In the previous chapter we investigated the ways in which a continuous body can deform and move. These deformations and motions are induced by three types of forces acting on the body, as explained in § 4.1. This chapter focuses on the quantification of the internal contact force, FC, and its relation to the stress tensor, denoted by S. The stress tensor describes the state of stress within a continuous body as explained in § 4.2; it is the dynamic counterpart to the strain and rate-of-strain tensors of kinematics.

In the study of kinematics and dynamics, we have many choices. For example, in kinematics we are free to choose the location and orientation of our reference coordinate system. This system may be inertial or non-inertial; it may be arbitrarily oriented or aligned with the principal axes of the strain or rate-of-strain tensor. Similarly, in dynamics we are free to define what is meant by a body. Most of the time, we think of a body as a complete physical entity, with a physically defined boundary. However, we are free to define as a “body” any arbitrary portion of a physical body, such as the parcel introduced in § 1.2. We will use this freedom to our advantage in what follows. In the process of defining an arbitrary portion of a physical body as a “body,” we must of necessity define its surfaces. The stresses on these defined surfaces are manifestations of the internal state of stress of the body at that location.

Forces

The deformations and motions of a continuous body are induced by three types of forces acting on the body: