# 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 | edit source]

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 | edit source]

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 3*G*.

## Simple shear[edit | edit source]

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 | edit source]

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

## Source[edit | edit source]

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