Neo-Hookean solid

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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 using invariants, though Mooney had already described a version in stretch form in 1940, and Wall had noted the equivalence in shear with the Hooke model in 1942.

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. In addition to being used to model physical materials, the stability and highly non-linear behaviour under compression has made neo-Hookean materials a popular choice for fictitious media approaches such as the third medium contact method.

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 The model is also inadequate for biaxial states of stress and has been superseded by the Mooney–Rivlin model.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 primary, and likely most widely employed, strain-energy function formulation is the Mooney–Rivlin model, which reduces to the widely known neo-Hookean model. The strain energy density function for an incompressible Mooney–Rivlin material is

${\displaystyle W=C_{10}(I_{1}-3)+C_{01}(I_{2}-3);~I_{3}=1}$

Setting ${\displaystyle C_{01}=0}$ reduces to the (incompressible) neo-Hookean strain energy function

${\displaystyle W=C_{1}(I_{1}-3)}$

where ${\displaystyle C_{1}}$ is a material constant, and ${\displaystyle I_{1}}$ is the first principal invariant (trace), of the left Cauchy-Green deformation tensor, i.e.,

${\displaystyle I_{1}=\mathrm {tr} (\mathbf {B} )=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}}$

where ${\displaystyle \lambda _{i}}$ are the principal stretches. Similarly, the second and third principal invariants are

${\displaystyle {\begin{aligned}I_{2}&={\frac {1}{2}}{\big (}I_{1}^{2}-\mathrm {tr} (\mathbf {B} )^{2}{\big )}\\I_{3}&=\mathrm {det} (\mathbf {B} )=\mathrm {det} (\mathbf {F} \mathbf {F} ^{T})=(\lambda _{1}\lambda _{2}\lambda _{3})^{2}=J^{2}\end{aligned}}}$

where ${\displaystyle \mathbf {F} }$ is the deformation gradient. Relaxing the incompressible assumption (${\displaystyle J=1}$), one can add a hydrostatic work term ${\displaystyle W_{H}(I_{3})}$ for a compressible material, but the first two terms must be adjusted to uncouple deviatoric and volumetric terms, resulting in

${\displaystyle W=C_{10}({\bar {I}}_{1}-3)+C_{01}({\bar {I}}_{2}-3)+D_{1}(J-1)^{2}}$

where

${\displaystyle {\bar {I}}_{1}=I_{1}J^{-2/3},~~~{\bar {I}}_{2}=I_{2}J^{-4/3}}$

Recall that a Mooney–Rivlin material with ${\displaystyle C_{01}=0}$ is a neo-Hookean material, so the compressible neo-Hookean strain energy density is given by

${\displaystyle W=C_{1}(I_{1}J^{-2/3}-3)+D_{1}(J-1)^{2}}$

where ${\displaystyle D_{1}}$ is a material constant.

Note that this is one of several strain energy functions employed in hyperelasticity measurements. For example, some neo-Hookean models contain an extra ${\displaystyle \ln J}$ term, namely

${\displaystyle W=C_{1}(I_{1}-3-2\ln J)+D_{1}(J-1)^{2}}$

Finally, for consistency with linear elasticity,

${\displaystyle C_{1}={\frac {\mu }{2}}~;~~D_{1}={\frac {\kappa }{2}}}$

where ${\displaystyle \kappa }$ is the bulk modulus and ${\displaystyle \mu }$ is the shear modulus or the second Lamé parameter. Alternative definitions of ${\displaystyle C_{1}}$ and ${\displaystyle D_{1}}$ are sometimes used, notably in commercial finite element analysis software such as Abaqus.