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and C10, C01, 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 three-dimensional Finger tensor is usually written
In this article, the trace a1 is written I1, the next coefficient a2 is written I2, and the determinant a3 would be written I3.)
The stress tensor depends upon Finger tensor by the following equation:
Additional recommended knowledge
The response of a neo-Hookean 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 neo-Hookean material does not. A hyperelastic material will initially be linear, but at a certain point, the stress-strain 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. Cross-linked 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.
Neo-Hookean Solid Model
The model of neo-Hookean solid assumes that the extra stresses due to deformation are proportional to 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.
For the case of uniaxial elongation, true stress can be calculated as:
and engineering stress can be calculated as:
The Mooney-Rivlin solid model usually fits experimental data better than Neo-Hookean solid does, but requires an additional empirical constant.
Elastic response of soft tissues like that in the brain is often modelled based on the Mooney--Rivlin model.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Mooney-Rivlin_solid". A list of authors is available in Wikipedia.|