Abstract A finite-strain tetrahedron with continuous stresses is proposed and analyzed. The complete stress tensor is now a nodal tensor degree-of-freedom, in addition to displacement. Specifically, stress conjugate to the relative Green-Lagrange strain is used within the framework of the Hellinger-Reissner variational principle. This is an extension of the Dunham and Pister element to arbitrary constitutive laws and finite strain. To avoid the excessive continuity shortcoming, outer faces can have null stress vectors. The resulting formulation is related to the nonlocal approaches popularized as smoothed finite element formulations. In contrast with smoothed formulations, the interpolation and integration domain is retained. Sparsity is also identical to the classical mixed formulations. When compared with variational multiscale methods, there are no parameters. Very high accuracy is obtained for four-node tetrahedra with incompressibility and bending benchmarks being successfully solved. Although the ad-hoc factor is removed and performance is highly competitive, computational cost is high, as each tetrahedron has 36 degrees-of-freedom. Besides the inf-sup test, four benchmark examples are adopted, with exceptional results in bending and compression with finite strains.

中文翻译：

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有限应变问题的连续应力四面体

摘要 提出并分析了具有连续应力的有限应变四面体。除了位移之外，完整的应力张量现在是一个节点张量自由度。具体而言，在 Hellinger-Reissner 变分原理的框架内使用与相对格林-拉格朗日应变的应力共轭。这是 Dunham 和 Pister 元素对任意本构定律和有限应变的扩展。为了避免过度连续性的缺点，外表面可以具有零应力向量。由此产生的公式与作为平滑有限元公式普及的非局部方法有关。与平滑公式相比，保留了插值和积分域。稀疏性也与经典的混合公式相同。与变分多尺度方法相比，没有参数。四节点四面体获得了非常高的精度，并且成功解决了不可压缩性和弯曲基准。尽管去除了临时因素并且性能具有很强的竞争力，但计算成本很高，因为每个四面体都有 36 个自由度。除了 inf-sup 测试之外，还采用了四个基准示例，在有限应变的弯曲和压缩方面取得了出色的结果。