A hyperelastic or Green elastic material is a type of constitutive model for 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 and incompressible. Hyperelasticity provides a means of modeling the stress–strain 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. In addition to being used to model physical materials, hyperelastic materials are also used as fictitious media, e.g. in the third medium contact method. 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. Other widely used hyperelastic material models include the Ogden model and the Arruda–Boyce model.

Hyperelastic material models

Saint Venant–Kirchhoff model

The simplest hyperelastic material model is the Saint Venant–Kirchhoff model which is just an extension of the geometrically linear elastic material model to the geometrically nonlinear regime. This model has the general form and the isotropic form respectively where \mathbin{:} is tensor contraction, is the second Piola–Kirchhoff stress, is a fourth order stiffness tensor and is the Lagrangian Green strain given by \lambda and \mu are the Lamé constants, and is the second order unit tensor. The strain-energy density function for the Saint Venant–Kirchhoff model is and the second Piola–Kirchhoff stress can be derived from the relation

Classification of hyperelastic material models

Hyperelastic material models can be classified as: Generally, a hyperelastic model should satisfy the Drucker stability criterion. Some hyperelastic models satisfy the Valanis-Landel hypothesis which states that the strain energy function can be separated into the sum of separate functions of the principal stretches :

Stress–strain relations

Compressible hyperelastic materials

First Piola–Kirchhoff stress

If is the strain energy density function, the 1st Piola–Kirchhoff 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 The above relation is also known as the Doyle-Ericksen formula in the material configuration.

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 The above expressions are valid even for anisotropic media (in which case, the potential function is understood to depend implicitly on reference directional quantities such as initial fiber orientations). In the special case of isotropy, the Cauchy stress can be expressed in terms of the left Cauchy-Green deformation tensor as follows:

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–Kirchhoff 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

Expressions for the Cauchy stress

Compressible isotropic hyperelastic materials

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 isotropic hyperelastic materials

For incompressible isotropic hyperelastic materials, the strain energy density function is. The Cauchy stress is then given by where p is an undetermined pressure. In terms of stress differences If in addition I_1 = I_2, then If, then

Consistency with linear elasticity

Consistency with linear elasticity is often used to determine some of the parameters of hyperelastic material models. These consistency conditions can be found by comparing Hooke's law with linearized hyperelasticity at small strains.

Consistency conditions for isotropic hyperelastic models

For isotropic hyperelastic materials to be consistent with isotropic linear elasticity, the stress–strain relation should have the following form in the infinitesimal strain limit: where are the Lamé constants. The strain energy density function that corresponds to the above relation is For an incompressible material and we have For any strain energy density function to reduce to the above forms for small strains the following conditions have to be met If the material is incompressible, then the above conditions may be expressed in the following form. These conditions can be used to find relations between the parameters of a given hyperelastic model and shear and bulk moduli.

Consistency conditions for incompressible

I{{sub|1}}

based rubber materials Many elastomers are modeled adequately by a strain energy density function that depends only on I_1. For such materials we have W = W(I_1). The consistency conditions for incompressible materials for may then be expressed as The second consistency condition above can be derived by noting that These relations can then be substituted into the consistency condition for isotropic incompressible hyperelastic materials.