Chapter 2 - Deformation: Displacements and Strains

Abstract

Abstract We begin development of the basic field equations of elasticity theory by first investigating the kinematics of material deformation. As a result of applied loadings, elastic solids will change shape or deform, and these deformations can be quantified by knowing the displacements of material points in the body. Under the continuum hypothesis, this concept will establish a displacement field at all points within the elastic solid. Using appropriate geometry, particular measures of deformation can be constructed leading to the development of the strain tensor. As expected, the strain components are related to the displacement field. The purpose of this chapter is to introduce the basic definitions of displacement and strain, establish relations between these two field quantities, and finally investigate requirements to ensure single-valued, continuous displacement fields. As appropriated for linear elasticity, these kinematical results will be developed under the conditions of small deformation theory. Developments in this chapter will lead to two fundamental sets of field equations: the strain–displacement relations and the compatibility equations.