Gent hyperelastic model

The Gent hyperelastic material model is a phenomenological model of rubber elasticity that is based on the concept of limiting chain extensibility. In this model, the strain energy density function is designed such that it has a singularity when the first invariant of the left Cauchy-Green deformation tensor reaches a limiting value ${\displaystyle I_{m}}$.

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The strain energy density function for the Gent model is

${\displaystyle W=-{\cfrac {\mu J_{m}}{2}}\ln \left(1-{\cfrac {I_{1}-3}{J_{m}}}\right)}$

where ${\displaystyle \mu }$ is the shear modulus and ${\displaystyle J_{m}=I_{m}-3}$.

In the limit where ${\displaystyle I_{m}\rightarrow \infty }$, the Gent model reduces to the Neo-Hookean solid model. This can be seen by expressing the Gent model in the form

${\displaystyle W=-{\cfrac {\mu }{2x}}\ln \left[1-(I_{1}-3)x\right]~;~~x:={\cfrac {1}{J_{m}}}}$

A Taylor series expansion of ${\displaystyle \ln \left[1-(I_{1}-3)x\right]}$ around ${\displaystyle x=0}$ and taking the limit as ${\displaystyle x\rightarrow 0}$ leads to

${\displaystyle W={\cfrac {\mu }{2}}(I_{1}-3)}$

which is the expression for the strain energy density of a Neo-Hookean solid.

Several compressible versions of the Gent model have been designed. One such model has the form (the below strain energy function yields a non zero hydrostatic stress at no deformation, refer for compressible Gent models).

${\displaystyle W=-{\cfrac {\mu J_{m}}{2}}\ln \left(1-{\cfrac {I_{1}-3}{J_{m}}}\right)+{\cfrac {\kappa }{2}}\left({\cfrac {J^{2}-1}{2}}-\ln J\right)^{4}}$

where ${\displaystyle J=\det({\boldsymbol {F}})}$, ${\displaystyle \kappa }$ is the bulk modulus, and ${\displaystyle {\boldsymbol {F}}}$ is the deformation gradient.