Wear equation
Content
Wear 318 (2014) 78–88
Contents lists available at ScienceDirect
Wear
journal homepage: www.elsevier.com/locate/wear
Experimental and theoretical analysis of wear mechanism
in hot-forging die and optimal design of die geometry
Wujiao Xu n, Wuhua Li, Yusong Wang
College of Material Science and Engineering, Chongqing University, Chongqing 400044, China
art ic l e i nf o
a b s t r a c t
Article history:
Received 27 February 2014
Received in revised form
21 June 2014
Accepted 22 June 2014
Available online 30 June 2014
The purpose of this study is to gain a better understanding of the wear mechanism in hot-forging die,
propose a new wear calculation model to predict the die wear more accurately and present a die
geometry optimization method to prolong the die lifespan. Taking the mandrel in wheel hub forging
process as example, the wear mechanism was studied by SEM examinations. It was found that three
competing wear mechanisms exist on the worn surface, namely adhesive wear, abrasive wear and
oxidation wear. A new wear calculation model was proposed with the consideration of these three wear
mechanisms. An optimal design of the mandrel geometry was carried out based on the proposed wear
calculation model, forging numerical simulation, BP neural network and Sequential Quadratic Programming (SQP) algorithm. Wear tests were performed to verify the wear calculation accuracy, which shows a
good agreement with the calculated result. This study will be beneficial to the accurate prediction of die
wear and the optimal design of die geometry in hot-forging process.
& 2014 Elsevier B.V. All rights reserved.
Keywords:
Geometry optimization
Wear mechanism
Wear calculation model
BP neural network
1. Introduction
Hot-forging die works under the condition of high temperature, strong contact pressure and intense friction, which contributes to the severe wear of die. Research shows that wear failure
accounts for more than 70% of the total hot-forging die failure [1].
According to the Archard wear model, wear rate is proportional
to the load and the sliding speed, inversely proportional to the
hardness of worn surface [2]. During plastic forming process,
the geometry of die directly affects the load, the sliding speed
and the temperature of contact area. Therefore, to reduce the die
wear, optimization design of the die geometry is crucial.
As stated before, the wear phenomenon is influenced by the
processing parameters such as temperature, sliding velocity, contact pressure and lubrication conditions [3–5]. The wear mechanism is extremely complicated and lots of studies have been
performed on it. Rapoprt [6] investigated the dominant and
competitive wear mechanisms of steel under different contact
conditions. It was found that three types of wear mechanisms
occurred in the plastic region. The wear mechanisms of iron based
metal matrix composites and the wear behaviors of various
microstructures were systematically studied by dry sliding wear
testing and SEM examination [7]. The experimental results
showed that three dominant wear mechanisms appeared in
n
Corresponding author. Tel.: þ 86 13368186252; fax: þ86 23 65111493.
E-mail address: [email protected] (W. Xu).
http://dx.doi.org/10.1016/j.wear.2014.06.021
0043-1648/& 2014 Elsevier B.V. All rights reserved.
succession with increasing normal load during dry sliding and
the transition of the wear mechanisms depended mainly upon the
conditions of testing.
Some studies paid attention to wear calculation method and put
forward the mathematical models considering the parameters which
influenced the wear process [8–10]. For example, the mathematical
model to predict the wear rate of Al–zircon composites had been
expressed in terms of reinforcement volume fraction, particle size,
applied load, sliding speed and sliding distance using five factors, five
level central composite design matrix approach by response surface
method. The morphology and topography of worn out surface wear
were studied using a scanning electron microscope [8]. However,
mathematical models proposed by these studies just gave the
mathematical relationship between wear amount and selected variables and ignored the mechanism of wear. Thus, they could only be
used in particular situation.
Yin et al. [11] established a calculation model to predict the
wear of fine-blanking die during its whole lifetime based on Back
Propagation (BP) neural network, Finite Element Method (FEM)
and experiments. The inherent law between wear of fine-blanking
die and its working parameters was revealed by utilizing the BP
neural network. Huang and Huang [12] constructed two BP neural
network models based on the experimental date. One was used for
predicting the length evolution of parison with its drop time, the
other was for predicting the swells along the parison.
This paper is organized as follows. Section 2 describes the
optimization problem and mathematical modeling for mandrel
W. Xu et al. / Wear 318 (2014) 78–88
79
Fig. 1. (a) The product of wheel hub, (b) wear condition of mandrel.
geometry optimization. In Section 3, the study of mandrel wear
mechanism during forging process and the establishment of a
new wear calculation model are presented. The BP neural network modeling and SQP algorithm optimization for the mandrel
geometry are presented in Section 4, respectively. Section 5
presents the experimental verification for the accuracy of proposed wear calculation model and the validity of the optimal
design of mandrel geometry. Finally, the last section contains the
conclusions.
2. Description of the optimization problem
2.1. Wear condition
Hot-forging die works under the condition of high temperature, strong contact pressure and intense friction, which contributes to the severe wear of die. For the wear failure of hot forging
die, the die geometry is an important factor which directly
influences the load, the sliding speed and the temperature of
contact area. Here we take the mandrel die of wheel hub forging
process as example. It was observed in the practical production
that the wear of mandrel fillet is serious, which greatly exceeds
that of other parts. Therefore, this paper aims to gain a better
understanding of the mandrel wear mechanism and try to make
the mandrel wear less and more uniform by optimizing the
geometry of mandrel. The wheel hub and wear condition of
mandrel are shown in Fig. 1.
2.2. Mathematical modeling for mandrel geometry optimization
Before optimizing the geometry of mandrel, a cubic spline
interpolation curve which has 19 control points is used to describe
the mandrel geometry and shown in Fig. 2. Considering the shape
feature of mandrel, control points in area A1 (yi þ 1 yi ¼0.2 mm) in
Fig. 2 are denser than that of other area (yi þ 1 yi ¼ 1 mm). In other
words, 11 control points are located in area A1 which has a larger
curvature and 8 control points are arranged in the other area.
Moreover, first boundary condition (the slopes of control point
1 and control point 19 are 0 and 8.1443, respectively, x1 ¼5.5 mm,
x19 ¼ 8.9266 mm, A1 ¼2 mm, A ¼10 mm) is used to draw the cubic
spline interpolation curve. Abscissa values of control points,
namely x1, x2, x3, …, x19, are design variables of this cubic spline
interpolation curve.
As observed in the practical production, the wear in mandrel
fillet is serious, which greatly exceeds that in other parts. For
quantitative analysis, the wear uniformity criteria defined by
Eq. (1) was proposed to evaluate the wear condition of mandrel.
Fig. 2. Description of the mandrel geometry by cubic spline interpolation curve.
Obviously the smaller the uniform wear f, the more uniform the
mandrel wear.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f¼
n
∑ ðwi wÞ2
i¼1
ð1Þ
where n is the number of total control points, wi is the wear of
control point i, w is the average wear amount of all the control
points (w ¼ ð1=nÞ∑ni¼ 1 wi ).
The flow chart of the mandrel geometry optimization is
illustrated in Fig. 3. First, the wear mechanism of mandrel was
studied by SEM examinations of the worn surfaces. Then, the wear
calculation formulas for these three wear mechanisms were
deduced, respectively. Based on the wear tests and SEM examinations of the wear debris, a new wear calculation model, which
takes these three wear mechanisms into consideration, was
proposed. Afterwards, an optimal design of the mandrel geometry
was carried out based on the proposed wear calculation model,
forging numerical simulation, BP neural network and SQP algorithm. Finally, to verify the accuracy of the proposed wear
calculation model and the validity of the optimal design of
mandrel geometry, experiments were performed on HSR-2M
wear-testing machine.
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W. Xu et al. / Wear 318 (2014) 78–88
Fig. 3. Flow chart of the mandrel geometry optimization process.
3. Wear mechanism and new wear calculation model
3.1. Wear mechanism
Archard [13] proposed a formula to calculate the wear of material
v¼k
Wx
H
ð2Þ
where v is wear amount, k is wear coefficient, W is load, x is sliding
distance, H is hardness of the worn surface.
This wear calculation formula was proposed based on classical
adhesive wear model. However, during the forging process of wheel
hub, a number of competing wear mechanisms (adhesive wear,
abrasive wear and oxidation wear) occur on the worn surface as
shown in Fig. 4. Therefore, it is not exact to calculate the wear of
mandrel by traditional Archard wear formula. On this condition, a
new wear calculation model which takes these wear mechanisms
into consideration is put forward to calculate the wear of mandrel
accurately.
Scanning electron microscopic examinations of the worn surfaces (Fig. 4) identified different wear mechanisms. They are
adhesive wear, abrasive wear and oxidation wear. It can be seen
from Fig. 4(a) that there are many spalling pits on the worn
surface. Spalling pits on the mandrel worn surface are typical
characteristics of the adhesive wear, which generated when two
solid surfaces slide over one another under load. Under the effect
of load, some asperities are plastically deformed and the two
surfaces will stick together. As sliding continues, some bonds are
broken and spalling pits (Fig. 4(b)) occur on the worn surface.
Meanwhile, wear particles generate on the other surface, which
will contribute to further wear of the surface. Fig. 4(c) and
(d) show that scratches which are mostly parallel to the sliding
direction generate on the worn surface. These scratches are typical
features of abrasive wear and indicate that the wear mechanism of
this area is abrasive wear. When two surfaces slide, under the
action of hard wear particles, the worn surface is cut and scratches
occur on the surface. In most case, these hard wear particles are
produced by adhesive wear and lead to the three-body abrasive
W. Xu et al. / Wear 318 (2014) 78–88
81
Fig. 4. Scanning electron microscopic examinations of the worn surfaces: (a) spalling pits information on the worn surface indicating adhesive wear, (b) partial enlarged
view of the block region in (a), (c) scratches information on the worn surface indicating abrasive wear, (d) partial enlarged view of the block region in (c), (e) oxide films
information on the worn surface indicating oxidation wear, (f) partial enlarged view of the block region in (e).
wear of the surface [14]. Oxide films are observed on the worn
surface in Fig. 4(e). These oxide films indicate the oxidation wear
of the mandrel. For two sliding surfaces under pressure, due to the
external heating and the frictional heat, the surface will be
oxidized and the oxide film grows slowly during the sliding
process. When the thickness of the oxide film reaches to a critical
value, it will break up and drop off as shown in Fig. 4(f).
In addition, we can find in Fig. 4(e) that the oxide film of contact
area is thicker than that of non-contact area.
Fig. 5(a) shows the Energy Dispersive Spectrometer (EDS)
analysis of the point 1 in Fig. 4(b). The result indicates that there
is almost no oxygen on the worn surface. Combined with the SEM
result of this area, the wear mechanism of this area is identified as
adhesive wear. The EDS analysis of point 2 in Fig. 4(f) is illustrated
in Fig. 5(b). It can be seen that there is much oxygen on this
shedding oxide film. This Phenomenon indicates that the wear
mechanism is oxidation wear.
3.2. Wear calculation formula under different wear mechanisms
3.2.1. Adhesive wear calculation formula
For two sliding surfaces under the load W, it is supposed that
the asperity is plastically deformed and will produce wear particle
with a certain probability. Besides, the contact area consists of two
asperities as shown in Fig. 6; and average radius of the two
asperities is a.
The wear depth of the worn surface can be calculated as [13]
Z t
PV
wadhesive ¼
dt
ð3Þ
k
HðTÞ
0
where P is the normal pressure on the contact area, V is the
relative sliding velocity, H(T) is the hardness of surface when the
temperature is T, t is the sliding time.
3.2.2. Abrasive wear calculation formula
For two sliding surfaces under the load W, it is assumed that
the conical asperity will cut the worn surface and lead to a scratch
along the sliding direction [15]. The schematic diagram of abrasive
wear model is shown in Fig. 7. In the figure, θ indicates the attack
angle of asperity; and 2b, d are the width and depth of the scratch,
respectively.
The wear depth of the worn surface can be expressed as
Z t
2 tan θ P V
wabrasive ¼
dt
ð4Þ
π
HðTÞ
0
3.2.3. Oxidation wear calculation formula
For two sliding surfaces, due to the external heating and the
frictional heat, the surface will oxidize and the oxide film of
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W. Xu et al. / Wear 318 (2014) 78–88
Fig. 5. EDS analysis results of the worn surface: (a) EDS of point 1 in Fig. 4(b), (b) EDS of point 2 in Fig. 4(f).
Fig. 6. The schematic diagram of adhesive wear model.
Fig. 8. The schematic diagram of oxidation wear model.
Therefore, in order to calculate the wear of mandrel accurately, a
new wear calculation model which takes these wear mechanisms
into consideration is proposed
Fig. 7. The schematic diagram of abrasive wear model.
contact area is thicker than that of non-contact area. It is assumed
that the oxide film grows slowly during the sliding process.
In addition, the oxide film will break instantly when its thickness
reach a critical value ξ. The schematic diagram of oxidation wear
model is shown in Fig. 8.
Therefore, the wear depth of the worn surface can be obtained
as [16]
Z t
Qp
PAp C 2
woxidation ¼
expð Þdt
ð5Þ
RT
0 HðTÞξ
where Ap is parabolic oxidation Arrhenius constant, Qp is parabolic
oxidation activation energy, R is molar gas constant, η is the
volume percentage of Fe3O4, ρ is the density of iron, MO and MFe
are the molar mass of oxygen and iron, respectively, C ¼
6M Fe =ð9 ηÞρM O .
3.3. Establishment of wear calculation model based on wear
mechanisms
According to the SEM examinations (Fig. 4) and EDS analysis
(Fig. 5) on the worn surface, it can be found that the mechanisms
of mandrel are the coexistence of adhesive wear, abrasive wear
and oxidation wear. In Section 3.2, the wear calculation formulas
of these three mechanisms have been obtained, respectively.
1
wtotal ¼ λadhesive wadhesive þλabrasive wabrasive þ λoxidation woxidation
R t PV
C
wadhesive ¼ 0 kHðTÞdt
C
C
R t 2 tan θ PV
C
C
wabrasive ¼ 0 π HðTÞdt
C
A
R t PAp C 2
Qp
woxidation ¼ 0 HðTÞξ expð RT Þdt
ð6Þ
where wtotal is the total wear depth of the worn surface; wadhesive,
wabrasive and woxidation are the wear depth which are caused by
adhesive wear, abrasive wear and oxidation wear, respectively;
λadhesive, λabrasive and λoxidation are the weight coefficients of
adhesive wear, abrasive wear and oxidation, repectively.
λadhesive, λabrasive and λoxidation are approximately equal to the
weight percentages of wear debris which are produced by adhesive wear, abrasive wear and oxidation wear, respectively. Therefore, in order to calculate the wear of mandrel by this wear
calculation model, wear tests were performed to obtain the related
variables in Eq. (6).
3.3.1. Materials in wear test
Fig. 9 illustrates the friction pairs of wear test. The material of
wear-resistant plate is AISI-H13, which is the same as mandrel.
And the material of pressure head is AISI-1045, which is the same
as wheel hub. The initial hardness of wear-resistant plate and
pressure head are HRC 54 and HRC 25, respectively. The chemical
composition of friction parts is shown in Table 1.
W. Xu et al. / Wear 318 (2014) 78–88
3.3.2. Wear test
The wear test was carried out on HSR-2M wear-testing
machine. During the forging process, the contact condition of
mandrel varied. However, almost no wear-testing machines can
make a real-time control of pressure and speed. The Deform
software is used to calculate the forging process in this paper.
Therefore, in order to approach the actual production as much as
possible, the normal pressure was obtained by FEA as shown in
Fig. 10. According to the obtained normal pressure, the wear
condition is divided into five stages, which is shown in Table 2.
The HSR-2M wear-testing machine and the schematic of wear
test are shown in Fig. 11. When the wear test was performed, the
wear-resistant plate was heated by a resistance furnace. The
heating temperature was controlled by the thermocouple and
the temperature controller. Then, the pressure head was loaded
and kept in contact with the wear-resistant plate on a constant
load by the loading device of wear-testing machine. Finally, the
wear-resistant plate did the reciprocating movement of 10 mm
length.
After the wear test, the wear debris was collected. According to
the density differences between wear-resistant plate (AISI-H13)
and pressure head (AISI-1045), the wear debris of wear-resistant
plate was separated. Fig. 12 illustrates SEM examinations of the
separated wear debris.
SEM examinations of the wear debris (Fig. 12) identified
different wear debris. They are adhesive wear debris, abrasive
wear debris and oxidation wear debris. Based on the weight of
collected adhesive wear debris, the variable k in Eq. (6) was
calculated. According to Fig. 12(c), the attack angle of asperity θ
was obtained. 50 Scratches in Fig. 4(c) were selected to calculate
the weighted average value of the whole asperities tan θ. The
oxidation wear debris (Fig. 12(d)) indicated the critical thickness of
oxide film ξ (10 μm) and the volume percentage of Fe3O4 η (58.2%).
Related variables of the proposed wear calculation model are
shown in Table 3.
Fig. 12 shows that three types of wear debris were found on the
worn surface. The adhesive wear debris was produced by plastic
deformation. Therefore, the adhesive wear debris has a rough
surface, an uneven thickness (Fig. 12(a)) and little oxygen. When
Fig. 9. Friction pairs of wear test: (a) wear-resistant plate, (b) pressure head.
83
two surfaces slide, due to the effect of hard wear particles, the
surface was cut and the abrasive wear debris was produced. Thus,
the shape of abrasive wear debris is banded (Fig. 12(b)). The
oxidation wear debris has a smooth surface, an even thickness
(Fig. 12(d)) and contains a lot of oxygen. According to the
morphology differences among adhesive wear debris, abrasive
Fig. 10. The normal pressure of mandrel.
Table 2
Wear test condition.
Stage number
Pressure (MPa)
Speed (rpm)
Temperature (1C)
Time (min)
Stage
Stage
Stage
Stage
Stage
280
390
330
280
1000
75
225
90
390
390
410
468
482
510
517
5
15
7.5
2.5
1
1
2
3
4
5
Fig. 11. The HSR-2M wear-testing machine and schematic diagram of wear test.
Table 1
Chemical composition of friction pairs.
Element
C
Si
Mn
P
S
Cr
Ni
Cu
Mo
V
Fe
AISI-H13 (wt%)
AISI-1045 (wt%)
0.4
0.45
0.97
0.24
0.38
0.67
0.025
0.028
0.003
0.016
5.09
0.15
0.11
0.03
0.16
0.08
1.24
–
0.96
–
90.662
98.336
84
W. Xu et al. / Wear 318 (2014) 78–88
Fig. 12. SEM examinations of the separated wear debris: (a) single adhesive wear debris, (b) single abrasive wear debris, (c) worn surface of abrasive wear, (d) single
oxidation wear debris, (e) one visual field of the whole wear debris, (f) the other visual field of the whole wear debris.
Table 3
Related variables of the proposed wear calculation model.
k
2 tan θ
π
C (mm3 kg 1)
ξ (μm)
TC (1C)
Ap (mg2 cm 4 h 1)
Qp [17] (J mol 1)
R (J mol 1 K 1)
2.85e 5
3.64e 3
3.168e 4
10
517
0.084
157539
8.314
Table 4
Features of each wear debris and weight coefficients of adhesive wear, abrasive wear and oxidation.
Type of wear debris
Features
Weight (mg)
Weight coefficient λ
Adhesive wear debris
Abrasive wear debris
Oxidation wear debris
Tearing, rough surface, uneven thickness and little oxygen
Long strip, little oxygen
Smooth surface, even thickness and a lot of oxygen
2.8
1.2
3.4
0.378
0.162
0.459
wear debris and oxidation debris, they were separated and
weighed. Table 4 illustrates the features of each wear debris and
the weight coefficients of adhesive wear, abrasive wear and
oxidation.
4. Optimization with BP neural network and SQP algorithm
4.1. BP neural network modeling
Our research aim is to do the optimization of the mandrel
geometry. Then we can reduce the mandrel wear and prolong the
die life. To optimize design variables according to the objective
function, it is very important to establish the internal relation
between design variables and objective function. However, due to
the highly nonlinear of the optimization problem, it is hard to use
a mathematical equation to establish the internal relation between
design variables (abscissa values of control points) and objective
function (uniform wear values) in this paper.
The BP neural network consists of input layer, hidden layer and
output layer, different transfer functions are used to link these
layers. And it is proved that a three-layer BP neural network can
approximate any nonlinear function with arbitrary precision.
Therefore, this paper uses a 19-20-1 BP neural network to reveal
the inherent law between abscissa values of control points and
uniform wear values. The network consists of one input layer with
19 neurons standing for the 19 abscissa values of control points;
the hidden layer with 20 neurons and the output layer with
1 neuron representing the uniform wear value. The structure of
the designed neural network can be seen in Fig. 13.
W. Xu et al. / Wear 318 (2014) 78–88
Based on the forging numerical simulation, the proposed wear
calculation model and uniform wear calculation formula, the sample
sets of BP neural network, which consist of uniform wear values of
the whole mandrel geometries, are obtained and shown in Table 5.
Among those, five samples (4th, 8th, 12th, 16th and 20th) are
selected to be test samples. The detailed generation procedure of
BP neural network sample data is shown in Appendix A.
The designed neural network should be trained by a number of
samples before it can acquire the functional relationship between
design variables (abscissa values of control points) and objective
function (uniform wear values). The BP neural network training
and testing process can be referenced in Appendix B. The testing
result shows that the predictive error is less than 5% on average.
Hence, the trained network can be used as the surrogate of the
functional relationship between uniform wear and abscissa values
of control points.
4.2. Optimization with SQP algorithm
SQP algorithm is particularly effective to solve the optimization
problem which is highly nonlinear. And the implementation of
SQP involves three main steps: (1) Hessian matrix update;
(2) quadratic programming problem solution; (3) line search and
merit function [18]. The details of optimization process are shown
in Appendix C.
Based on the SQP algorithm, the optimal mandrel geometry
which makes the wear of mandrel most uniform was obtained.
The design variables of which are [x1, x2, x3, x4, …, x19] ¼[5.5,
6.4661, 6.8511, 7.0974, 7.2853, 7.4371, 7.5611, 7.6703, 7.7643,
7.8484, 7.9334, 8.1894, 8.3523, 8.4712, 8.5652, 8.6487, 8.7315,
8.8186, 8.9266] mm. Fig. 14 illustrates the comparison of the
optimized and initial mandrel geometry.
Then, the optimized design variables were used as the input of
trained BP neural network. The uniform wear of the optimized
mandrel geometry was predicted to be 1.5963e 4 mm by the
trained BP neural network. However, the uniform wear of the
initial mandrel geometry is 2.6139e 4 mm. Compared with the
initial mandrel geometry, the uniform wear of the optimized
85
mandrel geometry decreased by 38.9%. Therefore, the wear of
the optimized mandrel geometry is more uniform than that of the
initial one. It means that the optimization of the mandrel geometry is effective. The wear condition of the optimized mandrel and
the initial mandrel is shown in Table 6.
5. Experimental verification
To verify the accuracy of the new wear calculation model and
the validity of the optimal design of mandrel geometry, experiments were performed on HSR-2M wear-testing machine. First,
the contact information (normal pressure, relative sliding velocity
and temperature) of initial and optimized mandrel geometry was
obtained by numerical simulations of wheel hub finish forging
process. Based on the proposed wear calculation model, the wear
depth of the whole control points was calculated. Then, according
to the contact information, wear tests representing 1000 forging
cycles were carried out. Fig. 15 illustrates the wear comparison of
the initial and optimized mandrel geometry.
The calculated wear depth of control point a, b, c, d, e and f
are 0.005172 mm, 0.126363 mm, 0.158423 mm, 0.065113 mm,
0.105326 mm and 0.089258 mm (Fig. 15(a)), respectively. It can
be seen from Fig. 15(c) that the experimental wear depth shows a
good agreement with the calculated wear depth. From Fig. 15
(b) and (d), a similar result is obtained. Therefore, it is proved that
the proposed wear calculation model is accurate and practical.
What's more, it is evident from Fig. 15(a) and (b) that the uniform
wear and the maximum wear depth of the optimized mandrel
geometry are less than those of the initial mandrel geometry. Thus,
the optimization of the mandrel geometry is effective.
6. Conclusion
In this paper, the wear mechanism of hot-forging die was
studied by SEM examinations and EDS analysis, a new wear
Fig. 13. Structure of the designed neural network.
Fig. 14. Comparison of the optimized and initial mandrel geometry.
Table 5
Uniform wear values of each sample.
Sample number
Uniform wear (e 4 mm)
1
2.01
2
2.22
3
2.07
4
2.32
5
1.92
6
1.90
7
1.78
8
1.92
9
2.15
Sample number
Uniform wear (e 4 mm)
10
2.12
11
1.97
12
2.16
13
1.89
14
1.92
15
1.88
16
2.22
17
2.04
18
1.74
Sample number
Uniform wear (e 4 mm)
19
2.25
20
1.87
21
1.85
22
2.00
23
2.22
24
2.33
25
2.11
86
W. Xu et al. / Wear 318 (2014) 78–88
calculation model was proposed to predict the die wear more
accurately and a die geometry optimization method was presented
to prolong the die lifespan.
Taking the mandrel die of wheel hub forging process as
example, the wear mechanism of mandrel was studied by scanning electron microscope (SEM) examinations of the worn surfaces. It was found that a number of competing wear mechanisms,
namely adhesive wear, abrasive wear and oxidation wear, occurred
on the worn surface of mandrel. Based on the wear tests and SEM
examinations of the wear debris, a new wear calculation model,
which takes these three wear mechanisms into consideration, was
proposed. Afterwards, an optimal design of the mandrel geometry
was carried out base on the proposed wear calculation model,
forging numerical simulation, BP neural network and SQP
algorithm.
To verify the accuracy of the proposed wear calculation model
and the validity of the optimal design of mandrel geometry,
experiments were performed on HSR-2M wear-testing machine.
By comparing the calculated wear depth and experimental wear
depth, it is proved that the proposed wear calculation model is
accurate and practical. In addition, the wear of optimized mandrel
geometry is less and more uniform than that of the initial one.
Therefore, the optimization of the mandrel geometry is effective.
This study will be beneficial to the accurate prediction of die wear
and the optimal design of die geometry in hot-forging process.
Acknowledgements
This work was financially supported by National Natural
Science Foundation of China under grant no. 51205427, Ministry
of Science and Technology Major Special Project of China (no.
2012ZX04010-081) and the Fundamental Research Funds for the
Central Universities (no. CDJZR13130087).
Table 6
Wear condition of the optimized mandrel and the initial mandrel.
Wear condition
Initial
geometry
Optimized
geometry
Reduction
(%)
Uniform wear (e 4 mm)
The maximum wear
(e 4 mm)
2.61
1.61
1.60
1.41
38.9
12.5
Fig. A.1. (a) 25 cubic spline interpolation curves, (b) the geometry of the 18th
mandrel.
Fig. 15. Wear comparison of the initial and optimized mandrel geometry: (a) calculated wear depth of the initial geometry, (b) calculated wear depth of the optimized
geometry, (c) wear tests of the initial geometry, (d) wear tests of the optimized geometry.
W. Xu et al. / Wear 318 (2014) 78–88
Appendix A. Generation of the BP neural network sample data
Based on MATLAB, 25 cubic spline interpolation curves were
generated randomly and shown in Fig. A.1(a). Then, these curves
were used to describe the geometry of mandrel and establish the
numerical simulation model. Taking the generation of the 18th
sample data as example, the geometry of the 18th mandrel
(corresponding to the 18th curve) was established by UG and
87
shown in Fig. A.1(b). Based on the 18th mandrel, the numerical
simulation model of wheel hub forging process was established
and shown in Fig. A.2.
After numerical simulation of wheel hub forging process, the
contact information (normal pressure P, relative sliding velocity V
and temperature T) of each control point in the 18th mandrel was
obtained. Since the wear depth calculation process of each control
point is the same, the wear depth calculation of control point
6 was taken as example. The contact information of control point
6, which is obtained by numerical simulation, is shown in Fig. A.3.
The related variables (Table 3) and the weight coefficients of
adhesive wear, abrasive wear and oxidation (Table 4) have been
obtained by wear test. Therefore, based on the new wear calculation model (Eq. (6)), the wear depth of control point 6 was
calculated. Similarly, the wear depth of the whole control points
in the 18th mandrel were calculated and illustrated in Table A.1.
Based on the uniform wear calculation formula (Eq. (1)), the
uniform wear of the 18th sample was calculated. Similarly, the
uniform wear values of all the samples could be obtained.
Appendix B. BP neural network training and testing
The 20 samples in Table 5 except test samples were used to
train the designed BP neural network. During training process, the
learn rate of the network, the mean square error of the training
date and the maximum number of training epochs were set as
Fig. A.2. The numerical simulation model of wheel hub forging process.
Fig. A.3. Contact information of control point 6: (a) normal pressure P, (b) relative sliding velocity V, (c) temperature T.
Table A.1
The wear depth of the whole control points in the 18th mandrel.
Control point number
Wear depth (e 4 mm)
1
0.09
2
0.46
3
0.76
4
0.99
5
1.15
6
1.26
7
1.35
8
1.37
9
1.4
Control point number
Wear depth (e 4 mm)
11
1.43
12
1.17
13
1.13
14
1.04
15
0.66
16
0.5
17
0.63
18
0.82
19
0.96
10
1.41
88
W. Xu et al. / Wear 318 (2014) 78–88
(2) Determine the related constraint conditions. The slopes of
control point 1 and control point 19 are 0 and 8.1443,
respectively. x1 ¼ 5.5 mm, x19 ¼ 8.9266 mm, 5:5 mm rxi r
xi þ 1 r 8:9266 mm. (i¼ 1, 2, 3, 4, …, 19).
(3) Set up the related calculation parameters. The maximum
iteration step and the maximum convergence error of objective function equation are set as 1000 and 1e 6, respectively.
(4) The algorithm begins to iterate. First, the Hessian matrix of
initial point is calculated by BFGS method. Then, the search
direction dk is obtained by the Hessian matrix and the step
length ak is determined through line search. Finally, a new
iteration point can be obtained as P k þ 1 ¼ P k þ ak dk .
(5) Judge the convergence of the iteration point. If the iteration
point is convergent, this iteration point is the optimal solution
of objective function. Otherwise, go back to the fourth step and
calculate the next iteration point until the calculation result is
convergent.
Fig. B.1. Training process of the network.
References
Fig. B.2. Testing of the 19-20-1 BP neural network.
0.001, 1e 5 and 2000, respectively. Fig. B.1 shows the training
process of the network, it can be seen that with the updating of
the connection weights, the mean square error between the
network prediction data and training data declines gradually and
reaches to 5.2e 6 at the 69th epoch.
Five test samples, which were not used in the training process,
were used to test the accuracy and reliability of the predictive
system. It can be seen from Fig. B.2 that the predictive values are in
good agreement with the calculated values.
Appendix C. Optimization process of SQP algorithm
The optimization steps are as follows:
(1) Select the initial point of algorithm iteration. This study selects
the 18th cubic spline interpolation curve as the initial point of
algorithm iteration, the design variables of which are [x1, x2, x3,
x4, …, x19] ¼[5.5, 6.5042, 6.8544, 7.0974, 7.2852, 7.4380, 7.5660,
7.6753, 7.7701, 7.8530, 7.9262, 8.1906, 8.3545, 8.4752, 8.5729,
8.6549, 8.7322, 8.8184, 8.9266] mm.
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modes. II: Adhesive and abrasive, Thin Solid Films 123 (1985) 113–126.
[15] E. Rabinowicz, Friction and Wear of Materials, second ed., Wiley, New York,
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Contents lists available at ScienceDirect
Wear
journal homepage: www.elsevier.com/locate/wear
Experimental and theoretical analysis of wear mechanism
in hot-forging die and optimal design of die geometry
Wujiao Xu n, Wuhua Li, Yusong Wang
College of Material Science and Engineering, Chongqing University, Chongqing 400044, China
art ic l e i nf o
a b s t r a c t
Article history:
Received 27 February 2014
Received in revised form
21 June 2014
Accepted 22 June 2014
Available online 30 June 2014
The purpose of this study is to gain a better understanding of the wear mechanism in hot-forging die,
propose a new wear calculation model to predict the die wear more accurately and present a die
geometry optimization method to prolong the die lifespan. Taking the mandrel in wheel hub forging
process as example, the wear mechanism was studied by SEM examinations. It was found that three
competing wear mechanisms exist on the worn surface, namely adhesive wear, abrasive wear and
oxidation wear. A new wear calculation model was proposed with the consideration of these three wear
mechanisms. An optimal design of the mandrel geometry was carried out based on the proposed wear
calculation model, forging numerical simulation, BP neural network and Sequential Quadratic Programming (SQP) algorithm. Wear tests were performed to verify the wear calculation accuracy, which shows a
good agreement with the calculated result. This study will be beneficial to the accurate prediction of die
wear and the optimal design of die geometry in hot-forging process.
& 2014 Elsevier B.V. All rights reserved.
Keywords:
Geometry optimization
Wear mechanism
Wear calculation model
BP neural network
1. Introduction
Hot-forging die works under the condition of high temperature, strong contact pressure and intense friction, which contributes to the severe wear of die. Research shows that wear failure
accounts for more than 70% of the total hot-forging die failure [1].
According to the Archard wear model, wear rate is proportional
to the load and the sliding speed, inversely proportional to the
hardness of worn surface [2]. During plastic forming process,
the geometry of die directly affects the load, the sliding speed
and the temperature of contact area. Therefore, to reduce the die
wear, optimization design of the die geometry is crucial.
As stated before, the wear phenomenon is influenced by the
processing parameters such as temperature, sliding velocity, contact pressure and lubrication conditions [3–5]. The wear mechanism is extremely complicated and lots of studies have been
performed on it. Rapoprt [6] investigated the dominant and
competitive wear mechanisms of steel under different contact
conditions. It was found that three types of wear mechanisms
occurred in the plastic region. The wear mechanisms of iron based
metal matrix composites and the wear behaviors of various
microstructures were systematically studied by dry sliding wear
testing and SEM examination [7]. The experimental results
showed that three dominant wear mechanisms appeared in
n
Corresponding author. Tel.: þ 86 13368186252; fax: þ86 23 65111493.
E-mail address: [email protected] (W. Xu).
http://dx.doi.org/10.1016/j.wear.2014.06.021
0043-1648/& 2014 Elsevier B.V. All rights reserved.
succession with increasing normal load during dry sliding and
the transition of the wear mechanisms depended mainly upon the
conditions of testing.
Some studies paid attention to wear calculation method and put
forward the mathematical models considering the parameters which
influenced the wear process [8–10]. For example, the mathematical
model to predict the wear rate of Al–zircon composites had been
expressed in terms of reinforcement volume fraction, particle size,
applied load, sliding speed and sliding distance using five factors, five
level central composite design matrix approach by response surface
method. The morphology and topography of worn out surface wear
were studied using a scanning electron microscope [8]. However,
mathematical models proposed by these studies just gave the
mathematical relationship between wear amount and selected variables and ignored the mechanism of wear. Thus, they could only be
used in particular situation.
Yin et al. [11] established a calculation model to predict the
wear of fine-blanking die during its whole lifetime based on Back
Propagation (BP) neural network, Finite Element Method (FEM)
and experiments. The inherent law between wear of fine-blanking
die and its working parameters was revealed by utilizing the BP
neural network. Huang and Huang [12] constructed two BP neural
network models based on the experimental date. One was used for
predicting the length evolution of parison with its drop time, the
other was for predicting the swells along the parison.
This paper is organized as follows. Section 2 describes the
optimization problem and mathematical modeling for mandrel
W. Xu et al. / Wear 318 (2014) 78–88
79
Fig. 1. (a) The product of wheel hub, (b) wear condition of mandrel.
geometry optimization. In Section 3, the study of mandrel wear
mechanism during forging process and the establishment of a
new wear calculation model are presented. The BP neural network modeling and SQP algorithm optimization for the mandrel
geometry are presented in Section 4, respectively. Section 5
presents the experimental verification for the accuracy of proposed wear calculation model and the validity of the optimal
design of mandrel geometry. Finally, the last section contains the
conclusions.
2. Description of the optimization problem
2.1. Wear condition
Hot-forging die works under the condition of high temperature, strong contact pressure and intense friction, which contributes to the severe wear of die. For the wear failure of hot forging
die, the die geometry is an important factor which directly
influences the load, the sliding speed and the temperature of
contact area. Here we take the mandrel die of wheel hub forging
process as example. It was observed in the practical production
that the wear of mandrel fillet is serious, which greatly exceeds
that of other parts. Therefore, this paper aims to gain a better
understanding of the mandrel wear mechanism and try to make
the mandrel wear less and more uniform by optimizing the
geometry of mandrel. The wheel hub and wear condition of
mandrel are shown in Fig. 1.
2.2. Mathematical modeling for mandrel geometry optimization
Before optimizing the geometry of mandrel, a cubic spline
interpolation curve which has 19 control points is used to describe
the mandrel geometry and shown in Fig. 2. Considering the shape
feature of mandrel, control points in area A1 (yi þ 1 yi ¼0.2 mm) in
Fig. 2 are denser than that of other area (yi þ 1 yi ¼ 1 mm). In other
words, 11 control points are located in area A1 which has a larger
curvature and 8 control points are arranged in the other area.
Moreover, first boundary condition (the slopes of control point
1 and control point 19 are 0 and 8.1443, respectively, x1 ¼5.5 mm,
x19 ¼ 8.9266 mm, A1 ¼2 mm, A ¼10 mm) is used to draw the cubic
spline interpolation curve. Abscissa values of control points,
namely x1, x2, x3, …, x19, are design variables of this cubic spline
interpolation curve.
As observed in the practical production, the wear in mandrel
fillet is serious, which greatly exceeds that in other parts. For
quantitative analysis, the wear uniformity criteria defined by
Eq. (1) was proposed to evaluate the wear condition of mandrel.
Fig. 2. Description of the mandrel geometry by cubic spline interpolation curve.
Obviously the smaller the uniform wear f, the more uniform the
mandrel wear.
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
f¼
n
∑ ðwi wÞ2
i¼1
ð1Þ
where n is the number of total control points, wi is the wear of
control point i, w is the average wear amount of all the control
points (w ¼ ð1=nÞ∑ni¼ 1 wi ).
The flow chart of the mandrel geometry optimization is
illustrated in Fig. 3. First, the wear mechanism of mandrel was
studied by SEM examinations of the worn surfaces. Then, the wear
calculation formulas for these three wear mechanisms were
deduced, respectively. Based on the wear tests and SEM examinations of the wear debris, a new wear calculation model, which
takes these three wear mechanisms into consideration, was
proposed. Afterwards, an optimal design of the mandrel geometry
was carried out based on the proposed wear calculation model,
forging numerical simulation, BP neural network and SQP algorithm. Finally, to verify the accuracy of the proposed wear
calculation model and the validity of the optimal design of
mandrel geometry, experiments were performed on HSR-2M
wear-testing machine.
80
W. Xu et al. / Wear 318 (2014) 78–88
Fig. 3. Flow chart of the mandrel geometry optimization process.
3. Wear mechanism and new wear calculation model
3.1. Wear mechanism
Archard [13] proposed a formula to calculate the wear of material
v¼k
Wx
H
ð2Þ
where v is wear amount, k is wear coefficient, W is load, x is sliding
distance, H is hardness of the worn surface.
This wear calculation formula was proposed based on classical
adhesive wear model. However, during the forging process of wheel
hub, a number of competing wear mechanisms (adhesive wear,
abrasive wear and oxidation wear) occur on the worn surface as
shown in Fig. 4. Therefore, it is not exact to calculate the wear of
mandrel by traditional Archard wear formula. On this condition, a
new wear calculation model which takes these wear mechanisms
into consideration is put forward to calculate the wear of mandrel
accurately.
Scanning electron microscopic examinations of the worn surfaces (Fig. 4) identified different wear mechanisms. They are
adhesive wear, abrasive wear and oxidation wear. It can be seen
from Fig. 4(a) that there are many spalling pits on the worn
surface. Spalling pits on the mandrel worn surface are typical
characteristics of the adhesive wear, which generated when two
solid surfaces slide over one another under load. Under the effect
of load, some asperities are plastically deformed and the two
surfaces will stick together. As sliding continues, some bonds are
broken and spalling pits (Fig. 4(b)) occur on the worn surface.
Meanwhile, wear particles generate on the other surface, which
will contribute to further wear of the surface. Fig. 4(c) and
(d) show that scratches which are mostly parallel to the sliding
direction generate on the worn surface. These scratches are typical
features of abrasive wear and indicate that the wear mechanism of
this area is abrasive wear. When two surfaces slide, under the
action of hard wear particles, the worn surface is cut and scratches
occur on the surface. In most case, these hard wear particles are
produced by adhesive wear and lead to the three-body abrasive
W. Xu et al. / Wear 318 (2014) 78–88
81
Fig. 4. Scanning electron microscopic examinations of the worn surfaces: (a) spalling pits information on the worn surface indicating adhesive wear, (b) partial enlarged
view of the block region in (a), (c) scratches information on the worn surface indicating abrasive wear, (d) partial enlarged view of the block region in (c), (e) oxide films
information on the worn surface indicating oxidation wear, (f) partial enlarged view of the block region in (e).
wear of the surface [14]. Oxide films are observed on the worn
surface in Fig. 4(e). These oxide films indicate the oxidation wear
of the mandrel. For two sliding surfaces under pressure, due to the
external heating and the frictional heat, the surface will be
oxidized and the oxide film grows slowly during the sliding
process. When the thickness of the oxide film reaches to a critical
value, it will break up and drop off as shown in Fig. 4(f).
In addition, we can find in Fig. 4(e) that the oxide film of contact
area is thicker than that of non-contact area.
Fig. 5(a) shows the Energy Dispersive Spectrometer (EDS)
analysis of the point 1 in Fig. 4(b). The result indicates that there
is almost no oxygen on the worn surface. Combined with the SEM
result of this area, the wear mechanism of this area is identified as
adhesive wear. The EDS analysis of point 2 in Fig. 4(f) is illustrated
in Fig. 5(b). It can be seen that there is much oxygen on this
shedding oxide film. This Phenomenon indicates that the wear
mechanism is oxidation wear.
3.2. Wear calculation formula under different wear mechanisms
3.2.1. Adhesive wear calculation formula
For two sliding surfaces under the load W, it is supposed that
the asperity is plastically deformed and will produce wear particle
with a certain probability. Besides, the contact area consists of two
asperities as shown in Fig. 6; and average radius of the two
asperities is a.
The wear depth of the worn surface can be calculated as [13]
Z t
PV
wadhesive ¼
dt
ð3Þ
k
HðTÞ
0
where P is the normal pressure on the contact area, V is the
relative sliding velocity, H(T) is the hardness of surface when the
temperature is T, t is the sliding time.
3.2.2. Abrasive wear calculation formula
For two sliding surfaces under the load W, it is assumed that
the conical asperity will cut the worn surface and lead to a scratch
along the sliding direction [15]. The schematic diagram of abrasive
wear model is shown in Fig. 7. In the figure, θ indicates the attack
angle of asperity; and 2b, d are the width and depth of the scratch,
respectively.
The wear depth of the worn surface can be expressed as
Z t
2 tan θ P V
wabrasive ¼
dt
ð4Þ
π
HðTÞ
0
3.2.3. Oxidation wear calculation formula
For two sliding surfaces, due to the external heating and the
frictional heat, the surface will oxidize and the oxide film of
82
W. Xu et al. / Wear 318 (2014) 78–88
Fig. 5. EDS analysis results of the worn surface: (a) EDS of point 1 in Fig. 4(b), (b) EDS of point 2 in Fig. 4(f).
Fig. 6. The schematic diagram of adhesive wear model.
Fig. 8. The schematic diagram of oxidation wear model.
Therefore, in order to calculate the wear of mandrel accurately, a
new wear calculation model which takes these wear mechanisms
into consideration is proposed
Fig. 7. The schematic diagram of abrasive wear model.
contact area is thicker than that of non-contact area. It is assumed
that the oxide film grows slowly during the sliding process.
In addition, the oxide film will break instantly when its thickness
reach a critical value ξ. The schematic diagram of oxidation wear
model is shown in Fig. 8.
Therefore, the wear depth of the worn surface can be obtained
as [16]
Z t
Qp
PAp C 2
woxidation ¼
expð Þdt
ð5Þ
RT
0 HðTÞξ
where Ap is parabolic oxidation Arrhenius constant, Qp is parabolic
oxidation activation energy, R is molar gas constant, η is the
volume percentage of Fe3O4, ρ is the density of iron, MO and MFe
are the molar mass of oxygen and iron, respectively, C ¼
6M Fe =ð9 ηÞρM O .
3.3. Establishment of wear calculation model based on wear
mechanisms
According to the SEM examinations (Fig. 4) and EDS analysis
(Fig. 5) on the worn surface, it can be found that the mechanisms
of mandrel are the coexistence of adhesive wear, abrasive wear
and oxidation wear. In Section 3.2, the wear calculation formulas
of these three mechanisms have been obtained, respectively.
1
wtotal ¼ λadhesive wadhesive þλabrasive wabrasive þ λoxidation woxidation
R t PV
C
wadhesive ¼ 0 kHðTÞdt
C
C
R t 2 tan θ PV
C
C
wabrasive ¼ 0 π HðTÞdt
C
A
R t PAp C 2
Qp
woxidation ¼ 0 HðTÞξ expð RT Þdt
ð6Þ
where wtotal is the total wear depth of the worn surface; wadhesive,
wabrasive and woxidation are the wear depth which are caused by
adhesive wear, abrasive wear and oxidation wear, respectively;
λadhesive, λabrasive and λoxidation are the weight coefficients of
adhesive wear, abrasive wear and oxidation, repectively.
λadhesive, λabrasive and λoxidation are approximately equal to the
weight percentages of wear debris which are produced by adhesive wear, abrasive wear and oxidation wear, respectively. Therefore, in order to calculate the wear of mandrel by this wear
calculation model, wear tests were performed to obtain the related
variables in Eq. (6).
3.3.1. Materials in wear test
Fig. 9 illustrates the friction pairs of wear test. The material of
wear-resistant plate is AISI-H13, which is the same as mandrel.
And the material of pressure head is AISI-1045, which is the same
as wheel hub. The initial hardness of wear-resistant plate and
pressure head are HRC 54 and HRC 25, respectively. The chemical
composition of friction parts is shown in Table 1.
W. Xu et al. / Wear 318 (2014) 78–88
3.3.2. Wear test
The wear test was carried out on HSR-2M wear-testing
machine. During the forging process, the contact condition of
mandrel varied. However, almost no wear-testing machines can
make a real-time control of pressure and speed. The Deform
software is used to calculate the forging process in this paper.
Therefore, in order to approach the actual production as much as
possible, the normal pressure was obtained by FEA as shown in
Fig. 10. According to the obtained normal pressure, the wear
condition is divided into five stages, which is shown in Table 2.
The HSR-2M wear-testing machine and the schematic of wear
test are shown in Fig. 11. When the wear test was performed, the
wear-resistant plate was heated by a resistance furnace. The
heating temperature was controlled by the thermocouple and
the temperature controller. Then, the pressure head was loaded
and kept in contact with the wear-resistant plate on a constant
load by the loading device of wear-testing machine. Finally, the
wear-resistant plate did the reciprocating movement of 10 mm
length.
After the wear test, the wear debris was collected. According to
the density differences between wear-resistant plate (AISI-H13)
and pressure head (AISI-1045), the wear debris of wear-resistant
plate was separated. Fig. 12 illustrates SEM examinations of the
separated wear debris.
SEM examinations of the wear debris (Fig. 12) identified
different wear debris. They are adhesive wear debris, abrasive
wear debris and oxidation wear debris. Based on the weight of
collected adhesive wear debris, the variable k in Eq. (6) was
calculated. According to Fig. 12(c), the attack angle of asperity θ
was obtained. 50 Scratches in Fig. 4(c) were selected to calculate
the weighted average value of the whole asperities tan θ. The
oxidation wear debris (Fig. 12(d)) indicated the critical thickness of
oxide film ξ (10 μm) and the volume percentage of Fe3O4 η (58.2%).
Related variables of the proposed wear calculation model are
shown in Table 3.
Fig. 12 shows that three types of wear debris were found on the
worn surface. The adhesive wear debris was produced by plastic
deformation. Therefore, the adhesive wear debris has a rough
surface, an uneven thickness (Fig. 12(a)) and little oxygen. When
Fig. 9. Friction pairs of wear test: (a) wear-resistant plate, (b) pressure head.
83
two surfaces slide, due to the effect of hard wear particles, the
surface was cut and the abrasive wear debris was produced. Thus,
the shape of abrasive wear debris is banded (Fig. 12(b)). The
oxidation wear debris has a smooth surface, an even thickness
(Fig. 12(d)) and contains a lot of oxygen. According to the
morphology differences among adhesive wear debris, abrasive
Fig. 10. The normal pressure of mandrel.
Table 2
Wear test condition.
Stage number
Pressure (MPa)
Speed (rpm)
Temperature (1C)
Time (min)
Stage
Stage
Stage
Stage
Stage
280
390
330
280
1000
75
225
90
390
390
410
468
482
510
517
5
15
7.5
2.5
1
1
2
3
4
5
Fig. 11. The HSR-2M wear-testing machine and schematic diagram of wear test.
Table 1
Chemical composition of friction pairs.
Element
C
Si
Mn
P
S
Cr
Ni
Cu
Mo
V
Fe
AISI-H13 (wt%)
AISI-1045 (wt%)
0.4
0.45
0.97
0.24
0.38
0.67
0.025
0.028
0.003
0.016
5.09
0.15
0.11
0.03
0.16
0.08
1.24
–
0.96
–
90.662
98.336
84
W. Xu et al. / Wear 318 (2014) 78–88
Fig. 12. SEM examinations of the separated wear debris: (a) single adhesive wear debris, (b) single abrasive wear debris, (c) worn surface of abrasive wear, (d) single
oxidation wear debris, (e) one visual field of the whole wear debris, (f) the other visual field of the whole wear debris.
Table 3
Related variables of the proposed wear calculation model.
k
2 tan θ
π
C (mm3 kg 1)
ξ (μm)
TC (1C)
Ap (mg2 cm 4 h 1)
Qp [17] (J mol 1)
R (J mol 1 K 1)
2.85e 5
3.64e 3
3.168e 4
10
517
0.084
157539
8.314
Table 4
Features of each wear debris and weight coefficients of adhesive wear, abrasive wear and oxidation.
Type of wear debris
Features
Weight (mg)
Weight coefficient λ
Adhesive wear debris
Abrasive wear debris
Oxidation wear debris
Tearing, rough surface, uneven thickness and little oxygen
Long strip, little oxygen
Smooth surface, even thickness and a lot of oxygen
2.8
1.2
3.4
0.378
0.162
0.459
wear debris and oxidation debris, they were separated and
weighed. Table 4 illustrates the features of each wear debris and
the weight coefficients of adhesive wear, abrasive wear and
oxidation.
4. Optimization with BP neural network and SQP algorithm
4.1. BP neural network modeling
Our research aim is to do the optimization of the mandrel
geometry. Then we can reduce the mandrel wear and prolong the
die life. To optimize design variables according to the objective
function, it is very important to establish the internal relation
between design variables and objective function. However, due to
the highly nonlinear of the optimization problem, it is hard to use
a mathematical equation to establish the internal relation between
design variables (abscissa values of control points) and objective
function (uniform wear values) in this paper.
The BP neural network consists of input layer, hidden layer and
output layer, different transfer functions are used to link these
layers. And it is proved that a three-layer BP neural network can
approximate any nonlinear function with arbitrary precision.
Therefore, this paper uses a 19-20-1 BP neural network to reveal
the inherent law between abscissa values of control points and
uniform wear values. The network consists of one input layer with
19 neurons standing for the 19 abscissa values of control points;
the hidden layer with 20 neurons and the output layer with
1 neuron representing the uniform wear value. The structure of
the designed neural network can be seen in Fig. 13.
W. Xu et al. / Wear 318 (2014) 78–88
Based on the forging numerical simulation, the proposed wear
calculation model and uniform wear calculation formula, the sample
sets of BP neural network, which consist of uniform wear values of
the whole mandrel geometries, are obtained and shown in Table 5.
Among those, five samples (4th, 8th, 12th, 16th and 20th) are
selected to be test samples. The detailed generation procedure of
BP neural network sample data is shown in Appendix A.
The designed neural network should be trained by a number of
samples before it can acquire the functional relationship between
design variables (abscissa values of control points) and objective
function (uniform wear values). The BP neural network training
and testing process can be referenced in Appendix B. The testing
result shows that the predictive error is less than 5% on average.
Hence, the trained network can be used as the surrogate of the
functional relationship between uniform wear and abscissa values
of control points.
4.2. Optimization with SQP algorithm
SQP algorithm is particularly effective to solve the optimization
problem which is highly nonlinear. And the implementation of
SQP involves three main steps: (1) Hessian matrix update;
(2) quadratic programming problem solution; (3) line search and
merit function [18]. The details of optimization process are shown
in Appendix C.
Based on the SQP algorithm, the optimal mandrel geometry
which makes the wear of mandrel most uniform was obtained.
The design variables of which are [x1, x2, x3, x4, …, x19] ¼[5.5,
6.4661, 6.8511, 7.0974, 7.2853, 7.4371, 7.5611, 7.6703, 7.7643,
7.8484, 7.9334, 8.1894, 8.3523, 8.4712, 8.5652, 8.6487, 8.7315,
8.8186, 8.9266] mm. Fig. 14 illustrates the comparison of the
optimized and initial mandrel geometry.
Then, the optimized design variables were used as the input of
trained BP neural network. The uniform wear of the optimized
mandrel geometry was predicted to be 1.5963e 4 mm by the
trained BP neural network. However, the uniform wear of the
initial mandrel geometry is 2.6139e 4 mm. Compared with the
initial mandrel geometry, the uniform wear of the optimized
85
mandrel geometry decreased by 38.9%. Therefore, the wear of
the optimized mandrel geometry is more uniform than that of the
initial one. It means that the optimization of the mandrel geometry is effective. The wear condition of the optimized mandrel and
the initial mandrel is shown in Table 6.
5. Experimental verification
To verify the accuracy of the new wear calculation model and
the validity of the optimal design of mandrel geometry, experiments were performed on HSR-2M wear-testing machine. First,
the contact information (normal pressure, relative sliding velocity
and temperature) of initial and optimized mandrel geometry was
obtained by numerical simulations of wheel hub finish forging
process. Based on the proposed wear calculation model, the wear
depth of the whole control points was calculated. Then, according
to the contact information, wear tests representing 1000 forging
cycles were carried out. Fig. 15 illustrates the wear comparison of
the initial and optimized mandrel geometry.
The calculated wear depth of control point a, b, c, d, e and f
are 0.005172 mm, 0.126363 mm, 0.158423 mm, 0.065113 mm,
0.105326 mm and 0.089258 mm (Fig. 15(a)), respectively. It can
be seen from Fig. 15(c) that the experimental wear depth shows a
good agreement with the calculated wear depth. From Fig. 15
(b) and (d), a similar result is obtained. Therefore, it is proved that
the proposed wear calculation model is accurate and practical.
What's more, it is evident from Fig. 15(a) and (b) that the uniform
wear and the maximum wear depth of the optimized mandrel
geometry are less than those of the initial mandrel geometry. Thus,
the optimization of the mandrel geometry is effective.
6. Conclusion
In this paper, the wear mechanism of hot-forging die was
studied by SEM examinations and EDS analysis, a new wear
Fig. 13. Structure of the designed neural network.
Fig. 14. Comparison of the optimized and initial mandrel geometry.
Table 5
Uniform wear values of each sample.
Sample number
Uniform wear (e 4 mm)
1
2.01
2
2.22
3
2.07
4
2.32
5
1.92
6
1.90
7
1.78
8
1.92
9
2.15
Sample number
Uniform wear (e 4 mm)
10
2.12
11
1.97
12
2.16
13
1.89
14
1.92
15
1.88
16
2.22
17
2.04
18
1.74
Sample number
Uniform wear (e 4 mm)
19
2.25
20
1.87
21
1.85
22
2.00
23
2.22
24
2.33
25
2.11
86
W. Xu et al. / Wear 318 (2014) 78–88
calculation model was proposed to predict the die wear more
accurately and a die geometry optimization method was presented
to prolong the die lifespan.
Taking the mandrel die of wheel hub forging process as
example, the wear mechanism of mandrel was studied by scanning electron microscope (SEM) examinations of the worn surfaces. It was found that a number of competing wear mechanisms,
namely adhesive wear, abrasive wear and oxidation wear, occurred
on the worn surface of mandrel. Based on the wear tests and SEM
examinations of the wear debris, a new wear calculation model,
which takes these three wear mechanisms into consideration, was
proposed. Afterwards, an optimal design of the mandrel geometry
was carried out base on the proposed wear calculation model,
forging numerical simulation, BP neural network and SQP
algorithm.
To verify the accuracy of the proposed wear calculation model
and the validity of the optimal design of mandrel geometry,
experiments were performed on HSR-2M wear-testing machine.
By comparing the calculated wear depth and experimental wear
depth, it is proved that the proposed wear calculation model is
accurate and practical. In addition, the wear of optimized mandrel
geometry is less and more uniform than that of the initial one.
Therefore, the optimization of the mandrel geometry is effective.
This study will be beneficial to the accurate prediction of die wear
and the optimal design of die geometry in hot-forging process.
Acknowledgements
This work was financially supported by National Natural
Science Foundation of China under grant no. 51205427, Ministry
of Science and Technology Major Special Project of China (no.
2012ZX04010-081) and the Fundamental Research Funds for the
Central Universities (no. CDJZR13130087).
Table 6
Wear condition of the optimized mandrel and the initial mandrel.
Wear condition
Initial
geometry
Optimized
geometry
Reduction
(%)
Uniform wear (e 4 mm)
The maximum wear
(e 4 mm)
2.61
1.61
1.60
1.41
38.9
12.5
Fig. A.1. (a) 25 cubic spline interpolation curves, (b) the geometry of the 18th
mandrel.
Fig. 15. Wear comparison of the initial and optimized mandrel geometry: (a) calculated wear depth of the initial geometry, (b) calculated wear depth of the optimized
geometry, (c) wear tests of the initial geometry, (d) wear tests of the optimized geometry.
W. Xu et al. / Wear 318 (2014) 78–88
Appendix A. Generation of the BP neural network sample data
Based on MATLAB, 25 cubic spline interpolation curves were
generated randomly and shown in Fig. A.1(a). Then, these curves
were used to describe the geometry of mandrel and establish the
numerical simulation model. Taking the generation of the 18th
sample data as example, the geometry of the 18th mandrel
(corresponding to the 18th curve) was established by UG and
87
shown in Fig. A.1(b). Based on the 18th mandrel, the numerical
simulation model of wheel hub forging process was established
and shown in Fig. A.2.
After numerical simulation of wheel hub forging process, the
contact information (normal pressure P, relative sliding velocity V
and temperature T) of each control point in the 18th mandrel was
obtained. Since the wear depth calculation process of each control
point is the same, the wear depth calculation of control point
6 was taken as example. The contact information of control point
6, which is obtained by numerical simulation, is shown in Fig. A.3.
The related variables (Table 3) and the weight coefficients of
adhesive wear, abrasive wear and oxidation (Table 4) have been
obtained by wear test. Therefore, based on the new wear calculation model (Eq. (6)), the wear depth of control point 6 was
calculated. Similarly, the wear depth of the whole control points
in the 18th mandrel were calculated and illustrated in Table A.1.
Based on the uniform wear calculation formula (Eq. (1)), the
uniform wear of the 18th sample was calculated. Similarly, the
uniform wear values of all the samples could be obtained.
Appendix B. BP neural network training and testing
The 20 samples in Table 5 except test samples were used to
train the designed BP neural network. During training process, the
learn rate of the network, the mean square error of the training
date and the maximum number of training epochs were set as
Fig. A.2. The numerical simulation model of wheel hub forging process.
Fig. A.3. Contact information of control point 6: (a) normal pressure P, (b) relative sliding velocity V, (c) temperature T.
Table A.1
The wear depth of the whole control points in the 18th mandrel.
Control point number
Wear depth (e 4 mm)
1
0.09
2
0.46
3
0.76
4
0.99
5
1.15
6
1.26
7
1.35
8
1.37
9
1.4
Control point number
Wear depth (e 4 mm)
11
1.43
12
1.17
13
1.13
14
1.04
15
0.66
16
0.5
17
0.63
18
0.82
19
0.96
10
1.41
88
W. Xu et al. / Wear 318 (2014) 78–88
(2) Determine the related constraint conditions. The slopes of
control point 1 and control point 19 are 0 and 8.1443,
respectively. x1 ¼ 5.5 mm, x19 ¼ 8.9266 mm, 5:5 mm rxi r
xi þ 1 r 8:9266 mm. (i¼ 1, 2, 3, 4, …, 19).
(3) Set up the related calculation parameters. The maximum
iteration step and the maximum convergence error of objective function equation are set as 1000 and 1e 6, respectively.
(4) The algorithm begins to iterate. First, the Hessian matrix of
initial point is calculated by BFGS method. Then, the search
direction dk is obtained by the Hessian matrix and the step
length ak is determined through line search. Finally, a new
iteration point can be obtained as P k þ 1 ¼ P k þ ak dk .
(5) Judge the convergence of the iteration point. If the iteration
point is convergent, this iteration point is the optimal solution
of objective function. Otherwise, go back to the fourth step and
calculate the next iteration point until the calculation result is
convergent.
Fig. B.1. Training process of the network.
References
Fig. B.2. Testing of the 19-20-1 BP neural network.
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Appendix C. Optimization process of SQP algorithm
The optimization steps are as follows:
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