A FINITE ELEMENT MODEL FOR ABRASIVE WATERJET … Benha... · Ashraf I. HASSAN, Jan KOSMOL ignored...

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Prace Naukowe Katedry Budowy Maszyn 1999 Ashraf I. HASSAN and Jan KOSMOL Department of Machine Technology, Technical University of Silesia, Gliwice, Poland A FINITE ELEMENT MODEL FOR ABRASIVE WATERJET MACHINING (AWJM) Abstract: Understanding the exact nature of erosion of workpiece materials by abrasive waterjet machining (AWJM) is still confused, although it is important for successful modeling of this promising process. This paper presents a first attempt to model the AWJM process using the powerful tool of the finite element method (FEM) in order to explain the abrasive particle-workpiece interaction. Also the model predicts the behaviour of the process. The main objective is to develop an FE model which would enable to predict the depth of cut without any cutting experiments. The new model takes into account the precise representation of the constitutive behaviour of the workpiece material under AWJ dynamic loading conditions which was ignored in previous AWJM models in which the flow stress was represented by a constant value. Additionally, deformations, stresses and strains occurring in the workpiece material in the vicinity of the cutting interface as a result of the erosion impact by AWJ, could be obtained. In the present model, forces acting on the abrasive particle need not be initially determined, as in previous AWJM studies, as they are automatically calculated at each time step. The results show that the finite element method is a useful tool in predicting abrasive-material interaction and AWJ depth of cut. 1. INTRODUCTION Abrasive waterjet machining (AWJM) has been successfully used in the machining of a wide variety of engineering materials. The capabilities and applications of this new promising process are discussed in detail elsewhere [1-3]. Due to the large number of parameters involved in AWJM, a predictive model is crucial for optimization of the process performance. To understand and predict AWJM performance, several models were developed. Most of these models concentrated on the prediction of the depth of cut and the surface quality, given the initial machining parameters. These models were based on either mathematical relationships describing the interaction between one abrasive particle and the workpiece in the cutting and deformation wear zones [4-8] or on experimental formulas based on statistical design principles [9-10]. The drawbacks of the early Hashish’s model were, the abrasive particle velocity was assumed constant along the whole depth of cut, the effects of the particle shape and size were

Transcript of A FINITE ELEMENT MODEL FOR ABRASIVE WATERJET … Benha... · Ashraf I. HASSAN, Jan KOSMOL ignored...

Page 1: A FINITE ELEMENT MODEL FOR ABRASIVE WATERJET … Benha... · Ashraf I. HASSAN, Jan KOSMOL ignored [5]. In a subsequent study [7], these effects were considered by expressing the sphericity

Prace Naukowe Katedry Budowy Maszyn 1999

Ashraf I. HASSAN and Jan KOSMOL

Department of Machine Technology, Technical University of Silesia, Gliwice, Poland

A FINITE ELEMENT MODEL FOR ABRASIVE WATERJET

MACHINING (AWJM)

Abstract: Understanding the exact nature of erosion of workpiece materials by abrasive

waterjet machining (AWJM) is still confused, although it is important for successful modeling

of this promising process. This paper presents a first attempt to model the AWJM process

using the powerful tool of the finite element method (FEM) in order to explain the abrasive

particle-workpiece interaction. Also the model predicts the behaviour of the process. The main

objective is to develop an FE model which would enable to predict the depth of cut without

any cutting experiments. The new model takes into account the precise representation of the

constitutive behaviour of the workpiece material under AWJ dynamic loading conditions which

was ignored in previous AWJM models in which the flow stress was represented by a constant

value. Additionally, deformations, stresses and strains occurring in the workpiece material in

the vicinity of the cutting interface as a result of the erosion impact by AWJ, could be obtained.

In the present model, forces acting on the abrasive particle need not be initially determined, as

in previous AWJM studies, as they are automatically calculated at each time step. The results

show that the finite element method is a useful tool in predicting abrasive-material interaction

and AWJ depth of cut.

1. INTRODUCTION

Abrasive waterjet machining (AWJM) has been successfully used in the machining of a

wide variety of engineering materials. The capabilities and applications of this new promising

process are discussed in detail elsewhere [1-3]. Due to the large number of parameters

involved in AWJM, a predictive model is crucial for optimization of the process performance.

To understand and predict AWJM performance, several models were developed. Most of these

models concentrated on the prediction of the depth of cut and the surface quality, given the

initial machining parameters. These models were based on either mathematical relationships

describing the interaction between one abrasive particle and the workpiece in the cutting and

deformation wear zones [4-8] or on experimental formulas based on statistical design principles

[9-10]. The drawbacks of the early Hashish’s model were, the abrasive particle velocity was

assumed constant along the whole depth of cut, the effects of the particle shape and size were

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Ashraf I. HASSAN, Jan KOSMOL

ignored [5]. In a subsequent study [7], these effects were considered by expressing the

sphericity and the roundness numbers of the abrasive particle. Moreover, the width of cut was

considered the same as the jet diameter. In this model, the flow stress was chosen equal to the

theoretical fracture strength i.e. (E/14) for mild steel. It was argued that in erosion, the impact

of a particle is very localized. The accuracy of the predictive AWJM models, e.g. Hashish’s

model, was only tested on materials with similar general behaviour. In addition, all published

formulas need specific coefficients, which have to be determined experimentally. This means

that the impact velocity, impact angle and grain size distribution are not accurately evaluated

[11]. The common drawback of the previous models is that the precise representation of the

constitutive behavior of the workpiece material, under AWJ dynamic loading conditions, was

ignored. Table 1 shows the main features which were used in the previous basic AWJ

theoretical models.

Table 1 Basic AWJ theoretical models

Analysis type Material

model

Failure

criterion

Loading Model

dimensions

Authors

Dynamic

analysis

Rigid plastic

strain rate,

strain

hardening

and large

strains

neglected

Von Mises:

flow stress is

constant =

2400-2800

MPa (from

erosion tests)

constant

forces are

assumed

2D plane

strain

Finnie [12]

Elastic-

plastic:

The previous

phenomena

are neglected

flow stress is

constant

Bitter [13,14]

Dynamic

analysis

Rigid plastic:

The previous

phenomena

are neglected

flow stress:

constant =

E/14

constant

forces were

assumed

Hashish [5,6]

The finite element method has been successfully applied in modeling of low-speed impact

problems. It was recognized in the early 1900 that an impact generates two different although

interconnected events: phenomena at or in the immediate vicinity of the contact position and

the propagation of a signal to distant points of the two objects. Both occurrences depend on

the properties and geometry of the impacting bodies and on the relative impact velocity. For

speeds less than 0.1 m/s, no significant waves will be produced, and hence the process can be

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A Finite Element Model for Abrasive Waterjet Machining (AWJM)

regarded as occurring under quasi-static conditions. This impact is characterized by completely

elastic behavior and the energy loss is a small portion of the total mechanical energy and can be

neglected [15]. The finite element method has also been successfully applied in the modeling of

high-speed impact problems. The quantitative description of impact processes utilizes the

conservation of linear and angular momentum, impulse momentum laws, and energy balance as

well as an appropriate material constitutive equation. Table 2 shows some of both low-speed

and high-speed impact models.

Up to date, there has been few attempts at analyzing the WJM using the powerful tool of

the finite element method. A preliminary effort of modeling pure waterjet using a finite element

model was carried out by Hassan and Kosmol [25]. Using this model, it was possible to predict

the shape of the AWJ kerf, workpiece deformations and stresses. The main drawback of the

model was that it could not be used in the prediction of the depth of cut as it is linear elastic.

Table 3 shows some other attempts to model WJM using FEM. These models fail to accurately

represent the actual erosion occurring in AWJM. They either oversimplified modeling, by

assuming linear elastic material models, or idealized the physical event, by assuming static

analysis.

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Table 2 FE models for impact problems

Process FEM

code

Analysis type Material

model

Elements Boundary

conditions

Time

step/Impact

speed

FE results Expl. work Authors

A projectile

impacting a

rigid surface

nonlinear

dynamic

projectile:

elastic/

plastic,

W.P :

rigid

15

lumped

mass

elements

impact load: up

to 50 kN/impact

area: 78.5

mm2/stand off

distance: 400 mm

-

50 to 250

m/s.

deformation

stress-strain

distributions

load-time

history and

the final

deformed

shape

Hamouda

and

Hashmi

[16]

Impact of a

square

hollow tube

MARC 2D nonlinear

dynamic/ implicit

updated

Lagrangian/

Newton Raphson

elastic-

plastic

with

strain

hardening

1344 four

nodded

shell

element

load is external

magnetic

pressure/Impact

area: 160

mm2/Impact

load: up to 8 kN

17.5 ms/

400 steps

/up to 4 m/s

-hollow

tube

deformation

- strains

deformation:

high-speed

camera -

strains:

strain

gauges.

Miyazaki

et al. [17]

Impact of a

hollow tube

MARC 2D nonlinear

dynamic/ implicit

updated

Lagrangian/

Newton Raphson

elastic-

plastic

116-240

quadilate-

ral

elements

magnetic

pressure

0.7 ms/

200 time

steps

hollow tube

deformation

deformation:

high-speed

camera

Hashimoto

et al. [18]

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Table 3 FE models for WJ

FEM code Analysis type Material

model

Elements Boundary

conditions

Time step/

speed

FE results Expl. work Authors

ASKA 3D static

nonlinear/total

Lagrangian

nonlinear

elastic-

plastic

forty 3D

solid

elements

calculated forces

specimen:1 mm *

1 mm

deformation AWJ kerf

deformation

Przyklenk &

Schlatter [19,

20]

ANSYS nonlinear

dynamic /

Lagrangian/2D

nonlinear

plastic

1960 four

nodded

axisymmetr-

ic (0.065

*0.08 mm

pressure

/specimen

dimensions: 2.5

mm* 3.2 mm

1 msec /

simulation

time : 0.33

msec

deformation

Maximum

deformation

: 0.5 mm

FEM depth

of cut

compared

with AWJ

kerfs

Li [21]

DYNA 3D explicit Linear

dynamic/Lagra-

ngian/3D

linear elastic 20,000 solid

element

contact elements

between the

waterdrop and the

w.p surface

< 1 ms

305 m/s

deformation

& stresses

- Alder [22]

NASTRAN 3D explicit

linear

dynamic/Lagra-

ngian

linear elastic shell

elements

< 1 ms

150-450 m/s

deformation

stresses: up

to 1600Mpa

comparison

with

literature

deBotton

[23]

SOLID dynamic fluid-

solid coupling /

Lagrangian

elastic-

plastic

pressure load

calculated by

numerical analysis

a fraction of

a micron

depth of

cut, stresses

AWJ kerfs Patella &

Reboud [24]

NASTRAN 2D linear static linear elastic WJ pressure - deformation

stresses

- Hassan &

Kosmol [25]

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Ashraf HASSAN, Jan KOSMOL

The objective of this paper is to conduct a finite element analysis of the workpiece material

under the action of AWJ. The new model takes into account the precise representation of the

constitutive behaviour of the workpiece material under AWJ dynamic loading conditions which

was ignored in previous AWJM models where the flow stress was represented by a constant

value. Additionally, deformations and stresses occurring in the workpiece material in the

vicinity of the cutting interface as a result of the erosion impact by AWJ could be obtained. In

the previous AWJM studies, forces acting on the abrasive particle are assumed in order to

solve for the depth of cut. In the present model these forces need not be initially determined, as

they are calculated at each time step. In the present model, the dynamic progress of the

abrasive-workpiece interaction will be tracked at small time increments.

2. FINITE ELEMENT MODEL

2.1 finite element mesh and modeling assummptions

It is critical that the finite element mesh must be fine enough to allow realistic deformation.

Meshes that are too coarse can prohibit actual deformation and yield erroneous results. In the

present study, the following assumptions are relevant:

1. The hydrostatic loading plays a secondary role in the cutting process, i.e. the effect of

the carrying fluid is only to accelerate the abrasive particle to high speeds [5].

2. The abrasive particle velocity is recalculated at the beginning of each time step. This is

contradicted to the assumed constant abrasive particle velocity in previous studies [5].

3. For the FE code to remain stable, the time step must subdivide the shortest natural

period in the mesh. This means that the time step must be smaller than the time taken for

a stress wave to cross the smallest element in the mesh. In order to get more accurate

results, the time step is chosen as 0.01 s.

4. Loading resulting from changes in motion is computed and applied internally.

5. It is established that the abrasive particle passing through a mixing chamber and a

focusing tube fail much more than it would correspond with the theoretical assumptions.

In this study, particles’ disintegration in the mixing chamber is neglected.

Figure 1 shows the finite element mesh adopted in the present study. Due to the symmetry of

geometry, we only need to model and analyze a quarter model of both the abrasive and

workpiece, which greatly reduces the computational requirements. The workpiece is modeled

by a three dimensional finite element model and is divided into 20 nodded higher order

nonlinear solid elements.

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A Finite Element Model for Abrasive Waterjet Machining (AWJM)

Fig. 1 Finite element model of AWJM

A total number of 250 elements were used to model the workpiece with size of each element of

0.1 mm * 0.1 mm * 0.1 mm. The number of nodal points is 1566.The overall workpiece

dimensions are 0.5 mm * 1 mm * 0.5 mm. The workpiece material data is shown in Table 4

below. In AWJM, the workpiece has nonlinear material behavior and geometric nonlinearity

under the applied impact loads, hence a nonlinear analysis is required. The constitutive model

used for the workpiece is chosen as Von Mises elastoplastic isotropic hardening with linear

strain hardening. This material model has more realistic response due to hardening and

provides a simple means of accounting for the material response due to a stress reversal. Using

this model, The workpiece erosion is witnessed simultaneously on the screen.

Table 4 Workpiece material data

Property

Material Carbon steel

Young's modulus 207 E 3 MPa

Poisson's ratio

Density

Yield stress

Strain hardening modulus

0.3

7850 Kg/m3

207 MPa

34.5 E3 MPa

The abrasive particle is modeled using 16 twenty nodded solid elements. Linear elastic

model is chosen for the abrasive particle material, Table 5. The dimensions of each element are

0.1 mm*0.1 mm* 0.1 mm. AWJ cutting conditions are shown in Table 6.

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Ashraf I. HASSAN, Jan KOSMOL

Table 5 Abrasive material data

Property

Material Garnet

Young's modulus 248 E 3 MPa

Poisson's ratio

Density

0.27

4325 Kg/m3

Table 6 AWJM conditions

Parameter

Abrasive particle velocity, m/s

Abrasive particle size, m

Mesh No

Stand off distance, mm

400

400

60

0.1

The boundary conditions include a fixed support of the workpiece from the bottom as it is

fixed on the table of the AWJ machine. The abrasive particle is allowed to move freely

downwards perpendicular to the workpiece surface. Preliminary investigations were carried out

on several models and a compromise between the excessive CPU time and accuracy of results

was considered.

2.2 modeling of abrasive-workpiece interaction

3D Contact elements, the specifications of which are shown in Table 7, are added between the

abrasive particle and the workpiece that allow for complete interaction including transfer of

momentum between the abrasive particle and the workpiece. Contact elements are

characterized by element nonlinearity where the stiffness matrix of an element will change as a

function of some specified variable. When the surfaces are in contact, the stiffness matrix of the

contact element is positive and the contact forces are transferred through the contact element.

When the surfaces are not in contact, the stiffness of the contact element is zero and no forces

are transferred.

Table 7 Specifications of contact elements

Parameter

Contact area, mm2

Compressive contact distance

Number of elements

Type of nonlinear analysis

Material model

0.01

0.01

9

Updated Lagrangian formulation

Bi-linear contact force-distance law

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A Finite Element Model for Abrasive Waterjet Machining (AWJM)

2.3 Solution procedure

The model was then analyzed on a Pentium II PC workstation and deformations and stresses in

the workpiece material were obtained using ALGOR Accupak/VE nonlinear dynamic stress

analysis and event simulation, Version 12 WIN [26], which models the nonlinear behavior by

incrementing the load and updating the geometric stiffness matrix. In Event Simulation, the

force is indeterminate and results from some type of action and motion. The deformation is

calculated directly from the governing equations. In AWJM, the advantage of this is that the

user does not have to assume a static situation or have to estimate values for AWJ impact

forces. In the present analysis, Newton’s second law and Hooke’s law are combined to

eliminate force as follows.

[M] {a} + [C] {V} + [k] {d} = 0 where,

M mass matrix

a acceleration vector

C damping constant matrix

V velocity vector

k stiffness matrix

d displacement vector

It is clearly seen from the previous equation that the combination of motion, damping and

mechanical deformation are combined and modeled. The analysis formulation used is Updated

Lagrangian Formulation, which is effective for modeling elastic-plastic analysis involving large

displacement, large rotation and large strain, Table 8. The constitutive relations are expressed

in terms of Jaumann stress rate and velocity strain tensors. At various critical loading points,

extreme smal time steps may be required in order to make an iterative process converge. At

large strain level, the generalized Hook’s law (constant E, etc) no longer represents any real

materials since the stress-strain relation is nonlinear. In large strain nonlinear analysis, the

constitutive tensor is given in terms of certain material parameters which are functions of

strains or stresses. Completely erroneous results may be generated if the material model is not

consistent with the strain level of the structure. It is also used where geometric stiffness

changes are expected. These conditions are typical for AWJM. The elastic-plastic model has

the advantages of handling the geometric nonlinearity and instability and calculation of residual

stresses to the contrary of the rigid-plastic model that does not account for geometric

nonlinearity and instability. The nonlinear iterative solution scheme used is the Combined Full

and Modified Newton-Raphson Iteration Algorithm. It is between the full Newton-Raphson

method, which has the advantages that it is usually more effective for problems with strong

nonlinearity such as AWJM, and that it converges quadratically with respect to the number of

iterations, and the Modified Newton-Raphson method, which is suitable for moderate

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Ashraf I. HASSAN, Jan KOSMOL

nonlinearity and it is not effective for sudden loading condition changes such as that

encountered in AWJM. This solution method allows to achieve convergent solutions for

problems involving motion. This solution method damps out common convergence problems

such as high frequencies, because they are just noise within the solution.

Table 8 Iteration Control for Nonlinear Element

Parameter

1. Maximum number of stiffness reformations

permitted in each matrix reform step

2. Convergence criteria code for equilibrium

iterations

3. Nonlinear iterative solution method

4. Contact wall tolerance

5. Displacement convergence tolerance

6. Maximum number of stiffness reformations

permitted in each time step

7. Number of time steps between equilibrium

iterations

8. Maximum number of equilibrium iterations

permitted

9. Displacement convergence tolerance

15

Displacement only

Combined full-

modified Newton

0.01

1E-15

15

1

15

1 E-15

3. RESULTS AND DISCUSSION

3.1 Workpiece deformation

As AWJ hits the workpiece surface, an erosion front develops. The cutting front is moving

through the workpiece material. The development of workpiece deformation as a result of

AWJ impact is shown in Fig. 2. The impact occurs after only 0.23 s, Fig. 2 (a), due to high

AWJ speed; 400 m/s. At the cutting interface, there is a strong interaction between AWJ and

the highly deformable layer of the workpiece. The abrasive particle interchanges its high

momentum with the workpiece surface and as a result, deformation of the workpiece surface

propagates. The severely deformed AWJ kerf continues displacement downwards and goes

into severe plastic deformation which is observed in Fig. 2 (b). It is clearly seen from the figure

that, the workpiece deformation rises sharply in a very small time from (a) to (b). As the

abrasive particle penetrates into the workpiece material, the AWJ kerf deepens and it becomes

narrower, Fig. 2 (c). It is clear from the figure that the plastic deformation is very localized in

the area of impact. This is considered one of the most important advantages of AWJM over

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A Finite Element Model for Abrasive Waterjet Machining (AWJM)

laser beam or plasma beam machining, which both induce a heat affected zone around the cut

area. However, the workpiece surface away from the jet impact region is not suffering from

severe deformation. The development of the depth of cut with time is shown in Fig. 3 below. It

is seen from the figure that, shortly after the initial impact period, which is characterized by a

large depth of cut, the rate of deformation decreases. As a result, the depth of cut then

increases at a slower rate. This observation occurs because AWJ loses its velocity as it

penetrates into the workpiece material, causing a decrease in both the workpiece erosion rate

and the surface quality towards the jet exit. Compared to the results of Hashish [5,6], the

present results provide better insight into the mechanism of interaction between the abrasive

particle and the workpiece surface at the early stage of impact. Hashish based his analysis on

macro-scale erosion, whose least time step used is 1 ms, whereas, here it is based on micro-

scale erosion in which the least time step is 0.01 s.

(a) t = 0.23 s

Fig. 2 Development of AWJ kerf with respect to time (abrasive particle is removed for

clarity)

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(b) t = 0.27 s

(c) t =0.37 s

Fig. 2 Development of AWJ kerf with respect to time (Cont’d) (abrasive particle is

removed for clarity)

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A Finite Element Model for Abrasive Waterjet Machining (AWJM)

Fig. 3 Development of the depth of cut with time

3.2 Workpiece stresses

The impact of the abrasive particle on the workpiece surface causes peak loads which cause

extremely high stresses to build up causing local plastic deformation at the point of impact.

Consequently, for a short instance, intensive stresses that frequently produce a local site of

damage develop in the workpiece material. In ductile materials, as in the present case, where

the primary failure mechanism is plastic flow, the damage appears in the form of a conical

crater around the impact site with the apex exactly in the center. Figure 4 shows the results of

the flow stresses acting at the cutting interface. The impact region is subject to high plastic

compressive stresses, which exceed the rupture strength of the workpiece material causing

local plastic deformation. It is seen that the maximum stress lies away from the center of the

impact site, Fig. 4 (a). As the abrasive particle penetrates into the workpiece material, the flow

stress region decreases and the position of the maximum flow stress shifts inwards to the center

of the impact site. The stress wave induced in the workpiece material as a result of the initial

impact is clearly seen, Fig. 4 (a). At a later stage, Fig. 4 (b), stress waves have propagated into

the workpiece causing further plastic flow.

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(a) t = 0.23 s

(b) t = 0.27 s

(c) t =0.37 s

Fig. 4 Development of Von Mises stresses as a result of AWJ impact

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A Finite Element Model for Abrasive Waterjet Machining (AWJM)

3.3 Workpiece strains

Figure 5 clearly shows that a comparatively wide region of compressive plastic strains develops

along AWJ kerf as soon as the abrasive particle impinges the workpiece surface, Fig. 5 (a).This

region moves to the center of the impact site, Fig. 5 (b). As the abrasive particle penetrates into

the workpiece material, Fig. 5 (c), plastic strains concentrate along the kerf especially in the

center of the site of impact which suffers excessive platic deformation. It is apparent that the

plastic deformation is very localized around the center of the impact site. This is consistent

with the previous findings which stated that the strains occurring in erosion must be very large,

and in addition the surface will become work-hardened by the eroding particles [27]. It is

clearly seen from the figure that a residual plastic strain appears in the area surrounding the

AWJ kerf. These strains are large on the inner edge of the kerf, and they decrease rapidly as we

go away from the kerf. This plastic strain distribution at various stages of abrasive particle

impact, shed light on the development of the AWJ kerf and the accumulation of plasticity.

ACKNOWLEDGMENTS

The authors would like to express their gratitude to the Polish State Committee for Scientific

Research (KBN) for its support of this work.

CONCLUSIONS

As a result of the present elastic-plastic dynamic FE model of the abrasive waterjet

machining (AWJM) impact, the following conclusions could be drawn :

1. There is a strong interaction between AWJ and the highly deformable layer of the

workpiece where the plastic deformation is very localized. The workpiece deformation

sharply rises in a very short time. Shortly after the initial impact period, the rate of

deformation decreases.

2. High compressive stresses are generated at AWJ cutting interface. These stresses reach

values that exceed the compressive rupture strength of the material under high strain rate

erosion loading causing material to flow. This leads to severe local plastic deformation.

3. As the abrasive particle penetrates into the workpiece material, the plastic strains

increase sharply along the AWJ kerf especially in the center of the site of impact which

suffers excessive plastic deformation. This deformation is very localized around the

center of the impact site.

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Ashraf I. HASSAN, Jan KOSMOL

(a) t = 0.23 s (step 23)

(b) t = 0.27 s

(c) t =0.37 s

Fig. 5 Development of strains with respect to time from the early impact

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A Finite Element Model for Abrasive Waterjet Machining (AWJM)

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