Curvature Regularization near Contacts with Stretched Elastic Tubes

Inserting a rigid object into a soft elastic tube produces conformal contact between the two, resulting in contact lines. The curvature of the tube walls near these contact lines is often large and is typically regularized by the finite bending rigidity of the tube. Here, it is demonstrated using experiments and a Föppl–von Kármán–like theory that a second, independent, mechanism of curvature regularization occurs when the tube is axially stretched. In contrast with the effects of finite bending rigidity, the radius of curvature obtained increases with the applied stretching force and decreases with sheet thickness. The dependence of the curvature on a suitably rescaled stretching force is found to be universal, independent of the shape of the intruder, and results from an interplay between the longitudinal stresses due to the applied stretch and hoop stresses characteristic of curved geometry. These results suggest that curvature measurements can be used to infer the mechanical properties of stretched tubular structures.

Figure 1

(a) Sketch of the setup showing an axisymmetric intruder in a tube (solid lines) that is cylindrical (radius $a$) in its undeformed reference configuration (dashed lines). The tube is stretched by applying equal and opposite forces per length of magnitude $f=F/(2\pi a)$ at the tube ends. The arclength coordinate $s$ in the undeformed configuration is a material label. The intersection of the undeformed tube and the intruder defines a geometric intersection length $2z_i$ while the region of contact (highlighted in green) has axial extent $2z_c$. (b)–(e) Experimental pictures of the shape of the tube of radius $a=3.25\mathrm{mm}$ around an intruder of radius $R_0=6.35\mathrm{mm}$ for increasing values of the applied force $F$. Scale bar is 10 mm.

Figure 2

(a) Experimental profiles obtained with a spherical intruders of dimensional radius $R_0/a=2.44$. The dimensionless force $\mathcal{F}$ is coded in color. (b) Exponential decay of the radial displacement $u_r-u_r^{\infty }$ with distance from the contact point $S-S_c$ for increasing dimensionless stretching forces. Solid lines are experiments for $\mathcal{F}=0.13$, 0.22, 0.27, 0.32, 0.38, 0.46, and 0.55, and symbols are theoretical predictions for $\mathcal{F}=0.1$, 0.2, 0.25, 0.3, 0.4, 0.55, and 0.8.

Figure 3

(a) Theoretical prediction of the tube shape for different stretches $\mathcal{F}$ with $\nu =1/2$ and a spherical intruder with $R_0/a=1.5$. As $\mathcal{F}$ increases, the tube relaxes more gradually to a cylindrical shape, while the size of the contact region shrinks. The curvature at the contact point diverges as ${\mathcal{F}}^{-1/2}$. The dashed line indicates the undeformed shape of the tube. (b) Theoretical prediction (in dashed blue lines) and experimental tube shape (in solid black lines) for $R_0/a=1.95$ and a small ($\mathcal{F}=0.001$) and large ($\mathcal{F}=0.5$) dimensionless stretching force. (c) Deviation of the contact length $z_c$ from its no-stretch value $z_i$ as a function of the applied stretching force $\mathcal{F}$. Symbols are numerical results for spherical intruders of different radii [$R_0=1.1$ (circles); $R_0=1.5$ (triangles); $R_0=2.0$ (squares)] and with different Poisson’s ratio [$\nu =0.5$ (filled); $\nu =0$ (open)].

Figure 4

Decay exponent $\lambda$ vs $\mathcal{F}$ for spherical intruders with radius $R_0/a=1.46$ (squares), 1.95 (circles), and 2.44 (diamonds), (axisymmetric) egg-shaped intruders ($▵$) with a half-width $w/a=1.96$ and (asymmetric) almond-shaped intruders ($▿$) with $w/a=2.35$. Numerical results for spheres (solid curves; $\nu =1/2$, $R_0$ matched with experiments) are insensitive to changes in intruder shape and size, and are in good agreement with the data. Both the experimental and the numerical results collapse onto the theoretical result (5) [dashed curve]. The horizontal dotted line at $\lambda ={(\sqrt{2}\beta )}^{-1}\approx 3.9$ corresponds to the bending-dominated limit with $\mathcal{F}\to 0$.