Elasticity/Hellinger-Reissner principle

Hellinger-Prange-Reissner Variational Principle
In this case, we assume that the elasticity field is invertible and $$s$$ is smooth on $$\mathcal{B}$$. We also assume that $$\mathcal{A}$$ is the set of all admissible states that satisfy the strain-displacement relations, the traction-stress relations and the balance of angular momentum.

Let $$\mathcal{A}$$ denote the set of all admissible states and let $$\mathcal{H}$$ be a functional on $$\mathcal{A}$$ defined by

{\mathcal H}[s] = \int_{\mathcal{B}} U^c(\boldsymbol{\sigma}) - \int_{\mathcal{B}} \boldsymbol{\sigma}:\boldsymbol{\varepsilon}~dV + \int_{\mathcal{B}} \mathbf{f}\bullet\mathbf{u}~dV + \int_{\partial{\mathcal{B}}^{u}} \mathbf{t}\bullet(\mathbf{u}-\widehat{\mathbf{u}})~dA + \int_{\partial{\mathcal{B}}^{t}} \widehat{\mathbf{t}}\bullet\mathbf{u}~dA $$ for every $$s = [\mathbf{u},\boldsymbol{\sigma}] \in \mathcal{A}$$.

Then,

\delta {\mathcal H}[s] = 0 $$ at an admissible state $$s\in\mathcal{A}$$ if and only if $$s$$ is a solution of the mixed problem.