# 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

The model of neo-Hookean solid assumes that the extra stresses due to deformation are proportional to Finger tensor:
    $\mathbf {T} =-p\mathbf {I} +G\mathbf {B}$,
where $\mathbf {T}$ - stress w:tensor, p - w:pressure, $\mathbf {I}$ - is the unity tensor, G is a constant equal to w:shear modulus, $\mathbf {B}$ is the w:Finger tensor.
The strain energy for this model is:
    $W={\frac {1}{2}}GI_{B}$,
where W is potential energy and $I_{B}=\mathrm {tr} (\mathbf {B} )$ is the trace (or first invariant) of w:Finger tensor $\mathbf {B}$.
Usually the model is used for incompressible media.
The model was proposed by w:Ronald Rivlin in 1948.

## Uni-axial extension

Under uni-axial extension from the definition of Finger tensor:
    $T_{11}=-p+G\alpha _{1}^{2}$
    $T_{22}=T_{33}=-p+{\frac {G}{\alpha _{1}}}$
where $\alpha _{1}$ is the elongation in the w:stretch ratio in the $1$-direction.
Assuming no traction on the sides, $T_{22}=T_{33}=0$, so:
    $T_{11}=G(\alpha _{1}^{2}-\alpha _{1}^{-1})=G{\frac {3\epsilon +3\epsilon ^{2}+\epsilon ^{3}}{1+\epsilon }}$,
where $\epsilon =\alpha _{1}-1$ 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:
    $T_{11eng}=G(\alpha _{1}-\alpha _{1}^{-2})$
For small deformations $\epsilon <<1$ we will have:
    $T_{11}=3G\epsilon$
Thus, the equivalent w:Young's modulus of a neo-Hookean solid in uniaxial extension is 3G.

## Simple shear

For the case of w:simple shear we will have:
    $T_{12}=G\gamma$
    $T_{11}-T_{22}=G\gamma ^{2}$
    $T_{22}-T_{33}=0$
where $\gamma$ 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

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

## Source

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