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 The model is also inadequate for biaxial states of stress and has been superseded by the Mooney-Rivlin model.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 formula_2 is a material constant, and formula_3 is the first invariant (trace), of the right Cauchy-Green deformation tensor, i.e.,
where formula_5 are the principal stretches.
For a compressible neo-Hookean material the strain energy density function is given by
where formula_7 is a material constant and formula_8 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 formula_11 is the first invariant of the isochoric part formula_12 of the right Cauchy–Green deformation tensor.
For consistency with linear elasticity,
where formula_14 is the shear modulus or the first Lamé parameters and formula_15 is the bulk modulus.
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 formula_17 is the left Cauchy-Green deformation tensor, and
For infinitesimal strains (formula_19)
and the Cauchy stress can be expressed as
Comparison with Hooke's law shows that formula_22 and formula_23.
Incompressible neo-Hookean material.
For an incompressible neo-Hookean material with formula_24
where formula_26 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
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 (Cauchy) stresses for a compressible neo-Hookean material are given by
The stress differences are given by
If the material is unconstrained we have formula_36. Then
Equating the two expressions for formula_38 gives a relation for formula_39 as a function of formula_40, 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, formula_43 and formula_44. Therefore,
Assuming no traction on the sides, formula_46, so we can write
where formula_48 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 cross-section). For the engineering stress the equation is:
For small deformations formula_51 we will have:
Thus, the equivalent Young's modulus of a neo-Hookean solid in uniaxial extension is formula_53, which is in concordance with linear elasticity (formula_54 with formula_55 for incompressibility).
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 formula_59 and we have
We also have a relation between formula_39 and formula_40:
or,
This equation can be solved for formula_39 using Newton's method.
Incompressible neo-Hookean material.
For an incompressible material formula_66 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 formula_71 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 formula_73 is the shear deformation. Therefore the left Cauchy-Green deformation tensor is
Compressible neo-Hookean material.
In this case formula_75. Hence, formula_76. 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 formula_26.