On general boundary value problems and duality in linear elasticity. I

Abstract

The equilibrium state of a deformable body under the action of body forces is described by the well known conditions of equilibrium, the straindisplacement relations, the constitutive law of the linear theory and the boundary conditions. The authors discuss in detail the boundary conditions. The starting point is the general relation between the vectors of stress and displacement on the boundary which can be expressed in terms of a subgradient relation. It is shown that this relation includes as special cases all known classical, bilateral and unilateral boundary conditions. Further, the principle of virtual displacements and the principle of minimum of the potential energy are established and it is shown that these principles are equivalent to the original boundary condition problem.