Integration of differential equations in elastostatics through-determination of the mean stress

Abstract

The paper is devoted to discussion and development of an analytical method, proposed by the author, for integration of differential equations in elastostatics. A homogeneous isotropic linear elastic body is considered. The differential equations are first formulated in terms of the displacement components (u, v, w) and the mean normal stress (p) for any real value of Poisson's ratio in the interval — 1 < v ≤ 1/2. By this means we obtain in a three-dimensional case a set of four equations in four unknown functions, and in a plane (two-dimensional) case – a set of three equations in three, both sets (unlike Navier's equations) being valid for incompressible bodies (v = 1/2) as well. Derivation of their solution is based on the condition of single-valuedness of the mean normal stress at any point of the body, with displacements presented by means of the Stokes-Helmholtz resolution of vector fields. Integration of this type of differential equations was first studied by the author in works on the deformation of nonhomogeneous elastic incompressible bodies, and carried further in works on elastostatic problems of incompressible and compressible bodies, both homogeneous and nonhomogeneous.