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, incompressible and generally independent of strain rate. 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.
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 formula_3 is the second Piola–Kirchhoff stress, formula_4 is a fourth order stiffness tensor and formula_5 is the Lagrangian Green strain given by
formula_7 and formula_8 are the Lamé constants, and formula_9 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:
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
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 formula_12:
Stress–strain relations.
Compressible hyperelastic materials.
First Piola–Kirchhoff stress.
If formula_14 is the strain energy density function, the 1st Piola–Kirchhoff stress tensor can be calculated for a hyperelastic material as
where formula_16 is the deformation gradient. In terms of the Lagrangian Green strain (formula_5)
In terms of the right Cauchy–Green deformation tensor (formula_19)
Second Piola–Kirchhoff stress.
If formula_3 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 formula_29. The incompressibility constraint is therefore formula_30. To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:
where the hydrostatic pressure formula_32 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 formula_35, 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 formula_37. The Cauchy stress is then given by
where formula_32 is an undetermined pressure. In terms of stress differences
If in addition formula_41, then
If formula_43, 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 formula_46 are the Lamé constants. The strain energy density function that corresponds to the above relation is
For an incompressible material formula_48 and we have
For any strain energy density function formula_50 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 formula_53 based rubber materials.
Many elastomers are modeled adequately by a strain energy density function that depends only on formula_53. For such materials we have formula_55.
The consistency conditions for incompressible materials for formula_56 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.