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.

Elasticity