Introduction to Elasticity/Hellinger-Reissner principle

Hellinger-Prange-Reissner Variational Principle

In this case, we assume that the elasticity field is invertible and ${\displaystyle s}$ is smooth on ${\displaystyle {\mathcal {B}}}$. We also assume that ${\displaystyle {\mathcal {A}}}$ is the set of all admissible states that satisfy the strain-displacement relations, the traction-stress relations and the balance of angular momentum.

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

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

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

Then,

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

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