Chapter 4 - Force and Stress

Abstract

Chapter 3 investigated the kinematics of continuum deformation regardless of the force or stress distribution that produced the motion or deformation. In this chapter, we now wish to examine how these forces and stresses can be quantitatively described. Following the classical continuum mechanics model, we assume a continuously distributed internal force system composed of body and surface forces. Each of these will be associated with a continuous density function that will represent the force per unit volume or per unit surface area. For surface forces, this will lead to the definition and use of the stress or traction vector and stress tensor. Each of these provides a quantitative method to describe both boundary and internal force distributions within a continuum. Since stress is related to force per unit area, we will have to keep track of whether we wish to use reference or current areas for large deformation problems. Stress is very important in continuum mechanics applications, because many materials exhibit some type of failure condition based on this variable. It should be noted that the developments in this chapter will not require a material constitutive assumption and thus they will apply to a broad class of material behavior.