Introduction

This chapter is dedicated to the study of deformation and stress in solid bodies under a prescribed set of forces and kinematic constraints. In a majority of problems, we assume that stresses and strains are small so that linear strain-displacement relations and Hooke's law are valid, and we use appropriate governing equations derived using the Lagrangian description in the previous chapters to solve them for stresses and displacements. In the linearized elasticity we assume that the geometric changes are so small that we neglect squares of the displacement gradients, that is, ∣∇u∣2 ≈ 0, and do not make a distinction between the deformed and undeformed geometries, between the second Piola–Kirchhoff stress tensor S and the Cauchy stress tensor σ, and between the current coordinates x and the material coordinates X (and use σ and x). Mathematically, we seek solutions to coupled partial differential equations over an elastic domain occupied by the reference (or undeformed) configuration of the body, subject to specified boundary conditions on displacements or forces. Such problems are called boundary value problems of elasticity.