Inversion Symmetry of the Solutions of Boundary-Value Problems of Elasticity for a Half-Space
- PDF / 188,041 Bytes
- 15 Pages / 595 x 794.048 pts Page_size
- 96 Downloads / 192 Views
International Applied Mechanics, Vol. 56, No. 5, September, 2020
INVERSION SYMMETRY OF THE SOLUTIONS OF BOUNDARY-VALUE PROBLEMS OF ELASTICITY FOR A HALF-SPACE
V. I. Ostrik
The inversion symmetry of the components of the displacement vector and stress tensor in the solution of the first boundary-value problem of elasticity for a half-space is studied. The case where one component of loading on the half-space boundary has inversion symmetry and the other two components are equal to zero is considered. Inversion symmetry is also studied for the mixed problem where normal forces act and tangential forces are equal to zero one part of the half-space boundary, while the conditions of smooth contact are prescribed on the other part, and the problem of the torsion of an elastic half-space with tangential stresses given on its boundary. Keywords: inversion, elastic half-space, Boussinesq and Cerruti problems, torsion, potentials Introduction. Inversion is used in some cases in the theory of elasticity [3, 6, 7, 13, 14] and mathematical physics [8, 13] to solve boundary-value problems in polar and spherical coordinates. This is because a harmonic function of two variables remains harmonic after inversion [4], while harmonic functions of three variables remain harmonic in the radial variable after inversion (Kelvin theorem) [8]. A biharmonic function of two variables remains biharmonic in the squared radial variable after such a transformation [13]. This circumstance allows us to solve, for example, the two-dimensional problem of elasticity for an elastic disk under concentrated forces using the solution of the similar problem for an elastic half-plane [6, 13, 14]. Inversion is also used to determine Green’s functions for a circle and a ball in the Dirichlet problem for the Laplace equation [12]. If after inversion, the domain of a boundary-value problem of potential theory remains the same (such as a wedge, a half-space, a cone) and so do the boundary conditions, the solution of this problem (with a certain functional factor in the case of three variables) transforms into itself, i.e., has inversion symmetry. For any boundary-value problem of elasticity, the inversion symmetry of the boundary conditions does not lead to the inversion symmetry of the solution. Nevertheless, as shown in [15] where the basic two-dimensional boundary-value problems for an elastic wedge were analyzed, individual components of the displacement vector and stress tensor have inversion symmetry either in the whole domain or on the edges of the wedge. In [9, 16], in solving two mixed problems of elasticity for a wedge in which boundary conditions have inversion symmetry, it was shown that the normal displacements and normal stresses on one of the edges of the wedge have inversion symmetry. In what follows, we will establish the inversion symmetry of individual components of the displacement vector and stress tensor in the solutions of the first basic problem, the mixed problem, and the torsion problem for an elastic half-space. 1. First Boundary-Va
Data Loading...
