View source for Physics:Gent (hyperelastic model)

{{Continuum mechanics|cTopic=[[Physics:Solid mechanics|Solid mechanics]]}}
The ''' Gent''' [[Physics:Hyperelastic material|hyperelastic material]] model <ref name=Gent/> is a phenomenological model of [[Physics:Rubber elasticity|rubber elasticity]] that is based on the concept of limiting chain extensibility.  In this model, the [[Physics:Strain energy density function|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 <math>I_m</math>.

The strain energy density function for the Gent model is <ref name=Gent>Gent, A.N., 1996, '' A new constitutive relation for rubber'', Rubber Chemistry Tech., 69, pp. 59-61.</ref>
:<math>
  W = -\cfrac{\mu J_m}{2} \ln\left(1 - \cfrac{I_1-3}{J_m}\right)
</math>
where <math>\mu</math> is the [[Physics:Shear modulus|shear modulus]] and <math>J_m = I_m -3</math>.

In the limit where <math>I_m \rightarrow \infty</math>, the Gent model reduces to the [[Physics:Neo-Hookean solid|Neo-Hookean solid]] model.  This can be seen by expressing the Gent model in the form
:<math>
   W =- \cfrac{\mu}{2x}\ln\left[1 - (I_1-3)x\right] ~;~~ x := \cfrac{1}{J_m}
 </math>
A Taylor series expansion of <math>\ln\left[1 - (I_1-3)x\right]</math> around <math>x = 0</math> and taking the limit as <math>x\rightarrow 0</math> leads to
:<math>
  W = \cfrac{\mu}{2} (I_1-3)
 </math>
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<ref>Mac Donald, B. J., 2007, '''Practical stress analysis with finite elements''', Glasnevin, Ireland.</ref> (the below strain energy function yields a non zero hydrostatic stress at no deformation, refer https://link.springer.com/article/10.1007/s10659-005-4408-x for compressible Gent models).
:<math>
    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
 </math>
where <math>J = \det(\boldsymbol{F})</math>, <math>\kappa</math> is the [[Physics:Bulk modulus|bulk modulus]], and <math>\boldsymbol{F}</math> is the [[Physics:Deformation gradient|deformation gradient]].

== Consistency condition ==
We may alternatively express the Gent model in the form
:<math>
  W = C_0 \ln\left(1 - \cfrac{I_1-3}{J_m}\right)
</math>
For the model to be consistent with [[Physics:Linear elasticity|linear elasticity]], the [[Physics:Hyperelastic material#Consistency conditions for incompressible I1 based rubber materials|following condition]] has to be satisfied:
:<math>
2\cfrac{\partial W}{\partial I_1}(3)  = \mu 
 </math>
where <math>\mu</math> is the [[Physics:Shear modulus|shear modulus]] of the material.
Now, at <math>I_1 = 3 (\lambda_i = \lambda_j = 1)</math>, 
:<math>
   \cfrac{\partial W}{\partial I_1} = -\cfrac{C_0}{J_m} 
 </math>
Therefore, the consistency condition for the Gent model is
:<math>
   -\cfrac{2C_0}{J_m} = \mu\, \qquad \implies \qquad C_0 = -\cfrac{\mu J_m}{2}
 </math>
The Gent model assumes that <math>J_m \gg 1</math>

== Stress-deformation relations ==
The Cauchy stress for the incompressible Gent model is given by
:<math>
   \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} 
 </math>

=== Uniaxial extension ===
[[Image:Hyperelastic.svg|thumb|350px|right|Stress-strain curves under uniaxial extension for Gent model compared with various hyperelastic material models.]]
For uniaxial extension in the <math>\mathbf{n}_1</math>-direction, the [[Physics:Finite strain theory|principal stretches]] are <math>\lambda_1 = \lambda,~ \lambda_2=\lambda_3</math>.  From incompressibility <math>\lambda_1~\lambda_2~\lambda_3=1</math>.  Hence <math>\lambda_2^2=\lambda_3^2=1/\lambda</math>. 
Therefore,
:<math>
   I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = \lambda^2 + \cfrac{2}{{\lambda}} ~.
 </math>
The [[Physics:Finite strain theory|left Cauchy-Green deformation tensor]] can then be expressed as
:<math>
   \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) ~.
 </math>
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
:<math>
     \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} ~.
 </math>
If <math>\sigma_{22} = \sigma_{33} = 0</math>, we have
:<math>
   p =  \cfrac{\mu J_m}{\lambda(J_m - I_1 + 3)}~.
 </math>
Therefore,
:<math>
   \sigma_{11} = \left(\lambda^2 - \cfrac{1}{\lambda}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.
 </math>
The [[Physics:Stress|engineering strain]] is <math>\lambda-1\,</math>.  The [[Physics:Stress|engineering stress]] is 
:<math>
  T_{11} = \sigma_{11}/\lambda = 
     \left(\lambda - \cfrac{1}{\lambda^2}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.
 </math>

=== Equibiaxial extension ===
For equibiaxial extension in the <math>\mathbf{n}_1</math> and <math>\mathbf{n}_2</math> directions, the [[Physics:Finite strain theory|principal stretches]] are <math>\lambda_1 = \lambda_2 = \lambda\,</math>.  From incompressibility <math>\lambda_1~\lambda_2~\lambda_3=1</math>.  Hence <math>\lambda_3=1/\lambda^2\,</math>.  
Therefore,
:<math>
   I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = 2~\lambda^2 + \cfrac{1}{\lambda^4} ~.
 </math>
The [[Physics:Finite strain theory|left Cauchy-Green deformation tensor]] can then be expressed as
:<math>
   \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 ~.
 </math>
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
:<math>
   \sigma_{11} = \left(\lambda^2 - \cfrac{1}{\lambda^4}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right) = \sigma_{22} ~.
 </math>
The [[Physics:Stress|engineering strain]] is <math>\lambda-1\,</math>.  The [[Physics:Stress|engineering stress]] is 
:<math>
  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}~.
 </math>

=== Planar extension ===
Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction.  For planar extension in the <math>\mathbf{n}_1</math> directions with the <math>\mathbf{n}_3</math> direction constrained, the [[Physics:Finite strain theory|principal stretches]] are <math>\lambda_1=\lambda, ~\lambda_3=1</math>.  From incompressibility <math>\lambda_1~\lambda_2~\lambda_3=1</math>.  Hence <math>\lambda_2=1/\lambda\,</math>.  
Therefore,
:<math>
   I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = \lambda^2 + \cfrac{1}{\lambda^2} + 1 ~.
 </math>
The [[Physics:Finite strain theory|left Cauchy-Green deformation tensor]] can then be expressed as
:<math>
   \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 ~.
 </math>
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
:<math>
   \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)~.
 </math>
The [[Physics:Stress|engineering strain]] is <math>\lambda-1\,</math>.  The [[Physics:Stress|engineering stress]] is 
:<math>
  T_{11} = \cfrac{\sigma_{11}}{\lambda} = 
     \left(\lambda - \cfrac{1}{\lambda^3}\right)\left(\cfrac{\mu J_m}{J_m - I_1 + 3}\right)~.
 </math>

=== Simple shear ===
The deformation gradient for a [[Physics:Simple shear|simple shear]] deformation has the form<ref name=Ogden>Ogden, R. W., 1984, '''Non-linear elastic deformations''', Dover.</ref>
:<math>
   \boldsymbol{F} = \boldsymbol{1} + \gamma~\mathbf{e}_1\otimes\mathbf{e}_2
 </math>
where <math>\mathbf{e}_1,\mathbf{e}_2</math> are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by
:<math>
   \gamma = \lambda - \cfrac{1}{\lambda} ~;~~ \lambda_1 = \lambda ~;~~ \lambda_2 = \cfrac{1}{\lambda} ~;~~ \lambda_3 = 1
 </math>
In matrix form, the deformation gradient and the left Cauchy-Green deformation tensor may then be expressed as
:<math>
   \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}
 </math> 
Therefore, 
:<math>
   I_1 = \mathrm{tr}(\boldsymbol{B}) = 3 + \gamma^2 
 </math>
and the Cauchy stress is given by
:<math>
   \boldsymbol{\sigma} = -p~\boldsymbol{\mathit{1}} + \cfrac{\mu J_m}{J_m - \gamma^2}~\boldsymbol{B} 
 </math>
In matrix form,
:<math>
   \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}
 </math>

==References==
<references/>

== See also ==
* [[Physics:Hyperelastic material|Hyperelastic material]]
* [[Physics:Strain energy density function|Strain energy density function]]
* Mooney-Rivlin solid
* [[Physics:Finite strain theory|Finite strain theory]]
* [[Physics:Stress measures|Stress measures]]

[[Category:Continuum mechanics]]
[[Category:Elasticity (physics)]]
[[Category:Non-Newtonian fluids]]
[[Category:Rubber properties]]
[[Category:Solid mechanics]]

{{Sourceattribution|Gent (hyperelastic model)|1}}