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# Hyperelastic material

A hyperelastic or Green elastic material is an ideally elastic material for which the stress-strain relationship derives from a strain energy density function. The hyperelastic material is a special case of a Cauchy elastic material. For many materials, linear elastic models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose stress-strain relationship can be defined as non-linearly elastic, isotropic, incompressible and generally independent of strain rate. Hyperelasticity provides a means of modeling the stress-strain behavior such materials. The behavior of unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal. Filled elastomers and biological tissues are also often modeled via the hyperelastic idealization.

Stress-strain relations

First Piola-Kirchhoff stress

If is the strain energy density function, the 1st Piola-Kirchoff stress tensor can be calculated for a hyperelastic material as:where is the deformation gradient. In terms of the Lagrangian Green strain ():In terms of the right Cauchy-Green deformation tensor ():

Second Piola-Kirchhoff stress

If is the second Piola-Kirchhoff stress tensor then:In terms of the Lagrangian Green strain:In terms of the right Cauchy-Green deformation tensor :

Cauchy stress

Similarly, the Cauchy stress is given by:In terms of the Lagrangian Green strain:In terms of the right Cauchy-Green deformation tensor :

Cauchy stress in terms of invariants

For isotropic hyperelastic materials, the Cauchy stress can be expressed in terms of the invariants of the left Cauchy-Green deformation tensor (or right Cauchy-Green deformation tensor). If the strain energy density function is , then:(See the page on the left Cauchy-Green deformation tensor for the definitions of these symbols).

:

Incompressible hyperelastic materials

For an incompressible material . The incompressibility constraint is therefore $J-1= 0$. To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form::where the hydrostatic pressure $p$ functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola-Kirchoff stress now becomes:This stress tensor can subsequently be converted into any of the other conventional stress tensors, such as the Cauchy Stress tensor which is given by:For incompressible isotropic hyperelastic materials, the strain energy density function is . The Cauchy stress is then given by:

Hyperelastic Models

The simplest hyperelastic material model is the Saint Venant-Kirchhoff model which is just an extension of the linear elastic material model to the nonlinear regime. This model has the form :where is the second Piola-Kirchhoff stress and is the Lagrangian Green strain, and $lambda$ and $mu$ are the Lamé constants.

The strain-energy density function for the St. Venant-Kirchhoff model is:and the second Piola-Kirchhoff stress can be derived from the relation:

Ronald Rivlin and Melvin Mooney developed the first hyperelastic models, the Neo-Hookean and Mooney-Rivlin solids. Many other hyperelastic models have since been developed. Models can be classified as:

1) phenomenological descriptions of observed behavior
*Mooney-Rivlin
*Ogden
*Polynomial
*Yeoh

2) mechanistic models deriving from arguments about underlying structure of the material
*Arruda-Boyce model
*Neo-Hookean

3) hybrids of phenomenological and mechanistic models
*Gent

References

* R.W. Ogden: "Non-Linear Elastic Deformations", ISBN 0-486-69648-0

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