Hyperelastic material explained

A hyperelastic or Green elastic material^[1] 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.^[2] The behavior of unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal. Filled elastomers and biological tissues^{[3] [4]} 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

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 respectivelywhere

n{:} is tensor contraction, \boldsymbol{S} is the second Piola–Kirchhoff stress, \boldsymbol{C} : \R ^{3} \to \R ^{3} is a fourth order stiffness tensor and \boldsymbol{E} is the Lagrangian Green strain given by$$\backslash mathbf\; E\; =\backslash frac\backslash left[(\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u)^\backslash textsf\{T\}\; +\; \backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u\; +\; (\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u)^\backslash textsf\{T\}\; \backslash cdot\backslash nabla\_\{\backslash mathbf\; X\}\backslash mathbf\; u\backslash right]\backslash ,\backslash !$$λ and \mu are the Lamé constants, and \boldsymbol{ I } is the second order unit tensor.

The strain-energy density function for the Saint Venant–Kirchhoff model is$$W(\backslash boldsymbol)\; =\; \backslash frac[\backslash text\{tr\}(\backslash boldsymbol\{E\})]^2\; +\; \backslash mu\; \backslash text\backslash mathord\backslash left(\backslash boldsymbol^2\backslash right)$$

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

Classification of hyperelastic material models

Hyperelastic material models can be classified as:

1. phenomenological descriptions of observed behavior
   • Fung
   • Mooney–Rivlin
   • Ogden
   • Polynomial
   • Saint Venant–Kirchhoff
   • Yeoh
   • Marlow
2. mechanistic models deriving from arguments about underlying structure of the material
   • Arruda–Boyce model^[5]
   • Neo–Hookean model^[1]
   • Buche–Silberstein model^[6]
3. 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

( λ _{1,} λ _{2,} λ _{3)} :$$W\; =\; f(\backslash lambda\_1)\; +\; f(\backslash lambda\_2)\; +\; f(\backslash lambda\_3)\; \backslash ,.$$

Stress–strain relations

Compressible hyperelastic materials

First Piola–Kirchhoff stress

If

W(\boldsymbol{F}) is the strain energy density function, the 1st Piola–Kirchhoff stress tensor can be calculated for a hyperelastic material as$$\backslash boldsymbol\; =\; \backslash frac\; \backslash qquad\; \backslash text\; \backslash qquad\; P\_\; =\; \backslash frac.$$where \boldsymbol{F} is the deformation gradient. In terms of the Lagrangian Green strain (\boldsymbol{E} )$$\backslash boldsymbol\; =\; \backslash boldsymbol\backslash cdot\backslash frac\; \backslash qquad\; \backslash text\; \backslash qquad\; P\_\; =\; F\_~\backslash frac\; ~.$$In terms of the right Cauchy–Green deformation tensor (\boldsymbol{C} )$$\backslash boldsymbol\; =\; 2~\backslash boldsymbol\backslash cdot\backslash frac\; \backslash qquad\; \backslash text\; \backslash qquad\; P\_\; =\; 2~F\_~\backslash frac\; ~.$$

Second Piola–Kirchhoff stress

If

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

Cauchy stress

Similarly, the Cauchy stress is given by$$\backslash boldsymbol\; =\; \backslash frac~\; \backslash frac\backslash cdot\backslash boldsymbol^\backslash textsf\; ~;~~\; J\; :=\; \backslash det\backslash boldsymbol\; \backslash qquad\; \backslash text\; \backslash qquad\; \backslash sigma\_\; =\; \backslash frac~\; \backslash frac~F\_\; ~.$$In terms of the Lagrangian Green strain$$\backslash boldsymbol\; =\; \backslash frac~\backslash boldsymbol\backslash cdot\backslash frac\backslash cdot\backslash boldsymbol^\backslash textsf\; \backslash qquad\; \backslash text\; \backslash qquad\; \backslash sigma\_\; =\; \backslash frac~F\_~\backslash frac~F\_\; ~.$$In terms of the right Cauchy–Green deformation tensor$$\backslash boldsymbol\; =\; \backslash frac~\backslash boldsymbol\backslash cdot\backslash frac\backslash cdot\backslash boldsymbol^\backslash textsf\; \backslash qquad\; \backslash text\; \backslash qquad\; \backslash sigma\_\; =\; \backslash frac~F\_~\backslash frac~F\_\; ~.$$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:^[7] $$\backslash boldsymbol\; =\; \backslash frac\backslash frac\backslash cdot~\backslash boldsymbol\; \backslash qquad\; \backslash text\; \backslash qquad\; \backslash sigma\_\; =\; \backslash frac~B\_~\backslash frac\; ~.$$

Incompressible hyperelastic materials

For an incompressible material

J : = \det\boldsymbol{F} = 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:$$W\; =\; W(\backslash boldsymbol)\; -\; p~(J-1)$$where the hydrostatic pressure p functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes$$\backslash boldsymbol=-p~J\backslash boldsymbol^\; +\; \backslash frac\; =\; -p~\backslash boldsymbol^\; +\; \backslash boldsymbol\backslash cdot\backslash frac\; =\; -p~\backslash boldsymbol^\; +\; 2~\backslash boldsymbol\backslash cdot\backslash frac\; ~.$$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$$\backslash boldsymbol=\backslash boldsymbol\backslash cdot\backslash boldsymbol^\backslash textsf\; =\; -p~\backslash boldsymbol\; +\; \backslash frac\backslash cdot\backslash boldsymbol^\backslash textsf\; =\; -p~\backslash boldsymbol\; +\; \backslash boldsymbol\backslash cdot\backslash frac\backslash cdot\backslash boldsymbol^\backslash textsf\; =\; -p~\backslash boldsymbol\; +\; 2~\backslash boldsymbol\backslash cdot\backslash frac\backslash cdot\backslash boldsymbol^\backslash textsf\; ~.$$

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 $$W(\backslash boldsymbol)=\backslash hat(I\_1,I\_2,I\_3)\; =\; \backslash bar(\backslash bar\_1,\backslash bar\_2,\; J)\; =\; \backslash tilde(\backslash lambda\_1,\backslash lambda\_2,\; \backslash lambda\_3),$$ 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

W(\boldsymbol{F}) = \hat{W}(I _{1,I} _{2)} . The Cauchy stress is then given bywhere p is an undetermined pressure. In terms of stress differences$$\backslash sigma\_\; -\; \backslash sigma\_\; =\; \backslash lambda\_1~\backslash frac\; -\; \backslash lambda\_3~\backslash frac~;~~\; \backslash sigma\_\; -\; \backslash sigma\_\; =\; \backslash lambda\_2~\backslash frac\; -\; \backslash lambda\_3~\backslash frac$$If in addition I _{1} = I _{2} , then$$\backslash boldsymbol\; =\; 2\backslash frac~\backslash boldsymbol\; -\; p~\backslash boldsymbol~.$$If λ _{1} = λ _{2} , then$$\backslash sigma\_\; -\; \backslash sigma\_\; =\; \backslash sigma\_\; -\; \backslash sigma\_\; =\; \backslash lambda\_1~\backslash frac\; -\; \backslash lambda\_3~\backslash frac$$

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:$$\backslash boldsymbol\; =\; \backslash lambda~\backslash mathrm(\backslash boldsymbol)~\backslash boldsymbol\; +\; 2\backslash mu\backslash boldsymbol$$where

λ , \mu are the Lamé constants. The strain energy density function that corresponds to the above relation is^[1] $$W\; =\; \backslash tfrac\backslash lambda~[\backslash mathrm\{tr\}(\backslash boldsymbol\{\backslash varepsilon\})]^2\; +\; \backslash mu~\backslash mathrm\backslash mathord\backslash left(\backslash boldsymbol^2\backslash right)$$For an incompressible material tr (\boldsymbol{\varepsilon}) = 0 and we have$$W\; =\; \backslash mu~\backslash mathrm\backslash mathord\backslash left(\backslash boldsymbol^2\backslash right)$$For any strain energy density function W( λ _{1,λ} _{2,λ} _{3)} to reduce to the above forms for small strains the following conditions have to be met^[1]

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

I _{1} . For such materials we have W = W(I _{1)} .The consistency conditions for incompressible materials for I _{1} = 3 , λ _{i} = λ _{j} = 1 may then be expressed as$$\backslash left.W(I\_1)\backslash right|\_\; =\; 0\; \backslash quad\; \backslash text\; \backslash quad\; \backslash left.\backslash frac\backslash right|\_\; =\; \backslash frac\; \backslash ,.$$The second consistency condition above can be derived by noting that$$\backslash frac\; =\; \backslash frac\backslash frac\; =\; 2\backslash lambda\_i\backslash frac\; \backslash quad\backslash text\backslash quad\; \backslash frac\; =\; 2\backslash delta\_\backslash frac\; +\; 4\backslash lambda\_i\backslash lambda\_j\; \backslash frac\backslash ,.$$These relations can then be substituted into the consistency condition for isotropic incompressible hyperelastic materials.

See also

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

Notes and References

1. R.W. Ogden, 1984, Non-Linear Elastic Deformations,, Dover.
2. Muhr . A. H. . 2005 . Modeling the stress–strain behavior of rubber . Rubber Chemistry and Technology . 78 . 3. 391–425 . 10.5254/1.3547890 .
3. 4278556 . 25319496 . 10.1002/cnm.2691 . 30 . A finite strain nonlinear human mitral valve model with fluid-structure interaction . Int J Numer Methods Biomed Eng . 1597–613 . Gao . H . Ma . X . Qi . N . Berry . C . Griffith . BE . Luo . X. 2014 . 12 .
4. 5332559 . 28228537 . 10.1098/rsif.2016.0596 . 14 . Morphoelasticity in the development of brown alga Ectocarpus siliculosus: from cell rounding to branching . J R Soc Interface . Jia . F . Ben Amar . M . Billoud . B . Charrier . B . 2017 . 127 . 20160596.
5. Arruda. E.M.. Boyce. M.C.. A three-dimensional model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids. 41. 389–412. 1993. 10.1016/0022-5096(93)90013-6. 136924401 .
6. Buche. M.R.. Silberstein. M.N.. Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble. Phys. Rev. E. 102. 012501. 2020. 1 . 10.1103/PhysRevE.102.012501. 32794915 . 2004.07874 . 2020PhRvE.102a2501B . 215814600 .
7. Y. Basar, 2000, Nonlinear continuum mechanics of solids, Springer, p. 157.