tailieunhanh - Elasticity Part 3

Tham khảo tài liệu 'elasticity part 3', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | This page intentionally left blank TLFeBOOK Sadd Elasticity Final Proof 2 59pm page 49 3 Stress and Equilibrium The previous chapter investigated the kinematics of deformation without regard to the force or stress distribution within the elastic solid. We now wish to examine these issues and explore the transmission of forces through deformable materials. Our study leads to the definition and 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 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. In this chapter we establish the definitions and properties of the traction vector and stress tensor and develop the equilibrium equations which become another set of field equations necessary in the overall formulation of elasticity theory. It should be noted that the developments in this chapter do not require that the material be elastic and thus in principle these results apply to a broader class of material behavior. Body and Surface Forces When a structure is subjected to applied external loadings internal forces are induced inside the body. Following the philosophy of continuum mechanics these internal forces are distributed continuously within the solid. In order to study such forces it is convenient to categorize them into two major groups commonly referred to as body forces and surface forces. Body forces are proportional to the