Symmetric unisolvent equations for linear elasticity purely in stresses

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 $H¹ \otimes \mathrm{Sym}(d)$ 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.