Boundary eigensolutions in elasticity. I. Theoretical development

Abstract

The theory of fundamental boundary eigensolutions for elastostatic boundary value problems is developed. The underlying fundamental eigenproblem is formed by inserting the eigenparameter and a tensor weight function into the boundary condition, rather than into the governing differential equation as is often done for vibration problems. The resulting spectra are real and the eigenfunctions (eigendeformations) are mutually orthogonal on the boundary, thus providing a basis for solutions. The weight function permits effective treatment of non-smooth problems associated with cracks, notches and mixed boundary conditions. Several ideas related to the behavior of eigensolutions in the domain, integral equation methods, variational methods, convergence characteristics, flexibility and stiffness kernels, and solutions to problems with body forces are also introduced. Of particular note are the integral equation and variational formulations that lead to the development of new computational formulations for boundary element and finite element methods, respectively. An example with closed form and numerical results is included to illustrate some aspects of the theory.