Gent hyperelastic model

Part of a series on
Continuum mechanics
${\displaystyle J=-D{\frac {d\varphi }{dx}}}$ Fick's laws of diffusion
Laws Conservations Mass Momentum Energy Inequalities Clausius–Duhem (entropy)
Solid mechanics Deformation Elasticity linear Plasticity Hooke's law Stress Strain Finite strain Infinitesimal strain Compatibility Bending Contact mechanics frictional Material failure theory Fracture mechanics
Fluid mechanics Fluids Statics · Dynamics Archimedes' principle · Bernoulli's principle Navier–Stokes equations Poiseuille equation · Pascal's law Viscosity (Newtonian · non-Newtonian) Buoyancy · Mixing · Pressure Liquids Adhesion Capillary action Chromatography Cohesion (chemistry) Surface tension Gases Atmosphere Boyle's law Charles's law Combined gas law Fick's law Gay-Lussac's law Graham's law Plasma
Rheology Viscoelasticity Rheometry Rheometer Smart fluids Electrorheological Magnetorheological Ferrofluids
Scientists Bernoulli Boyle Cauchy Charles Euler Fick Gay-Lussac Graham Hooke Newton Navier Noll Pascal Stokes Truesdell
v t e

The Gent hyperelastic material model ^[1] 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 ${\displaystyle I_{m}}$.

The strain energy density function for the Gent model is ^[1]

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

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

In the limit where ${\displaystyle J_{m}\rightarrow \infty }$, 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^[2] (the below strain energy function yields a non zero hydrostatic stress at no deformation, refer^[3] 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.

We may alternatively express the Gent model in the form

${\displaystyle W=C_{0}\ln \left(1-{\cfrac {I_{1}-3}{J_{m}}}\right)}$

For the model to be consistent with linear elasticity, the following condition has to be satisfied:

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

where ${\displaystyle \mu }$ is the shear modulus of the material. Now, at ${\displaystyle I_{1}=3(\lambda _{i}=\lambda _{j}=1)}$,

${\displaystyle {\cfrac {\partial W}{\partial I_{1}}}=-{\cfrac {C_{0}}{J_{m}}}}$

Therefore, the consistency condition for the Gent model is

${\displaystyle -{\cfrac {2C_{0}}{J_{m}}}=\mu \,\qquad \implies \qquad C_{0}=-{\cfrac {\mu J_{m}}{2}}}$

The Gent model assumes that ${\displaystyle J_{m}\gg 1}$

The Cauchy stress for the incompressible Gent model is given by

${\displaystyle {\boldsymbol {\sigma }}=-p~{\boldsymbol {\mathit {I}}}+2~{\cfrac {\partial W}{\partial I_{1}}}~{\boldsymbol {B}}=-p~{\boldsymbol {\mathit {I}}}+{\cfrac {\mu J_{m}}{J_{m}-I_{1}+3}}~{\boldsymbol {B}}}$

For uniaxial extension in the ${\displaystyle \mathbf {n} _{1}}$-direction, the principal stretches are ${\displaystyle \lambda _{1}=\lambda ,~\lambda _{2}=\lambda _{3}}$. From incompressibility ${\displaystyle \lambda _{1}~\lambda _{2}~\lambda _{3}=1}$. Hence ${\displaystyle \lambda _{2}^{2}=\lambda _{3}^{2}=1/\lambda }$. Therefore,

${\displaystyle I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=\lambda ^{2}+{\cfrac {2}{\lambda }}~.}$

The left Cauchy-Green deformation tensor can then be expressed as

${\displaystyle {\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\cfrac {1}{\lambda }}~(\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\mathbf {n} _{3}\otimes \mathbf {n} _{3})~.}$

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

${\displaystyle \sigma _{11}=-p+{\cfrac {\lambda ^{2}\mu J_{m}}{J_{m}-I_{1}+3}}~;~~\sigma _{22}=-p+{\cfrac {\mu J_{m}}{\lambda (J_{m}-I_{1}+3)}}=\sigma _{33}~.}$

If ${\displaystyle \sigma _{22}=\sigma _{33}=0}$, we have

${\displaystyle p={\cfrac {\mu J_{m}}{\lambda (J_{m}-I_{1}+3)}}~.}$

Therefore,

${\displaystyle \sigma _{11}=\left(\lambda ^{2}-{\cfrac {1}{\lambda }}\right)\left({\cfrac {\mu J_{m}}{J_{m}-I_{1}+3}}\right)~.}$

The engineering strain is ${\displaystyle \lambda -1\,}$. The engineering stress is

${\displaystyle T_{11}=\sigma _{11}/\lambda =\left(\lambda -{\cfrac {1}{\lambda ^{2}}}\right)\left({\cfrac {\mu J_{m}}{J_{m}-I_{1}+3}}\right)~.}$

For equibiaxial extension in the ${\displaystyle \mathbf {n} _{1}}$ and ${\displaystyle \mathbf {n} _{2}}$ directions, the principal stretches are ${\displaystyle \lambda _{1}=\lambda _{2}=\lambda \,}$. From incompressibility ${\displaystyle \lambda _{1}~\lambda _{2}~\lambda _{3}=1}$. Hence ${\displaystyle \lambda _{3}=1/\lambda ^{2}\,}$. Therefore,

${\displaystyle I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=2~\lambda ^{2}+{\cfrac {1}{\lambda ^{4}}}~.}$

The left Cauchy-Green deformation tensor can then be expressed as

${\displaystyle {\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+\lambda ^{2}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+{\cfrac {1}{\lambda ^{4}}}~\mathbf {n} _{3}\otimes \mathbf {n} _{3}~.}$

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

${\displaystyle \sigma _{11}=\left(\lambda ^{2}-{\cfrac {1}{\lambda ^{4}}}\right)\left({\cfrac {\mu J_{m}}{J_{m}-I_{1}+3}}\right)=\sigma _{22}~.}$

The engineering strain is ${\displaystyle \lambda -1\,}$. The engineering stress is

${\displaystyle T_{11}={\cfrac {\sigma _{11}}{\lambda }}=\left(\lambda -{\cfrac {1}{\lambda ^{5}}}\right)\left({\cfrac {\mu J_{m}}{J_{m}-I_{1}+3}}\right)=T_{22}~.}$

Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the ${\displaystyle \mathbf {n} _{1}}$ directions with the ${\displaystyle \mathbf {n} _{3}}$ direction constrained, the principal stretches are ${\displaystyle \lambda _{1}=\lambda ,~\lambda _{3}=1}$. From incompressibility ${\displaystyle \lambda _{1}~\lambda _{2}~\lambda _{3}=1}$. Hence ${\displaystyle \lambda _{2}=1/\lambda \,}$. Therefore,

${\displaystyle I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=\lambda ^{2}+{\cfrac {1}{\lambda ^{2}}}+1~.}$

The left Cauchy-Green deformation tensor can then be expressed as

${\displaystyle {\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\cfrac {1}{\lambda ^{2}}}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\mathbf {n} _{3}\otimes \mathbf {n} _{3}~.}$

If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

${\displaystyle \sigma _{11}=\left(\lambda ^{2}-{\cfrac {1}{\lambda ^{2}}}\right)\left({\cfrac {\mu J_{m}}{J_{m}-I_{1}+3}}\right)~;~~\sigma _{22}=0~;~~\sigma _{33}=\left(1-{\cfrac {1}{\lambda ^{2}}}\right)\left({\cfrac {\mu J_{m}}{J_{m}-I_{1}+3}}\right)~.}$

The engineering strain is ${\displaystyle \lambda -1\,}$. The engineering stress is

${\displaystyle T_{11}={\cfrac {\sigma _{11}}{\lambda }}=\left(\lambda -{\cfrac {1}{\lambda ^{3}}}\right)\left({\cfrac {\mu J_{m}}{J_{m}-I_{1}+3}}\right)~.}$

The deformation gradient for a simple shear deformation has the form^[4]

${\displaystyle {\boldsymbol {F}}={\boldsymbol {1}}+\gamma ~\mathbf {e} _{1}\otimes \mathbf {e} _{2}}$

where ${\displaystyle \mathbf {e} _{1},\mathbf {e} _{2}}$ are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by

${\displaystyle \gamma =\lambda -{\cfrac {1}{\lambda }}~;~~\lambda _{1}=\lambda ~;~~\lambda _{2}={\cfrac {1}{\lambda }}~;~~\lambda _{3}=1}$

In matrix form, the deformation gradient and the left Cauchy-Green deformation tensor may then be expressed as

${\displaystyle {\boldsymbol {F}}={\begin{bmatrix}1&\gamma &0\\0&1&0\\0&0&1\end{bmatrix}}~;~~{\boldsymbol {B}}={\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}={\begin{bmatrix}1+\gamma ^{2}&\gamma &0\\\gamma &1&0\\0&0&1\end{bmatrix}}}$

Therefore,

${\displaystyle I_{1}=\mathrm {tr} ({\boldsymbol {B}})=3+\gamma ^{2}}$

and the Cauchy stress is given by

${\displaystyle {\boldsymbol {\sigma }}=-p~{\boldsymbol {\mathit {1}}}+{\cfrac {\mu J_{m}}{J_{m}-\gamma ^{2}}}~{\boldsymbol {B}}}$

In matrix form,

${\displaystyle {\boldsymbol {\sigma }}={\begin{bmatrix}-p+{\cfrac {\mu J_{m}(1+\gamma ^{2})}{J_{m}-\gamma ^{2}}}&{\cfrac {\mu J_{m}\gamma }{J_{m}-\gamma ^{2}}}&0\\{\cfrac {\mu J_{m}\gamma }{J_{m}-\gamma ^{2}}}&-p+{\cfrac {\mu J_{m}}{J_{m}-\gamma ^{2}}}&0\\0&0&-p+{\cfrac {\mu J_{m}}{J_{m}-\gamma ^{2}}}\end{bmatrix}}}$

• Hyperelastic material
• Strain energy density function
• Mooney-Rivlin solid
• Finite strain theory
• Stress measures