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 $d \in \{2,3\}$ allow to directly approximate the stress tensor without any presupposed potential stress functions, and are shown to be well-posed in $\mathit{H}^1 \otimes \mathrm{Sym}(d)$ in the framework of functional analysis via the Lax-Milgram theorem, making their finite element implementation using $\mathit{C}^0$-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.

翻译：本文提出了新型纯应力线性弹性方程，重点关注利用加权残量法进行近似求解。针对三维实体及二维平面应力和平面应变问题，我们给出了四组纯应力边值问题。相关控制方程是通过对贝尔特拉米-米歇尔方程和纳维-柯西方程进行修正与组合推导得出的。在维度$d \in \{2,3\}$下，相应的变分形式能够直接逼近应力张量而无需预设应力势函数，并通过Lax-Milgram定理在泛函分析框架中证明其在$\mathit{H}^1 \otimes \mathrm{Sym}(d)$空间中的适定性，从而可以直接使用$\mathit{C}^0$连续单元进行有限元实现。此外，在有限元框架下，我们通过分布理论处理了常值及分段常值体积力问题。本文提供的算子和微分恒等式采用现代张量记法并依赖正合序列，使所得方程和微分关系更易理解。最后，通过数值收敛性测试和谱分析检验了方程的适用范围并识别了可行应用场景。