The explicit solution of the second main problem of elasticity for a torus (Q2755245)

This article deals with explicit solution to the boundary value problem of elasticity for a torus when on the torus surface the arbitrary displacements are given. The author proposes a general solution to the Lamé equation \(2{m-1\over m-2}{\text grad\,div}\,u- rot\,rot\,u=0\), where \(m\) is the Poisson ratio. The boundary value problem is reduced to a set of infinite algebraic systems with three-diagonal matrices, and a method for explicit analytical solution of such systems is presented. The problem on asymmetric displacement of a rigid toroidal inclusion in an elastic medium is considered, and the forces and displacements for different torus geometries and different values of Poisson ratio are calculated.