Advanced elasticity/Incompressible hyperelastic material
Incompressible hyperelastic materials
editFor an w:incompressible material . The incompressibility constraint is therefore . To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:
where the hydrostatic pressure functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola-Kirchhoff stress now becomes
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
For incompressible w:isotropic hyperelastic materials, the w:strain energy density function is . The Cauchy stress is then given by