Deformation of snow during a carved ski turn

Abstract

Skiing sports have experienced a significant boost in the 1990s when the classical ski geometry was replaced by a deep side cut ski, the carving ski. The geometrical and mechanical characteristics of modern carving skis are adjusted to typical turn radii, such that the skis cut through the snow surface with minimized skidding. As a ski passes over the snow surface in a carved turn the inclined skis penetrate the snow surface until the contact area suffices to allow snow of a given resistive strength to support the load exerted by the ski. In this study we measured the mean resistance pressure the snow exerts to an inclined plate penetrating the snow surface as a function of penetration depth. The tests were carried out on several groomed ski pistes of different snow types. We found that the mean penetration resistance pressure for homogeneous snow increased linearly with the penetration depth. Hence, the snow resistance of different snow types can be characterized by the slope A and the offset pressure B of this linear function. For groomed, homogeneous snow pistes values of the coefficient A lie typically between 5 and 40 MPa m^− 1, values of the coefficient B lie typically between 0 and 400 kPa. Moreover, the slope A increases with increasing edging angle, but if the edging angle surpasses a threshold angle found between 30° and 45° the mean snow resistance pressure falls due to frequent shear fractures, by which the snow is pressed out of the indent. The overall trend in snow resistance curves can still be approximated by a linear function, but the slope A falls to values in between 2 and 20 MPa m^− 1. The offset pressure B strongly depends on surface characteristics of the snow, which often deviate from the subjacent snow due to exposure to the sun's radiation or to skier's traffic. If the snow surface is considerably softer than lower snow layers, then the offset pressure B of the overall snow resistance function may even be negative. This study provides the so far missing experimental data for models of the ski–snow interaction in carved turns.

Introduction

In the development and design of modern Alpine skis numerical methods become more and more common tools. Nowadays, the finite element method (FEM) allows ski designers to calculate the mechanical properties of complex structures such as skis and bindings. In several previous studies the mechanical deformation of skis in the loading situation of a turn was calculated and used to predict the turn radius (Kaps et al., 2000) or the path of the skis on the snow surface (Casolo and Lorenzi, 2001). Clerc et al. (1989) used finite elements to predict the static pressure distribution under different kinds of skis. Snow was represented by non-linear springs with zero stiffness in tension and constant stiffness in compression. Renshaw and Mote (1991) considered the model of a stiff skier on one ski making a turn of constant speed and radius on a horizontal snow surface. An empirical formula for the skidding forces acting on the ski–snow interface was set up using experimental data published by Lieu (1982) and by Lieu and Mote, 1984, Lieu and Mote, 1985. Tada and Hirano (1999) measured the forces necessary for cutting ice and formulated empirical equations which they used to calculate the track of turning Alpine skis. In a more recent study they conducted the cutting force measurements on snow and implemented the results in their model (Tada and Hirano, 2002). Casolo and Lorenzi (2001) presented a sophisticated biomechanical model for the simulation of movements in skiing. For the ski–snow interaction they used a layer of springs and dashpots in parallel to simulate snow. Thus, the penetration depth into the snow could be calculated. Kaps et al. (2000) developed a simulation program in which the turn radius is calculated as a function of side cut, edging angle and penetration depth into the snow. The snow resistance pressure is considered proportional to the penetration depth as long as a threshold value, the so-called “yield pressure”, is not exceeded. Thus, all of the aforementioned studies relay either on data obtained in machining tests on snow carried out under laboratory conditions, or they use estimated models for the ski–snow interaction.

It is a characteristics of carving that the skis cut into the snow surface with minimized skidding. The ski–snow interaction in carved turns can be understood as a combination of friction, of dynamic effects, but predominantly of a penetration process of an inclined plate, i.e. the penetration direction is perpendicular to the contact surface between plate and snow. So far neither laboratory nor field tests on actual skiing slopes have been conducted, which resemble such a process. Machining test as carried out by Lieu (1982) and by Tada and Hirano (1999) are a suitable test to describe the ski–snow interaction during skidding. Mössner et al. (2003) have conducted penetration tests using inclined plates which penetrate the snow perpendicularly to the undisturbed snow surface. The resistance pressure of snow to plates vertically penetrating the snow surface was comprehensively investigated for other purposes by Fukue (1979), Abele and Gow (1976), Abele (1990) and Johnson et al. (1993) but penetration of inclined plates was not studied. It follows that in order to develop a model of the ski–snow interaction appropriate for ski carving new experimental data is needed.

One important result of the investigations of Fukue (1979), Abele and Gow (1976), Abele (1990) and Johnson et al. (1993) is that the snow strength depends strongly on the strain rate, i.e. the penetration speed of the penetrating device. In 1998 Johnson published a numerical investigation of the changes in the microstructure of snow during a compaction process (Johnson, 1998). He predicted several threshold compaction rates at which the physical processes, which determine the snow's compressive strength during compaction, change. Between the thresholds the snow strength does not change significantly with increasing compaction rates. The compaction rate of snow, which is penetrated by a passing ski, can only be estimated, since it is not possible to exactly determine to which depth the snow has actually been compacted. Therefore, it was first investigated in an earlier published study (Federolf et al., 2004), if the snow resistance pressure to a penetrating plate depends on the penetration speed of the plate in the range of penetration speeds, which are typical for the ski snow interaction. It was found that the penetration speed has a negligible impact on the snow resistance pressure. Compaction rates in skiing are not in the range of critical threshold rates reported by Johnson (1998). – This is reasonable since the rates are fast enough to cause elastic–brittle behaviour, but not fast enough for inertia to be important.

In this study the mean pressure acting at the interface between the snow and an inclined plate penetrating the snow similarly to an edged ski is measured and expressed as a function of the penetration depth. At the same time the impact of different edging angles on the penetration process is experimentally investigated and discussed. The measurements were carried out on actual ski pistes, providing the missing data for ski–snow interaction models.