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 $I_m$.

The strain energy density function for the Gent model is Gent, A.N., 1996, '' A new constitutive relation for rubber'', Rubber Chemistry Tech., 69, pp. 59-61.
:$W = -\cfrac \ln\left(1 - \cfrac\right)$
where $\mu$ is the shear modulus and $J_m = I_m -3$.

In the limit where $I_m \rightarrow \infty$, the Gent model reduces to the Neo-Hookean solid model. This can be seen by expressing the Gent model in the form
:
W =- \cfrac\ln\left - (I_1-3)x\right~;~~ x := \cfrac

A Taylor series expansion of \ln\left - (I_1-3)x\right/math> around $x = 0$ and taking the limit as $x\rightarrow 0$ leads to
:$W = \cfrac (I_1-3)$
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 formMac Donald, B. J., 2007, Practical stress analysis with finite elements, Glasnevin, Ireland. (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).
:$W = -\cfrac \ln\left(1 - \cfrac\right) + \cfrac\left(\cfrac - \ln J\right)^4$
where $J = \det(\boldsymbol)$, $\kappa$ is the bulk modulus, and $\boldsymbol$ is the deformation gradient.

Consistency condition

We may alternatively express the Gent model in the form
:$W = C_0 \ln\left(1 - \cfrac\right)$
For the model to be consistent with linear elasticity, the following condition has to be satisfied:
:$2\cfrac(3) = \mu$
where $\mu$ is the shear modulus of the material.
Now, at $I_1 = 3 (\lambda_i = \lambda_j = 1)$,
:$\cfrac = -\cfrac$
Therefore, the consistency condition for the Gent model is
:$-\cfrac = \mu\, \qquad \implies \qquad C_0 = -\cfrac$
The Gent model assumes that $J_m \gg 1$

Stress-deformation relations

The Cauchy stress for the incompressible Gent model is given by
:$\boldsymbol = -p~\boldsymbol + 2~\cfrac~\boldsymbol = -p~\boldsymbol + \cfrac~\boldsymbol$

Uniaxial extension

For uniaxial extension in the $\mathbf_1$-direction, the principal stretches are $\lambda_1 = \lambda,~ \lambda_2=\lambda_3$. From incompressibility $\lambda_1~\lambda_2~\lambda_3=1$. Hence $\lambda_2^2=\lambda_3^2=1/\lambda$.
Therefore,
:$I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = \lambda^2 + \cfrac ~.$
The left Cauchy-Green deformation tensor can then be expressed as
:$\boldsymbol = \lambda^2~\mathbf_1\otimes\mathbf_1 + \cfrac~(\mathbf_2\otimes\mathbf_2+\mathbf_3\otimes\mathbf_3) ~.$
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
:$\sigma_ = -p + \cfrac ~;~~ \sigma_ = -p + \cfrac = \sigma_ ~.$
If $\sigma_ = \sigma_ = 0$, we have
:$p = \cfrac~.$
Therefore,
:$\sigma_ = \left(\lambda^2 - \cfrac\right)\left(\cfrac\right)~.$
The engineering strain is $\lambda-1\,$. The engineering stress is
:$T_ = \sigma_/\lambda = \left(\lambda - \cfrac\right)\left(\cfrac\right)~.$

Equibiaxial extension

For equibiaxial extension in the $\mathbf_1$ and $\mathbf_2$ directions, the principal stretches are $\lambda_1 = \lambda_2 = \lambda\,$. From incompressibility $\lambda_1~\lambda_2~\lambda_3=1$. Hence $\lambda_3=1/\lambda^2\,$.
Therefore,
:$I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = 2~\lambda^2 + \cfrac ~.$
The left Cauchy-Green deformation tensor can then be expressed as
:$\boldsymbol = \lambda^2~\mathbf_1\otimes\mathbf_1 + \lambda^2~\mathbf_2\otimes\mathbf_2+ \cfrac~\mathbf_3\otimes\mathbf_3 ~.$
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
:$\sigma_ = \left(\lambda^2 - \cfrac\right)\left(\cfrac\right) = \sigma_ ~.$
The engineering strain is $\lambda-1\,$. The engineering stress is
:$T_ = \cfrac = \left(\lambda - \cfrac\right)\left(\cfrac\right) = T_~.$

Planar extension

Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the $\mathbf_1$ directions with the $\mathbf_3$ direction constrained, the principal stretches are $\lambda_1=\lambda, ~\lambda_3=1$. From incompressibility $\lambda_1~\lambda_2~\lambda_3=1$. Hence $\lambda_2=1/\lambda\,$.
Therefore,
:$I_1 = \lambda_1^2+\lambda_2^2+\lambda_3^2 = \lambda^2 + \cfrac + 1 ~.$
The left Cauchy-Green deformation tensor can then be expressed as
:$\boldsymbol = \lambda^2~\mathbf_1\otimes\mathbf_1 + \cfrac~\mathbf_2\otimes\mathbf_2+ \mathbf_3\otimes\mathbf_3 ~.$
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
:$\sigma_ = \left(\lambda^2 - \cfrac\right)\left(\cfrac\right) ~;~~ \sigma_ = 0 ~;~~ \sigma_ = \left(1 - \cfrac\right)\left(\cfrac\right)~.$
The engineering strain is $\lambda-1\,$. The engineering stress is
:$T_ = \cfrac = \left(\lambda - \cfrac\right)\left(\cfrac\right)~.$

Simple shear

The deformation gradient for a simple shear deformation has the formOgden, R. W., 1984, Non-linear elastic deformations, Dover.
:$\boldsymbol = \boldsymbol + \gamma~\mathbf_1\otimes\mathbf_2$
where $\mathbf_1,\mathbf_2$ are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by
:$\gamma = \lambda - \cfrac ~;~~ \lambda_1 = \lambda ~;~~ \lambda_2 = \cfrac ~;~~ \lambda_3 = 1$
In matrix form, the deformation gradient and the left Cauchy-Green deformation tensor may then be expressed as
:$\boldsymbol = \begin 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ~;~~ \boldsymbol = \boldsymbol\cdot\boldsymbol^T = \begin 1+\gamma^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end$
Therefore,
:$I_1 = \mathrm(\boldsymbol) = 3 + \gamma^2$
and the Cauchy stress is given by
:$\boldsymbol = -p~\boldsymbol + \cfrac~\boldsymbol$
In matrix form,
:$\boldsymbol = \begin -p +\cfrac & \cfrac & 0 \\ \cfrac & -p + \cfrac & 0 \\ 0 & 0 & -p + \cfrac \end{bmatrix}$

References

See also

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

Continuum mechanics
Elasticity (physics)
Non-Newtonian fluids
Rubber properties
Solid mechanics