Multi-wedge points and multi-wedge elements in computational mechanics: evaluation of exponent and angular distribution

There is growing interest in accounting for the internal structure of a material. This interest stimulates developing tools for the accurate evaluation of fields near common vertices of structural elements, in particular, grains. This paper presents a robust method to numerically evaluate the exponent which characterizes the asymptotic behaviour of stresses and displacements at a vertex of an arbitrary number of elastic wedges. The efficiency is achieved by (i) reduction of the problem to three-point matrix difference equations with appropriately normalized coefficients, and (ii) finding the roots of the determinant of the matrix by specially designed iterative and search procedures. This allows us to ensure convergence and not miss closely located significant roots. Numerical calculations for systems of two and three wedges, studied by other authors previously, show that the results agree to at least five digits. A number of new examples for three and four wedges with and without cracks reveal that the multi-wedge systems, which have more than one root generating singular stresses, are not rare; quite commonly such roots are closely located. We emphasize that this fact has important implications for the development of singular multi-wedge elements, intended to increase the accuracy of the BEM and FEM. The appendices serve to re-examine and clarify the relation between properties of the matrix of the system, the asymptotic behaviour of stresses and displacements, and the number of stress intensity factors. It is shown that the necessary condition, established by Dempsey and Sinclair for the logarithmic multiplier to be present in the asymptotic formulae, is also sufficient.