Finite element modelling of abrasive waterjet milled footprints

Journal of Materials Processing Technology 213 (2013) 180–193

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Finite element modelling of abrasive waterjet milled footprints S. Anwar, D.A. Axinte ∗ , A.A. Becker Faculty of Engineering, Department of Mechanical, Materials and Manufacturing Engineering, University of Nottingham, NG7 2RD, UK

a r t i c l e

i n f o

Article history: Received 23 May 2012 Received in revised form 3 September 2012 Accepted 13 September 2012 Available online 21 September 2012 Keywords: Abrasive waterjet machining/milling Finite element modelling Kerf profile

a b s t r a c t It is well-known that it is difficult to perform controlled-depth in abrasive waterjet (AWJ) milling, due to the dependency of the milled footprint not only on the jet kinematic parameters (e.g. jet traverse speed) but also on the jet energy parameters (e.g. pressure, abrasive mass flow, etc.). In this paper, an attempt has been made for modelling, simulation and validation of the AWJ footprint working in controlled depth (i.e. milling) mode at various jet traverse speeds and pump pressures at 90◦ incidence angles by using the finite element (FE) method. The proposed model is validated by comparing the material erosion rates and the profiles of the milled kerfs obtained by FE simulation to those generated from the experiments. The current model also simulates the effect of mass flow rate of the abrasive particles as well as the traverse rate of the AWJ plume across the workpiece. The abrasive particles (i.e. garnet) are modelled with various non-spherical shapes (rhombic, triangular and trapezoidal) and sharp cutting edges, while the workpiece material modelled is a titanium based superalloy (Ti–6Al–4V) extensively used in the aerospace industry. The workpiece material is modelled as elastic–plastic with a Johnson–Cook failure criterion, while a tensile failure criterion is used for the impacting garnet particles rather than considering them as rigid. The particles are arranged in a Gaussian distribution above the target surface to control the shape of the eroded footprint. To emulate the real interaction between the AWJ plume on the target surface, the material description in the FE model incorporates the effects of strain rate sensitivity, adiabatic heating and friction during the particles–workpiece interaction. The proposed modelling approach is capable of simulating the maximum depth of the AWJ footprint and the erosion rate at an acceptable level of accuracy (errors < 10%) when compared with experimentally generated data. Considering the possible sources of errors within the experimental data (e.g. non-constant traverse speed, particle flow and shapes), the results of this research are encouraging. © 2012 Elsevier B.V. All rights reserved.

1. Introduction With the development of materials with advanced properties (e.g. increased hardness, strength at high temperature) such as ceramics and aerospace super alloys, the need for advancements in cutting processes capable of coping with their difficult-to-cut properties is becoming of key importance for the utilisation of such materials. When compared with other conventional/nonconventional processing methods (e.g. laser, electro-discharge machining), abrasive waterjet (AWJ) machining can be considered a niche and versatile technology due to some key advantages; (i) material removal is done by attrition of high kinetic energy abrasives making possible the machining of any workpiece material regardless of their properties (e.g. advanced ceramics as presented by Srinivasu et al. (2009) and diamond composites as shown by Axinte et al. (2009)); (ii) the process requires very low specific cutting forces hence making it possible to machine low-stiffness

∗ Corresponding author. Tel.: +44 01159514117; fax: +44 01159513800. E-mail address: [email protected] (D.A. Axinte). 0924-0136/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jmatprotec.2012.09.006

components (Ojmertz, 1997); (iii) the process results in no heat affected zones and no thermal distortions making it suitable for machining heat sensitive materials and (iv) complex 3D shapes can be generated by manoeuvring the abrasive jet plume according to the required geometry. In the AWJ machining process, high pressure water is supplied by a pump at the orifice inside the cutting head from where it is converted into a high velocity jet. While passing through a mixing chamber, water creates a vacuum which draws the abrasive particles into a focussing tube where the abrasive waterjet (AWJ) mixture is formed as explained by Momber (1998). When the jet plume (mixture of abrasives and water droplets) impacts the target surface it results in the generation of a unique footprint (kerf). As a result of this, the workpiece material removal is mainly caused by the impact of a multitude of high velocities abrasive particles as discussed by Momber and Kovacevic (1998). While AWJ through cutting is an established technology, some significant challenges in performing controlled-depth AWJ machining (i.e. milling) are attributable to the following main facts: (i) the kerf not only depends on the jet specific energy but also on the properties (e.g. wear resistance) of the target material and jet kinematics

S. Anwar et al. / Journal of Materials Processing Technology 213 (2013) 180–193

Nomenclature d d ER LFE Lj Lj lp ˙a m mFE mL mPM nP nP1 nPM R P  SOD t Vf Vf Vimp Vw w

nozzle exit diameter (mm) jet diameter used in the model (mm) erosion rate (mg/mg) length to be traversed in the FE model (mm) length of the jet in real life for mFE (kg) mass of particles (mm) length of the jet in FE model for mFE (kg) mass of particles (mm) length of the pile-up material (mm) abrasive mass flow rate (kg/min) mass of particles required in the FE model (kg) mass of particles impinged per unit length (kg/mm) mass of one particles mix (mg) number of particles required in the FE model number of particles in one particle mix number of particles mix’s required in the FE model resultant angle of particles impact in the model (◦ ) water pump pressure (Pa) density of water (kg/m3 ) standoff distance (mm) time required by all the particles to hit the target surface in the model (s) traverse speed of the jet across the workpiece during experiments (m/s) traverse speed of the workpiece across the jet in the model (m/s) impacting velocity of the particles (m/s) waterjet velocity (m/s) width of the footprint (mm)

(e.g. traverse speed, tilt angle) as demonstrated by Fowler et al. (2005); (ii) stochastic nature of the process due to the fragmentation of the abrasive particles in the mixing chamber which alters their sizes (inertia/momentum) with which they impact the target surface as shown by Babu and Chetty (2003) and (iii) some uncontrolled variation in the process parameters such as fluctuation in water pressure and abrasive flow rate during the machining time. As a result of these issues it becomes difficult to control the amount of material to be removed during the AWJ milling process. The variation of both kinematic (e.g. feed speed) and energetic (e.g. water pump pressure, mass flow of abrasives) parameters of the jet plume lead to different amplitudes of the footprint/penetration depths of the milled trenches (along the jet traverse direction) on the target surfaces. Therefore, controlling the geometry of the footprint is of paramount importance in AWJ milling for the generation of desirable geometries. In these circumstances, finite element (FE) modelling for predicting the AWJ milled surface can improve the understanding of the key factors that affect the kerf generation and will enable its prediction in a scientific manner rather than the trial-and-error approaches that are used today. Thus, the aim of the presented work is to build and validate a FE model that will simulate the complete jet footprint during AWJ milling. To address this requirement, some models of jet footprint/particle impacts have been developed by using both analytical and FE techniques. From the analytical modelling point of view, a geometrical model for AWJ milling at 90◦ incidence angle is presented by Axinte et al. (2010). A similar erosional model is also manifested by Burzynski and Papini (2011) for predicting the footprint of a movable air powder-blasted jet. However, a drawback in these approaches is that they need special experimentally determined constants to calibrate/run the models. Although these calibration tests are simple in nature, they require special analysis

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(e.g. scanning of the footprint) unless an automated approach can be utilised. Furthermore, these modelling constants are not only dependent on the material wear-resistance properties but also on the process parameters, such as nozzle diameters, abrasive flow rate and pump pressure. This limits the use of these models to a certain set of material and process parameters and may require new experiments to recalibrate the models for new selected process parameters and/or materials. From the standpoint of FE modelling, various erosion models of single/multiple particles-target impact have been presented. Eltobgy et al. (2005) presented a three-dimensional (3D) FE model of erosion for multiple particles impact at a single location and incorporated (Johnson and Cook, 1983) plasticity and (Johnson and Cook, 1985) failure criteria to simulate material removal during the erosion process. (Wang and Yang, 2008) developed FE erosion models for both ductile and brittle materials using multiple (100) particles impact in groups of 10, each group impacting at the target centre area at random locations. A rhombic-shape single-particle erosion model was also presented by Takaffoli and Papini (2009) but they simplified the erosion problem using a 2D configuration in the FE model. One of the common deficiencies of all these FE models is that they use a rigid particle approach when modelling their impact on the target surface. However, in real life impact a significant amount of energy is absorbed by the impacting particles within their own deformation and fracturing phenomena even at lower impacting velocities as demonstrated by Salman et al. (2002). Models for a single particle impact have been reported for AWJ machining. However, there are some limitations in these approaches: (i) abrasive particles (e.g. garnet mesh no. 80, average particle size 180 ␮m), which are known to have irregular shapes and sharp cutting edges, were considered as cubic by Hassan (2001) and as spherical by Gudimetla and Yarlagadda (2007) and Junkar et al. (2006) in their respective FE models – this is likely to affect the erosion capability of the jet and (ii) the velocities (180–220 m/s) of the impacting particles used in the FE model by Junkar et al. (2006) are not within the real range (350–550 m/s) expected to happen in real AWJ processes as demonstrated by both Balz and Heiniger (2011) and Claude et al. (1998). These limitations are likely to lead to inaccuracies in the calculation of impacting particles momentum and hence will change the resulting dimensions of the jet footprints produced in the target materials. Another FE modelling work, at its initial stage of development, for a single particle impact during AWJ milling was reported by Anwar et al. (2011) in which an accurate validation criterion was used for the FE model, a nonrigid elastic–plastic with a failure criterion was considered in the impacting particle material model and an experimentally calculated velocity for the particles moving in the AWJ was used in the model. However, the model was initially reported for a single particle impact situation in which further research is needed to model the real AWJ controlled depth conditions. Due to the limitations of the presented models on AWJ machining, there is still a need for a model that can incorporate the elastic–plastic and failure response of both the target and impacting particles together with true impacting velocities obtained from the experiments, while being able to predict the AWJ milled footprints for a wide range of operating parameters, without the need for re-calibration. This research addresses these scientific challenges so that AWJ milled kerfs can be predicted with good accuracy at various jet traverse speeds, and water pump pressures.

2. Finite element modelling To build the FE model, the ABAQUS FE package (version 6.91) with explicit formulation was used to enable a more efficient solution of the current problem involving short term response, i.e.

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Fig. 1. (a) Symmetry of the AWJ milling process at 90◦ incidence angle. (b) 3D view of the model. (c) Meshing of the target and the particles. (d) Zoomed-in view of the elements used in the fine mesh area. (e) Gaps among the layers of the particles; in between these gaps the target will be traversed by some percentage of total distance to be covered across the jet. (f) Tilted top view of the model showing the length of the particles column.

high-speed impacts of the particles on the target surface. Because of the symmetry of the problem and to save the computational time, only a half model was built and evaluated as illustrated in Fig. 1. After completing the simulation, the half footprints are mirrored to be compared with the experimental footprints. The modelling procedure is discussed below.

Ti–6Al–4V target. Johnson and Cook (1983) (JC) proposed a model to calculate the flow stress in a target metal when subjected to high strain rate loading. The JC material model was employed for modelling the flow stress behaviour of Ti–6Al–4V. In the JC plasticity model, the flow stress  Y is assumed to be of the following form;





 T − T m  r

2.1. Material modelling

ε˙ Y = [A + Bε ] 1 + C ln ε˙ 0

2.1.1. Target material model According to the relationship among the impacting particle size, velocity of impact and the strain rate produced in the target as proposed by Hutchings (1977), strain rates of 105 –106 s−1 are expected in the current study during the impact of abrasive particles on the

where ε is the strain, ε˙ is the strain rate, ε˙ 0 is the reference strain rate at which material constants (A and B) are determined (Schwer, 2007), Tr , T and Tm are the ambient, workpiece, and melting temperature of the target material respectively. A, B, n, C and m are experimentally determined material constants whose values can be

n

1−

Tm − Tr

(1)

S. Anwar et al. / Journal of Materials Processing Technology 213 (2013) 180–193 Table 2 Material constants for garnet.

Table 1 Material constants for Ti–6Al–4V from Lesuer (2000). 1098 (MPa) 1092 (MPa) 0.93 0.014 1.1 5000 s−1 0.3 0.24 0.25 −0.5 0.014 3.87 4428 (kg/m3 ) 113.8 (GPa) 0.34 0.9 580 (J kg−1 K−1 ) 1878 (K)

2.1.2. Impacting particles material model The impacting particles material is modelled as garnet with density of 4120 kg/m3 (Gudimetla and Yarlagadda, 2007) for which a tensile failure criterion is used which considers the failure to occur when the stress in an element becomes more than the user defined hydrostatic cut-off stress ( cutoff ), i.e. the element is assumed to have achieved the required amount of energy to fracture and subsequently, it is removed from the mesh. The failure stress value for the garnet particles (see properties – Table 2) was estimated based on the work of Boud et al. (2010) and Shipway (1992).

obtained from the work of Lesuer (2000). The expression in the first brackets of Eq. (1) represents the flow stress as a function of strain, while the second bracket shows the increase in the yield stress at elevated strain rates; the third bracket represents the reduction in the yield stress due to thermal effects. For the erosion of the Ti–6Al–4V elements, the (Johnson and Cook, 1985) failure model is used. The expression for the material  pl failure strain ε is given as; f ε

pl f

= [d1 + d2 e

d3  ∗

 ] 1 + d4 ln



ε˙ pl ε˙ 0

  1 − d5

 T − T m  r Tm − Tr

(2)

where, * is a dimensionless pressure-effective stress ratio, ε˙ pl /ε˙ 0 is a non-dimensional plastic strain rate, and d1 –d5 are experimentally determined material constants. The failure/damage (i.e. removal from FE mesh) of an element occurs when the damage parameter D reaches the value of 1; as defined below. D=

(εpl ) pl

4120 kg/m3 248 GPa 0.3 150 MPa

Density Young’s modulus Poisson ratio Tensile failure stress

(3)

εf

where εpl is the cumulative plastic strain produced in all the increments during the analysis. At high strain rates, ductile materials exhibit adiabatic heating (Blazynski, 1987) i.e. heat produced during the plastic deformation is confined only to the deformation zone locally. Ti–6Al–4V has been reported to be prone to adiabatic heating (Lee and Lin, 1998), hence this effect has been incorporated in the model. The inelastic heating fraction was defined as 0.9 from Blazynski (1987), which assumes that 90% of the plastic work done on the material will be converted into heat. The material constants for the Ti–6Al–4V are listed in Table 1. It was found that the material constants for JC damage model adapted from the work of Lesuer (2000) produce excessive erosion in the target. This is due to the fact that these material constants have been determined by using larger sizes and different geometries (non-sharp, spherical/cylindrical) of the impacting projectiles generating relatively lower strain rates in the target material compared to the current study. A similar problem of excessive erosion was also encountered by Anderson et al. (2006). (Buyuk et al., 2009) pointed out the necessity of recalibrating the JC damage parameters to attain a better agreement between the simulation and the experimental results. Therefore the value of the JC damage parameter d1 was increased from −0.09 (as used Lesuer (2000)) to 0.24 based on good consistency with the current experimental results as presented in Section 4.

2.2. FE mesh Eight-noded linear brick elements are employed for the both the impacting particles and the Ti–6Al–4V target with viscous hourglass control which is suitable for high strain rate problems as discussed by Nam et al. (2008). A more refined mesh was used in the vicinity of the impact region on the target in the fine mesh region, while a relatively coarse mesh was employed away from the impact area (Fig. 1). All the elements in the fine mesh region are of the same size (15 ␮m × 22.5 ␮m × 7 ␮m). In the case of the high velocity impact problems where erosion occurs, it is not possible to obtain better solutions by simply refining the mesh as demonstrated by Buyuk et al. (2009) and Lim (2003) during the FE modelling of erosive impact. To address this, a preliminary study carried within this research shows that as the mesh is refined in the fine mesh region, a continuous increase in the erosion rate (ER) occurs, as shown in Fig. 2. The mesh size shown in Fig. 2 refers to the element size in the X-direction in the fine mesh region, while the size of the element in Y and Z-directions are changed accordingly keeping the same aspect ratio of the element in all the cases. Billon (1998) also found a similar behaviour in their FE model, i.e. the ballistic limit of their target decreases by refining the mesh i.e. the impacting projectiles penetrated through the same thickness of the target at lower impact velocities when the smaller mesh size is used in the target. This means that the target becomes more erodible by reducing the element size in it. Therefore, in the current study an optimum mesh size is selected after several trials i.e. the selected mesh size is still small enough to capture the fine details of footprint generated without consuming unnecessary computing resources.

0.031 0.029

ER in target (mg/mg)

A B N C M ε˙ 0 εf d1 d2 d3 d4 d5 Density Young’s modulus Poisson ratio Inelastic heat fraction Specific heat Melting temperature

183

0.027 0.025 0.023 0.021 0.019 0.017 0.015 5

7

9

11

13

15

Mesh size (µm) Fig. 2. Results for mesh convergence study at P = 138 MPa (Vimp = 368 m/s).

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Table 3 Process parameters used in the FE model. Water pressure (MPa)

TRS (mm/min)

Mass flow rate ˙a of garnet, m (kg/min)

Mass of garnet used in model, mFE (mg)

Velocity of particles used in the model (m/s)

Angle of incidence of AWJ (degrees)

138 138 207 207 276 276

2000 1000 2000 1000 2000 1000

0.02 0.02 0.02 0.02 0.02 0.02

4.1 8.2 4.1 8.2 4.1 8.2

368 368 450 450 520 520

90 90 90 90 90 90

2.3. Boundary conditions

the workpiece surface is assumed to be 0.1 based on the work of (Meo and Vignjevic, 2003).

The motion of the workpiece was constrained at the bottom plane in the Y and Z direction and at the symmetry plane in Ydirection. The diameter of the half-jet in the model is the same as the focussing nozzle diameter (d ’ = d) used in the experiments. In the experiments, the jet is moved across the target while in the FE model the target is traversed across the jet for convenience, which has the same effect as the former. The workpiece is traversed 1.5 × d mm in the X-direction in all the simulations to generate an area marked as BCFE (see Fig. 3) which receives the complete impact of the jet diameter. In the following, all the results related to jet footprint are referred to the eroded profile/area of this region. All the garnet particles are assigned the same velocity (Vimp ) corresponding to the selected water pump pressure. It is found by Balz and Heiniger (2011) and Roth et al. (2005) that the velocity of the abrasive particles is almost constant across the cross-section of the jet and is a fixed percentage of the pure waterjet velocity (Vw (from Bernoulli’s equation) see Eq. (4)). Eq. (5) gives the relationship between the pure waterjet velocity and the impacting velocity of the abrasive particles.



Vw =

2P P

Vimp = 0.7 × Vw

(4) (5)

Balz and Heiniger (2011) also reported that the abrasive particles form a Gaussian spatial distribution around the jet central axis, i.e. more particles are travelling in the inner radii of the jet compared to outer radii. This means that the shape of the eroded footprint is controlled by the distribution of the particles in the AWJ plume. For the first time, this aspect is captured by the current model while assembling the abrasive particles over the target by approximately following the same spatial distribution of the particles at the nozzle exit as presented in the work of Balz and Heiniger (2011). The process parameters used in the FE model are summarized in Table 3. 2.4. Contact The general-contact algorithm in ABAQUS is used to define the impact between the abrasive particles and the workpiece surface. When one layer of the elements fails during the impact (see failure criteria in Section 2.1), then the general-contact algorithm automatically redefines the contact between the newly exposed surfaces and the impacting particles. While running the simulation it was observed that when an element is removed from the mesh, its nodes act as free-floating point masses that are capable of transferring momentum either to the ingressive particles by colliding with them or to the target surface by bouncing back from the incoming particles. In order to avoid this situation, the nodal erosion capability of ABAQUS was used to eliminate the masses of these free flying nodes from the simulation and thus making them ineffective. The value of the coefficient of friction between the garnet particles and

2.5. Selecting the size and shapes of the garnet abrasive particles In the AWJ milling process, the jet footprint is mainly generated as a result of cumulative erosion caused by the impact of the high velocity abrasive particles upon the target surface during which the deformation/erosion behaviour is dominated by inertia (Blazynski, 1987) which is the property of the mass (size) of the impinging particles. Since the abrasive particles undergo fragmentation while passing through the focussing tube (Momber and Kovacevic, 1998), it is very important to carefully consider in the model the real sizes of the abrasive particles that impact the surface. This aspect is addressed by selecting the size distribution (SD) of the mass of the garnet particles after enduring a fragmentation process as reported by Babu and Chetty (2003) (see Table 4). In this way, the real particle sizes have been utilised in the proposed model; no similar attempts have been done before. Note that the considered SD by mass acquired in the model does not consider the garnet particles in the range of 355–400 ␮m, 315–355 ␮m, 63–90 ␮m and the pan (dust < 63 ␮m) which comprise 18.8% of the total mass of the garnet after the nozzle exit. The first two sizes have been ignored because they only add up to 1.8% of total mass and the last two have been discounted as a preliminary FE investigation indicated that the impact of the particles within these size ranges has negligible contribution in the overall erosion process compared to the larger size particles. This implies that whatever the mass of garnet is used in the experiments, approximately 82% of it will be used in the FE simulation. The term “particles mix” refers to a group of nineteen garnet particles in the model which contains approximately the same proportion of mass for each size of the particles as in real life AWJ machining. While calculating the number of particles for each size in one particles mix, the model aims to match the corresponding

Table 4 Details of the abrasive particles mix used in the FE model. A (mm)

B (mm)

C (mg)

D (%)

E (%)

F

G (mg)

H (%)

0.250–0.315 0.200–0.250 0.180–0.200 0.160–0.180 0.125–0.160 0.09–0.125 Total

0.275 0.225 0.190 0.170 0.140 0.125

0.041 0.029 0.014 0.010 0.006 0.004

0.14 0.24 0.10 0.13 0.12 0.09 0.82

0.17 0.29 0.13 0.16 0.14 0.11 1.00

1 2 2 3 5 6 19

0.041 0.057 0.028 0.031 0.029 0.025 0.211

0.19 0.27 0.13 0.15 0.14 0.12 1.00

A: size ranges adapted from Babu and Chetty (2003) (mm). B: approximated particle size in model (mm). C: mass of each particle based on a sphere of same diameter (mg). D: percentage by mass of each size in the whole garnet abrasive from Babu and Chetty (2003) (%). E: relative percentage of each size among selected sizes in real life (%). F: total number of particles used of this size in one particle mix. G: total mass of each size in one “particles mix” (mg). H: percentage of each size by mass in model (%).

S. Anwar et al. / Journal of Materials Processing Technology 213 (2013) 180–193

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Fig. 3. Movement of the target across the jet in the model. Points A and B show the initial position of the jet and points C and D show the final position. Hatched area “BCFE” illustrates the region where the jet footprint will be measured.

values in column D (percentage of each size by mass in the whole abrasive) and column H (percentage of each size by mass in model) of Table 4. This ensures that the amount of mass used in the FE model for a particular abrasive size is as close to the AWJ experiments as possible. Thirteen different layers (spatial distribution) of the particles mixes are utilised in the FE model which are then patterned over each other in the Z-direction to achieve the required total mass of the abrasive particles (mFE ) in the FE model. While assembling the first thirteen layers of the particles, it is tried to make sure that particles of different sizes and shapes should occur all around the half jet area and the particles are always placed in the new positions where they have not been placed in the previous layers. Also it is kept in consideration that there should not be concentration of any size or shape at one location in the jet (e.g. periphery or centre). The calculations for obtaining the number or mass of the particles in the FE model are presented below. ˙ a (kg/min) the Let Vf (mm/min) be the jet traverse speed and m abrasive mass flow rate used in the experiments. The mass of particles impinged per unit length, mL (kg/mm) is therefore given by: mL =

˙a m Vf

(6)

Let LFE be the length that the target surface will be traversed in the FE model. Since only a symmetrical half model will be used and 82% of the total mass used in the experiments (see column-D of Table 4) will be employed in the FE model, the mass of the particles required in the FE model (mFE ) can be calculated as follows. mFE = 0.5 · LFE · 0.82 · mL

(7)

Now, considering that mPM be the mass of one particles mix, i.e. 19 particles (0.211 mg) from Table 4. The number of particles mixes required in the FE model (nPM ) is therefore given by: nPM =

mFE mPM

(8)

which are selected from the generic shapes of the garnet particles presented in (Momber and Kovacevic, 1998). The sizes allotted to various shapes and numbers of particles used for each size in one particles mix are listed in Table 5. It should be noted that all different shaped abrasive particles are set in the layers in the FE model in approximately the same orientation as shown in Table 5 along the Z-direction. This due to the fact that in case of multiple impacts of sharp edged particles, once the target surface is eroded and it has formed a slope, the initial poster/orientation of the particles does not affect the erosion rate as explained by Chen and Li (2003). 2.6. Including the process kinematics into the model It is quite challenging to incorporate parameters such as dwell/exposure time of the target to the jet into the FE model which is controlled by the jet traverse speed across the target. The experimental values of the traverse speeds are not used in the model because these would result in prohibitively long simulation times. In real life, the abrasive particles are spaced quite apart from each other both in Y and Z directions (see Fig. 4 in Balz and Heiniger (2011)) and their impacting velocity is very high as compared to the jet traverse velocity across the target (Vimp  Vf ). On the other hand, in order to save computational time, the particles in the FE model are arranged in layers and these layers are spaced close to each other (50 ␮m), hence significantly reducing the length of the jet plume for the same amount of mass of the impacting particles. Fig. 4 shows the schematic illustration of the difference in how the abrasive particles are spaced in the real jet and in the FE model. In order to have the same number of particles impacting per unit area in the FE model and the real AWJ process, the target will need to be traversed much faster compared to the experiments. However, if a higher traverse velocity is applied to the target, the resultant angle of impact will deviate from 90◦ whereas it is well-known that for ductile materials, the erosion rate changes considerably due to

Hence, the total number of particles required in the FE model (np ) can be obtained as: nP = 19 · nPM

(9)

˙ a = 0.02 kg/min, For example, for Vf = 2000 mm/min and m the mass of the abrasive particles impinged per unit length, mL = 10 mg/mm. If LFE = 1.5 mm then mFE = 6.15 mg. This means twenty-nine (nPM ) particles mix will be required in the model i.e. 551 particles (nP ) of various selected shapes and sizes (as illustrated in Fig. 1) will need to be simulated. In all the previous studies reported for the FE modelling of the AWJ machining process (discussed in Section 1), only a spherical shape of the garnet particle was used, whereas in reality garnet particles are of irregular shapes with sharp cutting edges especially after fragmenting through the focussing nozzle. To address this aspect, three different shapes of the abrasive particles (rhombic, triangular and trapezoidal) are employed in the current model,

Fig. 4. Schematic representation of the differences between the particles spatial density in real and FE model jet. (a) Experimental jet with particles far apart from each other. (b) Jet in the FE model with closely spaced particles.

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Table 5 Sizes and number of particles assigned to different shapes. Particles are presented in the same colour as in figure.

Shapes used

Assigned sizes (␮m) No. of particles of this shape in one particle mix

275 1

225 2

190 2

170 3

140 5

125 6

Note: Particles shown in the table are not to scale.

a change in the angle of impact (Finnie, 1960). This is explained below. The total time (t) required by all the particles to hit the target surface is given by: 

t=

Lj

(10)

Vimp 

Let Vf be the traverse velocity used in the FE model as follows: 

Vf =

LFE t

(11)

The resultant angle of impact ( R ) can be obtained as: 

R = 90 − tan−1

Vf

(12)

Vimp

As an example, the calculations for  R at Vf = 2000 mm/min and Vimp = 368 m/s (i.e. P = 138 MPa) are presented as follows. From the  FE model Lj = 13.7 mm. Therefore from Eq. (10), t = 37.6 ␮s. Now 



Vf = 40 m/s. By substituting the values of Vf and Vimp in Eq. (12),  R is obtained as 83.7◦ . Therefore, to overcome this issue, i.e. keeping the normal angle of impact between the particles and the target, the target was moved across the jet in steps of equal magnitude only in the gaps present between the two adjacent layers of abrasive particles. Each step is a fixed proportion of the total length to be traversed (LFE ).

This is accomplished in the FE model by applying a displacement boundary condition to the target in the X-direction with an amplitude which divides the total displacement (LFE ) into a number of equal sub-displacements. It should be noted that water has not been included in the current study. This is due to the following facts: (i) Kong et al. (2010) has reported that pure waterjet does not have the capability to significantly erode Ti alloys until water pressure is more than 276 MPa (in the current study experiments 138–276 MPa pressures are utilised); (ii) preliminary FE investigations (not presented here) have revealed that at the jet traverse speeds of 1000 mm/min and 2000 mm/min, incorporating the water along with the abrasive particles in the FE model has negligible effect on the overall erosion rate.

3. Generation of experimental data AWJ milling trials for validating the FE model results were conducted on a 5-axis waterjet machine (Ormond) equipped with a streamline SL-V100D ultra-high pressure pump capable of providing a maximum water pressure (P) of 413.7 MPa (60,000 psi). The jet traverse speed (Vf ) can be varied in the range of 0–20,000 mm/min. An orifice diameter of 0.29 mm and focussing tube (nozzle) diameter of 1.02 mm were used throughout the experimental trials. The standoff distance (SOD), i.e. the distance between the nozzle exit

Fig. 5. Photograph of the experimental setup.

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Fig. 6. Procedure for extracting 2D profiles from an experimentally generated kerf.

and the workpiece, was set at 3 mm in all the tests. The experimental setup used during the trials is presented in Fig. 5. Experiments were performed on rectangular strips (140 mm × 35 mm × 3 mm) of an aerospace Titanium based superalloy (Ti–6Al–4V) with the following mechanical properties: average hardness 35 HRC, density 4428 kg/m3 , modulus of elasticity 113.8 GPa and Poisson ratio 0.342. In order to remove any scratches or micropits from the surface of the test specimens before generating the jet footprints, they were hand polished by sand papers starting with grit number 400 then 800 and finally 1200. Indian Garnet mesh 80 with an average particle size of 0.180 ␮m was used as an abrasive during the trials, which is the most common type of abrasive used in AWJ machining. The Indian origin of garnet and mesh size 80 have been selected in order to keep consistency with the data obtained from Babu and Chetty (2003). The water pump pressures used during the experiments were 138 MPa (20,000 psi), 207 MPa (30,000 psi) and 276 MPa (40,000 psi). Two jet traverse speeds (1000 mm/min and 2000 mm/min), suitable for controlled depth AWJ milling of Ti–6Al–4V workpiece (Hashish, 1998) were employed for the

trials while the mass flow rate (ma ) of the garnet abrasive was kept constant at 0.02 kg/min. To ensure repeatability and accuracy of the abrasive mass flow rate during the whole experiment, the abrasive was delivered by an analogue controlled mechanical abrasive metering system (FEEDLINE IV) which was calibrated before each start of the test. All the trials consisted of a single pass of AWJ over the target surface at 90◦ incidence angle. In order to measure the mass removed per AWJ pass, the weight of the workpiece was measured before and after each AWJ pass. This measurement was used to calculate the erosion rate (ER) in the target material, which is defined as in Eq. (13): Erosion rate (ER) =

Total mass removed in the target Cumulative mass of the impacting particles (13)

Once the AWJ milled trenches were generated, 3D surface scanning of the kerfs was performed using a Talysurf 300CLI laser scanner. In each scan an area of 1.8 mm × 30 mm (width (x) × length (y)) was evaluated along the jet traverse direction at a resolution

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Fig. 7. Equivalent plastic strain plot in the simulated half kerf; P = 276 Mpa, Vf = 1000 mm/min.

of x – 5 ␮m, y – 10 ␮m and gauge resolution Z – 0.168 ␮m. Then, a mean surface profile line was extracted out of each 3D scanned surface to enable the validation of the FE model results. A brief illustration of the scanning process is presented in Fig. 6. 4. Results and discussion It is evident from the 3D scanned surface in Fig. 6 that the surfaces of the milled footprints are not very uniform; rather they are uneven and have some pits in them. This is attributed to the facts that; (i) erosion is caused in the target by the impact of a number of particles which differ in masses (inertia), i.e. their capability to erode, hence causing a non-uniform erosion; (ii) although the abrasive particles were supplied by an analogue feeder, which can accurately control the total amount of abrasive supplied to the cutting head over a period of time, there is a degree of variation of this parameter as particles are entrained in the focussing tube/jet plume. Moreover, individually scanned 2D profiles differ from each other as shown by the cluster of the 2D profiles around the mean profile in Fig. 6. In order to overcome these issues, a longer length of footprint (30 mm) was scanned along the jet traverse direction so that a better overall picture of the eroded footprint could be captured by averaging all the individually scanned profiles. Fig. 7 presents a typical eroded surface from the FE simulations. Points “A” and “B” show the initial position of the jet while points “C” and “D” correspond to the jet final position. The dotted rectangle BCFE represents the same area BCFE as shown in Fig. 3 which has received one complete pass of the jet diameter in the X-direction. All the half 2D FE footprints are measured within this area by selecting a path of the nodes in the Y-direction and then the Z-coordinates of these nodes are recorded across the path. An average of these Zcoordinates for ten such nodal paths (half footprints) are taken for each of the eroded trench from the FE model and then mirrored as complete 2D footprints to be compared with the corresponding scanned experimental footprints. The simulation time varies between from 4 to 5 h for 2000 mm/min traverse speed (i.e. for

551 particles) and 16–17 h for 1000 mm/min traverse speed (i.e. for 1102 particles) on a cluster of eight nodes (3 GHz Intel quad-core each node) with 16 GB total ram depending on the pressure (velocity of the impacting particles) used in the FE model. It is noticed that keeping the same number of elements in the target and doubling the number of particles in the FE model increases the simulation time by four times, hence a considerable amount of computational time is saved by using the half model; however, the scope of this paper is not about the reducing the computational time but to show a new approach in FE modelling of AWJ milling. Hence, for the time being this computational approach has been found appropriate. In Fig. 8(a)–(e) the evolution of the erosion due to the impact of various different layers of the abrasive particles on the Ti–6Al–V4 target and the formation of the eroded footprint are illustrated. The different stages in Fig. 8 are expressed in terms of the percentage of the total mass of the particles (mFE ) that has impinged the target surface. Fig. 8(a) represents the initial stage before the particles started impinging the target surface. In Fig. 8(b)–(d), the particles impacted the target at various locations and both the target and the particles undergo erosion. Fig. 8(e) corresponds to the final eroded footprint after all the particles have impinged and the target has traversed the total distance LFE . It can be seen that the depth of cut increases with time as more and more particles impact the surface, and stabilizes only in the red highlighted region which has received one complete pass of the jet diameter. Fig. 9 shows the comparison of the simulated and experimental jet footprints at d ’ =1 mm. Although the depth of the simulated footprint is in good agreement at both traverse speeds, there are some differences in the widths. There is more difference in the width of the kerf at lower traverse speed (1000 mm/min); this is due to the fact that more particles strike per unit area on the target surface and displace more material away from the boundary compared to the higher traverse speed (2000 mm/min) as demonstrated by Srinivasu et al. (2009). It is a well established phenomenon that when a high velocity particle hits a surface, material is piled up at the boundaries of the

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Fig. 8. Stages of erosion of the target during the impact of abrasive particles. *Thirteen different layers of particles were used which were then patterned over each other in Z-direction to achieve the required mass of abrasive particles (mFE ) required in each simulation.

crater i.e. it is displaced away from the boundary of the impacting particle/projectile. This effect becomes more pronounced when the impacting velocity is higher (368–520 m/s) like used in the current study as shown by Dikshit et al. (1995). This implies that if “w” is the width of the jet footprint then the effective diameter of the

-1

-0.75

-0.5

-0.25

jet should be less than “w” (d < w), i.e. most of the abrasive particles in the AWJ plume travel inside the boundary (d < 1 mm) of the jet. It should be noted that the difference in the width is not due to the mesh size (element size) used in the Y-direction. This was verified by using three different mesh sizes in the Y-direction

0

0.25

0.5

0.75

1

Depth(µm)

20 -30 -80

Exp; Vf=1000mm/min FE; Vf'=1000mm/min Exp; Vf=2000mm/min FE; Vf'=2000mm/min

-130 -180 -230

Kerf width (mm)

Fig. 9. Differences between the simulated (FE) and experimental (Exp) width of the footprints at P = 276 MPa at d ’ =1 mm.

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Fig. 10. Investigating the effect of the mesh size on the width of the simulated footprint. (a) Particles with outer edge at diameter (d ’) = 1 mm in the jet causing the material to form a lip further away from the location of impact. (b) Comparison of width of the simulated footprints at various mesh sizes.

while keeping the same mesh size in the X and Z-directions in the fine mesh area. When the mesh size was changed, the value of the material constant d1 was also varied in such a way that the depth of the eroded footprint was approximately the same in all the cases (within ±2%). Fig. 10(a) demonstrates the fact that the lip on the target surface is formed away from the particle impact location. When more particles impact at the same location, the pile-up material is displaced further away. Fig. 10(b) shows the comparison of width of the simulated footprints at three different mesh sizes in the Y-direction where it can be seen that the width of the footprints remains almost the same for these mesh sizes, demonstrating that no mesh sensitivity is present. In order to eliminate the error in the width of the footprint, the diameter of the jet was therefore reduced, i.e. the particles were rearranged in a diameter d ’ =0.9 mm which is the effective diameter of the jet. Fig. 11 shows the simulated averaged jet footprints compared with the corresponding experimental footprints. The results of the FE model are quite encouraging both qualitatively (shape) and quantitatively (depth and width). The shape of the footprint is controlled by the distribution of the particles in the jet while the depth and the width are controlled by both the impacting velocity of the abrasive particles and traverse speed of the target across the jet. The shapes of the simulated kerfs at different pressures and traverse speeds are similar to the corresponding experimental ones, confirming that the particles were correctly distributed while assembling them into layers over the target surface. The depth and the width of the simulated kerfs are also in good agreement (error < 10%) with the experimental ones. Like the experiments, the FE model also predicted different depths of penetrations when the traverse speed was changed keeping the same water pressure. The depth reduces when the traverse speed is increased. This is due to the fact that when the traverse speed is increased, less particles impact per unit area on the target surface, i.e. reducing the exposure time of the target to the impacting particles leading to less erosion in the target. Fig. 12(a) shows that a linear relation exists between the pressure (in the used range 138–276 MPa) and the depth of the footprint at both traverse speeds used. Moreover it can be observed that the depth of the footprint is more sensitive to the change in pressure compared to the width of the footprint. Similar results were found by Srinivasu et al. (2009). The width of the footprint (w) is increased

Fig. 11. Comparison of FE and experimental footprints at given parameters.

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Vf'=2000mm/min

Vf'=1000mm/min

Linear (Vf'=2000mm/min)

Linear (Vf'=1000mm/min)

(a) 250 R² = 0.9964

Depth (um)

200 150

R² = 0.9969

100 50 0 100

150

200 Pressure (MPa)

250

300

Width, w (mm)

(b) 1.07

1.05 1.03 1.01

Vf'=2000mm/min Vf'=1000mm/min

0.99

100

150

200 250 Pressure (MPa)

300

Fig. 12. (a) Variation in depth of the footprint by change in water pump pressure. (b) Variation in width of the footprint (w) by change in water pump pressure.

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only by 1.04% when the water pump pressure (P) is doubled (see Fig. 12(b)). This implies that most of the particles in the AWJ plume are travelling close to the central axis of the jet and only fewer particles are travelling near the jet periphery. Furthermore the spatial distribution of the abrasive particles in AWJ column is not affected ˙ a) by changing the pressure, keeping the same mass flow rate (m based on the fact that a single spatial distribution of the particles in the FE model predicts the correct shapes of the footprints for all the cases. It should be noted that once the jet diameter in the FE model from nozzle exit diameter (d) is reduced to the effective jet diameter (d ’), the errors in the width of the simulated footprints becomes negligible for all the cases in Fig. 11(a)–(f). It can be observed in Fig. 13 that the height of the piled-up material at the boundaries of the simulated footprints is the same as the corresponding experimental ones, whereas there are differences in the length (lp ) of the piled up material in the Y-direction. In the simulated footprint, the target material is piled up over a smaller length compared to the experiment footprints. The reason for this difference can be explained by closely observing the boundaries (edges) of the scanned footprint within the dotted rectangles in Fig. 13(a). During the experiments, the abrasive particles on the boundaries of the AWJ plume were not always impacting the target surface at the maximum effective jet diameter (d ’); rather they hit the surface both inside and outside of the effective jet boundary as well. This means that the width of the experimental footprint was varying across the jet traverse direction and therefore when the average of all the scanned footprints is calculated over the entire trench length, the length of the piled up region also increases (see Fig. 13(b)). However, in the FE model all the particles impact at a constant diameter (d ’). On the other hand, the height of the lip remains the same in

Fig. 13. Details of piled-up material at the edges of the jet footprint at P = 276 MPa, Vf = 2000 mm/min.

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Erosion rate (mg/mg)

0.04

Traverse speed=2000mm/min

0.03 0.02

FE

0.01

Exp

0 100

150

200

250

300

Pressure(MPa)

Erosion rate (mg/mg)

0.04

Traverse speed=1000mm/min

diameter (d) as the jet diameter which can lead to over-prediction of the width of the kerf. The simulated jet footprints and the erosion rates revealed by the current FE model are consistent (errors < 10%) with the experimental results. The good agreement confirms the validity of the model and it might be used for process control as well as optimizing the operating parameters. Due to the stochastic nature of the AWJ milling process, such a model is required that can reliably predict the depth of the cut at a given pressure and traverse speeds. The reported FE modelling approach, by closely simulating the AWJ milling process, enables the prediction of accurate jet footprints and this lead to the possibility for further developments of models to simulate the generation of 3D surfaces on which the jet can move in arbitrary directions. Acknowledgements

0.03 0.02

The authors acknowledge that this work is performed with the support of EU-FP7 Conform-Jet Project (Grant No. 229155). The first author would like to thank the University of Engineering and Technology, Lahore, Pakistan and the University of Nottingham, UK for providing scholarship assistance.

FE Exp

0.01 0 100

150

200

250

300

References

Pressure(MPa) Fig. 14. Erosion rates obtained from FE simulations vs. experimental results.

both the simulated and experimental footprints due to fact that the numbers of particles impacting per unit area are identical in both cases. In order to further validate the model, the erosion rates (ER) are calculated from the FE model and compared with the corresponding experimental values as shown in Fig. 14. It should be noticed that only the abrasive particles impact the target in the FE model whereas in the experiments both water and the abrasive particles impact the target. The good agreement between the experimental and the simulated ER proves that the erosion produced in the target by the water present in the AWJ plume is negligible at the pressures and traverse speeds used in the current study. Thus, the current model can be used to reliably predict the AWJ milled footprints up to an operating pressure of 276 MPa. 5. Conclusions In this paper, a new FE model is developed to predict the abrasive waterjet milled footprints at different water pump pressures and traverse speeds. A method has been devised to include the mass flow rate of the abrasive particles in the FE model. Compared with the previously reported computational models which use spherical-shaped and rigid impacting particles, this FE model is much closer to the experimental conditions by considering, for the first time, the actual non-spherical shapes (rhombic, triangular and trapezoidal) and real size distribution of the abrasive particles. An elasto-plastic material model is also employed for the ductile Ti–6Al–4V target with material failure capability and a tensile failure criterion is utilised for the brittle garnet abrasive particles rather than considering them as rigid particles. The particles are arranged in the Gaussian distribution over the target surface, which is the key factor in controlling the shape of the eroded footprints. The model provides further understanding of the AWJ process by controlling the difference between the width of the simulated and experimental footprints by employing an effective jet diameter (d ’) contrary to the conventional approach of using the nozzle exit

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