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 approximate the stress tensor directly, without any presupposed potential stress functions, and are shown to be well-posed in $H^1 \otimes \mathrm{Sym}(d)$ in the framework of functional analysis via the Lax-Milgram theorem, making their finite element implementation using $C^0$-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：本文提出了全新的纯应力形式线性弹性方程，特别关注使用加权残量法对其近似求解。我们为三维固体的纯应力公式以及二维平面应力和平面应变问题，各给出了四组边值问题。相关控制方程是通过对Beltrami-Michell方程和Navier-Cauchy方程进行改进与组合推导得出的。维度$d \in \{2,3\}$的相应变分形式允许直接逼近应力张量，无需预先假定应力势函数，并且通过Lax-Milgram定理在泛函分析框架下证明了其在$H^1 \otimes \mathrm{Sym}(d)$中的适定性，这使得使用$C^0$连续单元进行有限元实现变得直接。此外，在有限元框架下，我们通过分布理论处理了恒定和分段恒定体力。本文中的算子和微分恒等式采用现代张量符号表示，并依赖于精确序列，使所得方程和微分关系易于理解。最后，通过收敛性数值基准测试和谱分析，检验了方程的限制并识别了可行的应用场景。