Introduction

So far we have studied the kinematics of deformation, the description of the state of stress and three basic conservation laws of continuum mechanics, viz., the conservation of mass (eqn. 3.11.8), the conservation of linear momentum (eqn. 4.1.14), and the conservation of moment of momentum (eqn. 4.1.15). All these relations are valid for every continuum, indeed no mention was made of any material in their derivations.

These equations are, however, not sufficient to describe the response of a body to a given loading. We know from experience that under the same loading conditions, identical specimens of steel and rubber deform differently. Furthermore, for a given body, the deformations vary with the loading conditions. For example, for moderate loadings, the deformation in steel caused by the application of loads disappears with the removal of loads. This aspect of the material behavior is known as elasticity. Beyond a certain loading, there will be permanent deformations, or even fracture exhibiting behavior quite different from that of elasticity. In this chapter, we shall study an idealized linear elastic material for which the stress-strain relationship is linear. Using this stress-strain law, we will then study some dynamic and static problems.

Henceforth in this chapter we will assume that the deformations are small so that $e_{AB}$ is an adequate measure of strain.