A Mooney-Rivlin solid is a generalization of the w:Neo-Hookean solid model, where the strain energy W is a linear combination of two invariants of the w:Finger tensor B {\displaystyle \mathbf {B} } : W = C 1 ( I  ̄ 1 − 3 ) + C 2 ( I  ̄ 2 − 3 ) {\displaystyle W=C_{1}({\overline {I}}_{1}-3)+C_{2}({\overline {I}}_{2}-3)} , where I  ̄ 1 {\displaystyle {\overline {I}}_{1}} and I  ̄ 2 {\displaystyle {\overline {I}}_{2}} are the first and the second invariant of w:deviatoric component of the w:Finger tensor: I 1 = λ 1 2 + λ 2 2 + λ 3 2 {\displaystyle I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}} , I 2 = λ 1 2 λ 2 2 + λ 2 2 λ 3 2 + λ 3 2 λ 1 2 {\displaystyle I_{2}=\lambda _{1}^{2}\lambda _{2}^{2}+\lambda _{2}^{2}\lambda _{3}^{2}+\lambda _{3}^{2}\lambda _{1}^{2}} , I 3 = λ 1 2 λ 2 2 λ 3 2 {\displaystyle I_{3}=\lambda _{1}^{2}\lambda _{2}^{2}\lambda _{3}^{2}} , where: C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} are constants. If C 1 = 1 2 G {\displaystyle C_{1}={\frac {1}{2}}G} (where G is the w:shear modulus) and C 2 = 0 {\displaystyle C_{2}=0} , we obtain a w:Neo-Hookean solid, a special case of a Mooney-Rivlin solid. The stress tensor T {\displaystyle \mathbf {T} } depends upon Finger tensor B {\displaystyle \mathbf {B} } by the following equation: T = − p I + 2 C 1 B + 2 C 2 B − 1 {\displaystyle \mathbf {T} =-p\mathbf {I} +2C_{1}\mathbf {B} +2C_{2}\mathbf {B} ^{-1}} The model was proposed by w:Melvin Mooney and w:Ronald Rivlin in two independent papers in 1952. For the case of uniaxial elongation, true stress can be calculated as: T 11 = ( 2 C 1 + 2 C 2 α 1 ) ( α 1 2 − α 1 − 1 ) {\displaystyle T_{11}=\left(2C_{1}+{\frac {2C_{2}}{\alpha _{1}}}\right)\left(\alpha _{1}^{2}-\alpha _{1}^{-1}\right)} and w:engineering stress can be calculated as: T 11 e n g = ( 2 C 1 + 2 C 2 α 1 ) ( α 1 − α 1 − 2 ) {\displaystyle T_{11eng}=\left(2C_{1}+{\frac {2C_{2}}{\alpha _{1}}}\right)\left(\alpha _{1}-\alpha _{1}^{-2}\right)} The Mooney-Rivlin solid model usually fits experimental data better than w:Neo-Hookean solid does, but requires an additional empirical constant. Elastic response of rubber-like materials are often modelled based on the Mooney-Rivlin model. C. W. Macosko Rheology: principles, measurement and applications, VCH Publishers, 1994, ISBN 1-56081-579-5 The characteristic polynomial of the linear operator corresponding to the second rank three-dimensional Finger tensor is usually written p B ( λ ) = λ 3 − a 1 λ 2 + a 2 λ − a 3 {\displaystyle p_{B}(\lambda )=\lambda ^{3}-a_{1}\,\lambda ^{2}+a_{2}\,\lambda -a_{3}} In this article, the trace a 1 {\displaystyle a_{1}} is written I 1 {\displaystyle I_{1}} , the next coefficient a 2 {\displaystyle a_{2}} is written I 2 {\displaystyle I_{2}} , and the determinant a 3 {\displaystyle a_{3}} would be written I 3 {\displaystyle I_{3}} . ← Neo-Hookean material Advanced elasticity Ogden material →
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