Solution of plane elasticity problems with Mathematica

Abstract

The classical complex-variable method of Muskhelishvili for the solution of general plane elasticity problems is revisited. The boundary conditions are satisfied automatically and the related functional equation of Muskhelishvili is approximately solved by selection of the collocation points outside the elastic medium. This ensures the approximate solution of the biharmonic equation of plane elasticity. This approach is completely different from boundary element and finite element methods, since the main attempt is to satisfy the analyticity of the complex potentials (almost equivalently, the biharmonic equation) and not the boundary conditions or both of these. The modern and powerful computer algebra system Mathematica was selected as an appropriate programming language for the related computer procedure, which is also given. Symbolic parameters can also be used in the computing environment offered by Mathematica. Numerical and semi-numerical results in a special application (elliptical region) are also presented and seen to converge rapidly. Both cases of the first and the second fundamental problems are considered.