Advanced elasticity/Incompressible hyperelastic material

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Incompressible hyperelastic materials[edit]

For 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