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A novel hybrid stress-function finite element method immune to severe mesh distortion

This paper introduces a hybrid stress-function finite element method proposed recently for developing 2D finite element models immune to element shapes. Deferent from the first version of the hybrid-stress element constructed by Pian, the stress function ϕ of 2D elastic or fracture problem is regarded as the functional variable of the complementary energy functional. Then, the basic analytical solutions of ϕ are taken as the trial functions for finite element models, and meanwhile, the corresponding unknown stress-function constants are introduced. By using the principle of minimum complementary energy, these unknown stress-function constants can be expressed in terms of the displacements along element edges. Finally, the complementary energy functional can be rewritten in terms of element nodal displacement vector, and thus, the element stiffness matrix of such hybrid-function element can be obtained. As examples, two (8- and 12-node) quadrilateral plane elements and an arbitrary polygonal crack element are constructed by employing different basic analytical solutions of different stress functions. Numerical results show that, the 8- and 12-node plane models can produce the exact solutions for pure bending and linear bending problems, respectively, even the element shape degenerates into triangle and concave quadrangle; and the crack element can also predict accurate results with very low computational cost in analysis of stress-singularity problems.