3 - Stress and Equilibrium

Abstract

This chapter examines the issue related to the kinematics of deformation without regard to the force or stress distribution within the elastic solid and explores the transmission of forces through deformable materials. The study leads to the definition and the use of the traction vector and stress tensor. Each provides a quantitative method to describe both boundary and internal force distributions within a continuum solid. Because it is commonly accepted that maximum stresses are a major contributing factor to material failure, the primary application of elasticity theory is used 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. For the static case, the summation of forces on an infinitesimal element is required to be zero, while for a dynamic problem the resultant force must equal the mass times the element's acceleration. This chapter also establishes the definitions and properties of the traction vector and stress tensor and develops the equilibrium equations, which become another set of field equations necessary in the overall formulation of elasticity theory.