On the theory of elasticity of inhomogeneous media: PMM, vol. 35, n≗ 5, 1971, pp. 853–860

Abstract

The general solution is obtained of the equilibrium equations in displacements for inhomogeneous isotropie media, whose elastic characteristics are differentiable functions of the Cartesian coordinates. It is shown that the components of the displacement vector in the three-dimensional problem of elasticity theory can always be expressed in terms of two functions which satisfy second and fourth order linear partial differential equations, respectively. Of the earlier research devoted to analogous problems, the paper [1] should first be noted in which an equation is derived for the Airy stress function in the two-dimensional problem of the theory of elasticity of an inhomogeneous medium. A general solution of the equilibrium equations in displacements is obtained in [2, 3] for the case of axisymmetric deformation of bodies of revolution whose elastic moduli vary exponentially as a function of the coordinate z. A powerlaw change in the elastic modulus was investigated in [4] with primary attention paid to the plane problem. General solutions of the equilibrium equations in displacements for the three-dimensional case and an arbitrary law of variation of the elastic characteristics of an inhomogeneous medium has apparently not been examined.