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== Incompressible hyperelastic materials ==
For an [[w:incompressible]] material <math>J := \det\boldsymbol{F} = 1</math>.  The incompressibility constraint is therefore <math>J-1= 0</math>.  To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:
:<math>
W = W(\boldsymbol{F}) - p~(J-1)
</math>
where the hydrostatic pressure <math>p</math> functions as a [[w:Lagrange_multipliers|Lagrangian multiplier]] to enforce the incompressibility constraint. The 1<sup>st</sup> Piola-Kirchhoff stress now becomes
:<math>
\boldsymbol{P}=-p~\boldsymbol{F}^{-T}+\frac{\partial W}{\partial \boldsymbol{F}}
  = -p~\boldsymbol{F}^{-T} + \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}}
  = -p~\boldsymbol{F}^{-T} + 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}} ~.
</math>
This stress tensor can subsequently be [[w:stress (physics)|converted]] into any of the other conventional stress tensors, such as the [[w:Cauchy_stress_tensor|Cauchy Stress tensor]] which is given by
:<math>
\boldsymbol{\sigma}=\boldsymbol{P}\cdot\boldsymbol{F}^T=
   -p~\boldsymbol{\mathit{1}} + \frac{\partial W}{\partial \boldsymbol{F}}\cdot\boldsymbol{F}^T
  = -p~\boldsymbol{\mathit{1}} + \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}}\cdot\boldsymbol{F}^T
  = -p~\boldsymbol{\mathit{1}} + 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^T ~.
</math>
For incompressible [[w:isotropic]] hyperelastic materials, the [[w:strain energy density function]] is <math>W(\boldsymbol{F})=\hat{W}(I_1,I_2)</math>.  The Cauchy stress is then given by
:<math>
   \boldsymbol{\sigma}  = -p~\boldsymbol{\mathit{1}} + 
     2\left[\left(\cfrac{\partial\hat{W}}{\partial I_1} + I_1~\cfrac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \cfrac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right]
 </math>

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[[Category:Continuum mechanics]]