STRESS FUNCTIONS IN ELASTICITY THEORY

Abstract—

The article is devoted to the study of stress functions that make it possible to identically satisfy the equilibrium equations of the classical elasticity theory and obtain a solution in stresses. To obtain the dependences between stresses and stress functions, the mathematical apparatus of the general theory of relativity is used, in particular, the property of the Einstein tensor to identically satisfy the equations of the conservation law, which, in relation to the theory of elasticity, are equilibrium equations. In this case, the metric coefficients of the Riemannian space, determined by the Einstein equations, are interpreted as functions of the stresses of the elasticity theory. As a result of the linearization of Einstein’s equations, general relations between stresses and stress functions in an orthogonal coordinate system are obtained. Stress functions corresponding to the Cartesian coordinate system are considered. The possibilities of satisfying the equilibrium equations with the help of various combinations of stress functions are analyzed – the known systems of Maxwell, Morer and other possible combinations formed from one, two and three functions. As a solvability criterion of the problem of the theory of elasticity in stresses, the correspondence of the number of stress functions to the number of mutually independent equations of compatibility of deformations in stresses is used.