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 around 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).

where , is the bulk modulus, and is the deformation gradient.

Consistency condition

edit

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

edit

The Cauchy stress for the incompressible Gent model is given by

 

Uniaxial extension

edit
 
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

edit

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

edit

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

edit

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,

 

References

edit
  1. ^ a b 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. ^ Horgan, Cornelius O.; Saccomandi, Giuseppe (2004-11-01). "Constitutive Models for Compressible Nonlinearly Elastic Materials with Limiting Chain Extensibility". Journal of Elasticity. 77 (2): 123–138. doi:10.1007/s10659-005-4408-x. ISSN 1573-2681.
  4. ^ Ogden, R. W., 1984, Non-linear elastic deformations, Dover.

See also

edit