In this work we introduce novel stress-only formulations of linear elasticity with special attention to their approximate solution using weighted residual methods. We present four sets of boundary value problems for a pure stress formulation of three-dimensional solids, and in two dimensions for plane stress and plane strain. The associated governing equations are derived by modifications and combinations of the Beltrami–Michell equations and the Navier–Cauchy equations. The corresponding variational forms of dimension allow to approximate the stress tensor directly, without any presupposed potential stress functions, and are shown to be well-posed in in the framework of functional analysis via the Lax–Milgram theorem, making their finite element implementation using -continuous elements straightforward. Further, in the finite element setting we provide a treatment for constant and piece-wise constant body forces via distributions. The operators and differential identities in this work are provided in modern tensor notation and rely on exact sequences, making the resulting equations and differential relations directly comprehensible. Finally, numerical benchmarks for convergence as well as spectral analysis are used to test the limits and identify viable use-cases of the equations.

中文翻译：

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纯应力线性弹性的对称不溶方程

在这项工作中，我们介绍了新颖的仅应力线性弹性公式，特别关注使用加权残差法的近似解。我们针对三维固体的纯应力公式以及二维平面应力和平面应变提出了四组边值问题。相关的控制方程是通过贝尔特拉米-米歇尔方程和纳维-柯西方程的修改和组合得出的。相应的维数变分形式允许直接近似应力张量，无需任何预设的潜在应力函数，并且通过 Lax-Milgram 定理在泛函分析框架中显示出良好的状态，从而使用以下方法进行有限元实现：连续元素简单明了。此外，在有限元设置中，我们通过分布提供恒定和分段恒定体积力的处理。这项工作中的算子和微分恒等式以现代张量表示法提供，并依赖于精确的序列，使得所得方程和微分关系可以直接理解。最后，使用收敛数值基准以及谱分析来测试极限并确定方程的可行用例。