Arruda–Boyce model

In continuum mechanics, an Arruda–Boyce model is a hyperelastic constitutive model used to describe the mechanical behavior of rubber and other polymeric substances. This model is based on the statistical mechanics of a material with a cubic representative volume element containing eight chains along the diagonal directions. The material is assumed to be incompressible. The model is named after Ellen Arruda and Mary Cunningham Boyce, who published it in 1993.

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The strain energy density function for the incompressible Arruda–Boyce model is given by

${\displaystyle W=Nk_{B}\theta {\sqrt {n}}\left[\beta \lambda _{\text{chain}}-{\sqrt {n}}\ln \left({\cfrac {\sinh \beta }{\beta }}\right)\right],}$

where ${\displaystyle n}$ is the number of chain segments, ${\displaystyle k_{B}}$ is the Boltzmann constant, ${\displaystyle \theta }$ is the temperature in kelvins, ${\displaystyle N}$ is the number of chains in the network of a cross-linked polymer,

${\displaystyle \lambda _{\mathrm {chain} }={\sqrt {\tfrac {I_{1}}{3}}},\quad \beta ={\mathcal {L}}^{-1}\left({\cfrac {\lambda _{\text{chain}}}{\sqrt {n}}}\right),}$

where ${\displaystyle I_{1}}$ is the first invariant of the left Cauchy–Green deformation tensor, and ${\displaystyle {\mathcal {L}}^{-1}(x)}$ is the inverse Langevin function which can be approximated by

${\displaystyle {\mathcal {L}}^{-1}(x)={\begin{cases}1.31\tan(1.59x)+0.91x&{\text{for}}\ |x|<0.841,\\{\tfrac {1}{\operatorname {sgn}(x)-x}}&{\text{for}}\ 0.841\leq |x|<1.\end{cases}}}$

For small deformations the Arruda–Boyce model reduces to the Gaussian network based neo-Hookean solid model. It can be shown that the Gent model is a simple and accurate approximation of the Arruda–Boyce model.