FUNDAMENTAL ASPECTS OF DISLOCATION INTERACTIONS

By A. KILAE

1 Introduction 1 11 Motivation 1 12 Research Objectives and Originality 3 13 State of the Art 5 14 Outline of the Thesis 11 15 Notation, Frequently Used Acronyms, Symbols, and Operators 13 2 Review on Extended Continua 17 21 The Notion of Classical and Generalized Continua 18 211 Classical continua 18 212 Generalized continua 26 22 Methods Based on an Extended Energy Balance 28 221 Invariance of energy balance 28 222 Covariance of energy balance 44 23 Extended Principle of Virtual Power 49 231 Fundamentals 49 232 Extended principle of virtual power 51 233 Extended principle of virtual work 54 24 Variational Approaches 56 241 Fundamentals 56 242 Extended continua specified by an extended Lagrangian 56 25 Postulates 57 26 Interim Conclusions 58 3 On Invariance Properties of an Extended Energy Balance 61 31 Additional Scalar-Valued DOF 62 311 Extended energy balance 62 312 Translational transformation 63 313 Rotational transformation 65 314 Nonlocal evolution equation for an additional DOF 68 315 Connection to an extended principle of virtual power 72 32 Additional Vector-Valued DOF 74 321 Extended energy balance 74 322 Translational transformation 75 323 Rotational transformation 75 33 Interim Conclusions 79 4 On Interface Conditions on a Material Singular Surface 81 41 Balance Equations and Dissipation Inequality 82 411 Extended energy balance 82 412 Mathematical preliminaries 85 413 Invariance considerations of an extended energy balance 87 414 Dissipation inequality 90 415 Potential relations and boundary conditions at the singular surface 91 42 Connection to an Extended Principle of Virtual Power 96 421 Weak forms 96 422 Extended principle of virtual power 97 43 Interim Conclusions 98 5 Application to a Slip Gradient Crystal Plasticity 101 51 General Remarks on Crystal Plasticity 101 511 Deformation behavior 101 512 Crystal plasticity 102 52 Bulk Material - Regular Points 102 521 Application to a two-phase laminate material 102 522 Solution of the considered boundary value problem 105 523 Influence of the defect parameter and the initial hardening modulus on the solution 107 524 Orientation of the slip system 111 53 Grain Boundary - Singular Points 112 531 Application to a three-phase laminate material 112 532 Solution of the considered boundary value problem 115 533 Influence of the defect parameter, the grain width, and the grain boundary strength on the solution 118 54 Interim Conclusions 129 6 Summary and Conclusions 131