Gent (hyperelastic model)

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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 Im.

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

W=−μJm2ln⁡(1−I1−3Jm)

where μ is the shear modulus and Jm=Im−3.

In the limit where Im->∞, the Gent model reduces to the Neo-Hookean solid model. This can be seen by expressing the Gent model in the form

W=−μ2xln⁡[1−(I1−3)x];x:=1Jm

A Taylor series expansion of ln⁡[1−(I1−3)x] around x=0 and taking the limit as x->0 leads to

W=μ2(I1−3)

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 https://link.springer.com/article/10.1007/s10659-005-4408-x for compressible Gent models).

W=−μJm2ln⁡(1−I1−3Jm)+κ2(J2−12−ln⁡J)4

where J=det⁡(F), κ is the bulk modulus, and F is the deformation gradient.

## Contents

* 1 Consistency condition
* 2 Stress-deformation relations
* 2.1 Uniaxial extension
* 2.2 Equibiaxial extension
* 2.3 Planar extension
* 2.4 Simple shear
* 3 References
* 4 See also

## Consistency condition

We may alternatively express the Gent model in the form

W=C0ln⁡(1−I1−3Jm)

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

2∂W∂I1(3)=μ

where μ is the shear modulus of the material. Now, at I1=3(λi=λj=1),

∂W∂I1=−C0Jm

Therefore, the consistency condition for the Gent model is

−2C0Jm=μ⟹C0=−μJm2

The Gent model assumes that Jm≫1

## Stress-deformation relations

The Cauchy stress for the incompressible Gent model is given by

σ=−pI+2∂W∂I1B=−pI+μJmJm−I1+3B

### Uniaxial extension

Stress-strain curves under uniaxial extension for Gent model compared with various hyperelastic material models.

For uniaxial extension in the 𝐧1-direction, the principal stretches are λ1=λ,λ2=λ3. From incompressibility λ1λ2λ3=1. Hence λ22=λ32=1/λ. Therefore,

I1=λ12+λ22+λ32=λ2+2λ.

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

B=λ2𝐧1⊗𝐧1+1λ(𝐧2⊗𝐧2+𝐧3⊗𝐧3).

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

σ11=−p+λ2μJmJm−I1+3;σ22=−p+μJmλ(Jm−I1+3)=σ33.

If σ22=σ33=0, we have

p=μJmλ(Jm−I1+3).

Therefore,

σ11=(λ2−1λ)(μJmJm−I1+3).

The engineering strain is λ−1. The engineering stress is

T11=σ11/λ=(λ−1λ2)(μJmJm−I1+3).

### Equibiaxial extension

For equibiaxial extension in the 𝐧1 and 𝐧2 directions, the principal stretches are λ1=λ2=λ. From incompressibility λ1λ2λ3=1. Hence λ3=1/λ2. Therefore,

I1=λ12+λ22+λ32=2λ2+1λ4.

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

B=λ2𝐧1⊗𝐧1+λ2𝐧2⊗𝐧2+1λ4𝐧3⊗𝐧3.

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

σ11=(λ2−1λ4)(μJmJm−I1+3)=σ22.

The engineering strain is λ−1. The engineering stress is

T11=σ11λ=(λ−1λ5)(μJmJm−I1+3)=T22.

### Planar extension

Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the 𝐧1 directions with the 𝐧3 direction constrained, the principal stretches are λ1=λ,λ3=1. From incompressibility λ1λ2λ3=1. Hence λ2=1/λ. Therefore,

I1=λ12+λ22+λ32=λ2+1λ2+1.

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

B=λ2𝐧1⊗𝐧1+1λ2𝐧2⊗𝐧2+𝐧3⊗𝐧3.

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

σ11=(λ2−1λ2)(μJmJm−I1+3);σ22=0;σ33=(1−1λ2)(μJmJm−I1+3).

The engineering strain is λ−1. The engineering stress is

T11=σ11λ=(λ−1λ3)(μJmJm−I1+3).

### Simple shear

The deformation gradient for a simple shear deformation has the form[3]

F=1+γ𝐞1⊗𝐞2

where 𝐞1,𝐞2 are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by

γ=λ−1λ;λ1=λ;λ2=1λ;λ3=1

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

F=[1γ0010001];B=F⋅FT=[1+γ2γ0γ10001]

Therefore,

I1=tr(B)=3+γ2

and the Cauchy stress is given by

σ=−p1+μJmJm−γ2B

In matrix form,

σ=[−p+μJm(1+γ2)Jm−γ2μJmγJm−γ20μJmγJm−γ2−p+μJmJm−γ2000−p+μJmJm−γ2]

## References

1. ↑ 1.0 1.1 Gent, A.N., 1996,  A new constitutive relation for rubber, Rubber Chemistry Tech., 69, pp. 59-61.
2. ↑ Mac Donald, B. J., 2007, Practical stress analysis with finite elements, Glasnevin, Ireland.
3. ↑ Ogden, R. W., 1984, Non-linear elastic deformations, Dover.

## See also

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

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