At least three invariants are necessary to model the mechanical response of incompressible, transversely isotropic materials

Abstract

The modelling of off-axis simple tension experiments on transversely isotropic nonlinearly elastic materials is considered. A testing protocol is proposed where normal force is applied to one edge of a rectangular specimen with the opposite edge allowed to move laterally but constrained so that no vertical displacement is allowed. Numerical simulations suggest that this deformation is likely to remain substantially homogeneous throughout the specimen for moderate deformations. It is therefore further proposed that such tests can be modelled adequately as a homogenous deformation consisting of a triaxial stretch accompanied by a simple shear. Thus the proposed test should be a viable alternative to the standard biaxial tests currently used as material characterisation tests for transversely isotropic materials in general and, in particular, for soft, biological tissue. A consequence of the analysis is a kinematical universal relation for off-axis testing that results when the strain-energy function is assumed to be a function of only one isotropic and one anisotropic invariant, as is typically the case. The universal relation provides a simple test of this assumption, which is usually made for mathematical convenience. Numerical simulations also suggest that this universal relation is unlikely to agree with experimental data and therefore that at least three invariants are necessary to fully capture the mechanical response of transversely isotropic materials.

Appendix

1.1 Numerical results

The following procedure was adopted to calculate the kinematical quantities $\lambda _{1},\lambda _{2},\kappa$ from the numerical results. The $x_{1}$ coordinates of the mid-points of the inclined faces were isolated from the rest of the output data at specified values of the prescribed axial stretch; call them $x_{1}^{l},x_{1}^{r}$, using an obvious notation. Referring to Fig. 3, let the origin coincide the bottom left corner of the undeformed block. Since the dimensions of the block were chosen to be $20\,\times \, 20\, \times \, 2\,\text{ mm }$, it follows from (9) that

$$\begin{aligned} x_1^l=\kappa \lambda _2 10, \qquad x_1^r=\lambda _1 20+ \kappa \lambda _2 10. \end{aligned}$$

Since $\lambda _{2}$ is controlled, $\kappa , \lambda _{1}$ are therefore obtained from

$$\begin{aligned} \kappa = \frac{x_1^l}{10\lambda _2}, \qquad \lambda _1=\frac{x_1^r-x_1^l}{20}. \end{aligned}$$

These calculated quantities to four decimal places are tabulated in Table 1.

Table 1 Kinematics for numerical experiments with a $45\,^{\circ }$ fibre angle

1.2 Comparative plots

Numerical results for a fibre angle of $45\,^{\circ }$ were presented in the main body of the paper. The simulations for this natural choice of fibre angle are supplemented below for an angle close to the horizontal and another close to the vertical. For the material parameters used here, it is clear from Figs. 8 and 9 that shear is negligible for a $20\,^{\circ }$ angle and much more pronounced for $80\,^{\circ }$. A comparison of these graphics with Fig. 3 shows that the amount of shear for $45\,^{\circ }$ is between these two limiting cases.

Fig. 8

Initial and final configurations with $\lambda _2=1.2$ for $20\,^{\circ }$ initial fibre orientation

Fig. 9

Initial and final configurations with $\lambda _2=1.2$ for $80\,^{\circ }$ initial fibre orientation