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MooneyRivlin solidIn continuum mechanics, a MooneyRivlin solid is a generalization of the NeoHookean solid model, where the strain energy W is a linear combination of two invariants of Finger tensor :
where and are the first and the second invariant of deviatoric component of the Finger tensor:
where: . and C_{10}, C_{01}, and d are constants. (Explanatory gloss for students of applied mathematics, physics, or other disciplines: the characteristic polynomial of the linear operator corresponding to the second rank threedimensional Finger tensor is usually written In this article, the trace a_{1} is written I_{1}, the next coefficient a_{2} is written I_{2}, and the determinant a_{3} would be written I_{3}.) If (where G is the shear modulus) and C_{2} = 0, we obtain a NeoHookean solid, a special case of a MooneyRivlin solid. The stress tensor depends upon Finger tensor by the following equation: The model was proposed by Melvin Mooney and Ronald Rivlin in two independent papers in 1952. Additional recommended knowledge
Uniaxial elongationNeoHookean solid model is an extension of Hooke's law for the case of large deformations. The model of neoHookean solid is usable for plastics and rubberlike substances. The response of a neoHookean material, or hyperelastic material, to an applied stress differs from that of a linear elastic material. While a linear elastic material has a linear relationship between applied stress and strain, a neoHookean material does not. A hyperelastic material will initially be linear, but at a certain point, the stressstrain curve will plateau due to the release of energy as heat while straining the material. Then, at another point, the elastic modulus of the material will increase again. This hyperelasticity, or rubber elasticity, is often observed in polymers. Crosslinked polymers will act in this way because initially the polymer chains can move relative to each other when a stress is applied. However, at a certain point the polymer chains will be stretched to the maximum point that the covalent cross links will allow, and this will cause a dramatic increase in the elastic modulus of the material. One can also use thermodynamics to explain the elasticity of polymers. NeoHookean Solid ModelThe model of neoHookean solid assumes that the extra stresses due to deformation are proportional to Finger tensor:
where  stress tensor, p  pressure,  is the unity tensor, G is a constant equal to shear modulus, is the Finger tensor. The strain energy for this model is:
where W is potential energy and is the trace (or first invariant) of Finger tensor . Usually the model is used for incompressible media. The model was proposed by Ronald Rivlin in 1948. Uniaxial extensionFor the case of uniaxial elongation, true stress can be calculated as: and engineering stress can be calculated as: The MooneyRivlin solid model usually fits experimental data better than NeoHookean solid does, but requires an additional empirical constant. Brain tissuesElastic response of soft tissues like that in the brain is often modelled based on the MooneyRivlin model. Source
Categories: Continuum mechanics  NonNewtonian fluids  Rubber properties 

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "MooneyRivlin_solid". A list of authors is available in Wikipedia. 