Material Inhomogeneities in Elasticity

Self contained, this book presents a thorough introduction to the complementary notions of physical forces and material (or configurational) forces. All the required elements of continuum mechanics, deformation theory and differential geometry are also covered. This book will be a great help to many, whilst revealing to others a rather new facet of continuum mechanics in general, and elasticity in particular. An organized exposition of continuum mechanics on the material manifold is given which allows for the consideration of material inhomogeneities in their most appropriate framework. In such a frame the nonlinear elasticity of anisotropic inhomogenous materials appears to be a true field theory. Extensions to the cases of electroelasticity and magnetelasticity are then straightforward. In addition, this original approach provides systematic computational means for the evaluation of characteristic parameters which are useful in various branches of applied mechanics and mathematical physics. This is the case for path-independent integrals and energy-release rates in brittle fracture, the influence of electromagnetic fields on fracture criteria (such as in ceramics), the notion of momentum of electromagnetic fields in matter in optics, and the perturbation of solitons propagating in elastic dispersive systems.

Gerard Maugin is Director of Research at CNRS, Paris, and teacher at the Universite Pierre et Marie Curie (Paris VI), France.

Preface -- 1 Newton's concept of physical force -- 1.1. Newton's viewpoint -- 1.2. D' Alembert's viewpoint -- 1.3. Point particles and continua 7 -- 1.4. The modern point of view: duality -- 1.5. Lagrange versus Euler -- 2 Eshelby's concept of material force -- 2.1. Ideas from solid state physics -- 2.2. Peach-Koehler force -- 2.3. Force on a singularity -- 2.4. Energy-release rate -- 2.5. Pseudomomentum -- 2.6. Relationship with phonon and photon physics -- 3 Essentials of nonlinear elasticity theory -- 3.1. Material continuum in motion -- 3.2. Elastic me----sures of strains -- 3.3. Compatibility of strains -- 3.4. Balance laws (Euler-Cauchy) -- 3.5. Balance laws (Piola-Kirchhoff) -- 3.6. Constitutive equations -- 3.7. Concluding remarks -- 4 Material balance laws and inhomogeneity -- 4.1. Fully material balance laws -- 4.2. Material inhomogeneity force and pseudomomentum -- 4.3. Interpretation of pseudomomentum -- 4.4. Four formulations of the balance of linear momentum -- 4.5. Other material balance laws -- 4.6. Comments -- 5 Elasticity as a field theory -- 5.1. Elements of field theory -- 5.2. Noether's theorem -- 5.3. Variational formulation (direct-motion description) -- 5.4. Variational formulation (inverse-motion description) -- 5.5. Other material balance laws -- 5.6. Canonical Hamiltonian formulation -- 5.7. Balance of total pseudomomentum -- 5.8. Nonsimple materals: second-gradient theory -- 5.9. Complementary-energy variational principle -- 5.10. Peach-Koehler force revisited -- 5.11. Concluding remarks -- 6 Geometrical aspects of elasticity theory -- 6.1. Material uniformity and inhomogeneity -- 6.2. Eshelby stress tensor -- 6.3. Covariant material balance law of momentum -- 6.4. Continuous distributions of dislocations -- 6.5. Variational formulation using two variations -- 6.6. Second-gradient theory -- 6.7. Continuous distributions of disclinations -- 6.8. Similarity to Einstein-Cartan gravitation theory -- 7 Material inhomogeneities and brittle fracture -- 7.1. The problem of fracture -- 7.2. Generalized Reynolds and Green-Gauss theorems -- 7.3. Global material force -- 7.4. J-integral in fracture -- 7.5. Dual I-integral in fracture -- 7.6. Variational inequality: fracture propagation criterion -- 7.7. Other material balance laws and related path-independent integrals -- 7.8. Remark on the dynamical case -- 8 Material forces in electromagnetoelasticity -- 8.1. Electromagnetic elastic solids -- 8.2. Reminder of electromagnetic equations -- 8.3. Material electromagnetic fields -- 8.4. Variational principles -- 8.5. Balance of pseudomomentum and material forces -- 8.6. Fracture in electroelasticity and magnetoelasticity -- 8.7. Geometrical aspects: material uniformity -- 8.8. Electric Peach-Koehler force -- 8.9. Example of application: piezoelectric ceramics -- 9 Pseudomomentum and quasi-particles -- 9.1. Pseudomomentum of photons and phonons -- 9.2. Electromagnetic pseudomomentum -- 9.3. Conservation laws in wave theory -- 9.4. Conservation laws in soliton theory -- 9.5. Sine-Gordon systems and topological solitons -- 9.6. Boussinesq crystal equation and pseudomomentum -- 9.7. Sine-Gordon-d'Alembert systems -- 9.8. Nonlinear Schrodinger and Zakharov systems -- 10 Material forces in anelastic materials -- 10.1. Internal variables and dissipation -- 10.2. Balance of pseudomomentum -- 10.3. Global material forces -- Bibliography and references -- Index .