# A numerical framework for discrete modelling of friction and wear using Voronoi polyhedrons

Guilhem Mollon

► To cite this version:

Guilhem Mollon. A numerical framework for discrete modelling of friction and wear using Voronoi polyhedrons. Tribology International, 2015, 90, pp.343-355. 10.1016/j.triboint.2015.04.011. hal-04710223

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Submitted on 27 Feb 2026

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
A numerical framework for discrete modelling of friction and wear using
Voronoi polyhedrons

Guilhem Mollon *

Université de Lyon, INSA Lyon, LaMCoS, UMR CNRS 5259, France

In this work, we present a numerical framework for the modelling of friction and wear of rough surfaces. This framework is based on the Discrete Element Method (DEM), and aims to mimic the inner microstructure of polycrystalline materials. For this purpose, a sample generation program has been developed, into which each grain of this microstructure is represented by a rigid 3D Voronoi cell. A typical sample generated by this program represents a body composed of a degradable and a non degradable parts, both of which presenting several well controlled features: size and geometry of the grains (controlled by the positions of the seeding points of the Voronoi tessellation), roughness of the surface and of the underlying substrate (controlled by the means of the frequency content and possible anisotropy of this roughness), and periodic or planar lateral boundaries. These features enable the modelling of a large number of contact situations with a great flexibility. After presenting in details the algorithmic routines developed for this sample generation, we detail some examples of generated samples and we provide insight of their implementation in the implicit discrete code LMGC90. An Illustrative simulation is then performed to demonstrate the ability of this framework to model diverse situations of friction and wear, by representing accurately the creation and the rheology of third bodies. © 2015 Elsevier Ltd. All rights reserved.

**1. Introduction**

The understanding and prediction of friction and wear occupy a
central position in tribology, and these phenomena may be efficiently
described in the context of the tribological triplet which was first
proposed by Godet and Berthier [1,2]. In this conceptual framework, a
frictional contact is investigated at three different scales which form
the tribological triplet: the scale of the mechanism surrounding the
contact, the scale of the two bodies in contact (called first bodies), and
the scale of the so called third body, i.e. the interfacial material coming
from the degradation of first bodies or entering into the contact from
the outside. This last scale is particularly complicated to study, because
it is the place for many mechanical transformations (damage, fracture,
fatigue, plasticity, phase change, etc.), many physical phenomena (heat
production, chemical reactions, electrostatic interactions, etc.) and
because its confinement prevents it from a convenient experimental
measurement. Observations of third body in various contact condi
tions [3,4] reveal that it may be more or less heterogeneous and
continuous, that its thickness may have various orders of magnitudes
(from a few nanometres to several micrometres), that a large part of its

content comes from the degradation of first bodies, and that it flows
into the contact with an unknown rheology. Continuous modelling
[5,6] of such a medium is very challenging, because it involves very
large deformations, frequent discontinuities, and complex constitutive
laws. Alternatively, successful attempts have been made by the
numerical tribology community to apply the concepts of Discrete
Element Modelling (DEM) to this specific problem. These techniques
are now rather common in the field of granular materials [7 9], and
offer the kinematic flexibility that lacks in continuum approaches
since they consist in representing the motion of each body composing
a granular assembly, and thus do not rely on any continuity assump
tion. Recent works on granular matter have consisted in introducing
more and more complex [10 14] and realistic [15 18] granular shapes
in such simulations, inspired by the progress of the experimental
assessment of granular morphologies [19,20]. In the field of tribology,
however, realistic shapes have not been very much investigated since
the DEM is more used as a convenient tool to represent the flow of
matter in a phenomenological way [21 23] than as a close reproduc
tion of the micromechanical behaviour of the third bodies. Rather,
efforts have been put in using more sophisticated or realistic contact
laws in order for the numerical phenomenology to approach the
experimental one [24 26]. In the last decade, DEM has been often
used in tribology to model the first body degradation and the third
body flow [27 29], and current trends consist in coupling discrete

<footnote>* Tel.: +33 472438937; fax: +33 478890980.</footnote>

<footnote>E-mail address: guilhem.mollon@gmail.com</footnote>
methods with finite elements to perform multi scale modelling [30],
or in enriching the models with other physical ingredients such as
heat or electricity [31 33].

Interestingly, the DEM has also been used in the field of fracture mechanics, to model the initiation and propagation of cracks in materials [34 37]. In particular, the work of Sadeghi and co workers on rolling contact fatigue [38 40] is of great interest, because it deals with microstructural cracks which develop at the scale of interest for tribologists. In these papers, the microstructure of a polycrystalline material is represented at small scale by a Voronoi tessellation, and each cell of this tessellation is then considered as an independent rigid body subjected to the laws of motions. This approach is then used in a DEM framework to study the microstructural damage of the material. In the present paper, we wish to use concepts devel oped in this set of papers to propose a DEM framework for the modelling of the superficial degradation of first bodies subjected to frictional contact, and then to follow the evolution of the damaged material as a third body within the tribological interface. The present work mostly focuses on methodological statements about the gen eration and implementation of such a model, and proposes several evolutions which might be implemented in further studies to improve our understanding and prediction of friction and wear.

## 2. Sample generation

### 2.1. *Principle of the method*

In this paper, the surface of a solid object is modelled in a simplified way, using the schematization presented in Fig. 1. We focus on a portion of this surface represented by a rectangular domain of dimensions Lₓ and Lᵧ. The z axis corresponds to the direction normal to the solid surface (this direction will be consid ered as vertical in this work). Since we are interested in the modelling of phenomena such as friction and wear, it is interesting to consider a certain degradable depth H of the material composing the solid surface. Hence, the model accounts for a certain volume of material, delimited by two surfaces Sₓ and Sᵧ along the x axis, two surfaces Sᵧ and S'ₓ along the y axis, and two surfaces S_z and S'_z along the z axis. The four lateral surfaces may be planar (i.e. the sample may be delimited by vertical walls), but they may also represent periodic geometric boundaries. In the most common case, the surface S_z might exhibit some roughness, as well as the lower horizontal boundary S'_z. In some situations, the average depth H of S'_z may represent a distance from the surface at which such phenomena as wear and degradation have a limited effect, and its choice will thus strongly depend on the considered material and on the contact conditions. In other situations, it may also represent the depth of a soft coating on a harder material, or a change of the mechanical properties of the material composing the solid. In any case, however, we suppose that the volume of interest (called "superficial layer" or

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Layout of the generated samples. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

"degradable first body" in this work, and represented in pink in Fig. 1) lies on an underlying material (called "substrate" or "non degradable first body", in light blue in Fig. 1), which is considered as perfectly rigid and will be used as a lower boundary condition. The lower border of this substrate is a surface S_z' with an arbitrary geometry.

In order to represent the evolution of the superficial layer submitted to a frictional contact and the possible creation of third body, it is desirable to introduce its microstructure in a geometrically relevant way. In this work, this microstructure is modelled by the means of a large number of polyhedral objects which form a complete partition of the superficial layer. The underlying assumption is that most of the deformation and degradation of the material will be related to relative motions located at the joints between these polyhedrons, which will thus be considered as individually rigid. Hence, the scope of this work is restricted to materials exhibiting a discrete microstructure, such as polycrystalline materials for example, and the case of amorphous materials may not be properly accounted for in the present model.

In order to generate these polyhedrons (which will be termed as "grains" in the remainder of this paper), a Voronoi tessellation is adopted. This common approach is implemented in many commercial computational packages, and provides a partition of an infinite space of arbitrary dimension, based on a given set of seeding points. Each "Voronoi cell" is attached to a certain seeding point P, and describes a portion of the space corresponding to the set of all the points which are closer to P than to any other seeding point. The Voronoi cells are convex, they do not overlap, and their union covers the totality of the considered space. In 3D, at the heart of the cloud of seeding point, they are convex polyhedrons and can readily represent the microstructure of a material. How ever, when approaching the border of the cloud of seeding points, most of these cells are "open" because they have some infinite dimensions. Hence, the method of sample generation proposed in this paper will consist in creating a set of seeding points in the superficial layer and in the substrate, perform a Voronoi tessellation to generate the corresponding cells, and apply some special techniques to deal with the cells located at the boundaries. The remainder of this section will detail the algorithms developed in order to generate the rough geometries of the surfaces S<sub>z</sub> and S'<sub>z</sub>, to generate a proper microstructure in the neighbourhood of these surfaces, and finally to deal with the lateral geometric conditions.

### 2.2. *Generation of rough surfaces*

For numerical purposes, the surfaces S<sub>z</sub> and S'<sub>z</sub> are defined in a discrete way, using a fixed mesh on the rectangular domain. More specifically, the domain of dimensions L<sub>x</sub> × L<sub>y</sub> is discretized into N<sub>x</sub> × N<sub>y</sub> identical rectangular elements, each of which is then divided into two triangles. Assigning a given elevation z(n<sub>x</sub>, n<sub>y</sub>) to each node of this grid thus defines an univocal triangulated surface, which represents an acceptable approximation of a real surface if the mesh is fine enough. In any case, each triangular facets should be small enough when compared to the typical scale of fluctuation of the surface elevation, and the appropriate mesh fineness is thus strongly problem dependant. In the present case, the values N<sub>x</sub> = N<sub>y</sub> = 100 are used.

There is a rich scientific literature about the characterization of morphologies of real surfaces at diverse scales (e.g. [41,42]). Without loss of generality, we employ in the present work a simple method of surface generation based on discrete random fields. Such fields are widely used in many scientific and engineering areas [43 45,17], and several mathematical methods of generation have been proposed in order, for example, to deal with non Gaussian probability density functions or with complicated autocorrelation patterns [46,47]. For the sake of simplicity, the one we use in this work is very basic and is based on the discrete 2D Fourier transform. The frequency content of
the surface roughness is controlled by the means of a user defined 2D discrete spectrum, defined on the abstract space of the 2D Fourier modes. More specifically, one has to choose the (real) amplitudes $A(n_x, n_y)$ of the Fourier modes $(n_x, n_y)$, with the following constraints:

$$1 \le n_x \le N_x + 1 \quad \text{and} \quad 1 \le n_y \le N_y + 1 \qquad (1)$$

$$A(n_x, N_y/2+1) = 0 \quad \forall n_x \quad \text{and} \quad A(N_x/2+1, n_y) = 0 \quad \forall n_y \qquad (2)$$

$$A(n_x, n_y) = A(n_x, N_y+2 \ n_y) \quad \forall n_x \qquad (3)$$

$$A(n_x, n_y) = A(N_x+2 \ n_x, n_y) \quad \forall n_y \qquad (4)$$

In other words, the amplitude of the 2D spectrum should be defined on the $(N_x+1) \times (N_y+1)$ discrete space, be null on the row $N_y/2+1$ and on the column $N_x/2+1$, and symmetric with respect to this row and this column. Again, for the sake of simplicity and without any loss of generality, we adopt here the following general formulation for the input spectrum:

$$A(n_x, n_y) = \exp[\lambda_x(n_x \ n_{x0}) \ \lambda_y(n_y \ n_{y0})] \\ \text{for } n_{x0} \le n_x \le N_x/2 \text{ and } n_{y0} \le n_y \le N_y/2 \\ A(n_x, n_y) = 0 \text{ elsewhere} \qquad (5)$$

hence, we set the amplitude of a fundamental mode $(n_{x0}, n_{y0})$ to 1, and we apply an exponential decay of parameters $\lambda_x$ in the x direction and $\lambda_y$ in the y direction, before applying the symmetry conditions (3) and (4). Each of these modes is then attributed a phase angle $\theta(n_x, n_y)$, randomly sampled between $\pi$ and $2\pi$. Once again, a specific symmetry condition is to be applied:

$$\theta(n_x, n_y) = \theta(N_x+2 \ n_x, N_y+2 \ n_y) \qquad (6)$$

Then, the complex Fourier spectrum is computed by

$$Z(n_x, n_y) = A(n_x, n_y) \exp[i \cdot \theta(n_x, n_y)] \qquad (7)$$

A random discrete 2D signal $z_c(n_x, n_y)$ is then obtained directly by a 2D Fast Fourier Transform (FFT) applied to $Z(n_x, n_y)$. If conditions (1) (4) and (6) are verified, this signal is real. Finally, the current standard deviation $\sigma_c$ of the field is computed, and the final surface $z(n_x, n_y)$ is obtained after scaling, in order to match a desired mean value $\mu$ and a desired standard deviation $\sigma$:

$$z(n_x, n_y) = z_c(n_x, n_y) \frac{\sigma}{\sigma_c} + \mu \qquad (8)$$

Three examples are provided in Fig. 2. For each one, the left hand column describes the discrete amplitude decay in the x and y directions, the central column provides the amplitude of the whole

<figure><img src="image_3.png" /><figcaption>Fig. 2. Examples of rough surface generation on the unit square, with $N_x$ $N_y$ 100, $n_{x0}$ $n_{y0}$ 2, $\mu$ 0, and $\sigma$ 0.03; (a) $\lambda_x$ $\lambda_y$ 0.15; (b) $\lambda_x$ $\lambda_y$ 0.6; (c) $\lambda_x$ 0.6 and $\lambda_y$ 0.15</figcaption></figure>
2D spectrum, and the right hand column provides an illustrative realization of the rough surface on the unit square, for a given set of random phase angles. From this figure, we observe that the gener ated surfaces are bi periodic, and that a good control is achieved on the amplitudes and frequencies of the roughness in both x and y directions. This methodology will be used in the remainder of this paper, but any other method of surface generation may be applied as well in the framework described hereafter.

2.3. Discretization of the upper face

As stated in a previous subsection, the use of Voronoi tessellation provides a nice set of conforming and non overlapping convex polyhedrons at the heart of the cloud of seeding points, but leads to open cells when close to a border of this cloud. Hence, if one wants to partition the volume of degradable first body presented in Fig. 1 with convex polyhedrons, special treatments have to be applied in the neighbourhood of the upper, lower, and lateral faces. These techniques are presented in this subsection and the next two ones. To illustrate the techniques described in these subsections, Figs. 3 5 provide illustrations of the employed algorithms in a 2D case. This simplifica tion is used here to ease understanding, but the actual methods that were developed are in 3D and rely on the same concepts.

Fig. 3a presents a random cloud of seeding points. A typical distance d between neighbouring points is computed in the following way:

$$
d = \sqrt{S_{tot}}/n \quad \text{in 2D} \tag{9}
$$

$$
d = \sqrt[3]{V_{tot}}/n \quad \text{in 3D} \tag{10}
$$

In these expressions, S<sub>tot</sub> (respectively V<sub>tot</sub>) represents the total surface (respectively volume) into which n seeding points have been generated. Fig. 3b presents a random curve crossing this cloud and representing in 2D the desired upper surface S<sub>z</sub>. This surface is discrete, i.e. it is a set of segments in the 2D case, and a set of triangular facets in the 3D case, as explained in Section 2.2. From Fig. 3c, it clearly appears that a direct application of a Voronoi tessellation of the cloud of points will not lead to acceptable results. A simple technique would consist in computing the intersections of the curve with the cells it crosses, and to keep only the portions of cells located below the curve. This approach, however, often leads to complicated intersection patterns with very small and very elongated cells (e.g. in black circles in Fig. 3c), which may lead to a superficial microstructure different from the one in the bulk of the material. To overcome this drawback, it is necessary to ensure that S<sub>z</sub> only crosses large cells close to their middle, in order for the lower parts of these cut cells to be large enough. To perform this, we first delete all the seeding points closer to S<sub>z</sub> than a certain distance d<sub>1</sub> = (α<sub>1</sub><sup>2</sup>d) (Fig. 3d). A typical value of α<sub>1</sub> = 1 seems to provide acceptable results. In 3D, the closest distance from a seeding point P to the surface S<sub>z</sub> is not difficult to compute since this surface is triangulated. Hence, a closest projection of a given point P on S<sub>z</sub> only requires a very fast discrete search. This projected point may be a node of the triangulated surface or lie on an edge or on a facet, and directly provides the distance from the seeding point P to S<sub>z</sub>.

After this stage, additional seeding points are input at random positions on S<sub>z</sub> (blue dots in Fig. 3e), and a Voronoi tessellation is finally performed (Fig. 3f). In this tessellation, S<sub>z</sub> only crosses large cells in their middle, and it is easy to determine their intersections with the rough curve/surface (green squares in Fig. 3g). In 2D, these intersection points are then linked by segments in order to close the lower parts of the cut cells, the upper parts are deleted as well as all the cells located above the curve, and these operations result in a satisfactory approximation of S<sub>z</sub> with a proper underlying microtruc ture (Fig. 3h). In 3D, the intersections of the cells edges with S<sub>z</sub> are also easy to compute since this surface is piecewise bilinear. Closing the

lower part of a given cut cell, however, is not straightforward since the different intersections of the cell's edges with S<sub>z</sub> are generally not coplanar. Hence, the vertices of the cell located below S<sub>z</sub> and its intersection points with S<sub>z</sub> are altogether submitted to a convex hull algorithm, which results in a new convex cell with all the vertices located below or exactly on S<sub>z</sub>.

2.4. Discretization of the lower face

This last observation (the non coplanarity of the points of intersection of a polyhedral cell with a non planar surface) is the reason why the method used to discretize the lower face S'z is slightly different from the one presented for S<sub>z</sub> in the previous subsection. Indeed, it is impossible to use a non planar surface to divide a given cell into two conforming convex polyhedrons (at least one of them will have to be non convex). This is not an issue in the case of S<sub>z</sub> because the upper part is discarded, but it prevents from using the same method to define the surface S'z separating the superficial layer and the substrate. An alternative method, illustrated in Fig. 4, is proposed. Fig. 4a c presents the same kind of situation than in the previous subsection, with the same difficulties in Fig. 4c if a Voronoi tessellation is performed directly in the initial cloud of seeding points. Instead of this, we propose to delete all the seeding points that are either (i) located below S'z and within a distance d<sub>2</sub> = (α'<sub>2</sub><sup>2</sup>d) or (ii) located above S'z and within a distance d<sub>3</sub> = (α'<sub>3</sub><sup>2</sup>d) (Fig. 4d). The values α<sub>2</sub> = 4 and α<sub>3</sub> = 0.1 seem to provide satisfactory results. Then, the remaining points located above S'z and within a distance d<sub>2</sub> (Blue dots in Fig. 4d) are "mirrored", i.e. copied in the lower part of the surface at a symmetrical position with respect to their closest projection on S'z (Fig. 4e). A Voronoi tessellation is then performed and results in a set of cells, with the desired surface being approximated by the joints between some of them (Fig. 4f). This approximation is better when the cloud of seeding points is dense and when the curvature of the target surface is limited, but will always result in a more approximate surface than with the procedure described in Section 2.3. However, it provides a separation of the considered space into two sets of conforming convex polyhedrons. Depending on the problem to be treated, the surface S'z may be taken as the one defined in the previous stages (the interface between the superficial layer and the substrate will then respect the input roughness, Fig. 4g), or as the one located above the first layer of cells (the interface will then present the same level of entangle ment than in the microstructure of the bulk material (Fig. 4h).

2.5. Discretization of the lateral faces

At the lateral faces S<sub>x</sub>, S<sub>y</sub>, S<sub>z</sub>, two different kinds of geometrical conditions may be introduced. These faces may be perfectly planar walls (which makes it possible to apply well controlled force or displacement boundary conditions on them), or may present a geometrical periodicity (which makes it possible to model an infinite surface with a controlled periodicity). This second case relies on the fact that the rough surfaces S<sub>x</sub> and S<sub>y</sub> were made bi periodic in the method described in Section 2.2. Moreover, these geometric boundary conditions are not coupled, which means that one may apply one type of condition in the x direction and the other one in the y direction. Fig. 5a presents such a situation, with a cloud of seeding points defined in a 2D L<sub>x</sub> × L<sub>y</sub> rectangular domain. Fig. 5b shows the Voronoi tessellation applied directly on this cloud of points, and illustrates the difficulties related to open cells at the domain boundaries. In Fig. 5c, a method is proposed to enforce a wall type geometric condition in the x direction. This method is similar to that used in the previous subsection, and was also used in [15, 17] for bounded Voronoi tessellation of a convex domain. Each seeding point closer to the plane S<sub>x</sub> than a given distance d<sub>4</sub> = (α'<sub>4</sub><sup>2</sup>d)
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** 2D illustration of the technique used to discretize the rough upper surface. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

is mirrored (i.e. copied at a symmetric position with respect to its orthogonal projection on $S_x$), and the same operation is performed for $S'_x$. On the other hand, Fig. 5d shows how to enforce a periodic geometric condition in the y direction. Each seeding point closer to the plane $S_y$ than the distance $d_4$ is copied at a new position, with the same x coordinate and a y coordinate shifted by a distance $L_y$ equal to the desired periodicity. The same operation is performed for $S'_y$, with a y shift equal to $L_y$. As shown in Fig. 5d, these operations need also to be performed on the seeding points added when previously dealing with the surfaces $S_x$ and $S'_x$. A typical value of $a_4 = 5$ is found to provide good results. A Voronoi tessellation is then performed (Fig. 5e), and only the Voronoi cells corresponding to the initial cloud of seeding points are kept (Fig. 5f). These cells present the desired geometric properties, i.e. in the present case they form planar walls in the x direction and a geometric periodicity in the y direction. The techniques presented in this subsection and in the two previous ones are implemented in the software MATLAB, and are applied on a 3D illustrative sample in Fig. 6, with various lateral geometric conditions.

# 3. Examples

## 3.1. Dimensions of sample and number of cells

Fig. 7 presents four illustrative samples generated on the unit square domain with bi periodic boundaries. The same parameters are used for the random generations of the upper surfaces (i.e. $n_{x0} = n_{y0} = 2$, $\sigma = 0.02$, and $\lambda_x = \lambda_y = 0.6$). The samples shown in Fig. 7a and b illustrate the effect of the number of seeding points introduced in the volume of the superficial layer with the same

average depth $H = 0.2$, since they respectively contain 1000 cells and 20,000 cells. It clearly appears that the number of cells should not be too small if one wants to render properly the desired statistical properties of the upper surface, especially when this surface has important local curvatures. Fig. 7c and d possess respectively 2500 and 7500 cells for respective average depths $H = 0.1$ and $H = 0.3$, and illustrate how the method of sample generation allows for various dimensions of the volume of degradable first body with similar grain size.

## 3.2. Microstructure of the material

In Fig. 8, several examples of possible microstructures are presented, all of which containing 5000 cells on a unit square domain with an average depth $H = 0.2$, and the same roughness parameters. Wall type lateral conditions are applied to represent the intersections of these microstructures with vertical planes. Fig. 8a and b illustrates the cases of regular microstructures, for which the seeding points are located on regular lattices (a cubic and a hexagonal lattice respectively). The case presented in Fig. 8c is analogous to that of Fig. 8b (i.e. regular hexagonal lattice), except that a small perturbation is applied to render it more realistic. Each seeding point is positioned on the lattice, and then moved from a small distance in a random direction of space. This distance is randomly chosen in the interval $[0, d_r]$, where $d_r = (a_r^2 d)$ is a fraction of the average distance between two seeding points as described in Eq. (10). In the case of Fig. 8c we took $a_r = 0.2$, but this parameter may be different and bring an additional degree of freedom in the definition of the material microstructure. Choosing large values of $a_r$ (typically larger than 1) leads to a distribution similar to that of a perfectly random cloud of points.
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** 2D illustration of the technique used to discretize the rough lower surface and the substrate. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** 2D illustration of the technique used to apply wall-type geometric conditions along the x-direction and periodic geometric conditions along the y-direction.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Example of micro-structured degradable first-body containing 5000 grains, with a rough surface and various lateral conditions; (a) walls along x and y; (b) x-periodicity; (c) y-periodicity; (d) bi-periodicity.</figcaption></figure>

Fig. 8d and e illustrates the cases of anisotropic microstructures which may occur in some particular materials. In Fig. 8d the grains are flat (their average vertical dimension is twice smaller than their average horizontal dimension), while in Fig. 8e they are vertically elongated (their average vertical dimension is twice larger than their average horizontal dimension). Such anisotropies cannot be easily introduced by simply acting on the initial set of seeding points, since any dilation of a random cloud of points will leave its isotropy unaltered. Alternatively, a simple technique is used here, based on a dilation of the microstructure itself. In the case of Fig. 8d, for example, the sample was generated using all the techniques presented earlier in this paper, but on a square domain of dimensions 0.5 × 0.5 instead of 1 × 1. Then, at the end of the generation, all the horizontal coordinates of the vertices of the cells were multiplied by a factor 2, resulting in this flat micro structure on the 1 × 1 domain. These operations did not disturb the frequency content of the upper surface, since all the modes introduced in the random surface generation are normalized with respect to the sample dimensions. On the contrary, in Fig. 8e, the

sample was first generated on a 2 × 2 domain and its horizontal dimensions were then divided by 2.

In further developments, it seems possible to implement techniques allowing the introduction of inclined anisotropic microstructures, of controlled distributions of the grain size (like already proposed in [15,17,48]), or making it possible to have an even better control on all the features of the microstructure (see e.g. [49]).

### 3.3. Surface roughness

Fig. 9 illustrates the effect of the parameters introduced during the generation of the rough surfaces $S_x$ and $S_z$ on the generated sample. Fig. 9a presents respectively a low frequency roughness, a high frequency roughness, and an anisotropic roughness of the upper surface $S_x$ of the degradable first body. These different situations seem to be handled well by the generation algorithm, although Fig. 9b shows that it is difficult to render properly a desired very rough surface with a limited number of convex grains. Fig. 9d presents the same kind of results for the surface
<figure><img src="image_8.png" /><figcaption>Fig. 7. Illustrative samples generated on the unit square domain with various average depths and numbers of cells.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 8. Illustrative samples generated on the unit square domain with various inner microstructures.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>**Fig. 9.** Illustrative samples generated on the unit square domain with various types of surface roughness; (a)–(c) roughness of the upper surface $S_z$; (d)–(f) roughness of the lower surface $S_z'$</figcaption></figure>

$S_z'$ separating the degradable and the non degradable parts of the first body. In Fig. 9d (flat interface) and e (low frequency roughness), the strategy described in Fig. 4g is applied, i.e. these interfaces respect the input statistics. In Fig. 9f, however, the strategy of Fig. 4h is applied, and an additional layer of cells is considered to belong to the non degradable part of the first body. This way, the interface is globally flat but the microstructure is entangled in the same way than in the bulk of the material.

# 4. Illustrative simulation

## 4.1. Implementation in the NSCD framework

The Discrete Element Method is a well known numerical model ling framework which was initially proposed by [7], and which is classically based on an explicit time integration of the laws of motion of each grain composing a granular assembly. These laws of motion depend on the forces applied on each grain, and thus require the definition and calibration of proper contact laws which provide the possible attractive, repulsive, and frictional forces between the contacting grains. An alternative approach for dealing with multi body multi contact problems was later proposed in [50] and further developed in [51,52]: the Non Smooth Contact Dynamics (NSCD) approach. In contrast with the classical DEM, the NSCD method is based on an implicit time integration and a non linear contact solver,

and does not require such numerical regularization parameters as normal or tangential stiffness at contact (hence the "non smooth" denomination). Among the advantages of the NSCD approach, one may mention the possibility to use larger time steps than in the classical DEM, and the ability to couple in the same system of equations both rigid discrete elements and deformable continuous elements (using FEM). It has been widely used in many scientific and engineering fields. In the present work we make use of the open source software LMGC90 [53], which implements all the features of the NSCD approach, together with a large choice of contact laws. Moreover, it is able to deal efficiently with complex particle shapes such as convex polyhedrons, by accounting for different types of contact such as node face, edge face, or edge edge.

The samples generated using the method described in Section 2 may thus be introduced in this software, provided that a few "cleaning" operations are performed. The first operation consists in deleting a few particles that may be too small (e.g. with a volume smaller than 1% of the average grain volume) since they would complicate the dynamic simulation without having a significant mechanical contribution. For the same reason, we also ignore the particles which are too elongated (e.g. with a ratio between the largest and the smallest dimension larger than 20), although such particles usually do not occur if the geometrical boundaries are properly accounted for (Section 2). The last operation consists in "collapsing" the too short edges (e.g. shorter than 1% of the average edge length of the given cell) and the too small angles (e.g. smaller
<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Sample generated for discrete modelling; (a) full sample; (b) non-degradable part of the lower first-body (2498 grains); (c) degradable part of the lower first-body (21,925 grains); (d) upper non-degradable first-body (2583 grains, flipped upside-down to show topography).</figcaption></figure>

than 1°) that often appear in the Voronoi cells. The algorithm developed for this purpose is very close to that proposed in [54] to overcome analogous difficulties.

## 4.2. Geometry, contact law and loading

The sample generated for simulation and introduced in LMGC90 is shown in Fig. 10a. It is composed of a lower first body, with a degradable part and a non degradable part, and of an upper first body which is considered as non degradable for the sake of simplicity. The horizontal dimensions are 500 µm × 500 µm, and the average depth of degradable material is $H = 80$ µm. The sample contains about 27,000 polyhedral grains, with an average dimension of the grains close to 10 µm. The contacting surfaces of the upper first body and of the degradable part of the lower first body are randomly generated with the same parameters (i.e. $n_{x0} = n_{y0} = 2$, $σ = 8$ µm, and $λ_x = λ_y = 0.5$, see Fig. 10c and d), and the interface between the degradable and the non degradable parts of the lower first body is globally flat but uses the approach of Fig. 4h to entangle properly the microstructure (Fig. 10b).

Since this sample is used here for illustrative purpose only, a very simple contact law is used for simulations. A non interpenetration condition is implemented by the means of a Signorini condition, and a normal attractive cohesive force is introduced between contacting grains. This force is equal to $n \cdot c$, where n is the number of contact points used by LMGC90 to deal with complex contacts between polyhedrons (for example, n ≈ 3 or 4 for a face face contact, n = 1 for an edge edge contact, etc.) and c is set to 0.1 N in the present case. When applied to a real case, such simulation might require less crude contact laws, as discussed in the last section of this paper.

The sample is bi periodic in the horizontal directions. The loading conditions applied to the sample are very classical, and similar to that usually applied when modelling this type of system. The six degrees of freedom of the non degradable part of the lower first body are fixed, while the upper first body is considered as a unique rigid body, and is fixed in all its rotational degrees of freedom and in the y direction translation. This rigid body is also submitted to an imposed velocity in the x direction and to a downward force in the

<figure><img src="image_12.png" /><figcaption>**Fig. 11.** Timing of the boundary conditions applied to the upper first-body.</figcaption></figure>

$z$ direction. This velocity and this force are applied following the timing detailed in Fig. 11: the vertical force is linearly increased from zero to a value $F_z$ between $t = 0$ and $t = t_1$ and then fixed, while the imposed velocity is equal to zero until $t = t_1$, linearly increased to a value $V_x$ until $t = t_2$, and then fixed. The chosen value for the vertical force is $F_z = 125$ N, corresponding to a moderate (500 MPa) local contact pressure. Besides, we set $V_x = 1$ m/s, $t_1 = 10$ µs, and $t_2 = 20$ µs. The time step for the implicit integration of the laws of motion in the NSCD framework is set to $10^{-9}$ s, the total simulated time is 120 µs, and the simulation is launched on 16 CPU on the computation cluster of the LaMCoS.

## 4.3. Numerical results

Every 0.1 µs, the total normal and tangential forces applied on the moving upper first body are computed, and an instantaneous mobilized friction is thus obtained. These "numerical measures" are plotted in Fig. 12, and appear to be strongly fluctuating. However, the application of a moving average to this discrete signal makes it possible to read the general trend of the system. Hence, we may observe that the mobilized friction increases
progressively until reaching a somewhat constant value (although very strongly fluctuating) at $t \approx 50$ µs. These fluctuations are to be related with local dynamic instabilities, force chains creations and breakages, etc. The average friction ratio corresponding to this established regime is close to 1, which is a very high value but may be realistic in some specific situations. In order to obtain a more realistic behaviour from a quantitative point of view, however, it is obvious that a proper calibration of the local contact laws seems necessary.

Fig. 13 provides a side view and a perspective view of the sample at $t = 100$ µs, with the colours of the grains corresponding

to their respective velocity magnitudes. It clearly appears that the grains belonging to the lower half of the degradable part of the lower first body seem to remain still, and that the microstructure remains mostly unaltered in this region. On the contrary, the grains located in the upper half of the degradable body are submitted to motion, which means that the microstructure and the initial roughness of the surface have been completely degraded by the friction. This observation is in good agreement with the third body theory, which states that the local friction is controlled by the rheology of the degraded material that friction itself has created. Fig. 13b also shows that the velocity field in the third body layer is not homogeneous at all, and that some clusters of particles seem to be moving at a velocity close to that of the upper first body while some others are moving more slowly. One may conclude that there is a high level of dynamic complexity in the third body flow patterns, which might be interesting to study in order to understand their contribution on the macroscopic friction.

Fig. 14 provides some quantitative support to the previous observations, by plotting the individual velocity of each grain as a function of its $z$ coordinate. The obtained velocity field is very heterogeneous and fluctuating, but the average velocity profile clearly shows that the microstructure of the lower first body is mostly unaltered on the first $60$ µm, and that the material is then submitted to a sheared flow on a depth of roughly $40$ µm, which corresponds to the third body layer. It also clearly appears that there is a velocity jump between the upper part of the third body (with an average velocity $V_x = 0.4$ m/s) and the upper first body (with a fixed velocity $V_x = 1$ m/s).

Obviously, the quantitative and qualitative results proposed in this section are not connected to the real behaviour of any system, since no calibration was performed on the contact laws, surface roughnesses, and material microstructures. It is clear that different modelling choices may lead to very different behaviours, which

<figure><img src="image_13.png" /><figcaption>**Fig. 12.** Mobilized friction (i.e. ratio of the tangential and normal forces) along time during the simulation: Instantaneous values, moving average (with a 10 µs time span), and average value in established regime (after 50 µs).</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>**Fig. 13.** Velocity magnitudes of the grains at $t = 100$ µs; (a) side view, including the upper substrate; (b) perspective view, with the upper substrate omitted for the sake of visibility.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Fig. 14.** Velocity magnitude of the grains at *t* 100 µs as a function of their z-coordinate, and corresponding average velocity profile.</figcaption></figure>

may help to understand the connexions between the local scale and the global scale. This will be the topic of future studies.

## 5. Perspectives

Thanks to its generality and its flexibility, the numerical framework presented in this paper may have an interesting potential in future modelling of friction and wear in dry contacts. One important condition for this type of model to be realistic, however, is the implementation and use of more physical contact laws. For the grains belonging to the degradable first body, such ingredients as normal and tangential contact stiffness (in order to represent more accurately the stress state in the bulk of the material and the Poisson effect, like for example in [38]) and a damage law (in order to represent the strength evolution at the grain joints submitted to complex loadings [26]) should be introduced in the model. On the other hand, for the grains belonging to the third body (i.e. completely separated from the first bodies), surface energy evolving with the physical environment of the grain (e.g. temperature, chemical reactivity [55] or gas contamination [56]) represents a promising trend.

From a purely mechanical point of view, a possible evolution of the model may account for the deformability of the non degradable parts of the first bodies (in order to investigate the possible stress redistributions induced by the degradation of the first bodies) or of the individual grains (in order to approach more realistically the phenomenology of the third body, which might not always be considered as a granular collection of rigid bodies). Such approach may be implemented in LMGC90 thanks to the natural ability of NSCD to couple DEM and FEM. These possible evolutions, however, will require a proper calibration based on experimental observations and measurements.

## Acknowledgement

The members of the team developing the LMGC90 software in the University of Montpellier, France (F. Dubois, R. Mozul, and

M. Renouf) have been very helpful during this project, and are gratefully acknowledged.

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