# Characterisation of the Contact between Cross-Country Skis and Snow: On the Multi-Scale Interaction between Ski Geometry and Ski-Base Texture

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theory

^{−1}they found the Young’s modulus of the snow, ${E}_{snow}$, to be scattered between 100 $\mathrm{M}$$\mathrm{Pa}$ and 250 $\mathrm{M}$$\mathrm{Pa}$, and residual strength to be linearly increasing (from 2 $\mathrm{M}$$\mathrm{Pa}$ to 8 $\mathrm{M}$$\mathrm{Pa}$) with the strain rate.

## 3. Method

## 4. Results and Discussion

## 5. Conclusions

- Adhesive friction: The difference in real contact area between the two ski-base textures was larger than ≈28%, while the difference between the skis was less than ≈4% (merely ≈0.4% for Grind 2). Hence, the ski-base texture seems to have a larger impact on the adhesive part of friction.
- Abrasive friction: Because of the coupling between the real contact area and the average contact pressure, the same can be said about the influence of the ski-base texture and the ski camber for the abrasive part of friction as for the adhesive part.
- Friction regime: The minimum average interfacial separation was strongly influenced by the ski-base texture, showing values larger than ≈318%, while the difference between the two ski-camber profiles was less than ≈8%. Though the minimum average interfacial separation is meaningful, a larger area with lower separation could potentially affect the transition between boundary and full-film lubrication. This might be avoided by reducing the apparent contact area.
- Viscous friction: The difference between the two ski-base textures, in terms of total average reciprocal interfacial separation, was larger than ≈59%, while the maximum difference between the skis was less than ≈46%. The differences for all combinations were, however, of comparable sizes, suggesting that the ski-base texture and the ski-camber profile have a similar impact on the viscous part of friction.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$X,Y$ | Macro-scale coordinates | $\mathrm{m}$ |

${X}_{m}$ | Equivalent load offset relative to $X=0$ | $\mathrm{m}$ |

F | Equivalent load | $\mathrm{N}$ |

${H}^{*}$ | ANN prediction of measured ski-camber height profile | $\mathrm{m}$ |

P | Apparent (macro-scale) contact pressure | $\mathrm{Pa}$ |

${P}_{n}$ | Effective (macro-scale) contact pressure | $\mathrm{Pa}$ |

${E}^{\prime}$ | Effective Young’s modulus | $\mathrm{Pa}$ |

${E}_{snow}$ | Young’s modulus of virtual snow | $\mathrm{Pa}$ |

${E}_{base}$ | Young’s modulus of ski-base material | $\mathrm{Pa}$ |

$\Delta $ | Rigid-body separation | $\mathrm{m}$ |

H | Apparent macro-scale clearance between ski base and virtual snow | $\mathrm{m}$ |

${U}_{e}$ | Macro-scale elastic deformation | $\mathrm{m}$ |

$x,y$ | Micro-scale coordinates | $\mathrm{m}$ |

${h}^{*}$ | Micro-scale topography height function | $\mathrm{m}$ |

f | Nominal load | $\mathrm{N}$ |

A | Nominal micro-scale area | ${\mathrm{m}}^{2}$ |

${A}_{n}$ | Effective micro-scale area | ${\mathrm{m}}^{2}$ |

n | Porosity | − |

$\delta $ | Rigid-body separation | $\mathrm{m}$ |

${p}_{n}$ | Micro-scale pressure distribution | $\mathrm{Pa}$ |

${E}_{ice}$ | Young’s modulus of ice | $\mathrm{Pa}$ |

${u}_{e}$ | Micro-scale elastic deformation | $\mathrm{m}$ |

${h}_{n}$ | Micro-scale clearance/interfacial separation | $\mathrm{m}$ |

${A}_{r,n}$ | Real contact area | ${\mathrm{m}}^{2}$ |

${\overline{h}}_{n}$ | Average interfacial separation | $\mathrm{m}$ |

${\overline{1/h}}_{n}$ | Average reciprocal interfacial separation | $\mathrm{m}$ |

$\nu $ | Poisson’s ratio | − |

$\omega $ | Micro-scale computational domain | ${\mathrm{m}}^{2}$ |

${\omega}_{g}$ | Part of $\omega $ where there is a gap | ${\mathrm{m}}^{2}$ |

${\omega}_{c}$ | Part of $\omega $ where there is solid–solid contact | ${\mathrm{m}}^{2}$ |

$|{\omega}_{*}|$ | Area of ${\omega}_{*}$ | ${\mathrm{m}}^{2}$ |

## Appendix A. Flow Chart of the Numerical Solution Procedure

**Figure A1.**A Flow chart of the solution procedure for solving the contact mechanics problem using the variational principle. Here, ${z}_{l}$ and ${z}_{u}$ describe the shape of the lower and upper surfaces, respectively; r is the relaxation coefficient; k is the tilting coefficient; and $\u03f5$ is the tolerance to determine points belonging to the contact plane.

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**Figure 1.**An illustration of the methodology used in the present work. At the macro-scale, the load (F), equivalent to the plantar pressure exerted by the skier, results in the apparent ski-camber height profile (H) with the corresponding apparent pressure distribution (P), with both being functions of macro-scale coordinate X; see the upper-left insert. At the micro-scale, the apparent pressure (P), distributed over the nominal micro-scale area (A), represents the nominal load (f). As such, it may be considered a coupling variable in the present multi-scale method, which relates the macro- and micro-scales. The lower-left insert presents a schematic of the micro-scale solution corresponding to macro-scale coordinate X, where the ski base and the worn snow particles (illustrated as shaved-off spheres) are in contact, i.e., $P>0$, and $H=0$. Similarly, the lower-right insert presents a schematic of the micro-scale solution, at macro-scale coordinate X, where the ski base is fully separated from the snow surface, i.e., $P=0$, and $H>0$. In both these inserts, the micro-scale solution is presented both in terms of contact area (red) and interfacial separation (in shades of blue) on top of the worn snow particles, and inverted topography of the in-contact surfaces (with peaks to valleys coloured in yellow to blue). The vertical plane, separating the two views of the solution, also shows a cross-section of the in-contact (

**lower left**) and out-of-contact surfaces (

**lower right**).

**Figure 2.**Macro-scale contact mechanical responses, in terms of H (grey) and P (red), of Ski A (

**left**) and Ski B (

**right**) sliding in the direction of the black arrow. The load is equivalent to 40 $\mathrm{k}$$\mathrm{g}$, and it is located $1.6$ $\mathrm{d}$$\mathrm{m}$ behind the balance point of the ski (red arrow). The resulting apparent contact area and mean apparent pressure for Ski A are $63.98$ $\mathrm{c}$$\mathrm{m}$${}^{2}$ and $61.40$ $\mathrm{k}$$\mathrm{Pa}$, and for Ski B, $121.84$ $\mathrm{c}$$\mathrm{m}$${}^{2}$ and $32.24$ $\mathrm{k}$$\mathrm{Pa}$. The elastic modulus of virtual snow is 200 MPa, and the skis are $4.4$ $\mathrm{c}$$\mathrm{m}$ wide.

**Figure 3.**Illustration of the two ski-base textures, Grind 1 (

**a**) and in Grind 2 (

**b**), considered in the present work. (1) The topography of the measured ski-base texture flipped upside down. The colour map goes from blue (at the bottom of the groves) to red (at the peak of the ridges), and while the range is set individually for the two surfaces, they share the same height scale. (2) An illustration of the ski-base texture’s topography being deformed (in situ) while in contact with virtual snow. (3) The same topography as in (2) but with a colour map in shades of blue representing interfacial separation (the darker, the larger), where red marks contact.

**Figure 4.**Illustration of how effective pressure ${P}_{n}$ is used as a coupling variable from the macro-scale (

**left**) to the micro-scale (

**right**).

**Figure 5.**The distributions of percentage real area of contact (${A}_{r}/A\times 100$) (blue and red) for Skis A and B sliding in the direction of the black arrow, with Grinds 1 and 2, under a load equivalent to 40 $\mathrm{k}$$\mathrm{g}$ located $1.6$ $\mathrm{d}$$\mathrm{m}$ behind the balance point of the ski (red arrow). The macro-scale contact mechanical response is also shown with the ski-camber profile (grey) and snow deformation (black).

**Figure 6.**The average interfacial separation ($\overline{h}$) (blue and red) for two skis (Skis A and B) sliding in the direction of the black arrowwith two different grinds (Grind 1 and 2) under 40 $\mathrm{k}$$\mathrm{g}$ of load located $1.6$ $\mathrm{d}$$\mathrm{m}$ behind the balance point of the ski (red arrow). The macro-scale contact mechanical response is also shown with the ski-camber profile (grey) and snow deformation (black).

**Figure 7.**The average reciprocal interfacial separation (${\overline{1/h}}_{n}$) (blue and red) for Skis A and B sliding in the direction of the black arrow, each with the different ski-base textures, Grinds 1 and 2, under 40 $\mathrm{k}$$\mathrm{g}$ of load located $1.6$ $\mathrm{d}$$\mathrm{m}$ behind the balance point of the ski (red arrow). The macro-scale contact mechanical response is also shown with the ski-camber profile (grey) and snow deformation (black).

**Table 1.**Surface roughness parameters for the ski-base textures shown in Figure 3.

Texture | ${\mathit{S}}_{\mathit{a}}$ ($\mathsf{\mu}$m) | ${\mathit{S}}_{\mathit{q}}$ ($\mathsf{\mu}$m) | ${\mathit{S}}_{\mathbf{sk}}$ (-) | ${\mathit{S}}_{\mathbf{ku}}$ (-) | ${\mathit{S}}_{\mathbf{dq}}$ ($\mathsf{\mu}$m/mm) | ${\mathit{S}}_{\mathbf{pk}}$ ($\mathsf{\mu}$m) | ${\mathit{S}}_{\mathit{k}}$ ($\mathsf{\mu}$m) | ${\mathit{S}}_{\mathbf{vk}}$ ($\mathsf{\mu}$m) |
---|---|---|---|---|---|---|---|---|

Grind 1 | 1.77 | 2.16 | −0.16 | 2.58 | 65.79 | 1.44 | 6.01 | 1.92 |

Grind 2 | 8.77 | 9.62 | −0.43 | 1.63 | 185.77 | 2.16 | 11.65 | 20.02 |

${\mathit{A}}_{\mathbf{tot}}$ | Grind 1 | Grind 2 |
---|---|---|

Ski A | 31.52 mm^{2} | 24.62 mm^{2} |

Ski B | 32.89 mm^{2} | 24.52 mm^{2} |

$\mathit{F}/{\mathit{A}}_{\mathbf{tot}}$ | Grind 1 | Grind 2 |
---|---|---|

Ski A | 12.46 MPa | 15.95 MPa |

Ski B | 11.94 MPa | 16.02 MPa |

**Table 4.**The minimum average interfacial separation along the ski for the 4 combinations of skis and grinds.

$min\left({\overline{\mathit{h}}}_{\mathit{n}}\right)$ | Grind 1 | Grind 2 |
---|---|---|

Ski A | 3.31 μm | 10.94 μm |

Ski B | 3.55 μm | 11.32 μm |

**Table 5.**The total average reciprocal interfacial separation for the 4 combinations of skis and grinds.

${\int}_{\mathbf{\Omega}}{\overline{1/\mathit{h}}}_{\mathit{n}}\left(\mathit{X}\right)\phantom{\rule{0.166667em}{0ex}}\mathit{dX}$ | Grind 1 | Grind 2 |
---|---|---|

Ski A | 5.47 km | 3.44 km |

Ski B | 7.97 km | 4.51 km |

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## Share and Cite

**MDPI and ACS Style**

Kalliorinne, K.; Hindér, G.; Sandberg, J.; Larsson, R.; Holmberg, H.-C.; Almqvist, A.
Characterisation of the Contact between Cross-Country Skis and Snow: On the Multi-Scale Interaction between Ski Geometry and Ski-Base Texture. *Lubricants* **2023**, *11*, 427.
https://doi.org/10.3390/lubricants11100427

**AMA Style**

Kalliorinne K, Hindér G, Sandberg J, Larsson R, Holmberg H-C, Almqvist A.
Characterisation of the Contact between Cross-Country Skis and Snow: On the Multi-Scale Interaction between Ski Geometry and Ski-Base Texture. *Lubricants*. 2023; 11(10):427.
https://doi.org/10.3390/lubricants11100427

**Chicago/Turabian Style**

Kalliorinne, Kalle, Gustav Hindér, Joakim Sandberg, Roland Larsson, Hans-Christer Holmberg, and Andreas Almqvist.
2023. "Characterisation of the Contact between Cross-Country Skis and Snow: On the Multi-Scale Interaction between Ski Geometry and Ski-Base Texture" *Lubricants* 11, no. 10: 427.
https://doi.org/10.3390/lubricants11100427