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 formula_1.
The strain energy density function for the Gent model is 
where formula_3 is the shear modulus and formula_4.
In the limit where formula_5, the Gent model reduces to the Neo-Hookean solid model. This can be seen by expressing the Gent model in the form
A Taylor series expansion of formula_7 around formula_8 and taking the limit as formula_9 leads to
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 https://link.springer.com/article/10.1007/s10659-005-4408-x for compressible Gent models).
where formula_12, formula_13 is the bulk modulus, and formula_14 is the deformation gradient.
Consistency condition.
We may alternatively express the Gent model in the form
For the model to be consistent with linear elasticity, the following condition has to be satisfied:
where formula_3 is the shear modulus of the material.
Now, at formula_18, 
Therefore, the consistency condition for the Gent model is
The Gent model assumes that formula_21
Stress-deformation relations.
The Cauchy stress for the incompressible Gent model is given by
Uniaxial extension.
For uniaxial extension in the formula_23-direction, the principal stretches are formula_24. From incompressibility formula_25. Hence formula_26. 
Therefore,
The left Cauchy-Green deformation tensor can then be expressed as
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
If formula_30, we have
Therefore,
The engineering strain is formula_33. The engineering stress is 
Equibiaxial extension.
For equibiaxial extension in the formula_23 and formula_36 directions, the principal stretches are formula_37. From incompressibility formula_25. Hence formula_39. 
Therefore,
The left Cauchy-Green deformation tensor can then be expressed as
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
The engineering strain is formula_33. The engineering stress is 
Planar extension.
Planar extension tests are carried out on thin specimens which are constrained from deforming in one direction. For planar extension in the formula_23 directions with the formula_46 direction constrained, the principal stretches are formula_47. From incompressibility formula_25. Hence formula_49. 
Therefore,
The left Cauchy-Green deformation tensor can then be expressed as
If the directions of the principal stretches are oriented with the coordinate basis vectors, we have
The engineering strain is formula_33. The engineering stress is 
Simple shear.
The deformation gradient for a simple shear deformation has the form
where formula_56 are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by
In matrix form, the deformation gradient and the left Cauchy-Green deformation tensor may then be expressed as
Therefore, 
and the Cauchy stress is given by
In matrix form,