Indentation of an elastic arch on a frictional substrate: Pinning, unfolding and snapping

TL;DR

This work analyzes a center-loaded, naturally curved elastic arch resting on a frictional substrate. It combines systematic numerical simulations with a linear planar-elastica theory under Amontons–Coulomb friction to derive a geometry-dependent indentation compliance and a predictive phase diagram distinguishing unfolding, folding, and pinning configurations; it identifies universal geometric thresholds $\Phi_c$ and $\mu_c$ (approximately $142^{\circ}$ and $0.424$, respectively). In the large-indentation regime, the arch undergoes a snapping transition with a discontinuous force drop, revealing nonlinear frictional mechanics beyond the linear theory. The findings provide a basis for understanding curvature-friction-elasticity interactions in slender structures and inform design ideas for energy-absorbing devices and complex shells.

Abstract

We investigate the morphology and mechanics of a naturally curved elastic arch loaded at its center and frictionally supported at both ends on a flat, rigid substrate. Through systematic numerical simulations, we classify the observed behaviors of the arch into three distinct types of configurations in terms of the arch geometry and the coefficient of static friction with the substrate. A linear theory is developed based on a planar elastica model combined with Amontons-Coulomb's frictional law, which quantitatively explains the numerically constructed phase diagram. The snapping transition of a loaded arch in a sufficiently large indentation regime, which involves a discontinuous force jump, is numerically observed. The proposed model problem allows a fully analytical investigation and demonstrates a rich variety of mechanical behaviors owing to the interplay between elasticity, geometry, and friction. This study provides a basis for understanding more common but complex systems, such as a cylindrical shell subjected to a concentrated load and simultaneously supported by frictional contact with surrounding objects.

Figures (5)

• Figure 1: Naturally curved elastic strip, which we call "arch," loaded at its center on a frictional substrate. (a) Definition of the coordinate system and variables in our theoretical analysis. (b)--(d) Photographs of the three characteristic configurations, i.e., unfolding, folding, and pinning, of a naturally curved plastic film ribbon, for illustration purposes only.
• Figure 2: Phase diagram, representative force--displacement curves, and configurations. (a) Numerically constructed phase diagram of the ($\Phi, \mu$) space. The blue, green, and red symbols represent the (i) unfolding, (ii) folding, and (iii) pinning phases, respectively. The solid lines represent our analytical predictions; see the main text. (b) Rescaled indentation force $PL^2/B$ as a function of the rescaled vertical displacement at the center of the arch, $\Delta_y/L$, for $\mu=0.2$. The inset displays the same data focusing on the small indentation regime, $\Delta_y/L < 0.05$. Note that the vertical axis in the inset shows $PL^2/B\Phi^2$ because our linear theory is developed under the basic assumption given by $PL^2/(B\Phi^2)\ll 1$ (see the main text). (c) Representative snapshots of an arch for increasing $\Delta_y/L$, obtained from the numerical simulations; (1)--(4) correspond to those shown in the horizontal axis in (b). The corresponding locations of (i)--(iii) on the phase diagram are also shown in (a).
• Figure 3: Analytical results of our linear theory for various values of the boundary stiffness $k$. (a) Normalized effective stiffness $K_{\rm el}/K_{\rm pin}$ plotted as a function of the opening angle $\Phi$ for various $k$ [indicated in the legend of (b)], based on Eq. (\ref{['eq_K_el']}). (b) Ratio of the vertical ($\Delta_y$) to horizontal ($\Delta_x$) displacements plotted as a function of $\Phi$ for various $k$ (indicated in the legend), based on Eq. (\ref{['eq_DelxDely']}).
• Figure 4: Snapping behavior and underlying friction mechanism along with the arch configurations. (a) Rescaled indentation force $PL^2/B$ plotted as a function of the rescaled vertical displacement $\Delta_y/L$ for $\Phi=110^{\circ}$ and for $\mu=0.2$ (red, unfolding) and $\mu=0.4$ (blue, pinning). The insets show several representative configurations from the numerical simulations. (b) Plot of $|F_x/F_y|$ as a function of $\Delta_y/L$ for the two cases shown in (a). The snap point indicated in the data for $\mu=0.4$ corresponds to the snap point of the significant force drop in (a). The dotted line is the prediction for the hinge--hinge (i.e., "completely pinned") boundary condition.
• Figure 5: Numerical simulation results on the hysteretic force response during a cycle process consisting of the forward (pushing) and backward (relaxing) indentations. Rescaled indentation force $PL^2/B$ plotted as a function of the rescaled vertical displacement $\Delta_y/L$ for $\Phi=110^{\circ}$, and (a) for $\mu=0.2$ (unfolding phase), and (b) $\mu=0.4$ (pinning phase). The dashed and solid lines represent the predictions from our linear theory. See Eq. (\ref{['eq_Kpm']}). (c) Schematics of off-center indentation geometry. A point of indentation is at a distance $e$ away from the center of the arch. (d) $PL^2/B$ plotted as a function of $\Delta_y/L$ observed in the off-center indentation of $e/R=0.077$, $\Phi=110^{\circ}$, and $\mu=0.4$, together with the representative configurations of the arch.