# Advanced elasticity/Incompressible hyperelastic material

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## Incompressible hyperelastic materials

For an w:incompressible material ${\displaystyle J:=\det {\boldsymbol {F}}=1}$. The incompressibility constraint is therefore ${\displaystyle J-1=0}$. To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:

${\displaystyle W=W({\boldsymbol {F}})-p~(J-1)}$

where the hydrostatic pressure ${\displaystyle p}$ functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola-Kirchhoff stress now becomes

${\displaystyle {\boldsymbol {P}}=-p~{\boldsymbol {F}}^{-T}+{\frac {\partial W}{\partial {\boldsymbol {F}}}}=-p~{\boldsymbol {F}}^{-T}+{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {E}}}}=-p~{\boldsymbol {F}}^{-T}+2~{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {C}}}}~.}$

This stress tensor can subsequently be converted into any of the other conventional stress tensors, such as the Cauchy Stress tensor which is given by

${\displaystyle {\boldsymbol {\sigma }}={\boldsymbol {P}}\cdot {\boldsymbol {F}}^{T}=-p~{\boldsymbol {\mathit {1}}}+{\frac {\partial W}{\partial {\boldsymbol {F}}}}\cdot {\boldsymbol {F}}^{T}=-p~{\boldsymbol {\mathit {1}}}+{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {E}}}}\cdot {\boldsymbol {F}}^{T}=-p~{\boldsymbol {\mathit {1}}}+2~{\boldsymbol {F}}\cdot {\frac {\partial W}{\partial {\boldsymbol {C}}}}\cdot {\boldsymbol {F}}^{T}~.}$

For incompressible w:isotropic hyperelastic materials, the w:strain energy density function is ${\displaystyle W({\boldsymbol {F}})={\hat {W}}(I_{1},I_{2})}$. The Cauchy stress is then given by

${\displaystyle {\boldsymbol {\sigma }}=-p~{\boldsymbol {\mathit {1}}}+2\left[\left({\cfrac {\partial {\hat {W}}}{\partial I_{1}}}+I_{1}~{\cfrac {\partial {\hat {W}}}{\partial I_{2}}}\right){\boldsymbol {B}}-{\cfrac {\partial {\hat {W}}}{\partial I_{2}}}~{\boldsymbol {B}}\cdot {\boldsymbol {B}}\right]}$