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Ahyperelastic orGreen elastic material^[1] is a type ofconstitutive model for ideallyelastic material for which the stress–strain relationship derives from astrain energy density function. The hyperelastic material is a special case of aCauchy 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, whosestress-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,vulcanizedelastomers often conforms closely to the hyperelastic ideal. Filled elastomers andbiological 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 thethird medium contact method.

Ronald Rivlin andMelvin Mooney developed the first hyperelastic models, theNeo-Hookean andMooney–Rivlin solids. Many other hyperelastic models have since been developed. Other widely used hyperelastic material models include theOgden model and theArruda–Boyce 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${\displaystyle :}$ is tensor contraction,${\displaystyle \mathit{S}}$ is the second Piola–Kirchhoff stress,${\displaystyle \mathit{C}\in {\mathbb{R}}^{3\times 3\times 3\times 3}}$ is a fourth orderstiffness tensor and${\displaystyle \mathit{E}}$ is the Lagrangian Green strain given by$${\displaystyle \mathbf{E}=\frac{1}{2}\left[({\mathrm{\nabla }}_{\mathbf{X}}\mathbf{u}{)}^{\mathsf{\text{T}}}+{\mathrm{\nabla }}_{\mathbf{X}}\mathbf{u}+({\mathrm{\nabla }}_{\mathbf{X}}\mathbf{u}{)}^{\mathsf{\text{T}}}\cdot {\mathrm{\nabla }}_{\mathbf{X}}\mathbf{u}\right]}$$${\displaystyle \lambda }$ and${\displaystyle \mu }$ are theLamé constants, and${\displaystyle \mathit{I}}$ is the second order unit tensor.

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

and the second Piola–Kirchhoff stress can be derived from the relation$${\displaystyle \mathit{S}=\frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{E}}.}$$

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 the 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 theDrucker stability criterion.Some hyperelastic models satisfy theValanis-Landel hypothesis which states that the strain energy function can be separated into the sum of separate functions of theprincipal stretches${\displaystyle ({\lambda }_1,{\lambda }_2,{\lambda }_3)}$:$${\displaystyle W=f({\lambda }_1)+f({\lambda }_2)+f({\lambda }_3).}$$

If${\displaystyle W(\mathit{F})}$ is the strain energy density function, the1st Piola–Kirchhoff stress tensor can be calculated for a hyperelastic material as$${\displaystyle \mathit{P}=\frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{F}}\text{or}{P}_{iK}=\frac{\mathrm{\partial }W}{\mathrm{\partial }{F}_{iK}}.}$$where${\displaystyle \mathit{F}}$ is thedeformation gradient. In terms of theLagrangian Green strain (${\displaystyle \mathit{E}}$)$${\displaystyle \mathit{P}=\mathit{F}\cdot \frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{E}}\text{or}{P}_{iK}=F_{iL}\frac{\mathrm{\partial }W}{\mathrm{\partial }{E}_{LK}}.}$$In terms of theright Cauchy–Green deformation tensor (${\displaystyle \mathit{C}}$)$${\displaystyle \mathit{P}=2\mathit{F}\cdot \frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{C}}\text{or}{P}_{iK}=2F_{iL}\frac{\mathrm{\partial }W}{\mathrm{\partial }{C}_{LK}}.}$$

If${\displaystyle \mathit{S}}$ is thesecond Piola–Kirchhoff stress tensor then$${\displaystyle \mathit{S}={\mathit{F}}^{-1}\cdot \frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{F}}\text{or}{S}_{IJ}=F_{Ik}^{-1}\frac{\mathrm{\partial }W}{\mathrm{\partial }{F}_{kJ}}.}$$In terms of theLagrangian Green strain$${\displaystyle \mathit{S}=\frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{E}}\text{or}{S}_{IJ}=\frac{\mathrm{\partial }W}{\mathrm{\partial }{E}_{IJ}}.}$$In terms of theright Cauchy–Green deformation tensor$${\displaystyle \mathit{S}=2\frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{C}}\text{or}{S}_{IJ}=2\frac{\mathrm{\partial }W}{\mathrm{\partial }{C}_{IJ}}.}$$The above relation is also known as theDoyle-Ericksen formula in the material configuration.

Similarly, theCauchy stress is given by$${\displaystyle \mathit{\sigma }=\frac{1}{J}\frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{F}}\cdot {\mathit{F}}^{\mathsf{\text{T}}};J:=det\mathit{F}\text{or}{\sigma }_{ij}=\frac{1}{J}\frac{\mathrm{\partial }W}{\mathrm{\partial }{F}_{iK}}{F}_{jK}.}$$In terms of theLagrangian Green strain$${\displaystyle \mathit{\sigma }=\frac{1}{J}\mathit{F}\cdot \frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{E}}\cdot {\mathit{F}}^{\mathsf{\text{T}}}\text{or}{\sigma }_{ij}=\frac{1}{J}{F}_{iK}\frac{\mathrm{\partial }W}{\mathrm{\partial }{E}_{KL}}{F}_{jL}.}$$In terms of theright Cauchy–Green deformation tensor$${\displaystyle \mathit{\sigma }=\frac{2}{J}\mathit{F}\cdot \frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{C}}\cdot {\mathit{F}}^{\mathsf{\text{T}}}\text{or}{\sigma }_{ij}=\frac{2}{J}{F}_{iK}\frac{\mathrm{\partial }W}{\mathrm{\partial }{C}_{KL}}{F}_{jL}.}$$The above expressions are valid even for anisotropic media (in which case, the potential function is understood to dependimplicitly on reference directional quantities such as initial fiber orientations). In the special case of isotropy, the Cauchy stress can be expressed in terms of theleft Cauchy-Green deformation tensor as follows:^[7]$${\displaystyle \mathit{\sigma }=\frac{2}{J}\frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{B}}\cdot \mathit{B}\text{or}{\sigma }_{ij}=\frac{2}{J}{B}_{ik}\frac{\mathrm{\partial }W}{\mathrm{\partial }{B}_{kj}}.}$$

For an incompressible material${\displaystyle J:=det\mathit{F}=1}$. The incompressibility constraint is therefore${\displaystyle J-1=0}$. To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form:$${\displaystyle W=W(\mathit{F})-p(J-1)}$$where the hydrostatic pressure${\displaystyle p}$ functions as aLagrangian multiplier to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes$${\displaystyle \mathit{P}=-pJ{\mathit{F}}^{-\mathsf{\text{T}}}+\frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{F}}=-p{\mathit{F}}^{-\mathsf{\text{T}}}+\mathit{F}\cdot \frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{E}}=-p{\mathit{F}}^{-\mathsf{\text{T}}}+2\mathit{F}\cdot \frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{C}}.}$$This stress tensor can subsequently beconverted into any of the other conventional stress tensors, such as theCauchy stress tensor which is given by$${\displaystyle \mathit{\sigma }=\mathit{P}\cdot {\mathit{F}}^{\mathsf{\text{T}}}=-p\mathit{1}+\frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{F}}\cdot {\mathit{F}}^{\mathsf{\text{T}}}=-p\mathit{1}+\mathit{F}\cdot \frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{E}}\cdot {\mathit{F}}^{\mathsf{\text{T}}}=-p\mathit{1}+2\mathit{F}\cdot \frac{\mathrm{\partial }W}{\mathrm{\partial }\mathit{C}}\cdot {\mathit{F}}^{\mathsf{\text{T}}}.}$$

Forisotropic hyperelastic materials, the Cauchy stress can be expressed in terms of the invariants of theleft Cauchy–Green deformation tensor (orright Cauchy–Green deformation tensor). If thestrain energy density function is$${\displaystyle W(\mathit{F})=\hat{W}(I_1,I_2,I_3)=\overline{W}({\overline{I}}_1,{\overline{I}}_2,J)=\tilde{W}({\lambda }_1,{\lambda }_2,{\lambda }_3),}$$ then(See the page onthe left Cauchy–Green deformation tensor for the definitions of these symbols).

For incompressibleisotropic hyperelastic materials, thestrain energy density function is${\displaystyle W(\mathit{F})=\hat{W}(I_1,I_2)}$. The Cauchy stress is then given bywhere${\displaystyle p}$ is an undetermined pressure. In terms of stress differences$${\displaystyle {\sigma }_{11}-{\sigma }_{33}={\lambda }_1\frac{\mathrm{\partial }W}{\mathrm{\partial }{\lambda }_1}-{\lambda }_3\frac{\mathrm{\partial }W}{\mathrm{\partial }{\lambda }_3};{\sigma }_{22}-{\sigma }_{33}={\lambda }_2\frac{\mathrm{\partial }W}{\mathrm{\partial }{\lambda }_2}-{\lambda }_3\frac{\mathrm{\partial }W}{\mathrm{\partial }{\lambda }_3}}$$If in addition${\displaystyle I_1=I_2}$, then$${\displaystyle \mathit{\sigma }=2\frac{\mathrm{\partial }W}{\mathrm{\partial }{I}_1}\mathit{B}-p\mathit{1}.}$$If${\displaystyle {\lambda }_1={\lambda }_2}$, then$${\displaystyle {\sigma }_{11}-{\sigma }_{33}={\sigma }_{22}-{\sigma }_{33}={\lambda }_1\frac{\mathrm{\partial }W}{\mathrm{\partial }{\lambda }_1}-{\lambda }_3\frac{\mathrm{\partial }W}{\mathrm{\partial }{\lambda }_3}}$$

Consistency with linear elasticity is often used to determine some of the parameters of hyperelastic material models. These consistency conditions can be found by comparingHooke's law with linearized hyperelasticity at small strains.

For isotropic hyperelastic materials to be consistent with isotropiclinear elasticity, the stress–strain relation should have the following form in theinfinitesimal strain limit:$${\displaystyle \mathit{\sigma }=\lambda \mathrm{t}\mathrm{r}(\mathit{\epsilon })\mathit{1}+2\mu \mathit{\epsilon }}$$where${\displaystyle \lambda ,\mu }$ are theLamé constants. The strain energy density function that corresponds to the above relation is^[1]$${\displaystyle W=\frac{1}{2}\lambda [\mathrm{t}\mathrm{r}(\mathit{\epsilon }){]}^2+\mu \mathrm{t}\mathrm{r}\left({\mathit{\epsilon }}^2\right)}$$For an incompressible material${\displaystyle \mathrm{t}\mathrm{r}(\mathit{\epsilon })=0}$ and we have$${\displaystyle W=\mu \mathrm{t}\mathrm{r}\left({\mathit{\epsilon }}^2\right)}$$For any strain energy density function${\displaystyle W({\lambda }_1,{\lambda }_2,{\lambda }_3)}$ to reduce to the above forms for small strains the following conditions have to be met^[1]

If the material isincompressible, 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.