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:

$W = \frac{1}{2} G I_B$,

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

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:

$T_{11}=-p+G \alpha_1^2$

$T_{22}=T_{33}=-p+ \frac {G} {\alpha_1}$

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