Advanced elasticity/Incompressible hyperelastic material

From Wikiversity

Jump to: navigation, search

[edit] Incompressible hyperelastic materials

For an w:incompressible material J := \det\boldsymbol{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(\boldsymbol{F}) - p~(J-1)

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

\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

\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 W(\boldsymbol{F})=\hat{W}(I_1,I_2). The Cauchy stress is then given by

\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]
Retrieved from "http://en.wikiversity.org/wiki/Advanced_elasticity/Incompressible_hyperelastic_material"