Table B1.

Comparison of parameters μ̃, φ̃, ζ̃ in transversely isotropic material models in the limit of infinitesimal strain relative to the undeformed reference configuration.

Reference	μ̃	φ̃	ζ̃	ψ
Spencer (1984)a,b	μT	$\frac{{\mu }_L}{{\mu }_T}-1$	$\frac{\beta }{4{\mu }_T}+\left(\frac{{\mu }_L}{{\mu }_T}-1\right)$	${\mu }_T\text{tr}({\mathbf{\epsilon }}^2)+2({\mu }_L+{\mu }_T)\mathbf{A}·{\mathbf{\epsilon }}^2·\mathbf{A}+\frac{\beta }{2}{(\mathbf{A}·\mathbf{\epsilon }·\mathbf{A})}^2(\text{inc}.)$
Qiu and Pence (1997)	μ	0	γ	$\frac{\mu }{2}({\overline{I}}_1-3)+\frac{\mu \gamma }{2}{\left(I_4-1\right)}^2(\text{inc}.)$
Merodio and Ogden (2005)	μ	0	4γ	$\frac{\mu }{2}({\overline{I}}_1-3)+\frac{\mu \gamma }{2}{(I_5-1)}^2(\text{inc}.)$
Velardi et al. (2006)	μ	0	$\frac{3}{4}k$	$\frac{2\mu }{2}({\lambda }_1^{\alpha }+{\lambda }_2^{\alpha }+{\lambda }_3^{\alpha }-3)+\frac{2k\mu }{{\alpha }^2}(I_4^{\alpha /2}+I_4^{-\alpha /4}-3)(\text{inc}.)$
Ning et al. (2006)	2C10	0	$\frac{\theta }{2C_{10}}$	$C_{10}({\overline{I}}_1-3)+\frac{\theta }{2}{({\overline{I}}_4-1)}^2+\frac{1}{D}{(J-1)}^2$
Gasser et al. (2006)c	μ	0	$\frac{{(1-3\kappa )}^2{k}_1}{\mu }$	$\frac{\mu }{2}({\overline{I}}_1-3)+\frac{k_1}{2k_2}\left[\mathit{exp}\{k_2{[\kappa {\overline{I}}_1+(1-3\kappa ){\overline{I}}_4-1]}^2\}-1\right](\text{inc}.)$
Chatelin et al. (2012)d	2(C10 + C01)	0	$\frac{C_3{C}_4}{6(C_{10}+C_{01})}$	$C_{10}({\overline{I}}_1-3)+C_{01}({\overline{I}}_2-3)+W_{\mathit{fibers}}^d({\overline{I}}_4)(\text{inc}.)$
Current example	μ	φ	ζ	$\frac{\mu }{2}\left[({\overline{I}}_1-3)+\zeta {({\overline{I}}_4-1)}^2+\varphi {\overline{I}}_5^{\ast }\right]+\frac{\kappa }{2}{(J-1)}^2$

^aε is the infinitesimal strain tensor; A is the fiber direction unit vector. This model is explicitly limited to small strains.

^bSpencer (1984) uses a stiffness matrix that relates the shear stress to the true strain vector, which introduces a factor of 2 in the elements of the stiffness matrix that govern shear in his Eq. (13).

^cFor the case of a single fiber family with a mean orientation in the e₁ direction, if the dispersion parameter κ equals zero, all fibers are aligned in the e₁ direction. When κ = 1/3, the fibers are isotropically distributed. In the exponential term the quantity (1–3 κ)I₄ is evaluated only if I₄≥1.

^d

The anisotropic strain energy term is defined by

$$\stackrel{\sim }{\lambda }\frac{\partial W_{\mathit{fibers}}^d}{\partial \stackrel{\sim }{\lambda }}=\{\begin{array}{ll}0,\hfill & 0<\stackrel{\sim }{\lambda }<1\hfill \\ C_3[\mathit{exp}\{C_4(\stackrel{\sim }{\lambda }-1)\}-1],\hfill & \stackrel{\sim }{\lambda }\ge 1\hfill \end{array}$$

where λ̃ is the stretch ratio in the fiber direction.