Hu-Washizu Variational Principle

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

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

${\displaystyle {\mathcal {W}}[s]=\int _{\mathcal {B}}U({\boldsymbol {\varepsilon }})-\int _{\mathcal {B}}{\boldsymbol {\sigma }}:{\boldsymbol {\varepsilon }}~dV-\int _{\mathcal {B}}({\boldsymbol {\nabla }}\bullet {\boldsymbol {\sigma }}+\mathbf {f} )\bullet \mathbf {u} ~dV+\int _{\partial {\mathcal {B}}^{u}}\mathbf {t} \bullet {\widehat {\mathbf {u} }}~dA+\int _{\partial {\mathcal {B}}^{t}}(\mathbf {t} -{\widehat {\mathbf {t} }})\bullet \mathbf {u} ~dA}$

for every ${\displaystyle s=[\mathbf {u} ,{\boldsymbol {\varepsilon }},{\boldsymbol {\sigma }}]\in {\mathcal {A}}}$.

Then,

${\displaystyle \delta {\mathcal {W}}[s]=0}$

at an admissible state ${\displaystyle s}$ if and only if ${\displaystyle s}$ is a solution of the mixed problem.