Mixed Finite Elements with Voids and Inclusions

For stress concentrations around voids and inclusions, one can derive appropriate stress and strain functions, which characterize the local fields very well, and use them for special problem adapted finite elements. For linear problems, such stress and strain functions can be obtained from complex solution representations. The functions can be derived in such a way that the equilibrium equations and the boundary conditions on the void surface or the continuity conditions along the matrix/inclusion boundary are satisfied a priori. For special finite elements with voids and inclusions, piecewise linear or quadratic boundary displacements have to be assumed for an appropriate coupling with other finite elements. There has to be a balance between the number of stress/strain parameters and the nodal displacements of the special elements with built-in voids or inclusions. The outer boundary of a twodimensional special finite element can be a polygon, and for the three-dimensional case the element boundary can be chosen as a polyhedron.