A hyperelastic or Green elastic material is an ideally elastic material for which the stressstrain 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 stressstrain relationship can be defined as nonlinearly elastic, isotropic, incompressible and generally independent of strain rate. Hyperelasticity provides a means of modeling the stressstrain behavior of 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.
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If is the strain energy density function, the 1^{st} PiolaKirchoff 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 CauchyGreen deformation tensor ()
If is the second PiolaKirchhoff stress tensor then
In terms of the Lagrangian Green strain
In terms of the right CauchyGreen deformation tensor
Similarly, the Cauchy stress is given by
In terms of the Lagrangian Green strain
In terms of the right CauchyGreen deformation tensor
For isotropic hyperelastic materials, the Cauchy stress can be expressed in terms of the invariants of the left CauchyGreen deformation tensor (or right CauchyGreen deformation tensor). If the strain energy density function is , then
(See the page on the left CauchyGreen deformation tensor for the definitions of these symbols).
Proof 1: 

The second PiolaKirchhoff stress tensor for a
hyperelastic material is given by
where is the right CauchyGreen deformation tensor and is the deformation gradient. The Cauchy stress is given by where . Let I_{1},I_{2},I_{ 3} be the three principal invariants of . Then The derivatives of the invariants of the symmetric tensor are Therefore we can write Plugging into the expression for the Cauchy stress gives Using the left CauchyGreen deformation tensor and noting that I_{3} = J^{2}, we can write 
Proof 2: 

To express the Cauchy stress in terms of the invariants
recall that
The chain rule of differentiation gives us Recall that the Cauchy stress is given by In terms of the invariants we have Plugging in the expressions for the derivatives of W in terms of , we have or, 
Proof 3: 

To express the Cauchy stress in terms of the stretches λ_{1},λ_{2},λ_{3}
recall that
The chain rule gives The Cauchy stress is given by Plugging in the expression for the derivative of W leads to Using the spectral decomposition of we have Also note that Therefore the expression for the Cauchy stress can be written as 
For an incompressible material . The incompressibility constraint is therefore J − 1 = 0. To ensure incompressibility of a hyperelastic material, the strainenergy function can be written in form:
where the hydrostatic pressure p functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1^{st} PiolaKirchoff 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
The simplest hyperelastic material model is the Saint VenantKirchhoff 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 PiolaKirchhoff stress and is the Lagrangian Green strain, and λ and μ are the Lamé constants.
The strainenergy density function for the St. VenantKirchhoff model is
and the second PiolaKirchhoff stress can be derived from the relation
Ronald Rivlin and Melvin Mooney developed the first hyperelastic models, the NeoHookean and MooneyRivlin solids. Many other hyperelastic models have since been developed. Models can be classified as:
1) phenomenological descriptions of observed behavior
2) mechanistic models deriving from arguments about underlying structure of the material
3) hybrids of phenomenological and mechanistic models
