View source for Elasticity/Hellinger-Reissner principle

== Hellinger-Prange-Reissner Variational Principle ==

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

Let <math>\mathcal{A}</math> denote the set of all admissible states and let <math>\mathcal{H}</math> be a functional on <math>\mathcal{A}</math> defined by
:<math>
  {\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
</math>
for every <math>s = [\mathbf{u},\boldsymbol{\sigma}] \in \mathcal{A}</math>.

Then,
:<math>
  \delta {\mathcal H}[s] = 0
</math>
at an admissible state <math>s\in\mathcal{A}</math> if and only if <math>s</math> is a solution of the mixed problem.

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[[Category:Elasticity]]