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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.
Additional recommended knowledge
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.
Under uni-axial extension from the definition of Finger tensor:
where α1 is the elongation in the stretch ratio in the 1-direction.
Assuming no traction on the sides, T22 = T33 = 0, so:
where ε = α1 − 1 is the strain.
The equation above is for the true stress (ratio of the elongation force to deformed cross-section), for engineering stress the equation is:
For small deformations ε < < 1 we will have:
Thus, the equivalent Young's modulus of a neo-Hookean solid in uniaxial extension is 3G.
For the case of simple shear we will have:
where γ is shear deformation. Thus neo-Hookean solid shows linear dependence of shear stresses upon shear deformation and quadratic first difference of normal stresses.
The most important generalisation of Neo-Hookean solid is Mooney-Rivlin solid.
|This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Neo-Hookean_solid". A list of authors is available in Wikipedia.|