Hu–Washizu 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. Let $f \in L^2(\Omega)^3$ be the body force, $g \in L^2(\Gamma_N)^3$ the surface traction, and $\bar u \in H^1(\Omega)^3$ a given displacement. For a displacement $v$ let $\varepsilon(v) = \tfrac12(\nabla v + \nabla v^T)$ be its linearised strain. Let $U = \set{v \in H^1(\Omega)^3 : v = \bar u \text{ on } \Gamma_D}$, 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 Hu–Washizu functional on $U \times L^2_{\mathrm{sym}} \times L^2_{\mathrm{sym}}$ is $$\Pi_{HW}(v, e, \tau) = \int_\Omega \left( \tfrac12\, \mathsf{C} e : e - \tau : \big(e - \varepsilon(v)\big) - f \cdot v \right) dx - \int_{\Gamma_N} g \cdot v \, dS.$$ Then a triple $(u, e, \sigma)$ is a stationary point of $\Pi_{HW}$, that is, its first variation vanishes in all directions of $V \times L^2_{\mathrm{sym}} \times L^2_{\mathrm{sym}}$, if and only if:

1. $e = \varepsilon(u)$ (kinematics);
2. $\sigma = \mathsf{C} e$ (constitutive law);
3. $\int_\Omega \sigma : \varepsilon(v)\, dx = \int_\Omega f \cdot v \, dx + \int_{\Gamma_N} g \cdot v \, dS$ for all $v \in V$ (weak equilibrium).

In that case $u$ is the unique weak solution of the problem of linear elasticity with these data, and $\sigma$ is its stress.