Hybrid equilibrium element with high-order stress fields for accurate elastic dynamic analysis

In the present article the two-dimensional hybrid equilibrium element formulation is initially developed, with quadratic, cubic, and quartic stress fields, for static analysis of compressible and quasi-incompressible elastic solids in the variational framework of the minimum complementary energy principle. Thereafter, the high-order hybrid equilibrium formulation is developed for dynamic analysis of elastic solids in the variational framework of the Toupin principle, which is the complementary form of the Hamilton principle. The Newmark time integration scheme is introduced for discretization of the stress fields in the time domain and dynamic analysis of both the compressible solid and quasi-incompressible ones. The hybrid equilibrium element formulation provides very accurate solutions with a high-order stress field and the results of the static and dynamic analyses are compared with the solution of the classic displacement-based quadratic formulation, showing the convergence of the two formulations to the exact solution and the very satisfying performance of the proposed formulation, especially for analysis of quasi-incompressible elastic solids.