A FINITE ELEMENT MODEL FOR ABRASIVE WATERJET … Benha... · Ashraf I. HASSAN, Jan KOSMOL ignored...
Transcript of A FINITE ELEMENT MODEL FOR ABRASIVE WATERJET … Benha... · Ashraf I. HASSAN, Jan KOSMOL ignored...
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
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
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.
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]
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]
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.
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.
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
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
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
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)
Ashraf I. HASSAN, Jan KOSMOL
(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)
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.
Ashraf HASSAN, Jan KOSMOL
(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
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.
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
A Finite Element Model for Abrasive Waterjet Machining (AWJM)
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