Seven Lectures on Finite Elasticity
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 knowled
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LECTURE! INTRODUCTION TO NONLINEAR ELASTICITYTHE GENERAL EQUATIONS OF THE THEORY 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.
1 The Cauchy Stress Principle and Equations of Motion A continuum body, briefly a body B, is a contiguous set of material points called particles. A reference frame is a set cp = { 0; ei} consisting of an origin 0 and an orthonormal vector basis ei = {e 1 , e 2 , e 3 }. The motion of a particle P relative to cp is the time locus of its position vector x(P, t) in cp. A typical particle P may be identified by its position vector X(P) in cp at some reference time tR, say. The domain ""R of X, the region occupied by Bat the time tR, is called a reference configuration of B. Then, relative to cp, the motion of a typical particle P from ""R is described by the vector function x = x(X, t). The domain "" of x, the region occupied by B at the time t, is called the current configuration of B. Hence, x denotes the place in the current configuration"" at timet, that is occupied by the particle P whose place was at X in the reference configuration ""R initially. When no confusion may result, we write x for the function *Acknowledgment: Preparation of these lectures was partially funded by Grant No. CSM-9634817 from the National Science Foundation. M. Hayes et al. (eds.), Topics in Finite Elasticity © Springer-Verlag Wien 2001
M.F. Beatty
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X· The velocity and acceleration of a particle P relative to c.p are defined by v(X, t) x(X, t) and a(X, t) :: v(X, t) = x(X, t), respectively. As usual, · :8f8t denotes the material time derivative, the time rate of change following the particle P. We begin with Euler's laws of motion
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