Introduction

In the previous chapter we introduced the idea of small deformation, which allowed us to neglect the distinction between the Lagrangian and the Eulerian description. Now we will apply the small-deformation hypothesis to the equation of motion (3.5.3). The resulting equation will include spatial derivatives of the stress tensor, the acceleration of the displacement, and body forces. The displacement, in turn, is related to the strain tensor via (2.4.1). Therefore, we have two systems of equations, one for stress and displacement and one for strain and displacement. Within the approximations that have been introduced, these equations are valid for any continuous medium. To apply them to a specific type of medium (e.g., solid, viscous fluid) it is necessary to establish a general relation (known as a constitutive equation) between stress and strain.

In the case of solids, when a body is subjected to external forces it becomes deformed (strained), and internal stresses are generated within the body. The relation between stress and strain depends on the nature of the deformation and other external factors, such as the temperature. If the deformation is such that the deformed body returns to its original state after the force that caused the deformation is removed, then the deformation is said to be elastic. If this is not the case, i.e., if part of the deformation remains, the deformation is known as plastic.