Slight asymmetry in the winding angles of reinforcing collagen can cause large shear stresses in arteries and even induce buckling

Abstract

Many models of the mechanical response of arteries assume a reinforcement with two families of helically wound fibres of collagen of opposite pitch. Motivated by experimental observations, the consequences for the internal pressurisation of arteries of a slight asymmetry in the winding angles is investigated here. It is shown that a torsional shear stress is generated as a result of this flaw, with some common models of the mechanical response of arteries exhibiting significant shear stresses. If the shear stress is significant, then the corresponding model would not seem to be robust, given that an infinitesimal change in a model parameter results in a large change in system response, although it is also shown that there is a ‘magic-angle’ for fibre winding that eliminates torsional shear stress for many of the commonly used models. Finite Element simulations are used to further illustrate the main consequences of fibre asymmetry for some of the more common models of arterial response. If the fibre asymmetry is localised in a region, then simulations show that there is the possibility of significant bending of the artery centred in this region at physiological blood pressure.

Appendix: Derivation of (15)

To the first order in $\epsilon$, the infinitesimal imbalance in the fibres,

$$\begin{aligned} W_k\left( I_1,I_2,I_4,I_5,I_6,I_7, I_8\right) &= W_k\left( I_1,I_2,I_4,I_5,I_4,I_5, S^2-\frac{r^2}{R^2}C^2 \right) \nonumber \\&+W_{k6}\left( I_1,I_2,I_4,I_5,I_4,I_5, S^2-\frac{r^2}{R^2}C^2 \right) \left( I_6-I_4\right) +W_{k7}\left( I_1,I_2,I_4,I_5,I_4,I_5, S^2-\frac{r^2}{R^2}C^2 \right) \left( I_7-I_5\right) \nonumber \\ & W_{k8}\left( I_1,I_2,I_4,I_5,I_4,I_5, S^2-\frac{r^2}{R^2}C^2\right) \left( I_8-S^2+\frac{r^2}{R^2}C^2\right) \end{aligned}$$

(28)

Let the ‘0’ superscript denote evaluation of the appropriate partial derivative at $\left( I_1,I_2,I_4,I_5,I_6,I_7, I_8\right) = \left( I_1,I_2,I_4,I_5,I_4,I_5, S^2-\frac{r^2}{R^2}C^2 \right)$. Then this truncated Taylor series expansion can be re-written more succinctly in the form

$$\begin{aligned} W_k\left( I_1,I_2,I_4,I_5,I_6,I_7, I_8\right)= W_k^0 +W_{k6}^0 2\epsilon SC\left( 1-\frac{r^2}{R^2}\right) +W_{k7}^0 2\epsilon SC\left( 1-\frac{r^4}{R^4}\right) + W_{k8}^0 \epsilon SC\left( 1+\frac{r^2}{R^2}\right) . \end{aligned}$$

(29)

If $W_k^1 \equiv W_{k6}^0 2SC\left( 1-\frac{r^2}{R^2}\right) +W_{k7}^0 2 SC\left( 1-\frac{r^4}{R^4}\right) + W_{k8}^0SC\left( 1+\frac{r^2}{R^2}\right)$, then (29) can be re-written in the form

$$\begin{aligned} W_k\left( I_1,I_2,I_4,I_5,I_6,I_7, I_8\right) = W_k^0+\epsilon W_k^1, \end{aligned}$$

which is (15).