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 $ℬ$. We also assume that $𝒜$ is the set of all admissible states that satisfy the strain-displacement relations, the traction-stress relations and the balance of angular momentum.

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

$\mathscr{H}[s]=\underset{ℬ}{\int }{U}^c(\mathit{\sigma })-\underset{ℬ}{\int }\mathit{\sigma }:\mathit{\epsilon }dV+\underset{ℬ}{\int }𝐟\bullet 𝐮dV+\underset{\partial {ℬ}^u}{\int }𝐭\bullet (𝐮-\widehat{𝐮})dA+\underset{\partial {ℬ}^t}{\int }\widehat{𝐭}\bullet 𝐮dA$

for every $s=[𝐮,\mathit{\sigma }]\in 𝒜$.

Then,

$\delta \mathscr{H}[s]=0$

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