Hybrid-Trefftz Finite Element Formulation for Simulation of Singular Stress Fields

An equilibrium hybrid-Trefftz formulation based on the direct approximation of the stress and boundary displacement fields is presented. The general solution of the governing differential equations is used to approximate the stress field and the boundary displacements are represented by polynomial functions. When singular solutions are implemented to model local high stress gradients due to concentrated loads or to the presence of wedges or cracks, rational functions are used to approximate the boundary displacements in the neighbourhood of such singular stress points. The equilibrium conditions and the kinematic boundary conditions are locally satisfied. The remaining fundamental relations—the compatibility conditions, the static boundary conditions and the constitutive relations—are enforced in a weighted residual form so designed as to preserve the duality and constitutive reciprocity. The resulting governing system is symmetric and all intervening structural operators have boundary integral expressions. Numerical applications are presented to illustrate the performance of the formulation.