Hyperelastic material

A hyperelastic or Green elastic material[1] is a type of constitutive model for ideally elastic material for which the stress–strain relationship derives from a strain energy density function.

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.

[2] The behavior of unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal.

Filled elastomers and biological tissues[3][4] are also often modeled via the hyperelastic idealization.

Ronald Rivlin and Melvin Mooney developed the first hyperelastic models, the Neo-Hookean and Mooney–Rivlin solids.

The strain-energy density function for the Saint Venant–Kirchhoff model is

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

is the strain energy density function, the 1st Piola–Kirchhoff stress tensor can be calculated for a hyperelastic material as

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:[7]

To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:

functions as a Lagrangian multiplier to enforce the incompressibility constraint.

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).

(See the page on the left Cauchy–Green deformation tensor for the definitions of these symbols).

is an undetermined pressure which acts as a Lagrange multiplier to enforce the incompressibility constraint.

, resulting in the isochoric deformation gradient having a determinant of 1, in other words it is volume stretch free.

Using this one can subsequently define the isochoric left Cauchy–Green deformation tensor

To express the Cauchy stress in terms of the invariants

is an undetermined pressure-like Lagrange multiplier term.

To express the Cauchy stress in terms of the stretches

[8][9] A rigorous tensor derivative can only be found by solving another eigenvalue problem.

For incompressible isotropic hyperelastic materials, the strain energy density function is

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.

The strain energy density function that corresponds to the above relation is[1]

to reduce to the above forms for small strains the following conditions have to be met[1]

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.

Many elastomers are modeled adequately by a strain energy density function that depends only on

These relations can then be substituted into the consistency condition for isotropic incompressible hyperelastic materials.