Solid Mechanics : Theory, Modeling, and Problems

"This textbook offers an introduction to modeling the mechanical behavior of solids within continuum mechanics and thermodynamics. To illustrate the fundamental principles, the book starts with an overview of the most important models in one dimension. Tensor calculus, which is called for in three-dimensional modeling, is concisely presented in the second part of the book. Once the reader is equipped with these essential mathematical tools, the third part of the book develops the foundations of continuum mechanics right from the beginning. Lastly, the book's fourth part focuses on modeling the mechanics of materials and in particular elasticity, viscoelasticity and plasticity. Intended as an introductory textbook for students and for professionals interested in self-study, it also features numerous worked-out examples to aid in understanding."--Page 4 of cover Preface 5 Contents 7 List of the Most Important Notations 10 Literature on the History of Mechanics 13 1 ONE-DIMENSIONAL MATERIAL THEORY 14 1.1 Deformations, Stresses, and Work 14 1.2 Elasticity 19 1.3 Viscoelasticity (Rheology) 22 1.3.1 The NEWTON Element 23 1.3.2 The MAXWELL Model 26 1.3.3 The KELVIN Model 30 1.3.4 The POYNTING Model 33 1.3.5 The BURGERS Model 35 1.3.6 Viscoelastic Models of Differential-Type 37 1.3.7 Viscoelastic Models of Integral-Type 38 1.3.8 Creep Damage 40 1.3.9 Fatigue 44 1.4 Plasticity 47 1.4.1 Rigid-Plastic Models 47 1.4.2 Elastic-Plastic Models 51 1.5 Viscoplasticity 53 2 INTRODUCTION TO TENSOR CALCULUS 56 2.1 Vector and Tensor Algebra 57 2.1.1 Summation Convention 57 2.1.2 Vectors 59 2.1.3 Dyads and Tensors 62 2.1.4 The Inverse of a Tensor 68 2.1.5 The Transpose of a Tensor 69 2.1.6 Square Forms and Tensor Surfaces 72 2.1.7 Cross-Product between Vectors and Tensors 72 2.1.8 Orthogonal Tensors 75 2.1.9 Transformations under Change of Basis 77 2.1.10 Eigenvalues and Eigenvectors 78 2.1.11 Spectral Forms of Symmetric Tensors 83 2.1.12 Time-Dependent Vectors and Tensors 94 2.1.13 Rigid Body Dynamics 95 2.1.14 Bending of Bars 102 2.1.15 Higher-Order Tensors 106 2.1.16 Tetrads 111 2.2 Vector and Tensor Analysis 117 2.2.1 The Directional Differential 117 2.2.2 The Nabla Operator 122 2.2.3 Cylindrical Coordinates 126 2.2.4 Integral Transformations 131 3 FOUNDATIONS OF CONTINUUM MECHANICS 134 3.1 Kinematics 134 3.1.1 Compatibility Conditions 147 3.2 Stress Analysis 153 3.2.1 The Principles of Mechanics 160 3.2.2 Stress Functions 167 4 THREE-DIMENSIONAL MATERIAL THEORY 177 4.1 Elasticity 178 4.1.1 Material Symmetry 182 4.1.2 Isotropic Elasticity 192 4.1.3 Projection Method 200 4.1.4 Identification of the Elastic Constants 203 4.1.5 Elastic Energy 205 4.1.6 Boundary Value Problems in Elastostatics 210 4.1.7 Variational Principles in Elastostatics 224 4.1.8 Displacement Functions 231 4.1.9 Wave Propagation in Elastic Media 257 4.2 Thermomechanics 262 4.2.1 Thermodynamic Balances 262 4.2.2 Thermoelasticity 265 4.2.3 Linear Thermoelasticity 268 4.2.4 Isotropic Linear Thermoelasticity 272 4.2.5 Boundary and Initial Value Problems of Thermoelasticity 274 4.3 Linear Viscoelasticity 279 4.4 Plasticity 284 4.4.1 Yield Criteria 285 4.4.2 Flow Rules 294 4.4.3 Hardening Rules 297 4.4.4 Consistency Condition 299 4.4.5 DRUCKER ́s Postulate 301 4.4.6 Thermoplasticity 314 5 INDEX 325 Front Matter....Pages i-xiii ONE-DIMENSIONAL MATERIAL THEORY....Pages 1-42 INTRODUCTION TO TENSOR CALCULUS....Pages 43-120 VECTOR AND TENSOR ANALYSIS....Pages 121-163 FOUNDATIONS OF CONTINUUM MECHANICS....Pages 164-311 Back Matter....Pages 312-318