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.
Highlights
The independent variables of conventional elastic mechanics problems are displacement, strain and stress, which are solved by governing equations, displacement boundary conditions and force boundary conditions
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
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
Summary
The independent variables of conventional elastic mechanics problems are displacement, strain and stress, which are solved by governing equations, displacement boundary conditions and force boundary conditions. Qing et al [25] considered general three-dimensional problems, based on the modified H–R variational principle and the symplectic theory of elastic mechanics, established a block mixed isoparametric element containing out-of-plane stress and displacement variables. Liu et al [26] established a parameter-containing, non-compatible sympletic element to solve the static problem of the steady-state temperature field based on the non-compatible symplectic element theory and the generalized H–R variational principle, and applied it to the stress analysis of the thermoelastic composite laminate. Numerical examples demonstrate that the element has good crack tracking capabilities He [29] gave a way to establish the variational principle through the semi-inverse method and applied this method to the thermodynamically coupled finite displacement problem [30]. The above references indicate that a variety of high-precision numerical methods can be established based on the H–R variational principle, but the current H–R variational principle does not consider the specified stress problem, and it is difficult to make the stress meet the specified condition in the result
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