Arruda-Boyce model

The Arruda-Boyce hyperelastic constitutive model.¹

Parameters

• The bulk modulus $\kappa$.
• The shear modulus $\mu$.
• The number of links $N_b$.

External variables

• The deformation gradient $\mathbf{F}$.

Internal variables

• None.

Notes

• The nondimensional end-to-end length per link of the chains is $\gamma =\sqrt{\mathrm{tr}({\mathbf{B}}^{\ast })/3N_b}$.
• The nondimensional force is given by the inverse Langevin function as $\eta ={\mathcal{L}}^{-1}(\gamma )$.
• The initial values are given by ${\gamma }_0=\sqrt{1/3N_b}$ and ${\eta }_0={\mathcal{L}}^{-1}({\gamma }_0)$.
• The Arruda-Boyce model reduces to the Neo-Hookean model when $N_b\to \infty$.

Helmholtz free energy density

$$a(\mathbf{F})=\frac{3\mu N_b{\gamma }_0}{{\eta }_0}\left[\gamma \eta -{\gamma }_0{\eta }_0-\ln \left(\frac{{\eta }_0\sinh \eta }{\eta \sinh {\eta }_0}\right)\right]+\frac{\kappa }{2}\left[\frac{1}{2}\left(J^2-1\right)-\ln J\right]$$

Cauchy stress

$$\mathbf{\sigma }(\mathbf{F})=\frac{\mu {\gamma }_0\eta }{J\gamma {\eta }_0}{{\mathbf{B}}^{\ast }}^{\prime }+\frac{\kappa }{2}\left(J-\frac{1}{J}\right)1$$

Cauchy tangent stiffness

$$\begin{array}{rl}{\mathcal{T}}_{ijkL}(\mathbf{F})=& \frac{\mu {\gamma }_0\eta }{J^{5/3}\gamma {\eta }_0}\left({\delta }_{ik}{F}_{jL}+{\delta }_{jk}{F}_{iL}-\frac{2}{3}{\delta }_{ij}{F}_{kL}-\frac{5}{3}{B}_{ij}^{\prime }{F}_{kL}^{-T}\right)\\ & +\frac{\mu {\gamma }_0\eta }{3J^{7/3}{N}_b{\gamma }^2{\eta }_0}\left(\frac{1}{\eta {\mathcal{L}}^{\prime }(\eta )}-\frac{1}{\gamma }\right)B_{ij}^{\prime }{B}_{km}^{\prime }{F}_{mL}^{-T}+\frac{\kappa }{2}\left(J+\frac{1}{J}\right){\delta }_{ij}{F}_{kL}^{-T}\end{array}$$

1. E.M. Arruda and M.C. Boyce, J. Mech. Phys. Solids 41, 389 (1993). ↩