Hyperelastic material

A hyperelastic or Green elastic materialR.W. Ogden, 1984, ''Non-Linear Elastic Deformations'', , Dover. 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.

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
$\begin \boldsymbol &= \boldsymbol : \boldsymbol \\ \boldsymbol &= \lambda~ \text(\boldsymbol)\boldsymbol + 2\mu\boldsymbol \text \end$
where $\mathbin$ is tensor contraction, $\boldsymbol$ is the second Piola–Kirchhoff stress, $\boldsymbol : \R^ \to \R^$ is a fourth order stiffness tensor and $\boldsymbol$ is the Lagrangian Green strain given by
\mathbf E =\frac\left (\nabla_\mathbf u)^\textsf + \nabla_\mathbf u + (\nabla_\mathbf u)^\textsf \cdot\nabla_\mathbf u\right,\!
$\lambda$ and $\mu$ are the Lamé constants, and $\boldsymbol$ is the second order unit tensor.

The strain-energy density function for the Saint Venant–Kirchhoff model is
W(\boldsymbol) = \frac text(\boldsymbol)2 + \mu \text\mathord\left(\boldsymbol^2\right)

and the second Piola–Kirchhoff stress can be derived from the relation
$\boldsymbol = \frac ~.$

Classification of hyperelastic material models

Hyperelastic material models can be classified as:

# phenomenological descriptions of observed behavior
#* Fung
#* Mooney–Rivlin
#* Ogden
#* Polynomial
#* Saint Venant–Kirchhoff
#* Yeoh
#* Marlow
# mechanistic models deriving from arguments about underlying structure of the material
#* Arruda–Boyce model
#* Neo–Hookean model
#* Buche–Silberstein model
# hybrids of phenomenological and mechanistic models
#* Gent
#* Van der Waals

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 $(\lambda_1, \lambda_2, \lambda_3)$:
$W = f(\lambda_1) + f(\lambda_2) + f(\lambda_3) \,.$

Stress–strain relations

Compressible hyperelastic materials

First Piola–Kirchhoff stress

If $W(\boldsymbol)$ is the strain energy density function, the 1st Piola–Kirchhoff stress tensor can be calculated for a hyperelastic material as
$\boldsymbol = \frac \qquad \text \qquad P_ = \frac.$
where $\boldsymbol$ is the deformation gradient. In terms of the Lagrangian Green strain ($\boldsymbol$)
$\boldsymbol = \boldsymbol\cdot\frac \qquad \text \qquad P_ = F_~\frac ~.$
In terms of the right Cauchy–Green deformation tensor ($\boldsymbol$)
$\boldsymbol = 2~\boldsymbol\cdot\frac \qquad \text \qquad P_ = 2~F_~\frac ~.$

Second Piola–Kirchhoff stress

If $\boldsymbol$ is the second Piola–Kirchhoff stress tensor then
$\boldsymbol = \boldsymbol^\cdot\frac \qquad \text \qquad S_ = F^_\frac ~.$
In terms of the Lagrangian Green strain
$\boldsymbol = \frac \qquad \text \qquad S_ = \frac ~.$
In terms of the right Cauchy–Green deformation tensor
$\boldsymbol = 2~\frac \qquad \text \qquad S_ = 2~\frac ~.$
The above relation is also known as the Doyle-Ericksen formula in the material configuration.

Cauchy stress

Similarly, the Cauchy stress is given by
$\boldsymbol = \frac~ \frac\cdot\boldsymbol^\textsf ~;~~ J := \det\boldsymbol \qquad \text \qquad \sigma_ = \frac~ \frac~F_ ~.$
In terms of the Lagrangian Green strain
$\boldsymbol = \frac~\boldsymbol\cdot\frac\cdot\boldsymbol^\textsf \qquad \text \qquad \sigma_ = \frac~F_~\frac~F_ ~.$
In terms of the right Cauchy–Green deformation tensor
$\boldsymbol = \frac~\boldsymbol\cdot\frac\cdot\boldsymbol^\textsf \qquad \text \qquad \sigma_ = \frac~F_~\frac~F_ ~.$
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:Y. Basar, 2000, Nonlinear continuum mechanics of solids, Springer, p. 157.
$\boldsymbol = \frac\frac\cdot~\boldsymbol \qquad \text \qquad \sigma_ = \frac~B_~\frac ~.$

Incompressible hyperelastic materials

For an incompressible material $J := \det\boldsymbol = 1$. The incompressibility constraint is therefore $J-1= 0$. To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:
$W = W(\boldsymbol) - p~(J-1)$
where the hydrostatic pressure $p$ functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes
$\boldsymbol=-p~J\boldsymbol^ + \frac = -p~\boldsymbol^ + \boldsymbol\cdot\frac = -p~\boldsymbol^ + 2~\boldsymbol\cdot\frac ~.$
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
$\boldsymbol=\boldsymbol\cdot\boldsymbol^\textsf = -p~\boldsymbol + \frac\cdot\boldsymbol^\textsf = -p~\boldsymbol + \boldsymbol\cdot\frac\cdot\boldsymbol^\textsf = -p~\boldsymbol + 2~\boldsymbol\cdot\frac\cdot\boldsymbol^\textsf ~.$

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 $W(\boldsymbol)=\hat(I_1,I_2,I_3) = \bar(\bar_1,\bar_2, J) = \tilde(\lambda_1,\lambda_2, \lambda_3),$ then
\begin
\boldsymbol & =
\frac\left left(\frac + I_1~\frac\right)\boldsymbol - \frac~\boldsymbol \cdot\boldsymbol \right+ 2\sqrt~\frac~\boldsymbol \\ pt & = \frac\left frac\left(\frac + \bar_1~\frac\right)\boldsymbol -
\frac~\frac~\boldsymbol \cdot\boldsymbol \right + \left frac - \frac \left(\bar_1~\frac + 2~\bar_2~\frac\right)\right~\boldsymbol \\ pt & = \frac \left left(\frac + \bar_1~\frac\right)\bar -
\frac~\bar \cdot\bar \right+ \left frac - \frac\left(\bar_1~\frac + 2~\bar_2~\frac\right)\right~\boldsymbol \\ pt & = \frac~\frac~\mathbf_1\otimes\mathbf_1 + \frac~\frac~\mathbf_2\otimes\mathbf_2 + \frac~\frac~\mathbf_3\otimes\mathbf_3
\end
(See the page on the left Cauchy–Green deformation tensor for the definitions of these symbols).

:=J^\boldsymbol, 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 $\bar := \bar\cdot\bar^T=J^\boldsymbol$.
The invariants of $\bar$ are
$\begin \bar I_1 &= \text(\bar) = J^\text(\boldsymbol) = J^ I_1 \\ \bar I_2 & = \frac\left(\text(\bar)^2 - \text(\bar^2)\right) = \frac\left( \left(J^\text(\boldsymbol)\right)^2 - \text(J^\boldsymbol^2) \right) = J^ I_2 \\ \bar I_3 &= \det(\bar) = J^ \det(\boldsymbol) = J^ I_3 = J^ J^2 = 1 \end$
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 $J$ into the fray to describe the volumetric behaviour.

To express the Cauchy stress in terms of the invariants $\bar_1, \bar_2, J$ recall that
$\bar_1 = J^~I_1 = I_3^~I_1 ~;~~ \bar_2 = J^~I_2 = I_3^~I_2 ~;~~ J = I_3^ ~.$
The chain rule of differentiation gives us
$\begin \frac & = \frac~\frac + \frac~\frac + \frac~\frac \\ & = I_3^~\frac = J^~\frac \\ \frac & = \frac~\frac + \frac~\frac + \frac~\frac \\ & = I_3^~\frac = J^~\frac \\ \frac & = \frac~\frac + \frac~\frac + \frac~\frac \\ & = - \frac~I_3^~I_1~\frac - \frac~I_3^~I_2~\frac + \frac~I_3^~\frac \\ & = - \frac~J^~J^~\bar_1~\frac - \frac~J^~J^~\bar_2~\frac + \frac~J^~\frac \\ & = -\frac~J^~\left(\bar_1~\frac+ 2~\bar_2~\frac\right) + \frac~J^~\frac \end$
Recall that the Cauchy stress is given by

\boldsymbol
= \frac~\left left(\frac +
I_1~\frac\right)~\boldsymbol -
\frac~\boldsymbol\cdot\boldsymbol\right+
2~\sqrt~\frac~\boldsymbol~.

In terms of the invariants $\bar_1, \bar_2, J$ we have

\boldsymbol
= \frac~\left left(\frac+
J^~\bar_1~\frac\right)~\boldsymbol -
\frac~\boldsymbol\cdot\boldsymbol\right+
2~J~\frac~\boldsymbol~.

Plugging in the expressions for the derivatives of $W$ in terms of $\bar_1, \bar_2, J$, we have
\begin
\boldsymbol
& = \frac~\left left(J^~\frac +
J^~\bar_1~\frac\right)~\boldsymbol -
J^~\frac~\boldsymbol\cdot\boldsymbol\right + \\
& \qquad
2~J~\left \frac~J^~\left(\bar_1~\frac+
2~\bar_2~\frac\right) +
\frac~J^~\frac\right\boldsymbol
\end
or,
\begin
\boldsymbol
& = \frac~\left frac~\left(\frac +
\bar_1~\frac\right)~\boldsymbol -
\frac~
\frac~\boldsymbol\cdot\boldsymbol\right\\
& \qquad + \left frac -
\frac\left(\bar_1~\frac+
2~\bar_2~\frac\right)\rightboldsymbol
\end
In terms of the deviatoric part of $\boldsymbol$, we can write
\begin
\boldsymbol
& = \frac~\left left(\frac +
\bar_1~\frac\right)~\bar -
\frac~\bar\cdot\bar\right\\
& \qquad + \left frac -
\frac\left(\bar_1~\frac+
2~\bar_2~\frac\right)\rightboldsymbol
\end

For an incompressible material $J = 1$ and hence $W = W(\bar_1,\bar_2)$.Then
the Cauchy stress is given by

\boldsymbol
= 2\left left(\frac +
I_1~\frac\right)~\bar -
\frac~\bar\cdot\bar\right- p~\boldsymbol~.

where $p$ is an undetermined pressure-like Lagrange multiplier term. In addition, if $\bar_1 = \bar_2$, we have $W = W(\bar_1)$ and hence
the Cauchy stress can be expressed as
$\boldsymbol = 2\frac~\bar - p~\boldsymbol~.$

= \frac~\boldsymbol^T\cdot(\mathbf_i\otimes\mathbf_i)\cdot\boldsymbol~;~~
i = 1,2,3 ~.

The chain rule gives
\begin
\frac & =
\frac~\frac +
\frac~\frac +
\frac~\frac \\
& = \boldsymbol^T\cdot\left frac~\frac~\mathbf_1\otimes\mathbf_1 +
\frac~\frac~\mathbf_2\otimes\mathbf_2 +
\frac~\frac~\mathbf_3\otimes\mathbf_3\rightcdot\boldsymbol
\end

The Cauchy stress is given by
$\boldsymbol = \frac~\boldsymbol\cdot \frac\cdot\boldsymbol^T = \frac~(\boldsymbol\cdot\boldsymbol)\cdot \frac\cdot(\boldsymbol^T\cdot\boldsymbol)$
Plugging in the expression for the derivative of $W$ leads to

\boldsymbol =
\frac~\boldsymbol\cdot
\left frac~
\frac~\mathbf_1\otimes\mathbf_1 +
\frac~
\frac~\mathbf_2\otimes\mathbf_2 +
\frac~
\frac~\mathbf_3\otimes\mathbf_3\right \cdot\boldsymbol

Using the spectral decomposition of $\boldsymbol$ we have
$\boldsymbol\cdot(\mathbf_i\otimes\mathbf_i)\cdot\boldsymbol = \lambda_i^2~\mathbf_i\otimes\mathbf_i ~;~~ i=1,2,3.$
Also note that
$J = \det(\boldsymbol) = \det(\boldsymbol)\det(\boldsymbol) = \det(\boldsymbol) = \lambda_1 \lambda_2 \lambda_3 ~.$
Therefore, the expression for the Cauchy stress can be written as

\boldsymbol =
\frac~
\left lambda_1~\frac~\mathbf_1\otimes\mathbf_1 +
\lambda_2~\frac~\mathbf_2\otimes\mathbf_2 +
\lambda_3~\frac~\mathbf_3\otimes\mathbf_3
\right
For an incompressible material $\lambda_1\lambda_2\lambda_3 = 1$ and hence $W = W(\lambda_1,\lambda_2)$. Following Ogden p. 485, we may write
$\boldsymbol = \lambda_1~\frac~\mathbf_1\otimes\mathbf_1 + \lambda_2~\frac~\mathbf_2\otimes\mathbf_2 + \lambda_3~\frac~\mathbf_3\otimes\mathbf_3 - p~\boldsymbol~$
Some care is required at this stage because, when an eigenvalue is repeated, it is in general only Gateaux differentiable, but not Fréchet differentiable.Friswell MI. ''The derivatives of repeated eigenvalues and their associated eigenvectors.'' Journal of Vibration and Acoustics (ASME) 1996; 118:390–397. A rigorous tensor derivative can only be found by solving another eigenvalue problem.

If we express the stress in terms of differences between components,
$\sigma_ - \sigma_ = \lambda_1~\frac - \lambda_3~\frac ~;~~ \sigma_ - \sigma_ = \lambda_2~\frac - \lambda_3~\frac$
If in addition to incompressibility we have $\lambda_1 = \lambda_2$ then a possible solution to the problem
requires $\sigma_ = \sigma_$ and we can write the stress differences as
$\sigma_ - \sigma_ = \sigma_ - \sigma_ = \lambda_1~\frac - \lambda_3~\frac$

Incompressible isotropic hyperelastic materials

For incompressible isotropic hyperelastic materials, the strain energy density function is $W(\boldsymbol)=\hat(I_1,I_2)$. The Cauchy stress is then given by
\begin
\boldsymbol & = -p~\boldsymbol +
2\left left(\frac + I_1~\frac\right)\boldsymbol - \frac~\boldsymbol \cdot\boldsymbol \right\\
& = - p~\boldsymbol + 2\left left(\frac +
I_1~\frac\right)~\bar -
\frac~\bar\cdot\bar\right\\
& = - p~\boldsymbol + \lambda_1~\frac~\mathbf_1\otimes\mathbf_1 +
\lambda_2~\frac~\mathbf_2\otimes\mathbf_2 + \lambda_3~\frac~\mathbf_3\otimes\mathbf_3
\end
where $p$ is an undetermined pressure. In terms of stress differences
$\sigma_ - \sigma_ = \lambda_1~\frac - \lambda_3~\frac~;~~ \sigma_ - \sigma_ = \lambda_2~\frac - \lambda_3~\frac$
If in addition $I_1 = I_2$, then
$\boldsymbol = 2\frac~\boldsymbol - p~\boldsymbol~.$
If $\lambda_1 = \lambda_2$, then
$\sigma_ - \sigma_ = \sigma_ - \sigma_ = \lambda_1~\frac - \lambda_3~\frac$

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:
$\boldsymbol = \lambda~\mathrm(\boldsymbol)~\boldsymbol + 2\mu\boldsymbol$
where $\lambda, \mu$ are the Lamé constants. The strain energy density function that corresponds to the above relation is
W = \tfrac\lambda~ mathrm(\boldsymbol)2 + \mu~\mathrm\mathord\left(\boldsymbol^2\right)
For an incompressible material $\mathrm(\boldsymbol) = 0$ and we have
$W = \mu~\mathrm\mathord\left(\boldsymbol^2\right)$
For any strain energy density function $W(\lambda_1,\lambda_2,\lambda_3)$ to reduce to the above forms for small strains the following conditions have to be met
$\begin & W(1,1,1) = 0 ~;~~ \frac(1,1,1) = 0 \\ & \frac(1,1,1) = \lambda + 2\mu\delta_ \end$

If the material is incompressible, then the above conditions may be expressed in the following form.
$\begin & W(1,1,1) = 0 \\ & \frac(1,1,1) = \frac(1,1,1) ~;~~ \frac(1,1,1) = \frac(1,1,1) \\ & \frac(1,1,1) = \mathrm~i,j\ne i \\ & \frac(1,1,1) - \frac(1,1,1) + \frac(1,1,1) = 2\mu ~~(i \ne j) \end$
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 $I_1$. For such materials we have $W = W(I_1)$.
The consistency conditions for incompressible materials for $I_1 = 3, \lambda_i = \lambda_j = 1$ may then be expressed as
$\left.W(I_1)\_ = 0 \quad \text \quad \left.\frac\_ = \frac \,.$
The second consistency condition above can be derived by noting that
$\frac = \frac\frac = 2\lambda_i\frac \quad\text\quad \frac = 2\delta_\frac + 4\lambda_i\lambda_j \frac\,.$
These relations can then be substituted into the consistency condition for isotropic incompressible hyperelastic materials.

References

See also

* Cauchy elastic material
*Continuum mechanics
*Deformation (mechanics)
*Finite strain theory
*Ogden–Roxburgh model
*Rubber elasticity
*Stress measures
*Stress (mechanics)

{{DEFAULTSORT:Hyperelastic Material
Continuum mechanics
Elasticity (physics)
Rubber properties
Solid mechanics