M. LI et al.: ESTABLISHMENT OF A MULTI-PARTICLE EROSION MODEL BASED ON THE LOW-CYCLE FATIGUE LAW ...
321–326
ESTABLISHMENT OF A MULTI-PARTICLE EROSION MODEL
BASED ON THE LOW-CYCLE FATIGUE LAW – AN
EXPERIMENTAL STUDY OF EROSION CHARACTERISTICS
UVEDBA MODELA EROZIJE, NA OSNOVI ZAKONA ZA
MALOCIKLI^NO UTRUJANJE, ZARADI UDARJANJA MNO@ICE
DELCEV NA JEKLENE STENE – EKSPERIMENTALNA [TUDIJA
ZNA^ILNOSTI EROZIJSKE OBRABE
Mingfei Li
1
, Yihua Dou
1
, Hong Li
2
, Jiarui Cheng
1*
, Wenlan Wei
1
1
School of Mechanical Engineering, Xi’an Shiyou University, Xi’an City, 710065, China
2
PetroChina Xinjiang Oilfield Company, Karamay City, 834000, China
Prejem rokopisa – received: 2019-06-06; sprejem za objavo – accepted for publication: 2020-02-12
doi:10.17222/mit.2019.120
In order to obtain a multi-particle erosion model for the prediction of the erosion law of plastic metal walls considering material
hardening, the low-cycle fatigue theory and a single-particle erosion model are theoretically combined. In addition, a gas–solid
jet experiment wherein the number of particle impacts can be controlled is performed to identify the variation in the material
hardness and erosion rate with particle impact times. The experiment tests the change of surface hardness and material loss rate,
and obtains the key coefficients in the model and critical impact times of material loss by varying the particle impact number
and the impact velocity. Results show that the local yield strength of the material surface will increase because of the hardening
effect exerted by the particle impact. Surface hardening results in a logarithmic reduction in the erosion rate with an increase in
particle impact number. In addition, there is a critical particle time, which corresponds to material loss and nonlinearly
decreases with the increase in the impact velocity, that leads to material loss.
Keywords: multi-particle erosion, hardening discipline, critical impact times, low-cycle fatigue
Avtorji tega ~lanka opisujejo {tudijo s katero so `eleli izdelati model erozije zaradi udarjanja velikega {tevila delcev v plasti~no
kovinsko steno z upo{tevanjem teorije utrjevanja materiala, teorije malocikli~nega utrujanja in erozijskega modela enega delca.
Izvedli so erozijski preizkus v katerem so {tevilni delci v curku plina kontrolirano udarjali v steno materiala. Analizirali so
spremembe trdote materiala in hitrost erozijske obrabe glede na ~ase udarjanja. S preizkusom so spremenili trdoto povr{ine
materiala in hitrost erozije (izgubo na masi). S spreminjanjem {tevila udarcev delcev in udarne hitrosti so dobili klju~ne
koeficiente modela in kriti~ne udarne ~ase glede na izgubo materiala. Rezultati so pokazali, da lokalna meja plasti~nosti
povr{ine materiala nara{~a zaradi u~inka utrjevanja v odvisnosti od {tevila udarcev delcev. Utrjevanje povr{ine materiala in
logaritmi~no zmanj{anje hitrosti erozije nara{~a z nara{~anjem {tevila udarcev delcev. Avtorji so ugotovili, da obstaja kriti~ni
~as delca, ki se ujema z izgubo materiala in se nelinearno zmanj{uje z nara{~anjem hitrosti udarcev, kar vodi do izgube (erozije)
materiala.
Klju~ne besede: erozija z udarjanjem velikega {tevila delcev, zakoni utrjevanja, kriti~ni udarni ~asi, malo cikli~no utrujanje
1 INTRODUCTION
The constant impact of particles in a multiphase flow
on walls may cause the deformation or mass loss of wall
material. This phenomenon is commonly referred to as
particle erosion. Severe particle erosion causes major
problems, such as wall-thickness reduction, structural
fracture, and equipment failure, and other issues that
affect production safety. Particle erosion may involve a
single (cutting failure) or multiple steps (fatigue fail-
ure).
1–3
Multistep particle erosion comprises extrusion,
stretching, fracture, and shedding processes. This
process could only be analyzed by simplifying the
fatigue abrasion behavior of plastic metal under repeated
impact load given its complexity.
4,5
Researchers have proposed more than 30 scouring
models for the prediction of particle erosion in plastic
and brittle materials, since Finnie
6
first systematically
proposed the theory of plastic material erosion in 1958.
Tilly
7
divided the scouring process into two steps on the
basis of experimental measurements. First, the impact of
particles on the material surface causes the material to
press towards the periphery. Then, the material is
squeezed again and peeled off through subsequent
particle impacts. Studies on multi-particle erosion can
refer to fatigue failure research given that the mass loss
of plastic materials is usually the result of multiple
particle impacts.
The fatigue life of plastic materials is usually divided
into short- and long-lifetime zones. Plastic strain plays a
major role in the long-lifetime zone. The material failure
loading cycle in the long-lifetime zone is short and
belongs to the high-load and low-cycle fatigue category.
Materiali in tehnologije / Materials and technology 54 (2020) 3, 321–326 321
UDK 620.193.1:620.178.3:620.193.95 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 54(3)321(2020)
*Corresponding author's e-mail:
cjr88112@163.com (Jiarui Cheng)

Elastic strain plays a leading role in the short-lifetime
zone. The small initial contact area between particles and
the wall results in remarkably high contact stress.
Therefore, multi-particle erosion is generally classified
in the low-cycle fatigue research category. Guo
8
per-
formed a round head impact test to measure the erosion
rate of materials. Their experimental results showed that
the minimum particle impact numbers that cause
material loss at impact velocities of 2 m·s
–1
and 15 m·s
–1
are 1,585- and 4-fold, respectively. This result shows that
the material-loss process approaches cutting failure when
the particle impact velocity is high. Hutchings
9
used the
low-cycle fatigue theory to establish an erosion model on
the basis of dynamic hardness and ductility and obtained
the velocity index of circular particles perpendicular to
the wall. Huang
10
also established a multi-particle
erosion-prediction model on the basis of the low-cycle
fatigue theory. This model includes numerous con-
siderations and a heavy experimental workload as it
contains multiple physical quantities and empirical
parameters.
We have also established a material deformation
model for calculating a single particle impact,
11
which
can successfully calculate the elastic-plastic deformation
of the material, the contact time between the particles
and the wall, and the size of the impact crater. However,
as most scholars have found, particle impact erosion is
an ongoing process. The whole process involves material
fatigue, fracture and material hardening, which requires
the establishment of a multi-particle impact erosion
model that considers these factors. Therefore, in this
work, we constructed a multi-particle erosion model that
considers the fatigue on the basis of our established
single-particle erosion model. We performed a gas–solid
jet flow experiment to identify the key parameters of the
model and identify the relationship between the erosion
rate and the particle impact times. Finally, we obtained a
linear section of the erosion rate and critical impact
times corresponding to the material loss.
2 FATIGUE DAMAGE ANALYSIS
As reported in our previous study on single-particle
impact erosion, the volume of the eroded crater is as
follows:
No sliding contact:() VL r hr h
s S p S p S
=+
1
3
12 32 2 //
(1)
Sliding contact: VL rh
s S p S
=
1
3
12 32 //
(2)
where h
S
and L
S
are the indentation depth and length of
an erosion crater, respectively; r
p
is particle radius.
As shown in Figure 1a, the traditional ten-
sion–compression fatigue loading is tensile force–pres-
sure alternate change. Metal surfaces usually experience
low-stress and long-life-strain fatigue damage when the
maximum loading is less than the materials’ yield limit.
However, when the maximum loading exceeds the yield
limit, metal surfaces will experience high-stress and
short-life strain fatigue damage. During multiple impact
fatigue processes, the load continuously acts upon the
surface in a discontinuous loading form, as shown in
Figure 1b, where Ti is the single loading cycle, and T
s
is
the loading-time interval.
(a) Tensile and compression tests
(b) Multiple impact test
Hutchings
9
reported that continuous particle impact
erosion can be regarded as a low-cycle fatigue fracture
process. The fatigue life of material loss can be
expressed as a power function of the plastic strain scope
 in accordance with the Coffin–Manson formula.
 =⋅ 2Δ N
b
(3)
where b indicates the material’s characteristic para-
meters, which are given by Coffin and Manson as –0.5
and –0.6, respectively. Given that the fatigue energy
analysis by Martin provided b = –0.5, the b value is set
as –0.5 in this work. The total indentation depth when
the particle impacts the same wall surface N times can
be expressed as follows:
hD h =
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ −
S

 2
2
(4)
No material loss can be observed in the region of a
smooth and flat metal surface under multiple particle
impacts when the particle impact time is within the
interval 0–N. However, cracks initiate and develop on the
surface. The surface material falls off when the particle
impact time is greater than N, and the lower layer ma-
terial undergoes continuous compressional deformation
to initiate alternate peeling. The weight loss due to
multiple particle erosions will stabilize after a certain
time if the material hardening is neglected.
The tangential velocity component for the surface
under multiple particle impacts is small. Thus, weight
loss due to erosion fatigue is expressed as follows:
M. LI et al.: ESTABLISHMENT OF A MULTI-PARTICLE EROSION MODEL BASED ON THE LOW-CYCLE FATIGUE LAW ...
322 Materiali in tehnologije / Materials and technology 54 (2020) 3, 321–326
Figure 1: Schematic of load changes during different cyclic loading
processes

VN N
VL r
s
s
=<
=
0
1
3
c
S p
()
12 32 2 //
hr hNN
S p S c
+≥
(5)
Guo
8
obtained the relational expression between N
and h by taking the logarithm. In contrast to the cyclic
loading mode, the metal surface will experience harden-
ing during multiple particle impact processes. Hardening
increases the yield limit of the local material to reduce
the erosion rate. Stress presents a nonlinear relation with
strain during plastic strain. The following expression can
be obtained in accordance with the Zener–Hollom
formula:
 =⋅ k
n
s
2
(6)
Meanwhile, the following equation can be obtained
by taking the logarithm of the two sides:
lg lg lg  =+ kn
s 2
(7)
The hardening index can be expressed in accordance
with the relationship between the hardening index and
the material strength inferred by Hu:
12
n
2
0341 005 =−− .( '/'). 
s b
(8)
where  s
' and  b
' are material yield strength and tensile
strength under cyclic loading, respectively. Equation (8)
implies that the deformation hardening index is a
nonconstant value that will change with particle impact
times and finally reach a stable value.
13
As a general
rule,  s
' and  b
' under impact times can be measured
experimentally to solve the indentation depth h at a
specific time. However, in actual operation, a single
measurement of yield strength or tensile strength of the
metal surface is difficult to obtain. Hence, a multi-
impact experiment is generally subject to the measure-
ment of changes in surface hardness. Many scholars
14
have obtained a linear relation between the yield
strength and the hardness through the numerical
regression of the experimental results.
 y
aHV b =+ (9)
Hence, the yield limit at a specific time can be solved
by measuring the value when the measured hardness is
stable. Then, the yield limit can be substituted into
Equation (9) to replace the vertical material yield limit
 y
. Subsequently,  y
is substituted into Equation (4) to
solve the material loss. If the erosion rate on the wall
surface is evaluated by using material loss within a unit
area and time, the erosion rate under a tangential
non-slipping particle impact can be expressed as follows:
C
E
= 0
()
c
E pc
b
pc
b
c
NN
CL rh Nrh NNN
<
=+≥
1
3
22
12 32 2 //
()() '
(10)
where N
ruQ
r
c
a
V
p
=
3
4
2
3
, h
BBA C
A
=
−+ −
2
1
2
2
,
L
BBA C
A
=
−+ −
2
2
4
2
,Ar
y
=
1
2
 π
p
, B
r
E
yp
=× 017
332
2
.
(* )
π  ,
Cm v
y 10
2
=−
p
andCm v
x 20
2
=−
p
The erosion rate of pipe flow under a tangential
slipping particle impact can be expressed as follows:
CN N
CL rh N
E c
E pc
=<
=
0
1
3
2
12 /
(
b
c
)'
/ 32
NN ≥
(11)
where N
ruQ
r
c
a
V
p
=
3
4
2
3
, h
BBA C
A
=
−+ −
2
2
2
,
Fr h
yy
= π
p
, F
v
vr h
x
x x
y yy
=
−
−
0
0
2
1
21
()
()
  π
p
,
Lr h
F
F
x
y
=−
⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 21
13
p
 /
,Ar
y
=
1
2
 π
p
, B
r
E
yp
=× 017
332
2
.
(* )
π  andCm v
y
=−
p 0
2
3 EXPERIMENTAL PART
Figure 2a shows the multiphase flow experimental
loop, which includes the screw pump, liquid flowmeter,
test chamber, sample holder, stirred tank in the liquid
flow system and air compressor, buffer tank, filter drier,
gas flowmeter, sand storage tank, and PLC feeder in the
gas flow system. Particles were added in the gas–solid
M. LI et al.: ESTABLISHMENT OF A MULTI-PARTICLE EROSION MODEL BASED ON THE LOW-CYCLE FATIGUE LAW ...
Materiali in tehnologije / Materials and technology 54 (2020) 3, 321–326 323
Table 1: Chemical compositions of P110 and 13Cr
Material C Si Mn Cr P S Ni Mo Cu
P110 0.28 0.26 1.68 0.03 0.013 0.0013 0.04 0.03 0.044
13Cr 0.03 0.18 0.44 12.5 0.02 0.01 5.82 2.0 0.51
Table 2: Mechanical properties of P110 and 13Cr
Material Tensile strength (MPa) Yield strength (MPa) Elongation (%) Hardness (HV)
P110 845 940 25 299
13Cr 855 970 20 323
Table 3: Particle impact time every mm
2
Particle mass (g) 50 100 200 400 800 1,600 3,200 6,400
Particle impact times 7 × 10
3
1.4×10
4
2.8×10
4
5.6×10
4
1.12 × 10
5
2.24 × 10
5
4.48 × 10
5
8.96 × 10
5

experiments using a PLC screw feeder during the
gas-flow process. Once the particles collided with the
sample surface, they were collected in the test section
and stirred tank because the sand would break after
impact and affect the experimental results.
Figure 2b shows the erosion test section. The sample
clamp can be used to adjust the angle and height of the
sample in a closed space. Square P110 and 13Cr samples
with dimensions of 20 mm × 20 mm×5mmwere used
in the experiment to study the particle erosion charac-
teristics of carbon and stainless steel. Tables 1 and 2
provide the chemical composition and mechanical
properties, respectively, of P110 and 13Cr. The surfaces
of the samples were ground before the experiment using
300-, 800-, and 1,200-mesh sandpapers to eliminate the
impurities and obtain smooth surfaces. Ceramic particles
(dp = 0.6 mm), which are weighed quantitative particles,
were used in each experiment. The sample surfaces were
examined through scanning electron microscopy (SEM,
JSM-6390, JEOL. Co., Japan). The sizes (radius R) of
the marginal impact craters in the independent impact
region on the P110 steel and 13Cr stainless steel under
an impact angle of 90° and impact velocity of 10 m·s
–1
were measured.
4 RESULTS
4.1 Experimental results for the changes in the
hardness of the target material
The particle impact time within a unit area was
calculated with the averaging method using the follow-
ing formula: N
unit
= N
c
÷A, where N
unit
is the number of
particle impacts within a unit area. Approximately 50 g
particles were obtained, and the number of particles was
calculated in accordance with the average mass of a
single particle. The particle mass in each experiment was
the same as that in a previous experiment, and the num-
bers of impacting particles every mm
2
were calculated,
as shown in Table 3.
Figure 3 displays the posti-mpact Vickers hardness
values on the impact surface. The hardness value of the
P110 steel changed within the range 296–303 HV when
the particle impact time was less than 2.8 × 10
4
. Mean-
while, the hardness values of the P110 and 13Cr steel
increased by 63.19 HV and 42.38 HV, respectively, when
the particle impact time was 8.96 × 10
4
. These results
indicate that the sensitivity of the P110 steel to impact
was higher than that of the 13Cr steel. In addition, the
increment in hardness gradually decreased as the
material hardness increased with impact time. Therefore,
nonlinear fitting can be conducted by using the loga-
rithmic function y = a – b × lg(x + c) to obtain the curve
of the relationship between the hardness and the particle
impact times. As shown in Figure 3, the calibrated
determination coefficients of both materials after fitting
exceeded 0.97, which is indicative of the favorable
regression of the fitted curve. These results show that the
change in surface hardness under unit impact times was
negligible in the initial particle impact phase (within 2 ×
10
4
times) and peaked in the early and middle impact
phases (2 × 10
4
to1×10
5
times). However, the change in
surface hardness under unit impact times in the middle
later phase (N
c
>1×10
5
) was minor. Thus, the surface
hardness reaches dynamic stability during long-term
particle impacts.
M. LI et al.: ESTABLISHMENT OF A MULTI-PARTICLE EROSION MODEL BASED ON THE LOW-CYCLE FATIGUE LAW ...
324 Materiali in tehnologije / Materials and technology 54 (2020) 3, 321–326
Figure 2: Jet flow experimental system: a) flow loop and b) erosion test section
Figure 3: Changes in the surface hardness of P110 and 13Cr steel
under multiple particle impacts

Figure 4 illustrates the relationship between the
particle impact times and the yield strength of the wall
material surface calculated using Equation (9). The yield
strengths of the surfaces of both materials were less than
900 MPa when the impact time was less than 20,000. At
this time, the difference in material deformation caused
by particle impact on the wall surface was small. Ma-
terial deformation will increase by a large margin
because of the large increment in yield strength when the
impact time was 50,000–400,000. Therefore, the
influence of surface-material hardening on the erosion
rate of a pipe during short-term service can be neglected
if the particle impact time is less than the scope of the
small deformation. If the particle impact time is within
the scope of a large deformation, the surface hardness
and yield strength at the current time should be com-
puted, and the surface weight eroded should then be
calculated. The cumulative material losses for the pipe
during long-term service within small- and large-defor-
mation intervals should be computed first. Then, the
material loss within the stable deformation interval
should be calculated in accordance with the service time.
4.2 Erosion experimental results
A 90° jet flow angle was selected for the particle ero-
sion experiment to ensure that the material surface
damage can be restricted by fatigue effect but not by the
cutting effect. The discrimination of surface failure was
subject to the generation of 0.1 mg weight loss and then
to the minimum particle impact time that caused fatigue
damage within the unit area (mm
2
). Specimen weight
loss in the low flow-velocity region (1–5 m·s
–1
)w a s
measured once every 100 g particle impact (impact times
within unit area: 1,000), and a subdivision was con-
ducted with 100 (10 g) impact times. The specimen
weight loss in the medium flow-velocity region
(5–15 m·s
–1
) was measured once every 10 g particle
impact (impact times within unit area: 100), and a
subdivision was performed with 10 (1 g) impact times.
The minimum particle impact time in the high flow-
velocity region (>15 m·s
–1
) was predicted through the
logarithmic fitting equation, and the results are shown in
Table 4. The exponential decline in the minimum
particle impact times that caused the peeling of the two
materials indicates that energy can easily accumulate
under high-speed particle impact to induce material to
momentarily peel off the surface.
Table 4: Critical impact times of material loss under different impact
velocities
Experimental
measurements
Fitted values
Impact
velocities /m·s
–1 1 2 4 8 12 16 20 24 30
Critical impact
times (13Cr)
4900 2400 600 210 60 32 13 5 1
Critical impact
times (P110)
3800 1700 200 80 30 14 3 1 –
Experimental results show that the minimum particle
time that caused the material loss of 13Cr steel under the
impact velocity of 30 m·s
–1
was approximately 1. This
finding indicates that particle impact can generate
material loss only once when the flow velocity exceeded
30 m·s
–1
. The corresponding impact velocity was close to
24 m·s
–1
when the P110 impact time was 1. This result
indicates that the threshold flow velocity that generated a
single impact failure in P110 steel was low. As shown in
Figure 5 and Table 5, the empirical coefficient D under
different impact velocities was calculated through the
experimental measurement of a total indentation depth h
(h=4.7×10
–6
m corresponds to 0.1 mg), an extrusion
depth h
S
of a single particle impact, and minimum
particle impact times N (De/2e = N) causing material
loss in accordance with Equation (4). The results are
shown in Figure 5. The indentation depths (thinning
quantity) of 13Cr and P110 steel under long-term service
can be expressed as follows:
13Cr steel:he h N
v
=×+⋅⋅
−
(. . )
/.
37 10 002
36 7
S c
(12)
P110 steel:he h N
v
=+××⋅
−
(. . )
/.
019 39 10
42 4
S c
(13)
M. LI et al.: ESTABLISHMENT OF A MULTI-PARTICLE EROSION MODEL BASED ON THE LOW-CYCLE FATIGUE LAW ...
Materiali in tehnologije / Materials and technology 54 (2020) 3, 321–326 325
Figure 4: Relationship between yield strength and particle impact
times
Figure 5: Coefficient D versus particle impact velocities

where N
c
=3 r
l
2·
u
a
·Q
v
/4r
p
3
, and vertical impact velocity
v
y
is adopted for the particle velocity. Therefore, the
uniform thinning quantity of the pipe wall can be solved
through Equation (12) or (13) when an actual two-phase
pipe flow was considered as the study object. However,
the precondition for calculation is that the particle-
carrying capacity of the continuous phases is strong.
Specifically, the interphase slipping velocity is small,
and the particle velocity is identical to the continuous
phase velocity by default. However, the influence of
interphase force and turbulent acceleration on particle
impact angle causes a continuous change in the vertical
impact velocity v
y
when particles impact the wall
surface in the actual processes.
Table 5: Variation in coefficient D with particle impact velocities
Coefficient
D
Particle impact velocity /m·s
–1
81 21 62 0
13Cr 590,485.34 32,087.08 6,840.11 902.70
P110 85,694.02 8,021.77 1,309.24 48.07
5 CONCLUSIONS
The single-particle impact erosion model combined
with the erosion-forecasting model for multi-particle
impact on metal wall surfaces was established in this
work on the basis of the low-cycle fatigue theory. A jet
flow-type erosion experiment was performed to identify
the trends shown by changes in the surface-hardness
values of P110 steel and 13Cr with particle impact times.
Moreover, key empirical coefficients were acquired.
Experimental results showed that the material hardness
logarithmically increased when the particles impacted
the two metal surfaces. An ideal corresponding material
erosion rate was obtained under long-term particle
impact when the surface hardness region was stable. In
addition, the local yield strength of the material surface
will increase because of the hardening effect exerted by
the particle impact. Consequently, the particle inden-
tation depth reduced. Given that particle impact energy
was absorbed during plastic material deformation,
material loss was the product of multi-particle impacts
under most circumstances. The minimum particle-impact
time that caused material peeling exponentially declined
with the impact velocity. This finding indicates that
material loss was equivalent to the linear superposition
of single losses when the impact velocity reached a
certain value, and the influence of material fatigue can
be neglected.
Acknowledgment
This research was funded by Basic Research Program
of Natural Science of Shaanxi Province (No.
2019JQ-809), and supported by National Natural
Science Foundation of China under grant number
51674199.
Appendix A
Nomenclature
CE
erosion rate, mm·s
–1
E* equivalent elastic modulus, Pa
Fx
tangential contact force, N
Fy
normal contact force, N
h total indentation depth, m
ks
intensity factor, Pa
L scratch length, m
mp
particle mass, kg
N
critical particle impact time corresponding to
material loss
N
c
actual particle impact time in a unit area
r
p
particle radius, m
r tube radius, m
u pipe flow velocity, m·s
–1
v
x
tangential velocity of the particle, m·s
–1
vy normal velocity of the particle, m·s
–1
s stress, Pa
sy
vertical material yield limit, Pa
De plastic strain scope
μ coefficient of friction, dimensionless
 x
velocity attenuation coefficient in tangential
direction, dimensionless
 y
velocity attenuation coefficient in normal direction,
dimensionless
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