Neo-Hookean solid


A neo-Hookean solid is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stress-strain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, the stress-strain curve of a neo-Hookean material is not linear. Instead, the relationship between applied stress and strain is initially linear, but at a certain point the stress-strain curve will plateau. The neo-Hookean model does not account for the dissipative release of energy as heat while straining the material and perfect elasticity is assumed at all stages of deformation.
The neo-Hookean model is based on the statistical thermodynamics of cross-linked polymer chains and is usable for plastics and rubber-like substances. Cross-linked polymers will act in a neo-Hookean manner 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. The neo-Hookean material model does not predict that increase in modulus at large strains and is typically accurate only for strains less than 20%. The model is also inadequate for biaxial states of stress and has been superseded by the Mooney-Rivlin model.
The strain energy density function for an incompressible neo-Hookean material in a three-dimensional description is
where is a material constant, and is the first invariant, of the right Cauchy-Green deformation tensor, i.e.,
where are the principal stretches.
For a compressible neo-Hookean material the strain energy density function is given by
where is a material constant and is the deformation gradient. It can be shown that in 2D, the strain energy density function is
Several alternative formulations exist for compressible neo-Hookean materials, for example
where is the first invariant of the isochoric part of the right Cauchy–Green deformation tensor.
For consistency with linear elasticity,
where and are the Lamé parameters.

Cauchy stress in terms of deformation tensors

Compressible neo-Hookean material

For a compressible Rivlin neo-Hookean material the Cauchy stress is given by
where is the left Cauchy-Green deformation tensor, and
For infinitesimal strains
and the Cauchy stress can be expressed as
Comparison with Hooke's law shows that and.
The Cauchy stress in a compressible hyperelastic material is given by
For a compressible Rivlin neo-Hookean material,
while, for a compressible Ogden neo-Hookean material,
Therefore, the Cauchy stress in a compressible Rivlin neo-Hookean material is given by
while that for the corresponding Ogden material is
If the part of the left Cauchy-Green deformation tensor is defined as, then we can write the Rivlin neo-Heooken stress as
and the Ogden neo-Hookean stress as
The quantities
have the form of pressures and are usually treated as such. The Rivlin neo-Hookean stress can then be expressed in the form
while the Ogden neo-Hookean stress has the form

Incompressible neo-Hookean material

For an incompressible neo-Hookean material with
where is an undetermined pressure.

Cauchy stress in terms of principal stretches

Compressible neo-Hookean material

For a compressible neo-Hookean hyperelastic material, the principal components of the Cauchy stress are given by
Therefore, the differences between the principal stresses are
For a compressible hyperelastic material, the principal components of the Cauchy stress are given by
The strain energy density function for a compressible neo Hookean material is
Therefore,
Since we have
Hence,
The principal Cauchy stresses are therefore given by

Incompressible neo-Hookean material

In terms of the principal stretches, the Cauchy stress differences for an incompressible hyperelastic material are given by
For an incompressible neo-Hookean material,
Therefore,
which gives

Uniaxial extension

Compressible neo-Hookean material

For a compressible material undergoing uniaxial extension, the principal stretches are
Hence, the true stresses for a compressible neo-Hookean material are given by
The stress differences are given by
If the material is unconstrained we have. Then
Equating the two expressions for gives a relation for as a function of, i.e.,
or
The above equation can be solved numerically using a Newton-Raphson iterative root finding procedure.

Incompressible neo-Hookean material

Under uniaxial extension, and. Therefore,
Assuming no traction on the sides,, so we can write
where is the engineering strain. This equation is often written in alternative notation as
The equation above is for the true stress. For the engineering stress the equation is:
For small deformations we will have:
Thus, the equivalent Young's modulus of a neo-Hookean solid in uniaxial extension is, which is in concordance with linear elasticity.

Equibiaxial extension

Compressible neo-Hookean material

In the case of equibiaxial extension
Therefore,
The stress differences are
If the material is in a state of plane stress then and we have
We also have a relation between and :
or,
This equation can be solved for using Newton's method.

Incompressible neo-Hookean material

For an incompressible material and the differences between the principal Cauchy stresses take the form
Under plane stress conditions we have

Pure dilation

For the case of pure dilation
Therefore, the principal Cauchy stresses for a compressible neo-Hookean material are given by
If the material is incompressible then and the principal stresses can be arbitrary.
The figures below show that extremely high stresses are needed to achieve large triaxial extensions or compressions. Equivalently, relatively small triaxial stretch states can cause very high stresses to develop in a rubber-like material. The magnitude of the stress is quite sensitive to the bulk modulus but not to the shear modulus.

Simple shear

For the case of simple shear the deformation gradient in terms of components with respect to a reference basis is of the form
where is the shear deformation. Therefore the left Cauchy-Green deformation tensor is

Compressible neo-Hookean material

In this case. Hence,. Now,
Hence the Cauchy stress is given by

Incompressible neo-Hookean material

Using the relation for the Cauchy stress for an incompressible neo-Hookean material we get
Thus neo-Hookean solid shows linear dependence of shear stresses upon shear deformation and quadratic dependence of the normal stress difference on the shear deformation. The expressions for the Cauchy stress for a compressible and an incompressible neo-Hookean material in simple shear represent the same quantity and provide a means of determining the unknown pressure.