Wear -
Elsevier Sequoia S.A., Lausanne
SOME OBSERVATIONS
- Printed
81
in the Netherlands
ON THE EROSION
OF DUCTILE
METALS
I. FINNIE
University of California, Berkeley, California (U.S.A.)
(Received July 2, 1971)
SUMMAkY
The factors which may influence the erosion of ductile metals are listed. It is
shown that the effect of some of these variables may be predicted on simple fundamental grounds. On this basis, quantitative predictions may be made for the erosion of
ductile metals by hard abrasive grains which strike at grazing angles. Other aspects of
erosion which are not as well understood at the present time, are also discussed.
INTRODUCTION
The erosion of a surface by a stream of solid particles has received considerable
attention in the past decade. When the present author discussed this topic about ten
years ago’, the primary motivation for erosion studies was the severe erosive wear
that occurred in the equipment used for the catalytic cracking of oil. Subsequently
other situations have arisen in which erosion has been a problem; for example, in
rocket nozzles and in helicopter engines. The economic importance of erosion in these
and other applications has led in recent years to many papers’-I2 which treat the
erosion of ductile metals from various points of view. Much useful experimental in-formation has been obtained but our understanding of the basic mechanisms by
which solid particles remove surface material does not appear to have been greatly
improved. For this reason it may be worthwhile to discuss, briefly, the aspects of
erosion which can be explained on simple fundamental grounds and then to point out
areas in which our present understanding is inadequate. In recent work’ 3 the erosion
of brittle solids has been discussed in detail so in the present paper only the case of
ductile metals will be treated. Although the concepts of ideally ductile and ideally
brittle behavior are oversimplifications, they do describe to a close approximation
the behavior of many real materials and allow analytical solutions to be developed.
Certainly, the mechanisms by which ductile and brittle solids erode must be understood before it will be possible to make a realistic analysis for some intermediate type
of behavior.
To start with, we list the factors which may influence the erosion of ductile
metals. For a reasonably complete understanding of erosion we should be able to
explain and predict the role of most of these factors. In making this list we exclude the
prediction, from the fluid flow conditions, of the number of particles striking the
surface in a given time and their velocity and direction relative to the surface. This
part of the erosion problem is common to both ductile and brittle materials and to a
Wear, 19 (1972) 81-90
I. FINNIE
82
first approximation
removal.
FACTORS
WHICH
may be treated separately from the mechanisms of material
MAY INFLUENCE
DUCTILE
EROSION
1. Angle of impingement.
2. Particle rotation at impingement.
3. Particle velocity at impingement.
4. Particle size.
5. Surface properties.
6. Shape of the surface.
7. Stress level in the surface.
8. Particle shape and strength.
9. Particle concentration in the fluid stream.
10. Nature of the carrier gas and its temperature.
Not all of these factors are controllable or even easily measurable during a test but we
should at least attempt to estimate their relative importance.
Compared to brittle materials, it is relatively easy to see how one might start
analyzing ductile erosion because material removal should occur by a cutting or
displacing process as in metal cutting or grinding. This led the author’* to write the
equations of motion for a rigid abrasive particle striking the surface of a ductile metal.
The analysis will be outlined here with some corrections and emphasis upon the
physical assumptions involved.
We consider the two dimensional case shown in Fig. 1 with an idealized particle
of unit width although the treatment can be extended to a particle of arbitrary shape’.
The volume displaced by this idealized particle is approximately the integral of yt dx,
(where x,, yr are the coordinates of the particle tip) taken over the period in which
Fig. 1. Idealized
picture
Wear, 19 (1972) 81-90
of an abrasive
particle
striking
a surface and removing
material
83
SOME OBSERVATIONS ON THE EROSION OF DUCTILE METALS
cutting occurs. So, we have to determine the trajectory of the tip of the particle and
estimate when cutting stops.
For simplicity, the following assumption were made:
(1) The displaced volume is the volume removed by the idealized particle.
(2) The particle is rigid and doesn’t fracture.
(3) No initial rotation of the particles, i.e., q$,= 0. In a sense, this is an average
condition. If the initial angular velocity distribution were known, it could be incorporated into the analysis.
(4) Rotation of the particle is small during the cutting period. This can be
confirmed’ and for polyhedral particles, such as shown in Fig. 1, implies that :
xtz.%+r6,
Yt”yG
(5) The conhguration of the particle and of the deformed material is assumed
to be geomet~cally similar during the cutting process. Thus we take the ratio of the
vertical force to the horizontal force on the particle as a constant K. Based on grinding
tests15 using abrasive grains and measurements made with single grains a reasonable
value is Kc= 2, i.e. a coefficient of friction = 0.5.
(6) Since large strains will be reached even at the beginning of the cutting
process, we would expect, based on metal cutting tests 16,that the plastic flow pressure
between particle and metal will be essentially constant. We denote the horizontal
component of this pressure by p.
(7) A final assumption, based on metal cutting observations, is that the area
over which the metal contacts the particle is about twice that given by the depth of cut.
That is, I N 2 y, in Fig. 1.
With the preceding assumptions the equations of motion may be written and
solved for x,, yr, As has been shown”i4 this leads to
v=
c
MU2
where
V
M
m
Z
r
tt
U
p
c
a!;
=
=
=
=
=
=
=
=
=
=
volume removed from surface,
mass of eroding particles,
mass of an individual particle,
moment of inertia of particle about its center of gravity,
average particle radius,
angle of impact,
particle velocity,
horizontal component of flow pressure,
fraction of particles cutting in idealized manner,
horizontal velocity of tip of particle when cutting ceases.
Eflect of angle of irn~inge~~~ on erosion
To start with, we examine the predictions for the effect of the angle of impingement CI.One possibility for the term .?; is that it is zero, i.e., cutting terminates when
the horizontal velocity of the particle tip is zero. This leads to the relation Vrxcos2 a.
Wear, f 9 (1972) 81-90
84
I. FINNIE
However, the particle may leave the surface while the tip is still moving horizontally
and in this case we need to determine 2: for yt = 0. It may be shown that this is given by :
2u
- --sinol
P
fj=Ucosci
where P = K + (1 + mr2/1)
Thus we obtain
v=
c MU2
2
[cos2a]
T----l
4p
1 + y--
cMU2
v=
4p
1+7
2
mr2
F
a
; 2_:for
Yt=O
T-----l
The maximum volume removal occurs at tan 2a = P while the two expressions are
equal at the slightly higher angle given by tan a=P/2. Typically, i-mr2/3 and for
# N 2, P LZ:0.5 and so the m~mum erosion should occur at about a = 13”. Figure 2
shows experimental data as well as the predicted behavior. It is seen that the agreement
is excellent at low values of a but becomes poorer as CIapproaches 90’. This discrepancy is to be expected for the idealized cutting mechanism on which the analysis is
based can hardly be applicable when the particle has no velocity component tangential
to the surface.
Mech~isms which might be invoked to explain the erosion of ductile metals
at angles near a = 90’ are :
(a) Once the surface is roughened, particles strike the surface locally at a
variety of angles and at grazing angles volume is removed.
(b) Particles follow the air flow and actually strike at grazing angles.
(c) The grains have initial rotation which leads to volume removal even at
a = 90’.
(d) The grains fracture on impact and by moving radially outward remove
surface material.
(e) Multiple impacts, battering the surface back and forth, eventually produce
fracture by low-cycle fatigue.
(r) The surface workhardens and eventually fails in a “brittle” manner.
Explanation (a} does not readily lend itself to analysis but would not appear
to be a major factor in producing erosion at angles near 90”. Explanation (b) can
easily be tested by calculating or observing the deflection of particles by a fluid stream
and in the present experiments it may be discounted. Explanation (c), as we will show,
does indeed lead to volume removal at large values of o(.However, it appears unlikely
that the particle angular velocities could be large enough to explain all of the observed
erosion at or near a = 90’. Explanation (d) has been studied by Tilly and Sage l2 and is
undoubtedly a major factor in influencing the form of the V-a curve when the
eroding particles are of a weak and friable nature. However, it appears unlikely that
particle fragmentation was a major factor in the tests shown in Fig. 2. Similar curves
Wear, I9 (1972) 81-90
SOME OBSERVATIONS ON THE EROSION OF DUCTILE METALS
0
0
I
I
I
30
I
I
I
60
ANGLE OF IMPINGEMENT
85
I
90
,
0’
Fig. 2. Predicted and observed values for erosion of commercially pure aluminum, by 120 mesh Sic particles
at 500 ft./set. The scale off (a) is chosen arbitrarily so that its maximum value’coincides with the experimental data. (From Sheldon and Finnie”).
were obtained when other ductile metals of varying hardness values were eroded by
silicon carbide grains. Explanation (e) has, to our knowledge, not been offered before.
However, it is clear that repeated impacts in adjacent areas will subject the surface to
alternating plastic strains which introduces the possibility of fracture by a low-cycle
fatigue mechanism. Explanation (l) has been given a number of times and usually it is
coupled with the assumption that a certain amount of energy is required to remove
surface material in perpendicular impact. That this explanation is, at best, a crude
approximation to actual behavior can be seen by eroding gold This material is
capable of absorbing enormous amounts of energy when it is beaten into thin sheets
of foil but shows the same type of volume removal Versus angle plot as other ductile
metals. In the present case it appears that factors (a) and (e) may be the main reason for
erosion at a = 90” with contributions also from (c) the initial rotation of the particles.
A number of authors have attempted to explain the discrepancy between the
solid and dashed lines in Fig. 2 by invoking two simultaneous mechanisms for volume
removal. For example, Bitter2 took the approach, which has been followed by a
number of other workers, of dividing erosion into “cutting wear” and “deformation
wear”. For “cutting wear” he took an expression similar to that derived here and in
earlier work14. A certain amount of the input energy is assigned to volume removal
by “deformation wear” which is taken as
V= E (U sin a-K)’
where ICand E are disposable constants. The mechanism by which this “deformation
wear” occurs is not at all clear but it is assumed that this is the mode by which volume
removal occurs in brittle solids. In fact, this type of analysis would appear to be
merely curve fitting. It is incapable of explaining the velocity or particle size dependence
of erosion rate in brittle solids and cannot predict the transition from a brittle to
Wear, 19 (1972) 81-90
86
I. FINNIE
ductile type of erosion that occurs in certain brittle solids when the grit size is very
small 13. In general, energy considerations appear to have been used very loosely in
the literature on erosion, and other types of wear. Of course, one can make use of the
specific energy (energy expended +- volume removed) to characterize, in an approximate manner, various types of wear or shaping processes. However, this approach
sheds little light on the physical processes involved in volume removal and is unlikely
to lead to precise predictions of the role of the various variables.
Efict of particle rotation
Initial rotation of the grains is easily incorporated into the analysis if we
consider the two dimensional case shown in Fig. 1. We replace U cos r: by U coso! i_ #+,r
or cos a by (cos tl + a) where a = 2 (ber/U. Clearly, erosion will now occur at a = 90”
to an extent which depends on the parameter do. Unfortunately, it would be difficult
to estimate or measure the distribution of & values characteristic of a given erosion
test apparatus. However, we will present some results based on a hypothetic~ distribution of (It, to show that this factor may, in some cases, be significant. In Fig. 3 we
show the result of assuming an “omega” distribution for initial rotation. The angle at
which maximum erosion occurs is slightly higher and erosion is now predicted for
a = 90”.
I.2
IO
4’
&
g 0.8
w
z
3
OMEGA
DISTRIBUTION
0.6
P
0
0
10
20
30
40
50
ANGLE
OF
IMPINGEMENT
60
70
80
90
Fig. 3. Iniluence of rotation on weight loss-angle relation. The assumed ~stribution
rotation parameter a= &r/V
is also shown.
for the dimensionless
Quite independently of whether the particles have initial rotation, we can see
in the equations for volume removal that it is important to allow for rotation in writing
the equations of motion. Typically, 1-S mr* so about $ of the initial kinetic energy
of translation is converted into rotational kinetic energy during impact.
Particle velocity
The equations predict T/cc U2 and this is observed experimentally to a first
approximation. However, more careful observation *J~*‘~ shows the relation to be
more nearly V4 in many cases. The reason or reasons for this discrepancy are not
clear. The explanations offered have been particle fragmentation at higher velocities’
Wear,19 (1972) 81-90
87
SOME OBSERVATIONS ON THE EROSION OF DUCTILE METALS
and size effect*’ (to be discussed later in connection with particle size). Curiously for
brittle solids, with no scatter in strength, the predicted relation&pi3
is VK U2.4.
Particle size
One of the most intriguing aspects of erosion is that the volume removed by a
given mass of abrasive grains is independent of particle size for particles larger than
about 100 pm. For particles below this size, the erosion process becomes less and less
efficient as the particle size is decreased I7 . This is a fortunate aspect of erosion because
as particles become smaller it becomes more difficult to separate them from the fluid
stream.
The physical reasons for this size effect are still not clear. Some of the factors
which one might consider are :
(a) Fragmentation of larger particles leading to more efficient cutting than
with smaller particles.
(b) Grain size of the metal eroded.
(c) Oxide coating on the metal eroded.
(d) Change in the geometry of the cutting process as smaller particles are used.
(e) A true physical size-effect such that regions below a certain size show an
increase in strength values.
Explanation (a) has been offered by Tilly and Sage12. However, it can be
discounted by the observation that the same size effect is observed in abrasion tests
on soft metals at very low speeds and in this case preferential fracture of large grains
could hardly be a signi~cant process. Explanations (b) and (c) appear to be possible
reasons for the size effect but were dismissed after a series of erosion tests were carried
out on single crystals of copper and on pure gold, which has no oxide coating. Explanation (d) also appears plausible but was dismissed after comparing the tests on
alum~ium shown in Fig. 4. The curves of weight loss versus angle for erosion by 127
pm and 9 pm particles are so similar (except for the scale factor of 4) that it appears
34
t
::
Le
2c1.
B
P
;
2E m
I
$
2 4
I
0
Iz
z
z
fl?
P
0
OW
30
0
PARTICLE
60
APPROACH
ANGLE
90
,
o”
Fig. 4. Weight removed (&g of abrasive particles) when commerciaIIy pure alu~num
particks of two sizes at 500 ft./set. (From SheIdon and Finnie”).
is eroded by Sic
Wear, 19 (1972) 81-90
88
I. FINNIF.
that the same process is occurring in both cases. We are left only with the explanation
(e) that there is a true physical size-effect. In this connection, it is interesting to note
that indentation tests made with tiny needles, and observed with the scanning electron
microscope show in some cases about a three-fold increase in hardness relative to
conventional tests’*.
Properties
of the surjbce
If we discount the size effect found with very small particles, we would expect
the stress state involved in erosion to be similar to metal cutting. Thus based on metal
cutting experience l9 the horizontal component of pressure p should be approximately
equal to the Vickers (or Brinell) hardness when cutting annealed metals. In previous
work we concluded that the fraction c of the particles which cut in the idealized manner
was about 0.5. However, more careful examination shows this to have been an overoptimistic assessment. Many particles only plow the surface and in fact remove little
if any material. Based on the careful study of Mulhearn and Samuel?’ a more realistic
figure for c is perhaps 0.1. Since the surface is being workhardened by many non-cutting
particles and since the strains produced by the cutting particles will be very high we
would not expect prior cold work of the material to influence erosion. Figure 5 shows
results obtained by the author and his colleagues 21. It is seen that annealed facecentered cubic materials follow the relationship VW_l/(Vickers Hardness) and prior
cold work was found to have no influence on erosion. Possible reasons for the slightly
different behavior of body-centered cubic and hexagonal metals are discussed in
reference 2 1. Here we examine the more important practical question : “Can we make
quantitative predictions of erosion ?” Taking c=O.l, p=Vickers Hardness, K=2,
I N 3 m?, c(= 20” and I/ = 250 ft. the preceding equations for volume removal lead to
the dashed line shown in Fig. 5. Thus, we conclude that for rigid abrasive grains, with
size larger than say 100 pm we can make order of magnitude estimates of volume
removal. For 10 pm particles the volume removal will be less by a factor of perhaps live.
IO
VICKERS
HARDNESS,
100
kg per sq mm,
IO00
VHN
Fig. 5. Volume removed (mm3/g abrasive) as a function of Vickers Hardness when annealed
eroded by 60 mesh SIC at o!= 20’ and U = 250 ft./set. (From Finnie, Wolak and Kabil”).
Wear, 19 (1972) 81-90
metals are
SOME OBSERVATIONS
ON THE EROSION OF DUCTILE
METALS
89
The shape of the surface
A curious feature of ductile erosion is that ripples appear on the surface when
materials are eroded at an angle at or near that for maximum erosion. The simple
analysis of plastic cutting that we have presented is also capable of explaining this
phenomenon22 and of predicting the effect of surface curvature on erosion rates.
Stress level in the surface
By contrast to brittle solids, we would expect that high residual streses would
have little influence on ductile erosion. In a few exploratory tests it was found that
residual stresses had no effect on erosion while applying the stress by external bending
moments during the test led to a barely detectable increase in erosion. As a result this
aspect of erosion was not pursued further.
Particle shape and strength
The simple analysis of ductile erosion which has been presented is based on
rigid abrasive particles that do not fracture during cutting. While most of our work
was carried out with angular silicon carbide grains, the same tests carried out with
rather more “blocky” aluminum oxide grains gave very similar results. The analysis
could be extended to rigid particles of other shapes, again excluding fracture of the
particle, by selecting appropriate values of K and 1. It is clear from the work5-* of
Tilly and colleagues that particle shape and strength play a role in erosion and in
particular it is seen that fracture of the particle may drastically change the shape of the
curve of volume removal as a function of angle.
Particle concentration in the fluid stream
A small effect of particle concentration on erosion has been reported several
times in the literature. To our knowledge no satisfactory explanation for this effect
has yet been offered.
Nature of the carrier gas and its temperature
The results ofelevated temperature tests have been reported in the literature6p”.
The most striking feature of this work is that the effect of temperature on erosion is
small in the normal range of operating temperatures for the alloys studied. In retrospect
this result is perhaps to be expected. Extremely high temperatures, of the order of the
melting point, can be computed for the material being removed in ordinary “room
temperature” erosion tests so the additional temperature imposed in “elevated temperature” testing may not be significant. In erosion the strain rates will also be extremely large and it is known from metal cutting tests that temperature and strain rate are
offsetting factors with the effect of elevated temperature being very greatly reduced at
high strain rates 23. More recent fundamental studies24 have shown that at very high
strain rates a thermal deformation process appears to be involved. Little has been
published on the effect of the carrier gas itself.
CONCLUSIONS
The model presented for ductile materials describes many features of the erosion
process. However, a number of aspects remain to be explained. Among these is the
Wear, 19 (1972) 81-90
90
1. FINNIE
size-effect observed in ductile materials. By contrast, there is no difficulty in explaining
the size-effect in brittle solids13. The observed dependence of erosion on velocity in
ductile &tals is somewhat different from that predicted and the reasons for this discrepancy remain obscure. The erosion of ductile metals at angles near CI= 90” can still
not be tre$ed in a quantitative manner although possible mechanisms for this type of
wear can be advanced.
From a design point of view, what is needed is an “order or magnitude” type
prediction for erosion damage since operating conditions can rarely be defined with
precision. For ductile materials eroded by hard abrasive grains at low angles of
impingement we have provided such an estimate based on the Vickers Hardness of
the annealed metal.
ACKNOWLEDGEMENT
This study was supported by the Solar Aircraft Division of International
Harvester Company. The effect of particle rotation was calculated by Dr. K. P. L. Oh
and Mr. S. Chibber conducted the erosion tests on single crystals and gold referred to
in the paper.
REFERENCES
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
I. FINNIE, Weur, 3 (1960) 87-103.
J. G. A. BITTER, Weur, 6 (1963) 5-21, 169-190.
J. H. NEIL~~NANDA. GILCHRLST,Wear, I1 (1968) 111-122.
J. H. NEILSONANLIA. GILCHRIST,Wear, 11 (1968) 122-143.
W. SAGE AND G. P. TILLY, Aeron. J. of Roy. Aeron. Sot., 73 (1969) 427428.
G. P. TILLY, Wear, 14 (1969) 63-79.
G. P. TILLY, Wear, 14 (1969) 241-248.
J. E. GOODWIN,W. SAGEANDG. P. TILLY, Proc. Inst. Mech. Eng., 184 (1969970) 279-292.
W. J. HEADAND M. E. HARR, Wear, 15 (1970) 146.
G. L. SHELWN, Truns.ASME, J. Basic. Eng., 920 (1970) 619-626.
C. E. SMELTZER,M. E. GULDENANDW. A. COMPTON,J. Basic Eng., 920 (1970) 639654.
G. P. TILLY ANDW. SAGE, Wear, 16 (1970) 447-465.
H. L. OH, K. P. L. OH, S. VAIDYANATHAN
ANDI. FINNIE,On the shaping of brittle solids by erosion and
ultrasonic cutting, Proc. Syrnp. Science of Ceramic Machining, to be published by U. S. Natl. Bureau
of Standards.
I. FINNIE, The Mechanism of Erosion of Ductile Metals, ASME, Proc. 3rd U. S. Nutl. Congress af
Applied Mechanics, 1958, pp. 527-532.
E. R. MARSHALLANDM. C. SHAW, Trans. ASME, 74 (1952) 51-59.
I. FINNIE ANDJ. WOLAK, Trans. ASME, 85B (1963) 351-356.
G. L. SHELDONANDI. FINNIE, Trans. ASME, 88B (1966) 387-392.
N. GANE ANDJ. M. Cox, Phil. Mug., 22 (1970) 881-891.
N. H. COOK, Manufacturing Analysis, Addison Wesley, Mass. 1966.
T. 0. MULHEARNANDL. E. SAMU~S, Wear, 5 (1962) 478-498.
I. FINNIE,J. WOLAK AND Y. H. KABIL, J. Mater., 2 (1967) 682-700.
I. FINNIE ANDY. H. KABIL, Wear, 8 (1965) 60-69.
M. C. SHAW ANDI. FINNIE, Trans. ASME, 77 (1955) 115-125.
M. P. VICTORIAet al., J. Appl. Phys., 41 (1970) 674-677.
Wear, 19 (1972) 81-90