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| {{continuum mechanics|cTopic=[[Solid mechanics]]}}
| | The author is recognized by the title of Figures Wunder. My working day occupation is a meter reader. Puerto Rico is exactly where he's been residing for years and he will never transfer. One of the issues she enjoys most is to do aerobics and now she is trying to make cash with it.<br><br>Check out my site; [http://facehack.ir/index.php?do=/blog/20/what-you-have-to-do-facing-candida/ facehack.ir] |
| In [[continuum mechanics]], a '''Mooney–Rivlin solid'''<ref name=Mooney>Mooney, M., 1940, ''A theory of large elastic deformation'', Journal of Applied Physics, 11(9), pp. 582-592.</ref><ref name=Rivlin>Rivlin, R. S., 1948, ''Large elastic deformations of isotropic materials. IV. Further developments of the general theory'', Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 241(835), pp. 379-397.</ref> is a [[hyperelastic material]] model where the [[strain energy density function]] <math>W\,</math> is a linear combination of two [[Invariants of tensors|invariants]] of the [[finite strain theory|left Cauchy–Green deformation tensor]] <math>\boldsymbol{B}</math>. The model was proposed by [[Melvin Mooney]] in 1940 and expressed in terms of invariants by [[Ronald Rivlin]] in 1948.
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| The strain energy density function for an [[incompressible]] Mooney–Rivlin material is<ref>Boulanger, P. and Hayes, M. A., 2001, '' Finite amplitude waves in Mooney–Rivlin and Hadamard materials'', in '''Topics in Finite Elasticity''', ed. M. A Hayes and G. Soccomandi, International Center for Mechanical Sciences.</ref><ref>C. W. Macosko, 1994, '''Rheology: principles, measurement and applications''', VCH Publishers, ISBN 1-56081-579-5.</ref>
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| :<math>W = C_{1} (\overline{I}_1-3) + C_{2} (\overline{I}_2-3), \, </math>
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| where <math>C_{1}</math> and <math>C_{2}</math> are empirically determined material constants, and <math>I_1</math> and <math>I_2</math> are the first and the second [[Invariants of tensors|invariant]] of the [[Unimodular matrix|unimodular]] component of the [[finite strain theory|left Cauchy–Green deformation tensor]]:<ref>The characteristic polynomial of the linear operator corresponding to the second rank three-dimensional Finger tensor (also called the left Cauchy–Green deformation tensor) is usually written
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| :<math>p_B (\lambda) = \lambda^3 - a_1 \, \lambda^2 + a_2 \, \lambda - a_3 \, </math>
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| In this article, the [[trace (linear algebra)|trace]] <math>a_1</math> is written <math>I_1</math>, the next coefficient <math>a_2</math> is written <math>I_2</math>, and the determinant <math>a_3</math> would be written <math>I_3</math>.</ref>
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| :<math>
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| \begin{align}
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| \bar{I}_1 & = J^{-2/3}~I_1 ~;~~ I_1 = \lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2 ~;~~ J = \det(\boldsymbol{F}) \\
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| \bar{I}_2 & = J^{-4/3}~I_2 ~;~~ I_2 = \lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_3^2 \lambda_1^2
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| \end{align}
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| </math>
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| where <math>\boldsymbol{F}</math> is the [[deformation gradient]]. For an [[incompressible]] material, <math>J=1</math>.
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| ==Derivation==
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| The Mooney–Rivlin model is a special case of the '''generalized Rivlin model''' (also called [[Polynomial (hyperelastic model)|polynomial hyperelastic model]]<ref name=Bower>{{cite book
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| |title=Applied Mechanics of Solids
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| |last=Bower |first=Allan
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| |year=2009
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| |publisher=CRC Press
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| |isbn=1-4398-0247-5
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| |url=http://solidmechanics.org/
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| |accessdate=January 2010}}</ref>) which has the form
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| :<math>
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| W = \sum_{p,q = 0}^N C_{pq} (\bar{I}_1 - 3)^p~(\bar{I}_2 - 3)^q +
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| \sum_{m = 1}^M D_m~(J-1)^{2m}
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| </math>
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| with <math>C_{00} = 0</math> where <math>C_{pq}</math> are material constants related to the distortional response and <math>D_m</math> are material constants related to the volumetric response. For a [[compressible]] Mooney–Rivlin material <math>N = 1, C_{01} = C_2, C_{11} = 0, C_{10} = C_1, M=1</math> and we have
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| :<math>
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| W = C_{01}~(\bar{I}_2 - 3) + C_{10}~(\bar{I}_1 - 3) + D_1~(J-1)^2
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| </math>
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| If <math>C_{01} = 0</math> we obtain a [[neo-Hookean solid]], a special case of a '''Mooney–Rivlin solid'''.
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| For consistency with [[linear elasticity]] in the limit of [[infinitesimal strain theory|small strains]], it is necessary that
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| :<math>
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| \kappa = 2 \cdot D_1 ~;~~ \mu = 2~(C_{01} + C_{10})
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| </math>
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| where <math>\kappa</math> is the [[bulk modulus]] and <math>\mu</math> is the [[shear modulus]].
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| ==Cauchy stress in terms of strain invariants and deformation tensors==
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| The [[stress (physics)|Cauchy stress]] in a [[compressible]] hyperelastic material with a stress free reference configuration is given by
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| :<math>
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| \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} -
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| \cfrac{1}{J^{4/3}}~\cfrac{\partial{W}}{\partial \bar{I}_2}~\boldsymbol{B} \cdot\boldsymbol{B} \right] + \left[\cfrac{\partial{W}}{\partial J} -
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| \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}}
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| </math>
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| For a compressible Mooney–Rivlin material,
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| :<math>
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| \cfrac{\partial{W}}{\partial \bar{I}_1} = C_1 ~;~~ \cfrac{\partial{W}}{\partial \bar{I}_2} = C_2 ~;~~ \cfrac{\partial{W}}{\partial J} = 2D_1(J-1)
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| </math>
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| Therefore, the Cauchy stress in a compressible Mooney–Rivlin material is given by
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| :<math>
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| \boldsymbol{\sigma} = \cfrac{2}{J}\left[\cfrac{1}{J^{2/3}}\left(C_1 + \bar{I}_1~C_2\right)\boldsymbol{B} -
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| \cfrac{1}{J^{4/3}}~C_2~\boldsymbol{B} \cdot\boldsymbol{B} \right] + \left[2D_1(J-1)-
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| \cfrac{2}{3J}\left(C_1\bar{I}_1 + 2C_2\bar{I}_2~\right)\right]\boldsymbol{\mathit{1}}
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| </math>
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| It can be shown, after some algebra, that the [[pressure]] is given by
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| :<math> | |
| p := -\tfrac{1}{3}\,\text{tr}(\boldsymbol{\sigma}) = -\frac{\partial W}{\partial J} = -2 D_1 (J-1) \,.
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| </math>
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| The stress can then be expressed in the form
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| :<math>
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| \boldsymbol{\sigma} = \cfrac{1}{J}\left[-p~\boldsymbol{\mathit{1}} + \cfrac{2}{J^{2/3}}\left(C_1 + \bar{I}_1~C_2\right)\boldsymbol{B} -
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| \cfrac{2}{J^{4/3}}~C_2~\boldsymbol{B}\cdot\boldsymbol{B} -\cfrac{2}{3}\left(C_1\,\bar{I}_1 + 2C_2\,\bar{I}_2\right)\boldsymbol{\mathit{1}}\right] \,.
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| </math>
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| The above equation is often written as
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| :<math>
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| \boldsymbol{\sigma} = \cfrac{1}{J}\left[-p~\boldsymbol{\mathit{1}} + 2\left(C_1 + \bar{I}_1~C_2\right)\bar{\boldsymbol{B}} -
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| 2~C_2~\bar{\boldsymbol{B}}\cdot\bar{\boldsymbol{B}} -\cfrac{2}{3}\left(C_1\,\bar{I}_1 + 2C_2\,\bar{I}_2\right)\boldsymbol{\mathit{1}}\right] \quad \text{where} \quad
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| \bar{\boldsymbol{B}} = J^{-2/3}\,\boldsymbol{B} \,.
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| </math>
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| For an '''incompressible''' Mooney–Rivlin material with <math> J = 1</math>
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| :<math>
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| \boldsymbol{\sigma} = 2\left(C_1 + I_1~C_2\right)\boldsymbol{B} -
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| 2C_2~\boldsymbol{B}\cdot\boldsymbol{B} -\cfrac{2}{3}\left(C_1\,\bar{I}_1 + 2C_2\,\bar{I}_2\right)\boldsymbol{\mathit{1}}\,.
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| </math>
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| Note that if <math>J=1</math> then
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| :<math>
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| \det(\boldsymbol{B}) = \det(\boldsymbol{F})\det(\boldsymbol{F}^T) = 1
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| </math>
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| Then, from the [[Cayley-Hamilton theorem]],
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| :<math>
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| \boldsymbol{B}^{-1} = \boldsymbol{B}\cdot\boldsymbol{B} - I_1~\boldsymbol{B} + I_2~\boldsymbol{\mathit{1}}
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| </math>
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| Hence, the Cauchy stress can be expressed as
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| :<math>
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| \boldsymbol{\sigma} = -p^{*}~\boldsymbol{\mathit{1}} + 2 C_1~\boldsymbol{B} - 2C_2~\boldsymbol{B}^{-1}
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| </math>
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| where <math>p^{*} := \tfrac{2}{3}(C_1~I_1 - C_2~I_2). \, </math>
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| ==Cauchy stress in terms of principal stretches==
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| In terms of the [[finite strain theory|principal stretches]], the Cauchy stress differences for an '''incompressible''' hyperelastic material are given by
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| :<math>
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| \sigma_{11} - \sigma_{33} = \lambda_1~\cfrac{\partial{W}}{\partial \lambda_1} - \lambda_3~\cfrac{\partial{W}}{\partial \lambda_3} ~;~~
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| \sigma_{22} - \sigma_{33} = \lambda_2~\cfrac{\partial{W}}{\partial \lambda_2} - \lambda_3~\cfrac{\partial{W}}{\partial \lambda_3}
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| </math>
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| For an '''incompressible''' Mooney-Rivlin material,
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| :<math>
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| W = C_1(\lambda_1^2 + \lambda_2 ^2+ \lambda_3 ^2 -3) + C_2(\lambda_1^2 \lambda_2^2 + \lambda_2^2 \lambda_3^2 + \lambda_3^2 \lambda_1^2 -3) ~;~~ \lambda_1\lambda_2\lambda_3 = 1
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| </math>
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| Therefore,
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| :<math>
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| \lambda_1\cfrac{\partial{W}}{\partial \lambda_1} = 2C_1\lambda_1^2 + 2C_2\lambda_1^2(\lambda_2^2+\lambda_3^2) ~;~~
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| \lambda_2\cfrac{\partial{W}}{\partial \lambda_2} = 2C_1\lambda_2^2 + 2C_2\lambda_2^2(\lambda_1^2+\lambda_3^2) ~;~~
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| \lambda_3\cfrac{\partial{W}}{\partial \lambda_3} = 2C_1\lambda_3^2 + 2C_2\lambda_3^2(\lambda_1^2+\lambda_2^2)
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| </math>
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| Since <math>\lambda_1\lambda_2\lambda_3=1</math>. we can write
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| :<math>
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| \begin{align}
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| \lambda_1\cfrac{\partial{W}}{\partial \lambda_1} & = 2C_1\lambda_1^2 + 2C_2\left(\cfrac{1}{\lambda_3^2}+\cfrac{1}{\lambda_2^2}\right) ~;~~
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| \lambda_2\cfrac{\partial{W}}{\partial \lambda_2} = 2C_1\lambda_2^2 + 2C_2\left(\cfrac{1}{\lambda_3^2}+\cfrac{1}{\lambda_1^2}\right) \\
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| \lambda_3\cfrac{\partial{W}}{\partial \lambda_3} & = 2C_1\lambda_3^2 + 2C_2\left(\cfrac{1}{\lambda_2^2}+\cfrac{1}{\lambda_1^2}\right)
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| \end{align}
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| </math>
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| Then the expressions for the Cauchy stress differences become
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| :<math>
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| \sigma_{11}-\sigma_{33} = 2C_1(\lambda_1^2-\lambda_3^2) - 2C_2\left(\cfrac{1}{\lambda_1^2}-\cfrac{1}{\lambda_3^2}\right)~;~~
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| \sigma_{22}-\sigma_{33} = 2C_1(\lambda_2^2-\lambda_3^2) - 2C_2\left(\cfrac{1}{\lambda_2^2}-\cfrac{1}{\lambda_3^2}\right)
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| </math>
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| ==Uniaxial extension==
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| For the case of an incompressible Mooney–Rivlin material under uniaxial elongation, <math>\lambda_1 = \lambda\,</math> and <math>\lambda_2 = \lambda_3 = 1/\sqrt{\lambda}</math>. Then the [[stress (physics)|true stress]] (Cauchy stress) differences can be calculated as:
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| :<math>
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| \begin{align}
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| \sigma_{11}-\sigma_{33} & = 2C_1\left(\lambda^2-\cfrac{1}{\lambda}\right) -2C_2\left(\cfrac{1}{\lambda^2} - \lambda\right)\\
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| \sigma_{22}-\sigma_{33} & = 0
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| \end{align}
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| </math>
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| ===Simple tension===
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| [[File:Mooney-Rivlin.svg|thumb|380px|right|Comparison of experimental results (dots) and predictions for [[Hooke's law]](1, blue line), [[neo-Hookean solid]](2, red line) and Mooney–Rivlin solid models(3, green line)]]
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| In the case of simple tension, <math>\sigma_{22} = \sigma_{33} = 0 </math>. Then we can write
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| :<math>
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| \sigma_{11} = \left(2C_1 + \cfrac {2C_2} {\lambda} \right) \left( \lambda^2 - \cfrac{1}{\lambda} \right)
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| </math>
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| In alternative notation, where the Cauchy stress is written as <math>\boldsymbol{T}</math> and the stretch as <math>\alpha</math>, we can write
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| :<math>T_{11} = \left(2C_1 + \frac {2C_2} {\alpha} \right) \left( \alpha^2 - \alpha^{-1} \right)</math>
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| and the [[engineering stress]] (force per unit reference area) for an incompressible Mooney–Rivlin material under simple tension can be calculated using
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| <math>T_{11}^{\mathrm{eng}} = T_{11}\alpha_2\alpha_3 = \cfrac{T_{11}}{\alpha} </math>. Hence
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| :<math>
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| T_{11}^{\mathrm{eng}}= \left(2C_1 + \frac {2C_2} {\alpha} \right) \left( \alpha - \alpha^{-2} \right)
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| </math>
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| If we define
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| :<math>
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| T^{*}_{11} := \cfrac{T_{11}^{\mathrm{eng}}}{\alpha - \alpha^{-2}} ~;~~ \beta := \cfrac{1}{\alpha}
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| </math>
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| then
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| :<math>
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| T^{*}_{11} = 2C_1 + 2C_2\beta ~.
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| </math>
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| The slope of the <math> T^{*}_{11}</math> versus <math>\beta</math> line gives the value of <math>C_2</math> while the intercept with the <math>T^{*}_{11}</math> axis gives the value of <math>C_1</math>. The Mooney–Rivlin solid model usually fits experimental data better than [[Neo-Hookean solid]] does, but requires an additional empirical constant.
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| ==Equibiaxial tension==
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| In the case of equibiaxial tension, the principal stretches are <math>\lambda_1 = \lambda_2 = \lambda</math>. If, in addition, the material is incompressible then <math>\lambda_3 = 1/\lambda^2</math>. The Cauchy stress differences may therefore be expressed as
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| :<math>
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| \sigma_{11}-\sigma_{33} = \sigma_{22}-\sigma_{33} = 2C_1\left(\lambda^2-\cfrac{1}{\lambda^4}\right) - 2C_2\left(\cfrac{1}{\lambda^2} - \lambda^4\right)
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| </math>
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| The equations for equibiaxial tension are equivalent to those governing uniaxial compression.
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| ==Pure shear==
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| A pure shear deformation can be achieved by applying stretches of the form <ref name=Ogden>Ogden, R. W., 1984, '''Nonlinear elastic deformations''', Dover</ref>
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| :<math>
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| \lambda_1 = \lambda ~;~~ \lambda_2 = \cfrac{1}{\lambda} ~;~~ \lambda_3 = 1
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| </math>
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| The Cauchy stress differences for pure shear may therefore be expressed as
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| :<math>
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| \sigma_{11} - \sigma_{33} = 2C_1(\lambda^2-1) - 2C_2\left(\cfrac{1}{\lambda^2}-1\right) ~;~~
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| \sigma_{22} - \sigma_{33} = 2C_1\left(\cfrac{1}{\lambda^2} -1\right) - 2C_2(\lambda^2 -1)
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| </math>
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| Therefore
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| :<math>
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| \sigma_{11} - \sigma_{22} = 2(C_1+C_2)\left(\lambda^2 - \cfrac{1}{\lambda^2}\right)
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| </math>
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| For a pure shear deformation
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| :<math>
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| I_1 = \lambda_1^2 + \lambda_2^2 + \lambda_3^2 = \lambda^2 + \cfrac{1}{\lambda^2} + 1 ~;~~
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| I_2 = \cfrac{1}{\lambda_1^2} + \cfrac{1}{\lambda_2^2} + \cfrac{1}{\lambda_3^2} = \cfrac{1}{\lambda^2} + \lambda^2 + 1
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| </math>
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| Therefore <math>I_1 = I_2</math>.
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| ==Simple shear==
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| The deformation gradient for a simple shear deformation has the form<ref name=Ogden/>
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| :<math>
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| \boldsymbol{F} = \boldsymbol{1} + \gamma~\mathbf{e}_1\otimes\mathbf{e}_2
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| </math>
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| where <math>\mathbf{e}_1,\mathbf{e}_2</math> are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by
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| :<math>
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| \gamma = \lambda - \cfrac{1}{\lambda} ~;~~ \lambda_1 = \lambda ~;~~ \lambda_2 = \cfrac{1}{\lambda} ~;~~ \lambda_3 = 1
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| </math>
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| In matrix form, the deformation gradient and the left Cauchy-Green deformation tensor may then be expressed as
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| :<math>
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| \boldsymbol{F} = \begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} ~;~~
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| \boldsymbol{B} = \boldsymbol{F}\cdot\boldsymbol{F}^T = \begin{bmatrix} 1+\gamma^2 & \gamma & 0 \\ \gamma & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
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| </math>
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| Therefore,
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| :<math>
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| \boldsymbol{B}^{-1} = \begin{bmatrix} 1 & -\gamma & 0 \\ -\gamma & 1+\gamma^2 & 0 \\ 0 & 0 & 1 \end{bmatrix}
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| </math>
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| The Cauchy stress is given by
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| :<math>
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| \boldsymbol{\sigma} = \begin{bmatrix} -p^* +2(C_1-C_2)+2C_1\gamma^2 & 2(C_1+C_2)\gamma & 0 \\ 2(C_1+C_2)\gamma & -p^* + 2(C_1 -C_2) - 2C_2\gamma^2 & 0 \\ 0 & 0 & -p^* + 2(C_1 - C_2)
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| \end{bmatrix}
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| </math>
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| For consistency with linear elasticity, clearly <math>\mu = 2(C_1+C_2)</math> where <math>\mu</math> is the shear modulus.
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| ==Rubber==
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| Elastic response of rubber-like materials are often modeled based on the Mooney—Rivlin model. The constants <math>C_1,C_2</math> are determined by the fitting predicted stress from the above equations to experimental data. The recommended tests are uniaxial tension, equibiaxial compression, equibiaxial tension, uniaxial compression, and for shear, planar tension and planar compression. The two parameter Mooney–Rivlin model is usually valid for strains less than 100%.
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| ==Notes and references==
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| <references/>
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| ==See also==
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| * [[Hyperelastic material]]
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| * [[Finite strain theory]]
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| * [[Continuum mechanics]]
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| * [[Strain energy density function]]
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| * [http://www.campoly.com/index.php/download_file/view/390/108/ Application note on experimentally determining Mooney Rivlin constants]
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| {{DEFAULTSORT:Mooney-Rivlin Solid}}
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| [[Category:Continuum mechanics]]
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| [[Category:Non-Newtonian fluids]]
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| [[Category:Rubber properties]]
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| [[Category:Solid mechanics]]
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