Linear elastostatics

In this chapter the basic field equations in terms of displacement, strain, and stress, and typical boundary conditions which are necessary to formulate a complete three-dimensional boundary-value-problem for linear elasticity in the static case (i.e., neglect of inertial terms) are stated. The uniqueness of a solution to such a boundary value problem is discussed. There are a number of alternate ways that one can approach the statement of an elastostatic boundary-value-problem. The first major approach is obtained by reducing the set of field equations by expressing them solely in terms of the displacement field --- the Navier equations --- while in the second major approach the general system of equations may be reformulated by eliminating the displacement and strain fields and casting the system solely in terms of the stress field --- the Beltrami-Michell equations. The special formulation of idealized two-dimensional boundary value problems is presented the two basic theories of plane strain and plane stress for isotropic materials are discussed.