3.73)

aX = 2∂CetrF p :

(F etrT s

F etr), (3.74)

where ∂X#F p is computed by Eq. (3.60) and ∂CetrF p is computed inmatrix notation as follows

(see 3.9.4):

∂CetrF p =

((I − 1

2hp

)−1

((I − 1

2hp

)−1

F pn

))(∑α

Mα(∂x∆γα)T)∂Cetrx,

(3.75)in which (Alipour et al. [2018])

∂Cetrx = −(∂xr)−1

(12fp−T s

fp−1

0

). (3.76)

Moreover, ∂X#Se and ∂Eetr Se in Eqns. (3.68) and (3.69) are calculated in matrix notation as

follows (see 3.9.8):

∂X#Se

=

((fp−1

s

fp−T, 0)− 2

∑α

M′sα(∂x∆γα)T

)∂X#x+ 2

∑α

M′sαM

Tα(∂τα∆γα),

(3.77)

∂Eetr Se = 2∂

Cetr Se

= 2

((fp−1

s

fp−T, 0)− 2

∑α

(M′α

)s

(∂x∆γα)T)∂Cetrx, (3.78)

M′α = d′α ⊗ n′α =

((I +

1

2hp

)−1

dα

)⊗

(fp−1Se

(I +

1

2hp

)−Tnα

). (3.79)

Note that ∂x∆γα is given in Eqns. (3.53) and (3.54) and ∂X#x is defined in Eq. (3.61).Further, the differential of GV

χ in Eq. (3.65) is written as follows:

dGVχ =

∫V0

(δαχ : dM − δF pχ : dXb

)dV, (3.80)

in which the differential ofXb is given by (see 3.9.7)

dXb = Ad : dd + ∂F pχXb : dF pχ, (3.81)


whereAd and ∂F pχXb are calculated by Eqns. (3.72) and (3.73), respectively.Moreover, the term GΓ

χ in the weak form (Eq. (3.65)), corresponding to the grain boundaries,is given as follows:

GΓχ =

∫Γ

mχ : δB dΓ, (3.82)

using the identity JmK = mχ = ∂BψΓ (see Eq. (3.25)). The linearization of GΓχ takes the

following form:dGΓ

χ =

∫Γ

(( dBmχ) : dB) : δB dΓ, (3.83)

in which dBmχ is calculated by (see 3.9.9)

dBmχ = Hχ

Γ

(I− dBB

Γ)

(3.84)

with dBBΓ = 0 in the elastic case (no GB yielding) and in the plastic case:

dBBΓ = Hχ

Γ

(∆γΓ

‖mχtr −XΓtr ‖I−

(∆γΓ

‖mχtr −XΓtr ‖− 1

HχΓ +Hh +Hk

)NN

T).

(3.85)In Eq. (3.85),mχtr andXΓtr are the trial-values ofmχ andXΓ, respectively.

Note that Eqns. (3.64) and (3.65) are discretized in space using the finite element method.The details can be found in 3.9.10. In the currentwork, quadrilateral elementswith 4 integrationpoints and two-noded line elements are implemented, respectively, for the bulk and the grainboundary.

3.7 Results

In this section, the implementation of the model is first verified by a comparison with ananalytical solution from the literature in the context of a well-known example, in which a shearload is applied to a single crystal channel when it is firmly connected to an elastic coating(example 1). Moreover, the performance of the model in geometrically nonlinear deformationas well as the effect of isotropic and kinematic GB hardening on the material cyclic behaviorare investigated for bicrystals under shear load (examples 2 to 4). The finite element softwareFEAP (Finite Element Analysis Program) [http://www.ce.berkeley.edu/projects/feap] is used.

3.7 Results 61

(a)

(b)

Figure 3.2: (a) Infinite channel with an elastic coating on the top (the Fig. is taken fromWulfinghoff [2017]) (b) shear stress-strain response for different sizes.

3.7.1 Example 1: plastic flow in a channel

An infinite crystaline channel with height s and one activated slip system is sheared. Thechannel is connected to an elastic coating on the top (see Fig. 3.2 (a), h = s/10). In addition,the bottom surface is assumed to be fixed in both x- and y-directions (ux = uy = 0) andimpenetrable for dislocations (B = 0). Material hardening inside the crystal is neglectedin this example (i.e., h0 = h∞ = 0 MPa) and a rate-independent behavior is assumed bychoosing p = 1 and τD → 0 (τD = 10−5 MPa in the numerical example) in Eq. (3.20). Allother material parameters are listed in Table 3.1, in which ν is Poisson’s ratio. Further, theelastic properties for both coating and crystal are assumed identical.

Fig. 3.2 (b) shows the shear stress-strain response of the model regarding two different sizes.Here, γ indicates the top displacement divided by the total height of the system. As it is


E [GPa] ν [-] τ c0 [MPa] l [µm] γ0 [1/s] Hχ [MPa]

210 0.3 50 0.1 10−3 107

Table 3.1: Material parameters for example 1 (taken from Wulfinghoff [2017]).

(a) (b)

Figure 3.3: (a) Schematical drawing of a bicrystal with one slip system in each grain (b)discretization of the grains and the GB by 100× 50 quadrilateral elements and 50line elements.

expected, the material becomes stiffer by decreasing the size of crystal. The results presentan excellent agreement with the analytical solution by Wulfinghoff [2017], in which the samemodel with the limitation to small deformations is used.

3.7.2 Example 2: bicrystal under shear load with one slip systemin each grain

In this example, an aluminum bicrystal is simulated. To keep the investigation simple thegrains with dimensions s × s have one slip system each, with the slip direction d pointing inx-direction (see Fig. 3.3). The displacement is prescribed at the upper side and the velocity onthe top surface is taken as 0.5 µm/s, whereas the bottom surface is fixed. Although Voce-typehardening is taken into account inside the grains, grain boundary hardening is neglected forsimplicity (i.e.,Hk = Hh = 0 N/m). It should also be mentioned that the grains and the grainboundary are discretized by 100×50 quadrilateral elements and 50 line elements, respectively,when s = 10 µm. All other material parameters are listed in Table 3.2.

Fig. 3.4 shows the shear stress-strain behavior of the sample regarding various values of thegrain boundary strength mc

o and the internal length scale l. Here, mco = 106 N/m represents

a micro-hard GB (since GB yielding never occurs), however mco = 0 N/m corresponds to a

micro-free grain boundary.

3.7 Results 63

E [GPa] 69 τ∞ [MPa] 49.51ν [-] 0.33 h0 [MPa] 541.48p [-] 20 h∞ [MPa] 1τD [MPa] 60 γ0 [1/s] 10−3

τ c0 [MPa] 0.84 Hχ [MPa] 107

HχΓ [MPa] 107

Table 3.2: Material parameters for examples 2 to 4 (mostly taken from Steinmann and Stein[1996]).

As it is shown in Fig. 3.4, increasing the grain boundary strength also increases the overallstrength of the bicrystal (compare e.g., the black and green curves in Fig. 3.4 (a)). On theother hand, increasing the internal length scale at a constant mc

o leads to an earlier activationof the grain boundary (i.e., at smaller strains, compare e.g., the dashed and solid lines in greenin Fig. 3.4 (b)). However, l has no influence on the shear behavior, after the GB activation.Interestingly, if only Fig. 3.4 (a) is considered, the influence of the internal length scale l isalmost negligible compared to the influence of the grain boundary strength mc

o. The effect ofl is thus restricted to small strains in this example.

Furthermore, Fig. 3.5 shows the distribution of accumulated plastic strain when the dis-placement of the top surface is 2 µm. Here, no dislocation transmission occurs on GB whenmc

o = 106 N/m (i.e., the grain boundary acts as an obstacle to dislocations crossing). Note thatthe finite value of the accumulated plastic strain on the GB withmc

o = 106 N/m is mainly dueto visualization reasons (the accumulated plastic strain is first averaged over the elements andthen projected to the nodes). In this example, dislocations are able to cross the GB for values ofthe grain boundary strength mc

o below 106 N/m. Further, the distribution of the accumulatedplastic strain is observed in Fig. 3.6 regarding four various values of the GB strength whichhave been used in Fig. 3.5.Fig. 3.7 (a) illustrates the stress-strain diagrams for different bicrystal sizes swhen l = 1 µm

andmco = 103 N/m. As it is shown, the overall strength of the bicrystal increases by decreasing

the size of grains. The circles on the curves show the points after which the plastic deformationdramatically increases in the bicrystal. The values of shear stress regarding the aforementionedpoints (i.e., σ2

o , σ5o and σ10

o ) are depicted in Fig. 3.7 (b) with respect to the bicrystal size.Moreover, the relative residual norms in Eqns. (3.42) and (3.44) as well as the global Newton

loop are listed in Table 3.3 when s = 10 µm, l = 1 µm,mco = 103 N/m and the displacement

of the top surface is 0.7 µm. As it is shown, a very good convergence is achieved on all threelevels.


(a)

(b)

Figure 3.4: (a) Shear stress-strain response of a bicrystal with one slip system in each grainunder shear load for s = 10 µm (b) Magnification of Fig. 3.4 (a) in the small strainregion (dashed box).

3.7 Results 65

Figure 3.5: Distribution of the accumulated plastic strain for s = 10 µm and l = 1 µm atdisplacement 2 µm on top surface.

(a) (b)

(c) (d)

Figure 3.6: Distribution of the accumulated plastic strain for s = 10 µm and l = 1 µm atdisplacement 2 µm when (a)mc

o = 0 N/m (b)mco = 102 N/m (c)mc

o = 103 N/m(d)mc

o = 106 N/m.


(a)

(b)

Figure 3.7: (a) Influence of the structure size on the mechanical response of bicrystal undershear load (b) Specified shear stresses in Fig. 3.7 (a) in various sizes.

Iteration Residual of Residual of Residual ofnumber Eq. (3.42) Eq. (3.44) the global problem

1 1 1 12 6.3× 10−1 6.8× 10−1 1.7× 10−2

3 7.9× 10−2 3.9× 10−1 3.7× 10−4

4 1.7× 10−3 1.6× 10−1 8.8× 10−7

5 8.8× 10−7 2.5× 10−2 4.4× 10−12

6 2.6× 10−13 1.5× 10−3

7 6.9× 10−6

8 1.5× 10−10

Table 3.3: Relative residual norms.

3.7 Results 67

Figure 3.8: Cyclic behavior of the bicrystal when there is no hardening in the material.

3.7.3 Example 3: bicrystal under cyclic shear load

In order to show the effect of isotropic and kinematic GB hardening on the material behavior,the same bicrystal as in example 2 is investigated (see Fig. 3.3) when a cyclic shear load isapplied on the top surface. The velocity magnitude of the upper side is assumed to be constant(i.e., 0.5 µm/s). Here, s = 10 µm, l = 1 µm, mc

o = 103 N/m and Voce-hardening inside thegrains is neglected. Other material parameters are taken from Table 3.2.

Fig. 3.8 shows the shear stress-strain curve when there is no hardening on the GB (i.e.,Hh = Hk = 0 N/m). In this Figure, the plastic deformation starts from point “A” wherethe grain boundary acts as an obstacle against the transmission of dislocations. The norm ofback stress ‖ Xb ‖ increases from point “A” to “B” such that the grain boundary activationoccurs from point “B” to “C”. In this region, ‖ Xb ‖ remains constant due to the neglect ofGB hardening. There is no slip in the system between point “C” and “D”, whereas plasticdeformation starts again from point “D”, indicating a strong Bauschinger effect. From “D”to “E”, the back stress first decreases regarding the reverse applied force and then increases.The grain boundary becomes activated again as soon as ‖ Xb ‖ reaches a sufficiently highvalue (i.e., from point “E” to “F”). Afterwards, a similar behavior occurs from “F” to “C”.This behavior remains identical for further cycles.

In addition, the influence of theGB hardening on thematerial behavior is depicted in Fig. 3.9.Fig. 3.9 (a) shows the cyclic shear deformation when there is only kinematic hardening on the


(a) (b)

Figure 3.9: Cyclic behavior of the bicrystal with (a) only GB kinematic hardening for Hh =0 N/m and Hk = 105 N/m (b) only GB isotropic hardening for Hh = 104 N/mand Hk = 0 N/m.

Figure 3.10: Schematical drawing of a bicrystal with specific two slip systems in each grain.

GB. Although the general material behavior in Fig. 3.9 (a) is similar to the curve in Fig. 3.8,the back stress remains no longer constant wherever the grain boundary is activated (e.g., frompoint “B” to “C”). This issue arises from the existence of the kinematic hardening on the GBthat increases the overall strength of the bicrystal.

Moreover, Fig. 3.9 (b) illustrates that the material becomes stiffer when GB isotropic hard-ening is assumed (i.e., cyclic hardening). Moreover, GB activation occurs in smaller regionsin each cycle compared to the former cycles. In the limit, the cycles converge to a final one inwhich no grain boundary activation is expected to occur. The observed results represent thetypical cyclic hardening behavior being experimentally found for manymetals (see e.g., Lambaand Sidebottom [1978a], Lamba and Sidebottom [1978b], Mroziński and Szala [2010]).

3.7 Results 69

Figure 3.11: Shear stress-strain response of a bicrystal with two slip systems in each grainunder shear load.

(a) (b)

(c) (d)

Figure 3.12: Distribution of dislocation density magnitude for s = 5 µm and l = 1 µm atdisplacement 0.15 µm (γ = 0.03) when (a)mc

o = 0 N/m (b)mco = 102 N/m (c)

mco = 103 N/m (d)mc

o = 106 N/m.


Figure 3.13: Mesh convergence for the bicrystal with two slip systems in each grain for s =5 µm,mc

o = 103 N/m and l = 1 µm.

3.7.4 Example 4: bicrystal under shear load with two slipsystems in each grain

In this example, a bicrystal with two specific slip systems in each grain is sheared (see Fig. 3.10).In addition to Voce-hardening inside the grains, isotropic and kinematic hardening on the GBare taken into account such that Hh = 104 N/m and Hk = 104 N/m. Moreover, the upperand lower surfaces are assumed to be impenetrable for dislocations (B = 0). Other materialparameters are identical to those in example 2 (see Table 3.2).

Fig. 3.11 shows the shear deformation of the bicrystal regarding different values of the GBstrengthmc

o and the internal length scale l. In this Figure, 76× 38 quadrilateral elements and38 line elements are implemented, when s = 5 µm. As it is shown, the material becomes stifferby increasing the value of mc

o at a constant l. The grain boundary acts as a micro-hard GB aslong as its strength is not overcome by the micro stresses. It is noted that increasing the value ofmc

o leads to a later activation of the grain boundary (i.e., at larger γ). Further, the distributionof the dislocation density magnitude (||αχ||) is shown in Fig. 3.12 regarding various valuesof the GB strength when γ = 0.03. Pile-ups of dislocations at the grain boundary are clearlyobserved which grow by increasing the GB strength. Note that the pile-ups of dislocationswhen mc

o = 0 N/m is only due to the misorientation between the slip systems in differentgrains. Moreover, the mesh convergence for the bicrystal with two slip systems in each grain

3.8 Summary 71

is shown in Fig. 3.13, in which the numbers refer to the quadrilateral elements in the sample.It is noteworthy that, in the current work, themain goal is to improve the crystal plasticitymodelregarding large plastic deformations and to develop a new grain boundary model. Therefore,the grain boundary strength has been assumed as a parameter which is not influenced by otherfactors, e.g., the mismatch between slip systems. However, one can evaluete the GB strengthusing a transmission criterion (see e.g., Bayerschen et al. [2016]) to take the effect of slipsystem orientations and the grain boundary orientation into account. This evaluation highlydepends on the criterion which is chosen (see e.g., the criterion suggested byWerner and Prantl[1990]).

3.8 Summary

A grain boundary model for geometrically nonlinear crystal plasticity has been presented. Themodel is based on the dislocation density tensor and surface related considerations (plasticsurface deformations). The implementation has been verified by a comparison with an analyt-ical solution regarding an investigation of the plastic flow in a crystaline channel under shearload. In order to show the performance of the model in geometrically nonlinear deformationsas well as the effect of isotropic and kinematic GB hardening on the material behavior, severalexamples on bicrystals have been examined. The finite element method has been implementedfor grains and grain boundaries with 4-noded quadrilateral elements and 2-noded line elements,respectively.

The shear stress-strain curves regarding the bicrystal with one slip system in each grain showthat increasing the value of mc

o leads to strengthening of the material, whereas increasing theinternal length scale results in an earlier GB activation. In addition, it has been shown thatthe grain size affects the material strength such that the bicrystal with smaller grains is stifferthan the one with larger grains. Compared to gradient plasticity models without special grainboundary models, the influence of the internal length scale parameter l on the results turnedout to be less pronounced.

In order to show the effect of GB hardening on the material performance, a cyclic shear loadhas been applied on the bicrystal. The results show that the grain boundary becomes activatedin specific regions when the back stress ‖ Xb ‖ reaches a sufficiently high value. The backstress remains constant in such regions if no hardening is assumed on the GB. However, itincreases in activation regions by assuming kinematic hardening on the grain boundary. Fur-thermore, assuming GB isotropic hardening leads to “cyclic hardening” phenomena where theGB activation regions decrease in each cycle. The cycles converge to a final one, in which no


grain boundary activation occurs.As a final example, a bicrystal with two specified slip systems in each grain has been investi-

gated where Voce-hardening inside the grains as well as isotropic and kinematic hardening onthe GB have been taken into account. The results show that increasing the value of mc

o leadsto a later GB activation. In future works, it is planned to implement the current GB model formore complicated micro-structures and to investigate the influence of misorientation betweentwo adjacent grains on the grain boundary strength.AcknowledgmentThe financial support of Aachen Institute for Advanced Study in Computational EngineeringScience (AICES) by the DFG through Grant GSC 111 is gratefully acknowledged. In addition,the authors acknowledge the financial support related to the project “Computational Methodsfor Microscale Plasticity StUpPD-180-15”.

3.9 Appendix

3.9.1 Calculation of the stress power

Using Eqns. (3.4) and (3.8) one writes:

m : B = M (N×)(N×) : Fpχ

(N×) = −M (N×)(N×)(N×) : Fpχ

= −M : Fpχ

(N×) = M : B = M : (FpχεN )

=(M : (F

pχε))·N .

(3.86)

Note that F pχ= H

pχ. Therefore, Eq. (3.7) is rewritten using Gauss’ theorem as follows:

Pext(3.86)=

∫V0

(Div(P Tu) + Div(M : (F

pχε)))

dV +

∫V0

ρ0b · u dV, (3.87)

in whichDiv(M : (F

pχε)) = −CurlT(M ) : F

pχ+M : αχ (3.88)

with αχ = CurlT(F pχ). Finally, the stress power is obtained by:

Pext =

∫V0

(P : F −Xb : F

pχ+M : αχ

)dV, (3.89)

whereXb = CurlT(M ).

3.9 Appendix 73

3.9.2 Evaluation of the dissipation inequality

By substituting Eqns. (3.13) and (3.14) in Eq. (3.6), the local dissipation takes the followingform:

D = P : F−∂Eeψe : Ee−qγacc−(Xb + ∂F pχψχ

): F

pχ+(M − ∂αχψg) : αχ−∂F pψχ : F p ≥ 0,

(3.90)with q = ∂γacc

ψh and Ee = sym(F eT ˙FF p−1). In addition, it is assumed thatXb = −∂F pχψχ

andM = ∂αχψg. Therefore,

D = P : F −(F e∂EeψeF

p−T) : F − ∂Eeψe :

(F eTF

˙F p−1

)− qγacc − ∂F pψχ : F p ≥ 0.

(3.91)The dissipation is assumed to vanish for virtually ′frozen′ plastic deformation:

D∣∣γα=0

=(P − F e∂EeψeF

p−T) : F = 0. (3.92)

Since F is arbitrary, the first Piola-Kirchhoff stress can be obtained by

P = F e∂EeψeFp−T, (3.93)

which must be equal to its definition as

P = τF−T = F eSeF eTF−T. (3.94)

Equality of Eqns. (3.93) and (3.94) leads to the second Piola-Kirchhoff stress tensor withrespect to the intermediate configuration as

Se = ∂Eeψe = C : Ee. (3.95)

Finally, the local dissipation (Eq. (3.91)) can be written as (with F p ˙F p−1 = −F pF p−1 =

−Lp)D = (CeSe) : Lp −Xb : F p − qγacc ≥ 0. (3.96)


3.9.3 Fourth-order box products in matrix notation

Assuming three arbitrary second order tensorsA,B andC, one can write Eq. (3.58) in indexnotation as follows: (

A B)ijkl

Ckl = AikCklBlj, (3.97)

inwhich thematrixVoigt-formofA B in 2Dproblem iswritten as (withC = (C11, C12, C21, C22)T):

A B =

A11B11 A11B21 A12B11 A12B21

A11B12 A11B22 A12B12 A12B22

A21B11 A21B21 A22B11 A22B21

A21B12 A21B21 A22B12 A22B22

. (3.98)

Moreover,AT

B in Eq. (3.59) is obtained by exchanging the columns 2 and 3 in Eq. (3.98).Now, we assume thatB is a symmetric tensor. Therefore,D = ATBA = (AT

s

A) : B isobtained as a symmetric tensor with the normalized Voigt notation of:

B =

B11

B22√2B12

, D =

D11

D22√2D12

. (3.99)

Therefore, the matrix form ofAs

AT in Eq. (3.70) regarding a 2D case reads:

As

AT =

A211 A2

21

√2A11A21

A212 A2

22

√2A12A22√

2A12A11

√2A21A22 (A11A22 + A21A12)

. (3.100)

3.9.4 Derivation of dF p in Eqns. (3.57) and (3.74)

Using Eqns. (3.31) and (3.32), the differential of F p is obtained by

dF p =

(I − 1

2hp

)−1

dhp

(I − 1

2hp

)−1

F pn, (3.101)

in which hp = ∆tLp anddhp (3.2)

=∑α

d∆γαdα ⊗ nα (3.102)

withd∆γα = ∂x∆γα dx+ ∂X#

α∆γαMα : dX#. (3.103)

3.9 Appendix 75

Note that (∂X#α

∆γα) = −(∂τα∆γα). Therefore, Eq. (3.101) is written in matrix notation asfollows:

dF p (3.102)=

((I − 1

2hp

)−1

((I − 1

2hp

)−1

F pn

))(∑α

Mα(∂x∆γα)T)

dx

+

((I − 1

2hp

)−1

((I − 1

2hp

)−1

F pn

))∂X#hp dX#.

(3.104)

Finally, substituting dx = (∂X#x) dX#+(∂Cetrx) dCetr and dX#hp =

∑α

Mα(∂X#α

∆γα)MTα

in Eq. (3.104) leads to:

∂CetrF p =

((I − 1

2hp

)−1

((I − 1

2hp

)−1

F pn

))(∑α

Mα(∂x∆γα)T)∂Cetrx,

(3.105)

∂X#F p =

((I − 1

2hp

)−1

((I − 1

2hp

)−1

F pn

))((∑

α

Mα(∂x∆γα)T)∂X#x−

∑α

Mα(∂τα∆γα)MTα

).

(3.106)

3.9.5 Derivation of ∂X#x in Eq. (3.60)

The differential of the nonlinear system of Eqns. (3.38) and (3.39) takes the form

dr = (∂xr) dx+ (∂X#r) dX# = 0, (3.107)

which leads to∂X#x = −(∂xr)−1 (∂X#r

), (3.108)

where ∂xr is defined in Eq. (3.45). In Eq. (3.108), ∂X#r is computed by:

∂X#rs(3.38)= ∂X#Ee, (3.109)

∂X#rτ(3.39)= −

(∂γaccτ

cf

)(∑α

sgn(τα − X#α )(∂τα∆γα)MT

α

). (3.110)

In order to obtain ∂X#Ee in Eq. (3.109), the differential of Ee is given as follows (Alipouret al. [2018]):

dEe = −∑α

( d∆γα)Ms

α, (3.111)


in which d∆γα and M s

α are defined in Eq. (3.103) and (3.50), respectively . Consequently,

∂X#rs = ∂X#Ee (3.111),(3.103)=

∑α

(∂τα∆γα)Ms

αMTα. (3.112)

3.9.6 Calculation of the weak form of Eq. (3.11)

The weak form of Eq. (3.11) has the following form:∫V0

(CurlT(M)−Xb

): δF pχ dV = 0, (3.113)

which can be written as follows (see Eq. (3.9)):

Gχ =

∫V0

(M : δαχ −Xb : δF

pχ)

dV −∫∂Vm

m : δB dA+

∫Γ

JmK : δB dΓ = 0. (3.114)

3.9.7 Derivation of F etr

dSeF etrT

in Eq. (3.66)

According to Se= S

e(Eetr

,X#) one writes:

F etr

dSeF etrT

= (F etr s

F etrT) : ∂

Eetr Se : dEetr

+ (F etr s

F etrT) : ∂X#Se : dX#.

(3.115)Furthermore, (sinceX# = XbF pT)

dX# = dXbF pT +Xb dF pT (3.18)= dXbF pT +Xb

(dF pχT +

1

HχdXbT

)=

((I F pT

)+Xb

Hχ

T

I

)︸︷︷︸

AX

: dXb +

(Xb

T

I

)︸︷︷︸∂F pχX#

: dF pχ.(3.116)

In order to obtain dXb in Eq. (3.116), the differential of rχ = Xb − Hχ (F p − F pχ) (seeEq. (3.18)) is obtained by

drχ = dXb −Hχ(∂X#F p : dX# + ∂

CetrF p : dCetr)

+HχdF pχ, (3.117)

3.9 Appendix 77

in which ∂CetrF p and ∂X#F p are computed by Eqns. (3.105) and (3.106). Therefore,

Eq. (3.117) takes the following form:

drχ =

(I− ∂X#F p :

(Hχ(I F pT) + (Xb

T

I)

))︸︷︷︸

∂Xbrχ

: dXb

+Hχ

(I− ∂X#F p : (Xb

T

I)

)︸︷︷︸

∂F pχrχ

: dF pχ −Hχ(

2∂CetrF p :

(F etrT s

F etr))

︸︷︷︸aX

: dd = 0,

(3.118)in which dd = sym( dFF−1) and the product A

s

AT is defined in Eq. (3.70). FromEq. (3.118), dXb is calculated by

dXb =(−(∂Xbrχ)−1 : ∂F pχrχ

)︸︷︷︸∂F pχXb

: dF pχ

+((∂Xbrχ)−1 : HχaX

)︸︷︷︸Ad

: dd.(3.119)

Substituting Eq. (3.116) in Eq. (3.115) and using Eq.(3.119) lead to:

F etr

dSeF etrT=(

(F etr s

F etrT) : ∂

Eetr Se : (F etrT s

F etr

) + (F etr s

F etrT) : ∂X#Se : AX : Ad

)︸︷︷︸

cd

: dd

+

((F etr s

F etrT) : ∂X#Se : (AX : ∂F pχXb +Xb

T

I)

)︸︷︷︸

cχ

: dF pχ.

(3.120)

3.9.8 Derivation of dSe

The differential dSe

= d(fp−1Sefp−T) can be decomposed into three parts (see Alipour et al.[2018]):

dfp−1Sefp−T = −(I +

1

2hp

)−1

dhp

(I +

1

2hp

)−1

Sefp−T = −∑α

M′α d∆γα,

(3.121)


fp−1Se dfp−T =(

dfp−1Sefp−T)T

= −∑α

(M′α

)Td∆γα, (3.122)

fp−1 dSefp−T =(fp−1

s

fp−T, 0)

dx. (3.123)

Therefore,

dSe

=

((fp−1

s

fp−T, 0)

dx− 2∑α

(M ′

α

)s

( d∆γα)T), (3.124)

where d∆γα is calculated in Eq. (3.103). Substituting Eq. (3.103) and the relation dx

= (∂X#x) dX# +(∂Cetrx) dCetr in Eq. (3.124) leads to

dSe

=

(((fp−1

s

fp−T, 0)− 2

∑α

(M′α

)s

(∂x∆γα)T)∂X#x+ 2

∑α

M′sαM

Tα(∂τα∆γα)

)︸︷︷︸

∂X#

˜Se

dX#

+

(((fp−1

s

fp−T, 0)− 2

∑α

(M′α

)s

(∂x∆γα)T)∂Cetrx

)︸︷︷︸

∂Cetr

˜Se

dCetr

.

(3.125)

3.9.9 Derivation of dBmχ in Eq. (3.83)

Regarding the assumptionmχ = ∂BψΓ = HχΓ

(B −BΓ

), the differential ofmχ is given by

dBmχ = Hχ

Γ

(I− dBB

Γ)

(3.126)

Further, the differential ofBΓ reads (see Eq. (3.29)):

dBΓ = dγΓN + ∆γΓ dN . (3.127)

In order to calculate dγΓ in Eq. (3.127), the differential of fΓ in Eq. (3.26) is given by

dfΓ = N :(

dmχ − dXΓ)− dqh. (3.128)

3.9 Appendix 79

Implementing XΓ = HkBΓ, qh = Hhβ and Eqns. (3.29) and (3.126), Eq. (3.128) takes the

following form (in the plastic case dfΓ = 0):

dfΓ = HχΓN : dB − (Hχ

Γ +Hk) N : dBΓ −Hh dγΓ = 0, (3.129)

which leads to (with Eq. (3.127))

dγΓ =Hχ

ΓN

HχΓ +Hh +Hk

: dB. (3.130)

Moreover, dN in Eq. (3.127) reads:

dN = dNtr

= d

(mχtr −XΓtr

‖mχtr −XΓtr ‖

)

=dmχtr

‖mχtr −XΓtr ‖− mχtr −XΓtr

‖mχtr −XΓtr ‖3⊗(mχtr −XΓtr

): dmχtr,

(3.131)in which the notation (•)tr represents the trial-value of a tensor. Using dmχtr = Hχ

Γ dB inEq. (3.131), dN is obtained by

dN =Hχ

Γ

‖mχtr −XΓtr ‖

(I− N ⊗ N

): dB. (3.132)

Finally, substituting Eqns. (3.130) and (3.132) in Eq. (3.127) leads to:

dBBΓ = Hχ

Γ

(∆γΓ

‖mχtr −XΓtr ‖I−

(∆γΓ

‖mχtr −XΓtr ‖− 1

HχΓ +Hh +Hk

)NN

T).

(3.133)


After the linearization of Eqns. (3.64) and (3.65), they are discretized in space using the finiteelement method. To this end, lδ and ld are discretized using the shape functionsNI as follows:

lδ = δFF−1 = grad(δu) =

nI∑I=1

δuI ⊗∇xNI , (3.134)

ld = grad( du) =

nI∑I=1

duI ⊗∇xNI , (3.135)


in which nI is the total number of nodes in each element. In the current work, quadrilateralelements with 4 integration points are implemented for the bulk (i.e., nI = 4). In addition, dd

is discretized as:d d = Bε du, (3.136)

where

Bε =

N1,x 0 N2,x 0 N3,x . . . NnI ,x 0

0 N1,y 0 N2,y 0 . . . 0 NnI ,y√2

2N1,y

√2

2N1,x

√2

2N2,y

√2

2N2,x

√2

2N3,y . . .

√2

2NnI ,y

√2

2NnI ,x

. (3.137)

Here, NI,x and NI,y are the first and second components of the shape function gradient,respectively, with respect to the current configuration. Furthermore, F pχ and the dislocationdensity matrix for each element are given by

F pχ = (F pχ11 F pχ

12 F pχ21 F pχ

22 )T, αχ(3.10)= (αχ13 αχ23)T (3.138)

and they can bewritten in terms of the nodal degrees of freedom Hpχ

by (with Hpχ

= Fpχ−I)

F pχ = NχFpχ, αχ=BαF

pχ, (3.139)

in which

Nχ =

N1 0 0 0 0 N2 . . . NnI 0 0 0

0 N1 0 0 0 0 N2 . . . NnI 0 0

0 0 N1 0 0 0 0 N2 . . . NnI 0

0 0 0 N1 0 0 0 0 N2 . . . NnI

, (3.140)

Fpχ

=(F pχ

111F pχ

112F pχ

121F pχ

122. . . F pχ

nI11F pχnI12

F pχnI21

F pχnI22

)T, (3.141)

Bα =

(−N1,Y N1,X 0 0 . . . −NnI ,Y NnI ,X 0 0

0 0 −N1,Y N1,X . . . 0 0 −NnI ,Y NnI ,X

).

(3.142)Note that NI,X and NI,Y are the first and second components of the shape function gradient,respectively, with respect to the initial configuration.

3.9 Appendix 81

The discretized forms of Gu and dGu in Eqns. (3.64) and (3.66) take the following forms:

Gu ≈ne∑e=1

δueT

np∑p=1

(BTεpτ pWpJp

)e, (3.143)

dGu ≈ne∑e=1

δueT(Ke

uudue +Ke

uχdFpχe), (3.144)

where the element stiffness matrices are calculated by summation over the integration points pas follows:

Keuu = Ke

uug+

np∑p=1

(BTεpcdpBεp)WpJp, (3.145)

Keuχ =

np∑p=1

BTεpcχpNχpWpJp, (3.146)

in which

Keuug

=

g11 0 g12 . . . g1np 0

0 g11 0 . . . 0 g1np

g12 0 g22 . . . g2np 0... ... ... ... ... ...0 g1np 0 . . . 0 gnpnp

, (3.147)

gIJ =

np∑p=1

(∇xNI · (τ∇xNJ))WpJp. (3.148)

It should be noted thatWp and Jp are, respectively, the integration weight and determinant ofthe Jacobian matrix regarding the integration point p.Similar to the previous section, the discretized forms ofGV

χ and dGVχ in Eqns. (3.65) and (3.80)

are given by

GVχ ≈

ne∑e=1

δFpχe

T ∑p

(BTαpM p −NT

χpXbp

)eWpJp, (3.149)

dGVχ ≈

ne∑e=1

δFpχe

T (Ke

χu due +Keχχ dF

pχe), (3.150)

where the element stiffness matrices are obtained as follows:

Keχu = −

np∑p=1

NTχpAdpBεpWpJp, (3.151)


Keχχ =

np∑p=1

(BTαp(∂αχM )pBαp −NT

χp(∂F pχXb)pNχp

)WpJp, (3.152)

in which ∂αχM = El2I .In the current work, two-noded line elements are implemented for the grain boundaries.Therefore, Eqns. (3.82) and (3.83) are discretized as follows:

GΓχ ≈

ne∑e=1

Le2

2∑p=1

((mχ

p )TδBp

)e, (3.153)

dGΓχ ≈

ne∑e=1

Le2

2∑p=1

((δF

pχ

p )TN

T

p

(∂Bpm

χp

)N p dF

pχ

p

)e, (3.154)

in whichLe is the element length in the reference configuration and p represents the integrationpoint index of the line elements. Here, the integration point positions are chosen identical tothose of the nodes. In addition, δBe in Eq. (3.153) is calculated as follows (see Eq. (3.4)):

δB = −δF pχ(N×) =⇒ δBe = NeδF

pχe

, (3.155)

Ne

=

(−N2 N1 0 0

0 0 −N2 N1

)e

, (3.156)

in which Ni is the ith component of the normal vector corresponding to the element.Finally, the internal element force vector and the element stiffness matrix are written, respec-tively, using Eqns. (3.153) and (3.154) as follows:

f e =Le2

(0 0

((mχ)Tp=1 N

)0 0

((mχ)Tp=2 N

) )e, (3.157)

Ke =Le2

02×2 02×4 02×2 02×4

04×2 NT(∂Bm

χ)p=1 N 04×2 04×4

02×2 02×4 02×2 02×4

04×2 04×4 04×2 NT(∂Bm

χ)p=2 N

e

. (3.158)

4 Article 3:A grain boundary modelconsidering the grainmisorientation within ageometrically nonlineargradient-extended crystalviscoplasticity theory


Alipour, A., Reese, S., Svendsen, B. and Wulfinghoff, S. [2020], ‘A grain boundary modelconsidering the grainmisorientationwithin a geometrically nonlinear gradient-extended crystalviscoplasticity theory’, Proceedings of The Royal Society A. 476: 20190581.


A. Alipour extended the crystal plasticity framework by Alipour et al. (2019) by evaluatingthe grain boundary strength according to the misorientation between neighboring grains.She implemented the model into the academic finite element software FEAP (as an ownelement subroutine), set up and performed all simulations, interpreted the results and wrotethe manuscript. S. Wulfinghoff had the main idea to investigate the effect of misorienation onthe GB strength. S. Wulfinghoff, B. Svendsen and S. Reese read the article, contributed to thediscussion of the results and gave valuable suggestions for improvement. All authors approvedthe publication of the final version of the manuscript.

83

84 4 A grain boundary model considering the grain misorientation within a geometrically . . .

4.1 Abstract

The main goal of the current work is to present a grain boundary model based on the mis-match between adjacent grains in a geometrically nonlinear crystal viscoplasticity frameworkincluding the effect of the dislocation density tensor. To this end, the geometrically non-linear crystal viscoplasticity theory by Alipour et al. [2019] is extended by a more complexfree energy and a geometrical transmissibility parameter is used to evaluate the dislocationtransmission at the grain boundaries which includes the orientations of slip directions and slipplane normals. Then, the grain boundary strength is evaluated based on the misorientationbetween neighboring grains using the transmissibility parameter. In some examples, the effectof mismatch in adjacent grains on the grain boundary strength, the dislocation transmission atthe grain boundaries and the Hall-Petch slope is discussed by a comparison of two-dimensionalrandom-oriented polycrystals and textured polycrystals under shear deformation.

4.2 Introduction

Metals are used for a wide range of industrial applications mostly due to their high strengthand good processability. The movement of dislocations through the crystal lattice and theirinteractions with each other represent the fundamental mechanisms of plastic deformation incrystalline materials. To investigate the plasticity in crystals, numerous atomistic and discretesimulation approaches such as discrete dislocation dynamics have been developed. However,such discrete theories are often computationally expensive due to large numbers of the inter-actions of the individual dislocations. Therefore, their implementations are limited (or at leasttime-consuming) when they are used for large-scale problems.

In contrast, continuum approaches model dislocations in an averaged sense. Among the con-tinuum approaches, size-independent plasticity models (e.g., von Mises [1913], Hill [1966],Klusemann et al. [2012], Klusemann et al. [2013], Zhang et al. [2015], Alipour et al. [2018])are typically used in case of large-scale applications on the millimeter- to meter-scale and showgood agreement with experimental data. However, their accuracy is limited when stronglyinhomogeneous plastic deformations of materials on the microscale are investigated. Thisinhomogeneity arises from the microstructural characteristics such as grain boundaries whichlead to pile ups of dislocations and local stress concentrations. To investigate small-scaleplasticity, studies have shown that strain-gradient theories (e.g., Aifantis [1984], Gurtin et al.[2007], Cordero et al. [2013], Bayerschen et al. [2015], Wulfinghoff [2017]) are in betteragreement with the available experimental data (e.g., Yao et al. [2014], Ziemann et al. [2015],

4.2 Introduction 85

Mu et al. [2016]) since they introduce an internal length scale into the theory. Furthermore,extended crystal plasticity models for geometrically nonlinear crystal plasticity have been stud-ied by numerous authors (Bargmann et al. [2010], Gurtin [2010], Pouriayevali and Xu [2017],Alipour et al. [2019]).

In addition to the investigation of plasticity inside the grains, grain boundary (GB) modelingis also important since the presence of GBs leads to pile-ups of dislocations and consequently tostrengthening of the material, which is called Hall-Petch effect (Hall [1951] and Petch [1953]).Therefore, GBs might be considered as obstacles which prevent dislocation transmission (seee.g., Gottschalk et al. [2016], Aifantis et al. [2019]).

Although grain boundaries are mostly assumed to be ‘micro-hard’ (i.e., impenetrable todislocations) or ‘micro-free’ in some computational models (see e.g., Ekh et al. [2007], Wulf-inghoff et al. [2013], Bayerschen et al. [2015]), transmission of dislocations at the GBs dependson various factors, e.g., the grain boundary type, the dislocations reactions and the misorien-tation of adjacent grains (see e.g., Zaefferer et al. [2003], Gemperle et al. [2005], Bayerschenet al. [2016]). Therefore, numerous authors proposed various GB models to investigate dislo-cation transmission more precisely (e.g., Ashmawi and Zikry [2001], Ma et al. [2006], Gurtin[2008], Li et al. [2009], van Beers et al. [2013], McBride et al. [2016], Gottschalk et al. [2016],Wulfinghoff [2017], Rezaei et al. [2019]).

Regarding the misorientation of adjacent grains, many geometric slip transmission crite-ria have been developed using experimental data (e.g., Lee et al. [1989], Luster and Morris[1995], Ravi Kumar [2010], Bieler et al. [2014]) and have been implemented into computa-tional methods (e.g., Shi and Zikry [2011], Spearot and Sangid [2014], Hamid et al. [2017]).For example, Livingston and Chalmers [1957] investigated a geometric slip criterion, includingthe orientations of slip directions and the slip plane normals, to find the activated slip systemsclose to the location of dislocation pile-ups. Later, Werner and Prantl [1990] evaluated themismatch between the slip systems in adjacent grains via a sum over all existing slip systemcombinations (see also Ravi Kumar [2010], Beyerlein et al. [2012]). A review on slip trans-mission criteria is found in Bayerschen et al. [2016]. In computational methods, Ekh et al.[2011] implemented a transmission factor according to the degree of mismatch between theslip systems in neighboring grains. More recently, Hamid et al. [2017] applied a combinationof the misorientation of adjacent grains and resolved shear stress on relative slip planes. Theyimplemented the criterion ofWerner and Prantl [1990] and assumed a linear function to predictthe grain boundary strength.

In this work, the main goal is to investigate the effect of mismatch between the adjacentgrains on the grain boundary strength. To this end, a transmissibility parameter suggested by


Werner and Prantl [1990] is implemented into the grain boundary model proposed by Alipouret al. [2019] which is recalled at the beginning of the text. The performance of the GBmodel isthen examined in periodic polycrystals in which the strength of each GB is evaluated accordingto the misorientation between neighboring grains. A comparison between randomly orientedpolycrystals (with high average misorientation) and textured polycrystals (with smaller averagemisorientation) shows the influence of mismatch between the slip systems in adjacent grainson the GB strength.Notation. A direct tensor notation is preferred throughout the text. Vectors and 2nd-ordertensors are represented by small and capital bold letters, e.g. a or A, respectively. In addi-tion, AT and A−1 represent, respectively, the transpose and inverse of a tensor. Moreover,a⊗ b = aibj ei ⊗ ej and a · b = aibi (using summation convention) denote, respectively, thedyadic product and the inner product of two vectors and A : B = tr(ATB) = AijBij desig-nates the double contraction of two second order tensors. The double contraction of a fourthorder tensor with a second order tensor is indicated by C : A = CklijAij . Furthermore, thenorm of a vector (or a tensor) is represented by ‖ a ‖ (or ‖ A ‖) and sgn(•) is the sign of anexpression.

4.3 Gradient-extended geometrically nonlinear crystal

viscoplasticity

In this section, after showing the basic kinematical formulations, the plasticity theory outlinedin Alipour et al. [2019] is extended by a more complex free energy and a grain boundarystrength (and a related GB yield criterion) which depends on the misorientation of adjacentgrains.

4.3.1 Kinematics

Based on a widely accepted continuum theory of plasticity, the deformation gradient F isdecomposed multiplicatively, i.e., F = F eF p into elastic and plastic parts (Kröner [1959]and Lee [1969]). Furthermore, the elastic right Cauchy-Green tensor Ce and the elasticGreen-Lagrangian strain tensor Ee with respect to the intermediate configuration are definedby

Ce = F eTF e, Ee =1

2(Ce − I). (4.1)

4.3 Gradient-extended geometrically nonlinear crystal viscoplasticity 87

In addition, the velocity gradient l is given as follows:

l = F F−1 = F eF e−1 + F eLpF e−1, (4.2)

where, regarding the continuum model of crystal plasticity, the second order tensor Lp =

F pF p−1 is given as a superposition of the contribution of the individual slip systems α:

Lp =∑α

γαMα, (4.3)

where α runs from 1 to N (e.g., N = 12 for fcc crystals) and γα are the slip rates. Further-more,Mα = dα ⊗nα represents the crystal geometry where dα and nα are the slip directionand slip plane normal vectors in the intermediate configuration, respectively. An accumulatedplastic slip can be obtained by

γacc =∑α

∫t

|γα|dt. (4.4)

4.3.2 Plastic surface deformation

Microscopic studies on the plastic deformation of crystal surfaces show that dislocationscrossing the crystal surface result in numerous discrete slip lines on it. On the continuum level,one may represent the Burgers-vector flow across the surface A (see Fig. 4.1) with the normalvectorN as follows (Gurtin [2008] and Wulfinghoff [2017]):

B = −F pχ(N×) (4.5)

to investigate the effects of discrete slip lines. We constrainF pχ to be equal toF p (F pχ = F p)by a penalty method in the sequel. The tensor (N×) rotates and projects any vector into theplane defined byN and is calculated as follows:

(N×) = −εN , (4.6)

where ε denotes the permutation tensor. It is noteworthy to mention thatB represents only theplastic deformation of surface line elements (i.e., in-plane line elements dx, see Fig. 4.1) whichare related to dislocations crossing the crystal surface leaving behind slip lines. The plasticdeformation of out-of-plane line elements (i.e., line elements perpendicular to the surface) donot describe the formation of slip lines and are neglected. In addition,B = 0 means that thereis no plastic deformation on the crystal boundary (i.e., a micro-hard surface).


Figure 4.1: Volume V with external surface ∂V and in-plane line element dx (taken fromWulfinghoff [2017]).

4.3.3 Dissipation

Neglecting thermal effects and body forces, the total dissipationDtot, expressed in the referenceconfiguration V0, is assumed to be given by

Dtot =

∫V0

D dV +

∫Γ

DΓ dA =

∫∂V0

(t · u+m : B) dA−∫V0

ψ dV −∫

Γ

ψΓ dV . (4.7)

In addition, t = PN and m = −M(N×)(N×) are the reference traction vector andgeneralized surface traction, respectively. Here, P and M are defined as the first Piola-Kirchhoff stress and a generalized stress tensor, respectively. The tractions t and m areprescribed on the Neumann boundaries ∂Vt and ∂Vm. Moreover, the displacement u and Bare assumed to be given on the Dirichlet boundaries ∂Vu and ∂Vχ, respectively, where ∂V0 =

∂Vt ∪ ∂Vu = ∂Vm ∪ ∂Vχ. Here, ψ is the free energy density of the bulk and ψΓ represents thefree energy density of the grain boundary Γ. Note that the termm : B accounts for plasticdeformations on the surface which are developed as the result of dislocations crossing thecrystal surface (i.e., slip lines on the crystal surface in microscopic view).

Using Gauss’ theorem and assuming [[B]] = 0 (for a motivation, see Alipour et al. [2019]),one can show that (see also Levkovitch and Svendsen [2006]):∫

∂V0

m : B dA−∫Γ

JmK : B dA =

∫V0

(M : αχ −Xb : F

pχ)

dV, (4.8)

in which αχ = CurlT(F pχ) is well-known as the dislocation density tensor (Nye [1953])andXb = CurlT(M) is the back stress tensor. It is noticeable that the equality ofm : B =

M : B has been used in Eq. (4.8) (4.6.1).Given [[u]] = 0 (no GB sliding), [[B]] = 0 (see Bayerschen et al. [2015] and Wulfinghoff


[2017]), as well as (quasi-static) mechanical equilibrium in V0 and on Γ,∫∂V0

(t · u+m : B) dA =

∫V0

(P : F +M : CurlT Fpχ − CurlTM : F

pχ) dV

+

∫Γ

[[m]] : B dA.

(4.9)Thus, the local bulk and GB dissipation densities read

D = P : F +M : CurlT Fpχ − CurlTM : F

pχ − ψ, (4.10)

DΓ = [[m]] : B − ψΓ. (4.11)

4.3.4 Free energy density

4.3.4.1 Free energy density of the bulk ψ

The free energy per unit volume is assumed to be of the additive form:

ψ = ψe (Ee) + ψg (αχ) + ψh (γacc) +1

2Hχ ‖ F pχ − F p ‖2︸︷︷︸

ψχ

, (4.12)

including the elastic energy (ψe), the defect energy (ψg) and the hardening energy (ψh) insidegrains. Furthermore, the free energy is extended by a penalty term in spirit of themicromorphictheory of Forest [2009]. Note that the coefficientHχ is assumed as large as possible to guaranteethat F pχ ≈ F p.

Consequently, by substituting Eq. (4.12) in Eq. (4.10), the local dissipation (i.e., dissipationper unit volume) takes the following form:

D =∑α

(τα −X#

α

)γα − qγacc ≥ 0, (4.13)

where q = ∂γaccψh. Moreover, τα = dα · (Σenα) and X#

α = dα ·(X#nα

)represent,

respectively, the resolved shear stress and the back stress in slip system α in the intermediateconfiguration, where X# = XbF pT and Σe = CeSe represents the Mandel stress tensor.Here, Se is the elastic second Piola-Kirchhoff stress tensor in the intermediate configuration.The following assumptions have been considered:

Se = ∂Eeψe, M = ∂αχψg, Xb = −∂F pχψχ. (4.14)


In the current work, the following assumptions regarding the free energy terms in Eq. (4.12)are considered. The elastic part of free energy is assumed as follows:

ψe(Ee) =

1

2Ee : C : Ee, (4.15)

where C denotes the fourth-order elastic stiffness tensor. In addition, the defect energy ψg isassumed to take a regularized logarithmic form (see Svendsen and Bargmann [2010], Forestand Guéninchault [2013] and Wulfinghoff et al. [2015]), originally motivated based on resultsby Groma et al. [2007]:

ψg(αχ) =

1

2

c0

bl2 ‖ αχ2 ‖ if ‖ αχ ‖< αL

c0 ‖ αχ ‖ ln‖ αχ ‖α0

+W0 else.(4.16)

Here, b is the Burgers vector magnitude and α0 is a constant. Moreover, c0 is given by

c0 =Gbβ

2π(1− ν), (4.17)

in which G and ν are, respectively, the macroscopic shear modulus and Poisson’s ratio. Here,β is of order unity. The internal length scale l, the transition density αL and the offset energyW0 are chosen as follows (see Wulfinghoff et al. [2015]):

l2 =b

α0

, αL = α0, W0 =c0α0

2, (4.18)

such that α0 is the only free parameter in Eq. (4.16). This parameter can be approximatelyrelated to the density of statistically stored dislocations (see Wulfinghoff et al. [2015]).Furthermore, the hardening part in the free energy ψh(γacc) is assumed as [compare, e.g.,Steinmann and Stein, 1996]:

ψh(γacc) =(τ∞ − τ c

0)2

h0 − h∞ln

(cosh


τ∞ − τ c0

))+

1

2h∞γ

2acc, (4.19)

where h0 and h∞ are the initial and saturation hardeningmoduli and τ∞ represents an additionalmaterial parameter.


4.3.4.2 Free energy density of the grain boundary ψΓ

The free energy density of the grain boundary ψΓ is assumed to be as follows:

ψΓ =1

2Hχ

Γ ‖ B −BΓ ‖2, (4.20)

inwhichBΓ is the total surface plastic deformation on the grain boundary and the coefficientHχΓ

is, in this work, chosen as large as possible such that B ≈ BΓ. By substituting Eq. (4.20) inEq. (4.11), the local dissipation of the GB is given by

DΓ = (JmK− ∂BψΓ) : B − ∂BΓψΓ : BΓ

= mχ : BΓ ≥ 0, (4.21)

where it has been assumed that JmK = ∂BψΓ andmχ = ∂BψΓ.

4.3.5 Crystal plasticity flow rule

To satisfy the local dissipation inequality (Eq. (4.13)), a thermodynamically consistent flowrule may be assumed to take the form:

γα = sgn(τα −X#α )γ0

⟨|τα −X#

α | − (τ c0 + q)

τD

⟩p, (4.22)

in which γ0, τ c0 , τD and p are, respectively, the reference shear rate, the initial yield stress, the

drag stress and the strain rate sensitivity parameter. In addition, τ c0 + q is the shear flow stress.

4.3.6 Grain boundary yield criterion

Motivated by Eq. (4.21), one assumes the following grain boundary yield criterion:

fΓ =‖mχ ‖ −mc ≤ 0, (4.23)

in which mc is the grain boundary yield strength. Moreover, assuming the validity of theprinciple of maximum dissipation (see, e.g., Simo and Hughes [1998] and Eq. (4.21)), oneobtains an associative grain boundary flow rule:

BΓ

= γΓ∂mχfΓ = γΓ mχ

‖mχ ‖, (4.24)


in which γΓ is a consistency parameter satisfying the Kuhn-Tucker conditions:

γΓ ≥ 0, fΓ ≤ 0, γΓfΓ = 0. (4.25)

Remark concerning the implementation:To evaluate the GB yield condition, a standard return mapping algorithm can be applied. Forthis purpose, all quantities from the previous time step are assumed to be known. By computingthe trial-value of the grain boundary yield criterion f tr

Γ in the usual way (see Eq. (4.23)), thegrain boundary condition is evaluated. When f tr

Γ < 0, the GB is not activated (i.e., it does notallow the dislocations to be transmitted and acts as an obstacle). Otherwise, the GB is assumedto be activated (i.e., dislocations cross the GB).

4.3.6.1 Evaluation of GB strength mc

In the current work, we aim to investigate the effect of the grain misorientation on the grainboundary strength. To this end, a geometric criterion is used (Werner and Prantl [1990]):

λ =∑α,β

cos

(π

2ωcarccos(nAα · nBβ )

)cos

(π

2κcarccos(dAα · dBβ )

)(4.26)

which accounts for the slip directions dAα and dBβ and the slip plane normals nAα and nBβin adjacent grains A and B. The aforementioned geometric criterion represents the overallmeasure of slip transferability by summing over the individual components of all possibleslip system combinations. In addition, ωc and κc are the critical angles above which no sliptransfer is expected to occur on the associated slip systems and are obtained from experimentalobservations by Werner and Prantl [1990]. Here, ωc = 45 and κc = 45 are assumed for 2Dproblems.

Figure 4.2: Schematical drawing of a bicrystal with one slip systems in each grain.

Example 1: Assume a 2-dimensional bicrystal with one slip system in each grain and the


Figure 4.3: Distribution of λ in the bicrystal with one slip system in each grain.

Figure 4.4: Schematical drawing of a bicrystal with two slip systems in each grain.

misorientation angle of θ (see Fig. 4.2). The distribution of λ as a function of themisorientationangle is shown in Fig. 4.3. As it is seen, the maximum and minimum values of λ are 1 and 0,respectively, and there is no dislocation transmission between θ = 45 and θ = 135. However,the maximum transmission occurs when the slip directions in neighboring grains are parallelto each other (θ = 0 and θ = 180).Example 2: Figure 4.4 shows a bicrystal with two slip systems in each grain and a 60 anglebetween them. The distribution of λ with respect to the misorietation angle θ is shown inFig. 4.5. As it is observed, λ = 2 is the maximum value for which the dislocations are assumedto cross the GB freely, since both slip systems in ‘Grain 1’ are parallel to the slip systems in‘Grain 2’. In contrast, the GB acts as a barrier to dislocation transmission when λ < 2 whichleads to an increased GB strength and dislocation pile-ups at the GB. It is noteworthy that the


Figure 4.5: Distribution of λ in bicrystal with two slip systems in each grain.

minimum value of λ is 0.5, when θ = 90, for which the slip systems in adjacent grains havethe smallest effective interaction with each other. In this case, each slip direction in ‘Grain1’ has a 30 angle with one slip direction in ‘Grain 2’. In addition to the absolute maximumand minimum values of λ, some further extrema are observed in Fig. 4.5. The minima atθ = 38 and θ = 142 indicate that there is little effective interaction among slip systems ofneighboring grains while the maxima at θ = 60 and θ = 120 illustrate that one slip systemin ‘Grain 1’ is parallel to one slip system in ‘Grain 2’.

To formulate the GB strengthmc (see Eq. (4.23)) as a function of λ, it should be noted thatmc is supposed to take a very high value, if λ = 0 (i.e., micro-hard GB). In contrast, mc = 0,if λ takes the maximum value (λmax) (i.e., micro-free GB). Consequently, one can assume thefollowing relation between the GB strengthmc and the geometric parameter λ:

mc = Aλmax − λ

λ, (4.27)

where A is a constant.Example 3: The distribution ofmc versus λ is depicted in Fig. 4.6 for a bicrystal with two slipsystems in each grain (see Fig. 4.4) when A = 10 N/m. As it is shown, decreasing λ leads toan increased grain boundary strength mc and a later activation of the GB. Moreover, the GBis free when λ = 2 (mc = 0 N/m). Note that the GB is never hard in the bicrystal shown inFig. 4.4 since there is at least one misorientation angle below 45, regardless of θ.

4.4 Results 95

Figure 4.6: Grain boundary strengthmc versus the geometric parameter λ.

4.4 Results

In this section, the performance of the grain boundary model is investigated for polycrystals onthe micro-scale. The implementation is done using the finite element software FEAP (FiniteElement Analysis Program) [http : //www.ce.berkeley .edu/projects/feap].

4.4.1 Comparison between textured polycrystals andrandom-oriented polycrystals

Figure 4.7 shows a unit cell of a 2-dimensional polycrystal with periodic boundary condi-tions (BCs) and the average grain size d (i.e., the average square root of the grains’ area). Twoslip systems in each grain are assumed such that the angle between them is set to 60. More-over, two kinds of orientation distribution are investigated in this work: 1) random polycrystalsin which the first slip system in each grain is chosen randomly, and 2) textured polycrystalsin which the grains follow a preferred orientation. All material parameters are listed in Ta-ble 4.1. Furthermore, a macroscopic shear deformation is prescribed with a maximum valueof H12 = 0.5 (H = F − I is the displacement gradient).

The average macroscopic shear stress-strain response of a polycrystal with random crystalorientations is shown in Fig. 4.8 (a), in which the influence of parameter A (see Eq. (4.27))on the material behavior can be observed. Generally, increasing A leads to a higher grain


Figure 4.7: Schematical drawing of a 2-dimensional polycrystal with two slip systems in eachgrain.

E [GPa] 70 τ∞ [MPa] 49.51ν [-] 0.34 h0 [MPa] 541.48p [-] 10 h∞ [MPa] 1τD [MPa] 60 γ0 [1/s] 10−3

τ c0 [MPa] 0.84 Hχ [MPa] 106

HχΓ [MPa] 106 b [nm] 0.286

α0/b [µm−2] 1 β [-] 1

Table 4.1: Material parameters (taken from Steinmann and Stein [1996] and Wulfinghoff et al.[2015]).

boundary strength and a later activation of the GBs. Note that the blue and red curves in thisfigure represent the material behavior when all grain boundaries are assumed to be micro-free(i.e.,mc = 0 N/m) and mico-hard, respectively.

Moreover, Fig. 4.8 (b) shows the macroscopic stress-strain behavior in case of a strongtexture, where the grain orientation follows a normal distribution function with mean valueof 0 and standard deviation of 3 (it is recalled that the second slip system in each grainexhibits a 60 angle with the first one in the same grain). Similar to the random polycrystal,the parameterA controls the grain boundary strength such that the GBs are activated later (i.e.,higher yield stress) by increasing the value of A (Fig. 4.8 (b)). It should be mentioned thatthe transmissibility parameter λ is different on various GBs with respect to the misorientionbetween adjacent grains which leads to a specified strength at each GB.

4.4 Results 97

(a) (b)

Figure 4.8: Macroscopic shear stress-strain response of (a) random polycrystals (b) stronglytextured polycrystals when d = 0.1µm.

The activation of GBs in random-oriented and textured polycrystals in a specific deformationis shown in Fig. 4.9. Note that the activated GBs are shown in red color while the blue linesdesignate the inactive GBs. As it is observed, more GBs are activated in strongly texturedpolycrystals compared to random polycrystals due to lower misorientation between adjacentgrains.

Fig. 4.10 illustrates the distribution of the accumulated strain in random polycrystals withdifferent GBmodels. As it is shown, the maximum value of γacc in polycrystals with intermedi-ate GBs (Fig. 4.10 (b)) is between those for free and hard GBs, since, in this model, some GBsare activated while others are still inactive. Dislocations enter and leave the GBs freely in the‘free-GBs’ model, neglecting the grain misorientation (Fig. 4.10 (a)). In contrast, dislocationsremain in each grain during the deformation process when the GBs are hard (Fig. 4.10 (c)).

The effective shear yield stress at 0.2% elastic strain of different random polycrystals andpolycrystals with strong texture are depicted in Fig. 4.11 as a function of the grain size anddifferent values of A. As it is shown, decreasing the grain size leads to increased values ofyield stress. Since the grain boundaries are activated later by increasing the value of A in bothrandom and textured polycrystals. However, the influence of parameter A in the strongly tex-tured cases is less pronounced due to the small misorientation between adjacent grains whichleads to the simultaneous activation of many GBs (see Fig. 4.9). Accordingly, the yield stressvalues in the random polycrystal are larger than those in the textured polycrystal. Furthermore,Fig. 4.11 shows that hard and free GBs are unable to properly model a significant texture effect,while the results are strongly texture-dependent using the proposed GB model.


(a) (b)

Figure 4.9: Activation of GBs in (a) random polycrystals (b) strongly textured polycrystalswhen F12 = 0.003, A = 10 N/m and d = 0.1µm.

(a) (b)

(c)

Figure 4.10: Distribution of the accumulated plastic strain in random polycrystals with (a) freeGBs (b) intermediate GBs (A = 10 N/m) (c) hard GBs when F12 = 0.01 and d =0.1µm. Note the different scales of γacc.

4.4 Results 99

(a) (b)

Figure 4.11: Shear yield stress versus the grain size of (a) random polycrystals (b) stronglytextured polycrystals.

To compare random polycrystals with textured polycrystals with respect to their yield stresses,Fig. 4.12 presents the shear yield stress values in random polycrystals, a strongly texturedpolycrystal with mean value 0 and standard deviation (std in Fig. 4.12) 3 as well as a texturedpolycrystal with mean value 0 and standard deviation 20. Three different grain boundarymodels are compared: free GBs (a), intermediate GBs (b) in which the misorientation ofadjacent grains is taken into account (here, A = 10 N/m) and hard GBs (c). Note that freeand hard GB strengths are obtained when A = 0 N/m and A → ∞, respectively. As it isshown in Fig. 4.12 (b), smaller misorientations between neighboring grains lead to a smallerHall-Petch slope as well as smaller yield stresses due to the simultaneous activation of manyGBs and easier plasticity transmission. However, the orientation distribution of grains doesnot have a major influence on the Hall-Petch slope when hard and free GBs are assumed inpolycrystals (Fig. 4.12 (a) and (c)). In fact, Fig. 4.12 illustrates that hard and free GBs cannotproperly capture the texture effects. In contrast, the results using the proposed GB model (in-termediate GBs) show a smaller Hall-Petch slope in textured polycrystals compared to randompolycrystals which is in qualitative agreement with other experimental works (see Yu et al.[2018]), where of course also other effects play a role. Yu et al. [2018] introduced a geometricalcompatibility factorm = cos(α) · cos(β) where α and β are the angles between the slip planesand slip directions of neighboring grains, respectively. The authors mentioned that a lowervalue ofm (high mismatch of adjacent grains) results in a higher boundary obstacle effect andthe resultant higher Hall-Petch slope, which are in agreement with the present findings.


(a) (b)

(c)

Figure 4.12: Shear yield stress versus the grain size of random polycrystal and two kinds oftextured polycrystal with (a) free (b) intermediate (c) hard GBs.

4.4.2 Comparison between textured polycrystals with elongatedgrain shape and random-oriented polycrystals

As in reality textures usually result from deformation processes like rolling, which also leads toa grain shape modification, we aim to apply the shear deformation to a textured polycrystal withelongated grains and compare the results with random polycrystals with almost circular grainshape. To obtain reasonable texture-data, a 2-dimensional unit cell of a randomly-orientedpolycrystal including 12 grains with periodic BCs is deformed as shown in Fig. 4.13. Note thatthe deformation is volume-conserving. After the simulation, a texture with elongated grainsand the preferred orientation 169 for the first slip system has been captured (Fig. 4.13 (b)).

4.4 Results 101

The crossed signs in each grain in Fig. 4.13 represent the slip systems’ directions which havebeen altered during the deformation process.

(a)

(b)

Figure 4.13: Schematical drawing corresponding to a 2D unit cell of a (a) random polycrystalwith 12 grains (b) textured polycrystal with elongated grain shape.

Fig. 4.14 shows the effective shear yield stress values concerning random polycrystals withalmost circular grain shapes in comparison with textured polycrystals with elongated grainswhere again three kinds of grain boundaries are investigated. Again, it can be observed that thepolycrystal with smaller grain size has a higher yield stress (i.e., the size effect phenomenon),regardless of the GB type. Moreover, the textured polycrystals exhibit smaller yield stresses incomparison with the random polycrystals since there is less misorientation between adjacentgrains leading to an earlier activation of GBs. Comparing Figs. 4.12 and 4.14, it is foundthat the model with hard GBs gives reasonable results only when elongated grain shapes areinvestigated. However, one can capture proper material behavior by considering the modelwith intermediate GBs, regardless of the grain shape.

It should be noted that in the current work the effects of the GB orientation has beenneglected. Also other factors which have influence on the GB deformation are neglected here,e.g., temperature and metal purity. However, they may have major influence on the materialbehavior, as it can be seen in experimental data. As an example, the work of Wyrzykowskiand Grabski [1986] shows various outcomes for aluminum polycrystals with the same grainsize during plastic deformation at room temperature. They conclude that the effects of GBorientation, metal purity and the condition of annealing are themain reasons to capture differentexperimental results. Moreover, Godon et al. [2010] studied two factors of coincidencesite lattice as well as grain misorientation to compare random polycrystals with texturedpolycrystals. Their results show that the Hall-Petch slope highly depends on the grain size aswell as the texture type.


(a) (b)

(c)

Figure 4.14: Shear yield stress versus the grain size of a random polycrystal and a textured poly-crystal with elongated grain shape when GBs are (a) free (b) intermediate (c) hard.

4.4.3 Statistical analysis

To present a statistical investigation of the material behavior using the current grain boundarymodel, random-oriented polycrystals as well as textured polycrystals with elongated grains(see Fig. 4.13) are examined with 10 different orientation sets such that the mean and standarddeviation values are kept as in the example of section 4.4.2 (i.e., mean value 169 and standarddeviation 121 in textured polycrystals). Fig. 4.15 shows the statistical results (mean valuesand standard deviations of the yield stress) for various orientation sets for two different grainsizes using intermediate grain boundaries. As it can be seen, the same trend as in Fig. 4.14 (b)is found. However, the magnitude of the yield stress standard deviation shows that a signifi-

4.5 Summary 103

cantly larger number of grains would be required to make the microstructures statistically morerepresentative.

Moreover, a statistical analysis on the textured polycrystal with elongated grains has beenconsidered in which 10 different grain shape sets were assumed such that d = 0.14 µm in allsets. According to the results, the mean value 205 MPa and standard deviation 8 MPa hasbeen obtained. As it was expected, the grain shape has less influence on the scatter of theaverage macroscopic shear stress, compared to the grain orientation distribution.

Figure 4.15: Statistical analysis on random-oriented and textured polycrystals with differentorientation distributions.

4.5 Summary

A grain boundary model based on the misorientation between adjacent grains has been pre-sented which evaluates the grain boundary strength using a geometric parameter proposed byWerner and Prantl [1990]. In order to show the performance of the model, first a comparisonbetween random-oriented polycrystals and textured polycrystals with the same grain shape hasbeen examined. Later, the model has been applied to a textured polycrystal with elongatedgrain shape. In all simulations, a 2-dimensional unit cell of a periodic polycrystal is used.The results show that the GB strength increases by increasing the parameter A which leads


to a delayed GB activation. Moreover, the effective shear yield stress (at 0.2% elastic strain)has been studied for these two kinds of polycrystaline structure. The size effect phenomenonhas been observed by changing the grain size d such that the yield stress value decreased byincreasing d. In addition, the effective yield stress increases by increasing the value of A dueto later GBs activation. It is noteworthy that the yield stress as well as the Hall-Petch slope forrandom polycrystals were predicted to be larger than for textured polycrystals.

In another example, random polycrytals with almost circular grain shape have been com-pared to textured polycrystals with elongated grains. The size effect phenomenon has beencaptured and the results show that the effective yield stress of textured polycrystal is smallerthan in the random case due to smaller misorientation angles between adjacent grains andeasier activation of GBs. Furthermore, the Hall-Petch slope is larger in random polycrystalscompared to textured polycrystals with elongated grains.

It should be mentioned that the observed behavior in random and textured polycrystals isqualitatively in agreement with the experimental study of Yu et al. [2018]. However, there arestill several factors e.g., GB orientation, temperature and coincidence site lattice which havebeen neglected in the current GB model.AcknowledgmentThe financial support of Aachen Institute for Advanced Study in Computational Engineer-ing Science (AICES) by the German Science Foundation (DFG) through Grant GSC 111 isgratefully acknowledged. Furthermore, the financial support provided by the DFG related tothe project “Computational homogenization of inelastic conventional and gradient-extendedmicrostructures by a shear band approach” (WU 847/1-1) is thankfully acknowledged.

4.6 Appendix

4.6.1 Proof of m : B = M : B

The normal-transverse split ofM with respect toN is as follows:

M = M (N ⊗N ) +M (I −N ⊗N ) = M (N ⊗N ) +M (N×)(N×)T︸︷︷︸=m

, (4.28)

in which caseM (N ⊗N ) : B = 0. Therefore,m : B = M : B.

5 Conclusions and Outlook

The current cumulative dissertation was concerned with the theoretical development and nu-merical investigation of the geometrically nonlinear plastic deformation of face centered cubicplastic materials on large- and small-scale using continuum approaches.

In Chapter 2, a classical approach (size-independent model) was used where there is no sizeeffect in the material. Due to severe problems in convergence of equations including the power-law-type flow rule, a regularization method for the power law with high value of sensitivityexponent up to 1000 in finite single crystal viscoplasticity was implemented. Moreover, a newconcept for hybrid discontinuous Galerkin methods –control points– was presented, leading toa numerically efficient, robust and locking free model. The element stiffness matrix for the ap-plied hybrid DG framework by Wulfinghoff et al. [2017] was implemented analytically whichhad to be computed numerically before. The performance of the framework has been examinedby a planar single crystal and a 2D oligocrystal under uniaxial load. It has been shown thatthe effect of the penalty parameter θ is almost negligible for the investigated examples. Inaddition, secondary shear bands are formed in each quadrant of planar single crystal as a resultof refining the mesh for high values of sensitivity exponent. In the 2D oligocrystal-example,a negligible influence of the penalty parameter on the material behavior has been obtained fora larger range of θ. Furthermore, the effects of the strain rate sensitivity exponent are lesssignificant in the investigated oligocrystal deformation compared to the planar single crystal.

In Chapter 3, the role of grain boundaries on heterogeneous plasticity, pile-ups of dis-locations at the grain boundaries as well as transmission of dislocations through the grainboundaries were investigated by a gradient-extended approach using the concept of geomet-rically nonlinear viscoplasticity. Motivated by Wulfinghoff [2017], this model was based onthe dislocation density tensor and plastic surface deformation which leads to a grain boundaryyield criterion with isotropic and kinematic hardening. A micromorphic approach was usedwhich makes it rather easy to differentiate between elastic and plastic regions in the body. Tothis end, the micromorphic variable has been constrained to be (almost) equal to its coun-terpart by choosing a sufficiently large penalty parameter. Moreover, a three-level solutionalgorithm was implemented which allows to extend an already existing local single crystal

105

106 5 Conclusions and Outlook

material subroutine, leading to decreasing the implementation effort. In addition to micro-hard and micro-free grain boundaries, intermediate grain boundaries with specific strengthswere investigated which are activated when the back stress resulting from dislocation pile-upsreaches a sufficiently high value. Since the main goal in Chapter 3 was to develop a new grainboundary model, the grain boundary strength has been assumed as a parameter which is notinfluenced by other factors, e.g., the mismatch between slip systems and the orientation of thegrain boundary.

The performance of the model in geometrically nonlinear deformations as well as the effectof isotropic and kinematic GB hardening on the material behavior have been examined inseveral examples on bicrystals. The results showed that increasing the grain boundary strengthleads to strengthening of the material. In contrast, increasing the internal length scale resultsin an earlier GB activation. Moreover, the Hall-Petch effect was observed in material behaviorsuch that the bicrystal with smaller grains is stiffer than the bicrystal with larger grains. Acyclic shear load on the bicrystal showed that the grain boundary becomes activated when theback stress reaches a sufficiently high value. The back stress increases in activated regionsby assuming kinematic hardening on the grain boundary. However, assuming GB isotropichardening leads to an overall “cyclic hardening” phenomenon where the GB activation regionsdecrease in each cycle such that the cycles converge to a final one, in which there is no grainboundary activation. In another example, a bicrystal with two specified slip systems in eachgrain, assuming Voce-hardening inside the grains and isotropic and kinematic hardening onthe GB, has been considered. Based on the results, increasing the GB strength leads to a laterGB activation as well as growing pile-ups of dislocations at the grain boundary.

The effect of orientation mismatch between the adjacent grains on the grain boundarystrength was investigated in Chapter 4 using a geometric parameter proposed by Werner andPrantl [1990] in the grain boundary model by Alipour et al. [2019] using the concept of ge-ometrically nonlinear crystal plasticity. A unit cell of polycrystals with periodic boundaryconditions has been considered to examine the performance of the GB model. The strengthof each GB was evaluated according to the misorientation between neighboring grains. Ac-cording to the results, the overall yield stress decreases by increasing the grain size (size effectphenomenon). Moreover, the effective yield stress increases by increasing the GB strength dueto later GBs activation. The yield stress as well as the Hall-Petch slope for random polycrystalswere predicted to be larger than for textured polycrystals.

A comparison between random polycrytals with almost circular grain shape and texturedpolycrystals with elongated grains showed that the effective yield stress of textured polycrystalis smaller than in the random case due to smaller misorientation angles between adjacent

107

grains. In addition, the Hall-Petch slope is larger in random polycrystals.One of the major areas of future work is to include other effective factors on the grain bound-

aries e.g., GB orientation, temperature and coincidence site lattice which have been neglectedin the current GB model.

List of Figures

2.1 Left: Division of the body into subdomains. Right: Illustration with shrunksubdomains (Fig. taken from the work by Wulfinghoff et al. [2017]). . . . . . 13

2.2 Tractions, normals and weights (Fig. taken from the work byWulfinghoff et al.[2017]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 (a) The power law (with steep slope for large shear stresses) results in non-resolvable answers. (b) Regularization of the power law. . . . . . . . . . . . 26

2.4 Left: Boundary conditions. Right: Distribution of the accumulated plasticstrain at elongations (a) 2.5 mm (b) 5 mm (c) 5 mm via new DG method withcontrol points, where θ = 10−1|∂Ωe|−1 MPa and p = 200. . . . . . . . . . . 30

2.5 Load-displacement curves for different penalty parameters with p = 200 (20×40 elements in each quadrant). . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.6 Load-displacement curves for different penalty parameters with (a) p = 200

(b) p = 1000 (110× 220 elements in each quadrant). . . . . . . . . . . . . . 332.7 Distribution of the accumulated plastic strain for p = 1000 at elongations (a)

3 mm (b) 5 mm, where θ = 10−1|∂Ωe|−1 MPa (110× 220 elements). . . . . 332.8 Mesh convergence for the planar single crystal for θ = 1|∂Ωe|−1 MPa and p =

1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.9 The geometry of oligocrystal. . . . . . . . . . . . . . . . . . . . . . . . . . . 342.10 Load-displacement curves for the oligocrystal for (a) various penalty pa-

rameters (p = 200) and (b) various strain rate sensitivity exponents (θ =

103|∂Ωe|−1 MPa). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352.11 Distribution of accumulated plastic strain for (a) p = 20 (b) p = 1000, where

the elongation is 7 mm and θ = 10−1|∂Ωe|−1 MPa. . . . . . . . . . . . . . . 36

3.1 Volume V with external surface ∂V and in-plane line element dx (taken fromWulfinghoff [2017]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 (a) Infinite channel with an elastic coating on the top (the Fig. is taken fromWulfinghoff [2017]) (b) shear stress-strain response for different sizes. . . . . 61

109

110 List of Figures

3.3 (a) Schematical drawing of a bicrystal with one slip system in each grain (b)discretization of the grains and the GB by 100× 50 quadrilateral elements and50 line elements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4 (a) Shear stress-strain response of a bicrystal with one slip system in each grainunder shear load for s = 10 µm (b) Magnification of Fig. 3.4 (a) in the smallstrain region (dashed box). . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.5 Distribution of the accumulated plastic strain for s = 10 µm and l = 1 µm atdisplacement 2 µm on top surface. . . . . . . . . . . . . . . . . . . . . . . . 65

3.6 Distribution of the accumulated plastic strain for s = 10 µm and l = 1 µm

at displacement 2 µm when (a) mco = 0 N/m (b) mc

o = 102 N/m (c) mco =

103 N/m (d)mco = 106 N/m. . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.7 (a) Influence of the structure size on the mechanical response of bicrystal undershear load (b) Specified shear stresses in Fig. 3.7 (a) in various sizes. . . . . . 66

3.8 Cyclic behavior of the bicrystal when there is no hardening in the material. . . 673.9 Cyclic behavior of the bicrystal with (a) only GB kinematic hardening for

Hh = 0 N/m and Hk = 105 N/m (b) only GB isotropic hardening forHh = 104 N/m and Hk = 0 N/m. . . . . . . . . . . . . . . . . . . . . . . . 68

3.10 Schematical drawing of a bicrystal with specific two slip systems in each grain. 683.11 Shear stress-strain response of a bicrystal with two slip systems in each grain

under shear load. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.12 Distribution of dislocation density magnitude for s = 5 µm and l = 1 µm at

displacement 0.15 µm (γ = 0.03) when (a) mco = 0 N/m (b) mc

o = 102 N/m

(c)mco = 103 N/m (d)mc

o = 106 N/m. . . . . . . . . . . . . . . . . . . . . 693.13 Mesh convergence for the bicrystal with two slip systems in each grain for

s = 5 µm,mco = 103 N/m and l = 1 µm. . . . . . . . . . . . . . . . . . . . 70

4.1 Volume V with external surface ∂V and in-plane line element dx (taken fromWulfinghoff [2017]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4.2 Schematical drawing of a bicrystal with one slip systems in each grain. . . . . 924.3 Distribution of λ in the bicrystal with one slip system in each grain. . . . . . 934.4 Schematical drawing of a bicrystal with two slip systems in each grain. . . . . 934.5 Distribution of λ in bicrystal with two slip systems in each grain. . . . . . . . 944.6 Grain boundary strengthmc versus the geometric parameter λ. . . . . . . . . 954.7 Schematical drawing of a 2-dimensional polycrystal with two slip systems in

each grain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

List of Figures 111

4.8 Macroscopic shear stress-strain response of (a) randompolycrystals (b) stronglytextured polycrystals when d = 0.1µm. . . . . . . . . . . . . . . . . . . . . 97

4.9 Activation of GBs in (a) random polycrystals (b) strongly textured polycrystalswhen F12 = 0.003, A = 10 N/m and d = 0.1µm. . . . . . . . . . . . . . . . 98

4.10 Distribution of the accumulated plastic strain in randompolycrystalswith (a) freeGBs (b) intermediateGBs (A = 10 N/m) (c) hardGBswhen F12 = 0.01 andd =

0.1µm. Note the different scales of γacc. . . . . . . . . . . . . . . . . . . . . 984.11 Shear yield stress versus the grain size of (a) random polycrystals (b) strongly

textured polycrystals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.12 Shear yield stress versus the grain size of random polycrystal and two kinds of

textured polycrystal with (a) free (b) intermediate (c) hard GBs. . . . . . . . . 1004.13 Schematical drawing corresponding to a 2Dunit cell of a (a) randompolycrystal

with 12 grains (b) textured polycrystal with elongated grain shape. . . . . . . 1014.14 Shear yield stress versus the grain size of a random polycrystal and a textured

polycrystal with elongated grain shape when GBs are (a) free (b) intermedi-ate (c) hard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.15 Statistical analysis on random-oriented and textured polycrystals with differentorientation distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

List of Tables

2.1 The gradual increase of p for the large value of p = 1000 . . . . . . . . . . . 312.2 The number of time steps for different p at elongation 5 mm (≈ 16.67% average

tensile strain) where θ = 10−1|∂Ωe|−1 MPa (110× 220 elements). . . . . . . 322.3 The number of time steps for different meshes at elongation 5 mm for θ =

1|∂Ωe|−1 MPa and p = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.1 Material parameters for example 1 (taken from Wulfinghoff [2017]). . . . . . 623.2 Material parameters for examples 2 to 4 (mostly taken from Steinmann and

Stein [1996]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.3 Relative residual norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.1 Material parameters (taken from Steinmann and Stein [1996] and Wulfinghoffet al. [2015]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

113

Bibliography

Aifantis, E. C. [1984], ‘On the microstructural origin of certain inelastic models’, Journal ofEngineering Materials and Technology 106(4), 326–330.

Aifantis, E. C. [1987], ‘The physics of plastic deformation’, International Journal of Plasticity3(3), 211–247.

Aifantis, K. E., Deng, H., Shibata, H., Tsurekawa, S., Lejček, P. and Hackney, S. A. [2019],‘Interpreting slip transmission through mechanically induced interface energies: a Fe–3%Sicase study’, Journal of Materials Science 54(2), 1831–1843.

Aifantis, K. E. andWillis, J. R. [2005], ‘The role of interfaces in enhancing the yield strength ofcomposites and polycrystals’, Journal of the Mechanics and Physics of Solids 53(5), 1047–1070.

Alberty, J. andCarstensen, C. [2002], ‘DiscontinuousGalerkin time discretization in elastoplas-ticity: motivation, numerical algorithms, and applications’, Computer Methods in AppliedMechanics and Engineering 191(43), 4949–4968.

Alipour, A., Reese, S., Svendsen, B. and Wulfinghoff, S. [2020], ‘A grain boundary modelconsidering the grain misorientation within a geometrically nonlinear gradient-extendedcrystal viscoplasticity theory’, Proceedings of the Royal Society A 471: 20190581.

Alipour, A., Reese, S. and Wulfinghoff, S. [2019], ‘A grain boundary model for gradient-extended geometrically nonlinear crystal plasticity: Theory and numerics’, InternationalJournal of Plasticity 118, 17–35.

Alipour, A., Wulfinghoff, S., Bayat, H. R., Reese, S. and Svendsen, B. [2018], ‘The con-cept of control points in hybrid discontinuous galerkin methods-application to gometricallynonlinear crystal plasticity’, International Journal for Numerical Methods in Engineering114(5), 557–579.

115

116 Bibliography

Arnold, D. N. [1982], ‘An interior penalty finite element method with discontinuous elements’,SIAM journal on numerical analysis 19(4), 742–760.

Asaro, R. J. [1983], ‘Crystal plasticity’, ASME. Journal of Applied Mechanics 50(4b), 921–934.

Asaro, R. J. and Rice, J. R. [1977], ‘Strain localization in ductile single crystals’, Journal ofthe Mechanics and Physics of Solids 25(5), 309–338.

Ashby, M. F. [1970], ‘The deformation of plastically non-homogeneous materials’, ThePhilosophical Magazine: A Journal of Theoretical Experimental and Applied Physics21(170), 399–424.

Ashmawi, W. M. and Zikry, M. A. [2001], ‘Prediction of grain-boundary interfacial mecha-nisms in polycrystalline materials’, ASME Journal of EngineeringMaterials and Technology124(1), 88–96.

Bargmann, S., Ekh, M., Runesson, K. and Svendsen, B. [2010], ‘Modeling of polycrys-tals with gradient crystal plasticity: A comparison of strategies’, Philosophical Magazine90(10), 1263–1288.

Bassi, F. and Rebay, S. [1997], ‘A high-order accurate discontinuous finite element methodfor the numerical solution of the compressible Navier-Stokes equations’, Journal ofcomputational physics 131(2), 267–279.

Baumann, C. E. and Oden, J. T. [1999], ‘A discontinuous hp finite element method forconvection-diffusion problems’, Computer Methods in Applied Mechanics and Engineering175(3–4), 311–341.

Bayerschen, E., McBride, A., Reddy, B. and Böhlke, T. [2016], ‘Review on slip transmis-sion criteria in experiments and crystal plasticity models’, Journal of Materials Science51(5), 2243–2258.

Bayerschen, E., Stricker, M., Wulfinghoff, S., Weygand, D. and Böhlke, T. [2015], ‘Equivalentplastic strain gradient plasticity with grain boundary hardening and comparison to discretedislocation dynamics’, Proceedings of the Royal Society A. 471: 20150388.

Berdichevsky, V. L. [2006], ‘On thermodynamics of crystal plasticity’, Scripta Materialia54(5), 711–716.

Bibliography 117

Beyerlein, I. J., Mara, N. A., Wang, J., Carpenter, J. S., Zheng, S. J., Han, W. Z., Zhang,R. F., Kang, K., Nizolek, T. and Pollock, T. M. [2012], ‘Structure–property–functionality ofbimetal interfaces’, JOM 64(10), 1192–1207.

Bieler, T. R., Eisenlohr, P., Zhang, C., Phukan, H. J. and Crimp, M. A. [2014], ‘Grainboundaries and interfaces in slip transfer’, Current Opinion in Solid State and MaterialsScience 18(4), 212–226.

Bilby, B. A., Bullough, R. and de Grinberg, D. K. [1964], ‘General theory of surface disloca-tions’, Discussions of the Faraday Society 38, 61–68.

Bilby, B. A., Bullough, R. and Smith, E. [1955], ‘Continuous distributions of dislocations:a new application of the methods of non-riemannian geometry’, Proceedings of the RoyalSociety London A 231, 263–273.

Burgers, J. M. [1939], Some considerations on the fields of stress connected with dislocationsin a regular crystal lattice, I. Koninklijke Nederlandse Akademie van Wetenschappen.

Caminero, M. Á., Montáns, F. J. and Bathe, K. J. [2011], ‘Modeling large strain anisotropicelasto-plasticity with logarithmic strain and stress measures’, Computers and Structures89(11–12), 826–843.

Cermelli, P. and Gurtin, M. E. [2002], ‘Geometrically necessary dislocations in viscoplasticsingle crystals and bicrystals undergoing small deformations’, International Journal of Solidsand Structures 39(26), 6281–6309.

Chang, Y. W. and Asaro, R. J. [1981], ‘An experimental study of shear localization inaluminum-copper single crystals’, Acta Metallurgica 29(1), 241–257.

Cockburn, B., Kanschat, G. and Schötzau, D. [2004], ‘The local discontinuous Galerkinmethod for the Oseen equations’, Mathematics of Computation 73(246), 569–593.

Cockburn, B., Kanschat, G., Schötzau, D. and Schwab, C. [2002], ‘Local discontinuousGalerkin methods for the Stokes system’, SIAM Journal on Numerical Analysis 40(1), 319–343.

Cockburn, B. and Shu, C.-W. [1998a], ‘The local discontinuous Galerkin method for time-dependent convection-diffusion systems’, SIAMJournal onNumericalAnalysis 35(6), 2440–2463.

118 Bibliography

Cockburn, B. and Shu, C.-W. [1998b], ‘The Runge-Kutta discontinuous Galerkin methodfor conservation laws V: Multidimensional systems’, Journal of Computational Physics141(2), 199 – 224.

Cordero, N. M., Forest, S. and Busso, E. P. [2013], ‘Micromorphic modelling of grain sizeeffects in metal polycrystals’, GAMMMitteilungen 36(2), 186–202.

Dimiduk, D. M., Uchic, M. D. and Parthasarathy, T. A. [2005], ‘Size-affected single-slipbehavior of pure nickel microcrystals’, Acta Materialia 53(15), 4065–4077.

Dumoulin, S., Hopperstad, O. S. andBerstad, T. [2009], ‘Investigation of integration algorithmsfor rate-dependent crystal plasticity using explicit finite element codes’, ComputationalMaterials Science 46(4), 785–799.

Ekh, M., Bargmann, S. and Grymer, M. [2011], ‘Influence of grain boundary conditions onmodeling of size-dependence in polycrystals’, Acta Mechanica 218, 103–113.

Ekh, M., Grymer, M., Runesson, K. and Svedberg, T. [2007], ‘Gradient crystal plasticity aspart of the computational modelling of polycrystals’, International Journal for NumericalMethods in Engineering 72(2), 197–220.

Engel, G., Garikipati, K., Hughes, T. J. R., Larson, M. G., Mazzei, L. and Taylor, R. L.[2002], ‘Continuous/discontinuous finite element approximations of fourth-order ellipticproblems in structural and continuum mechanics with applications to thin beams and plates,and strain gradient elasticity’, Computer Methods in Applied Mechanics and Engineering191(34), 3669–3750.

Fleck, N. A. and Hutchinson, J. W. [1993], ‘A phenomenological theory for strain gradienteffects in plasticity’, Journal of the Mechanics and Physics of Solids 41(12), 1825–1857.

Fleck, N. A., Hutchinson, J. W. and Willis, J. R. [2015], ‘Guidelines for constructing straingradient plasticity theories’, ASME. Journal of Applied Mechanics 82(7), 071002.

Fleck, N. A. and Willis, J. R. [2009], ‘A mathematical basis for strain-gradient plasticitytheory-Part I: Scalar plastic multiplier’, Journal of the Mechanics and Physics of Solids57(1), 161–177.

Forest, S. [2009], ‘Micromorphic approach for gradient elasticity, viscoplasticity, and damage’,Journal of Engineering Mechanics 135(3), 117–131.

Bibliography 119

Forest, S. [2016], ‘Nonlinear regularization operators as derived from the micromorphic ap-proach to gradient elasticity, viscoplasticity and damage’, Proceedings of the Royal Societyof London A: Mathematical, Physical and Engineering Sciences 472(2188).

Forest, S. and Guéninchault, N. [2013], ‘Inspection of free energy functions in gradient crystalplasticity’, Acta Mechanica Sinica 29(6), 763–772.

Gemperle, A., Zárubová, N. and Gemperlová, J. [2005], ‘Reactions of slip dislocations withtwin boundary in fe-si bicrystals’, Journal of Materials Science 40(12), 3247–3254.

Godon, A., Creus, J., Cohendoz, S., Conforto, E., Feaugas, X., Girault, P. and Savall, C. [2010],‘Effects of grain orientation on the Hall-Petch relationship in electrodeposited nickel withnanocrystalline grains’, Scripta Materialia 62(6), 403–406.

Gottschalk, D., McBride, A., Reddy, B. D., Javili, A., Wriggers, P. and Hirschberger, C. B.[2016], ‘Computational and theoretical aspects of a grain-boundary model that accountsfor grain misorientation and grain-boundary orientation’, Computational Materials Science111, 443–459.

Groma, I., Györgyi, G. andKocsis, B. [2007], ‘Dynamics of coarse grained dislocation densitiesfrom an effective free energy’, Philosophical Magazine 87(8–9), 1185–1199.

Gurtin, M. E. [2000], ‘On the plasticity of single crystals: free energy, microforces, plastic-strain gradients’, Journal of the Mechanics and Physics of Solids 48(5), 989–1036.

Gurtin, M. E. [2002], ‘A gradient theory of single-crystal viscoplasticity that accounts for geo-metrically necessary dislocations’, Journal of the Mechanics and Physics of Solids 50(1), 5–32.

Gurtin, M. E. [2006], ‘The Burgers vector and the flow of screw and edge dislocations infinite-deformation single-crystal plasticity’, Journal of the Mechanics and Physics of Solids54(9), 1882–1898.

Gurtin, M. E. [2008], ‘A theory of grain boundaries that accounts automatically for grainmisorientation and grain-boundary orientation’, Journal of the Mechanics and Physics ofSolids 56(2), 640–662.

Gurtin, M. E. [2010], ‘A finite-deformation, gradient theory of single-crystal plasticitywith free energy dependent on the accumulation of geometrically necessary dislocations’,International Journal of Plasticity 26(8), 1073–1096.

120 Bibliography

Gurtin, M. E., Anand, L. and Lele, S. P. [2007], ‘Gradient single-crystal plasticity with freeenergy dependent on dislocation densities’, Journal of the Mechanics and Physics of Solids55(9), 1853–1878.

Gurtin, M. E. and Needleman, A. [2005], ‘Boundary conditions in small-deformation, single-crystal plasticity that account for the Burgers vector’, Journal of the Mechanics and Physicsof Solids 53(1), 1–31.

Ha, S., Jang, J.-H. and Kim, K. [2017], ‘Finite element implementation of dislocation–density–based crystal plasticity model and its application to pure aluminum crystalline materials’,International Journal of Mechanical Sciences 120, 249–262.

Hall, E. O. [1951], ‘The deformation and ageing of mild steel: II characteristics of the Lüdersdeformation’, Proceedings of the Physical Society B. 64(9), 742–747.

Halphen, B. and Nguyen, Q. [1975], ‘Generalized standard materials’, Journal de mécanique14(1), 39–63.

Hamid, M., Lyu, H., Schuessler, B. J., Wo, P. C. and Zbib, H. M. [2017], ‘Modeling and char-acterization of grain boundaries and slip transmission in dislocation density-based crystalplasticity’, Crystals 7(6).

Hill, R. [1950], The mathematical theory of plasticity, Clarendon Press, Oxford.

Hill, R. [1966], ‘Generalized constitutive relations for incremental deformation of metal crys-tals by multislip’, Journal of the Mechanics and Physics of Solids 14(2), 95–102.

Hirsch, P. B., Horne, R. W. and Whelan, M. J. [1956], ‘Lxviii. direct observations of thearrangement and motion of dislocations in aluminium’, The Philosophical Magazine: AJournal of Theoretical Experimental and Applied Physics 1(7), 677–684.

Hirsch, P. B., Horne, R. W. and Whelan, M. J. [2006], ‘Direct observations of the arrangementand motion of dislocations in aluminium’, Philosophical Magazine 86(29-31), 4553–4572.

Hirth, J. P. [1985], ‘A brief history of dislocation theory’, Metallurgical Transactions A16(12), 2085–2090.

Hirth, J. P. and Lothe, J. [1982], ‘Theory of dislocations’, 2nd edition, John Wiley & Sons,New York. .

Hull, D. and Bacon, D. J. [1984], ‘Introduction to dislocations’, Pergamon Press, Oxford. 257.

Bibliography 121

Hurtado, D. E. and Ortiz, M. [2012], ‘Surface effects and the size-dependent hardeningand strengthening of nickel micropillars’, Journal of the Mechanics and Physics of Solids60(8), 1432–1446.

Hurtado, D. E. and Ortiz, M. [2013], ‘Finite element analysis of geometrically necessary dis-locations in crystal plasticity’, International Journal for Numerical Methods in Engineering93(1), 66–79.

Hutchinson, J. W. [2000], ‘Plasticity at the micron scale’, International Journal of Solids andStructures 37(1), 225–238.

Izadbakhsh, A., Inal, K., Mishra, R. K. and Niewczas, M. [2011], ‘New crystal plasticity con-stitutive model for large strain deformation in single crystals of magnesium’, ComputationalMaterials Science 50(7), 2185–2202.

Klusemann, B., Svendsen, B. and Vehoff, H. [2012], ‘Investigation of the deformation behaviorof Fe-3%Si sheet metal with large grains via crystal plasticity and finite-element modeling’,Computational Materials Science 52(1), 25–32.

Klusemann, B., Svendsen, B. and Vehoff, H. [2013], ‘Modeling and simulation of deformationbehavior, orientation gradient development and heterogeneous hardening in thin sheets withcoarse texture’, International Journal of Plasticity 50, 109–126.

Knezevic, M., Zecevic, M., Beyerlein, I. J. and Lebensohn, R. A. [2016], ‘A numericalprocedure enabling accurate descriptions of strain rate-sensitive flow of polycrystals withincrystal visco-plasticity theory’, Computer Methods in Appled Mechanics and Engineering308, 468–482.

Kröner, E. [1959], ‘Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen’,Archive for Rational Mechanics and Analysis 4 pp. 273–334.

Kweon, S. and Raja, D. S. [2017], ‘Comparison of anisotropy evolution in BCC and FCC met-als using crystal plasticity and texture analysis’, European Journal of Mechanics–A/Solids62, 22–38.

Lamba, H. S. and Sidebottom, O. M. [1978a], ‘Cyclic plasticity for nonproportional paths:Part 1: Cyclic hardening, erasure of memory, and subsequent strain hardening experiments’,ASME Journal of Engineering Materials and Technology 100(1), 96–103.

122 Bibliography

Lamba, H. S. and Sidebottom, O. M. [1978b], ‘Proportional biaxial cyclic hardening of an-nealed oxygen-free high-conductivity copper’, Journal of Testing and Evaluation 6(4), 260–267.

Lee, B.-J., Shim, J.-H. and Baskes, M. I. [2003], ‘Semiempirical atomic potentials for thefcc metals Cu, Ag, Au, Ni, Pd, Pt, Al, and Pb based on first and second nearest-neighbormodified embedded atom method’, Phys. Rev. B 68(14), 144112, 1–11.

Lee, E. H. [1969], ‘Elastic-plastic deformation at finite strains’, Journal of Applied Mechanics,ASME 36(1), 1–6.

Lee, T. C., Robertson, I. M. and Birnbaum, H. K. [1989], ‘Prediction of slip transfer mecha-nisms across grain boundaries’, Scripta Metallurgica 23(5), 799–803.

Levkovitch, V. and Svendsen, B. [2006], ‘On the large-deformation- and continuum-basedformulation of models for extended crystal plasticity’, International Journal of Solids andStructures 43(24), 7246–7267.

Li, Z., Hou, C., Huang, M. and Ouyang, C. [2009], ‘Strengthening mechanism in micro-polycrystals with penetrable grain boundaries by discrete dislocation dynamics simulationand hall-petch effect’, Computational Materials Science 46(4), 1124–1134.

Liebe, T., Menzel, A. and Steinmann, P. [2003], ‘Theory and numerics of geometrically non-linear gradient plasticity’, International Journal of Engineering Science 41(13), 1603–1629.

Lim, H., Battaile, C. C., Bishop, J. E. and Foulk, J. W. [2019], ‘Investigating mesh sensitiv-ity and polycrystalline RVEs in crystal plasticity finite element simulations’, InternationalJournal of Plasticity .

Livingston, J. D. and Chalmers, B. [1957], ‘Multiple slip in bicrystal deformation’, ActaMetallurgica 5(6), 322–327.

Lomtev, I. and Karniadakis, G. [1997], ‘Simulations of viscous supersonic flows on unstruc-tured h-p meshes’, Proceedings of 35th Aerospace Sciences Meeting, Reno, Nevada .

Lomtev, I., Quillen, C. B. and Karniadakis, G. E. [1998], ‘Spectral/hp methods for vis-cous compressible flows on unstructured 2d meshes’, Journal of Computational Physics144(2), 325–357.

Bibliography 123

Luster, J. and Morris, M. A. [1995], ‘Compatibility of deformation in two-phase Ti-Al alloys:Dependence on microstructure and orientation relationships’, Metallurgical and MaterialsTransactions A 26(7), 1745–1756.

Ma, A., Roters, F. and Raabe, D. [2006], ‘On the consideration of interactions betweendislocations and grain boundaries in crystal plasticity finite element modeling- theory,experiments, and simulations’, Acta Materialia 54, 2181–2194.

McBride, A. T., Gottschalk, D., Reddy, B. D., Wriggers, P. and Javili, A. [2016], ‘Computa-tional and theoretical aspects of a grain-boundary model at finite deformations’, TechnischeMechanik 36(1–2), 92–109.

Mergheim, J., Kuhl, E. and Steinmann, P. [2004], ‘A hybrid discontinuous Galerkin/interfacemethod for the computational modelling of failure’, Communications in numerical methodsin engineering 20(7), 511–519.

Mesarovic, S. D. [2010], ‘Plasticity of crystals and interfaces: from discrete dislocations tosize-dependent continuum theory’, Theoretical and Applied Mechanics 37(4), 289–332.

Mesarovic, S. D., Forest, S. and Jaric, J. P. [2015], ‘Size-dependent energy in crystal plasticityand continuum dislocation models’, Proceedings of the Royal Society A 471: 20140868.

Miehe, C. [2002], ‘Strain-driven homogenization of inelastic microstructures and compos-ites based on an incremental variational formulation’, International Journal for NumericalMethods in Engineering 55(11), 1285–1322.

Miehe, C. [2014], ‘Variational gradient plasticity at finite strains. part I: Mixed potentialsfor the evolution and update problems of gradient-extended dissipative solids’, ComputerMethods in Applied Mechanics and Engineering 268, 677–703.

Mishin, Y., Farkas, D., Mehl, M. J. and Papaconstantopoulos, D. A. [1998], ‘Interatomicpotentials for Al andNi from experimental data and ab initio calculations’, MRS Proceedings538, 535.

Mroziński, S. and Szala, J. [2010], ‘Problem of cyclic hardening or softening in metalsunder programmed loading’, Scientific Problems of Machines Operation and Maintenance45(4), 83–96.

Mu, Y., Zhang, X., Hutchinson, J. W. andMeng, W. J. [2016], ‘Dependence of confined plasticflow of polycrystalline Cu thin films on microstructure’, MRS Communications 6(3), 289–294.

124 Bibliography

Needleman, A., Asaro, R. J., Lemonds, J. and Peirce, D. [1985], ‘Finite element analysis ofcrystalline solids’, ComputerMethods in AppliedMechanics and Engineering 52(1–3), 689–708.

Needleman, A. and Tvergaard, V. [1993], ‘Comparison of Crystal Plasticity and Isotropic Hard-ening Predictions for Metal-Matrix Composites’, Journal of Applied Mechanics 60(1), 70–76.

Nitsche, J. [1971], ‘Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Ver-wendung von Teilräumen, die keinen Randbedingungen unterworfen sind: Abhandlungenaus dem Mathematischen Seminar der Universität Hamburg’, 36(1), 9–15.

Nix, W. D. and Gao, H. [1998], ‘Indentation size effects in crystalline materials: A law forstrain gradient plasticity’, Journal of the Mechanics and Physics of Solids 46(3), 411–425.

Nye, J. F. [1953], ‘Some geometrical relations in dislocated crystals’, Acta Metallurgica1(2), 153–162.

Orowan, E. [1934], ‘Zur Kristallplastizität. III’, Zeitschrift für Physik 89(9), 634–659.

Ortiz, M. and Repetto, E. A. [1999], ‘Nonconvex energy minimization and dislocation struc-tures in ductile single crystals’, Journal of the Mechanics and Physics of Solids 47(2), 397–462.

Ortiz, M. and Stainier, L. [1999], ‘The variational formulation of viscoplastic constitutiveupdates’, Computer Methods In Applied Mechanics and Engineering 171, 419–444.

Peirce, D., Asaro, R. J. and Needleman, A. [1982], ‘An analysis of nonuniform and localizeddeformation in ductile single crystals’, Acta Metallurgica 30(6), 1087–1119.

Peirce, D., Asaro, R. J. and Needleman, A. [1983], ‘Material rate dependence and localiseddeformation in crystalline solids’, Acta Metallurgica 31(12), 1951–1976.

Petch, N. J. [1953], ‘The cleavage strength of polycrystals’, The Journal of the Iron and SteelInstitute 174, 25–28.

Polanyi, M. [1934], ‘Über eine Art Gitterstörung, die einen Kristall plastisch machen könnte’,Zeitschrift für Physik 89(9), 660–664.

Pouriayevali, H. and Xu, B. X. [2017], ‘A study of gradient strengthening based on a finite-deformation gradient crystal-plasticity model’, ContinuumMechanics and Thermodynamics29(6), 1389–1412.

Bibliography 125

Quey, R., Dawson, P. R. and Barbe, F. [2011], ‘Large-scale 3D random polycrystals for thefinite element method: Generation, meshing and remeshing’, Computer Methods in AppliedMechanics and Engineering 200(17-20), 1729–1745.

Rashid, M. M. and Nemat-Nasser, S. [1992], ‘A constitutive algorithm for rate-dependentcrystal plasticity’, Computer Methods in Applied Mechanics and Engineering 94(2), 201–228.

Ravi Kumar, B. [2010], ‘Influence of crystallographic textures on tensile properties of 316laustenitic stainless steel’, Journal of Materials Science 45(10), 2598–2605.

Read,W. T. and Shockley,W. [1950], ‘Dislocationmodels of crystal grain boundaries’, PhysicalReview 78, 275–289.

Reed, W. H. and Hill, T. R. [1973], ‘Triangular mesh methods for the neutron transportequation’, Technical Report LA-UR-73-479, Los Alamos .

Rezaei, S., Jaworek, D., Mianroodi, J. R., Wulfinghoff, S. and Reese, S. [2019], ‘Atomisticallymotivated interface model to account for coupled plasticity and damage at grain boundaries’,Journal of the Mechanics and Physics of Solids 124, 325–349.

Shi, J. and Zikry, M. A. [2011], ‘Modeling of grain boundary transmission, emission, absorp-tion and overall crystalline behavior in Σ1, Σ3, and Σ17b bicrystals’, Journal of MaterialsResearch 26(14), 1676–1687.

Simo, J. C. and Hughes, T. J. R. [1998], ‘Computational inelasticity’, Springer, New York .

Spearot, D. E. and Sangid,M.D. [2014], ‘Insights on slip transmission at grain boundaries fromatomistic simulations’, Current Opinion in Solid State andMaterials Science 18(4), 188–195.

Steinmann, P. and Stein, E. [1996], ‘On the numerical treatment and analysis of finite de-formation ductile single crystal plasticity’, Computer Methods in Applied Mechanics andEngineering 129(3), 235–254.

Svendsen, B. and Bargmann, S. [2010], ‘On the continuum thermodynamic rate variationalformulation of models for extended crystal plasticity at large deformation’, Journal of theMechanics and Physics of Solids 58(9), 1253–1271.

Taylor, G. I. [1934], ‘The mechanism of plastic deformation of crystals. Part I. Theoretical’,Proceedings of the Royal Society of London A 145, 362–387.

126 Bibliography

van Beers, P., McShane, G., Kouznetsova, V. and Geers, M. [2013], ‘Grain boundary interfacemechanics in strain gradient crystal plasticity’, Journal of the Mechanics and Physics ofSolids 61(12), 2659–2679.

von Mises, R. [1913], ‘Mechanics of solid bodies in the plastically-deformablestate’, Nachrichten von der Koeniglichen Gesellschaft der Wissenschaften Göttingen,Mathematical physics 4, 582–592.

Wells, G. N., Garikipati, K. and Molari, L. [2004], ‘A discontinuous Galerkin formulation fora strain gradient-dependent damage model’, Computer Methods in Applied Mechanics andEngineering 193(33–35), 3633–3645.

Werner, E. and Prantl, W. [1990], ‘Slip transfer across grain and phase boundaries author linksopen overlay panel’, Acta Metallurgica et Materialia 38(3), 533–537.

Weygand, D., Friedman, L. H., Van der Giessen, E. and Needleman, A. [2002], ‘Aspects ofboundary-value problem solutions with three-dimensional dislocation dynamics’, Modellingand Simulation in Materials Science and Engineering 10(4), 437–468.

Weygand, D., Senger, J., Motz, C., Augustin, W., Heuveline, V. and Gumbsch, P. [2009], ‘Highperformance computing and discrete dislocation dynamics: Plasticity of micrometer sizedspecimens’, Springer Berlin Heidelberg pp. 507–523.

Wulfinghoff, S. [2017], ‘A generalized cohesive zone model and a grain boundary yieldcriterion for gradient plasticity derived from surface- and interface-related arguments’,International Journal of Plasticity 92, 57–78.

Wulfinghoff, S., Bayat, H., Alipour, A. and Reese, S. [2017], ‘A low-order locking-free hybriddiscontinuous Galerkin element formulation for large deformations’, Computer Methods inApplied Mechanics and Engineering 323, 353–372.

Wulfinghoff, S., Bayerschen, E. and Böhlke, T. [2013], ‘A gradient plasticity grain boundaryyield theory’, International Journal of Plasticity 51, 33–46.

Wulfinghoff, S. andBöhlke, T. [2012], ‘Equivalent plastic strain gradient enhancement of singlecrystal plasticity: theory and numerics’, Proceedings of theRoyal SocietyA 468, 2682–2703.

Wulfinghoff, S. and Böhlke, T. [2013], ‘Equivalent plastic strain gradient crystal plasticity-enhanced power law subroutine’, GAMMMitteilungen 36(2), 134–148.

Bibliography 127

Wulfinghoff, S. and Böhlke, T. [2015], ‘Gradient crystal plasticity including dislocation-basedwork-hardening and dislocation transport’, International Journal of Plasticity 69, 152–169.

Wulfinghoff, S., Forest, S. and Böhlke, T. [2015], ‘Strain gradient plasticity modeling of thecyclic behavior of laminate microstructures’, Journal of the Mechanics and Physics of Solids79, 1–20.

Wyrzykowski, J. W. and Grabski, M. W. [1986], ‘The Hall-Petch relation in aluminium and itsdependence on the grain boundary structure’, Philosophical Magazine A 53(4), 505–520.

Yao,W. Z., Krill III, C. E., Albinski, B., Schneider, H. C. andYou, J. H. [2014], ‘Plasticmaterialparameters and plastic anisotropy of tungsten single crystal: a spherical micro-indentationstudy’, Journal of Materials Science 49(10), 3705–3715.

Yu, H., Xin, Y., Wang, M. and Liu, Q. [2018], ‘Hall-Petch relationship inMg alloys: A review’,Journal of Materials Science and Technology 34(2), 248–256.

Zaafarani, N., Raabe, D., Singh, R. N., Roters, F. and Zaefferer, S. [2006], ‘Three-dimensionalinvestigation of the texture andmicrostructure below a nanoindent in a Cu single crystal using3D EBSD and crystal plasticity finite element simulations’, Acta Materialia 54(7), 1863–1876.

Zaefferer, S., Kuo, J. C., Zhao, Z., Winning, M. and Raabe, D. [2003], ‘On the influence ofthe grain boundary misorientation on the plastic deformation of aluminum bicrystals’, ActaMaterialia 51(16), 4719–4735.

Zecevic, M., Beyerlein, I. J., Mccabe, R. J., Mcwilliams, B. A. and Knezevic, M. [2016],‘Transitioning rate sensitivities across multiple length scales : Microstructure-propertyrelationships in the Taylor cylinder impact test on zirconium’, International Journal ofPlasticity 84, 138–159.

Zeng, W., Larsen, J. M. and Liu, G. R. [2015], ‘Smoothing technique based crystal plasticityfinite element modeling of crystalline materials’, International Journal of Plasticity 65, 250–268.

Zhang, K., Holmedal, B., Hopperstad, O. S., Dumoulin, S., Gawad, J., Van Bael, A. andVan Houtte, P. [2015], ‘Multi-level modelling of mechanical anisotropy of commercialpure aluminium plate: Crystal plasticity models, advanced yield functions and parameteridentification’, International Journal of Plasticity 66, 3–30.

Ziemann, M., Chen, Y., Kraft, O., Bayerschen, E., Wulfinghoff, S., Kirchlechner, C., Tamura,N., Böhlke, T., Walter, M. and Gruber, P. A. [2015], ‘Deformation patterns in cross-sectionsof twisted bamboo-structured Au microwires’, Acta Materialia 97, 216–222.

Publications

2020

• A. Alipour, S. Reese, B. Svendsen, S.Wulfinghoff; ‘A grain boundary model consideringthe grain misorientation within a geometrically nonlinear gradient-extended crystal vis-coplasticity theory’; Proceedings ofTheRoyal SocietyA; 2020; DOI: 10.1098/rspa.2019.0581.

2019

• A. Alipour, S. Reese, S. Wulfinghoff; ‘A grain boundary model for gradient-extendedgeometrically nonlinear crystal plasticity: Theory and numerics’; International Journalof Plasticity; 2019; 118, 17-35.

2018

• A. Alipour, S. Wulfinghoff, H. R. Bayat, S. Reese, B. Svendsen; ‘The concept of controlpoints in hybrid discontinuous galerkin methods-application to gometrically nonlinearcrystal plasticity’; International Journal for Numerical Methods in Engineering; 2018;114 (5), 557-579.

2017

• S. Wulfinghoff, H. R. Bayat, A. Alipour, S. Reese; ‘A low-order locking-free hybriddiscontinuous Galerkin element formulation for large deformations’; Computer Methodsin Applied Mechanics and Engineering; 2017; 323, 353-372.

• A. Alipour, S. Wulfinghoff, H. R. Bayat, B. Svendsen, S. Reese; ‘Geometrically non-linear single crystal viscoplasticity implemented into a hybrid discontinuous Galerkinframework’; Proceedings of the 7th GACM Colloquium on Computational Mechanicsfor Young Scientists from Academia and Industry; 2017; 795-798.

• A. Alipour, S. Wulfinghoff, H. R. Bayat, S. Reese; ‘Geometrically nonlinear crystalplasticity implemented into a discontinuous Galerkin element formulation’; Proceedingsin Applied Mathematics and Mechanics (PAMM); Wiley; 2017; ISSN: 1617-7061; Vol.17, 753-754, DOI: 10.1002/pamm.201710344.

Conferences

2018

• A.Alipour, S.Wulfinghoff, S. Reese, B. Svendsen; ‘A grain boundarymodel for gradient-extended geometrically nonlinear crystal plasticity’; European Solid Mechanics Confer-ence (ESMC), Bologna (Italy).

• A.Alipour, S.Wulfinghoff, S.Reese; ‘Control-point based hybrid discontinuousGalerkinmethod applied to geometrically nonlinear crystal plasticity’; European Mechanics ofMaterials Conference (EMMC16), Nantes (France).

2017

• A. Alipour, S. Wulfinghoff, S. Reese, B. Svendsen; ‘Geometrically nonlinear singlecrystal viscoplasticity implemented into a hybrid discontinuous Galerkin framework’;GACM Colloquium on computational mechanics, Stuttgart (Germany).

• A. Alipour, S. Wulfinghoff, S. Reese, B. Svendsen; ‘A regularization technique forgeometrically nonlinear crystal plasticity implemented into a newdiscontinuousGalerkinelement formulation’; GAMM Annual Meeting, Weimar (Germany).

crystal plasticity and grain boundaries on small scales

Documents