# Arruda–Boyce model

Template:Continuum mechanics 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 strain energy density function for the incompressible Arruda–Boyce model is given by

$W=Nk_{B}\theta {\sqrt {n}}\left[\beta \lambda _{\mathrm {chain} }-{\sqrt {n}}\ln \left({\cfrac {\sinh \beta }{\beta }}\right)\right]$ $\lambda _{\mathrm {chain} }={\sqrt {\tfrac {I_{1}}{3}}}~;~~\beta ={\mathcal {L}}^{-1}\left({\cfrac {\lambda _{\mathrm {chain} }}{\sqrt {n}}}\right)$ where $I_{1}$ is the first invariant of the left Cauchy–Green deformation tensor, and ${\mathcal {L}}^{-1}(x)$ is the inverse Langevin function which can approximated by

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

## Alternative expressions for the Arruda–Boyce model

An alternative form of the Arruda–Boyce model, using the first five terms of the inverse Langevin function, is

$W=C_{1}\left[{\tfrac {1}{2}}(I_{1}-3)+{\tfrac {1}{20N}}(I_{1}^{2}-9)+{\tfrac {11}{1050N^{2}}}(I_{1}^{3}-27)+{\tfrac {19}{7000N^{3}}}(I_{1}^{4}-81)+{\tfrac {519}{673750N^{4}}}(I_{1}^{5}-243)\right]$ where $C_{1}$ is a material constant. The quantity $N$ can also be interpreted as a measure of the limiting network stretch.

If $\lambda _{m}$ is the stretch at which the polymer chain network becomes locked, we can express the Arruda–Boyce strain energy density as

$W=C_{1}\left[{\tfrac {1}{2}}(I_{1}-3)+{\tfrac {1}{20\lambda _{m}^{2}}}(I_{1}^{2}-9)+{\tfrac {11}{1050\lambda _{m}^{4}}}(I_{1}^{3}-27)+{\tfrac {19}{7000\lambda _{m}^{6}}}(I_{1}^{4}-81)+{\tfrac {519}{673750\lambda _{m}^{8}}}(I_{1}^{5}-243)\right]$ We may alternatively express the Arruda–Boyce model in the form

$W=C_{1}~\sum _{i=1}^{5}\alpha _{i}~\beta ^{i-1}~(I_{1}^{i}-3^{i})$ If the rubber is compressible, a dependence on $J=\det({\boldsymbol {F}})$ can be introduced into the strain energy density; ${\boldsymbol {F}}$ being the deformation gradient. Several possibilities exist, among which the Kaliske–Rothert extension has been found to be reasonably accurate. With that extension, the Arruda-Boyce strain energy density function can be expressed as

$W=D_{1}\left({\tfrac {J^{2}-1}{2}}-\ln J\right)+C_{1}~\sum _{i=1}^{5}\alpha _{i}~\beta ^{i-1}~({\overline {I}}_{1}^{i}-3^{i})$ ## Consistency condition

For the incompressible Arruda–Boyce model to be consistent with linear elasticity, with $\mu$ as the shear modulus of the material, the following condition has to be satisfied:

${\cfrac {\partial W}{\partial I_{1}}}{\biggr |}_{I_{1}=3}={\frac {\mu }{2}}\,.$ From the Arruda–Boyce strain energy density function, we have,

${\cfrac {\partial W}{\partial I_{1}}}=C_{1}~\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\,.$ $\mu =2C_{1}~\sum _{i=1}^{5}i\,\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\,.$ Substituting in the values of $\alpha _{i}$ leads to the consistency condition

$\mu =C_{1}\left(1+{\tfrac {3}{5\lambda _{m}^{2}}}+{\tfrac {99}{175\lambda _{m}^{4}}}+{\tfrac {513}{875\lambda _{m}^{6}}}+{\tfrac {42039}{67375\lambda _{m}^{8}}}\right)\,.$ ## Stress-deformation relations

The Cauchy stress for the incompressible Arruda–Boyce model is given by

${\boldsymbol {\sigma }}=-p~{\boldsymbol {\mathit {1}}}+2~{\cfrac {\partial W}{\partial I_{1}}}~{\boldsymbol {B}}=-p~{\boldsymbol {\mathit {1}}}+2C_{1}~\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]{\boldsymbol {B}}$ ### Uniaxial extension Stress-strain curves under uniaxial extension for Arruda–Boyce model compared with various hyperelastic material models.
$I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=\lambda ^{2}+{\cfrac {2}{\lambda }}~.$ The left Cauchy–Green deformation tensor can then be expressed as

${\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\cfrac {1}{\lambda }}~(\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\mathbf {n} _{3}\otimes \mathbf {n} _{3})~.$ If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

{\begin{aligned}\sigma _{11}&=-p+2C_{1}\lambda ^{2}\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]\\\sigma _{22}&=-p+{\cfrac {2C_{1}}{\lambda }}\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]=\sigma _{33}~.\end{aligned}} $p={\cfrac {2C_{1}}{\lambda }}\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]~.$ Therefore,

$\sigma _{11}=2C_{1}\left(\lambda ^{2}-{\cfrac {1}{\lambda }}\right)\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]~.$ $T_{11}=\sigma _{11}/\lambda =2C_{1}\left(\lambda -{\cfrac {1}{\lambda ^{2}}}\right)\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]~.$ ### Equibiaxial extension

$I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=2~\lambda ^{2}+{\cfrac {1}{\lambda ^{4}}}~.$ The left Cauchy–Green deformation tensor can then be expressed as

${\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+\lambda ^{2}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+{\cfrac {1}{\lambda ^{4}}}~\mathbf {n} _{3}\otimes \mathbf {n} _{3}~.$ If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

$\sigma _{11}=2C_{1}\left(\lambda ^{2}-{\cfrac {1}{\lambda ^{4}}}\right)\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]=\sigma _{22}~.$ $T_{11}={\cfrac {\sigma _{11}}{\lambda }}=2C_{1}\left(\lambda -{\cfrac {1}{\lambda ^{5}}}\right)\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]=T_{22}~.$ ### Planar extension

$I_{1}=\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}=\lambda ^{2}+{\cfrac {1}{\lambda ^{2}}}+1~.$ The left Cauchy–Green deformation tensor can then be expressed as

${\boldsymbol {B}}=\lambda ^{2}~\mathbf {n} _{1}\otimes \mathbf {n} _{1}+{\cfrac {1}{\lambda ^{2}}}~\mathbf {n} _{2}\otimes \mathbf {n} _{2}+\mathbf {n} _{3}\otimes \mathbf {n} _{3}~.$ If the directions of the principal stretches are oriented with the coordinate basis vectors, we have

$\sigma _{11}=2C_{1}\left(\lambda ^{2}-{\cfrac {1}{\lambda ^{2}}}\right)\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]~;~~\sigma _{22}=0~;~~\sigma _{33}=2C_{1}\left(1-{\cfrac {1}{\lambda ^{2}}}\right)\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]~.$ $T_{11}={\cfrac {\sigma _{11}}{\lambda }}=2C_{1}\left(\lambda -{\cfrac {1}{\lambda ^{3}}}\right)\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~I_{1}^{i-1}\right]~.$ ### Simple shear

The deformation gradient for a simple shear deformation has the form

${\boldsymbol {F}}={\boldsymbol {1}}+\gamma ~\mathbf {e} _{1}\otimes \mathbf {e} _{2}$ where $\mathbf {e} _{1},\mathbf {e} _{2}$ are reference orthonormal basis vectors in the plane of deformation and the shear deformation is given by

$\gamma =\lambda -{\cfrac {1}{\lambda }}~;~~\lambda _{1}=\lambda ~;~~\lambda _{2}={\cfrac {1}{\lambda }}~;~~\lambda _{3}=1$ In matrix form, the deformation gradient and the left Cauchy–Green deformation tensor may then be expressed as

${\boldsymbol {F}}={\begin{bmatrix}1&\gamma &0\\0&1&0\\0&0&1\end{bmatrix}}~;~~{\boldsymbol {B}}={\boldsymbol {F}}\cdot {\boldsymbol {F}}^{T}={\begin{bmatrix}1+\gamma ^{2}&\gamma &0\\\gamma &1&0\\0&0&1\end{bmatrix}}$ Therefore,

$I_{1}=\mathrm {tr} ({\boldsymbol {B}})=3+\gamma ^{2}$ and the Cauchy stress is given by

${\boldsymbol {\sigma }}=-p~{\boldsymbol {\mathit {1}}}+2C_{1}\left[\sum _{i=1}^{5}i~\alpha _{i}~\beta ^{i-1}~(3+\gamma ^{2})^{i-1}\right]~{\boldsymbol {B}}$ ## Statistical mechanics of polymer deformation

{{#invoke:main|main}} The Arruda–Boyce model is based on the statistical mechanics of polymer chains. In this approach, each macromolecule is described as a chain of $N$ segments, each of length $l$ . If we assume that the initial configuration of a chain can be described by a random walk, then the initial chain length is

$r_{0}=l{\sqrt {N}}$ If we assume that one end of the chain is at the origin, then the probability that a block of size $dx_{1}dx_{2}dx_{3}$ around the origin will contain the other end of the chain, $(x_{1},x_{2},x_{3})$ , assuming a Gaussian probability density function, is

$p(x_{1},x_{2},x_{3})={\cfrac {b^{3}}{\pi ^{3/2}}}~\exp[-b^{2}(x_{1}^{2}+x_{2}^{2}+x_{3}^{2})]~;~~b:={\sqrt {\cfrac {3}{2Nl^{2}}}}$ The configurational entropy of a single chain from Boltzmann statistical mechanics is

$s=c-k_{B}b^{2}r^{2}$ $\Delta S=-{\tfrac {1}{2}}nk_{B}(\lambda _{1}^{2}+\lambda _{2}^{2}+\lambda _{3}^{2}-3)=-{\tfrac {1}{2}}nk_{B}(I_{1}-3)$ where an affine deformation has been assumed. Therefore the strain energy of the deformed network is

$W=-\theta \,dS={\tfrac {1}{2}}nk_{B}\theta (I_{1}-3)$ ## Notes and references

1. Arruda, E. M. and Boyce, M. C., 1993, A three-dimensional model for the large stretch behavior of rubber elastic materials,, J. Mech. Phys. Solids, 41(2), pp. 389–412.
2. Bergstrom, J. S. and Boyce, M. C., 2001, Deformation of Elastomeric Networks: Relation between Molecular Level Deformation and Classical Statistical Mechanics Models of Rubber Elasticity, Macromolecules, 34 (3), pp 614–626, Template:Hide in printTemplate:Only in print.
3. Horgan, C.O. and Saccomandi, G., 2002, A molecular-statistical basis for the Gent constitutive model of rubber elasticity, Journal of Elasticity, 68(1), pp. 167–176.
4. Hiermaier, S. J., 2008, Structures under Crash and Impact, Springer.
5. Kaliske, M. and Rothert, H., 1997, On the finite element implementation of rubber-like materials at finite strains, Engineering Computations, 14(2), pp. 216–232.
6. Ogden, R. W., 1984, Non-linear elastic deformations, Dover.