Nonlinear Solid Mechanics a Continuum Approach for Engineering Gerhard A. Holzapfel Graz University of Technology, Austria With a modern, comprehensive approach directed towards computational mechanics, this book covers a unique combination of subjects at present unavailable in any other text. It includes vital information on 'variational principles' constituting the cornerstone of the finite element method. In fact this is the only method by which Nonlinear Solid Mechanics is utilized in engineering practice. The book opens with a fundamental chapter on vectors and tensors. The following chapters are based on nonlinear continuum mechanics - an inevitable prerequisite for computational mechanicians. In addition, continuum field theory (applied to a representative sample of hyperelastic materials currently used in nonlinear computations such as incompressible and compressible materials) is presented, as are transversely isotropic materials, composite materials, viscoelastic materials and hyperelastic materials with isotropic damage. Another central chapter is devoted to the thermodynamics of materials, covering both finite thermoelasticity and finite thermoviscoelasticity. Also included are:
* an up-to-date list of almost 300 references and a comprehensive index
* useful examples and exercises for the student
* selected topics of statistical and continuum thermodynamics.
Furthermore, the principle of virtual work (in both the material and spatial descriptions) is compared with two and three-field variational principles particularly designed to capture kinematic constraints such as incompressibility. All of the features combined result in an essential text for final year undergraduates, postgraduates and researchers in mechanical, civil and aerospace engineering and applied maths and physics.
Author(s): Gerhard A. Holzapfel
Publisher: John Wiley & Sons
Year: 2001
Language: English
Pages: 469
City: Chichester
1 Introduction to Vectors and Tensors 1 --
1.1 Algebra of Vectors 1 --
1.2 Algebra of Tensors 9 --
1.3 Higher-order Tensors 20 --
1.4 Eigenvalues, Eigenvectors of Tensors 24 --
1.5 Transformation Laws for Basis Vectors and Components 28 --
1.6 General Bases 32 --
1.7 Scalar, Vector, Tensor Functions 40 --
1.8 Gradients and Related Operators 44 --
1.9 Integral Theorems 52 --
2 Kinematics 55 --
2.1 Configurations, and Motions of Continuum Bodies 56 --
2.2 Displacement, Velocity, Acceleration Fields 61 --
2.3 Material, Spatial Derivatives 64 --
2.4 Deformation Gradient 70 --
2.5 Strain Tensors 76 --
2.6 Rotation, Stretch Tensors 85 --
2.7 Rates of Deformation Tensors 95 --
2.8 Lie Time Derivatives 106 --
3 Concept of Stress 109 --
3.1 Traction Vectors, and Stress Tensors 109 --
3.2 Extremal Stress Values 119 --
3.3 Examples of States of Stress 123 --
3.4 Alternative Stress Tensors 127 --
4 Balance Principles 131 --
4.1 Conservation of Mass 131 --
4.2 Reynolds' Transport Theorem 138 --
4.3 Momentum Balance Principles 141 --
4.4 Balance of Mechanical Energy 152 --
4.5 Balance of Energy in Continuum Thermodynamics 161 --
4.6 Entropy Inequality Principle 166 --
4.7 Master Balance Principle 174 --
5 Some Aspects of Objectivity 179 --
5.1 Change of Observer, and Objective Tensor Fields 179 --
5.2 Superimposed Rigid-body Motions 187 --
5.3 Objective Rates 192 --
5.4 Invariance of Elastic Material Response 196 --
6 Hyperelastic Materials 205 --
6.1 General Remarks on Constitutive Equations 206 --
6.2 Isotropic Hyperelastic Materials 212 --
6.3 Incompressible Hyperelastic Materials 222 --
6.4 Compressible Hyperelastic Materials 227 --
6.5 Some Forms of Strain-energy Functions 235 --
6.6 Elasticity Tensors 252 --
6.7 Transversely Isotropic Materials 265 --
6.8 Composite Materials with Two Families of Fibers 272 --
6.9 Constitutive Models with Internal Variables 278 --
6.10 Viscoelastic Materials at Large Strains 282 --
6.11 Hyperelastic Materials with Isotropic Damage 295 --
7 Thermodynamics of Materials 305 --
7.1 Physical Preliminaries 306 --
7.2 Thermoelasticity of Macroscopic Networks 311 --
7.3 Thermodynamic Potentials 321 --
7.4 Calorimetry 325 --
7.5 Isothermal, Isentropic Elasticity Tensors 328 --
7.6 Entropic Elastic Materials 333 --
7.7 Thermodynamic Extension of Ogden's Material Model 337 --
7.8 Simple Tension of Entropic Elastic Materials 343 --
7.9 Thermodynamics with Internal Variables 357 --
8 Variational Principles 371 --
8.1 Virtual Displacements, Variations 372 --
8.2 Principle of Virtual Work 377 --
8.3 Principle of Stationary Potential Energy 386 --
8.4 Linearization of the Principle of Virtual Work 392 --
8.5 Two-field Variational Principles 402 --
8.6 Three-field Variational Principles 409.
