A combined mixed functional is proposed for analysis of linear elastic problems. The functional is a modification of the Hellinger-Reissner functional and a generalization of the Slivker's mixed functional. It is constructed by linearly combining the hellinger-Reissner functional and the total potential energy. The resulting bilinear form is shown to be positive definite and V-elliptic or V-coercive according to the combination parameter, which guarantee the existence and uniqueness of the solution. The equivalence theorem between mixed elements and reduced/selective integration elements is applied and the stabilization matrices are obtained for the continuum elements. By performing the error analysis, the combined mixed model is shown to have optimal convergence rate. This means that the stabilization methods have the same convergence rate as the displacement-based model.
The combined mixed functional is applied to the Mindlin plate problem. Existence and uniqueness of the solution of the proposed mixed model are proven. The stabilization matrix of Belytschko is obtained for the four-node plate element based on the equivalence theorem between mixed elements and reduced/selectively integrated elements. Using the present method, stabilization matrices can be obtained for higher-order elements and triangular elements without any difficulty. The combined mixed model is shown to have optimal convergence rate from the error analysis under the assumption of the finite thickness. The stabilization parameter of Belytschko is modified and numerically tested.
For the numerical verification of the combined mixed model, eigenvalues of the stiffness matrix of a typical rectangular element are obtained for various combinations of approximating polynomials. The structure of them is shown to conform to that theoretically predicted. From the numerical analyses of the rectangular plates under the uniformly distributed loads, the stabilization parameter is shown to be insensitive to Young's modulus, Poisson's ratio, thickness and area. It is also shown that the combined mixed element is not sensitive to stabilization parameter with reasonable meshes and that the element is free from the shear locking and spurious zero energy modes.
