Variational multifield modeling of the formation and evolution of laminate microstructure

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(1)Variational Multifield Modeling of the Formation and Evolution of Laminate Microstructure. Felix Eberhard Hildebrand. ˙ t , p) ˙ inf inf ΠB (ϕt , p; ϕ ˙t ϕ. p˙. ˙ t , γ˙ α ) ΠB (ϕt , γ α , γ +α ; ϕ inf inf α ˙ t γ˙ ϕ. Bericht Nr.: I-25 (2013) Institut f¨ ur Mechanik (Bauwesen), Lehrstuhl I Professor Dr.-Ing. C. Miehe Stuttgart 2013.

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(5) Variational Multifield Modeling of the Formation and Evolution of Laminate Microstructure. Von der Fakultät Bau- und Umweltingenieurwissenschaften der Universität Stuttgart und dem Stuttgart Research Center for Simulation Technology zur Erlangung der W¨ urde eines Doktor-Ingenieurs (Dr.-Ing.) genehmigte Abhandlung. von Felix Eberhard Hildebrand aus T¨ ubingen a. N.. Hauptberichter Mitberichter. : Prof. Dr.-Ing. Christian Miehe : Prof. Dr. rer. nat. Klaus Hackl. Tag der m¨ undlichen Pr¨ ufung: 26. Oktober 2012. Institut f¨ ur Mechanik (Bauwesen) der Universität Stuttgart 2013.

(6) Herausgeber: Prof. Dr.-Ing. habil. C. Miehe. Organisation und Verwaltung: Institut f¨ ur Mechanik (Bauwesen) Lehrstuhl I Universität Stuttgart Pfaffenwaldring 7 70550 Stuttgart Tel.: +49(0)711/685-66378 Fax: +49(0)711/685-66347. c. Felix E. Hildebrand Institut f¨ ur Mechanik (Bauwesen) Lehrstuhl I Universität Stuttgart Pfaffenwaldring 7 70569 Stuttgart Tel.: +49(0)711/685-66377 Fax: +49(0)711/685-66347. ¨ Alle Rechte, insbesondere das der Ubersetzung in fremde Sprachen, vorbehalten. Ohne Genehmigung des Autors ist es nicht gestattet, dieses Heft ganz oder teilweise auf fotomechanischem Wege (Fotokopie, Mikrokopie) zu vervielfältigen.. ISBN 3-937859-13-6 (D 93 Stuttgart).

(7) to Lovis, Malin & Julia.

(8) Abstract The optimization of material properties and the design of new materials with tailored material behavior are among the greatest challenges in the field of computational continuum mechanics. Since the macroscopic material behavior of many technically relevant materials is very closely linked to their microstructure, a profound physical and mathematical understanding and a reliable computational prediction of the formation and evolution of this microstructure is the necessary basis for any optimization or material design. In this work, we focus on the physical and mathematical understanding and the modeling and simulation of laminate microstructure and use the modeling framework of gradient-extended standard-dissipative solids to construct a phase field model for martensitic laminate microstructure in two-variant martensitic CuAlNi and a gradient crystal plasticity model for laminate deformation microstructure in Copper with two active slip systems on the same slip plane. We derive rate- and incremental-variational as well as finite element formulations for the two models and carry out numerical simulations. Basis for our modeling are the modeling framework of gradient-extended standard-dissipative solids on the one hand, and the continuum theory of non-material sharp interfaces with interface energy on the other hand, from which we derive the condition of kinematic compatibility, jump conditions in analogy to the balance equations and the dissipation postulate for the moving interface. We consider the variational origin of the formation of laminate microstructure and identify gradient-extended modeling approaches as the suitable choice for the modeling of the formation and dissipative evolution of laminate microstructure with interface energy. Based on these considerations, we propose a phase field model for the formation and evolution of laminate microstructure in two-variant martensitic CuAlNi that is based on the variational smooth approximation of sharp topologies and contains a coherencedependent interface energy. We show that an internal mixing approach for the bulk energy allows a clear separation of interface and bulk energy and that the model is capable of predicting the formation and dissipative evolution of martensitic laminate microstructure and size effects. Furthermore, we propose a gradient crystal plasticity model for Copper with two active slip systems on the same slip plane that allows a prediction of both the formation and evolution of plastic laminate microstructure and incorporates the effect of geometrically necessary dislocations (GNDs). The model contains a biquadratic non-convex latent hardening function and a gradient contribution based on the dislocation density tensor. The evolution equations of the plastic slips and the accumulated plastic slips are obtained by use of a rate regularization that makes use of the approximation |x| ≈ ν ln(cosh(x/ν)) for ν ≪ 1. The model is shown to be capable of predicting the formation and evolution of deformation laminate microstructure together with length-scale effects related to GNDs..

(9) Zusammenfassung Die Optimierung von Materialeigenschaften und die Entwicklung neuer Materialien mit maßgeschneiderten Eigenschaften zählen zu den größten Herausforderungen im Bereich der computerorientierten Kontinuumsmechanik. Da das makroskopische Materialverhalten vieler technisch relevanter Materialien eng mit ihrer Mikrostruktur verbunden ist, sind ein fundiertes physikalisches und mathematisches Verständnis sowie eine zuverlässige numerische Vorhersage der Entstehung und Evolution von Mikrostrukturen die nötige Basis f¨ ur jegliche Art von Optimierung und Materialentwicklung. Diese Arbeit befasst sich speziell mit den physikalischen und mathematischen Grundlagen und der Modellierung und Simulation von Laminatmikrostruktur. Im Rahmen der Materialklasse der gradientenerweiterten standarddissipativen Festkörper formulieren wir ein Phasenfeldmodell f¨ ur Laminatmikrostruktur in martensitischem CuAlNi mit zwei martensitischen Varianten sowie ein gradientenerweitertes Kristallplastizitätsmodell f¨ ur plastische Laminatmikrostruktur in Kupfer mit zwei aktiven Gleitsystemen auf derselben Gleitebene. F¨ ur die Modelle werden ratenbasiert und inkrementell variationelle Prinzipe sowie Finite Element Formulierungen hergeleitet und zur Betrachtung numerischer Beispiele verwendet. Basis f¨ ur die Modellierung ist auf der einen Seite der Modellierungsrahmen der Materialklasse der gradientenerweiterten standarddissipativen Festkörper und auf der anderen Seite die Kontinuumsmechanik scharfer nichtmaterieller Grenzflächen mit Grenzflächenenergie, mit deren Hilfe wir die Bedingung der kinematischen Kompatibilität, die Sprungbedingungen in Analogie zu den Bilanzgleichungen sowie das Dissipationspostulat der Grenzfläche herleiten. Wir betrachten den variationellen Ursprung von Laminatmikrostrukturen und identifizieren gradientenerweiterte Ansätze als den am Besten geeigneten Weg zu deren Modellierung unter Einbeziehung von Grenzflächenenergie und Dissipation. ¨ Ausgehend von diesen Uberlegungen entwickeln wir ein Phasenfeldmodell f¨ ur die Enstehung und Evolution von Laminatmikrostruktur in martensitischem CuAlNi mit zwei martensitischen Varianten, das auf der variationellen und glatten Approximation scharfer Grenzflächen basiert und eine kohärenzabhängige Grenzflächenenergie beinhalten. Wir zeigen, dass ein innerer Mischungsansatz der elastischen Energie eine saubere Trennung zwischen Grenzflächen- und elastischer Energie erlaubt und dass das Modell die Enstehung und dissipative Evolution von martensitischer Laminatmikrostruktur vorhersagen kann. Des Weiteren konstruieren wir ein gradientenerweitertes Kristallplastizitätsmodell f¨ ur plastische Laminatmikrostruktur in Kupfer mit zwei aktiven Gleitsystemen auf derselben Gleitebene, das die Vorhersage sowohl der Enstehung und Evolution von Laminatmikrostruktur als auch des Effekts von geometrisch notwendigen Versetzungen erlaubt. Das Modell beinhaltet eine biquadratische Funktion f¨ ur latente Verfestigung und einen Gradiententerm, der auf dem Versetzungsdichtetensor basiert. Die Evolutionsgleichungen f¨ ur die plastischen Gleitungen und die akkumulierten plastischen Gleitungen basieren auf Ratenregularisierungen mit Hilfe der Approximation |x| ≈ ν ln(cosh(x/ν)) f¨ ur ν ≪ 1. Wir zeigen die Fähigkeit unseres Modells, die Enstehung und Entwicklung von Laminatmikrostruktur zusammen mit Längenskaleneffekten aufgrund geometrisch notwendiger Versetzungen vorherzusagen..

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(11) Acknowledgments This thesis is the result of four challenging and exciting years at the Institute of Applied Mechanics (Civil Engineering) at the University of Stuttgart. Many people have contributed to this work in countless ways and I would like to sincerely thank all of them. First and foremost, I am very deeply indebted to my supervisor Professor Christian Miehe for his thoughtful guidance, his support in academic and personal matters and his great confidence. His vast knowledge, his very structured way of thinking and his absolute commitment to academic excellence have been a constant source of inspiration for me and have decisively contributed to the successful completion of this work. I am also grateful to Professor Klaus Hackl for his interest in my research, his very helpful comments and for his acceptance to become the co-referee for this thesis. I would like to take this opportunity to thank those who were my mentors and teachers before I came to Stuttgart: I will always remember the exciting and brilliant lectures of Professor Peter Gummert, who first raised my interest in the field of mechanics during my time in Berlin. Furthermore, I want to extend my sincere gratitude to Professor Rohan Abeyaratne in Boston, who has taught me how to do research with joy, curiosity and an open mind and how to boldly face new challenges. Finally, I want to thank Professor Sanjay Govindjee for an unforgettable time in Zurich. This work has greatly profited from fruitful interactions with my colleagues at the institute. I especially thank my officemates Ilona Zimmermann and Lukas Böger for their patience and encouragement during the many ups and downs of academic life. I would further like to thank Fabian Welschinger, Dominic Zäh and specifically Steffen Mauthe for their contributions to this work, it was a real pleasure to work with them. To Gautam Ethiraj and Daniele Rosato I am truly grateful for many good conversations, philosophical coffee breaks and for their friendship beyond the realms of the university. I would like to gratefully acknowledge the financial support of this work through the German Research Foundation (DFG) within the SimTech Cluster of Excellence (EXC 310/1). The interdisciplinary environment of SimTech was very pleasant and inspiring and the numerous lunch and coffee breaks with my SimTech colleagues and friends Tille Rupp, Jonas Offtermatt and Katherina Baber were a great and sometimes necessary distraction from my work. I will miss these very joyful occasions a lot. I thank my parents for their unconditional love, everlasting support and unwavering faith in me. They have been there for me when I needed them on countless occasions and have helped me in many ways to become the person that I am today. Most importantly, I owe heartfelt gratitude to my dear family: To little Lovis, without whom I would still be fine tuning the typesetting of the formulae in this work, and to Malin for her tolerance when I was stressed or absorbed in my work and especially for her words of motivation during the writing process when she would often sweetly say “Felix?” and then very strictly add “Arbeiten!!!”. Finally, I am most deeply indebted to my beloved wife Julia for her limitless support and constant encouragement, her understanding, her patient advice, her never-failing confidence and for simply being there. Stuttgart, February 2013. Felix Hildebrand.

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(13) i. Contents. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Motivation and State of the Art . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. Laminate Microstructure in Martensitic Shape Memory Alloys 1.1.2. Laminate Microstructure in Gradient Crystal Plasticity . . . . . 1.2. Objectives and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 1 1 2 4 6. 2. Continuum Mechanics . . . . . . . . . . . . . . . . . . . . . . . 2.1. Kinematics of Finite Deformation . . . . . . . . . . . . . 2.1.1. Description of Motion . . . . . . . . . . . . . . . . 2.1.2. Deformation Gradient . . . . . . . . . . . . . . . . 2.1.3. Polar Decomposition . . . . . . . . . . . . . . . . . 2.2. Integral Identities and Canonical Balance Principle . 2.2.1. Rate of Volume Integrals . . . . . . . . . . . . . . 2.2.2. Divergence Theorem . . . . . . . . . . . . . . . . . 2.2.3. Canonical Balance Principle . . . . . . . . . . . . 2.3. Physical Balance Principles . . . . . . . . . . . . . . . . . 2.3.1. Isothermal and Quasistatic Conditions . . . . . 2.3.2. Mechanical and Thermal Fluxes . . . . . . . . . 2.3.3. Balance of Mass . . . . . . . . . . . . . . . . . . . . . 2.3.4. Balance of Linear Momentum . . . . . . . . . . . 2.3.5. Balance of Angular Momentum . . . . . . . . . . 2.3.6. Balance of Energy . . . . . . . . . . . . . . . . . . . 2.3.7. Balance of Entropy . . . . . . . . . . . . . . . . . . 2.4. Dissipation Postulate . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. 9 9 9 11 13 13 13 13 14 14 14 15 16 17 17 18 19 19. 3. Modeling Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1. Mechanical Initial Boundary Value Problem . . . . . . . . . . . . . 3.1.2. Necessity of Constitutive Relations . . . . . . . . . . . . . . . . . . . 3.2. Constitutive Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1. Principle of Determinism . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Concept of Internal Variables . . . . . . . . . . . . . . . . . . . . . . . 3.2.3. Principle of Local Action . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4. Principle of Material Frame Invariance . . . . . . . . . . . . . . . . . 3.2.5. Principle of Material Symmetry . . . . . . . . . . . . . . . . . . . . . . 3.2.6. Evaluation of Dissipation Postulate . . . . . . . . . . . . . . . . . . . 3.2.7. Concept of Dissipation Potentials . . . . . . . . . . . . . . . . . . . . 3.3. Modeling Ingredients for Elastic and Dissipative Behavior . . . . . . . . 3.3.1. Ingredients for the Free Energy Function . . . . . . . . . . . . . . . 3.3.2. Ingredients for the Dissipation and Dual Dissipation Potential. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. 21 21 21 22 23 23 23 24 24 26 27 28 32 32 33. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . ..

(14) ii. Contents. 4. Variational and Algorithmic Framework . . . . . . . . . . . . . . . . . . . 4.1. Modified Initial Boundary Value Problem . . . . . . . . . . . . . . . . . . 4.1.1. Governing Field Equations . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2. Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . 4.1.3. Thermodynamic Consistency of Micro-Boundary Conditions 4.2. Continuous Rate-Variational Formulation . . . . . . . . . . . . . . . . . . 4.2.1. Rate Potential Functionals . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Continuous Minimum Principle . . . . . . . . . . . . . . . . . . . . . 4.2.3. Euler Equations of the Continuous Minimum Principle . . . . 4.3. Time-Discrete Incremental-Variational Formulation . . . . . . . . . . . . 4.3.1. Time Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Incremental Potential Functionals . . . . . . . . . . . . . . . . . . . 4.3.3. Time-Discrete Minimum Principle . . . . . . . . . . . . . . . . . . . 4.3.4. Euler Equations of the Time-Discrete Minimum Principle . . 4.4. Space-Time-Discrete Finite Element Formulation . . . . . . . . . . . . . 4.4.1. Finite Element Approximation . . . . . . . . . . . . . . . . . . . . . 4.4.2. Finite Element Potential Functionals . . . . . . . . . . . . . . . . . 4.4.3. Space-Time-Discrete Minimum Principle . . . . . . . . . . . . . . 4.4.4. Solution of the Space-Time-Discrete Minimum Principle . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .. 39 39 39 40 41 41 41 43 43 44 44 45 46 46 47 48 50 52 52. 5. Continuum Mechanics of Sharp Interfaces . . . . . . . . . . . . 5.1. Kinematics of Non-Material Sharp Interfaces . . . . . . . . . . 5.1.1. Kinematics of Non-Material Interfaces . . . . . . . . . . 5.1.2. Hadamard Jump Condition . . . . . . . . . . . . . . . . . . 5.1.3. Kinematic Compatibility . . . . . . . . . . . . . . . . . . . 5.2. Integral Identities and Canonical Jump Condition . . . . . . . 5.2.1. Rate of Integral over Moving Interface . . . . . . . . . . 5.2.2. Rate of Integral over Volume with Moving Interface 5.2.3. Modified Divergence Theorem . . . . . . . . . . . . . . . . 5.2.4. Canonical Jump Condition . . . . . . . . . . . . . . . . . . 5.3. Physical Balance Principles at the Interface . . . . . . . . . . . 5.3.1. Continuity of Temperature at the Interface . . . . . . . 5.3.2. Balance of Mass at the Interface . . . . . . . . . . . . . . 5.3.3. Balance of Linear Momentum at the Interface . . . . 5.3.4. Balance of Angular Momentum at the Interface . . . 5.3.5. Balance of Energy at the Interface . . . . . . . . . . . . . 5.3.6. Balance of Entropy at the Interface . . . . . . . . . . . . 5.4. Dissipation Postulate at the Interface . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. 55 55 55 57 58 59 59 60 61 61 62 62 63 63 63 64 64 65. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. 6. Motivation: Idealized Energetic Approach . . . . . . . . . . . . . . . . . . . . . . 67 6.1. Elastostatic Boundary Value Problem with Discontinuities . . . . . . . . . . . 67 6.1.1. Governing Field Equations and Jump Conditions . . . . . . . . . . . . . 67.

(15) iii. Contents . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . .. 68 69 69 69 70 71 72 72 76 80 80 81 82 83 84 84 84 85. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. 89 89 90 90 90 91 92 92 95 96 99 99 100 101. 8. Phasefield Modeling of Martensitic Laminates . . . . . . . . . . . . . . . . . 8.1. Phase Field Approximation of Sharp Interfaces . . . . . . . . . . . . . . . . . . 8.1.1. Approximation of One-Dimensional Sharp Interfaces . . . . . . . . . 8.1.2. Approximation of Two- and Three-Dimensional Sharp Interfaces 8.1.3. Finite Element Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4. Numerical Example of Sharp Interface Approximation . . . . . . . . 8.1.5. Sharp Interface Kinematics and Phase Field Evolution . . . . . . . 8.2. Constitutive Phase Field Modeling of Martensitic CuAlNi . . . . . . . . . .. . . . . . . . .. .105 . 105 . 105 . 107 . 109 . 110 . 112 . 113. 6.2.. 6.3.. 6.4.. 6.5.. 6.1.2. Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3. Relation to Microstructure . . . . . . . . . . . . . . . . . . . . . . . . Variational Formulation of Nonlinear Elastostatics . . . . . . . . . . . . 6.2.1. Elastostatic Functionals . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2. Variational Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3. Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Examples for the Formation of Microstructure . . . . . . . . . . . 6.3.1. Formation of Microstructure in One Dimension . . . . . . . . . . 6.3.2. Formation of Laminate Microstructure in Three Dimensions Conditions for the Formation of Microstructure . . . . . . . . . . . . . . 6.4.1. Common Properties of Examples . . . . . . . . . . . . . . . . . . . . 6.4.2. Loss of Sequentially Weak Lower Semi-Continuity . . . . . . . . 6.4.3. Average Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4. Decreasing Frustration . . . . . . . . . . . . . . . . . . . . . . . . . . . Challenges and Modeling Approaches . . . . . . . . . . . . . . . . . . . . . 6.5.1. Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2. Interface Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3. Modeling Approaches for Microstructure . . . . . . . . . . . . . .. 7. Fundamentals of Martensitic Laminates . . . . . . . . . . . . 7.1. Continuum Theory of Crystalline Solids . . . . . . . . . . . . . 7.1.1. Description of Crystal Lattices . . . . . . . . . . . . . . 7.1.2. Cauchy-Born Hypothesis . . . . . . . . . . . . . . . . . . 7.1.3. Free Energy of Crystal Lattices . . . . . . . . . . . . . . 7.1.4. Symmetry of Crystal Lattices . . . . . . . . . . . . . . . 7.2. Basic Properties of Single Crystal CuAlNi . . . . . . . . . . . 7.2.1. Cubic to Orthorhombic Transformation in CuAlNi 7.2.2. Multiwell Free Energy of CuAlNi . . . . . . . . . . . . . 7.2.3. Twinning of Martensitic Variants in CuAlNi . . . . . 7.2.4. Non-Quasiconvexity of the Free Energy of CuAlNi 7.2.5. Microstructure in CuAlNi . . . . . . . . . . . . . . . . . . 7.2.6. Interface Energy of Twin-Interfaces in CuAlNi . . . 7.2.7. Dissipative Twin-Interface Motion . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . ..

(16) iv. Contents. 8.3.. 8.4.. 8.5.. 8.6.. 8.7.. 8.8.. 8.2.1. General Form of Constitutive Functions . . . . . . . . . . . . . . . . . . . 8.2.2. Constitutive Modeling of Energy Storage . . . . . . . . . . . . . . . . . . 8.2.3. Constitutive Modeling of Dissipation . . . . . . . . . . . . . . . . . . . . . Phase Field Initial Boundary Value Problem . . . . . . . . . . . . . . . . . . . . 8.3.1. Governing Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2. Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3. Thermodynamic Consistency of Phase Field-Boundary Conditions Continuous Rate-Variational Formulation . . . . . . . . . . . . . . . . . . . . . . 8.4.1. Rate Potential Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2. Continuous Minimum Principle . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3. Euler Equations of the Continuous Minimum Principle . . . . . . . . Time-Discrete Incremental-Variational Formulation . . . . . . . . . . . . . . . . 8.5.1. Incremental Potential Functional . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2. Time-Discrete Minimum Principle . . . . . . . . . . . . . . . . . . . . . . . 8.5.3. Euler Equations of the Time-Discrete Minimum Principle . . . . . . Space-Time-Discrete Finite Element Formulation . . . . . . . . . . . . . . . . . 8.6.1. Finite Element Potential Functionals . . . . . . . . . . . . . . . . . . . . . 8.6.2. Space-Time-Discrete Minimum Principle . . . . . . . . . . . . . . . . . . 8.6.3. Solution of the Space-Time-Discrete Minimum Principle . . . . . . . Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1. General Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2. Summary of Material Parameters . . . . . . . . . . . . . . . . . . . . . . . 8.7.3. Simulations of Single Crystal CuAlNi in Two Dimensions . . . . . . 8.7.4. Simulations of Single Crystal CuAlNi in Three Dimensions . . . . . 8.7.5. Simulations of Poly-Crystalline CuAlNi in Three Dimensions . . . . Key Features of Phase Field Models for Laminates . . . . . . . . . . . . . . . . 8.8.1. Non-Convexity of Energy with respect to Internal Variable . . . . . 8.8.2. Rank-One Connection of Associated Deformation States . . . . . . . 8.8.3. Anisotropic Gradient Energy Favoring Rank-One Normal . . . . . . 8.8.4. Bulk-Energy-Free Rank-one Connection of Minima . . . . . . . . . . .. 9. Gradient Plasticity Modeling of Plastic Laminates . . . . . 9.1. Modeling Assumptions and Preliminary Definitions . . . . . . 9.1.1. Double Slip with common slip normal . . . . . . . . . . 9.1.2. Double Slip Plastic Deformation Gradient . . . . . . . 9.1.3. Double Slip Dislocation Density Tensor . . . . . . . . . 9.1.4. Accumulated Plastic Slips . . . . . . . . . . . . . . . . . . . 9.1.5. Rank-One Connection of Plastic Deformation States 9.2. Constitutive Modeling of Energy Storage and Dissipation . 9.2.1. General Form of Constitutive Functions . . . . . . . . . 9.2.2. Constitutive Modeling of Stored Energy . . . . . . . . . 9.2.3. Constitutive Modeling of Dissipation . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 113 113 118 120 120 121 122 123 123 123 123 124 124 124 125 125 126 126 127 128 128 129 129 134 138 139 140 140 140 140. .141 . 142 . 142 . 142 . 143 . 144 . 144 . 144 . 144 . 145 . 147.

(17) v. Contents 9.3. Gradient Plasticity Initial Boundary Value Problem . . . . . . . . . . . . . 9.3.1. Governing Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Boundary and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . 9.3.3. Thermodynamic Consistency of Slip-Boundary Conditions . . . . 9.4. Continuous Rate-Variational Formulation . . . . . . . . . . . . . . . . . . . . 9.4.1. Rate Potential Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2. Continuous Minimum Principle . . . . . . . . . . . . . . . . . . . . . . . 9.4.3. Euler Equations of the Continuous Minimum Principle . . . . . . 9.5. Time-Discrete Incremental-Variational Formulation . . . . . . . . . . . . . . 9.5.1. Incremental Potential Functional . . . . . . . . . . . . . . . . . . . . . . 9.5.2. Time-Discrete Minimum Principle . . . . . . . . . . . . . . . . . . . . . 9.5.3. Euler Equations of the Time-Discrete Minimum Principle . . . . 9.6. Space-Time-Discrete Finite Element Formulation . . . . . . . . . . . . . . . 9.6.1. Finite Element Potential Functionals . . . . . . . . . . . . . . . . . . . 9.6.2. Space-Time-Discrete Minimum Principle . . . . . . . . . . . . . . . . 9.6.3. Solution of the Space-Time-Discrete Minimum Principle . . . . . 9.7. Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.1. Double Slip Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.2. Material Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7.3. Laminate Deformation Microstructure in Single Crystal Copper. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . .. 148 148 149 150 150 151 151 151 152 152 153 153 154 155 155 156 156 157 158 158. 10. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .163.

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(19) 1. 1. Introduction The behavior of many technically relevant materials with interesting material properties is governed by the formation and evolution of microstructure. The prediction of such microstructure and the understanding of its effects on the macroscopic behavior are the key to a reliable physically informed constitutive modeling, provide a basis for the optimization of material properties and are the precondition for the design of materials with tailored behavior. This work focuses on the modeling and numerical prediction of the formation and evolution of laminate microstructure. The main objectives are thereby a deeper understanding of the interplay of energetic and dissipative effects, the construction of suitable variational-based models and their application to numerical experiments.. 1.1. Motivation and State of the Art Laminate deformation microstructure occurs in a number of technically relevant materials. Examples are shape memory alloys, plastically deformed single crystals and ferroelectric ceramics, see Figure 1.1.. a). b). c). Figure 1.1: Optical micrographs of laminate microstructures a) due to martensitic transformation in the shape memory alloy CuAlNi (1.25 mm × 0.86 mm, taken from Abeyaratne et al. [1]), b) due to plastic deformation in Copper (taken from Dmitrieva et al. [42]) and c) due to electromechanical loading in BaTiO3 (taken from Arlt [9]).. The formation and evolution of laminate microstructure in these materials strongly influences their material behavior on the macroscale: In shape memory alloys, the formation and evolution of martensitic laminates is one of the key ingredients to the pseudo-plastic behavior and the shape memory effect. In plastically deformed single crystals, the formation of plastic laminates can strongly influence the hardening behavior. Finally, in ferroelectric ceramics, the evolution of laminate microstructure plays an important role for macroscopic hysteretic effects. A reliable prediction of the behavior of these materials on the macroscale is thus not possible without a deeper understanding and physically informed modeling of the formation and evolution of laminate microstructure on the mesoscale. The aim of this work is to develop the necessary physical and mathematical understanding and to derive models for the formation and evolution of such laminate microstructure that resolve the microstructure and contain all necessary physical modeling ingredients. We will thereby focus on the modeling of the formation and evolution of martensitic laminate microstructure in two-variant martensitic CuAlNi and of deformation laminate microstructure in Copper with two active slip systems on one slip plane..

(20) 2. Introduction. 1.1.1. Laminate Microstructure in Martensitic Shape Memory Alloys An example for materials whose microstructure is closely linked to extraordinary thermomechanical properties are shape memory alloys. These materials exhibit unique macroscopic behavior such as superelasticity or the shape memory effect and are often denoted as a class of smart or functional materials. They are very attractive for a wide range of applications e.g. as actuators, stents or medical prosthetics, see Duerig et al. [43]. The ability of shape memory alloys to form microstructure stems from their ability to undergo martensitic transformations, and the most basic building block for this microstructure are martensitic laminates, see Bhattacharya [22]. Martensitic transformations are displacive, diffusionless first-order solid to solid phase transformations between the high temperature austenite and the low temperature martensite. Generally, the austenitic lattice structure has greater crystallographic symmetry than the martensitic lattice structure, see e.g. Wayman [153]. This gives rise to multiple symmetry-related variants of martensite which can form the typical coherent twinned laminates. Together, the austenite and the variants of martensite can form complex patterns at length scales much smaller than the size of the specimen that consist of regions of different phases separated by a collection of both coherent and incoherent sharp phase boundaries and that evolve under thermomechanical loading. While the mechanical response within each phase is considered to be elastic, the interface evolution is considered to be a dissipative mechanism. We refer to Bhattacharya [22] and Abeyaratne & Knowles [4], for an overview of crystallographic, energetic and kinetic aspects of microstructure in shape memory materials. Experimentally, microstructure in shape memory alloys has been observed in countless cases, see e.g. Otsuka & Shimizu [133, 134], Ichinose et al. [78], Chu & James [31], Ball et al. [15], Abeyaratne et al. [1] (see Figure 1.1a) or Tan & Xu [150]. Mathematically, the formation of microstructure in shape memory alloys is closely linked to the non-quasiconvexity of their free energy functions that stems from the existence of multiple martensitic variants. Consequently, all models that fully or partly resolve the microstructure contain some non-convex contribution. One approach to the treatment of this non-convexity is the minimization of the energy with respect to microstructural parameters describing sequential laminates. This results in a rank-one convexification of the free energy also referred to as relaxation, see e.g. Kohn & Strang [89, 90, 91] or Dacorogna [38], and allows a prediction of the formation of sequential laminates while neglecting interface energy and dissipative interface motion. Examples of the application of this approach to the modeling of shape memory alloys are given in Luskin [105], Aubry et al. [13] and Kruˇ z´ık et al. [98]. A modification of the relaxation that allows the incorporation of dissipative effects is based on a dissipative evolution of certain microstructural parameters and a minimization with respect to the rest. This approach is referred to as partial relaxation and allows a prediction of the formation and dissipative evolution of sequential laminates while neglecting interfacial energy. Examples for an application of the partial relaxation to shape memory alloys are given in Bartel [18] and Bartel & Hackl [19, 20]. The ideas of partial relaxation can also be extended to incorporate size effects by including parametrized approximations of interface and misfit energies based on analytical estimates for specific forms of sequential laminates, see e.g. Petryk et al. [136, 137]. Both relaxation-based approaches lead to a certain convexification of the energy but also result in a restriction of the possible microstructure and do not account for interface and.

(21) 1.1 Motivation and State of the Art. 3. misfit energies. One way to allow a formation of unrestricted microstructure and to incorporate additional energy primarily in interface regions is to use a strain gradient approach that leads to a smooth approximation of the jump between the strain states in the interface region, see e.g. Barsch & Krumhansl [17]. It can be shown by Γ-convergence that suitable (time-independent) strain gradient formulations approach the sharp interface limit of elasticity with sharp interfaces, see e.g. Conti et al. [35]. Examples for the application of strain gradient approaches to shape memory alloys are given in Jacobs et al. [79, 80] and Rasmussen et al. [141] in the small strain setting and Finel et al. [52] in the large strain setting. The drawback of strain gradient formulations is that the incorporation of dissipation is not limited to the motion of phase boundaries but also affects the elastic behavior and leads to effectively viscoelastic models. An alternative approach in the spirit of strain gradient approaches that allows the formation of unrestricted microstructure and incorporates interface energy and dissipative effects is obtained by phase field approaches that lead to a smooth description of microstructure by use of order parameters for which nonconvex and gradient energy contributions are specified. Basis for these approaches are analytical considerations as carried out in Fried & Gurtin [57, 58] and Fried & Grach [56]. The resulting formulations generally represent extensions of the classical Ginzburg-Landau equation as described in Allen & Cahn [8] to coupled problems of phase transformation and elastic deformation. It can be shown by Γ-convergence that suitable static phase field formulations approach the sharp interface limit of elasticity with sharp interfaces, see e.g. Garcke [60]. For the time-dependent evolution, a similar relation can be established by asymptotic analysis, see e.g. Fried & Gurtin [57] and Alber & Zhu [5]. Examples for phase field models for martensitic transformations are given in Levitas et al. [100, 101, 103], Artemev et al. [10, 11] and Jin et al. [82] in the small strain setting and Levitas et al. [102] in the large strain setting. In this context, we propose a new phase field model for the smooth description of the formation and evolution of laminate microstructure consisting of two orthorhombic variants of CuAlNi, see also Hildebrand & Miehe [71, 72]. The model is based on the variational approximation of sharp interface geometries, see e.g. Alberti et al. [6]. This allows the introduction of a coherence-dependent interface energy, see Murr [126] and Porter & Easterling [139], by use of a referential anisotropy, see McFadden et al. [107]. A bulk energy is proposed that allows a clear separation of bulk and interface energy and a viscous evolution finally yields a generalized Ginzburg-Landau model, see Gurtin [65]. Making use of the framework outlined in Miehe [111, 112], we construct rate- and incremental variational formulations as well as a variational-based finite element formulation. Finally, we present numerical results that demonstrate the modeling capabilities of our formulation. In summary, the main features of the model are the following: • Variational approximation of sharp topologies: Basis of the phase field model is a variational approximation of sharp interface geometries estimating the surface area. • Phase Field Modeling at Finite Strains: Finite strains are necessary due to the large rotations of the variants and allows the introduction of referential anisotropy. • Coherence-dependent interface energy: The phase field model allows both coherent and incoherent interfaces and accounts for the different energetics..

(22) 4. Introduction • Concise separation of bulk and interface energy: Concise modeling of the energy storage allowing a clear separation between bulk and interface energy. • Variational nature: Continuous rate-type and finite-step-sized incremental-variational principles for a phase field description of martensitic transformations. • Symmetric FE solver: Variational time- and space-discrete finite element formulation, resulting in symmetric algebraic structures for the monolithic problem. • Prediction of laminates: The model allows the prediction of the formation and evolution of laminate microstructure with misfit, length-scale effects and hysteresis.. 1.1.2. Laminate Microstructure in Gradient Crystal Plasticity Another example for an occurrence of laminate microstructure that has recently gained more and more attention is the formation and evolution of laminate deformation microstructure in crystal plasticity, see e.g. Ortiz & Repetto [131]. These laminates are triggered by a dominance of latent hardening over self hardening, see e.g. Kocks [87] and Franciosi et al. [54]. Even though in contrast to martensitic laminates, the prediction of plastic laminates is not as decisive for the prediction of the related material behavior, the formation and evolution of plastic laminates can have a strong effect on the hardening properties of single crystals. Besides the formation of laminates, another phenomenon that plays an important role on the microscale and introduces a physical length scale is that of geometrically necessary dislocations (GNDs) that arise due to inhomogeneity ¨ ner [97]. To our knowledge, a forof plastic deformation, see e.g. Nye [130] and Kro mulation that combines the effect of laminate formation due to latent hardening with a physically motivated modeling of GNDs is lacking in the literature. The formation of laminate deformation microstructure during plastic deformation of single crystals has been experimentally observed by Dmitrieva et al. [42], see Figure 1.1b for a shear deformation of Copper. Mathematically, the formation of laminate microstructure is triggered by a non-convexity of the hardening function that favors an inhomogeneous state of laminate microstructure of alternating single slip states over a homogeneous multislip state for certain slip combinations due to a dominance of latent over self hardening. One approach to the modeling of plastic laminates is that of a minimization of the condensed non-convex free energy function with respect to a parametrized laminate microstructure. Again, this approach is referred to as relaxation, restricts the possible microstructure and neglects interface energy. Since the relaxation is applied to the condensed energy, this approach does include dissipation. Examples for applications of relaxation to single crystal plasticity can be found in Ortiz & Repetto [131], Carstensen et al. [28], miehe et al. [116], Kochmann [85] or Kochmann & Hackl [86]. An alternative approach that allows the formation of unrestricted microstructure and incorporates both interface energy and dissipation is obtained by a suitable slip gradient regularization together with a nonconvex hardening function formulated directly in the slips. Simplified examples for this approach in one dimension are given in Yalcinkaya et al. [156] and very similarly in Klusemann et al. [84]. A two-dimensional small strain approach with an isotropic gradient term is outlined in Yalcinkaya et al. [156]. The notion of geometrically necessary dislocations (GNDs) goes back to Cottrell [36] and Ashby [12] and is related to the lattice incompatibility arising due to an inhomogeneity of plastic deformation. Based on the ideas of Kondo [93, 94], Nye [130], and.

(23) 1.1 Motivation and State of the Art. 5. ¨ ner [97], Cermelli & Gurtin [29] have suggested a physically sound macroscopic Kro measure for the density of GNDs, namely the geometric dislocation density tensor, which is directly connected to the notion of the Burgers vector. Models that account for GNDs are generally connected to slip gradient contributions that introduce a certain length scale energetically, see e.g. Evers et al. [50], Ekh et al. [45], Kuroda & Tvergaard [99] and Svendsen et al. [149], or via the evolution of the critical resolved shear stresses, see e.g. Becker [21]. A model that uses the energetic approach and is directly based on the dislocation density tensor and thereby on the curl of the plastic deformation is proposed in Gurtin [66]. In this work, we propose a new gradient crystal plasticity model capable of predicting the formation and evolution of plastic laminates and the length scale effects induced by geometrically necessary dislocations in a single crystal with two active slip systems on the same slip plane, see also Miehe et al. [113] and Hildebrand & Miehe [73]. The model is based on a double slip version of the multi-slip framework outlined in Miehe et al. [117]. Consequently, both the dissipation potential that governs the evolution of the slips as well as the evolution equations of the accumulated slips are regularized by use of the approximation |x| ≈ ν ln(cosh(x/ν)) for ν ≪ 1 and the model contains a gradient term that is inspired by the model of Gurtin [66] and based on the dislocation density tensor as introduced in Cermelli & Gurtin [29]. The latent hardening is modeled by hardening function that is a nonconvex in the accumulated slips that favors single slip over multi-slip and constitutes a biquadratic version of the function proposed by Hutchinson [77]. The laminates triggered by the non-convexity of the hardening function are regularized using an additional gradient term. In summary, the main features of the model are: • Rate regularized slip evolution: The approximation |x| ≈ ν ln(cosh(x/ν)) for ν ≪ 1 allows the approximative implementation of slip evolutions with thresholds without the use of active sets. • Rate regularized accumulated slip evolution: Rate regularization of the evolution of the accumulated plastic slips allows setup with only one slip per slip system. • Latent hardening with realistic features: Non-convex latent hardening contribution that punishes multi-slip over single slip states and triggers laminates. • Physically motivated modeling of geometrically necessary dislocations: Modeling of GNDs by use of the dislocation density tensor induces physical length scale. • Smooth approximation of laminate interfaces: Smooth approximation of sharp interfaces in plastic laminate microstructure using artificial length scale. • Variational nature: Continuous rate-type and finite-step-sized incremental-variational principles for the gradient crystal plasticity model. • Symmetric FE solver: Variational time- and space-discrete finite element formulation, resulting in symmetric algebraic structures for the monolithic problem. • Combined modeling of plastic laminates and GNDs: Unique combined prediction of the formation and evolution of laminate microstructure and the effect of GNDs..

(24) 6. Introduction. 1.2. Objectives and Overview The general aim of the presented thesis is the variational-based gradient-extended multifield formulation of models for the formation and evolution of laminate microstructure based on a thorough understanding of both the underlying mathematical structure and the involved physical effects. In summary, the major contributions of this work are • the derivation of modeling, variational and algorithmic frameworks for gradientextended standard-dissipative solids in combination with a new rate-regularization of non-smooth dissipation potentials with threshold. • the understanding of the physics of non-material interfaces with interface energy. • the construction of gradient-extended models for laminate microstructure based on anisotropic gradient terms, non-convexity in the internal variables and a bulk-energyfree rank-one connection of the deformation states associated with these minima. In particular, we propose modeling frameworks for two cases of laminates, i.e. for • the energetically consistent modeling of the formation and evolution of laminate microstructure in martensitic CuAlNi with coherence-dependent interface energy. • the modeling of laminate microstructure in same plane double slip f.c.c. Copper using a new rate regularization for the evolution of accumulated plastic slips and allowing a combination of plastic laminates with GND-length scale effects. Following this introduction, Chapter 2 outlines the continuum mechanical fundamentals and the employed notation that will be the basis for all further considerations. In particular, we introduce the employed large strain formulation, derive a canonical balance principle, comment on our assumptions of quasistatic and isothermal conditions and derive the balance principles and the dissipation inequality. In Chapter 3, we present a consistent derivation of our modeling framework. Starting from the assumptions of quasistatic and isothermal conditions, we introduce internal variables, apply the standard constitutive principles and obtain the framework of gradient-extended standard-dissipative solids, for which we identify the free energy and the dissipation potential as the necessary constitutive functions. For both functions, we present restrictions and some useful modeling ingredients. Specifically, we propose a new rate regularization for dissipation potentials representing evolutions with thresholds. Chapter 4 presents the initial boundary value problem of gradient-extended standarddissipative solids that is governed by the balance of linear momentum and a partial differential evolution equation of the internal variable. The initial boundary value problem is expressed in terms of a continuous rate-variational formulation. Time-discretization allows the derivation of a consistent incremental variational framework and space-discretization yields a finite element formulation with symmetric structure for the monolithic solution of the coupled problem of deformation and gradient-extended internal variable evolution. In Chapter 5, we consider the kinematics of non-material sharp interfaces and state the resulting condition of kinematic compatibility. We derive canonical and physical jump conditions in analogy to the balance equations of Chapter 2 taking into account interfacial energy. The dissipation inequality of the moving interface links the interface kinetics to the Eshelby tensor and imposes a restriction on the interface velocity..

(25) 1.2 Objectives and Overview. 7. Chapter 6 outlines the origin of laminate microstructure based on an elastostatic variational approach that does not account for interface energy. Well-known analytical solutions for boundary value problems with a two-well energy in one and a rank-one connected two-well energy in three dimensions are presented. Specifically, the three dimensional example will be the basis for the numerical initial boundary value problems in Chapters 8 and 9. The main ingredients to the formation of microstructure and the deficiencies of the discussed approach are outlined and a number of modeling approaches for (laminate) microstructure are discussed. The phase field approach is identified as the only approach that resolves the microstructure and includes both interface energy and dissipation. In Chapter 7, we summarize the important properties of CuAlNi relevant for the constitutive modeling of the formation and evolution of martensitic laminate microstructure. This includes the introduction of the Cauchy-Born hypothesis and symmetry properties of lattices, the cubic to orthorhombic transformation that CuAlNi undergoes between austenite and martensite and the existence of martensitic variants. We then state the basic properties of the free energy of CuAlNi, i.e. the rank-one connection of its martensitic minima, the capability to form twins and the loss of quasiconvexity. Recalling the analytical considerations of Chapter 6, we expect the formation of laminate microstructure, which is indeed observed in numerous experiments. We summarize the main ingredients related to coherence-dependent interface energy and dissipative interface motion. Chapter 8 outlines an application of the modeling framework of gradient-extended standard-dissipative solids: a new phase field model for the modeling of the formation and evolution of martensitic laminate microstructure in two-variant CuAlNi. The model is based on a variational approximation of sharp interface geometries and incorporates both a coherence-dependent interface energy and dissipative interface evolution. We consider two approaches to the construction of the elastic bulk energy (external and internal mixing), state the initial boundary value problem, derive suitable rate-variational and incremental-variational principles and construct a finite element formulation. Finally, we show that the model is capable of predicting the formation and evolution of laminate microstructure under the boundary conditions considered in Chapter 6. In particular, we show that the internal mixing leads to an energetically concise separation between bulk and interface energy and is thus suitable for the prediction of size effects. Finally, in Chapter 9 we outline a second application of the modeling framework of gradient-extended standard-dissipative solids: we derive a new model for the formation and evolution of laminate deformation microstructure in same plane double slip gradient crystal plasticity. The model is based on the continuum slip theory and accounts for geometrically necessary dislocations (GNDs) by use of the dislocation density tensor. The effect of latent hardening is accounted for by the introduction of accumulated plastic slips and a biquadratic non-convex latent hardening term that favors single slip over multi-slip states. The resulting sharp laminate interfaces are regularized by a gradient term acting in the direction of the laminate interface normals and the evolution is prescribed by a dissipation potential combining a threshold with a viscous contribution. Both the non-smooth evolution of the accumulated slips as well as the non-smooth dissipation potential are regularized using the approximation |x| ≈ ν ln(cosh(x/ν)) for ν ≪ 1. Numerical examples demonstrate the capability of the model to predict the formation and evolution of laminate microstructure. In particular, the model allows a simulation of regularized laminate microstructure in combination with length scale effects stemming from GNDs..

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(27) 9. 2. Continuum Mechanics In this chapter, a short introduction to the fundamentals of continuum mechanics will be given. This is not meant to be comprehensive, but intended to introduce the notation and lay the foundations for the considerations in the remainder of this work. The results ¨ ller [123], Holzapfel [74], are summarized in Box 1. For further reading, see e.g. Mu ˇ ´ [146] or Truesdell & Noll [152]. Note that we do not address Liu [104], Silhav y the often very helpful connection to differential geometry here, which can be found e.g. in Miehe [108, 109] or Marsden & Hughes [106].. 2.1. Kinematics of Finite Deformation We start our considerations by discussing the basic kinematic quantities used to describe and analyze the motion and deformation of solids at finite strain. 2.1.1. Description of Motion Basis for the kinematic description of motion at finite strain is the introduction of material bodies, their configurations and the associated deformation map. 2.1.1.1. Definition of a Material Body. A material body B is a physical object with properties such as color, texture or mass density. Mathematically, the body B consists of a fixed set of an infinite number of particles P ∈ B that occupy a certain region Bt ⊂ R3 in the Euclidean space R3 at time t ∈ R+ . 2.1.1.2. Material Description of the Motion of a Body. At a given point in time t ∈ T = [to , tΩ ] ⊆ R+ , each particle P ∈ B is mapped to a Euclidean point x ∈ Bt ⊂ R3 by the placement χt . The domain resulting from the placement of the body B in the Euclidean space Bt ⊂ R3 is called a configuration of the body B. χt is defined by χt :. . B × T → Bt ⊂ R3 P 7→ x = χt (P ) ,. (2.1). P ∈B χt1 χt2. χt3. 3. R. xt1 Bt1. xt2 Bt2. χt4. xt3 Bt3. xt4 Bt4. Figure 2.1: The motion of a material body B in the Euclidean space R3 is a series of placements χt of the physical body B by a family of configurations χt : B → Bt ⊂ R3 ..

(28) 10. Continuum Mechanics. B. P ∈B χt. χ0 R3. ϕt (X) X∈B. x∈S. B. S. Figure 2.2: Placements of the physical body B in the Euclidean space R3 . Reference configuration B with positions X and current configuration S with positions x at time t.. see Figure 2.1. Since χt is a one-to-one mapping, it is invertible and excludes interpenetration of matter. We can thus both say that at time t, the particle P ∈ B occupies the point x = χt (P ) ∈ Bt or that at time t, the point x ∈ Bt is occupied by the particle P = χ−1 t (x) ∈ B. The set of subsequent placements χt in a given interval of time T ⊂ R+ defines a set of subsequent configurations of B and thereby completely describes the motion of the body B. 2.1.1.3. Point Map. To be able to identify the particles P ∈ B, we label them by their position as defined by an arbitrary reference placement χo . This placement defines a reference or material position X for each particle P and places the body B in its reference or Lagrangian configuration B . It is defined by B → B ⊂ R3 (2.2) χo : P 7→ X = χo (P ) , We can now identify any particle P by its referential position X using the inverse reference placement P = χ−1 o (X) ∈ B. Denoting the current or spatial position of a particle P at time t by x and the current or Eulerian configuration of the body B at time t by S, (2.1) and (2.2) allow the establishment of the relation −1 x = χt (P ) = χt χ−1 (2.3) 0 (X) = χt ◦ χ0 (X) =: ϕt (X) , where f ◦ g denotes the composition of the two functions f and g. The function B × T → S ⊂ R3 ϕt : (X, t) 7→ x = ϕt (X). (2.4). is referred to as the deformation map or point map. It maps the referential position X ∈ B of a particle P ∈ B to its current position x ∈ S at time t ∈ T ⊂ R+ , see Figure 2.2. Usually, the reference configuration B ⊂ R3 is chosen to be an unloaded (stress-free) natural reference state. To distinguish reference and spatial quantities also in index notation, we introduce the orthonormal Cartesian reference basis E A and the orthonormal Cartesian spatial basis ea . We can then e.g. express reference positions, spatial positions and the deformation map by X = XA E A ,. x = xa ea. and ϕt = ϕa ea ,. (2.5).

(29) 11. 2.1 Kinematics of Finite Deformation ϕt (X) x. X F t (X) Tˆ B. ˆt ˆ C(Θ). ˆ(Θ) c Θ. S. R. Figure 2.3: The deformation gradient F t maps material tangent vectors Tˆ at Lagrangian ˆ ˆ(Θ). curves C(Θ) onto spatial tangent vectors ˆt = F t Tˆ at deformed material curves c. where equal Latin indices imply summation over {1, 2, 3} according to the summation convention. 2.1.2. Deformation Gradient The material gradient (the gradient with respect to the reference positions X ∈ B) of the deformation map (2.4) is called the deformation gradient. It is defined as F t (X) := ∇X ϕt (X) = Grad(ϕt ) =. ∂ϕa ea ⊗ E B = ϕa,B ea ⊗ E B , ∂XB. (2.6). where ∇X (·) and Grad(·) denote the referential gradient of (·) with respect to X, where ⊗ defines the dyadic product and where ϕa,B is shorthand for the partial derivative of ϕa with respect to XB . Since ϕt constitutes a one-to-one mapping between reference and current positions, the deformation gradient is invertible, yielding the property that det(F t ) 6= 0. And since negative values of det(F t ) would be associated with the interpenetration of matter, the determinant must also be non-negative. Together we have det(F t ) > 0 .. (2.7). Throughout this work, the deformation gradient F t will play the central role in the description of local strains during deformation. ˆ 2.1.2.1. Tangent Map. Consider a material curve C(Θ) ∈ B and the associated deˆ(Θ) ∈ S that are both parametrized by Θ ∈ R, and point to equal formed spatial curve c particles P ∈ B for equal arguments Θ, see Figure 2.3. If the curves are differentiable ˆ with respect to Θ, the tangent vector Tˆ to the material curve at C(Θ) is given by the d ˆ ˆ ˆ ˆ(Θ) derivative T = dΘ C(Θ) and the tangent vector t to the deformed spatial curve at c d ˆ ˆ(Θ). Since the curves are imprinted on the same particles, the points on the by t = dΘ c ˆ ˆ(Θ) = ϕt (C(Θ)). two curves are connected by c Using the chain rule and (2.6) yields d ˆ d ˆ ˆt = d c ˆ(Θ) = = ∇X ϕt · C(Θ) = F t · Tˆ . ϕt C(Θ) dΘ dΘ dΘ. (2.8). From (2.8) it can be seen that the deformation gradient F t linearly maps material tangent vectors to associated deformed spatial tangent vectors. It is thus also referred to as the tangent map. Note that (2.8) generally does not preserve the length of Tˆ ..

(30) 12. Continuum Mechanics ϕt (X) n. N X. F −T t (X). x. B C(X) = const.. S c(x) = const.. Figure 2.4: The inverse transpose of the deformation gradient F −T maps material normal t vectors N at Lagrangian isosurfaces C(X) onto spatial normal vectors n = F −T t N at deformed isosurfaces c(x).. 2.1.2.2. Normal Map. Consider a material scalar density function C(X) ∈ R and an associated deformed spatial density function c(x) ∈ R that assign equal scalar values to equal particles P ∈ B and are both differentiable with respect to X and x, respectively. The normal vector N to the material level set C(X) = const. at X ∈ B is then given by N = ∇X C(X), the normal vector n to the associated deformed level set c(x = ϕt (X)) = const. at x ∈ S by n = ∇x c(x), see Figure 2.4. Since the two density functions yield equal values for equal particles, they are connected by C(X) = c(ϕt (X)). Using the chain rule and (2.6), we can use this to write N = ∇X C(X) = ∇X c ϕt (X) = ∇x c(x) · ∇X ϕt (X) = n · F t ⇔ n = F −T t N . (2.9). linearly maps material normal vectors to From (2.9) it can be seen that the tensor F −T t associated deformed spatial normal vectors. It is thus also referred to as the normal map. Note that (2.9) generally does not preserve the length of N . ˆ 2.1.2.3. Area map. Now consider two intersecting material curves C(Θ) ∈ B and ¯ ˆ(Θ) ∈ S and C(Θ) ∈ B and the two associated deformed intersecting spatial curves c ˆ ¯ ¯(Θ) ∈ S. At the intersection of C(Θ) and C(Θ), the cross product of the two material c tangent vectors Tˆ and T¯ defines a material area vector A = Tˆ × T¯ , while at the inter¯(Θ), the cross product of the two spatial tangent vectors ˆt and ¯t section of cˆ(Θ) and c defines the associated deformed area vector a = ˆt × ¯t. From Section 2.1.2.1 we know that the tangent vectors satisfy the relations ˆt = F t Tˆ and ¯t = F t T¯ . Now recall the definition of the determinant of a tensor B ∈ R3×3 by the relation (Bu) · (Bv) × (Bw) = det[B]u · (v × w),. ∀u, v, w ∈ R3 .. (2.10). Equation (2.10) can be rewritten to yield u · [B T (Bv × Bw)] = u · [det(B)(v × w)]. Since this should hold ∀u ∈ R3 , we obtain B T (Bv ×Bw) = det(B)(v ×w), ∀v, w ∈ R3 . Setting B = F t , multiplying with F −T and choosing v = Tˆ and w = T¯ we can write t ˆ ¯ a = ˆt × ¯t = (F t Tˆ ) × (F t T¯ ) = det(F t )F −T t T × T = cof(F t )A .. (2.11). From (2.11) it can be seen that material area vectors are mapped to the associated deformed spatial area vectors by the cofactor of the deformation gradient cof(F t ) = det(F t )F −T t . The cofactor of F t is thus also referred to as the area map. The area map is also often used to relate infinitesimal area elements, yielding da = cof(F t ) dA..

(31) 2.2 Integral Identities and Canonical Balance Principle. 13. ˆ 2.1.2.4. Volume map. Finally, consider three material material curves C(Θ) ∈ B, ¯ ˜ C(Θ) ∈ B and C(Θ) ∈ B intersecting at a single point X ∈ B and also consider the three ˆ(Θ) ∈ S, c ¯(Θ) ∈ S and c ˜(Θ) ∈ S that intersect at associated deformed spatial curves c ˆ ¯ ˜ x = ϕt (X) ∈ S. At the intersection of C(Θ), C(Θ) and C(Θ), the triple scalar product ˆ ¯ ˜ between the tangents T , T and T defines a material volume V = Tˆ · (T¯ × T˜ ), while at the ˆ(Θ), c ¯(Θ) and c ˜(Θ), the associated deformed spatial volume is defined by intersection of c v = ˆt · (¯t × ˜t). From Section 2.1.2.1 we know that the tangent vectors satisfy the relations ˆt = F t Tˆ , ¯t = F t T¯ and ˜t = F t T˜ . Recalling (2.10), we can use this to write v = ˆt · [¯t × ˜t] = (F t Tˆ ) · (F t T¯ ) × (F t T˜ ) = det(F t ) Tˆ · [T¯ × T˜ ] = det(F t ) V . (2.12). From (2.12) it can be seen that the determinant of the deformation gradient det(F t ) linearly maps material volumes to associated deformed spatial volumes. The determinant of F t is thus also referred to as the volume map. The volume map is also often used to relate infinitesimal volume elements, yielding dv = cof(F t ) dV . 2.1.3. Polar Decomposition Locally, the deformation gradient defined in (2.6) can be uniquely decomposed into a rotation and a stretch according to F t = RU = V R ,. (2.13). where R ∈ SO(3) is the proper orthogonal rigid rotation tensor and where U = U T and V = V T are the symmetric and positive definite material stretch tensor and spatial stretch tensor, respectively.. 2.2. Integral Identities and Canonical Balance Principle As a basis for the derivation of the physical balances in Section 2.3, we now consider some integral identities and formulate a canonical balance principle that will be specified in the ¨ ller [123] and will be specifically sequel. The employed notation follows the work of Mu useful in the context of balance principles in bodies with sharp interfaces, see Section 5. 2.2.1. Rate of Volume Integrals Let ξVX (X, t) be a continuous material density field quantity per unit reference volume defined in the material domain BP ⊆ B. Then the related quantity XBP associated with the domain BP and the rate of this quantity dtd XBP are given by the relations Z Z Z d d X X ξV (X, t) dV = ξ˙VX (X, t) dV , XBP = ξV (X, t) dV ⇒ X˙BP = XBP = dt dt BP BP BP (2.14) d ˙ where we denote by (·) = dt (·) the total time derivative of (·). Equation (2.14) exploits the fact that BP ⊆ B is a material domain that is connected to a set of particles. Since these particles have a fixed reference placement χo , the domain BP does not change in time and we can interchange time derivatives and integration over BP . Note that (2.14) works for ξVX , X ∈ R, for ξVX , X ∈ R3 and for higher dimensional tensors. 2.2.2. Divergence Theorem Let f (X, t) be a continuously differentiable material vector field quantity per unit reference area defined in the domain BP ⊆ B. Then the divergence theorem relates the flux.

(32) 14. Continuum Mechanics. of f through the surface ∂BP of BP to the behavior of f inside BP by Z Z Z f (X, t) · dA = f (X, t) · N dA = Div f (X, t) dV , ∂BP. ∂BP. (2.15). BP. where N is the outward normal on ∂BP and where Div f (X, t) = f (X, t) · ∇X is the right divergence of f (X, t). Note that the divergence theorem (2.15) works for f ∈ R3 , f ∈ R3×3 and higher dimensional tensors. 2.2.3. Canonical Balance Principle As a starting point for the balance principles in Section 2.3, we now construct a canonical balance principle for the material field density ξVX (X, t). To do so, we consider a material domain BP ⊆ B and formulate the balance of the global quantity XBP as defined in (2.14). Assuming that the flux of X over the boundary ∂BP is given by f X (X, t), the production X of X in BP is given by pX V (X, t) and the supply of X in BP is given by sV (X, t), we can write down the balance equation Z Z Z Z d d X X X X˙BP = XBP = ξ dV = sV dV + pV dV + f X · N dA , (2.16) dt dt BP V } | BP {z } | BP {z } | ∂BP {z supply in BP. production in BP. flux through ∂BP. X where the difference between supply sX V and production pV is that the supply can be controlled from the outside (e.g. by application of external volume forces) and where we X X X 3 can use sX V , pV ∈ R, sV , pV ∈ R and higher dimensions. Application of relation (2.14) and the divergence theorem (2.15) yields Z Z X BP →dV ˙X X X X X ξ˙V dV = sV + pX ⇒ ξ V = sX (2.17) V + Div(f ) dV V + pV + Div(f ) , BP. BP. where (2.17)2 can be deduced as (2.17)1 has to hold for all subdomains BP ⊆ B such that BP can be shrunk to an infinitesimal volume dV around any given point X ∈ B. This argument that allows the construction of a local form (2.17)2 from a global form (2.17)1 is called localization and is very useful in the context of the derivation of balance principles.. 2.3. Physical Balance Principles We now specify the canonical balance principle introduced in 2.2.3 to the balances of mass, linear momentum, angular momentum, energy and entropy. To be able to specify the related supplies and fluxes, we first apply Euler’s cut principle to a subdomain BP and specify the relevant quantities acting on BP , specifically the stress and heat flux. 2.3.1. Isothermal and Quasistatic Conditions Throughout this work, we will assume that all occurring processes are isothermal and quasistatic. The assumption of isothermal conditions is based on the idea that the body B is surrounded by an infinite heat bath of constant temperature θ = θo such that the body will continuously try to reach the resulting equilibrium state θ(X, t) = θo through heat exchange with the heat bath. If this heat exchange and the heat conduction within the body occur much faster and the heat production much slower than all other relevant.

(33) 15. 2.3 Physical Balance Principles. ϕt (X). BP. SP. X γo. x. B. ∂BP dA N. BP. ϕt (X). q. dA X γo. S. ∂SP. da x. t T SP. ∂SP. ∂BP. da n. Figure 2.5: Euler’s cut principle: The material part BP ⊆ B cut out from the reference domain B and the corresponding deformed spatial part SP ⊆ S cut out from the current domain S. The mechanical action of the rest of the body is replaced by the traction t(x, t) acting on ∂SP , the thermal action is replaced by the heat flux Q(X, t) acting on ∂BP .. processes (e.g. mechanic deformation), we can assume a constant temperature distribution in the body, leading to the relations θ(X, t) ≈ θo = const.. ⇒. ˙ θ(X, t) ≈ 0 and. Grad θ(X, t) ≈ 0. in B .. (2.18). The assumption of quasistatic conditions used here is based on the idea that the mechanical loading is applied extremely slowly such that the body is always in mechanical equilibrium. As a consequence, the energy and the forces related to inertial effects will be much smaller than e.g. the energy and the forces related to the deformation of the body. We can hence neglect all contributions related to inertia, leading to ˙ tk ≈ 0 kϕ. ¨ tk ≈ 0 and kϕ. in B .. (2.19). It should be noted here that (2.19) does not rule out gradients of ϕ˙ t , such that e.g. F˙ t 6= 0 also in quasistatic processes. Furthermore, since we will account for dissipation where it occurs (i.e. during phase transformations or plastic deformation), the assumption of quasistatic processes does not imply reversibility (even though the reverse holds). To guarantee consistency, all derivations in this section are carried out taking into account both temperature and inertia and then specialized using (2.18) and (2.19). 2.3.2. Mechanical and Thermal Fluxes To introduce the notion of stress and heat flux, we consider a body B whose particles are mapped from referential positions X ∈ B to spatial positions x ∈ S by the point map ϕt (X). Applying Euler’s cut principle, we “cut out” a referential subdomain BP ⊆ B and the associated deformed spatial subdomain SP = ϕt (BP ) ⊆ S and consider the mechanical and thermal fluxes over the referential and spatial boundaries ∂BP and ∂SP . 2.3.2.1. Stress Tensors. To replace the mechanical action of the remainder of the body S \ SP on the cut out part SP , we introduce a spatial traction vector t(x, t) acting on its surface ∂SP , see Figure 2.5. The Cauchy theorem then states that at every point.

(34) 16. Continuum Mechanics. on the surface x ∈ ∂SP , the traction vector is a linear function of the spatial outward unit normal n(x, t) to ∂SP , t(x, t) = σ(x, t) n(x, t) ,. (2.20). where σ(x, t) is called the Cauchy stress. While t measures the current force per unit spatial area da, we can also introduce a nominal traction vector T parallel to t that measures the current force per unit reference area dA. For this nominal traction vector, we postulate a Cauchy-like theorem similar to (2.20), T (X, t) = P (X, t) N (X) ,. (2.21). where P (X, t) is called the first Piola-Kirchhoff stress and where N (X) is the outward reference unit normal to ∂BP . The link between the two stress tensors is established by noting that P N dA = σn da. Recalling (2.11) this yields P dA = σ da. ⇔. P dA = σ cof(F t ) dA. ⇔. P = σ cof(F t ) .. (2.22). Note that the Cauchy stress is often referred to as true stress, whereas the first PiolaKirchhoff stress is denoted as the nominal stress. 2.3.2.2. Heat Flux. To replace the thermal action of the remainder of the body B \ BP on the cut out part BP , we introduce a heat flux Q(X, t) per unit area of the reference surface ∂BP . We then assume that at every point on the surface X ∈ ∂BP , the heat supply is a linear function of the referential outward unit normal N (X) to ∂BP , Q(X, t) = q(X, t) · N (X) ,. (2.23). where q(X, t) is called the referential heat flux vector. In analogy to (2.21), q is sometimes ˇ ´ [146]. also called the Piola-Kirchhoff heat flux vector, see e.g. Silhav y 2.3.3. Balance of Mass The mass MBP of the material subdomain BP ⊆ B is defined in terms of the referential R mass density ρo (X) as MBP = BP ρo (X) dV . In the absence of mass supply, production and transport, the mass of a material subdomain is conserved during deformation. Recalling the canonical balance principle introduced in Section 2.2.3, we can state ξVM sM V pM V fM. = ρo =0 =0 =0. – – – –. density of mass M in BP , supply of mass M in BP , production of mass M in BP , flux of mass M over ∂BP ,. (2.24). where we confirm from pM V = 0 that the mass is a conserved quantity. From (2.16) and (2.17), we can deduce the global and local forms of the balance of mass, Z d d ρo (X, t) dV = 0 ⇒ ρ˙ o = 0 . (2.25) MBP = dt dt BP Equation (2.25) tells us that the mass MBP of any material subdomain as well as the referential mass density ρo (X) are constant. Considering the mass MSP of the related deformed spatial domain SP = ϕt (BP ) with spatial mass density ρ(x, t), we can write Z Z Z M SP = ρ(x, t) dv = ρ ϕt (X), t det(F t ) dV = ρo (X) dV = MBP , (2.26) SP. BP. BP.