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Wikipedia / Hyperelastic material

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

is a fourth order stiffness tensor are the Lamé constants is the second order unit tensor.

Classification of hyperelastic material models

Hyperelastic material models can be classified as:

phenomenological descriptions of observed behavior

mechanistic models deriving from arguments about the underlying structure of the material

hybrids of phenomenological and mechanistic models

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

is the strain energy density function, the 1st Piola–Kirchhoff stress tensor is the deformation gradient. In terms of the Lagrangian Green strain In terms of the right Cauchy–Green deformation tensor

Second Piola–Kirchhoff stress

is the second Piola–Kirchhoff stress tensor 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 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

functions as a Lagrangian multiplier This stress tensor can subsequently be converted into any of the other conventional stress tensors, such as the Cauchy stress tensor

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 (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

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 are the Lamé constants. The strain energy density function that corresponds to the above relation is to reduce to the above forms for small strains the following conditions have to be met

If the material is incompressible, 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 I1 based rubber materials

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

References

See also