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 is the second Piola–Kirchhoff stress, is a fourth order stiffness tensor and is the Lagrangian Green strain given by
and 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:
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 :
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
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.
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 . To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:
where the hydrostatic pressure functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes
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).
Proof 1: The second Piola–Kirchhoff stress tensor for a hyperelastic material is given by
where . Let 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 Cauchy–Green deformation tensor and noting that , we can write
For an incompressible material and hence .Then
Therefore, the Cauchy stress is given by
where is an undetermined pressure which acts as a Lagrange multiplier to enforce the incompressibility constraint.
If, in addition, , we have and hence
In that case the Cauchy stress can be expressed as
Proof 2: The isochoric deformation gradient is defined as , 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 .
The invariants of are The set of invariants which are used to define the distortional behavior are the first two invariants of the isochoric left Cauchy–Green deformation tensor tensor, (which are identical to the ones for the right Cauchy Green stretch tensor), and add into the fray to describe the volumetric behaviour.
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 in terms of , we have
In terms of the deviatoric part of , we can write
For an incompressible material and hence .Then the Cauchy stress is given by
where is an undetermined pressure-like Lagrange multiplier term. In addition, if , we have and hence the Cauchy stress can be expressed as
Proof 3: To express the Cauchy stress in terms of the stretches recall that
The chain rule gives
The Cauchy stress is given by
Plugging in the expression for the derivative of leads to
Using the spectral decomposition of we have
Also note that
Therefore, the expression for the Cauchy stress can be written as
Some care is required at this stage because, when an eigenvalue is repeated, it is in general only Gateaux differentiable, but not Fréchet differentiable. A rigorous tensor derivative can only be found by solving another eigenvalue problem.
If we express the stress in terms of differences between components,
If in addition to incompressibility we have then a possible solution to the problem requires and we can write the stress differences as
Incompressible isotropic hyperelastic materials
where is an undetermined pressure. In terms of stress differences
If in addition , 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 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 based rubber materials
Many elastomers are modeled adequately by a strain energy density function that depends only on . For such materials we have . 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.
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