Hellinger–Reissner principle

Statement

Let $\Omega \subset \R^3$ be a bounded domain with Lipschitz boundary $\partial\Omega = \overline{\Gamma_D} \cup \overline{\Gamma_N}$, where $\Gamma_D$ and $\Gamma_N$ are disjoint and $\Gamma_D$ has positive surface measure. Let $\mathsf{C}$ be a bounded symmetric elasticity tensor field that is uniformly positive definite on symmetric $3 \times 3$ matrices, and let $\mathsf{S} = \mathsf{C}^{-1}$ be the compliance tensor. Let $f \in L^2(\Omega)^3$ be the body force and $g \in L^2(\Gamma_N)^3$ the surface traction. For a displacement $v$ let $\varepsilon(v) = \tfrac12(\nabla v + \nabla v^T)$ be its linearised strain. Let $V = \set{v \in H^1(\Omega)^3 : v = 0 \text{ on } \Gamma_D}$ and let $L^2_{\mathrm{sym}}$ be the space of square-integrable fields of symmetric $3 \times 3$ matrices. The Hellinger–Reissner functional on $V \times L^2_{\mathrm{sym}}$ is $$\Pi_{HR}(v, \tau) = \int_\Omega \left( \tau : \varepsilon(v) - \tfrac12\, \mathsf{S}\tau : \tau - f \cdot v \right) dx - \int_{\Gamma_N} g \cdot v \, dS.$$ Then a pair $(u, \sigma) \in V \times L^2_{\mathrm{sym}}$ is a stationary point of $\Pi_{HR}$, that is, its first variation vanishes in all directions, if and only if:

1. $\mathsf{S}\sigma = \varepsilon(u)$, that is, $\sigma = \mathsf{C}\varepsilon(u)$;
2. $\int_\Omega \sigma : \varepsilon(v)\, dx = \int_\Omega f \cdot v \, dx + \int_{\Gamma_N} g \cdot v \, dS$ for all $v \in V$.

In that case $u$ is the unique weak solution of the problem of linear elasticity with these data, and $\sigma$ is its stress. Moreover $(u, \sigma)$ is a saddle point: $\Pi_{HR}(u, \tau) \le \Pi_{HR}(u, \sigma) \le \Pi_{HR}(v, \sigma)$ for all $v \in V$ and $\tau \in L^2_{\mathrm{sym}}$.