Recent Researches in Applied Mathematics and Economics
Variation of Kinetic Friction Coefficient with respect to Impact Velocity
in Tube Type Energy Absorbers
W. M. CHOI
New Transportation Systems Research Center
Korea Railroad Research Institute
360-1 Woram-dong, Uiwang, Gyeonggi
KOREA
[email protected]
T. S. KWON
New Transportation Systems Research Center
Korea Railroad Research Institute
360-1 Woram-dong, Uiwang, Gyeonggi
KOREA
[email protected]
Abstract: - The expansion tube used as a crash element dissipates crash energy through internal
deformation energy of the tube and frictional energy. In this paper, in order to study the effects
of impact velocity on the characteristics of energy absorption of an expansion tube, friction
quasi-static and dynamic experiments were carried out and an inverse method using finite
element analysis and the least square method was introduced to calculate the friction coefficient
at each impact velocity. The results of the experiments and the finite element analyses show that
an increase of the impact velocity increased the strain rate of the tube. Energy absorbed into tube
was increased due to the strain rate effect. The friction coefficient in the quasi-static state was
higher than that in the dynamic state. The results of a comparison of mean forces and absorbed
energies with experimental results indicate that the least square method is appropriate to
calculate the friction coefficient in the case of an expansion tube.
Key-Words: Expanding of tube; Slip rate; Inverse method; Strain rate
practice, it is very difficult to measure the frictional
coefficient between punch and tube.
Recently, various studies on expansion tubes have
been carried out due to their excellent characteristics
in terms of specific characteristics and deceleration.
Shakeri et al. [1] identified the effects of three
contact conditions (blasting, coating, and dry) on the
uniform force of an expansion tube with static
experiments and suggested a simple numerical
model to estimate the force. In addition, Lucanin
and Tanaskovic [2] conducted a study on the energy
absorption characteristics of an expansion tube
through quasi-static and dynamic experiments.
Karrech and Seibi [3] created a numerical model to
predict the reaction force and absorbed energy and
suggested the optimum punch shape. Yang et al. [4]
performed a quasi-static experiment by changing the
thickness of an aluminum expansion tube and the
punch angle and carried out finite element analysis
1 Introduction
An expansion tube dissipates kinetic energy through
the frictional energy and plastic deformation energy
generated when a punch whose external diameter is
larger than the internal diameter of the tube expands
the tube. The expansion tube has high energy
absorption efficiency, as more than 80% of its
length can be used for energy absorption. In
addition, it has good characteristics in terms of
deceleration, as uniform force occurs during the
expansion of the tube. Given that an expansion tube
absorbs crash energy with frictional energy and
plastic strain energy, accurate friction coefficient
and the dynamic properties of the material,
depending on the strain rate, should be applied so as
to precisely estimate, via the finite element method,
the energy absorbed into the tube. The dynamic
properties of a material can be identified by a tensile
test using a high-speed testing machine but, in
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Recent Researches in Applied Mathematics and Economics
coefficient, the results of the finite element analyses,
applying the calculated frictional coefficient, were
compared with experimental results in terms of
mean force and absorbed energy.
under the same conditions as the experiment to
ascertain the effects of the tube thickness and punch
angle on the force and absorbed energy. The authors
also suggested a numerical analysis model to predict
energy absorbed into the tube. Choi et al. [5]
conducted quasi-static experiments with three types
of the punch angle in order to investigate the effects
of the external punch shapes on energy absorption
characteristics of the tube and conducted finite
element analysis under the same conditions as those
of the experiments. The authors found that the
punch angle has a significant effect on the frictional
coefficient and the plastic deformation energy of the
tube and on the frictional energy and expansion ratio.
Choi et al. [6] investigated the effects of the angle
and diameter of the punch expanding the tube on the
failure instability of the expansion tube. The authors
carried out a tensile test with three kinds of tensile
specimens having different notch shapes in order to
calculate the failure strain depending on the stress
triaxiality and conducted finite element analysis
with a damage model. As a result, the failure
instability of the tube was found to be more
sensitive to the punch angle than to the punch
diameter.
Previous studies mainly focused on the contact
conditions under the quasi-static state, along with
the energy absorption characteristic, failure
instability, and strain rate effect. However, there
have been few studies on the effects of impact
velocity on frictional coefficient and energy
absorption characteristics. As an expansion tube
dissipates
kinetic
energy
through
plastic
deformation energy and frictional energy, the effect
of impact velocity should be considered in the
design process of the expansion tube so as to
precisely predict the absorbed energy. Therefore, a
study on the effect of impact velocity on the energy
absorption characteristic of expansion tubes needed
to be performed.
In this study, quasi-static experiments using a
hydraulic press and dynamic experiments using an
impacting wagon were carried out to examine the
effects of the impact velocity on the energy
absorption characteristics and frictional coefficient.
In addition, an inverse method using finite element
analysis and the least square method was
implemented in order to calculate the frictional
coefficient at each impact velocity. Finite element
analyses were performed under the same conditions
as those of the experiments by changing the
frictional coefficient and were used to calculate the
approximate frictional coefficient that had the
minimum error. In order to verify the use of the
least square method for calculating the frictional
ISBN: 978-1-61804-076-3
2 Specimens and material
In order to produce the expansion tube, first, a deep
hole drilling process was conducted with circularly
forged S20C. Next, heat treatment and annealing
were applied to enhance the elongation and to obtain
a homogenous material. Finally, the production of
the expansion tube was completed through precision
mechanical processing. Fig. 1 shows an expansion
tube with total length L=560.2 mm, inner diameter
D=186 mm, first thickness t1=12.7 mm and second
thickness t2=15 mm. The force generated during the
expansion process of the tube depends on the
thickness of the tube, because the outer diameter of
the punch that expands the tube and the inner
diameter of the tube do not change. Since the
thickness of the tube increases from 12.7 mm to 15
mm, it is possible for a single tube to generate two
different forces.
As the expansion tube works under dynamic
conditions, change of flow stress, depending on the
strain rate, should be considered in order to predict
the absorbed energy using finite element analysis.
Fig. 2 shows the true stress-true strain relationship
of S20C with respect to strain rate in the range of
0.003/sec to 300/sec obtained from a high speed
material testing machine.
3 Experimental testing
3.1 Test setup
Quasi-static and dynamic experiments were
implemented to identify the effects of impact
velocity on the characteristics of energy absorption
of the tube. A hydraulic press with maximum
displacement of 2,000 mm and maximum compress
load of 5,000 kN was used in the quasi-static
experiments. Reaction force was measured at a load
cell located between an actuator and a test jig. The
displacement was measured by an LVDT (Linear
Variable Differential Transformer) installed inside
the actuator. Fig. 3 shows a picture of the test jig
installed in the hydraulic press for the quasi-static
experiments. The moving speed of the actuator was
10 mm/sec.
As shown in Fig. 5, four load cells and a support
plate were installed in an impacting wagon and a
conduct the dynamic experiments. A locomotive
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Recent Researches in Applied Mathematics and Economics
Fig. 1 Sketch and primary dimensions of expansion
tube (mm)
Fig. 3 Test setup for quasi-static experiments
600
True stress(MPa)
500
400
300
Fig. 4 View of impacting wagon and test jig for
dynamic experiments
0.001/sec
0.1/sec
1/sec
10/sec
100/sec
300/sec
200
100
is the initiation of expansion. In this stage the
impacting force gradually increases. Second is first
uniform force section corresponding to the thickness
of t1. Third is a force transfer section in which the
thickness increases from t1 to t2. Fourth is the
second uniform force section corresponding to a
thickness of t2. Fig. 6 illustrates the comparison of
the results for the quasi-static and dynamic
experiments, indicating that an increase of the
impact velocity increases the impact force because
the strain rate in the circumferential direction of the
tube increases with respect to the impact velocity.
Fig. 7 provides a graph comparing the absorbed
energies with respect to the impact velocity. As
shown in Fig. 6, since impact force increases with
the increase of the impact velocity, the absorbed
energy at the same crash distance increases. Fig. 8
shows pictures that compare the deformed tube
shapes before and after the quasi-static experiment.
Fig. 9 presents high-speed camera still images that
show the energy absorption process in the time
sequence.
0
0.00
0.05
0.10
0.15
0.20
True strain
Fig. 2 True stress-strain curve of S20C with strain
rate
test jig was attached on the support plate in order
toaccelerated the impacting wagon up to the target
speed. Next, the impacting wagon was separated
and crashed into a rigid wall. Impacting force was
measured at the four load cells installed between the
impacting wagon and the support plate, and the data
measured were stored in a DAS (Data Acquisition
System) installed in the upper part of the wagon at a
sampling rate of 100,000 Hz. Additionally, an image
taken by a high-speed camera at a speed of 1,000
f/sec was analyzed by TEMA3.1 image analysis
software, in order to measure the displacement of
the tube. The tube was lubricated with Molykote
lubricant including MoS2 in order to prevent the
relatively weak material of the tube from sticking to
the punch when the tube was expanded by the punch.
4 Influence of impact velocity
4.1 Finite element model
3.2 Experimental results
In order to study the effect of impact velocity on the
kinetic friction coefficient, ABAQUS, a widely used
finite element analysis code, was introduced to
perform a quasi-static analysis and a dynamic
Fig. 5 shows the variation of impacting forces
measured during the expanding process of the tube.
According to the graph, the impacting force
experiences force changes through four stages. First
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Recent Researches in Applied Mathematics and Economics
Fig. 8 Initial and final shapes of tubes in quasi-static
experiment
Fig. 5 Variation of force with respect to
displacement (v=5.69m/s)
Fig. 9 Deformation process of tube in dynamic
experiment
2000
1750
1500
Force(kN)
1250
1000
Quasi-static 1
Quasi-static 2
Quasi-static 3
Quasi-static 4
Dynamic 1(v=4.49m/s)
Dynamic 2(v=5.63m/s)
Dynamic 3(v=5.69m/s)
Dynamic 4(v=9.95m/s)
750
500
250
0
Fig. 10 Finite element models and mesh systems
(axisymmetric)
-250
-100
0
100
200
300
400
500
600
700
Displacement(mm)
Fig. 6 Effect of impact velocity
apply the strain rate effect to the dynamic analysis.
The range of the measured strain rate was from
0.003/sec to 300/sec. The material properties were
assumed to be those of an isotropic material and
were applied as a piecewise linear form, as shown in
Fig. 2. The friction model between the punch and
the tube is a coulomb friction model. In order to
prevent penetration of punch nodes into segments of
tube elements, the surface discretization method was
applied [7].
Quasi-static analyses were conducted with an
identical friction coefficient of 0.05 in all of the
analysis models by increasing the element layers
from two (6.35mm) to six (2.1mm) layers in the
direction of thickness of t1, as can be seen in Fig. 11,
in order to determine the element size of the tube.
The results of the finite element analysis indicate
that an increase of the element layer leads to an
increase of the force. Fig. 12 illustrates the
relationship between the number of element layers
and the mean force. When the number of element
layers is more than four, the mean force did not
increase. However, in the uniform force section, the
force failed to increase further only when the
1200
Absorbed energy(kJ)
1000
800
600
Quasi-static 1
Quasi-static 2
Quasi-static 3
Quasi-static 4
Dynamic 1(v=4.49m/s)
Dynamic 2(v=5.63m/s)
Dynamic 3(v=5.69m/s)
Dynamic 4(v=9.95m/s)
400
200
0
-100
0
100
200
300
400
500
600
700
Displacement(mm)
Fig. 7 Comparison of absorbed energy
analysis. Fig. 10 presents finite element model for
the expansion tube, in which the punch was
modeled as a rigid body and the tube was
constituted as four-nodal-point axisymmetric
elements with one reduced integration point. The
dynamic material properties of the S20C were
measured using a high-speed testing machine to
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Recent Researches in Applied Mathematics and Economics
the decay coefficient, and γ&eq is the relative velocity
of the contact surfaces [10].
1600
1400
Force(kN)
1200
E total =
1000
V
σ ε&dV + ∫
S
f
∫
ur
(τ f du r ) dS
(2)
800
600
Eq. (2) is the energy equation of an expansion tube.
The first term denotes the internal plastic
deformation energy of the tube and the second term
the frictional energy. In this equation, τ f refers to
2layers in t1(6.35mm)
3layers in t1(4.2mm)
4layers in t1(3.2mm)
5layers in t1(2.54mm)
6layers in t1(2.1mm)
400
200
0
the shear stress between the punch and the tube and
µ r to the relative velocity of the contact surfaces. As
can be seen in Eq. (2), the expansion tube absorbs
the kinetic energy through internal deformation
energy and frictional energy. Therefore, the friction
coefficient and dynamic properties of materials are
very important in predicting the energy absorbed
into the expansion tube using finite element analysis.
While the dynamic properties of a material can be
measured with a high speed material testing
machine, it is impossible to directly measure the
friction coefficient. In this study, an inverse method
using the finite element method was applied to
calculate the static friction coefficient and the
kinetic friction coefficient related to the impact
velocity.
In order to calculate the friction coefficient during
the expanding process of the tube, finite element
analyses were carried out, changing the friction
coefficient and the least square method was used to
calculate the error, defined as the difference
between the finite element analysis results and the
experimental results [11]. In addition, the error was
approximated using a cubic polynomial, as shown in
Fig. 14, and the approximated friction coefficient,
which has a minimum error, was calculated. The
least square method is expressed as the following.
-200
0
100
200
300
400
Displacement(mm)
Fig. 11 Effect of element size
1300
1250
1200
Force(kN)
∫
1150
1100
1050
1000
2
3
4
5
6
Number of layers in t1
Fig. 12 Variation of mean force with respect to
element size
number of element layers exceeded five. Hence, in
this study, six (2.1 mm) layers were applied for all
finite element analyses.
4.2 Kinetic friction coefficient
µ opt ∈ Λ , such as Φ ( µ opt ) = Min Φ ( µ )
The friction when initiation of slipping from
sticking condition occurred by external force is
typically referred to as the static friction coefficient.
In contrast, the friction under the slipping condition
is called the kinetic friction condition. According to
previous study results, the static friction coefficient
is larger than the kinetic friction coefficient [8][9].
The relationship of the friction coefficient to the slip
rate can be expressed as an exponential function, as
in Eq. (1).
µ = µ k + ( µ s − µ k )e
− d c γ& eq
Φ (µ ) =
FEM
i
( µ ) − D iexp ) 2
(3)
i =1
Here, Φ is the error and D iFEM means the th value
calculated by the finite element analysis. In addition,
Diexp is the i th value measured in the experiments
and Λ is a constraint space.
In order to a calculate friction coefficient having a
minimum error, finite element analysis was
performed with five initial friction coefficients (0.03,
0.04, 0.05, 0.06, 0.07), as shown in Fig. 13, and Eq.
[3] was used to calculate the error ( Φ ). The
minimum error was located in the hatched region
(1)
Here, µ k and µ s are the kinetic friction coefficient
and the static friction coefficient respectively. d c is
ISBN: 978-1-61804-076-3
n
∑ (D
34
Recent Researches in Applied Mathematics and Economics
Table 1 Result of least square methods
Impact velocity (m/s)
Quasi-static
4.49
5.63
5.69
9.95
Friction coefficient
0.0436
0.0308
0.0264
0.0267
0.0204
0.045
(0.03~0.05) of the graph (Fig.13), and therefore the
hatched region was segmented further into seven
friction coefficients and the finite element analysis
was implemented again. The error for each friction
coefficient was then calculated and those errors
were approximated by a cubic polynomial, as can be
seen in Fig. 14. From the results, the friction
coefficient of 0.0438 under the quasi-static state was
calculated at the minimum error. The friction
coefficient for each impact velocity was calculated
in the same manner and the results are summarized
in Table 1. Fig. 15 illustrates the results of a curve
fitting of the friction coefficient specified in Table 1
with an exponential function of Eq. (1). The results
Friction coefficient(µ)
0.040
0.035
0.030
0.025
0.020
0.015
0
2
4
6
8
10
Impact velocity(m/s)
Fig. 15 Curve fitted to experiment data with
exponential function
2800
2600
show good agreement in the case in which
µ k =0.00959, µ s =0.04368, and d c =0.11729.
ABAQUS offers a friction model that can express
static/kinetic friction coefficients by using Equation
(1). In this study, the value calculated from the
curve fitting seen in Fig. 15 was applied to the
ABAQUS input file and dynamic analyses were
conducted. The mean force and absorbed energy
were calculated and compared with the experimental
results. Fig. 16 ~20 provides graphs that compare
the finite element analyses at each impact velocity
with the experimental data. Those graphs show the
good agreement between the finite element analyses
and the experimental data. Table 2 shows a
comparison of the finite element analyses results
with the experimental data in terms of the mean
force and absorbed energy. The results of the
comparison indicate that the differences of the mean
force and the absorbed energy were approximately
0.41% ~ 1.76% and 0.35 ~ 2.67%, respectively,
which values are in good agreement with the
experimental results.
2400
φ (µ)
Experiments
Fitted curve
2200
2000
1800
1600
0.03
0.04
0.05
0.06
0.07
Friction coefficient(µ)
Fig. 13 Result of least square method (quasi-static)
2100
2000
φ (µ)
1900
Approximation
FEM
1800
1700
5 Result
Minimum value(0.0436)
1600
0.030
0.035
0.040
0.045
Quasi-static and dynamic experiments were carried
out to identify the effects of impact velocity on the
energy absorbing characteristic of an expansion tube.
In addition, in order to study the relationship
between the impact velocity and the friction
0.050
Friction coefficient(µ)
Fig. 14 Determination of minimum value of error
(quasi-static)
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Recent Researches in Applied Mathematics and Economics
Table 2 Comparison of FEM with experiment
Velocity
(µ )
quasi-static
(0.0436)
Fm
(kN)
1,188
1,194
0.51
Exp.
FEM
Diff.(%)
4.49
(0.0308)
E
(kJ)
463.5
466.2
0.58
Fm
(kN)
1,268
1,277
0.71
5.63
(0.0264)
E
(kJ)
383.7
385.8
0.55
Fm
(kN)
1,366
1,390
1.76
E
(kJ)
587.5
603.2
2.67
Fm
(kN)
1,381
1,396
1.09
Experiment
FEM
1750
Fm
(kN)
1,453
1,459
0.41
E
(kJ)
850.6
853.6
0.35
Experiment
FEM
1750
1500
1250
1250
Force(kN)
1500
1000
750
1000
750
500
500
250
250
0
0
0
100
200
300
0
400
100
200
300
400
500
Displacement(mm)
Displacement(mm)
Fig. 18 Comparison of FEM to experiment
(v=5.63m/s)
Fig. 16 Comparison of FEM to experiment (quasistatic)
2000
2000
Experiment
FEM
1750
Experiment
FEM
1750
1500
1500
1250
1250
Force(kN)
Force(kN)
E
(kJ)
618.8
619.0
0.03
9.95
(0.0204)
2000
2000
Force(kN)
5.69
(0.0267)
1000
750
1000
750
500
500
250
250
0
0
0
50
100
150
200
250
300
0
350
200
300
400
Displacement(mm)
Displacement(mm)
Fig. 19 Comparison of FEM to experiment
(v=5.69m/s)
Fig. 17 Comparison of FEM to experiment
(v=4.49m/s)
ISBN: 978-1-61804-076-3
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500
Recent Researches in Applied Mathematics and Economics
2000
Experiment
FEM
1750
References:
[1] M. Shakeri, S. Salehghaffari and R. Mirzaeifar,
Expansion of circular tubes by rigid tubes as
impact energy absorbers: Experimental and
theoretical investigation, International Journal
of Crashworthiness, Vol.12, 2007, pp.493-501.
[2] Vojkan Lucanin, Jovan Tanaskovic, Dragan
Milkovic and Snezana Golubovic, Experimental
Research of the Tube Absorbers of Kinetic
Energy During Collision, FEM Transactions,
Vol.35, 2007, pp.201-204.
[3] A. Karrech and A. Seibi, Analytical model for
the expansion of tubes under tension, Journal of
Materials Processing Technology, Vol.210,
2010, pp.356-362.
[4] Jialing Yang, Min Luo, Yunlong Hua and
Guoxiing Lu, Energy Absorption of Expansion
Tubes using a Conical-Cylindrical Die:
Experiments and Numerical Simulation,
International Journal of Mechanical Sciences,
Vol. 52, 2010, pp.716-725.
[5] W. M. Choi, J. S. Kim, H. S. Jung and T. S.
Kwon, Effect of Punch Angle on Energy
Absorbing Characteristics of Tube-Type Crash
Element:, International Journal of Automotive
Technology, Vol. 12, No. 3, 2011, pp. 383-389.
[6] Won Mok Choi, Tae Su Kwon, Hyun Sung Jung
and Jin Sung Kim, Study on Rupture of Tube
Type Crash Energy Absorber using Finite
Element Method, World Academy of Science,
Engineering and Technology, Vol. 76, 2011,
pp.537-542.
[7] ABAQUS Verson 6.6, User's Manual, Hibbitt
Karlsson and Sorensen Inc.
[8] S. Ozaki and K. Hashguchi, Numerical analysis
of stick-slip instability by a rate-dependent
elastoplastic formulation for friction, Tribology
International, Vol. 43, 2010, pp.2120-2133.
[9] Fuping Yuan, Nai-Shang Liou and Vikas
Prakash, High-speed frictional slip at metal-onmetal interfaces, International Journal of
Plasticity, Vol. 25, 2009, pp.612-634.
[10]
Oden. J. T and J. A. C. Martins, Models and
Computational Methods for Dynamic Friction
Phenomena, Computer Methods in Applied
Mechanics and Engineering, Vol. 52, 1985,
pp.527-634.
[11]
Zhou, J. M, Qi, L. H and Chen, G. D, New
inverse method for identification of constitutive
parameters, Transactions of Nonferrous
Materials Society of China, Vol. 16, 2006,
pp.148-152.
1500
Force(kN)
1250
1000
750
500
250
0
0
100
200
300
400
500
600
Displacement(mm)
Fig. 20 Comparison of FEM to experiment
(v=9.95m/s)
coefficient, the friction coefficient at each impact
velocity was calculated using the finite element
method and an inverse method based on the least
square method. The results are summarized as
follows:
- When the impact velocity increased, the force
increased, and accordingly the absorbed energy
calculated at the same crash distance also increased.
This phenomenon can be explained by the following
reason: an increase in the impact velocity leads to a
rising strain rate in the circumferential direction of
the tube, which results in an increase of yield stress.
- Finite element analyses were conducted by
increasing the number of element layers in the
direction of the tube’s thickness at the same friction
coefficient. As a result, the force increased when the
number of element layers increased. However, when
the number of layers exceeded six, the force did not
increase any further. Hence, in order to conduct
finite element analysis of the tube, the number of
element layers in the thickness direction should be
at least six.
- As it is impossible to directly measure the friction
coefficient between the punch and the tube, the
friction coefficients at each velocity were calculated
with an inverse method using the least square
method. Additionally, as a result of comparing the
mean force and the absorbed energy with the
experimental results, the error was found to be
below 3%, thereby indicating that the inverse
method using finite element analysis and the least
square method is appropriate for the calculation of
the friction coefficient for an expansion tube.
- Based on the results of a comparison of the friction
coefficients at each velocity, it can be seen that
when the velocity increases, the friction coefficient
decreases exponentially.
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