Chapter 3 - Stress and Equilibrium

Abstract

We now examine the force or stress distribution within the elastic solid. Our study will lead to the definition and use of the traction vector and stress tensor. Each of these provides a quantitative method to describe both boundary and internal force distributions within a continuum solid. Since it is commonly accepted that maximum stresses are a major contributing factor for material failure, primary application of elasticity theory is to determine the distribution of stress within a given structure. Related to these force distribution issues is the concept of equilibrium. Within a deformable solid, the force distribution at each point must be balanced, and thus the summation of forces on an infinitesimal element is required to be zero. In this chapter we establish the definitions and properties of the traction vector and stress tensor, and develop the equilibrium equations, which will become another set of field equations necessary in the overall formulation of elasticity theory. It should be noted that the developments in this chapter will not require that the material be elastic, and thus in principle these results would apply to a broader class of material behavior.