Hyperelastic material

{{Short description|Constitutive model for ideally elastic material}} {{Use dmy dates|date=November 2017}} thumb|upright=1.5|Stress–strain curves for various hyperelastic material models. {{continuum mechanics|cTopic=Solid mechanics}} A '''hyperelastic''' or '''Green elastic''' material<ref name=Ogden>R.W. Ogden, 1984, ''Non-Linear Elastic Deformations'', {{ISBN|0-486-69648-0}}, Dover.</ref> 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.<ref>{{cite journal | last1 = Muhr | first1 = A. H. | year = 2005 | title = Modeling the stress–strain behavior of rubber | journal = Rubber Chemistry and Technology | volume = 78 | issue = 3| pages = 391–425 | doi = 10.5254/1.3547890 }}</ref> The behavior of unfilled, vulcanized elastomers often conforms closely to the hyperelastic ideal. Filled elastomers and biological tissues<ref>{{cite journal | pmc= 4278556 | pmid=25319496 | doi=10.1002/cnm.2691 | volume=30 | title=A finite strain nonlinear human mitral valve model with fluid-structure interaction | journal=Int J Numer Methods Biomed Eng | pages=1597–613 | last1 = Gao | first1 = H | last2 = Ma | first2 = X | last3 = Qi | first3 = N | last4 = Berry | first4 = C | last5 = Griffith | first5 = BE | last6 = Luo | first6 = X| year=2014 | issue=12 }}</ref><ref>{{cite journal | pmc= 5332559 | pmid=28228537 | doi=10.1098/rsif.2016.0596 | volume=14 | title=Morphoelasticity in the development of brown alga ''Ectocarpus siliculosus'': from cell rounding to branching | journal=J R Soc Interface | last1 = Jia | first1 = F | last2 = Ben Amar | first2 = M | last3 = Billoud | first3 = B | last4 = Charrier | first4 = B | year=2017 | issue=127 | article-number=20160596}}</ref> 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 respectively <math display="block">\begin{align} \boldsymbol{S} &= \boldsymbol{C} : \boldsymbol{E} \\ \boldsymbol{S} &= \lambda~ \text{tr}(\boldsymbol{E})\boldsymbol{\mathit{I}} + 2\mu\boldsymbol{E} \text{.} \end{align}</math> where <math>\mathbin{:}</math> is tensor contraction, <math>\boldsymbol{S}</math> is the second Piola–Kirchhoff stress, <math>\boldsymbol{C} \in \R^{3 \times 3 \times 3 \times 3}</math> is a fourth order stiffness tensor and <math>\boldsymbol{E}</math> is the Lagrangian Green strain given by <math display="block">\mathbf E =\frac{1}{2}\left[ (\nabla_{\mathbf X}\mathbf u)^\textsf{T} + \nabla_{\mathbf X}\mathbf u + (\nabla_{\mathbf X}\mathbf u)^\textsf{T} \cdot\nabla_{\mathbf X}\mathbf u\right]\,\!</math> <math>\lambda</math> and <math>\mu</math> are the Lamé constants, and <math>\boldsymbol{\mathit{I}}</math> is the second order unit tensor.

The strain-energy density per unit volume (of the reference configuration) function for the Saint Venant–Kirchhoff model is <math display="block">W(\boldsymbol{E}) = \frac{\lambda}{2}[\text{tr}(\boldsymbol{E})]^2 + \mu \text{tr}\mathord\left(\boldsymbol{E}^2\right)</math>

and the second Piola–Kirchhoff stress can be derived from the relation <math display="block"> \boldsymbol{S} = \frac{\partial W}{\partial \boldsymbol{E}} ~. </math>

=== 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 the underlying structure of the material #* Arruda–Boyce model<ref name=AB>{{cite journal|last1=Arruda|first1=E.M.|last2=Boyce|first2=M.C.|title=A three-dimensional model for the large stretch behavior of rubber elastic materials|journal=J. Mech. Phys. Solids|volume=41|pages=389–412|year=1993|doi=10.1016/0022-5096(93)90013-6|s2cid=136924401 |url=https://hal.archives-ouvertes.fr/hal-01390807/file/MACB.pdf }}</ref> #* Neo–Hookean model<ref name=Ogden/> #* Buche–Silberstein model<ref name=BS>{{cite journal|last1=Buche|first1=M.R.|last2=Silberstein|first2=M.N.|title=Statistical mechanical constitutive theory of polymer networks: The inextricable links between distribution, behavior, and ensemble|journal=Phys. Rev. E|volume=102|article-number=012501|year=2020|issue=1 |doi=10.1103/PhysRevE.102.012501|pmid=32794915 |arxiv=2004.07874 |bibcode=2020PhRvE.102a2501B |s2cid=215814600 }}</ref> # 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 <math>(\lambda_1, \lambda_2, \lambda_3)</math>: <math display="block"> W = f(\lambda_1) + f(\lambda_2) + f(\lambda_3) \,. </math>

== Stress–strain relations ==

=== Compressible hyperelastic materials ===

==== First Piola–Kirchhoff stress ==== If <math>W(\boldsymbol{F})</math> is the strain energy density function, the 1st Piola–Kirchhoff stress tensor can be calculated for a hyperelastic material as <math display="block"> \boldsymbol{P} = \frac{\partial W}{\partial \boldsymbol{F}} \qquad \text{or} \qquad P_{iK} = \frac{\partial W}{\partial F_{iK}}. </math> where <math>\boldsymbol{F}</math> is the deformation gradient. In terms of the Lagrangian Green strain (<math>\boldsymbol{E}</math>) <math display="block"> \boldsymbol{P} = \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}} \qquad \text{or} \qquad P_{iK} = F_{iL}~\frac{\partial W}{\partial E_{LK}} ~. </math> In terms of the right Cauchy–Green deformation tensor (<math>\boldsymbol{C}</math>) <math display="block"> \boldsymbol{P} = 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}} \qquad \text{or} \qquad P_{iK} = 2~F_{iL}~\frac{\partial W}{\partial C_{LK}} ~. </math>

==== Second Piola–Kirchhoff stress ==== If <math>\boldsymbol{S}</math> is the second Piola–Kirchhoff stress tensor then <math display="block"> \boldsymbol{S} = \boldsymbol{F}^{-1}\cdot\frac{\partial W}{\partial \boldsymbol{F}} \qquad \text{or} \qquad S_{IJ} = F^{-1}_{Ik}\frac{\partial W}{\partial F_{kJ}} ~. </math> In terms of the Lagrangian Green strain <math display="block"> \boldsymbol{S} = \frac{\partial W}{\partial \boldsymbol{E}} \qquad \text{or} \qquad S_{IJ} = \frac{\partial W}{\partial E_{IJ}} ~. </math> In terms of the right Cauchy–Green deformation tensor <math display="block"> \boldsymbol{S} = 2~\frac{\partial W}{\partial \boldsymbol{C}} \qquad \text{or} \qquad S_{IJ} = 2~\frac{\partial W}{\partial C_{IJ}} ~. </math> The above relation is also known as the '''Doyle-Ericksen formula''' in the material configuration.

==== Cauchy stress ==== Similarly, the Cauchy stress is given by <math display="block"> \boldsymbol{\sigma} = \frac{1}{J}~ \frac{\partial W}{\partial \boldsymbol{F}}\cdot\boldsymbol{F}^\textsf{T} ~;~~ J := \det\boldsymbol{F} \qquad \text{or} \qquad \sigma_{ij} = \frac{1}{J}~ \frac{\partial W}{\partial F_{iK}}~F_{jK} ~. </math> In terms of the Lagrangian Green strain <math display="block"> \boldsymbol{\sigma} = \frac{1}{J}~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}}\cdot\boldsymbol{F}^\textsf{T} \qquad \text{or} \qquad \sigma_{ij} = \frac{1}{J}~F_{iK}~\frac{\partial W}{\partial E_{KL}}~F_{jL} ~. </math> In terms of the right Cauchy–Green deformation tensor <math display="block"> \boldsymbol{\sigma} = \frac{2}{J}~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^\textsf{T} \qquad \text{or} \qquad \sigma_{ij} = \frac{2}{J}~F_{iK}~\frac{\partial W}{\partial C_{KL}}~F_{jL} ~. </math> 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:<ref name=Basar>Y. Basar, 2000, Nonlinear continuum mechanics of solids, Springer, p. 157.</ref> <math display="block"> \boldsymbol{\sigma} = \frac{2}{J}\frac{\partial W}{\partial \boldsymbol{B}}\cdot~\boldsymbol{B} \qquad \text{or} \qquad \sigma_{ij} = \frac{2}{J}~B_{ik}~\frac{\partial W}{\partial B_{kj}} ~. </math>

=== Incompressible hyperelastic materials === For an incompressible material <math>J := \det\boldsymbol{F} = 1</math>. The incompressibility constraint is therefore <math>J-1= 0</math>. To ensure incompressibility of a hyperelastic material, the strain-energy function can be written in form: <math display="block">W = W(\boldsymbol{F}) - p~(J-1)</math> where the hydrostatic pressure <math>p</math> functions as a Lagrangian multiplier to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes <math display="block"> \boldsymbol{P}=-p~J\boldsymbol{F}^{-\textsf{T}} + \frac{\partial W}{\partial \boldsymbol{F}} = -p~\boldsymbol{F}^{-\textsf{T}} + \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}} = -p~\boldsymbol{F}^{-\textsf{T}} + 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}} ~. </math> 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 <math display="block"> \boldsymbol{\sigma}=\boldsymbol{P}\cdot\boldsymbol{F}^\textsf{T} = -p~\boldsymbol{\mathit{1}} + \frac{\partial W}{\partial \boldsymbol{F}}\cdot\boldsymbol{F}^\textsf{T} = -p~\boldsymbol{\mathit{1}} + \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}}\cdot\boldsymbol{F}^\textsf{T} = -p~\boldsymbol{\mathit{1}} + 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^\textsf{T} ~. </math>

== 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 <math display="block">W(\boldsymbol{F})=\hat{W}(I_1,I_2,I_3) = \bar{W}(\bar{I}_1,\bar{I}_2, J) = \tilde{W}(\lambda_1,\lambda_2, \lambda_3),</math> then <math display="block">\begin{align} \boldsymbol{\sigma} & = \frac{2}{\sqrt{I_3}}\left[\left(\frac{\partial\hat{W}}{\partial I_1} + I_1~\frac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \frac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] + 2\sqrt{I_3}~\frac{\partial\hat{W}}{\partial I_3}~\boldsymbol{\mathit{1}} \\[5pt] & = \frac{2}{J}\left[\frac{1}{J^{2/3}}\left(\frac{\partial\bar{W}}{\partial \bar{I}_1} + \bar{I}_1~\frac{\partial\bar{W}}{\partial \bar{I}_2}\right)\boldsymbol{B} - \frac{1}{J^{4/3}}~\frac{\partial\bar{W}}{\partial \bar{I}_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] + \left[\frac{\partial\bar{W}}{\partial J} - \frac{2}{3J} \left(\bar{I}_1~\frac{\partial\bar{W}}{\partial \bar{I}_1} + 2~\bar{I}_2~\frac{\partial\bar{W}}{\partial \bar{I}_2}\right)\right] ~\boldsymbol{\mathit{1}} \\[5pt] & = \frac{2}{J} \left[\left(\frac{\partial\bar{W}}{\partial \bar{I}_1} + \bar{I}_1~\frac{\partial\bar{W}}{\partial \bar{I}_2}\right)\bar{\boldsymbol{B}} - \frac{\partial\bar{W}}{\partial \bar{I}_2}~\bar{\boldsymbol{B}} \cdot\bar{\boldsymbol{B}} \right] + \left[\frac{\partial\bar{W}}{\partial J} - \frac{2}{3J}\left(\bar{I}_1~\frac{\partial\bar{W}}{\partial \bar{I}_1} + 2~\bar{I}_2~\frac{\partial\bar{W}}{\partial \bar{I}_2}\right)\right] ~\boldsymbol{\mathit{1}} \\[5pt] & = \frac{\lambda_1}{\lambda_1\lambda_2\lambda_3}~\frac{\partial\tilde{W}}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 + \frac{\lambda_2}{\lambda_1\lambda_2\lambda_3}~\frac{\partial\tilde{W}}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \frac{\lambda_3}{\lambda_1\lambda_2\lambda_3}~\frac{\partial\tilde{W}}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3 \end{align} </math> (See the page on the left Cauchy–Green deformation tensor for the definitions of these symbols).

{{math proof | title = Proof 1 | proof = The second Piola–Kirchhoff stress tensor for a hyperelastic material is given by <math display="block"> \boldsymbol{S} = 2~\frac{\partial W}{\partial \boldsymbol{C}} </math> where <math>\boldsymbol{C} = \boldsymbol{F}^T\cdot\boldsymbol{F}</math> is the right Cauchy–Green deformation tensor and <math>\boldsymbol{F}</math> is the deformation gradient. The Cauchy stress is given by <math display="block"> \boldsymbol{\sigma} = \frac{1}{J}~\boldsymbol{F}\cdot\boldsymbol{S}\cdot\boldsymbol{F}^T = \frac{2}{J}~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^T </math> where <math>J = \det\boldsymbol{F}</math>. Let <math>I_1, I_2, I_3</math> be the three principal invariants of <math>\boldsymbol{C}</math>. Then <math display="block"> \frac{\partial W}{\partial \boldsymbol{C}} = \frac{\partial W}{\partial I_1}~\frac{\partial I_1}{\partial \boldsymbol{C}} + \frac{\partial W}{\partial I_2}~\frac{\partial I_2}{\partial \boldsymbol{C}} + \frac{\partial W}{\partial I_3}~\frac{\partial I_3}{\partial \boldsymbol{C}} ~. </math> The derivatives of the invariants of the symmetric tensor <math>\boldsymbol{C}</math> are <math display="block"> \frac{\partial I_1}{\partial \boldsymbol{C}} = \boldsymbol{\mathit{1}} ~;~~ \frac{\partial I_2}{\partial \boldsymbol{C}} = I_1~\boldsymbol{\mathit{1}} - \boldsymbol{C} ~;~~ \frac{\partial I_3}{\partial \boldsymbol{C}} = \det(\boldsymbol{C})~\boldsymbol{C}^{-1} </math> Therefore, we can write <math display="block"> \frac{\partial W}{\partial \boldsymbol{C}} = \frac{\partial W}{\partial I_1}~\boldsymbol{\mathit{1}} + \frac{\partial W}{\partial I_2}~(I_1~\boldsymbol{\mathit{1}} - \boldsymbol{F}^T\cdot\boldsymbol{F}) + \frac{\partial W}{\partial I_3}~I_3~\boldsymbol{F}^{-1}\cdot\boldsymbol{F}^{-T} ~. </math> Plugging into the expression for the Cauchy stress gives <math display="block"> \boldsymbol{\sigma} = \frac{2}{J}~\left[\frac{\partial W}{\partial I_1}~\boldsymbol{F}\cdot\boldsymbol{F}^T+ \frac{\partial W}{\partial I_2}~(I_1~\boldsymbol{F}\cdot\boldsymbol{F}^T - \boldsymbol{F}\cdot \boldsymbol{F}^T \cdot \boldsymbol{F}\cdot \boldsymbol{F}^T) + \frac{\partial W}{\partial I_3}~I_3~\boldsymbol{\mathit{1}}\right] </math> Using the left Cauchy–Green deformation tensor <math>\boldsymbol{B}=\boldsymbol{F}\cdot\boldsymbol{F}^T</math> and noting that <math>I_3 = J^2</math>, we can write <math display="block"> \boldsymbol{\sigma} = \frac{2}{\sqrt{I_3}}~\left[\left(\frac{\partial W}{\partial I_1} + I_1~\frac{\partial W}{\partial I_2}\right)~\boldsymbol{B} - \frac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] + 2~\sqrt{I_3}~\frac{\partial W}{\partial I_3}~\boldsymbol{\mathit{1}}~. </math> For an incompressible material <math>I_3 = 1</math> and hence <math>W = W(I_1,I_2)</math>.Then <math display="block"> \frac{\partial W}{\partial \boldsymbol{C}} = \frac{\partial W}{\partial I_1}~\frac{\partial I_1}{\partial \boldsymbol{C}} + \frac{\partial W}{\partial I_2}~\frac{\partial I_2}{\partial \boldsymbol{C}} = \frac{\partial W}{\partial I_1}~\boldsymbol{\mathit{1}} + \frac{\partial W}{\partial I_2}~(I_1~\boldsymbol{\mathit{1}} - \boldsymbol{F}^T\cdot\boldsymbol{F}) </math> Therefore, the Cauchy stress is given by <math display="block"> \boldsymbol{\sigma} = 2\left[\left(\frac{\partial W}{\partial I_1} + I_1~\frac{\partial W}{\partial I_2}\right)~\boldsymbol{B} - \frac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] - p~\boldsymbol{\mathit{1}}~. </math> where <math>p</math> is an undetermined pressure which acts as a Lagrange multiplier to enforce the incompressibility constraint.

If, in addition, <math>I_1 = I_2</math>, we have <math> W = W(I_1) </math> and hence <math display="block"> \frac{\partial W}{\partial \boldsymbol{C}} = \frac{\partial W}{\partial I_1}~\frac{\partial I_1}{\partial \boldsymbol{C}} = \frac{\partial W}{\partial I_1}~\boldsymbol{\mathit{1}} </math> In that case the Cauchy stress can be expressed as <math display="block"> \boldsymbol{\sigma} = 2\frac{\partial W}{\partial I_1}~\boldsymbol{B} - p~\boldsymbol{\mathit{1}}~. </math> }}

{{math proof | title = Proof 2 | proof = The isochoric deformation gradient is defined as <math>\bar{\boldsymbol{F}}:=J^{-1/3}\boldsymbol{F}</math>, 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 <math>\bar{\boldsymbol{B}} := \bar{\boldsymbol{F}}\cdot\bar{\boldsymbol{F}}^T=J^{-2/3}\boldsymbol{B}</math>. The invariants of <math>\bar{\boldsymbol{B}}</math> are <math display="block">\begin{align} \bar I_1 &= \text{tr}(\bar{\boldsymbol{B}}) = J^{-2/3}\text{tr}(\boldsymbol{B}) = J^{-2/3} I_1 \\ \bar I_2 & = \frac{1}{2}\left(\text{tr}(\bar{\boldsymbol{B}})^2 - \text{tr}(\bar{\boldsymbol{B}}^2)\right) = \frac{1}{2}\left( \left(J^{-2/3}\text{tr}(\boldsymbol{B})\right)^2 - \text{tr}(J^{-4/3}\boldsymbol{B}^2) \right) = J^{-4/3} I_2 \\ \bar I_3 &= \det(\bar{\boldsymbol{B}}) = J^{-6/3} \det(\boldsymbol{B}) = J^{-2} I_3 = J^{-2} J^2 = 1 \end{align}</math> 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 <math>J</math> into the fray to describe the volumetric behaviour.

To express the Cauchy stress in terms of the invariants <math>\bar{I}_1, \bar{I}_2, J</math> recall that <math display="block"> \bar{I}_1 = J^{-2/3}~I_1 = I_3^{-1/3}~I_1 ~;~~ \bar{I}_2 = J^{-4/3}~I_2 = I_3^{-2/3}~I_2 ~;~~ J = I_3^{1/2} ~. </math> The chain rule of differentiation gives us <math display="block">\begin{align} \frac{\partial W}{\partial I_1} & = \frac{\partial W}{\partial \bar{I}_1}~\frac{\partial \bar{I}_1}{\partial I_1} + \frac{\partial W}{\partial \bar{I}_2}~\frac{\partial \bar{I}_2}{\partial I_1} + \frac{\partial W}{\partial J}~\frac{\partial J}{\partial I_1} \\ & = I_3^{-1/3}~\frac{\partial W}{\partial \bar{I}_1} = J^{-2/3}~\frac{\partial W}{\partial \bar{I}_1} \\ \frac{\partial W}{\partial I_2} & = \frac{\partial W}{\partial \bar{I}_1}~\frac{\partial \bar{I}_1}{\partial I_2} + \frac{\partial W}{\partial \bar{I}_2}~\frac{\partial \bar{I}_2}{\partial I_2} + \frac{\partial W}{\partial J}~\frac{\partial J}{\partial I_2} \\ & = I_3^{-2/3}~\frac{\partial W}{\partial \bar{I}_2} = J^{-4/3}~\frac{\partial W}{\partial \bar{I}_2} \\ \frac{\partial W}{\partial I_3} & = \frac{\partial W}{\partial \bar{I}_1}~\frac{\partial \bar{I}_1}{\partial I_3} + \frac{\partial W}{\partial \bar{I}_2}~\frac{\partial \bar{I}_2}{\partial I_3} + \frac{\partial W}{\partial J}~\frac{\partial J}{\partial I_3} \\ & = - \frac{1}{3}~I_3^{-4/3}~I_1~\frac{\partial W}{\partial \bar{I}_1} - \frac{2}{3}~I_3^{-5/3}~I_2~\frac{\partial W}{\partial \bar{I}_2} + \frac{1}{2}~I_3^{-1/2}~\frac{\partial W}{\partial J} \\ & = - \frac{1}{3}~J^{-8/3}~J^{2/3}~\bar{I}_1~\frac{\partial W}{\partial \bar{I}_1} - \frac{2}{3}~J^{-10/3}~J^{4/3}~\bar{I}_2~\frac{\partial W}{\partial \bar{I}_2} + \frac{1}{2}~J^{-1}~\frac{\partial W}{\partial J} \\ & = -\frac{1}{3}~J^{-2}~\left(\bar{I}_1~\frac{\partial W}{\partial \bar{I}_1}+ 2~\bar{I}_2~\frac{\partial W}{\partial \bar{I}_2}\right) + \frac{1}{2}~J^{-1}~\frac{\partial W}{\partial J} \end{align} </math> Recall that the Cauchy stress is given by <math display="block"> \boldsymbol{\sigma} = \frac{2}{\sqrt{I_3}}~\left[\left(\frac{\partial W}{\partial I_1} + I_1~\frac{\partial W}{\partial I_2}\right)~\boldsymbol{B} - \frac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] + 2~\sqrt{I_3}~\frac{\partial W}{\partial I_3}~\boldsymbol{\mathit{1}}~. </math> In terms of the invariants <math>\bar{I}_1, \bar{I}_2, J</math> we have <math display="block"> \boldsymbol{\sigma} = \frac{2}{J}~\left[\left(\frac{\partial W}{\partial I_1}+ J^{2/3}~\bar{I}_1~\frac{\partial W}{\partial I_2}\right)~\boldsymbol{B} - \frac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] + 2~J~\frac{\partial W}{\partial I_3}~\boldsymbol{\mathit{1}}~. </math> Plugging in the expressions for the derivatives of <math>W</math> in terms of <math>\bar{I}_1, \bar{I}_2, J</math>, we have <math display="block">\begin{align} \boldsymbol{\sigma} & = \frac{2}{J}~\left[\left(J^{-2/3}~\frac{\partial W}{\partial \bar{I}_1} + J^{-2/3}~\bar{I}_1~\frac{\partial W}{\partial \bar{I}_2}\right)~\boldsymbol{B} - J^{-4/3}~\frac{\partial W}{\partial \bar{I}_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] + \\ & \qquad 2~J~\left[-\frac{1}{3}~J^{-2}~\left(\bar{I}_1~\frac{\partial W}{\partial \bar{I}_1}+ 2~\bar{I}_2~\frac{\partial W}{\partial \bar{I}_2}\right) + \frac{1}{2}~J^{-1}~\frac{\partial W}{\partial J}\right]~\boldsymbol{\mathit{1}} \end{align}</math> or, <math display="block"> \begin{align} \boldsymbol{\sigma} & = \frac{2}{J}~\left[\frac{1}{J^{2/3}}~\left(\frac{\partial W}{\partial \bar{I}_1} + \bar{I}_1~\frac{\partial W}{\partial \bar{I}_2}\right)~\boldsymbol{B} - \frac{1}{J^{4/3}}~ \frac{\partial W}{\partial \bar{I}_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] \\ & \qquad + \left[\frac{\partial W}{\partial J} - \frac{2}{3J}\left(\bar{I}_1~\frac{\partial W}{\partial \bar{I}_1}+ 2~\bar{I}_2~\frac{\partial W}{\partial \bar{I}_2}\right)\right]\boldsymbol{\mathit{1}} \end{align} </math> In terms of the deviatoric part of <math>\boldsymbol{B}</math>, we can write <math display="block">\begin{align} \boldsymbol{\sigma} & = \frac{2}{J}~\left[\left(\frac{\partial W}{\partial \bar{I}_1} + \bar{I}_1~\frac{\partial W}{\partial \bar{I}_2}\right)~\bar{\boldsymbol{B}} - \frac{\partial W}{\partial \bar{I}_2}~\bar{\boldsymbol{B}}\cdot\bar{\boldsymbol{B}}\right] \\ & \qquad + \left[\frac{\partial W}{\partial J} - \frac{2}{3J}\left(\bar{I}_1~\frac{\partial W}{\partial \bar{I}_1}+ 2~\bar{I}_2~\frac{\partial W}{\partial \bar{I}_2}\right)\right]\boldsymbol{\mathit{1}} \end{align} </math> For an incompressible material <math>J = 1</math> and hence <math>W = W(\bar{I}_1,\bar{I}_2)</math>.Then the Cauchy stress is given by <math display="block"> \boldsymbol{\sigma} = 2\left[\left(\frac{\partial W}{\partial \bar{I}_1} + I_1~\frac{\partial W}{\partial \bar{I}_2}\right)~\bar{\boldsymbol{B}} - \frac{\partial W}{\partial \bar{I}_2}~\bar{\boldsymbol{B}}\cdot\bar{\boldsymbol{B}}\right] - p~\boldsymbol{\mathit{1}}~. </math> where <math>p</math> is an undetermined pressure-like Lagrange multiplier term. In addition, if <math>\bar{I}_1 = \bar{I}_2</math>, we have <math> W = W(\bar{I}_1) </math> and hence the Cauchy stress can be expressed as <math display="block"> \boldsymbol{\sigma} = 2\frac{\partial W}{\partial \bar{I}_1}~\bar{\boldsymbol{B}} - p~\boldsymbol{\mathit{1}}~. </math> }}

{{math proof | title = Proof 3 | proof = To express the Cauchy stress in terms of the stretches <math>\lambda_1, \lambda_2, \lambda_3</math> recall that <math display="block"> \frac{\partial \lambda_i}{\partial\boldsymbol{C}} = \frac{1}{2\lambda_i}~\boldsymbol{R}^T\cdot(\mathbf{n}_i\otimes\mathbf{n}_i)\cdot\boldsymbol{R}~;~~ i = 1,2,3 ~. </math> The chain rule gives <math display="block">\begin{align} \frac{\partial W}{\partial\boldsymbol{C}} & = \frac{\partial W}{\partial \lambda_1}~\frac{\partial \lambda_1}{\partial\boldsymbol{C}} + \frac{\partial W}{\partial \lambda_2}~\frac{\partial \lambda_2}{\partial\boldsymbol{C}} + \frac{\partial W}{\partial \lambda_3}~\frac{\partial \lambda_3}{\partial\boldsymbol{C}} \\ & = \boldsymbol{R}^T\cdot\left[\frac{1}{2\lambda_1}~\frac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 + \frac{1}{2\lambda_2}~\frac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \frac{1}{2\lambda_3}~\frac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3\right]\cdot\boldsymbol{R} \end{align} </math> The Cauchy stress is given by <math display="block"> \boldsymbol{\sigma} = \frac{2}{J}~\boldsymbol{F}\cdot \frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^T = \frac{2}{J}~(\boldsymbol{V}\cdot\boldsymbol{R})\cdot \frac{\partial W}{\partial \boldsymbol{C}}\cdot(\boldsymbol{R}^T\cdot\boldsymbol{V}) </math> Plugging in the expression for the derivative of <math>W</math> leads to <math display="block"> \boldsymbol{\sigma} = \frac{2}{J}~\boldsymbol{V}\cdot \left[\frac{1}{2\lambda_1}~ \frac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 + \frac{1}{2\lambda_2}~ \frac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \frac{1}{2\lambda_3}~ \frac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3\right] \cdot\boldsymbol{V} </math> Using the spectral decomposition of <math>\boldsymbol{V}</math> we have <math display="block"> \boldsymbol{V}\cdot(\mathbf{n}_i\otimes\mathbf{n}_i)\cdot\boldsymbol{V} = \lambda_i^2~\mathbf{n}_i\otimes\mathbf{n}_i ~;~~ i=1,2,3. </math> Also note that <math display="block"> J = \det(\boldsymbol{F}) = \det(\boldsymbol{V})\det(\boldsymbol{R}) = \det(\boldsymbol{V}) = \lambda_1 \lambda_2 \lambda_3 ~. </math> Therefore, the expression for the Cauchy stress can be written as <math display="block"> \boldsymbol{\sigma} = \frac{1}{\lambda_1\lambda_2\lambda_3}~ \left[\lambda_1~\frac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 + \lambda_2~\frac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \lambda_3~\frac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3 \right] </math> For an incompressible material <math>\lambda_1\lambda_2\lambda_3 = 1</math> and hence <math>W = W(\lambda_1,\lambda_2)</math>. Following Ogden<ref name=Ogden/> p. 485, we may write <math display="block"> \boldsymbol{\sigma} = \lambda_1~\frac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 + \lambda_2~\frac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \lambda_3~\frac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3 - p~\boldsymbol{\mathit{1}}~ </math> Some care is required at this stage because, when an eigenvalue is repeated, it is in general only Gateaux differentiable, but not Fréchet differentiable.<ref>Fox & Kapoor, ''Rates of change of eigenvalues and eigenvectors'', '''AIAA Journal''', 6 (12) 2426–2429 (1968)</ref><ref>Friswell MI. ''The derivatives of repeated eigenvalues and their associated eigenvectors.'' '''Journal of Vibration and Acoustics''' (ASME) 1996; 118:390–397.</ref> A rigorous tensor derivative can only be found by solving another eigenvalue problem.

If we express the stress in terms of differences between components, <math display="block"> \sigma_{11} - \sigma_{33} = \lambda_1~\frac{\partial W}{\partial \lambda_1} - \lambda_3~\frac{\partial W}{\partial \lambda_3} ~;~~ \sigma_{22} - \sigma_{33} = \lambda_2~\frac{\partial W}{\partial \lambda_2} - \lambda_3~\frac{\partial W}{\partial \lambda_3} </math> If in addition to incompressibility we have <math>\lambda_1 = \lambda_2</math> then a possible solution to the problem requires <math>\sigma_{11} = \sigma_{22}</math> and we can write the stress differences as <math display="block"> \sigma_{11} - \sigma_{33} = \sigma_{22} - \sigma_{33} = \lambda_1~\frac{\partial W}{\partial \lambda_1} - \lambda_3~\frac{\partial W}{\partial \lambda_3} </math> }}

=== Incompressible isotropic hyperelastic materials === For incompressible isotropic hyperelastic materials, the strain energy density function is <math>W(\boldsymbol{F})=\hat{W}(I_1,I_2)</math>. The Cauchy stress is then given by <math display="block">\begin{align} \boldsymbol{\sigma} & = -p~\boldsymbol{\mathit{1}} + 2\left[\left(\frac{\partial\hat{W}}{\partial I_1} + I_1~\frac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \frac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] \\ & = - p~\boldsymbol{\mathit{1}} + 2\left[\left(\frac{\partial W}{\partial \bar{I}_1} + I_1~\frac{\partial W}{\partial \bar{I}_2}\right)~\bar{\boldsymbol{B}} - \frac{\partial W}{\partial \bar{I}_2}~\bar{\boldsymbol{B}}\cdot\bar{\boldsymbol{B}}\right] \\ & = - p~\boldsymbol{\mathit{1}} + \lambda_1~\frac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 + \lambda_2~\frac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \lambda_3~\frac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3 \end{align} </math> where <math>p</math> is an undetermined pressure. In terms of stress differences <math display="block"> \sigma_{11} - \sigma_{33} = \lambda_1~\frac{\partial W}{\partial \lambda_1} - \lambda_3~\frac{\partial W}{\partial \lambda_3}~;~~ \sigma_{22} - \sigma_{33} = \lambda_2~\frac{\partial W}{\partial \lambda_2} - \lambda_3~\frac{\partial W}{\partial \lambda_3} </math> If in addition <math>I_1 = I_2</math>, then <math display="block"> \boldsymbol{\sigma} = 2\frac{\partial W}{\partial I_1}~\boldsymbol{B} - p~\boldsymbol{\mathit{1}}~. </math> If <math>\lambda_1 = \lambda_2</math>, then <math display="block"> \sigma_{11} - \sigma_{33} = \sigma_{22} - \sigma_{33} = \lambda_1~\frac{\partial W}{\partial \lambda_1} - \lambda_3~\frac{\partial W}{\partial \lambda_3} </math>

== 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: <math display="block"> \boldsymbol{\sigma} = \lambda~\mathrm{tr}(\boldsymbol{\varepsilon})~\boldsymbol{\mathit{1}} + 2\mu\boldsymbol{\varepsilon} </math> where <math>\lambda, \mu</math> are the Lamé constants. The strain energy density function that corresponds to the above relation is<ref name=Ogden/> <math display="block"> W = \tfrac{1}{2}\lambda~[\mathrm{tr}(\boldsymbol{\varepsilon})]^2 + \mu~\mathrm{tr}\mathord\left(\boldsymbol{\varepsilon}^2\right) </math> For an incompressible material <math>\mathrm{tr}(\boldsymbol{\varepsilon}) = 0</math> and we have <math display="block"> W = \mu~\mathrm{tr}\mathord\left(\boldsymbol{\varepsilon}^2\right) </math> For any strain energy density function <math>W(\lambda_1,\lambda_2,\lambda_3)</math> to reduce to the above forms for small strains the following conditions have to be met<ref name=Ogden/> <math display="block">\begin{align} & W(1,1,1) = 0 ~;~~ \frac{\partial W}{\partial \lambda_i}(1,1,1) = 0 \\ & \frac{\partial^2 W}{\partial \lambda_i \partial \lambda_j}(1,1,1) = \lambda + 2\mu\delta_{ij} \end{align} </math>

If the material is '''incompressible,''' then the above conditions may be expressed in the following form. <math display="block">\begin{align} & W(1,1,1) = 0 \\ & \frac{\partial W}{\partial \lambda_i}(1,1,1) = \frac{\partial W}{\partial \lambda_j}(1,1,1) ~;~~ \frac{\partial^2 W}{\partial \lambda_i^2}(1,1,1) = \frac{\partial^2 W}{\partial \lambda_j^2}(1,1,1) \\ & \frac{\partial^2 W}{\partial \lambda_i \partial \lambda_j}(1,1,1) = \mathrm{independent of}~i,j\ne i \\ & \frac{\partial^2 W}{\partial \lambda_i^2}(1,1,1) - \frac{\partial^2 W}{\partial \lambda_i \partial \lambda_j}(1,1,1) + \frac{\partial W}{\partial \lambda_i}(1,1,1) = 2\mu ~~(i \ne j) \end{align} </math> 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 {{math|''I''<sub>1</sub>}} based rubber materials === Many elastomers are modeled adequately by a strain energy density function that depends only on <math>I_1</math>. For such materials we have <math> W = W(I_1) </math>. The consistency conditions for incompressible materials for <math>I_1 = 3, \lambda_i = \lambda_j = 1</math> may then be expressed as <math display="block"> \left.W(I_1)\right|_{I_1=3} = 0 \quad \text{and} \quad \left.\frac{\partial W}{\partial I_1}\right|_{I_1=3} = \frac{\mu}{2} \,. </math> The second consistency condition above can be derived by noting that <math display="block"> \frac{\partial W}{\partial \lambda_i} = \frac{\partial W}{\partial I_1}\frac{\partial I_1}{\partial \lambda_i} = 2\lambda_i\frac{\partial W}{\partial I_1} \quad\text{and}\quad \frac{\partial^2 W}{\partial \lambda_i \partial \lambda_j} = 2\delta_{ij}\frac{\partial W}{\partial I_1} + 4\lambda_i\lambda_j \frac{\partial^2 W}{\partial I_1^2}\,. </math> These relations can then be substituted into the consistency condition for isotropic incompressible hyperelastic materials.

== References ==

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