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INSTITUTE OF PHYSICS PUBLISHING MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING

Modelling Simul. Mater. Sci. Eng. 12 (2004) 11591170 PII: S0965-0393(04)78666-0

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

E-mail: [email protected]

Received 1 April 2004, in final form 2 April 2004

Published 16 September 2004

Online at stacks.iop.org/MSMSE/12/1159doi:10.1088/0965-0393/12/6/010

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 theimpact of a particular abrasivegrain. The

geometrical characteristicsof theunit eventare 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 atypical 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 vj:

vj =

2p

w, (1)

0965-0393/04/061159+12$30.00 2004 IOP Publishing Ltd Printed in the UK 1159
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1160 A Lebar and M Junkar

ABRASIVEINLET

MIXINGTUBE

ORIFICE

TUBEMIXING

HIGH-PRESSURE WATER

5 mm

Figure 1. AWJ cutting head scheme.

where p is the water pressure, w is the density of water and is the discharge coefficient,

which is a measure of the disagreement with the theoretical jet velocity. The coefficient 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, amixing 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 thejet, waterdroplets andthe mixingtube innerwall. Theabrasive

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 4.46 rotationsper 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 hso 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 themachined 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.

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Simulation of abrasive water jet cutting process 1161

x

z

h

ldr

t

hso

hsc

Figure 2. Schematic overview of the geometry related AWJ parameters: traverse velocity of AWJcutting head, vt , angle of incidence, , AWJ cutting head stand-off distance, hso, depth of cut, h,

depth of smooth cutting zone, hsc, and jet lag, ldr.

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 hsc. In contrast, the lower zone is often referred to as therough 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 materialjet 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 Bitters 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 hasbeen 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 Bitters 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,

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drive type, control system

VELOCITY MODEL

MIXING TUBE TRAVERSE

mesh, distribution type

DIAMETER VALUES

MODEL OF ABRASIVE GRAINS

UNIT EVENT VOLUME

REMOVAL MODEL

abrasive grains velocity vector, grain sizematerial hardness, impact angle

REBOUND MODEL

ABRASIVE GRAINS

workpiece topographyneighborhood type

DISTRIBUTION OF GRAINS

AT MIXING TUBE EXIT

cutting head type

MODEL OF AWJ MACHINING

water pressure, mixing efficiency, cutting head design

MODEL OF INITIAL

VELOCITY VECTOR OF ABRASIVE GRAINS

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 ofmachining 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 structureof themodular 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.

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The macroscopic features are not revealed until the computer simulation is done, and at that

time the simulation results can be also verified.

In thecomputersimulation, theworkpiece topography is representedbya matrixof 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 anumerical 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.

2.1. Abrasive grain diameter, initial velocity and position modules

The grains used in AWJ machining are produced by sieving crystalline hard rock deposits. Inorder 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 function distribution of grain size diameters and therefore also their

mass. In order to calculate thekinetic energyof theabrasive, its velocity should be determined.

According to the described model we supposed that all abrasive grains have mainly thevertical

velocity component vz vx , vy and that only small random components (x , y ) in the

orthogonal direction exist, v = (x , y , vz). The amount of defocusing (x , 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 jets core and the inner surface of the mixing tube.

2.2. 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]. Thefunctions for the abrasive wear, m (equations (2) and (3)), which was derived by Finnie, are

usually referred to in the domain of AWJ modelling as

m =mv2

p

sin 2 3sin2

, 18, 5, (2)

m =mv2

p6cos2 , > 18, 5, (3)

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0 10 20 30 40 50 60 70 80 900

0.2

0.4

0.6

0.8

1

1.2

1.4

x 10-7

250 MPa

200 MPa

180 MPa

150 MPa

IMPACT ANGLE [deg]

UNITEVENTWEAR[g] DRY

EROSION

Figure 4. Results of the measured dependence of unit event mass removal versus abrasive impact

angle at different water pressures [17].

where 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, is the ratio of the vertical component of the force on the

particle to the horizontal force component and 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 beobserved in figure 4. The curve labelled dry erosionwasobtained 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 200MPa.

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.

Thesurfacesof thetest specimenswere afterwards subjectedto a microscopic examination

in order to obtain information on the size and shape of the craters on the surface. It can be

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= 80 = 20

0.1 mm0.1 mm

Figure 5. Craters eroded at a low grazing angle = 20 and at nearly perpendicular impact

at = 80.

s

econdaryremoval

-

+

+

i=i+1

INITIALISATION

UNIT EVENT

FULFILLED

PEEK NEW GRAIN

GRAIN STACK

RESULTS

EXHAUSTED?

abrasive-grain-i

CONDITIONS

EVENT

FOR SECONDARY

CONDITIONS

FOR SECONDARY

}

EVENT

kinetic energy

first and third component

-

-of velocity vector

higher than threshold energy

smaller than zero

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 [79] and it has to be rebounded

in a direction opposite to the cutting head movement direction v = [vx < 0, vy , vz < 0].

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n Si+1, j1

Si+1, j+1

Si1, j+1

Si+1, j

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,

cos

2

=

v n

|v||n|. (4)

We estimated a normal vector to the surface S at the point of impact, Si,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 = Si+1,jSi,j Si+1,j1Si,j. (5)

Using the calculated impact angle, , 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.

2.3. 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 impactsrebounds. In our simulation we assumed that the rebound

angle is equal to the impact angle = as showed in figure 8.

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w

n

T1

Figure 8. Abrasive grain impact situation. Impact velocity vector, v, and angle, , rebound grain

velocity vector, w, and rebound angle, . In this paper, = 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: vy = wz and vx = wx . The transformation that

satisfies this relations is

A =

1 0 0

0 0 0

0 0 1

. (6)

The transformation A works only in the transformed coordinate system , in which the

x z plane is coplanar with the grain impactvelocity vector and the rebounded velocity vector.

We have to finda transformationTthat could image the transformationA into and theresults

back to system

Tv = v, (7)

ATv = w,

T1w = w,

T1ATv = w,

A = T1AT. (8)

In order to express the components of the transformation T, we have to calculate how the

basis vectors of system are expressed in the coordinate system . The first prerequisite is

that the new basis vector be identical with the normal vector to the plane after transformation.

Thesecond conditionis that it is perpendicular to theplane after the transformation,determined

by the vectors and

k = n, (9)

v n = j, (10)

n j = i. (11)

Here all the vectors with a hat are considered to be unit vectors. When the transformation

A isknown,it isnottoo difficult toderive theprojectionsof thevelocity vectors on theworkpiece

surface (equation (7)) and to calculate the coordinates of possible collisions with the surface.

To determinethepositionof thesecondaryimpact, additionalcriteriaare required. Thedistance

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.

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surface

workspiece

waviness marks(striations)

cutting kerfbottom

7,4 mm

AWJ impact directioncutting direction

surface

surfacesimulated

(b)(a)

machined

image of AWJ

machinedworkpiece

1 mm

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. Figure9 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 inAWJcutting experimentson 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 andcompared with the results of the simulation.

The velocity of the AWJ cutting head, vt, was varied in the interval from 5 to 30mm s1.

The water pressure was kept constant at 200 MPa. The abrasive mass flow rate, ma, was set

at 0.31 g s1 in the first set of experiments and doubled to 0.62g s1 in the second set. The

abrasive material was garnet, Barton mesh 150. The diameter of the orifice was 0.356 mm

and the diameter of the collimating nozzle was 0.88mm. The stand-off distance of the

collimating nozzle from the workpiece, hso, was 14mm. The dimensions of the aluminum

alloy (SEA Al6061-T6) workpieces were 60 50mm2.

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 loglog 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 headtraversal velocity. That is why a straight line is expected in the loglog 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 thesimulation results are in quite good agreementwith themeasured

results. Therefore the simulation can be used for optimizing the AWJ machining process.

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100

101

102

10-1

100

massremovalm

[g]

ma= 0.62 g/s

ma= 0.31 g/s

simulation

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 surfaceswereobtained thatexhibit characteristicstypical

for AWJ surfaces; these are striations, surface roughnessand evidenceof 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, Thry B and Buisine D 2004 Flow Meas. Instrum. 15 3748

[4] Colosimo B M, Monno M and Semeraro Q 2000 Int. J. Mater. Product. Technol. 15 1019

[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 J G A 1963 Wear6 521

[8] Bitter J G A 1963 Wear6 16990

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[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. 42 134554

[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 Westkamper 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 217 699703

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