Hyperelastic material - AbsoluteAstronomy.com

A hyperelastic or Green elastic material is a type of constitutive model

Constitutive equation

In physics, a constitutive equation is a relation between two physical quantities that is specific to a material or substance, and approximates the response of that material to external forces...

for ideally elastic material for which the stress-strain relationship derives from a strain energy density function

Strain energy density function

A strain energy density function or stored energy density function is a scalar valued function that relates the strain energy density of a material to the deformation gradient....

. The hyperelastic material is a special case of a Cauchy elastic material

Cauchy elastic material

A Cauchy-elastic material is one in which the Cauchy stress at each material point is determined only by the current state of deformation . A Cauchy-elastic material is one in which the Cauchy stress at each material point is determined only by the current state of deformation (with respect to an...

.

For many materials, linear elastic

Elasticity (physics)

In physics, elasticity is the physical property of a material that returns to its original shape after the stress that made it deform or distort is removed. The relative amount of deformation is called the strain....

models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose stress

Stress (physics)

In continuum mechanics, stress is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces are a reaction to external forces applied on the body...

-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

Ronald Rivlin

Ronald Samuel Rivlin was a British-American physicist, mathematician, rheologist and a noted expert on rubber.-Life:Rivlin was born in London in 1915. He studied physics and mathematics at St John's College, Cambridge, being awarded a BA in 1937 and a ScD in 1952...

and Melvin Mooney

Melvin Mooney

Melvin Mooney was an American physicist and rheologist.He developed the Mooney Viscometer and other testing equipment used in the rubber industry. He also proposed the Mooney-Rivlin solid constitutive law describing the hyperelastic stress-strain behavior of rubber...

developed the first hyperelastic models, the Neo-Hookean

Neo-Hookean solid

A Neo-Hookean solid is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, a the...

and Mooney–Rivlin solids. Many other hyperelastic models have since been developed. Other widely used hyperelastic material models include the Ogden

Ogden (hyperelastic model)

The Ogden material model is a hyperelastic material model used to describe the non-linear stress-strain behaviour of complex materials such as rubbers, polymers, and biological tissue. The model was developed by Ray W. Ogden in 1972...

model and the Arruda–Boyce model.

Saint Venant–Kirchhoff model

The simplest hyperelastic material model is the Saint Venant–Kirchhoff model which is just an extension of the linear elastic material model to the nonlinear regime. This model has the form

where

is the second Piola–Kirchhoff stress and

is the Lagrangian Green strain, and

and

are the Lamé constants.

The strain-energy density function for the St. 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

Phenomenology (science)

The term phenomenology in science is used to describe a body of knowledge that relates empirical observations of phenomena to each other, in a way that is consistent with fundamental theory, but is not directly derived from theory. For example, we find the following definition in the Concise...

descriptions of observed behavior

Fung
Mooney–Rivlin
Ogden
Ogden (hyperelastic model)
The Ogden material model is a hyperelastic material model used to describe the non-linear stress-strain behaviour of complex materials such as rubbers, polymers, and biological tissue. The model was developed by Ray W. Ogden in 1972...
Polynomial
Polynomial (hyperelastic model)
The polynomial hyperelastic material model is a phenomenological model of rubber elasticity. In this model, the strain energy density function is of the form of a polynomial in the two invariants I_1,I_2 of the left Cauchy-Green deformation tensor....
Saint Venant–Kirchhoff
Yeoh
Yeoh (hyperelastic model)
300px|right|thumb|Yeoh model prediction versus experimental data for natural rubber. Model parameters and experimental data from The Yeoh hyperelastic material model is a phenomenological model for the deformation of nearly incompressible, nonlinear elastic materials such as rubber...

2) mechanistic models

Rubber Elasticity

Rubber elasticity, a well-known example of hyperelasticity, describes the mechanical behavior of many polymers, especially those with crosslinking. Invoking the theory of rubber elasticity, one considers a polymer chain in a crosslinked network as an entropic spring. When the chain is stretched,...

deriving from arguments about underlying structure of the material

Arruda–Boyce model
Neo-Hookean
Neo-Hookean solid
A Neo-Hookean solid is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, a the...

3) hybrids of phenomenological and mechanistic models

Gent
Gent (hyperelastic model)
The Gent hyperelastic material model is a phenomenological model of rubber elasticity that is based on the concept of limiting chain extensibility...

First Piola–Kirchhoff stress

is the strain energy density function, the 1st Piola–Kirchoff 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

is the second Piola–Kirchhoff stress tensor then

In terms of the Lagrangian Green strain

In terms of the right Cauchy–Green deformation tensor

Cauchy stress

Similarly, the Cauchy stress

Stress (physics)

is given by

In terms of the Lagrangian Green strain

In terms of the right Cauchy–Green deformation tensor

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

Lagrange multipliers

In mathematical optimization, the method of Lagrange multipliers provides a strategy for finding the maxima and minima of a function subject to constraints.For instance , consider the optimization problem...

to enforce the incompressibility constraint. The 1st Piola–Kirchoff stress now becomes

This stress tensor can subsequently be converted

Stress (physics)

into any of the other conventional stress tensors, such as the Cauchy Stress tensor which is given by

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

Strain energy density function

A strain energy density function or stored energy density function is a scalar valued function that relates the strain energy density of a material to the deformation gradient....

, 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

Stress (physics)

for a hyperelastic material is given by

where

is the right Cauchy–Green deformation tensor and

is the deformation gradient. The Cauchy stress

Stress (physics)

is given by

where

. Let

be the three principal invariants of

. Then

The derivatives of the invariants

Tensor derivative (continuum mechanics)

The derivatives of scalars, vectors, and second-order tensors with respect to second-order tensors are of considerable use in continuum mechanics. These derivatives are used in the theories of nonlinear elasticity and plasticity, particularly in the design of algorithms for numerical...

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

Isochoric

Isochoric may refer to:*cell-transitive, in geometry*isochoric process, in chemistry or thermodynamics...

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 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

or,

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

For an incompressible material

and hence

. Following Ogden p. 485, we may write

Some care is to required at this stage because, when an eigenvalue is repeated, it is in general only Gateaux differentiable, but not Frechet differentiable. A rigorous tensor derivative

Tensor derivative (continuum mechanics)

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

For incompressible isotropic hyperelastic materials, the strain energy density function

Strain energy density function

A strain energy density function or stored energy density function is a scalar valued function that relates the strain energy density of a material to the deformation gradient....

. The Cauchy stress is then given by

where

is an undetermined pressure. In terms of stress differences

If in addition

, 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

Hooke's law

In mechanics, and physics, Hooke's law of elasticity is an approximation that states that the extension of a spring is in direct proportion with the load applied to it. Many materials obey this law as long as the load does not exceed the material's elastic limit. Materials for which 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

Linear elasticity

Linear elasticity is the mathematical study of how solid objects deform and become internally stressed due to prescribed loading conditions. Linear elasticity models materials as continua. Linear elasticity is a simplification of the more general nonlinear theory of elasticity and is a branch of...

, the stress-strain relation should have the following form in the infinitesimal strain limit: