Simulation of abrasive water jet cutting process: Part 1. Unit event approach

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1 INSTITUTE OF PHYSICS PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING Modelling Simul. Mater. Sci. Eng. 12 (2004) PII: S (04) Simulation of abrasive water jet cutting process: Part 1. Unit event approach Andrej Lebar and Mihael Junkar Faculty of Mechanical Engineering, University of Ljubljana, Askerceva 6, 1000 Ljubljana, Slovenia Received 1 April 2004, in final form 2 April 2004 Published 16 September 2004 Online at stacks.iop.org/msmse/12/1159 doi: / /12/6/010 Abstract Abrasive water jet (AWJ) machined surfaces exhibit the texture typical of machining with high energy density beam processing technologies. It has a superior surface quality in the upper region and rough surface in the lower zone with pronounced texture marks called striations. The nature of the mechanisms involved in the domain of AWJ machining is still not well understood but is essential for AWJ control improvement. In this paper, the development of an AWJ machining simulation is reported on. It is based on an AWJ process unit event, which in this case represents the impact of a particular abrasive grain. The geometrical characteristics of the unit event are measured on a physical model of the AWJ process. The measured dependences and the proposed model relations are then implemented in the AWJ machining process simulation. The obtained results are in good agreement in the engraving regime of AWJ machining. To expand the validity of the simulation further, a cellular automata approach is explored in the second part of the paper. 1. Introduction Abrasive water jet (AWJ) cutting is a non-conventional machining process in which abrasive grains entrained in a high speed water jet collide with the workpiece and erode it. A water jet is used to accelerate the abrasive grains and to assist the material removal process. The velocity of the water jet is up to 900 m s 1. It is obtained by a high pressure water pump with a typical pressure value of 400 MPa. The pressurized water is forced through an orifice made of sapphire. The orifice, also called the water nozzle, is a part of an AWJ cutting head as shown in figure 1. The water is thereby accelerated in the orifice according to the Bernoulli equation to a high velocity v j : v j = µ 2p ρ w, (1) /04/ $ IOP Publishing Ltd Printed in the UK 1159

2 1160 A Lebar and M Junkar HIGH-PRESSURE WATER MIXING TUBE ABRASIVE INLET ORIFICE MIXING TUBE 5 mm Figure 1. AWJ cutting head scheme. where p is the water pressure, ρ w is the density of water and µ is the discharge coefficient, which is a measure of the disagreement with the theoretical jet velocity. The coefficient µ is always less than 1, with a typical value of 0.86 [15]. Downstream of the water nozzle the water jet expands and becomes unstable due to the several forces acting on the jet: friction, surface tension and turbulence. Following the jet, a mixing chamber, where the abrasive grains are added, is placed below the water nozzle. Due to the friction between the high speed water jet and the air, a suction pressure is generated that sucks abrasive particles and air through the abrasive inlet into the mixing chamber. Collisions between the jet and the abrasive grains increase the droplet formation process, causing a fog curtain. Beneath the mixing chamber the jet enters the mixing tube. The abrasive grains rebound repeatedly between the jet, water droplets and the mixing tube inner wall. The abrasive grains are thereby accelerated in the longitudinal direction to nearly one-third of the velocity of the water jet and to a rotation speed of up to rotations per minute [2]. The AWJ exits the mixing tube collimated, but with a complex distribution of water speed and abrasive speed [3]. The workpiece is placed in stand-off distance h so 2.5 mm beneath the mixing tube, which traverses in the direction of the cutting. The erosive action of the AWJ removes the material of the workpiece, and shortly a kerf with a cutting front is formed. The quality of the machined workpiece is determined by the AWJ process control parameters and the material properties of the workpiece. AWJ machining is superior in performance to other similar machining processes, regardless of the brittleness, ductility or composition; however, a workpiece cut with AWJ exhibits a rather random character, which limits its use for accurate machining operations, for example, operations with tolerances of less then 0.05 mm. In figure 2 some of the geometry related process parameters and the resulting surface are schematically presented.

3 Simulation of abrasive water jet cutting process 1161 υ t h so x z h h sc l dr Figure 2. Schematic overview of the geometry related AWJ parameters: traverse velocity of AWJ cutting head, v t, angle of incidence, φ, AWJ cutting head stand-off distance, h so, depth of cut, h, depth of smooth cutting zone, h sc, and jet lag, l dr. A machined surface can be roughly divided into two zones with respect to the surface roughness and texture. The upper zone is called the smooth cutting zone and spans from the top of the workpiece to the depth h sc. In contrast, the lower zone is often referred to as the rough cutting zone, where a characteristic texture can be observed. The ability to predict the topography of AWJ machined workpieces, especially the inaccuracies at the bottom of the cut, would enable AWJ machining to be used also for more precise machining, but depends largely on the correct definition of the mechanisms involved. In the case of AWJ machining only indirect measuring methods are available for measuring the cutting front advance in real time, because the action of the high velocity water jet hides the material jet interface zone. One of the few possibilities left for exploring this process is to assume a process model, implement it in a simulation and validate it with experiments. There have been some previous attempts to model AWJ machined surface topography. Kobayashi et al [9, 10] have numerically simulated AWJ cut surface topography. Their work made use of Bitter s theory [7, 8] on erosion processes for predicting the striations on the cut surface. Although the capability of the model in predicting the AWJ cut surface has been demonstrated by obtaining the jet lag and striations, the background theory as well as simulation process have not been clearly presented [15]. Vikram and Babu [11] have tried a similar approach. They have also used Bitter s theory for predicting material removal model and the theory of ballistics for predicting the trajectory of jet penetration into the material. In their simulation they obtained both striations and jet lag and then by employing surface generation theory have generated the surface topography. Yong and Kovacevic [12] have developed a numerical model for AWJ machining that includes several aspects of the process,

4 1162 A Lebar and M Junkar MODEL OF AWJ MACHINING UNIT EVENT VOLUME REMOVAL MODEL abrasive grains velocity vector, grain size material hardness, impact angle ABRASIVE GRAINS REBOUND MODEL workpiece topography neighborhood type MODEL OF INITIAL VELOCITY VECTOR OF ABRASIVE GRAINS water pressure, mixing efficiency, cutting head design DISTRIBUTION OF GRAINS AT MIXING TUBE EXIT cutting head type MODEL OF ABRASIVE GRAINS DIAMETER VALUES mesh, distribution type MIXING TUBE TRAVERSE VELOCITY MODEL drive type, control system Figure 3. AWJ machining modular model. such as simulation of multiphase pipe flow, tracer records of abrasive particles and energy transformation in a so-called memory cell. The workpiece surface area is divided into a network of cells. After the kinematics of the abrasive grains is calculated, each cell records the number of abrasive particles striking a small area of the workpiece in order to predict the depth of the cut at the point where the particular cell is situated. The joint result of all memory cells gives the resulting surface of the cut. Ditzinger et al [13] have studied the non-linear dynamics of AWJs. They have derived a partial differential equation that describes the development of the cutting front in time. This two-part paper presents two alternative approaches to computer simulation of machining with an AWJ. In the first part a unit event approach and in the second part a cellular automata (CA) approach to the simulation is presented. The unit event model studies the impact of each individual abrasive grain (unit event) and gives a cumulative result of all impacts in the form of the machined surface topography. Although the model is numerically very intensive, it exhibits considerable flexibility in the sense that different process scenarios can be tested and verified using it. The CA models the AWJ cutting process on a mesoscopic level and was found to be faster because of its lower complexity. The material removal process is modelled by considering the energy of an AWJ together with its impact angle and the erosion resistance of the workpiece material. 2. AWJ machining model and simulation With modelling, based on the unit event of the machining process a development of the machined surface can be observed on the micro and macro scales. In order to avoid a priori simplifications of the process model, a modular model has been introduced [17, 18]. In this study, six modules were identified. The structure of the modular model is presented in figure 3. The modular model of AWJ machining presented here is based on the AWJ machining process unit event. Therefore, there are no functional or other relations in the model between the machining process parameters and the macroscopic features of the workpiece.

5 Simulation of abrasive water jet cutting process 1163 The macroscopic features are not revealed until the computer simulation is done, and at that time the simulation results can be also verified. In the computer simulation, the workpiece topography is represented by a matrix of equally spaced elements. Each matrix element is a real number that exhibits the workpiece depth at a particular location. The size of the matrix is determined by the size of the workpiece and a numerical resolution. It is important that the resolution is high enough since craters vary in size from zero volume up to the maximal value, which is defined by the control parameters of AWJ machining. In this simulation craters smaller than two-by-two elements were neglected. The resolution selected was as high as 300 elements per millimetre. The core module was a model of the AWJ machining unit event. By using the unit event approach, it was possible to determine the process results on the microscopic scale and generate a virtual AWJ machined surface topography with characteristic macroscopic features through a computer simulation Abrasive grain diameter, initial velocity and position modules The grains used in AWJ machining are produced by sieving crystalline hard rock deposits. In order to obtain the distribution of abrasive grain diameter values, the size of the grains were measured by means of microscopy and image processing. The results were then compared with the data provided by the producer of the abrasive. A beta function was fitted on the measured data. It was the most suitable function for describing the distribution of the abrasive grain size [17]. The abrasive grain size is generated during simulation initialization. We used a random generator that gives a β function distribution of grain size diameters and therefore also their mass. In order to calculate the kinetic energy of the abrasive, its velocity should be determined. According to the described model we supposed that all abrasive grains have mainly the vertical velocity component v z v x,v y and that only small random components (ε x,ε y ) in the orthogonal direction exist, v = (ε x,ε y,v z ). The amount of defocusing (ε x,ε y ) was estimated based on visual observations using CCD camera [17]. As regards the position of abrasive grains at the mixing tube exit, it was assumed that the abrasive grains were uniformly distributed over the jet. The abrasive grains were sucked into the mixing tube by a stream of air driven by air-jet friction in the mixing tube. Before they were entrained in the jet and were accelerated in it, they were subjected to several rebounds from the jet s core and the inner surface of the mixing tube Unit event The unit event module describes to what extent the workpiece topography is modified by the impact of the particular abrasive grain. In the case of ductile materials, the functional dependence of the erosion wear on the impact angle of the single abrasive grain, its speed and mutual material properties are known from the work of Finnie [6] and Bitter [7, 8]. The functions for the abrasive wear, ɛ m (equations (2) and (3)), which was derived by Finnie, are usually referred to in the domain of AWJ modelling as ɛ m = ρmv2 ( sin 2α 3 sin 2 α ), α 18, 5, (2) σ p κ ɛ m = mv2 σ p 6 cos2 α, α > 18, 5, (3)

6 1164 A Lebar and M Junkar 1.4 x UNIT EVENT WEAR [g] DRY EROSION MPa 200 MPa MPa 150 MPa IMPACT ANGLE [deg] Figure 4. Results of the measured dependence of unit event mass removal versus abrasive impact angle at different water pressures [17]. where σ p is the horizontal component of the stress on the particle face, is the ratio of the depth of contact to the depth of cut, κ is the ratio of the vertical component of the force on the particle to the horizontal force component and α is the impact angle. The major drawback of equations (2) and (3) is that the predicted erosion diminishes to zero for perpendicular impact, which is not the case in reality, but the equations can be corrected with an additional linear term, as suggested in the literature by Finnie [6]. The graph of equations (2) and (3) can be observed in figure 4. The curve labelled dry erosion was obtained numerically and corresponds to equations (2) and (3), but is corrected at higher angles. The curve is normalized to the results of the experiment at 200 MPa. The model of the unit event used in our study is based on experimentally obtained data, presented in figure 4, in which the four solid lines correspond to experiments with different water pressures. The material removal rate was measured on an aluminium alloy test specimen as a function of the abrasive jet impact angle by weighing the workpiece mass before and after the machining. It can be observed in figure 4 that the maximum wear for dry erosion is at a much lower angle, than the measured ones. The difference between the curves based on equations (2) and (3) and the measurements is in our opinion due to the fact that the action of high velocity water emphasizes the portion of wear that is due to crack propagation and brittle fracture. The jet velocity, abrasive flow rate and jet transverse velocity were kept constant during these experiments. Afterwards, the polynomial was fitted to a set of measurement results as can be seen in figure 4. It was found that the closest fit was obtained with the polynomial of the fourth degree. The four curves in figure 4 correspond to four sets of experiments with different pump pressures. The surfaces of the test specimens were afterwards subjected to a microscopic examination in order to obtain information on the size and shape of the craters on the surface. It can be

7 Simulation of abrasive water jet cutting process 1165 α = 20 α = mm 0.1 mm Figure 5. Craters eroded at a low grazing angle α = 20 and at nearly perpendicular impact at α = 80. INITIALISATION abrasive-grain-i UNIT EVENT CONDITIONS FOR SECONDARY EVENT } secondary removal CONDITIONS FOR SECONDARY EVENT - kinetic energy higher than threshold energy - first and third component of velocity vector FULFILLED + PEEK NEW GRAIN i=i+1 GRAIN STACK EXHAUSTED? + - smaller than zero RESULTS Figure 6. Surface generation flowchart. observed in figure 5 that the craters mainly exhibit the orientation determined by the direction of the abrasives velocity vector and very random orientation at higher impact angles. After the first collision of the abrasive grain with the workpiece, i.e. primary unit event, the abrasive grains rebound. They are re-entrained in the AWJ and are subjected to several consecutive impacts, i.e. secondary events within the cutting kerf, as long as the required conditions are fulfilled as presented in figure 6. With our model, the following conditions were taken into account: the abrasive grain must have a kinetic energy that is higher than the threshold energy [7 9] and it has to be rebounded in a direction opposite to the cutting head movement direction v = [v x < 0,v y,v z < 0].

8 1166 A Lebar and M Junkar n S i+1, j 1 S i+1, j S i+1, j+1 S i 1, j+1 Figure 7. The normal vector to the surface is estimated at the point of impact. After initialization, simulation of the material removal procedure starts. When the actual coordinates of the abrasive grain impact are known, the impact angle can be determined by the scalar product of the impact velocity vector and the normal vector to a workpiece surface S, ( π ) cos 2 α = v n v n. (4) We estimated a normal vector to the surface S at the point of impact, S i,j, as the average of normals to the eight triangles in the neighbourhood of the point of impact (figure 7). The size of the triangles can be varied, so that they match the size of the cutting interface zone. The normal to the particular triangle is calculated by the cross product: n = S i+1,j S i,j S i+1,j 1 S i,j. (5) Using the calculated impact angle, α, and the unit event feature measurements on the physical model, the volume removal and crater parameters can be determined. A procedure is called that calculates the matrix representation of the eroded crater C. Subsequently the matrix C is subtracted from S and the workpiece surface after the impact of one grain is obtained. Before subtraction we used a blurr filter on the corresponding sub-matrix of S. The blurr filter performs a convolution of the original matrix and the filter matrix, which is in our case a three-by-three matrix of 1s. This operation is considered to be physically correct, because abrasive grains cause continuous traces after cutting. Overall up to ten million abrasive grain impacts are evaluated. After each set of ten thousand primary impacts the workpiece representation matrix, S, and the images of several graphs can be saved for later analysis Abrasive grains rebound model Shortly after exiting the collimating nozzle, the abrasive grains hit the workpiece surface, causing material wear. They rebound and are entrained back to the jet and are thus subjected to several consecutive impacts rebounds. In our simulation we assumed that the rebound angle is equal to the impact angle α = α as showed in figure 8.

9 Simulation of abrasive water jet cutting process 1167 υ α n T 1 α w Figure 8. Abrasive grain impact situation. Impact velocity vector, v, and angle, α, rebound grain velocity vector, w, and rebound angle, α. In this paper, α = α is assumed. The velocity vector of the rebounded abrasive grain is calculated using an operator of reflection, A, which transforms the impact velocity vector to the rebounded velocity vector. The operator has to satisfy two relations: v y = w z and v x = w x. The transformation that satisfies this relations is A = (6) The transformation A works only in the transformed coordinate system π, in which the x z plane is coplanar with the grain impact velocity vector and the rebounded velocity vector. We have to find a transformation T that could image the transformation A into π and the results back to system π Tv = v, (7) A Tv = w, T 1 w = w, T 1 A Tv = w, A = T 1 A T. (8) In order to express the components of the transformation T, we have to calculate how the basis vectors of system π are expressed in the coordinate system π. The first prerequisite is that the new basis vector be identical with the normal vector to the plane after transformation. The second condition is that it is perpendicular to the plane after the transformation, determined by the vectors and ˆk = ˆn, (9) ˆv ˆn = ĵ, (10) ˆn ĵ = î. (11) Here all the vectors with a hat are considered to be unit vectors. When the transformation A is known, it is not too difficult to derive the projections of the velocity vectors on the workpiece surface (equation (7)) and to calculate the coordinates of possible collisions with the surface. To determine the position of the secondary impact, additional criteria are required. The distance from the primary impact has to be big enough for the abrasive grain to be accelerated by the jet again. It is also required that the location of the secondary impact be in the area described with the matrix S.

10 1168 A Lebar and M Junkar (a) machined surface (b) simulated surface AWJ impact direction cutting direction surface workspiece 7,4 mm image of AWJ machined workpiece 1 mm waviness marks (striations) cutting kerf bottom Figure 9. An example of an AWJ machined surface, left, and a simulated surface. After the secondary impact, a third impact and a fourth impact were calculated, until there was no energy left for the abrasive grain to produce noticeable wear at the selected resolution or the process was cut off by the limiting conditions. 3. AWJ machining simulation and model verification By using the model proposed and the numerical simulation, virtual surfaces are obtained that were typical for AWJ machining. Figure 9 shows that the machined surface is much finer in the upper cutting zone than in the lower cutting zone. In the lower zone the abrasive grains have a lower velocity due to energy dissipation in the upper part, and this effect yields a rougher surface in the lower zone. Additionally, a step formation can be observed on the simulated cutting front, which is also visible in AWJ cutting experiments on transparent materials [14,15]. In order to verify the presented model and the simulation of AWJ machining, two sets of experiments were performed with different abrasive mass flows. The relation between the traversal velocity of the cutting head and the workpiece mass decrease was measured and compared with the results of the simulation. The velocity of the AWJ cutting head, v t, was varied in the interval from 5 to 30 mm s 1. The water pressure was kept constant at 200 MPa. The abrasive mass flow rate, ṁ a, was set at 0.31 g s 1 in the first set of experiments and doubled to 0.62 g s 1 in the second set. The abrasive material was garnet, Barton mesh 150. The diameter of the orifice was mm and the diameter of the collimating nozzle was 0.88 mm. The stand-off distance of the collimating nozzle from the workpiece, h so, was 14 mm. The dimensions of the aluminum alloy (SEA Al6061-T6) workpieces were mm 2. Such a combination of AWJ control parameters was selected in order to machine the workpiece in the engraving regime. The results are presented in the log log plot in figure 10. They show a good correlation between the experiments and the simulations. According to available evidence [16], the depth of the cut is inversely proportional to the AWJ cutting head traversal velocity. That is why a straight line is expected in the log log plot. It can be seen from the measurements presented in figure 10 that the amount of workpiece mass removal agrees with the straight line and that the experimental data correspond to the results of the simulation [17].The simulation was performed at the same setup parameters. The results of the experimental validation are shown in figure 9. As shown in figure 10 the simulation results are in quite good agreement with the measured results. Therefore the simulation can be used for optimizing the AWJ machining process.

11 Simulation of abrasive water jet cutting process m a = 0.62 g/s m a = 0.31 g/s simulation mass removal m [g] Figure 10. Comparison of the experiments and simulation results. 4. Conclusions An AWJ cutting model has been developed to simulate the workpiece topography after being machined by an AWJ. The concept of the presented model is modular so as to enable flexibility of modelling in the future. With the model proposed, virtual surfaces were obtained that exhibit characteristics typical for AWJ surfaces; these are striations, surface roughness and evidence of multiple cutting steps formation. It is believed that they are the consequences of the non-linear nature of the AWJ machining process. The results obtained so far are promising, but the model has to be tuned more precisely. In future work, the dependence of the abrasive grain rebound angle, mutual material hardness and angle of impact on grain velocity after the rebound will have to be included in the model as additional elements. References [1] Momber A W and Kovacevic R 1998 Principles of Abrasive Water Jet Machining (London: Springer) [2] Swanson R K, Kilman M, Cerwin S and Carver W 1987 Proc. 4th Am. Water Jet Conf. (St. Louis, MO, USA: Waterjet Technology Association) p 163 [3] Osman A H, Mabrouki T, Théry B and Buisine D 2004 Flow Meas. Instrum [4] Colosimo B M, Monno M and Semeraro Q 2000 Int. J. Mater. Product. Technol [5] Pacifique Harmsze F A 2000 A modular structure for scientific articles in an electronic environment PhD Thesis Universiteit van Amsterdam [6] Finnie I 1958 Proc. 3rd US Natl Congress of Applied Mechanics p 527 [7] Bitter JGA1963 Wear [8] Bitter JGA1963 Wear

12 1170 A Lebar and M Junkar [9] Fukunishi Y, Kobayashi R and Uchida K 1995 Proc. 8th Am. Water Jet Conf. (Waterjet Technology Association) p 657 [10] Sawamura T and Fukunishi Y 1997 Proc. 9th Am. Water Jet Conf. (Waterjet Technology Association) p 15 [11] Vikram G and Ramesh Babu N 2002 Int. J. Mach. Tools Manuf [12] Yong Z and Kovacevic R 1996 Jetting Technology (Bury St Edmunds, UK: Professional Engineering Publishing Limited) p 73 [13] Ditzinger T, Friedrich R, Henning A and Radons G 1999 Proc. 10th Am. Water Jet Conf. (Waterjet Technology Association) p 15 [14] Hashish M 1983 Proc. 2nd Am. Water Jet Conf. (Waterjet Technology Association) p 402 [15] Momber A W and Kovacevic R 1998 Principles of Abrasive Water Jet Machining (London: Springer) [16] Henning A and Westkämper E 2000 Jetting Technology (Bury St Edmunds, UK: Professional Engineering Publishing Limited) p 309 [17] Lebar A 2002 PhD Thesis University of Ljubljana, Ljubljana, Slovenia [18] Lebar A and Junkar M 2003 Proc. I. Mech. E., J. Eng. Manuf. B