|
|
| Line 1: |
Line 1: |
| {{continuum mechanics|cTopic=[[Solid mechanics]]}}
| | Irwin Butts is what my wife loves to call me although I don't really like becoming known as like that. My family lives in Minnesota and my family members enjoys [http://www.ninfeta.tv/blog/99493 click the up coming web page] std home test it. The [http://www.womansday.com/sex-relationships/sex-tips/everything-you-need-to-know-about-lubricants-101625 preferred pastime] for my kids and me is to perform baseball and I'm attempting to make it a occupation. Since she was eighteen she's been operating as a meter reader but [http://www.youporntime.com/user/KBiaggini at home std testing] she's always needed her personal company.<br><br>Also visit my home std test kit blog post ... [http://youulike.com/profile/116412 http://youulike.com] |
| [[Image:Hyperelastic.svg|thumb|290px|right|Stress-strain curves for various hyperelastic material models.]]
| |
| 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 equation|constitutive model]] for ideally [[elastic (solid mechanics)|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, [[Elasticity (physics)|linear elastic]] models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose [[stress (physics)|stress]]-[[strain (physics)|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.<ref>Muhr, A. H. (2005). Modeling the stress-strain behavior of rubber. Rubber chemistry and technology, 78(3), 391-425. [http://dx.doi.org/10.5254/1.3547890]</ref> 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 solid|Neo-Hookean]] and [[Mooney–Rivlin solid|Mooney–Rivlin]] solids. Many other hyperelastic models have since been developed. Other widely used hyperelastic material models include the [[Ogden (hyperelastic model)|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 linear elastic material model to the nonlinear regime. This model has the form
| |
| :<math>
| |
| \boldsymbol{S} = \lambda~ \text{tr}(\boldsymbol{E})\boldsymbol{\mathit{1}} + 2\mu\boldsymbol{E}
| |
| </math>
| |
| where <math>\boldsymbol{S}</math> is the second Piola–Kirchhoff stress and <math>\boldsymbol{E}</math> is the Lagrangian Green strain, and <math>\lambda</math> and <math>\mu</math> are the [[Lame constants|Lamé constants]].
| |
| | |
| The strain-energy density function for the St. Venant–Kirchhoff model is
| |
| :<math>
| |
| W(\boldsymbol{E}) = \frac{\lambda}{2}[\text{tr}(\boldsymbol{E})]^2 + \mu \text{tr}(\boldsymbol{E}^2)
| |
| </math>
| |
| and the second Piola–Kirchhoff stress can be derived from the relation
| |
| :<math>
| |
| \boldsymbol{S} = \cfrac{\partial W}{\partial \boldsymbol{E}} ~.
| |
| </math>
| |
| | |
| === Classification of hyperelastic material models ===
| |
| Hyperelastic material models can be classified as:
| |
| | |
| 1) [[Phenomenology (science)|phenomenological]] descriptions of observed behavior
| |
| *[[Soft tissue#Fung-elastic material|Fung]]
| |
| *[[Mooney–Rivlin solid|Mooney–Rivlin]]
| |
| *[[Ogden (hyperelastic model)|Ogden]]
| |
| *[[Polynomial (hyperelastic model)|Polynomial]]
| |
| *Saint Venant–Kirchhoff
| |
| *[[Yeoh (hyperelastic model)|Yeoh]]
| |
| *[[Marlow (hyperelastic model)|Marlow]]
| |
| | |
| 2) [[Rubber elasticity|mechanistic models]] deriving from arguments about underlying structure of the material
| |
| *[[Arruda–Boyce model]]
| |
| *[[Neo-Hookean solid|Neo-Hookean]]
| |
| | |
| 3) hybrids of phenomenological and mechanistic models
| |
| *[[Gent (hyperelastic model)|Gent]]
| |
| *[[Van der Waals (hyperelatic model)|Van der Waals]]
| |
| | |
| Generally, a hyperelastic model should satisfy the [[Drucker stability]] criterion.
| |
| | |
| == Stress-strain relations ==
| |
| | |
| === Compressible hyperelastic materials ===
| |
| | |
| ==== First Piola–Kirchhoff stress ====
| |
| If <math>W(\boldsymbol{F})</math> is the strain energy density function, the [[Piola–Kirchhoff stress tensor|1st Piola–Kirchhoff stress tensor]] can be calculated for a hyperelastic material as
| |
| :<math>
| |
| \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 [[Finite_strain_theory#Finite_strain_tensors|Lagrangian Green strain]] (<math>\boldsymbol{E}</math>)
| |
| :<math>
| |
| \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 [[finite strain theory|right Cauchy–Green deformation tensor]] (<math>\boldsymbol{C}</math>)
| |
| :<math>
| |
| \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 [[Piola–Kirchhoff stress tensor|second Piola–Kirchhoff stress tensor]] then
| |
| :<math>
| |
| \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 [[Finite_strain_theory#Finite_strain_tensors|Lagrangian Green strain]]
| |
| :<math>
| |
| \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 [[finite strain theory|right Cauchy–Green deformation tensor]]
| |
| :<math>
| |
| \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 [[stress (physics)|Cauchy stress]] is given by
| |
| :<math>
| |
| \boldsymbol{\sigma} = \cfrac{1}{J}~ \cfrac{\partial W}{\partial \boldsymbol{F}}\cdot\boldsymbol{F}^T ~;~~ J := \det\boldsymbol{F} \qquad \text{or} \qquad
| |
| \sigma_{ij} = \cfrac{1}{J}~ \cfrac{\partial W}{\partial F_{iK}}~F_{jK} ~.
| |
| </math>
| |
| In terms of the [[Finite_strain_theory#Finite_strain_tensors|Lagrangian Green strain]]
| |
| :<math>
| |
| \boldsymbol{\sigma} = \cfrac{1}{J}~\boldsymbol{F}\cdot\cfrac{\partial W}{\partial \boldsymbol{E}}\cdot\boldsymbol{F}^T \qquad \text{or} \qquad
| |
| \sigma_{ij} = \cfrac{1}{J}~F_{iK}~\cfrac{\partial W}{\partial E_{KL}}~F_{jL} ~.
| |
| </math>
| |
| In terms of the [[finite strain theory|right Cauchy–Green deformation tensor]]
| |
| :<math>
| |
| \boldsymbol{\sigma} = \cfrac{2}{J}~\boldsymbol{F}\cdot\cfrac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^T \qquad \text{or} \qquad
| |
| \sigma_{ij} = \cfrac{2}{J}~F_{iK}~\cfrac{\partial W}{\partial C_{KL}}~F_{jL} ~.
| |
| </math>
| |
| The above expression can also be expressed in terms of the ''left'' Cauchy-Green deformation tensor. In that case <ref name=Basar>Y. Basar, 2000, Nonlinear continuum mechanics of solids, Springer, p. 157.</ref>
| |
| :<math>
| |
| \boldsymbol{\sigma} = \cfrac{2}{J}~\boldsymbol{B}\cdot\cfrac{\partial W}{\partial \boldsymbol{B}} \qquad \text{or} \qquad
| |
| \sigma_{ij} = \cfrac{2}{J}~B_{ik}~\cfrac{\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>
| |
| W = W(\boldsymbol{F}) - p~(J-1)
| |
| </math>
| |
| where the hydrostatic pressure <math>p</math> functions as a [[Lagrange multipliers|Lagrangian multiplier]] to enforce the incompressibility constraint. The 1st Piola–Kirchhoff stress now becomes
| |
| :<math>
| |
| \boldsymbol{P}=-p~\boldsymbol{F}^{-T}+\frac{\partial W}{\partial \boldsymbol{F}}
| |
| = -p~\boldsymbol{F}^{-T} + \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}} | |
| = -p~\boldsymbol{F}^{-T} + 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}} ~.
| |
| </math>
| |
| This stress tensor can subsequently be [[stress (physics)|converted]] into any of the other conventional stress tensors, such as the [[Cauchy stress tensor|Cauchy Stress tensor]] which is given by
| |
| :<math>
| |
| \boldsymbol{\sigma}=\boldsymbol{P}\cdot\boldsymbol{F}^T=
| |
| -p~\boldsymbol{\mathit{1}} + \frac{\partial W}{\partial \boldsymbol{F}}\cdot\boldsymbol{F}^T
| |
| = -p~\boldsymbol{\mathit{1}} + \boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{E}}\cdot\boldsymbol{F}^T
| |
| = -p~\boldsymbol{\mathit{1}} + 2~\boldsymbol{F}\cdot\frac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^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 [[Finite strain theory#The_Left_Cauchy–Green_deformation_tensor|left Cauchy–Green deformation tensor]] (or [[Finite strain theory#The_Right_Cauchy-Green_deformation_tensor|right Cauchy–Green deformation tensor]]). If the [[strain energy density function]] is <math>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>
| |
| \begin{align}
| |
| \boldsymbol{\sigma} & =
| |
| \cfrac{2}{\sqrt{I_3}}\left[\left(\cfrac{\partial\hat{W}}{\partial I_1} + I_1~\cfrac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \cfrac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] + 2\sqrt{I_3}~\cfrac{\partial\hat{W}}{\partial I_3}~\boldsymbol{\mathit{1}} \\
| |
| & = \cfrac{2}{J}\left[\cfrac{1}{J^{2/3}}\left(\cfrac{\partial\bar{W}}{\partial \bar{I}_1} + \bar{I}_1~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}\right)\boldsymbol{B} -
| |
| \cfrac{1}{J^{4/3}}~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] \\
| |
| & \qquad \qquad + \left[\cfrac{\partial\bar{W}}{\partial J} - \cfrac{2}{3J}\left(\bar{I}_1~\cfrac{\partial\bar{W}}{\partial \bar{I}_1} + 2~\bar{I}_2~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}\right)\right] ~\boldsymbol{\mathit{1}} \\
| |
| & = \cfrac{2}{J}\left[\left(\cfrac{\partial\bar{W}}{\partial \bar{I}_1} + \bar{I}_1~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}\right)\bar{\boldsymbol{B}} -
| |
| \cfrac{\partial\bar{W}}{\partial \bar{I}_2}~\bar{\boldsymbol{B}} \cdot\bar{\boldsymbol{B}} \right] + \left[\cfrac{\partial\bar{W}}{\partial J} - \cfrac{2}{3J}\left(\bar{I}_1~\cfrac{\partial\bar{W}}{\partial \bar{I}_1} + 2~\bar{I}_2~\cfrac{\partial\bar{W}}{\partial \bar{I}_2}\right)\right] ~\boldsymbol{\mathit{1}} \\
| |
| & = \cfrac{\lambda_1}{\lambda_1\lambda_2\lambda_3}~\cfrac{\partial\tilde{W}}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 + \cfrac{\lambda_2}{\lambda_1\lambda_2\lambda_3}~\cfrac{\partial\tilde{W}}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \cfrac{\lambda_3}{\lambda_1\lambda_2\lambda_3}~\cfrac{\partial\tilde{W}}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3
| |
| \end{align}
| |
| </math>
| |
| (See the page on [[Finite strain theory#The_Left_Cauchy–Green_deformation_tensor|the left Cauchy–Green deformation tensor]] for the definitions of these symbols).
| |
| | |
| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
| |
| !Proof 1:
| |
| |-
| |
| | The [[stress (physics)|second Piola–Kirchhoff stress tensor]] for a hyperelastic material is given by
| |
| :<math>
| |
| \boldsymbol{S} = 2~\cfrac{\partial W}{\partial \boldsymbol{C}}
| |
| </math>
| |
| where <math>\boldsymbol{C} = \boldsymbol{F}^T\cdot\boldsymbol{F}</math> is the [[finite strain theory|right Cauchy–Green deformation tensor]] and <math>\boldsymbol{F}</math> is the [[finite strain theory|deformation gradient]]. The [[stress (physics)|Cauchy stress]] is given by
| |
| :<math>
| |
| \boldsymbol{\sigma} = \cfrac{1}{J}~\boldsymbol{F}\cdot\boldsymbol{S}\cdot\boldsymbol{F}^T
| |
| = \cfrac{2}{J}~\boldsymbol{F}\cdot\cfrac{\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>
| |
| \cfrac{\partial W}{\partial \boldsymbol{C}} =
| |
| \cfrac{\partial W}{\partial I_1}~\cfrac{\partial I_1}{\partial \boldsymbol{C}} +
| |
| \cfrac{\partial W}{\partial I_2}~\cfrac{\partial I_2}{\partial \boldsymbol{C}} +
| |
| \cfrac{\partial W}{\partial I_3}~\cfrac{\partial I_3}{\partial \boldsymbol{C}} ~.
| |
| </math>
| |
| The [[tensor derivative (continuum mechanics)|derivatives of the invariants]] of the symmetric tensor <math>\boldsymbol{C}</math> are
| |
| :<math>
| |
| \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>
| |
| \cfrac{\partial W}{\partial \boldsymbol{C}} =
| |
| \cfrac{\partial W}{\partial I_1}~\boldsymbol{\mathit{1}} +
| |
| \cfrac{\partial W}{\partial I_2}~(I_1~\boldsymbol{\mathit{1}} - \boldsymbol{F}^T\cdot\boldsymbol{F}) +
| |
| \cfrac{\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>
| |
| \boldsymbol{\sigma}
| |
| = \cfrac{2}{J}~\left[\cfrac{\partial W}{\partial I_1}~\boldsymbol{F}\cdot\boldsymbol{F}^T+
| |
| \cfrac{\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) +
| |
| \cfrac{\partial W}{\partial I_3}~I_3~\boldsymbol{\mathit{1}}\right]
| |
| </math>
| |
| Using the [[Finite strain theory#The_Left_Cauchy–Green_deformation_tensor|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>
| |
| \boldsymbol{\sigma}
| |
| = \cfrac{2}{\sqrt{I_3}}~\left[\left(\cfrac{\partial W}{\partial I_1} +
| |
| I_1~\cfrac{\partial W}{\partial I_2}\right)~\boldsymbol{B} -
| |
| \cfrac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] +
| |
| 2~\sqrt{I_3}~\cfrac{\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>
| |
| \cfrac{\partial W}{\partial \boldsymbol{C}} =
| |
| \cfrac{\partial W}{\partial I_1}~\cfrac{\partial I_1}{\partial \boldsymbol{C}} +
| |
| \cfrac{\partial W}{\partial I_2}~\cfrac{\partial I_2}{\partial \boldsymbol{C}}
| |
| = \cfrac{\partial W}{\partial I_1}~\boldsymbol{\mathit{1}} +
| |
| \cfrac{\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>
| |
| \boldsymbol{\sigma}
| |
| = 2\left[\left(\cfrac{\partial W}{\partial I_1} +
| |
| I_1~\cfrac{\partial W}{\partial I_2}\right)~\boldsymbol{B} -
| |
| \cfrac{\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>
| |
| \cfrac{\partial W}{\partial \boldsymbol{C}} =
| |
| \cfrac{\partial W}{\partial I_1}~\cfrac{\partial I_1}{\partial \boldsymbol{C}} = \cfrac{\partial W}{\partial I_1}~\boldsymbol{\mathit{1}}
| |
| </math>
| |
| In that case the Cauchy stress can be expressed as
| |
| :<math>
| |
| \boldsymbol{\sigma} = 2\cfrac{\partial W}{\partial I_1}~\boldsymbol{B} - p~\boldsymbol{\mathit{1}}~.
| |
| </math>
| |
| |}
| |
| | |
| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
| |
| !Proof 2:
| |
| |-
| |
| |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 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>
| |
| \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>
| |
| \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>
| |
| \begin{align}
| |
| \cfrac{\partial W}{\partial I_1} & =
| |
| \cfrac{\partial W}{\partial \bar{I}_1}~\cfrac{\partial \bar{I}_1}{\partial I_1} +
| |
| \cfrac{\partial W}{\partial \bar{I}_2}~\cfrac{\partial \bar{I}_2}{\partial I_1} +
| |
| \cfrac{\partial W}{\partial J}~\cfrac{\partial J}{\partial I_1} \\
| |
| & = I_3^{-1/3}~\cfrac{\partial W}{\partial \bar{I}_1}
| |
| = J^{-2/3}~\cfrac{\partial W}{\partial \bar{I}_1} \\
| |
| \cfrac{\partial W}{\partial I_2} & =
| |
| \cfrac{\partial W}{\partial \bar{I}_1}~\cfrac{\partial \bar{I}_1}{\partial I_2} +
| |
| \cfrac{\partial W}{\partial \bar{I}_2}~\cfrac{\partial \bar{I}_2}{\partial I_2} +
| |
| \cfrac{\partial W}{\partial J}~\cfrac{\partial J}{\partial I_2} \\
| |
| & = I_3^{-2/3}~\cfrac{\partial W}{\partial \bar{I}_2}
| |
| = J^{-4/3}~\cfrac{\partial W}{\partial \bar{I}_2} \\
| |
| \cfrac{\partial W}{\partial I_3} & =
| |
| \cfrac{\partial W}{\partial \bar{I}_1}~\cfrac{\partial \bar{I}_1}{\partial I_3} +
| |
| \cfrac{\partial W}{\partial \bar{I}_2}~\cfrac{\partial \bar{I}_2}{\partial I_3} +
| |
| \cfrac{\partial W}{\partial J}~\cfrac{\partial J}{\partial I_3} \\
| |
| & = - \cfrac{1}{3}~I_3^{-4/3}~I_1~\cfrac{\partial W}{\partial \bar{I}_1}
| |
| - \cfrac{2}{3}~I_3^{-5/3}~I_2~\cfrac{\partial W}{\partial \bar{I}_2}
| |
| + \cfrac{1}{2}~I_3^{-1/2}~\cfrac{\partial W}{\partial J} \\
| |
| & = - \cfrac{1}{3}~J^{-8/3}~J^{2/3}~\bar{I}_1~\cfrac{\partial W}{\partial \bar{I}_1}
| |
| - \cfrac{2}{3}~J^{-10/3}~J^{4/3}~\bar{I}_2~\cfrac{\partial W}{\partial \bar{I}_2}
| |
| + \cfrac{1}{2}~J^{-1}~\cfrac{\partial W}{\partial J} \\
| |
| & = -\cfrac{1}{3}~J^{-2}~\left(\bar{I}_1~\cfrac{\partial W}{\partial \bar{I}_1}+
| |
| 2~\bar{I}_2~\cfrac{\partial W}{\partial \bar{I}_2}\right) +
| |
| \cfrac{1}{2}~J^{-1}~\cfrac{\partial W}{\partial J}
| |
| \end{align}
| |
| </math>
| |
| Recall that the Cauchy stress is given by
| |
| :<math>
| |
| \boldsymbol{\sigma}
| |
| = \cfrac{2}{\sqrt{I_3}}~\left[\left(\cfrac{\partial W}{\partial I_1} +
| |
| I_1~\cfrac{\partial W}{\partial I_2}\right)~\boldsymbol{B} -
| |
| \cfrac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] +
| |
| 2~\sqrt{I_3}~\cfrac{\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>
| |
| \boldsymbol{\sigma}
| |
| = \cfrac{2}{J}~\left[\left(\cfrac{\partial W}{\partial I_1}+
| |
| J^{2/3}~\bar{I}_1~\cfrac{\partial W}{\partial I_2}\right)~\boldsymbol{B} -
| |
| \cfrac{\partial W}{\partial I_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] +
| |
| 2~J~\cfrac{\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>
| |
| \begin{align}
| |
| \boldsymbol{\sigma}
| |
| & = \cfrac{2}{J}~\left[\left(J^{-2/3}~\cfrac{\partial W}{\partial \bar{I}_1} +
| |
| J^{-2/3}~\bar{I}_1~\cfrac{\partial W}{\partial \bar{I}_2}\right)~\boldsymbol{B} -
| |
| J^{-4/3}~\cfrac{\partial W}{\partial \bar{I}_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right]
| |
| + \\
| |
| & \qquad
| |
| 2~J~\left[-\cfrac{1}{3}~J^{-2}~\left(\bar{I}_1~\cfrac{\partial W}{\partial \bar{I}_1}+
| |
| 2~\bar{I}_2~\cfrac{\partial W}{\partial \bar{I}_2}\right) +
| |
| \cfrac{1}{2}~J^{-1}~\cfrac{\partial W}{\partial J}\right]~\boldsymbol{\mathit{1}}
| |
| \end{align}
| |
| </math>
| |
| or,
| |
| :<math>
| |
| \begin{align}
| |
| \boldsymbol{\sigma}
| |
| & = \cfrac{2}{J}~\left[\cfrac{1}{J^{2/3}}~\left(\cfrac{\partial W}{\partial \bar{I}_1} +
| |
| \bar{I}_1~\cfrac{\partial W}{\partial \bar{I}_2}\right)~\boldsymbol{B} -
| |
| \cfrac{1}{J^{4/3}}~
| |
| \cfrac{\partial W}{\partial \bar{I}_2}~\boldsymbol{B}\cdot\boldsymbol{B}\right] \\
| |
| & \qquad + \left[\cfrac{\partial W}{\partial J} -
| |
| \cfrac{2}{3J}\left(\bar{I}_1~\cfrac{\partial W}{\partial \bar{I}_1}+
| |
| 2~\bar{I}_2~\cfrac{\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>
| |
| \begin{align}
| |
| \boldsymbol{\sigma}
| |
| & = \cfrac{2}{J}~\left[\left(\cfrac{\partial W}{\partial \bar{I}_1} +
| |
| \bar{I}_1~\cfrac{\partial W}{\partial \bar{I}_2}\right)~\bar{\boldsymbol{B}} -
| |
| \cfrac{\partial W}{\partial \bar{I}_2}~\bar{\boldsymbol{B}}\cdot\bar{\boldsymbol{B}}\right] \\
| |
| & \qquad + \left[\cfrac{\partial W}{\partial J} -
| |
| \cfrac{2}{3J}\left(\bar{I}_1~\cfrac{\partial W}{\partial \bar{I}_1}+
| |
| 2~\bar{I}_2~\cfrac{\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>
| |
| \boldsymbol{\sigma}
| |
| = 2\left[\left(\cfrac{\partial W}{\partial \bar{I}_1} +
| |
| I_1~\cfrac{\partial W}{\partial \bar{I}_2}\right)~\bar{\boldsymbol{B}} -
| |
| \cfrac{\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>
| |
| \boldsymbol{\sigma} = 2\cfrac{\partial W}{\partial \bar{I}_1}~\bar{\boldsymbol{B}} - p~\boldsymbol{\mathit{1}}~.
| |
| </math> | |
| |}
| |
| | |
| :{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
| |
| !Proof 3:
| |
| |-
| |
| | To express the Cauchy stress in terms of the [[finite strain theory|stretches]] <math>\lambda_1, \lambda_2, \lambda_3</math> recall that
| |
| :<math>
| |
| \cfrac{\partial \lambda_i}{\partial\boldsymbol{C}} = \cfrac{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>
| |
| \begin{align}
| |
| \cfrac{\partial W}{\partial\boldsymbol{C}} & =
| |
| \cfrac{\partial W}{\partial \lambda_1}~\cfrac{\partial \lambda_1}{\partial\boldsymbol{C}} +
| |
| \cfrac{\partial W}{\partial \lambda_2}~\cfrac{\partial \lambda_2}{\partial\boldsymbol{C}} +
| |
| \cfrac{\partial W}{\partial \lambda_3}~\cfrac{\partial \lambda_3}{\partial\boldsymbol{C}} \\
| |
| & = \boldsymbol{R}^T\cdot\left[\cfrac{1}{2\lambda_1}~\cfrac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +
| |
| \cfrac{1}{2\lambda_2}~\cfrac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 +
| |
| \cfrac{1}{2\lambda_3}~\cfrac{\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>
| |
| \boldsymbol{\sigma} = \cfrac{2}{J}~\boldsymbol{F}\cdot
| |
| \cfrac{\partial W}{\partial \boldsymbol{C}}\cdot\boldsymbol{F}^T =
| |
| \cfrac{2}{J}~(\boldsymbol{V}\cdot\boldsymbol{R})\cdot
| |
| \cfrac{\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> | |
| \boldsymbol{\sigma} =
| |
| \cfrac{2}{J}~\boldsymbol{V}\cdot
| |
| \left[\cfrac{1}{2\lambda_1}~
| |
| \cfrac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +
| |
| \cfrac{1}{2\lambda_2}~
| |
| \cfrac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 +
| |
| \cfrac{1}{2\lambda_3}~
| |
| \cfrac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3\right]
| |
| \cdot\boldsymbol{V}
| |
| </math>
| |
| Using the [[finite strain theory|spectral decomposition]] of <math>\boldsymbol{V}</math> we have
| |
| :<math>
| |
| \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>
| |
| 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>
| |
| \boldsymbol{\sigma} =
| |
| \cfrac{1}{\lambda_1\lambda_2\lambda_3}~
| |
| \left[\lambda_1~\cfrac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +
| |
| \lambda_2~\cfrac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 +
| |
| \lambda_3~\cfrac{\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>
| |
| \boldsymbol{\sigma} =
| |
| \lambda_1~\cfrac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +
| |
| \lambda_2~\cfrac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 +
| |
| \lambda_3~\cfrac{\partial W}{\partial \lambda_3}~\mathbf{n}_3\otimes\mathbf{n}_3
| |
| - p~\boldsymbol{\mathit{1}}~
| |
| </math>
| |
| Some care is to required at this stage because, when an eigenvalue is repeated, it is in general only [[Gâteaux derivative|Gâteaux 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 (continuum mechanics)|tensor derivative]] can only be found by solving another eigenvalue problem.
| |
| | |
| If we express the stress in terms of differences between components,
| |
| :<math>
| |
| \sigma_{11} - \sigma_{33} = \lambda_1~\cfrac{\partial W}{\partial \lambda_1} - \lambda_3~\cfrac{\partial W}{\partial \lambda_3} ~;~~
| |
| \sigma_{22} - \sigma_{33} = \lambda_2~\cfrac{\partial W}{\partial \lambda_2} - \lambda_3~\cfrac{\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>
| |
| \sigma_{11} - \sigma_{33} = \sigma_{22} - \sigma_{33} = \lambda_1~\cfrac{\partial W}{\partial \lambda_1} - \lambda_3~\cfrac{\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>
| |
| \begin{align}
| |
| \boldsymbol{\sigma} & = -p~\boldsymbol{\mathit{1}} +
| |
| 2\left[\left(\cfrac{\partial\hat{W}}{\partial I_1} + I_1~\cfrac{\partial\hat{W}}{\partial I_2}\right)\boldsymbol{B} - \cfrac{\partial\hat{W}}{\partial I_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] \\
| |
| & = - p~\boldsymbol{\mathit{1}} + 2\left[\left(\cfrac{\partial W}{\partial \bar{I}_1} +
| |
| I_1~\cfrac{\partial W}{\partial \bar{I}_2}\right)~\bar{\boldsymbol{B}} -
| |
| \cfrac{\partial W}{\partial \bar{I}_2}~\bar{\boldsymbol{B}}\cdot\bar{\boldsymbol{B}}\right] \\
| |
| & = - p~\boldsymbol{\mathit{1}} + \lambda_1~\cfrac{\partial W}{\partial \lambda_1}~\mathbf{n}_1\otimes\mathbf{n}_1 +
| |
| \lambda_2~\cfrac{\partial W}{\partial \lambda_2}~\mathbf{n}_2\otimes\mathbf{n}_2 + \lambda_3~\cfrac{\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>
| |
| \sigma_{11} - \sigma_{33} = \lambda_1~\cfrac{\partial W}{\partial \lambda_1} - \lambda_3~\cfrac{\partial W}{\partial \lambda_3}~;~~
| |
| \sigma_{22} - \sigma_{33} = \lambda_2~\cfrac{\partial W}{\partial \lambda_2} - \lambda_3~\cfrac{\partial W}{\partial \lambda_3}
| |
| </math>
| |
| If in addition <math>I_1 = I_2</math>, then
| |
| :<math>
| |
| \boldsymbol{\sigma} = 2\cfrac{\partial W}{\partial I_1}~\boldsymbol{B} - p~\boldsymbol{\mathit{1}}~.
| |
| </math>
| |
| If <math>\lambda_1 = \lambda_2</math>, then
| |
| :<math>
| |
| \sigma_{11} - \sigma_{33} = \sigma_{22} - \sigma_{33} = \lambda_1~\cfrac{\partial W}{\partial \lambda_1} - \lambda_3~\cfrac{\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 theory|infinitesimal strain]] limit:
| |
| :<math>
| |
| \boldsymbol{\sigma} = \lambda~\mathrm{tr}(\boldsymbol{\varepsilon})~\boldsymbol{\mathit{1}} + 2\mu\boldsymbol{\varepsilon}
| |
| </math>
| |
| where <math>\lambda, \mu</math> are the [[Lame constants]]. The strain energy density function that corresponds to the above relation is<ref name=Ogden/>
| |
| :<math>
| |
| W = \tfrac{1}{2}\lambda~[\mathrm{tr}(\boldsymbol{\varepsilon})]^2 + \mu~\mathrm{tr}(\boldsymbol{\varepsilon}^2)
| |
| </math>
| |
| For an incompressible material <math>\mathrm{tr}(\boldsymbol{\varepsilon}) = 0</math> and we have
| |
| :<math> | |
| W = \mu~\mathrm{tr}(\boldsymbol{\varepsilon}^2)
| |
| </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>
| |
| \begin{align}
| |
| & W(1,1,1) = 0 ~;~~
| |
| \cfrac{\partial W}{\partial \lambda_i}(1,1,1) = 0 \\
| |
| & \cfrac{\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>
| |
| \begin{align}
| |
| & W(1,1,1) = 0 \\
| |
| & \cfrac{\partial W}{\partial \lambda_i}(1,1,1) = \cfrac{\partial W}{\partial \lambda_j}(1,1,1) ~;~~
| |
| \cfrac{\partial^2 W}{\partial \lambda_i^2}(1,1,1) = \cfrac{\partial^2 W}{\partial \lambda_j^2}(1,1,1) \\
| |
| & \cfrac{\partial^2 W}{\partial \lambda_i \partial \lambda_j}(1,1,1) = \mathrm{independent of}~i,j\ne i \\
| |
| & \cfrac{\partial^2 W}{\partial \lambda_i^2}(1,1,1) - \cfrac{\partial^2 W}{\partial \lambda_i \partial \lambda_j}(1,1,1) + \cfrac{\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_1</math> 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> | |
| W(I_1)\biggr|_{I_1=3} = 0 \quad \text{and} \quad \cfrac{\partial W}{\partial I_1}\biggr|_{I_1=3} = \frac{\mu}{2} \,.
| |
| </math>
| |
| The second consistency condition above can be derived by noting that
| |
| :<math>
| |
| \cfrac{\partial W}{\partial \lambda_i} = \cfrac{\partial W}{\partial I_1}\cfrac{\partial I_1}{\partial \lambda_i} = 2\lambda_i\cfrac{\partial W}{\partial I_1} \quad\text{and}\quad
| |
| \cfrac{\partial^2 W}{\partial \lambda_i \partial \lambda_j} = 2\delta_{ij}\cfrac{\partial W}{\partial I_1} + 4\lambda_i\lambda_j \cfrac{\partial^2 W}{\partial I_1^2}\,.
| |
| </math>
| |
| The can then be substituted into the consistency condition for isotropic incompressible hyperelastic materials.
| |
| | |
| == References ==
| |
| | |
| <references/>
| |
| | |
| == See also ==
| |
| *[[Cauchy elastic material]]
| |
| *[[Continuum mechanics]]
| |
| *[[Deformation (mechanics)]]
| |
| *[[Finite strain theory]]
| |
| *[[Rubber elasticity]]
| |
| *[[Stress measures]]
| |
| *[[Stress (mechanics)]]
| |
| | |
| {{DEFAULTSORT:Hyperelastic Material}}
| |
| [[Category:Continuum mechanics]]
| |
| [[Category:Elasticity (physics)]]
| |
| [[Category:Rubber properties]]
| |
| [[Category:Solid mechanics]]
| |