Gent hyperelastic model
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
The strain energy density function for the Gent model is [1] where
is the shear modulus and
In the limit where
, 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
and taking the limit as
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[2] (the below strain energy function yields a non zero hydrostatic stress at no deformation, refer[3] for compressible Gent models).
is the bulk modulus, and
is the deformation gradient.
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.
, Therefore, the consistency condition for the Gent model is The Gent model assumes that
The Cauchy stress for the incompressible Gent model is given by For uniaxial extension in the
-direction, the principal stretches are
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 For equibiaxial extension in the
directions, the principal stretches are
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 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
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 The deformation gradient for a simple shear deformation has the form[4] 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,
