Contact Surface of the Elastic Orthogonal Clasp

Fig. 1A presents the experimental setup that we designed to study elastic orthogonal clasps systematically. We clamped two homogeneous elastic rods, rod A and rod B, to two rigid walls, the distance between which (H) is varied to bring the rods into contact. The two rods have equal rest diameters D, respective rest lengths LA and LB, and their extremities are clamped at distances WA and WB apart. As the imposed wall-to-wall distance H is varied, the overall geometry of the elastic clasp changes. Each wall applies a total vertical force of 2Fz to the elastic clasp (Fz per extremity). We define the normalized vertical force fz=Fz/EA, where E is Young’s modulus of the material and A=πD2/4 is the unloaded rod cross-section area. The experiments enabled us to investigate the contact geometry between the touching rods by analyzing the μCT volumetric images of tomographically scanned elastic clasps. To enable such a three-dimensional (3D) image analysis, we customized a rod-fabrication protocol (detailed in SI Appendix, section 1.A) to produce (coaxial) composite rods. These rod designs, together with μCT, enabled the precise determination of the material centerlines of each rod, along with the contact-surface geometry between the rods.

In Fig. 1B, we show a representative example of a rendered 3D μCT image of an elastic clasp, along with rod centerlines and contact surface (Materials and Methods, SI Appendix, sections 1.B and 1.C, and Movies S1 and S2). The saddle-shaped contact surface, physically hidden in between the two rods, is the primary object of our study. In parallel to our experimental investigation, we conducted full 3D simulations using the FEM to extract quantities that were not readily available from the experiment, with particular focus on the contact pressure between the rods (color bar in Fig. 1C). In SI Appendix, section 3, we detail our validation procedure of the FEM numerics using the force-displacement curves, Fz(H), for the clasp configuration presented Fig. 1A (see also Movie S3). Beyond the precise localization of the saddle-shaped contact surface, our computational FEM procedure reveals a highly heterogeneous pressure field between the two rods, which we will analyze in detail. One of the central goals of our study is to obtain a qualitative rationalization of the features observed in this pressure field using the highly simplified, purely geometric theory of idealized orthogonal clasps.

The Geometrical Theory of Idealized Orthogonal Clasps

The geometrical problem of predicting contact sets between tightly interwound filaments subject to a nonpenetration constraint has been addressed previously in the context of ideal knot shapes—for example, refs. 27 and 28—where it was observed that it is common for double-contact lines to arise. Ideal shapes involve filaments that, by assumption, have undeformable, circular, orthogonal cross-sections of finite radius. Typically, ideal knot problems are formulated as a purely geometrical problem with no mechanics, which corresponds to their centerlines being considered as inextensible, yet perfectly flexible, so that they can support no bending moment. Starostin (29) first considered the particular case of the ideal shape of an orthogonal clasp, but only for the fourfold symmetric case, where the two components are congruent. In our notation, Starostin assumed that the two local opening angles (defined in Fig. 1D) were equal θA=θB. In Materials and Methods and SI Appendix, section 4.D, we present a self-contained variant of Starostin’s analysis that allows for an explicit computation of the contact set (e.g., Fig. 1D), including the cases θA ≠ θB, via the numerical solution of a set of ordinary differential equations (see additional computation results in Movie S4). For each 0≤θA,θB<π, the contact set is a closed, curvilinear, diamond-shaped line, which stands in contrast to the surface-patch contact set observed in both the experimental μCT and numerical FEM data for the elastic orthogonal clasp (where cross-sections are deformable; Fig. 1 B and C). A significant part of our study seeks to explain the connection between the contact patches observed for elastic orthogonal clasps and the contact lines predicted by the ideal orthogonal clasp model.

Double-contact lines arise in ideal knot shapes when, at arc-length s1 along the tube centerline, the corresponding circular arc in the tube surface has two contact points with other surface arcs, each of which correspond to another two distinct centerline arc-lengths, s2 and s3, say. The straight line segment connecting two points on the knot centerline with such pairs of arc lengths (s1,s2), or (s1,s3), is referred to as a contact chord (cf. SI Appendix, Fig. S15). This double-contact feature is found in the present case of the ideal orthogonal clasp, where the phenomenon is perhaps less surprising due to the presence of two planes of reflectional symmetry. Moreover, due to the reflection symmetry, the set of all contact chords decomposes into a one-parameter family of closed equilateral tetrahedra. It is the one-parameter family of midpoints of these four tetrahedral edges that traces out the curvilinear, diamond-shaped contact line. The corners of the diamond arise when the family of open tetrahedra collapses to two double-covered straight line segments. In turn, this coalescent limit arises at first and last contact points along the curve centerlines (or touch-down and lift-off points; cf. SI Appendix, section 4.D for more details). The diamond-shaped contact line surrounds the two tips of the clasp equilibria, and, consequently, there is always an enclosed gap, or physical space, separating the two tubes close to their tips, except in degenerate cases, where one or both tubes have vanishing curvature at their tip.

Heuristically, the connection between the contact lines of the ideal orthogonal clasp and the contact patches of the elastic orthogonal clasp can be explained, in both experiment and FEM, by cross-section deformation. Consequently, the idealized contact lines are, in reality, “fattened” to become surface patches. In general, the deformation of the cross-sections can be sufficiently large that the small gap between the tips of the ideal orthogonal clasp can close in the corresponding elastic orthogonal clasp. Nevertheless, in most cases, we do observe an overall resemblance between the shape of the ideal contact line and the boundary of the elastic clasp surface patch, as in Fig. 1 B–D (see Movie S1 for a full 3D view of the contact region). In our consideration of the four cases illustrated in Fig. 2, we observed one elastic clasp configuration in which the tip gap did persist (Fig. 3B2). Furthermore, we will show below that, in many cases, the pressure distribution in our FEM simulations of the elastic orthogonal clasp are highly concentrated on the diamond-shaped contact lines in the corresponding ideal orthogonal clasp and that these high pressures arise toward the boundary of the contact patch, with comparatively low contact pressures close to the central tip regions.

Local Opening Angles of the Elastic Clasp

Next, we make a more detailed comparison between corresponding equilibria of the ideal and elastic orthogonal clasp configurations. To identify corresponding cases, we use the local opening angles θA and θB, defined as the angle described by the tangents to the centerlines of either the tube (for the ideal case) or rod (for the elastic case) at the touch-down and lift-off contact points (Fig. 1 B and D). Since these angles appear in both cases, we can take them as the common denominator to identify corresponding equilibria. In the ideal orthogonal clasp, these two local opening angles are the only model input parameters; the tube diameter and the magnitude of the tensions applied at the tube ends are scale factors. Moreover, because segments of ideal equilibria that are not in contact have straight centerlines, the local opening angles equal the constant, global opening angles between tail segments far from contact. However, this is no longer true in the elastic orthogonal clasp, where the presence of bending stiffness implies that the global angle between two tail regions is not constant, and, in particular, the global and local opening angles can differ significantly. In the elastic orthogonal clasp, the inputs are the applied vertical load (or the vertical imposed displacement) and the horizontal displacement boundary conditions enforced at the ends of both rods. For the elastic orthogonal clasp, the two local opening angles are observables, which can be measured experimentally from the μCT data or computed as part of the FEM simulation. As we will discuss further below, it is also the local opening angles that are of key importance when describing the frictional mechanics of touching filaments, because they reflect the extent of the contact region.

In Fig. 2 A–D, we present four different configurations of displacement boundary conditions that we employed to explore a range of local opening angles (the details of each configuration are provided in the legend of Fig. 2). The corresponding curves for θA and θB, as functions of the normalized vertical load fz, for these four configurations are provided in Fig. 2 E and F for experiment (datapoints) and simulation (solid lines; the small squiggles in the FEM data are due to the discrete nature of the underlying numerical mesh). We note that the local opening angles computed using FEM are in excellent agreement with the experimental measurement. As expected, in all four cases, the local opening angles of rods A and B decrease monotonically as fz increases; the clasp configurations become tighter.

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Fig. 2.

Local opening angles of the elastic orthogonal clasp, as a function of applied load, for four specific configurations of displacement boundary conditions. (A) The elastic rods of configuration (Conf.) 1 have the following normalized geometric attributes: rod A, lA=LA/D=20, wA=WA/D=2; rod B, lB=LB/D=7.5, wB=WB/D=7.5 (rod B is straight when no load is applied). (B) Configuration 2: rod A, lA=20, wA=2; rod B, lB=20, wB=2. (C) Configuration 3: rod A, lA=20, wA=2; rod B, lB=7.5, wB=2. (D) Configuration 4: rod A, lA=20, wA=14; rod B, lB=wB=7.5 (rod B is straight when load is applied). (Length of the lines representing the x, y, z directions in A–D: 10 mm.) (E and F) Local opening angles θA (E) and θB (F) plotted versus the normalized (Norm.) applied vertical force fz. The experimental local opening angles (data points, measured through μCT tomography) and the FEM simulations (solid lines) are in excellent agreement. In all cases, the local opening angles systematically decrease as the load fz increases. A simplified capstan computation of local opening angles derived from a single elastic rod wrapped around a rigid straight cylinder in the absence of friction (dashed lines) delivers rather poor predictions, which reflects the need for a more detailed model of the contact geometry.

Interestingly, the θA(fz) response is largely independent of the geometric conditions imposed on rod B and vice versa. Indeed, in configurations 1, 2, and 3, rod A has the same imposed boundary conditions; even though it is in contact with rods B of different lengths subject to different boundary conditions, it exhibits a nearly identical θA(fz) curve across these three configurations. Motivated by this observation, we sought to predict the local opening-angle response based on a simplified theory of a single Kirchhoff elastic rod wrapped around a rigid, right circular cylinder of diameter D. Full details of this simplified theory are provided in Materials and Methods and in SI Appendix, sections 4.A and 4.C. The Kirchhoff-based predictions, dashed lines in Fig. 2 E and F, show a significant mismatch with the measured local opening angles (experiments and FEM). We conclude that this simple theoretical approximation is inadequate, which will be consequential when we model the frictional sliding clasp below.

The standard Kirchhoff rod model is based on the assumptions (or approximations) that cross-sections are undeformable and unshearable (30). In the SI Appendix, section 2, we quantify the cross-section deformation, the centerline curvature, and the shear strain along the centerline of rod A, taking configuration 3 as a representative case. We observe that, even at the relatively low load fz=0.05, all of these quantities reach substantial values in the neighborhood of contact. First, the cross-section can deform significantly, flattening from its original circular shape by up to 10%. Secondly, the radius of the centerline curvature approaches the rod diameter. Finally, the shear strain reaches values up to 10%. Therefore, in the elastic clasp, the Kirchhoff rod approximations are not accurate. Nevertheless, as we demonstrate next, the alternative 1D geometric theory of the idealized orthogonal clasp, with its detailed treatment of the contact geometry, does provide a framework within which to successfully construct a qualitative explanation of the skeleton of the contact-pressure distribution that is observed for the 3D elastic orthogonal clasp.

Distributions of the Contact Pressure

We proceed by analyzing the pressure distribution in the contact region of elastic orthogonal clasps. By way of example, we focus on the three representative cases of imposed boundary conditions shown in Fig. 3, each at three different values of the normalized applied loads (fz={0.005, 0.05, 0.15}). The three configurations considered are: a pair of long rods (lA=lB=20 and wA=wB=2; Fig. 3A1), a pair of short rods (lA=lB=7.5 and wA=wB=2; Fig. 3B1), and a combination of the long and short rods (lA=20, lB=7.5 and wA=wB=2; Fig. 3C1), where the physical configurations shown correspond to the low applied load fz≃0.005. For each of the nine cases, the contact-pressure maps are computed by using FEM simulation and then compared with both the extent of the corresponding contact surface patch measured from the μCT experiments and the contact line predicted by the ideal orthogonal clasp analysis.

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Fig. 3.

Contact-pressure maps for nine different elastic orthogonal clasp equilibria. FEM-computed contact-pressure fields of elastic orthogonal clasps subject to three different sets of geometrical boundary conditions at three different values of the applied load: lA=lB=20 (A1), lA=lB=7.5 (B1), and lA=20, lB=7.5 (C1) (with the three physical configurations illustrated corresponding to the applied load fz=0.005). The pressure-field data for different values of normalized load fz={0.005, 0.05, 0.15} are presented in (A2, B2, C2), (A3, B3, C3), and (A4, B4, C4), respectively. Blue solid lines, ridges extracted from the FEM-computed pressure field; dashed lines, experimentally (Exp.) observed boundaries of the contact surfaces; black solid lines, prediction of contact curves from ideal orthogonal clasp theory.

The contact pressure is computed from the FEM data as the normal force per unit area on the contact surface patch embedded in 3D. However, for visualization purposes, this scalar field is plotted as a color map in the two-dimensional parameter space (sA=SA/D, sB=SB/D); the normalized arc lengths along the two rod centerlines provide a coordinate system for the contact surface patch (Fig. 3 A2–A4, B2–B4, and C2–C4). The point sA=sB=0 corresponds to the apices of the rods. With this parameterization, it is possible to overlay the boundaries of the contact region extracted from the μCT experiments of the elastic clasp configuration on the analogous FEM computations with the same local opening angles (dashed lines); excellent agreement is found between the two. In these plots, we also juxtapose the contact line predicted by the ideal orthogonal clasp theory (solid black lines in Fig. 3 A2–A4, B2–B4, and C2–C4). All three equilibria for the first two sets of boundary conditions exhibit fourfold symmetric contact-pressure maps (θA=θB). By contrast, there is only twofold symmetry for the third configuration (θA ≠ θB). The contact pressures are highly heterogeneous. For instance, Fig. 3C3 shows four pressure peaks near the entrance and exit regions of contact along the two rods, connected by ridges of high-pressure values (solid blue lines). To construct points along the ridge in the pressure map, we located 1D local maxima in pressure when evaluated along 720 radial straight lines emanating from the (0, 0) origin point.

Remarkably, for all three cases of moderate loading fz≃0.05 presented in Fig. 3, the pressure ridges closely follow the curvilinear, diamond-shaped contact line predicted by the ideal clasp geometric theory (solid black lines) for the same pair of local opening angles; the four peaks in pressure are close to the four corners of the diamond. In these three cases, the purely geometric ideal clasp model provides an accurate skeleton of the pressure map in the physical system. Due to the deformability of the rod cross-sections, the singular, linear pressure field (force per unit length) predicted from geometry is smoothed out to yield the ridges in the surface pressure (force per unit area) for the elastic orthogonal clasp. In the case of higher loading (fz≃0.15; Fig. 3 A4, B4, and C4), where there is significant deformation in the rod cross-sections, the surface pressure becomes more uniform over the interior of the region of contact, and the ridges are no longer observed.

The case of small loading (fz≃0.005; Fig. 3 A2, B2, and C2) reveals a variety of contact-pressure distributions depending on the global geometry of the equilibrium. Fig. 3A2 shows a highly localized pressure map, characterized by a single peak close to the apices sA=sB=0. However, in Fig. 3B2, both pressure ridges and a central region of vanishing pressure are observed, corresponding closely to the predictions of the ideal orthogonal clasp with its predicted gap between the tube surfaces close to the two centerline tips. A clarification of precisely which loading regimes give rise to a gap between the tip regions of the elastic orthogonal clasp would require further study.

The hybrid long/short rod configuration in Fig. 3C2 exhibits a twofold symmetric pressure corresponding to the asymmetry between rod A and rod B. There are two pressure peaks along rod B, separated by a saddle with low pressure between the apices. Even in this extreme case, the ideal clasp diamond contact line yields a reasonable prediction of the extent of the actual contact surface in the elastic orthogonal clasp.

The Elastic Clasp with Sliding Friction

Leveraging the physical understanding gained above for the elastic orthogonal clasp with negligible friction, we next impose a relative motion between two elastically deformable rods in a clasp configuration, with finite friction, to extend the problem to obtain the frictional sliding clasp. There is a direct analogy between this system and the capstan problem, in which one deformable filament is wrapped in a planar configuration around a rigid drum, or capstan. The classic Euler–Eytelwein version of this problem (21, 31, 32) assumes that the capstan is a rigid cylinder and that the filament is perfectly flexible, with a negligible radius of curvature compared to the radius of the capstan. Then, the maximal possible ratio of high tension T1 at one end to low tension T0 at the other end is predicted by the well-known capstan relation,T1T0=eμφ,[1]where φ is the winding angle around the capstan, and μ is the friction coefficient (either static or dynamic depending on context) between the filament and the drum.

Our frictional sliding clasp load-displacement experiment can be viewed as a generalized capstan problem, in which the static, rigid drum is replaced by a second deformable rod, thereby adding elastic deformation in a nontrivial way. Motivated by this analogy, we seek to measure the change in the tension between the two ends of a rod sliding in steady state while wrapped around a second static, but deformable, rod. In our experiments (Fig. 4A and Movie S5), rod A is guided by two frictionless pulleys to thread around the clamped rod B. The rods are surface-treated to ensure robust Amontons–Coulomb frictional behavior with μ=0.32±0.03 (see details in SI Appendix, section 1.F). We measured the vertical force T necessary to displace the upper extremity of rod A at a constant velocity subject to a dead load T0=Mg, applied at the other extremity (where M is the mass of dead weight and g is the gravitational acceleration). Upon pulling with a prescribed velocity, the measured tension T first increases and then reaches a plateau value T1 (Fig. 4C and Materials and Methods). We then quantified the steady tension ratio K=T1/T0 between the high- and low-tension strands, as a function of the nondimensional lower tension t0=T0/EA, for different geometric configurations. Specifically, we adopt the same four sets of geometric parameters imposed in the global boundary conditions as for the static elastic orthogonal clasp experiments, namely, configurations 1 to 4 described in Fig. 2 A–D), but with the presence of the pulleys instead of clamps for rod A (Fig. 4 B1–B4). In SI Appendix, section 4.B, we show that our long sliding rod (rod A) mainly carries an internal tensile force (i.e., parallel to the tangent of the rod centerline) far from the contact region with rod B.

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Fig. 4.

Friction-induced tension gain in the sliding elastic clasp. (A) Experimental apparatus used to measure the tension ratio in different geometric configurations (Conf.). (Scale bar: 20 mm.) (B1–B4) Adaptation of configurations 1, 2, 3, and 4 described in Fig. 2 to allow for the sliding of rod A around the static rod B. (C) Experimentally measured upward force T in the upper segment of the sliding rod A as a function of the upward displacement d of the upper tip. (D) Ratio K=T1/T0 between the steady-state value of T (plateau in C) and the value of the dead load T0 applied to the lower end of rod A, plotted as a function of the normalized dead load t0=T0/EA for the four different configurations. Experimental and FEM data are presented. The shaded regions correspond to the sensitivity of the FEM computation to the experimental uncertainty of the measured friction coefficient (μ=0.32±0.03). The horizontal dashed lines are the predictions from the capstan relation Eq. 1 applied to the global opening angle. (E) Representative contact-pressure map in the contact region obtained from the FEM simulation of configuration 3, at a normalized dead load of t0=0.05 (the pressure is normalized by EA/D2). (F) Linear pressure map of the ideal clasp, where it can be computed analytically that for a maximum tension growth equilibrium, the linear pressure vanishes at the touch-down point (black circle), but has a singular Delta function at the lift-off corner (represented by a black arrow).The qualitative features of the linear pressure field (force per unit length) shown in F match those of the pressure field (force per unit area) evaluated along its ridge lines, as shown in E.

In Fig. 4D, we plot the experimental tension ratio K (data points) as a function of the dimensionless lower tension t0 imposed by the dead load. We observe that all K(t0) increase monotonically with t0, albeit with significant differences for the various imposed geometrical configurations in either rod A or rod B. In parallel to the experiments, we carried out FEM numerical simulations of the steady-state frictional sliding elastic clasp (solid lines; see Materials and Methods, SI Appendix, section 3, and Movie S6 for details of the simulations), finding good agreement with the experimental data (mean difference of ≈4.2%). The experimental and numerical data highlight the importance of the global opening angle imposed on the sliding rod; i.e., the angle between the tangents to rod A in the nearly straight tail regions far from contact, which are set experimentally by the distance between the pulleys. Indeed, we observed that K(t0) is lower when the pulleys are far apart (configuration 4, orange triangles) than when they are close to each other (configuration 1, green circles). The overall ordering of this difference in tension ratio for varying global opening angle in rod A is correctly predicted in the capstan Eq. 1 when applied with the wrapping angle φ interpreted as the global opening angle of rod A (dashed lines). (Note that the local opening angles, i.e., the angles between centerline tangents at touch-down and lift-off points, cannot be accessed experimentally because μCT data are not available for this experimental setup.) However, the simple capstan equation interpreted in this way is inadequate to explain our data because it predicts that the tension ratio should be independent of both t0 and the imposed geometry on the static rod B; neither of these predictions is close to being accurate. Specifically, our results reveal the essential role of the static rod B; the tension-ratio response is significantly larger when the static rod wraps further around the sliding rod. For example, the tension ratio of configuration 3 (where the static rod is initially short with ends close to each other) is larger than the tension ratio for configuration 1, by 21% at t0=0.05.

The limitations of the simple capstan relation in predicting the experimental data are consistent with the observation made above that a single elastic rod wrapped around a rigid cylindrical capstan, with its simple, single contact line, is inadequate to accurately describe local opening angles of an elastic clasp configuration. It is the local opening angles that are strongly related to the detailed contact geometry, which sets the maximal tension ratio K(t0). The mismatch between the capstan equation and our results stems from the nontrivial dependence of the local opening angles of the sliding and static rods on the applied tension t0 and the associated changes in the contact set.

The detail of the contact-region interactions, including sliding friction, can be further probed via FEM. In Fig. 4E, we plot the pressure field between the two rods for the frictional sliding clasp configuration of Fig. 4B3 (configuration 3). Just as in the static, frictionless case, the support of the contact-pressure map in the presence of friction displays the familiar diamond shape, and, again, there are strong heterogeneities. Now, however, we observe only three pressure peaks in the (sA,sB)-plane: two at the extreme values of sB and one at the high-tension, or lift-off, end along sA, where the pressure is largest. The symmetry along rod A is now broken, and the pressure at the touch-down end of the contact set is extremely low. As a first step toward understanding these qualitative features in the contact-pressure distribution for the sliding elastic clasp, in SI Appendix, section 4.F, we present a hybrid analysis of the ideal orthogonal clasp. In this framework, the shape of tube B is frozen to form a rigid tubular capstan with its own intricate geometry and local opening angle, around which the flexible tube A can reptate, with the contact set remaining identical with the static case (i.e., a closed, curvilinear, diamond-shaped line). In this idealized context, and as a consequence of moment balance on the positive radius flexible tube A, it can be derived (SI Appendix, section 4.E) that, in the presence of friction, the linear pressure distribution giving rise to the maximal possible tension ratio vanishes at the touch-down point, and has a Dirac delta function singularity at the lift-off point, as illustrated in Fig. 4F. Despite the strong simplifying assumptions in this ideal geometric capstan model, the numerical predictions for its linear pressure density have a striking qualitative similarity to the surface-pressure distributions for the elastic orthogonal clasp as computed in FEM simulations in Fig. 4E.