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 $𝒜$ denote the set of all admissible states and let $𝒲$ be a functional on $𝒜$ defined by

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

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

Then,

$\delta 𝒲[s]=0$

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

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