The effectiveness of the energy method for the numerical-analytical solution of a mixed problem of elasticity theory

Abstract

The process of constructing a numerical-analytical solution to the basic mixed boundary value problem of the static theory of elasticity (TE) is considered. The defining relations of a linear homogeneous isotropic elastic medium are equivalent to a system of three resolving Lame equations, each linear of the second order with respect to partial derivatives. By means of the energetic method of boundary states (MBS), by decomposing the desired state into a Fourier series according to the elements of the separable basis of the Hilbert space of states, the boundary value problem of mathematical physics is reduced to a system of linear algebraic equations (SLAE). The construction of the basis is based on a variant of the general solution of a system of elliptic type equations. Each basic state is created by using harmonic polynomials. Orthogonalization is performed by the Gram-Schmidt algorithm. The main mixed task involves dividing the body boundary into two classes. Point movements are set on one part of the border, and surface movements are set on the remaining part. The accuracy of the solution is assessed by two factors: 1) Bessel's inequality; 2) integral quadratic residual of the restored boundary state with BC. The use of both factors leads to the effect of self-sufficiency of the MBS: there is no need to compare trial solutions with reference ones based on other methods. Specific calculations have been performed for two classes of mixed tasks: 1) a bounded bicavous body (a ball with two symmetrically arranged spherical cavities. A rigid counter-displacement of the surfaces of the cavities is set, the outer boundary of the body is free from load; 2) an elastic circular cylinder is fixed with a rigid rod along part of one base. The side surface is loaded with tangential forces directed along the axis of the cylinder. In the second problem, a singular jump in the characteristics of the boundary state along the interface line belonging to the boundary of the body is tested. Numerical-analytical solutions to both problems are constructed, the fields of characteristics of stress-strain states are illustrated, comments on states are made, and conclusions are done.