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.

Elasticity