Elasticity/Hu-Washizu principle

Hu-Washizu Variational Principle
In this case, the admissible states are not required to meet any of the field equations or boundary conditions.

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

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

Then,

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