Introduction

Slacklines are elastic and to a first appoximation behave like an idealized spring with a characteristic stiffness (modulus, or elasticity constant) \(k\) so that their stretch is a linear function of the applied tension. However, this is a crude approximation. In reality, the stiffness, determined here as the derivative \(k=\frac{dT}{de}\), where \(T\) is the tension of the webbing and \(e\) its relative elongation, is not a constant. For static loads, i.e. when tension changes very slowly with time, the respective relative elongation follows a characteristic stress-strain curve, which generally is determined experimentally by measuring the webbing tension while applying a very low and costant strain rate (typically smaller than 6 %  min\(^{-1}\)), hereafter referred as slow-pull experiment. For most webbings in the market such measurements are performed by the manufacturers and can be found plotted by the Austrian Slackline Assocciation. Nevertheless, the stiffness of a webbing changes significantly with its history of use and the present environmental conditions (temperature, humidity) and therefore the above-mentioned curves may be considered as indicative for static tensions but are not applicable to dynamic loads (strain rates larger than 10 %  min\(^{-1}\)).

Slackline dynamics strongly depend upon the elastic response of the used webbing. When the tension changes quickly with time, viscoelasticity effects can no longer be neglected. The latter include, creep, stress relaxation, internal friction and consequently hysteresis. Through a series of carefully executed experiments, this study examines the behavior of slackline webbing under dynamic loads determining the departures from the respective static response (stress-strain curves). Such knowledge is fundamental for the accurate simulation of slackline dynamics so as to predict peak forces and aid safe rigging.

The results of the present study (here is a link to the published article) show that the effective modulus during leash falls is significantly higher than the slope of the respective stress-strain curve, indicating a stiffer response. Also, the effective modulus is found to increase with the applied pretension. Using the moduli determined experimentally for the rigged slacklines with different types of webbing, the respective leash falls were simulated numerically with high accuracy. A standardized test is proposed, to be adopted by the International Slackline Association (ISA) and the slackline webbing manufacturers, so as to provide key information on the response of each webbing available in the market under typical dynamic loads, similarly to the impact force test designed for dynamic ropes by the International Climbing and Mountaineering Federation (UIAA).

Experimental setup

An experiment was designed to measure continuously and simultaneously the tension (\(T\)) and the relative elongation (\(\epsilon\)) of the line during leash falls. The experimental setup included a mass (\(m\)) attached with a static leash of length (\(d\)) at the middle point of a horizontal slackline. The mass was let to fall directly under that point so that its trajectory was almost perfectly vertical, at least during the first bounce. The line tension was measured very accurately (error \(<\)0.03 %) with a load cell taking recordings at a frequency of 500  Hz. The latter was found to be more than sufficient for resolving the phenomenon. The leash falls were video-recorded in slow motion from the side against a flat vertical white board parallel to the line (Fig. 1). The vertical displacement of the line versus time (\(y\)) was determined through feature-tracking performed on the slow-motion videos. Using \(y\), the relative elongation of the line versus time (\(\epsilon\)) was determined through simple geometric considerations. The first few bounces of the mass were recorded and analyzed. Figure 2 provides a general view of the experimental setup, while Figs. 2 and 3 show some respective details.

Figure 1:  A drawing of the experimental setup. The slackline (red) was anchored on two columns at a height of 7  m. A load cell with a temporal resolution of 500  Hz was installed at one end (purple box) between the anchor and the line-lock. A video-camera operating at 500  fps was placed at the side at a distance of 12  m pointing perpendicularly to the slackline at mid-height. A flat board (gray rectangle) was erected 1  m behind the slackline. The latter served as a uniform background to facilitate video-tracking. A vertical meter was painted on the board for scaling purposes. A solid cylindrical weight (blue) was connected with a static leash (green) to the slackline at the midpoint. A light, colored ball (yellow) was fixed next to the leash ring to aid feature tracking. This way, during the leash falls, the vertical displacement (\(y\)) and the tension of the slackline (\(T\)) were accurately determined.

Figure 2:  Specific components of the rigged system are indicated. A metal cylinder of variable weight (1}) was attached to a static leash made of flexible wire-rope (2). The latter was attached to the slackline (3) with a leash ring. A light, colored ball (4) was fixed next to the leash ring to aid video-tracking of the slackline position. The flat vertical board (5) served as a uniform background. An anchor on the ceiling (6) was used for raising the weight with a rope (7) and a locking pulley (8). Then, the weight was suspended directly below the slackline by means of a release hook (9) operated with a string from the floor.

Twelve (12) different rigs were tested representing selected combinations of the following: (i) two different anchor-to-anchor distances: \(\,D_1=\,\)22  m   and   \(\,D_2=\,\)11  m, (ii) two different webbings: high-stretch nylon (NY)   and   low-stretch polyester (PE), (iii) four different leashes: \(\,d_1=\,\)2.0  m, \(\,d_2=\,\)2.5  m, \(\,d_3=\,\)3.4  m (wire rope) and \(\,d_4=\,\)2.0  m (climbing rope), (iv) two different line tensions (\(T_1 < T_2\), dependent on distance and webbing), and (v) two different masses: \(\,m_1=\,\)58  kg   and   \(\,m_2=\,\)78  kg. To test reproducibility and to increase statistical robustness, three seemingly identical leash falls were performed for each individual rig (3\(\times\)11\(=\)33 leash falls). Table 1 gives an overview of the recorded falls, each identified by a code name.

Table 1:  The code names in the first column signify the different rigs tested. Each rig (row) corresponds to a unique combination of slackline distance (\(D\)), webbing type (NY / PE), leash length (\(d\)), pretension (\(T_{\!o}\)), and falling mass (\(m\)). To assess reproducibility, three leash falls were performed and recorded for each rig. In total 3\(\times\)11\(=\)33 leash falls.

Measuring methods and analysis

Having the leash ring exactly at the midpoint of a perfectly horizontal line ensured plane symmetry. This way, during the experiments, the line tension at both sides of the leash ring is expected to be the same. Arguably, when dealing with tensions of several kN, the effect of the mass of the webbing (\(<\)1.5  kg) can be ignored and thus, at any instant, its tension can be assumed to be uniform along its entire length. Therefore, a single load cell (attached to one anchor, Fig. 1) was considered sufficient.

Hence, measuring the line tension was relatively simple from an instrumental view point. In contrast, simultaneously measuring the line elongation was quite challenging. Ignoring any vibrations / waves induced to the webbing due to transverse accelerations, the webbing between the leash ring and either anchor was assumed to remain straight. Under these justified approximations, the total elongation of the webbing can be determined accurately by measuring the vertical displacement of the center of the line and using the Pythagorean theorem.

For each leash fall experiment, determining the actual vertical displacement of the leash ring versus time, \(y(t)\), was accomplished in two steps. First, the apparent displacement against the background board, \(h(t)\), was determined from the respective slow-motion video (Fig. 3). This was done in an automated way using a feature-tracking program program and the vertical reference length printed on the board (Fig. 1). Because the latter program employs an algorithm based on the chromatic contrast between the tracked feature and the background, a unicolor, feather-light ball was attached to the leash ring (Fig. 2) to aid tracking of the line.

Then, a parallax correction was applied to account for the projection effect, converting the apparent displacement (\(h\)) to the actual one (\(y\)). This small correction is explained in Fig. 3. Practically, the video camera (Sony RX100-IV, focal length of 50 mm) has infinitesimal barrel distortion, thus any regular grid in the video frames corresponds to a real regular grid on the flat board behind the vertical plane containing the slackline. Therefore, the feature-tracking performed on the video frames provided accurately the apparent displacement \(h(t)\). Then, using the intersect theorem (see Fig. 3) the actual vertical displacement \(y\) was determined at a high frequency (500  fps), equal to the load cell measuring frequency (500  Hz).

Figure 3:  When the slackline is perfectly horizontal, the leash ring represented by the yellow ball is at the point A. Seen from the camera (horizontal distance \(D_c\)), this point projects to A\(^\prime\) on the flat board (horizontal distance \(D_b\)). Then, an actual vertical displacement of the leash ring, \(y\), corresponds to an apparent displacement \(h\). Using the intersect theorem it is easy to show that \(y=\frac{D_c}{D_c+D_b}h\). In the text, this is referred to as parallax correction.

Comparing to the standard UIAA (International Climbing and Mountaineering Federation) test for dynamic ropes used to determine the so-called impact force [EN 892], slackline leash falls induce much lower strain rates to the slackline webbing. This is mainly because of two reasons: (i) the slackline is generally pretensioned in contrast to the rope in a climbing fall and (ii) the effective fall-factor (fall length divided by the material length) is much lower in a leash fall compared to the UIAA impact-force test. A typical strain rate in the UIAA test (300 %  s\(^{-1}\)) is about 30 times larger than that occurring in leash falls, and thus the time-dependent response of the slackline webbing in a leash fall is expected to be quite different to that of dynamical ropes absorbing serious climbing falls (e.g. Bedogni and Manes, 2011).

To examine the above-mentioned dynamical response for slackline webbings we plot the line tension (\(T\)) versus its relative elongation (\(\epsilon\)) for the first bounce (down and up) of the falling mass. Figure 4 (upper panel) shows this for representative leash falls with the NY webbing. The static tension versus elongation as measured in the CAI laboratory and that given by the manufacturer for a new piece of NY webbing are also shown for comparison. Figure 4 also shows the corresponding results for the PE webbing (lower panel). It is found that during leash falls the line tension as a function of its relative elongation exhibits a steeper gradient (higher effective modulus) compared to the static tension curve. This difference seems to increase with increasing pretension, a finding that is corroborated by the quantitative analysis presented in Fig. 5. Also, hysteresis effects (assessed by the area enclosed by the curves) appear weaker as pretension increases. Finally, the assumption of a constant effective modulus during leash falls is acceptable as linearity appears to be a good approximation. This last finding is verified by further analysis presented in the next section.

Figure 4:  Tension-elongation plots showing the slow-pull curves determined by the webbing manufacturer (dashed lines), the respective curves for our used webbings as determined at the CAI laboratory, and some dynamical tension-elongation loops corresponding to the first bounce during leash falls (color curves). Top plot for NY webbing (E01, E02, E03, E04), bottom plot for PE webbing (E05, E06, E07, E08). Red curves for long leash and blue curves for short leash. The green curve corresponds to a regular rope-leash but for a shorter line and a heavier mass (E11). The small dots along the loops indicate equal time intervals (100  ms) assisting the estimation of the strain rate (Fig. 5). In the absence of hysteresis effects, the branches of each open loop (elongation phase and contraction phase) would collapse to the same curve.

Figure 5 aims to demonstrate more clearly the dependence of the effective modulus on the pretension and the strain rate. The markers in the plot show the instantaneous effective modulus computed for each experiment in time intervals of 100  ms (the time elapsed between two successive dots in Fig. 4), while colors indicate different pretension values as documented in Table 1. The evident separation between markers of different color clearly indicates the effect of the applied pretension (higher pretension gives higher effective modulus). In contrast, the strain rate (horizontal axis) appears to have a smaller, yet noticeable, effect. To conclude, the effective modulus increases significantly with pretension and weakly with the applied strain rate. Arguably, the dependence on the strain rate is expected to be more dominant at even high strain rates, such as those occurring in the UIAA impact force test for dynamic ropes.

Between the triplet of seemingly identical leash falls performed for every different rig, the response of the webbing was not found to be identical. Hysteresis appeared to be stronger at the first respective leash fall, likely due to fiber alignment/ adjustment within the webbing itself. Diabatic effects and the time intervals between the leash falls possibly influenced these differences. In any case, the latter did not seem to represent random errors as there was always some hysteresis detected (even if very small at certain occasions).

Figure 5:  The instantaneous effective modulus is shown for each experiment with the NY webbing (E01, E02, E03, E04, E9, E10) against the respective strain rate. The markers show the respective values in time slices of 100  ms (pairs of successive dots in Fig. 4). The colors indicate different values of pretension \(T_{\!o}\): purple for higher, orange for lower and green for intermediate pretension (as in Table 1). The straight lines show the slope of the static T–E curves (solid line: manufacturer, dashed line: measured) at the point corresponding to the respective applied pretension. Evidently, the effective modulus depends strongly on the applied pretension.

Given the significant differences seen between the experimentally determined slow-pull T–E curve for the PE webbing and the corresponding curve based on data provided by the manufacturer (Fig. 4, lower panel), it should be mentioned that all leash falls were performed before the slow-pull tests in the CAI laboratory and always with the same two pieces of webbing. For reference, the PE webbing (Core 2 Low-Stretch by Landcruising Slacklines) was 3-year old, it was tensioned 5-6 times before at about 12-14  kN, and had been stored in a cool and dry place. More importantly, the respective strain rates applied in the CAI laboratory (\(<\)0.01 %  s\(^{-1}\)) were significantly lower than those used by the manufacturer (\(\sim\)0.1 %  s\(^{-1}\)), and considering the associated stress relaxation over time, a slower pull is expected to give higher elongation values for each given tension. The NY webbing (Sonic 2 by Landcruising Slacklines) was 2-year old, and before the experiments had been used approximately 10-12 times in highlines and at the park with typical tensions not exceeding 7  kN.

Figure 6 displays the timeseries of line tension and relative elongation during two leash falls (E11 for NY webbing, E05 for PE webbing). The tension was measured directly with the load cell, while the elongation was computed from the vertical displacement of the line determined through video-tracking. The hysteresis effect is evident in the mismatch between the two curves in the contraction phase and in the remaining loss of tension. The intervals in which the tension remains quasi constant correspond to the periods of time when the leash is not engaged (mass in free flight). The evident upward trend in the lower panel is due to the fact that the mass remains suspended by the slackline, thus increasing its tension. This effect is not seen in the upper panel because even after the 5th bounce of the mass, the slackline momentarily returns to its initial horizontal position. Very similar curves have been obtained through numerical simulations.

Figure 6:  The line tension (purple lines) and the relative elongation (green lines) versus time during leash falls. The two curves refer to different vertical axes (left for tension, right for elongation) and have been scaled to assist their comparison. The mismatch between the two curves, more evident at the end of the first contraction phase and between the first two bounces, is another clear indication of the hysteresis effect. Top plot for the NY webbing with a rope leash (E11), bottom plot for the PE webbing (E05).

As mentioned earlier, in the experiments the mass was released directly below the line so as to ensure a vertical trajectory. This makes the leash falls easier to analyse so that numerical simulations can be compared to real experiments. Moreover, only for a vertical trajectory is the applied parallax correction valid. However, in real leash falls the slackliner is usually walking on the line before falling, typically with a leash shorter than 1.5  m. For this reason, but also for simulating higher strain rates, longer leashes were used in the experiments. Furthermore, in a real leash fall, as the slackliner falls to the side, the line gets free to move upwards, and it does so limited by the length of the leash. This partial recovery of the line towards its horizontal position causes a quick drop in line tension, which may be significant for lines with a walking sag comparable to the leash length. In any case, for reference, it should be clear that such a recovery does not occur in the experiments presented here as the line is initially free of weight (the mass was suspended from the ceiling anchor with a release hook). It should be noted that in real leash falls it is literally impossible to have a purely vertical trajectory, and thus there is always some left-over kinetic energy (associated to the horizontal velocity at the lowest point of the trajectory), which effectively reduces the occurring peak leash tension.

Effective modulus: a linear approximation

In Fig. 4, the part of the curves   /  loops corresponding to the elongation phase (increasing tension) can be approximated with with a straight line. This is equivalent to assuming a constant effective modulus (linear elasticity). Such a fit can be made for each individual leash fall experiment so that the associate error can be determined via a simple dynamical model described below.

Figure 7:  Illustration of the linear elasticity model corresponding to a constant effective modulus. The dashed line represents the static T–E curve of the webbing, while the steeper straight line represents the linear fit to a recorded leash fall with pretension \(T_{\!o}\). The shaded areas represent the elastic energy absorbed by the webbing for an additional elongation \(\,x=\varepsilon\!-\!\varepsilon_o\).

Suppose that \(L\) is the length of the webbing at zero tension that has been stretched to fit the anchor-to-anchor distance \(D\). Let \(\,\varepsilon_o\!=\!D\!-\!L\,\) be the initial elongation of the webbing, and \(\;T_{\!o}\;\) the respective pretension satisfying the associated static T–E curve. The mass \(\,m\,\) is attached to the middle of the line with a static leash of length \(\,d\,\) (as in Fig. 1) and is let to fall directly below the line so that it covers an equal vertical distance (\(d\)) until the leash is engaged. At that point the latter starts pulling down the slackline, which undergoes additional elongation \(\,x=\varepsilon\!-\!\varepsilon_o\). As the mass continues its downward motion, the vertical displacement of the slackline \(\,y\,\) is uniquely determined by \(\,x\). In fact, for \(\,x>0\), \(\,y(x)\,\) is a one-to-one and monotonically increasing function determined through simple geometric considerations: \begin{equation} \label{eqn:Yx} y(x) \:=\: \frac{1}{2}\sqrt{x\,(2D+x)}. \end{equation} Assuming that the webbing tension increases linearly with \(\,x\,\) means that: \begin{equation} \label{eqn:Tx} T(x) \:=\: T_{\!o} + k\,x, \end{equation} where \(\,k=\) tan\(\alpha\,\) (Fig. 7) represents the elasticity constant related to the effective modulus of the webbing. Note that the modulus \(K_{eff}\) is defined here as the force (tension) divided by the strain (relative elongation) and thus it characterizes the webbing without depending on the webbing length. \(K_{eff}\) is estimated through a linear fit to the respective experimental data (Fig. 4). \(\,K_{eff}\,\) and its length \(\,L\), namely: \begin{equation} \label{eqn:Keff} K_{eff} \:=\: k\,L. \end{equation} The force exerted on the mass (leash tension) increases nonlinearly with \(\,y\,\) due to the increasing webbing tension, as well as due to the increasing angle \(\,\theta\,\) (Fig. 1). The conservation of mechanical energy requires that: \begin{equation} \label{eqn:energy_conserve} \frac{1}{2}\,k\,{x}^{2} \:+\: T_{\!o}\,x \:+\: \frac{1}{2}\,m v^{2} \:=\: mg( y+d), \end{equation} where the right-hand-side term represents the change in the gravitational energy of the mass in respect to its initial position, and \(\,v\,\) is the velocity of the mass at \(\,y(x)\). The first two left-hand-side terms represent the elastic energy absorbed by the webbing as illustrated in Fig. 7.

The kinetic energy of the mass will be zero when the latter reaches the lowest point of its trajectory (assumed vertical) at \(\,y_{max} + d\). Hence, this point can be determined through Eq. \ref{eqn:Yx} and the value of \(\,x>0\,\) that satisfies the following equation: \begin{equation} \label{eqn:x_solve} \frac{k}{mg}\,{x}^{2} \:+\: \frac{2T_{\!o}}{mg}\,x \:-\: \sqrt{x\,(2D+x)} \:-\: 2d \:=\: 0. \end{equation} The above equation is equivalent to a forth-degree polynomial with roots given by analytical formulas, though solving this equation numerically (e.g. using SciPy) is probably easier and faster.

Using the relative elongation \(\,\epsilon = x/D\),  the previous equation can be transformed to a non-dimensional form: \begin{equation} \label{eqn:e_solve} \frac{K_{\scriptscriptstyle{eff}}}{mg}\,{\epsilon}^{2} \:+\: \frac{2T_{\!o}}{mg}\,\epsilon \:-\: \sqrt{\epsilon\,(2+\epsilon)} \:=\: 2\frac{d}{D}. \end{equation} This is insightful as it makes evident that when the ratio \(\,d/D\,\) is kept constant, or for \(\,d=0\), the solution to this equation does not depend on the anchor-to-anchor distance. In that case, it is easy to see that the angle \(\,\theta_{max}\,\) and the tension \(\,T_{max}\,\) corresponding to the lowest point of the trajectory are also independent of \(\,D\).

It is straightforward to express the velocity of the mass as a function of its vertical position. Eliminating \(\,x\,\) between Eq. \ref{eqn:Yx} and Eq. \ref{eqn:energy_conserve} leads to: \begin{equation} \label{eqn:Vy} v(y) \:=\: \pm\,\sqrt{\,2g\left(y+d\right)-\left(\sqrt{D^2+4y^2}-D\right)\left(\frac{k}{m}\sqrt{D^2+4y^2}-\frac{kD}{m}+\frac{2T_{\!o}}{m}\right)}. \end{equation} The time-dependent solution of the above-described dynamical system can be derived via the respective differential equations of motion, nevertheless the time-dependent behavior can also be determined by making use of the Eq. \ref{eqn:Vy} and the definition of vertical velocity: \begin{equation} \label{eqn:time} dt \:=\: \frac{dy}{v} \quad\Longrightarrow\quad t(y) \:=\: t_o \:+\: \int_{0}^{y}\frac{dy}{v(y)}\,. \end{equation} If \(t\!=\!0\) when the mass is released, then \(t_o = \sqrt{2d\;\!/g}\). \(\;t(y)\,\) can be computed numerically. Consequently, \(\,y(t)\,\) can be considered as known, and using Eqs. \ref{eqn:Yx} and \ref{eqn:Tx}, respectively, \(\,x\,\) and \(\,T\,\) can be expressed as functions of time.

Using this simple linear model, effectively assuming a constant effective modulus and the absence of dissipative processes, the recorded experiments were simulated with very high accuracy for the first bounce of the mass. The results for two example leash falls are shown in Figs. 8 and 9 exhibiting errors smaller than 4 % in respect to the recorded peak tension and maximum vertical displacement. The simulated values are always slightly higher than those recorded, which is evidenced by the red strips in Fig. 8 and the small mismatch of the curves in Fig. 9. This is rather expected considering the absence of energy dissipation in the linear model. In reality, dissipative processes are always present in the form of visco-elasticity effects in the webbing and aerodynamic friction acting on all moving components.

Figure 8:  Two leash falls (left: E08 for PE webbing, right: E03 for NY webbing) are simulated numerically using the effective modulus determined by linear fit (red lines) to the observed data (blue curves, representing the elongation segments of the respective loops in Fig. 4). The shaded areas represent the energy absorbed by the webbing when the mass reaches the lowest point of its trajectory, theoretically with zero kinetic energy. The simulations, performed as described in Section 5, lead to a small overestimation of the peak tension (red dots). The respective errors are smaller than 4 %.

For the first bounce of the mass (elongation and contraction cycle of the webbing), the area enclosed by the respective loop in Fig. 8 represents the energy consumed by irreversible visco-elastic processes in the webbing. In contrast, the blue-shaded areas in Fig. 8 represent the total energy transferred to the webbing by the end of the elongation phase. For E03, this turns to be about 6 % smaller than the corresponding change in gravitational energy \(mg\;\!(y_{max}\!+\!d\;\!)\). This small difference is likely due to any left-over kinetic energy of the mass (body rotation and horizontal velocity) and other dissipative processes external to the webbing.

The conducted experiments and the simulations did not aim to reproduce realistic leash falls but to study of the dynamical behavior of the webbings. Hence, no results are presented here regarding the maximum leash tension. In leash falls that happen on the field there are various other damping effects, including energy absorption by the leash knots and the human body, while the kinetic energy is never eliminated completely. These effects, acting together, significantly reduce the peak forces encountered.

The leash fall simulations presented here, using a linear approximation to the empirically determined effective modulus, exhibit unprecedented accuracy. For reference, if the simulations are performed using a modulus matching the respective static T–E curve (black lines in Fig. 8) the errors turn to be as large as 25 %. Therefore, it becomes evident that the accurate simulation of leash falls and backup falls requires knowledge of the dynamical response of the slackline webbing in hand.

Figure 9:  The vertical displacement of the webbing (\(y\), left axis) and the webbing tension (\(T\), right axis) as functions of time for the experiment E03 shown in Fig. 4. The solid (dashed) lines show the recorded (simulated) values. The sign of \(\,y\,\) is reversed for clearer illustration. After the release of the mass at \(\,t\!=\!0\,\) the latter experiences a free fall until the leash is engaged, approximately at \(\,t_o\!=\!0.7\)  s. In the simulation the duration of a complete bounce (peak-to-peak distance) is slightly longer than the observed period (\(\sim\)2  s) because, without dissipation, the mass returns to its original height and thus it passes more time "flying" up and down. For the first bounce the errors in the simulation are smaller than 4 %.

Discussion and conclusions

The original questions that motivated the present study have been adequately answered. The first question was: can the static T–E curves (e.g. as provided by slackline webbing manufacturers) be used to accurately simulate dynamic loads in slacklining, and in particular leash falls? The answer to this question is negative. Even if hysteresis and dissipative effects are completely ignored, the effective modulus of the slackline webbing in a real leash fall differs significantly from the slope of the T–E curve. As expected, the effective modulus was found to be significantly higher than the static modulus determined by a slow-pull test.

The second question regarded the factors that control the dynamic response of slackline webbings. The answer to this is that the effective modulus appears to depend more strongly on the applied pretension and less so on the strain rate. Finally, it was demonstrated that using an empirically determined constant effective modulus permits highly accurate numerical simulations of controlled leash falls. In view of these findings, a new research question arose regarding the possibility to predict  /  estimate the effective modulus by knowing the pretension and the applied strain rate.

The observed hysteresis effects in slackline webbings may relate to various physical mechanisms. Research on the material properties of polymers is still ongoing, while various books and monographs have been written on this subject. Yet, it is beyond the scope of this study to discuss such processes at the molecular level. Leuthausser (2016) discusses time-dependent internal friction as key in explaining the observed behavior of climbing ropes under heavy falls. The nearly adiabatic stretching at the beginning of the elongation phase raises the temperature of the polymer fibers, consequently modifying their dynamic response (as the modulus of polymers strongly depends on temperature). In earlier studies modeling damping in polymers —a review can be found in Bert (1973)— viscous effects have been parametrized as a linear drag (internal friction proportional to the strain rate, the so-called dash-pot approach). However, the results presented here indicate that non-conservative processes (which are present and evidenced by hysteresis) are less dominant in slackline systems compared to climbing falls on dynamic ropes.

The increased effective modulus detected and its dependence on the pretension may be understood considering a characteristic behavior of polymers referred to as stress relaxation. Early studies, e.g. Meredith and Hsu (1962) and Ko (1976), have demonstrated that immediately after rapid permanent elongation of nylon fibers, their stress decreases sharply with time before following a slower exponential decay. It was also found that this sharp initial decrease in stress is faster for larger applied strains (initial tensions). In this regard, the effective modulus during rapid elongation of slackline webbings is expected to be higher than the slope of the respective static T–E curve because when the elongation occurs very slowly the webbing has time to relax.

Considering that climbing ropes and slackline webbings are characterized by a complex wave structure, one should also consider that the elongation of a kernmandle rope or a slackline webbing is partly related to the alignment of the fibers (rather than their elongation). This effect is referred to as constructional elongation and may be particularly important during the first elongation cycle. In fact, for each individual rig the experiments showed a larger loss of tension after the very first leash fall, indicating an adjustment process.

For dynamic climbing ropes, UIAA asks manufacturers to provide an indication of the dynamical behavior of each rope. This is done through a standardized experiment (EN 892) determining the so-called impact force. Here it is suggested that a similar norm should be established also for slackline webbings. For example, the manufacturers could provide an indicative value for each webbing corresponding to the effective modulus as determined through a standardized test (e.g. new webbing, anchor to anchor distance: \(D=\)30  m, static leash length: \(d=\)2  m, stable pretension: \(T_{\!o}=3\)  kN, solid mass: \(m=\)70  kg, to fall the full length of the leash before the line is engaged). Alternatively, because the effective modulus is not directly measurable, the respective peak tension of the line could be provided. Of course, this indicative effective modulus will be slightly different for leash falls characterized by different parameters (\(D\), \(d\), \(T_{\!o}\), \(m\)) and ways of falling. Still, the latter can be very useful for comparing the dynamical response of different webbings and in getting rough theoretical estimates of the peak tension in real leash falls and accidental backup falls. The International Slackline Association (ISA) has seen this proposition with great interest, yet the details of the test remain to be decided.

Acknowledgements:
This study would not had been possible without the kind collaboration of the Centro Studi Materiali e Tecniche of the Club Alpino Italiano (CAI). In particular, Giuliano Bressan, Massimo Polato and Claudio Melchiorri are gratefully thanked for their assistance in conducting the experiments. The author is thankful to Damian Jörren (Landcruising Slacklines) and Jerry Miszewski (Balance Community: Slackline Outfitters) for providing details on the procedure of determining the static tension-elongation curves of their products. The LineGrip Corporation is also acknowledged for supporting this study with a generous promotion discount for the LineGrip, which proved essential for conducting the experiments accurately. Stefano Righetti and Gianluca Sgariglia are kindly thanked for their help in conducting the experiments and providing comments on the manuscript (GS). Last but not least, the author is grateful to the OZ Bologna sports and cultural center for providing the necessary ambient space.

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