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 .

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]

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[3]

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. ^ Ogden, R. W., 1984, Non-linear elastic deformations, Dover.

See also[edit]