Using computer algebra to construct analytical solutions for elastostatic problems

Abstract

The article gives a brief overview of elastostatic methods. It notes successful attempts of building analytical solutions. It is necessary to “legalize” a number of parameters of the problem in the full-parameter solution (FPS): elasticity, boundary conditions (BCs), volume forces, “geometry” of the body. To provide stiffness and geometry parameters figuring, the Lagrange interpolation and Chebyshev approximation are applicable (the FPS of the first main problem for a ball is constructed). The perturbation method has claimed to be a resource-saving technology for the “legalization” of the Poisson’s ratio. To construct the state of the body under the action of volume forces, two efficient algorithms from a wide class of polynomial ones are proposed. They allow strict writing out of a particular solution. There is shown the FPS of a problem of deformation of the ball layer pinched on the surface by such forces. An approach based on the construction “reference” solutions is recommended for the “legalization” of BC parameters and volume forces.