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.

中文翻译：

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具有高阶应力场的混合平衡单元，用于精确的弹性动力学分析

在本文中，最初开发了二维混合平衡单元公式，其中包含二次、三次和四次应力场，用于在最小互补能量原理的变分框架中对可压缩和准不可压缩弹性固体进行静态分析。此后，在 Toupin 原理的变分框架中开发了高阶混合平衡公式，用于弹性固体的动力学分析，这是 Hamilton 原理的补充形式。引入 Newmark 时间积分方案用于时域中应力场的离散化以及可压缩固体和准不可压缩固体的动态分析。