Advanced elasticity/Incompressible hyperelastic material

Incompressible hyperelastic materials

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

$W=W(𝑭)-p(J-1)$

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

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

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

$\mathit{\sigma }=𝑷\cdot {𝑭}^T=-p1+\frac{\partial W}{\partial 𝑭}\cdot {𝑭}^T=-p1+𝑭\cdot \frac{\partial W}{\partial 𝑬}\cdot {𝑭}^T=-p1+2𝑭\cdot \frac{\partial W}{\partial 𝑪}\cdot {𝑭}^T.$

For incompressible w:isotropic hyperelastic materials, the w:strain energy density function is $W(𝑭)=\hat{W}(I_1,I_2)$. The Cauchy stress is then given by

$\mathit{\sigma }=-p1+2[(\frac{\partial \hat{W}}{\partial I_1}+I_1\frac{\partial \hat{W}}{\partial I_2})𝑩-\frac{\partial \hat{W}}{\partial I_2}𝑩\cdot 𝑩]$