 NeoHookean solid

Continuum mechanics LawsScientistsA NeoHookean solid^{[1]}^{[2]} is a hyperelastic material model, similar to Hooke's law, that can be used for predicting the nonlinear stressstrain behavior of materials undergoing large deformations. The model was proposed by Ronald Rivlin in 1948. In contrast to linear elastic materials, a the stressstrain curve of a neoHookean material is not linear. Instead, the relationship between applied stress and strain is initially linear, but at a certain point the stressstrain curve will plateau. The neoHookean 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 neoHookean model is based on the statistical thermodynamics of crosslinked polymer chains and is usable for plastics and rubberlike substances. Crosslinked polymers will act in a neoHookean 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 neoHookean material model does not predict that increase in modulus at large strains and is typically accurate only for strains less than 20%^{[3]}. The model is also inadequate for biaxial states of stress and has been superseded by the MooneyRivlin model.
The strain energy density function for an incompressible neoHookean material is
where C_{1} is a material constant, and I_{1} is the first invariant of the left CauchyGreen deformation tensor, i.e.,
where λ_{i} are the principal stretches. For a compressible neoHookean material the strain energy density function is given by
where D_{1} is a material constant, is the first invariant of the deviatoric part of the left CauchyGreen deformation tensor, and is the deformation gradient. Several alternative formulations exist for compressible neoHookean materials, for example ^{[1]}
For consistency with linear elasticity,
where μ is the shear modulus and κ is the bulk modulus.
Contents
Cauchy stress in terms of deformation tensors
Compressible neoHookean material
For a compressible Rivlin neoHookean material the Cauchy stress is given by
where is the left CauchyGreen deformation tensor, and
For infinitesimal strains ()
and the Cauchy stress can be expressed as
Comparison with Hooke's law shows that μ = 2C_{1} and κ = 2D_{1}.

Proof: The Cauchy stress in a compressible hyperelastic material is given by
For a compressible Rivlin neoHookean material,
while, for a compressible Ogden neoHookean material,
Therefore, the Cauchy stress in a compressible Rivlin neoHookean material is given by
while that for the corresponding Ogden material is
If the isochoric part of the left CauchyGreen deformation tensor is defined as , then we can write the Rivlin neoHeooken stress as
and the Ogden neoHookean stress as
The quantities
have the form of pressures and are usually treated as such. The Rivlin neoHookean stress can then be expressed in the form
while the Ogden neoHookean stress has the form
Incompressible neoHookean material
For an incompressible neoHookean material with J = 1
where p is an undetermined pressure.
Cauchy stress in terms of principal stretches
Compressible NeoHookean material
For a compressible neoHookean hyperelastic material, the principal components of the Cauchy stress are given by
Therefore, the differences between the principal stresses are

Proof: 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 J = λ_{1}λ_{2}λ_{3} we have
Hence,
The principal Cauchy stresses are therefore given by
Incompressible NeoHookean material
In terms of the principal stretches, the Cauchy stress differences for an incompressible hyperelastic material are given by
For an incompressible neoHookean material,
Therefore,
which gives
Uniaxial extension
Compressible neoHookean material
For a compressible material undergoing uniaxial extension, the principal stretches are
Hence, the true (Cauchy) stresses for a compressible neoHookean material are given by
The stress differences are given by
If the material is unconstrained we have σ_{22} = σ_{33} = 0. Then
Equating the two expressions for σ_{11} gives a relation for J as a function of λ, i.e.,
or
The above equation can be solved numerically using a NewtonRaphson iterative root finding procedure.
Incompressible neoHookean material
Under uniaxial extension, and . Therefore,
Assuming no traction on the sides, σ_{22} = σ_{33} = 0, so we can write
where ε_{11} = λ − 1 is the engineering strain. This equation is often written in alternative notation as
The equation above is for the true stress (ratio of the elongation force to deformed crosssection). For the engineering stress the equation is:
For small deformations we will have:
 σ_{11} = 6C_{1}ε = 3με
Thus, the equivalent Young's modulus of a neoHookean solid in uniaxial extension is 3μ.
Equibiaxial extension
Compressible NeoHookean material
In the case of equibiaxial extension
Therefore,
The stress differences are
If the material is in a state of plane stress then σ_{33} = 0 and we have
We also have a relation between J and λ:
or,
This equation can be solved for J using Newton's method.
Incompressible NeoHookean material
For an incompressible material J = 1 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 neoHookean material are given by
If the material is incompressible then λ^{3} = 1 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 rubberlike material. Note also that 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 ^{[1]}
where γ is the shear deformation. Therefore the left CauchyGreen deformation tensor is
Compressible NeoHookean material
In this case . Hence, . Now,
Hence the Cauchy stress is given by
Incompressible NeoHookean material
Using the relation for the Cauchy stress for an incompressible neoHookean material we get
Thus neoHookean solid shows linear dependence of shear stresses upon shear deformation and quadratic dependence of the normal stress difference on the shear deformation. Note that the expressions for the Cauchy stress for a compressible and an incompressible neoHookean material in simple shear represent the same quantity and provide a means of determining the unknown pressure p.
References
 ^ ^{a} ^{b} ^{c} Ogden, R. W. , 1998, Nonlinear Elastic Deformations, Dover.
 ^ C. W. Macosko, 1994, Rheology: principles, measurement and applications, VCH Publishers, ISBN 1560815795.
 ^ Gent, A. N., ed., 2001, Engineering with rubber, Carl Hanser Verlag, Munich.
See also
Categories: Continuum mechanics
 NonNewtonian fluids
 Rubber properties
 Solid mechanics

Wikimedia Foundation. 2010.