Correction to: The effect of residual stress on the stability of a circular cylindrical tube

1 Correction to: J Eng Math (2021) 127:9 https://doi.org/10.1007/s10665-021-10097-4

In this note we wish to correct an error in our paper ‘The effect of residual stress on the stability of a circular cylindrical tube’, which appeared in the Journal of Engineering Mathematics 127, 9 (2021). The code used in the computation of Figs. 7 and 8 in [1] had a small error in computing the values of $\bar{\nu }$ and $\bar{a}$ for which bifurcation is possible. This affected the accuracy of the figures, which are replaced with Fig. 1 herein. The difference is not noticeable for a thin walled tube with $\bar{B}=1.1$, but becomes more evident with increasing wall thickness. The panels in the first and second columns of Fig. 1 correspond to Figs. 7 and 8 in [1], respectively.

The discussion of the results in [1] remains largely valid except that now there are no bifurcation curves in the region where $\bar{a} \ge 1$. Figure 1 shows that all bifurcation curves in the strongly elliptic region are in the region defined by $\bar{a} <1$, which is consistent with the situation where there is no residual stress [2]. In particular, within the strongly elliptic region bifurcation for an unloaded cylinder is not possible.

We also clarify that Eqs. (75) and (76) in [1] show the explicit forms of the first derivative of $\alpha$ and the first and second derivatives of $\gamma$ for the case $\kappa =0$ only. The corresponding expressions for $\bar{\kappa } \ne 0$ were not included, but are easily obtained.

Fig. 1

Plots of $\bar{\nu }$ versus $\bar{a}$ for which bifurcation is possible for $\lambda _z=1$. The mode numbers n are indicated in each panel by their adjacent numerical values. The panels a, c, e in the left column correspond to Fig. 7 in [1], with $\kappa =0$, and panels b, d, f in the right column correspond to Fig. 8 in [1], with $\bar{\kappa }=0.5$. Panels a and b are for $\bar{B}=1.1$, c and d for $\bar{B}=1.2$, e and f for $\bar{B}=1.5$. For $\kappa =0$ the region of strong ellipticity lies between the two horizontal red lines, while for $\bar{\kappa }=0.5$ the region is between the two red curves