Seven Lectures on Finite Elasticity

Abstract

This is the first of two introductory lectures on the equations of nonlinear elasticity theory and some example problems for both general and special material models. The kinematics of finite deformations is discussed by others, so we shall assume knowledge of the polar decomposition theorem and of various related deformation tensors. Relations essential to my presentation, however, will be recorded again as the need arises but without details. We shall begin with Euler’s laws of balance from which the Cauchy stress principle and Cauchy’s laws of motion are obtained. The Cauchy and engineering stress tensors are described. The theory of elasticity of materials for which there exists an elastic potential energy function is known as hyperelasticity. While much of our work emphasizes hyperelasticity theory, some results within the general theory of elasticity that do not require existence of a strain energy function will be noted here and there. The general constitutive equation for hyperelastic materials is derived from the mechanical energy principle. Implications of frame indifference and of material symmetry on the form of the strain energy function are sketched. This leads to constitutive equations for compressible and incompressible, isotropic hyperelastic materials. The empirical inequalities are introduced for use in subsequent applications. Discussion of special constitutive equations is reserved for another lecture.