Advanced elasticity/Neo-Hookean material

A Neo-Hookean model is an extension of w:Hooke's law for the case of large w:deformations. The model of neo-Hookean solid is usable for w:plastics and w:rubber-like substances.

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 w: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 edit

The model of neo-Hookean solid assumes that the extra stresses due to deformation are proportional to Finger tensor:


where   - stress w:tensor, p - w:pressure,   - is the unity tensor, G is a constant equal to w:shear modulus,   is the w:Finger tensor.

The strain energy for this model is:


where W is potential energy and   is the trace (or first invariant) of w:Finger tensor  .

Usually the model is used for incompressible media.

The model was proposed by w:Ronald Rivlin in 1948.

Uni-axial extension edit

Comparison of experimental results (dots) and predictions for w:Hooke's law(1), w:Neo-Hookean solid(2) and w:Mooney-Rivlin solid models(3)

Under uni-axial extension from the definition of Finger tensor:


where   is the elongation in the w:stretch ratio in the  -direction.

Assuming no traction on the sides,  , so:


where   is the strain.

The equation above is for the true stress (ratio of the elongation force to deformed cross-section), for w:engineering stress the equation is:


For small deformations   we will have:


Thus, the equivalent w:Young's modulus of a neo-Hookean solid in uniaxial extension is 3G.

Simple shear edit

For the case of w:simple shear we will have:


where   is shear deformation. Thus neo-Hookean solid shows linear dependence of shear stresses upon shear deformation and quadratic w:first difference of normal stresses.

Generalization edit

The most important generalisation of Neo-Hookean solid is w:Mooney-Rivlin solid.

Source edit

  • C. W. Macosko Rheology: principles, measurement and applications, VCH Publishers, 1994, ISBN 1-56081-579-5