Advanced elasticity/Neo-Hookean material

A Neo-Hookean model is an extension of Hooke's law for the case of large deformations. The model of neo-Hookean solid is usable for plastics and 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 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 tensor, p - pressure, $$\mathbf {I} $$ - is the unity tensor, G is a constant equal to shear modulus, $$\mathbf {B} $$ is the Finger tensor.

The strain energy for this model is:
 * $$W = \frac{1}{2} G I_B$$,

where W is potential energy and $$I_B=\mathrm{tr}(\mathbf{B})$$ is the trace (or first invariant) of Finger tensor $$\mathbf {B} $$.

Usually the model is used for incompressible media.

The model was proposed by 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 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 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 Young's modulus of a neo-Hookean solid in uniaxial extension is 3G.

Simple shear
For the case of 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 first difference of normal stresses.

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

Source

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