Modelling of Grain Motion for Three-body Abrasion

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ScienceDirect Procedia CIRP 31 (2015) 282 – 286

15th CIRP Conference on Modelling of Machining Operations

Modelling of grain motion for three-body abrasion I. Lorescha,*, O. Riemera a

Laboratory for Precision Machining LFM, Badgasteiner Str. 2, 28359 Bremen, Germany

* Corresponding author. Tel.: +49-421-218-51169; fax: +49-421-218-51119. E-mail address: [email protected]

Abstract Surface formation for three-body abrasion is essentially more difficult to model than for two-body abrasion. In contrast to grains embedded in the polishing pad and therefore time-invariant tool geometry for two-body abrasion, here a variable tool geometry is prevailing as a result of the rolling abrasive grains. Nevertheless, three-body abrasion is an important abrasive process in the surface finishing of glass and ceramics and the modelling of the surface formation for mechanical abrasive machining with loose grains is required for a better process understanding and enhanced process design. The modelling of the surface formation for three-body abrasion is dealt with by splitting the integral removal volume under the polishing pad into the removal of single grains and distributing the total polishing force according to the interaction of individual grains with the workpiece material. This new methodical approach intends to determine the material removal in discrete time steps for every grain and further on the generated new surface topography. Data from polishing experiments like concentration and form of abrasives, polishing load, surface topography and integral removal are applied to set up this analytic-empiric model. Crucial elements of this model are the geometry of single grains, their rolling motion under load and the principle of material removal. In this paper the simulation of the rolling motion of single abrasive grains using a numerical time step model is introduced.

© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license © 2015 The Authors. Published by Elsevier B.V. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of The International Scientific Committee of the “15th Conference on Modelling of Machining Operations”. Peer-review under responsibility of the International Scientific Committee of the “15th Conference on Modelling of Machining Operations Keywords: Modelling, material removal, surface topography, three-body abrasion

1. Introduction For abrasive polishing, the abrasive grains are admixed to a paste or fluid and then supplied to the polishing process. Depending on process kinematics and geometrical conditions as well as the governing process parameters, the abrasive grains can principally either be embedded in the polishing pad and slide across the workpiece surface or roll freely in the active polishing zone. From these general grain-workpiece interactions different material removal mechanisms occur – the so called two- or three-body abrasion; also mixed conditions are possible. The nature of the removal mechanism strongly influences the resulting surface topography and therefore surface quality as well as removal rate, which are decisive criteria for the effectiveness and economic efficiency of the machining method. The abrasion by solid particles in a fluid stream (also erosion) was investigated amongst others by Finnie and Bitter extensively [1-3], also with regard to twoand three-bodies abrasion [4]. For mechanical abrasion

Tresona et al. and Adachi and Hutchings carried out investigations, to find out the dependence of the removal mechanism on the relevant process parameters [5-6]. Heisel has derived the dependence of the removal mechanism on the grain form [7]. Due to its occurrence in grinding processes two-body abrasion is still investigated to much greater depth than three-body abrasion, e.g. [8-10]. The research of threebody abrasion is substantially more complicated, because on the one hand it is difficult to separate it completely from twobody abrasion and on the other hand the rolling motion of the grains causes an ever changing tool geometry. Indeed, the trajectory of single grains is directed by the movement of the polishing pad, nevertheless, the relative strength and the interaction between grains and their surroundings (pad, work piece, as well as other grains) can induce a deflection. A model which considers an entire polishing system including material removal and the simulation of surface formation with rolling grains (three-body abrasion) is not known from the current literature. In this paper a numerical model and a

2212-8271 © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the International Scientific Committee of the “15th Conference on Modelling of Machining Operations doi:10.1016/j.procir.2015.03.083

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simulation of the rolling movement of single grains is presented, which considers the geometry and initial position of the grains in every step, as well as the surface topography of the workpiece. This simulation is understood as the first step in the development of a comprehensive model for the surface formation through three-body abrasion. Fig. 2. The five Platonic solids: tetrahedron, hexahedron (cube), octahedron, dodecahedron and icosahedron (from left to right).

2. Modelling The abrasive grains are often modeled as spheres, cf. [11], what simplifies their mathematical description substantially. For more realistic approaches the grains can rather be described by polyhedrons, cf. Fig. 1. Pellegrin and Stachowiak have introduced the threedimensional description of abrasive grains in their work [12]. Therefore, they sliced oft polygonal faces, starting from a simple polyhedral primary solid, e.g. an unit cube. With this approach polishing grains can be reproduced most realistically. Nevertheless, the simulation of the rolling motion of such grains, in particular over a rough surface, gets extremely complicated.

The five kinds of Platonic solids offer a reasonable variety for the simulation close to reality; nevertheless, the program complexity remains manageable. Thus, the real grain mixture is reproduced only by information on the distribution of the grain form and on the distribution of the grain size.

2.1. Rolling grains on flat surfaces The movement of the grains during three-body abrasion is predefined by the overall movement of the polishing pad (here declared as feed direction). If the grains are modeled as a sphere, the trajectory of the grain conforms to the path of the polishing pad. For the rolling motion of a polyhedron the trajectory is, in addition, determined by the starting alignment of the polyhedron. The following three starting alignments are possible: the grain rests on a vertex, the grain rests on an edge or the grain lies on one of the faces. The different initial positions cause different tilting behavior. Thus, an octahedron, which rests on a vertex, has four edges and four faces as next possible positions. However, an octahedron, which lies on a face, only has three vertexes and three edges as the next possible position, cf. Fig 3.

Fig. 1. Light microscopy of polishing slurry with abrasives exhibiting diamond grains with a nominal, average grain diameter of dg = 30 μm.

Diamond grains applied in real machining processes like grinding or polishing can be classified from blunt to sharp types and associated with simple geometric bodies like a cube or a tetrahedron. To receive the regularity and mathematical simplicity of a sphere, for imaging the grains, while complying with realistic conditions, in this work the abrasives were modeled as completely regular convex polyhedrons, which in geometry are called Platonic solids (Fig. 2).

Fig. 3. Rolling behavior according to the starting alignment of hexahedron (cube) and octahedron.

For the simulation the next tilting direction of the grain is determined randomly from all possible directions. To bring this in compliance with the given movement of the polishing pad, some restrictions are introduced for this decision. If vector ݀ ‫ א‬Թଶ  is the feed direction, and ݀ଵ ǡ ǥ ǡ ݀௡ ‫ א‬Թଶ are the possible rolling directions of the polyhedron, with the condition ‫݀ۃ‬௜ ǡ ݀‫ ۄ‬൐ Ͳ for all ݅ ‫ א‬ሼͳǡ ǥ ǡ ݊ሽ all rolling directions are excluded, which are opposite to the feed direction; thus, backward motions are not permitted. As next step for every rolling direction ݀௜ , a probability ‫݌‬௜ ‫ א‬ሾͲǡͳሿ is determined, which gives the probability that the

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polyhedron rolls in this specific direction. Hereby, all ‫݌‬௜ should fulfil the conditions: 1.

n

¦p

1

i

i 1

2. pi

1 œ di

3. pi

p j œ di , d

d dj,d

With ~ pi :

– 1  n

j 1 iz j

d j,d

2



pi :

¦

~ pi n j 1

~ pj

(1)

all conditions are fulfilled. In Fig. 4 examples are given for weighted probabilities, which are marked by red arrows, after the principle described above.

Fig. 4. Probability of tilting directions (expressed by length of the arrows) for hexahedron (cube), icosahedron and octahedron.

2.2. Rolling grains on structured surface For simulating the rolling motion of abrasive grains on a rough surface the feed direction is fixed at first. Secondly, the surface is defined by modelling first of all regular structures, e.g. flat, sine wave, corner cubes, double sine wave or others including their orientation. In the next step the grain mixture is defined by specifying the distributions of grain form and grain size; start positions of grains are randomly chosen. The tilting behavior of single grains is additionally influenced by the surface topography and feed velocity when rolling over a rough surface. In Fig. 5 surface topography is modelled in a first approach by a linear sine wave structure with varying orientation (α) and the rolling behavior of single gains is shown for a low feed velocity.

Fig. 5. Trajectories of rolling grains on a structured surface for a constant, low feed velocity (݇ ൌ ʹͲ).

The grains are deflected as soon as they engage with structures, which do not run in the feed direction. So further weightings, which consider the obstructions imposed by the surface topography, are required here for each tilting direction of the polyhedron. For this ݄ଵ ǡ ǥ ǡ ݄௡ ‫ א‬Թ are the height differences, which have to be overcome by the polyhedrons when rolling in the directions ݀ଵ ǡ ǥ ǡ ݀௡ . Here ݄௜ ൏ Ͳ means that the polyhedrons roll downhill in the direction of ݀௜ Ǥ First, the probabilities ‫݌‬௜ ǡ ǥ ǡ ‫݌‬௡ are calculated according to equation (1). Now the probabilities ܲଵ ǡ ǥ ǡ ܲ௡ ‫ א‬ሾͲǡͳሿ are determined, which consider the height differences. If pi Pi : n (2) k (h h ) ¦ p je i j j 1

then n

¦P i

1

i 1

is valid as desired. For the case that the height in all rolling directions is the same, which means ݄ଵ ൌ ݄ଶ ൌ ‫ ڮ‬ൌ ݄௡ then ܲ௜ ൌ ‫݌‬௜ is valid appropriately for all ݅ ‫ א‬ሼͳǡ ǥ ǡ ݊ሽ. In addition, the limit behavior is reasonable for the probabilities ܲ௜ : lim Pi 0 for pi z 1 hi o f

lim Pi

1

for pi z 0

lim Pj

0

j z i for pi z 0.

hi o f hi o f

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Fig. 7. Trajectory of rolling grains on a structured surface for a high feed velocity ݇ ൌ ʹ (a); and low feed velocity ݇ ൌ ʹͲ (b).

This effect becomes more evident, when grain size and structure height are in the same order of magnitude, cf. Fig 8. Here the obstructions are avoided only when ݇ is large (Fig. 8 (b)), otherwise they show only a minor influence on the rolling direction (Fig. 8(a)). Fig. 6. Trajectories of rolling grains on a structured surface for a constant, high feed velocity (݇ ൌ ʹ).

In equation (2) the factor ݇, a scalar, is introduced. This factor determines how strong the surface topography influences the probability regarding the tilting direction of the rolling grains. If the factor remains low, obstructions in the surface are nearly neglected and the grains roll in the feed direction (Fig. 6, ݇ ൌ ʹ ), as for high velocities in the polishing process. If the factor grows, the grains avoid the obstructions (Fig. 5, ݇ ൌ ʹͲ), reflecting the behavior of the abrasive grains at low feed velocities. The feed velocity is included in the model for the decision on the tilting behavior of single grains through the factor ݇ which is invers proportional to the feed velocity. The actual correlation of feed velocity and ݇ has to be adapted to the respective real process. In Figs. 7 and 8 (here surface topography is modelled as corner cubes and double sine wave respectively) the effect of the factor ݇ on the grains rolling over surface structures is illustrated; with a factor of ݇ ൌ ʹͲ the grains avoid the obstructions and remain in the valleys, while with ݇ ൌ ʹ the grains partially roll over the peaks.

Fig. 8. Trajectory of rolling grains on a structured surface for a high feed velocity ݇ ൌ ʹ (a); and low feed velocity ݇ ൌ ʹͲ (b).

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3. Summary and Outlook In this work the simulation of the rolling motion of abrasive grains close to a real polishing process in three-body abrasion was accomplished. The grains – here mimicked as Platonic solids – are specified by the distributions of grain form and grain size in the simulation according to the actual grain mixture in the polishing process. The rolling behavior of the grains is determined, apart from the workpiece surface, by the process parameters like feed velocity of the polishing pad. These information have influence on the rolling direction with respect to obstructions steered by the factor ݇. The rolling movement of the abrasives was reproduced in detail by a time step model - e.g. the movement of an individual grain from one of its faces over the vertex and edge over to the next face. Hu et al. [13] have shown that different grain forms (sharp, rounded, flat) lead to different removal behavior, because stress distribution under a grain’s vertex differs substantially from the stress distribution under the grain which lies on a face. The introduced model delivers knowledge about the position and orientation of each single abrasive at every time step. From this (position and orientation) and the governing load on each single grain the stress distribution under each grain in each single time step will be derived in the future. From the knowledge about the effect of the single abrasives on the surface and the volume of material removal under the whole polishing pad, the removal, which is achieved by single grains, can be determined. In next steps polishing experiments will be carried out with diamond abrasives on electroless nickel specimens to determine the integral polishing load and total material removal. From this a material removal function will be derived and applied to the single grain removal. All in all this will allow for the modelling of the surface formation in three-body abrasion. Acknowledgements The authors greatly acknowledge the funding of the project RI 1108/3-1 by the German Research Foundation (DFG). References [1] Finnie I. Some reflection on the past and future of erosion. Wear 186-187 (1995), pp. 1-10

[2] Bitter JGA. A study of erosion phenomena part I. Wear, Volume 6, Issue 1, 1963, pp. 5-21 [3] Bitter JGA. A study of erosion phenomena part II. Wear, Volume 6, Issue 3, 1963, pp. 169-190 [4] Misra A, Finnie I. An experimental study of three-body abrasive wear. Wear, Volume 85, Issue 1, 1983, pp. 57-68 [5] Trezona RI, Allsopp DN, Hutchings IM. Transition between two-body and three-body abrasive wear: Influence of test conditions in the microscale abrasive wear test. Wear 225-229 (1999), pp. 205-214 [6] Adachi K, Hutchings IM. Wear-mode mapping for the micro-scale abrasion test. Wear 255 (2003), pp. 23-29 [7] Heisel U, Avroutine J. Process Analysis for the Evaluation of the Surface Formation and Removal Rate in Lapping. Annals of the CIRP Vol. 50/1/2001, pp. 229-232 [8] Jacobsen S, Wallén P, Hogmark S. Correlation Between Groove Size, Wear Rate and Topography of Abraded Surfaces. Wear 115 (1987), pp. 83-93 [9] Jacobsen S, Wallén P, Hogmark S. Fundamental Aspects of Abrasive Wear Studied by a New Numerical Simulation Model. Wear 123 (1988), pp. 207-223 [10] Zitt U. Modellierung und Simulation von Hochleistungsschleifprozessen. Dissertation, Technische Universität Kaiserslautern, 1999 [11] Fang L, Liu W, Duc D, Zhang X, Xueb Q. Predicting three-body abrasive wear using Monte Carlo methods. Wear 256 (2004), pp. 685-694 [12] Pellegrin DV, Stachowiak GW. Simulation of three-dimensional abrasive particles. Wear 258 (2005), pp. 208-216 [13] Hu J, Li DY, Llewellyn R. Synergetic effects of microstructure and abrasion condition on abrasive wear of composites – A modeling study. Wear, Vol. 263, 2007, pp. 218-227