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
is the shear modulus of the material.
Now, at
,

Therefore, the consistency condition for the Gent model is

The Gent model assumes that
Stress-deformation relations
The Cauchy stress for the incompressible Gent model is given by

Uniaxial extension
Stress-strain curves under uniaxial extension for Gent model compared with various hyperelastic material models.
For uniaxial extension in the
-direction, the principal stretches are
. From incompressibility
. Hence
.
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
, we have

Therefore,

The engineering strain is
. The engineering stress is

Equibiaxial extension
For equibiaxial extension in the
and
directions, the principal stretches are
. From incompressibility
. Hence
.
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
. 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
directions with the
direction constrained, the principal stretches are
. From incompressibility
. Hence
.
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
. The engineering stress is

Simple shear
The deformation gradient for a simple shear deformation has the form

where
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,
