On the construction of general solutions of the elasticity theory equations for transversely isotropic inhomogeneous bodies: PMM vol. 40, n≗ 5, 1976, pp. 956–958

Abstract

The solution is presented for the three-dimensional problem of the theory of elasticity of transversely isotropic elastic bodies, where the elastic characteristics vary arbitrarily along the axis of symmetry of the elastic properies of the medium. The solution is written in orthogonal curvilinear cylindrical coordinates and is represented by using two independent functions. The question of separation of the boundary conditions in the plane of isotropy is examined. A number of investigations, which examine primarily the equilibrium of Isotropic bodies with an exponential law of variation of the elastic modulus and a constant Poisson's ratio, is devoted to the solution of two-dimensional problems of the theory of elasticity of inhomogeneous bodies. One of the first works in this area is apparently [1]. Under analogous assumptions, the general solution of the three-dimensional problem of the theory of elasticity of transversely isotropic bodies and isotropic bodies adapted for the analysis of laminar media has been constructed in [2]. It is also represented by using two independent functions for which the conditions of the boundaries of the layers are separated. The dependences obtained are used in [2] for the general solution of the equilibrium problem for half-spaces comprised of layers which are not homogeneous with depth under the effect of surface forces. A solution of the three-dimensional problem of the theory of elasticity of an inhomogeneous isotropic body, constructed by a scheme similar to that elucidated, but without the constraints imposed on the elastic characteristics and taking account of the volume forces is presented in [3].