Sliding of cylindrical shell into a rigid hole

TL;DR

This work addresses how an open, naturally curved cylindrical shell slides into a rigid circular hole under friction, a canonical snap-fit scenario. The authors develop a centerline elastica model with Amontons–Coulomb friction, validated by experiments and centerline-based simulations, to predict deformation and force response. They identify three sliding modes—Folding, Pinning, and Unfolding—and map their phase boundaries in terms of the dimensionless opening angles $\tilde{\Phi}$ and $\tilde{\Psi}$, providing a predictive framework for geometry–friction–elastic interactions in slender structures. The results offer a geometry-driven, predictive approach for contact-based elastic systems with friction, informing the design of snap-fit components and related soft-actuated devices.

Abstract

Fitting two different materials is a versatile methodology in manufacturing complex structures. One of the canonical models for fitting is the snap-fit model, in which flexible materials and rigid structures are assembled by pushing their interlocking components together. The assembly via snap-fit is often accompanied by large deformations of flexible structures and abrupt force drops, highlighting the role of elasticity, geometry, and contact friction. Despite several model studies revealing fundamental mechanics for snap-fit, the current snap-fit design relies on prototyping and empirical rules. In this paper, we analyze a snap-fit model in which a naturally curved beam slips into a rigid hole. We construct an analytical model based on the theory of elastica with contact friction and demonstrate that its predictions are in excellent quantitative agreement with both simulations and experiments. We find three distinct sliding modes: Folding, Pinning, and Unfolding. The classification is systematically organized in a phase diagram based on the geometric parameters of the shells and the hole. Our study complements existing approaches by providing a predictive framework for contact-based structures that involve friction, elasticity, and geometry, and sheds light on a unified understanding of the interactions between an elastic and a rigid body.

Figures (4)

• Figure 1: An elastic strip with spontaneous curvature is inserted into a hole in a rigid body. (a) Experimental setup. The naturally-curved elastic strip of total length, $2L$, and radius of curvature, $R$, is pressed by a rigid indenter from above by $\Delta_y$ and is slid into the rigid hole underneath. (b)–(d) Snapshots of three different sliding modes obtained from the experiments for three characteristic configurations: (b) Folding, (c) Pinning, and (d) Unfolding
• Figure 2: Rescaled force-displacement curve measured in experiments(solid line) and simulations($\bigcirc$). The dashed lines represent the linear response theory. (a) Folding, $(\Phi, \Psi) \approx (0.80\pi, 0.15\pi)$, (b) Pinning, $(\Phi, \Psi) \approx (0.6\pi, 0.15\pi)$, and (c) Unfolding, $(\Phi, \Psi) \approx (0.75\pi, 0.15\pi)$. In both experiments and simulations ($\mu_{\mathrm i}=\mu_{\mathrm e}=0.2$), cross symbols ($\times$) indicate the slip onset at the tip, $s = L$, corresponding to the condition such that the friction force at the indenter reaches the maximum as $|\mathcal{F}_{L,x}/\mathcal{F}_{L, y}| = \mu_{\mathrm i}$. The simulation results for different friction coefficients of $\mu_{\rm i}~(0.18\leq\mu_{\rm i}\leq0.22)$ are shown in (a) and (c). (a) The $\times$ marks represent the corresponding slip onset, while (c) the corresponding force displacement curves are shown as the upper and lower bounds of the shaded region. The dashed lines represent the prediction from the linear response theory.
• Figure 3: (a)Phase diagram constructed from theory, simulations, and experiment summarized on the $(\tilde{\Psi},\tilde{\Phi})$ parameter space. The background color and phase boundaries are predicted from linear-response theoretical analysis. Analytical phase boundaries are indicated by different line styles: solid and chain lines for slip in the closing and opening directions at the indenter as $\mathcal{F}_{L,x}/\mathcal{F}_{L,y} = \pm \mu_{\mathrm i}$, respectively, and dotted lines for slipping at the the edge of the hole as $\mathcal{F}_{s_c,t}/\mathcal{F}_{s_c,n} = \mu_{\mathrm e}$. Both theoretical and simulation results are obtained with the friction coefficients $\mu_{\mathrm i} = \mu_{\mathrm e} = 0.2$. The corresponding filled and empty symbols represent the experimental and simulation results, respectively. (b) Coordinate system and variable definitions for theoretical analysis. Due to the left-right symmetry of the natural shape, half of the elastic strip is analyzed.
• Figure 4: The effect of friction coefficients on the sliding behavior of the strip based on linear response theory. The force ratio at (a) $s = L$ and (b) $s = s_c$ predicted from the linear response as a function of $(\tilde{\Psi},\tilde{\Phi})$. (a) The intersections of the surface with the planes $\eta_{\mathrm i}=\pm\mu_{\mathrm i}=\pm0.2$ correspond to the slip conditions at the indenter (either opening or closing the tip, respectively). (b) The intersection with the plane $\eta_{\mathrm e}=\mu_{\mathrm e}=0.2$ describes the slip condition at the edge of the hole. (c)–(e) Phase diagrams constructed from linear response theory for different friction coefficients: (c) $\mu_{\mathrm e}=0.3,~\mu_{\mathrm i} = 0.1$, (d) $\mu_{\mathrm e}=0.3,~\mu_{\mathrm i}= 0.3$, and (e) $\mu_{\mathrm e}=0.3,~\mu_{\mathrm i}= 0.5$.