The First Basic Problem of Elasticity Theory

Abstract

Primary equations are derived and the problem of elasticity theory in stresses is given. First, the equations of equilibrium of an infinitesimal element of the body are derived. A set of normal and tangential stresses is applied to the element, taking into account the inhomogeneity of the stress state. The conditions of equilibrium of forces and moments give three differential equations. Then Saint-Venant identities are obtained, which establish differential dependencies between the strain components. Replacing deformations through stresses according to Hooke’s law turns Saint-Venant identities into the so-called compatibility conditions. The equilibrium conditions of an infinitesimal pyramid with one face on the surface of the body give three relations, called boundary conditions. As a result, we come to the following. Mathematically, the task is to define six functions (stress components) from the system of nine differential dependencies: three equilibrium equations, three equations of compatibility of strains, and three boundary conditions. Conditions of unambiguity of solving this problem are discussed.