Hellinger–Reissner variational principle for a class of specified stress problems

Abstract

Abstract Aiming at the problem of the specified stress condition in a partial region of the structure, the Hellinger–Reissner (H–R) variational principle is studied to provide a theoretical basis for the finite element numerical analysis. By introducing the unknown non-elastic strain as an additional unknown quantity to fulfill the specified stress condition, the elastic mechanics governing equations for the specified stress problem are given. Both stress and unknown non-elastic strain are taken as independent variables to establish the complementary energy principle and virtual work equation which are equivalent to the elastic mechanical control equation of the specified stress problem. Based on the conventional H–R variational principle, using displacement, stress and unknown non-elastic strain as independent variables, a H–R variational functional that satisfies the specified stress conditions is established by using Lagrange multiplier method. Also the variational functional with displacement, elastic strain and unknown non-elastic strain as independent variables is deduced by transforming the stress into the elastic strain. The corresponding finite formulae are derived based on an intra-element stress hybridization method. The H–R variational principle for the specified stress problem takes non-elastic strain as an independent variable, so that the stress explicitly appears in the equilibrium equation of the element and structure, which expands the application range and capabilities of the existing variational principle and finite element method. The correctness and accuracy of the theory and algorithm are verified by numerical examples.