1. Introduction to Mechanics of Anisotropic Materials

This book is focused on constitutive description of mechanical behavior of engineering materials: both conventional (e.g., polycrystalline homogeneous isotropic or anisotropic metallic materials) and nonconventional ones (e.g., heterogeneous multicomponent usually anisotropic composite materials) fabricated by modern material engineering. Effective material properties at the macrolevel depend on both the material microstructure (isotropic or originally anisotropic in general case) and on dissipative phenomena occurred on fabrication and consecutive loading phase resulting in irreversible microstructure changes (acquired anisotropy). The material symmetry is a background and anisotropy is a core around which the book is formed. In this way a revision of classical rules of enhanced constitutive description of materials is required. The aim of this introductory chapter lies in providing, apart from classical definitions of tensor single invariants, also the choice of state variables necessary to describe irreversible microstructure changes accompanying coupled dissipative phenomena, and basic definitions of common invariants of either two second-order tensors (e.g., stress/strain and damage tensors) or two different-order tensors (e.g., stress/strain and fourth-order structural tensors). Concise classification of anisotropic materials with respect to symmetry of elastic matrices as referred to the crystal lattice symmetry is given, and extended analogy between symmetries of constitutive material matrices (elastic and yield/failure) is also discussed. Next, strain and complementary energy as function of either stress/strain invariants (initial elastic isotropy) or common stress/strain—damage invariants (damage acquired anisotropy) are mentioned. Constitutive equation of linear elasticity in terms of common invariants of strain and structural orthotropic tensors is given. The scope of this chapter provides necessary tools for more extended constitutive description of materials which exhibit either virgin anisotropy or damage or phase change acquired anisotropy following microstructure changes.