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Milahin N., Li Q. / Физическая мезомеханика 18 4 (2015) 38-41
УДК 531.43
Трение и износ сферического индентора под действием нормальных к плоскости контакта ультразвуковых колебаний
N. МПаЫп, Q. Li
Берлинский технический университет, Берлин, 10623, Германия
В статье экспериментально и теоретически исследованы трение и износ сферического индентора. Проведены испытания по схеме «палец - диск», в которых на скользящий индентор воздействовали колебаниями в нормальном к плоскости скольжения направлении. Колебания приводят к снижению коэффициента трения как функции скорости скольжения и амплитуды колебаний. В процессе скольжения площадь контакта индентора с поверхностью увеличивается вследствие износа материала. После каждого периода скольжения измеряли радиус изнашиваемого наконечника. Показано, что радиус площадки износа индентора растет с увеличением пути трения согласно степенному закону с показателем степени 1/4 и не зависит от скорости скольжения. Износ также практически не зависит от наличия колебания. Проведенный теоретический анализ и численное моделирование на основе метода редукции размерности хорошо согласуются с экспериментальными данными.
Ключевые слова: коэффициент трения скольжения, износ, ультразвуковые колебания, метод редукции размерности
Friction and wear of a spherical indenter under influence of out-of-plane ultrasonic oscillations
N. Milahin and Q. Li
Berlin University of Technology, Berlin, 10623, Germany
This paper presents an experimental and theoretical investigation of friction and wear of a spherical indenter. With the pin-on-disc-tribometer the out-of-plane oscillations are applied to the sliding indenter. Oscillations lead to a decrease of the coefficient of friction, and this effect is also related to the sliding velocity and oscillation amplitude. During the sliding movement, the contact area of indenter increases due to the wear of material. This radius of the worn spherical cap is measured after each sliding period. It is found that the radius of the wear flat increases with sliding distance according to a power law with the power 1/4 and is independent of the sliding velocity. It further is practically insensitive to the presence of oscillations. A theoretical analysis and a numerical simulation based on the method of dimensionality reduction are carried out, both describing the experimental data very well.
Keywords: coefficient of sliding friction, wear, ultrasonic oscillation, method of dimensionality reduction
1. Introduction
Dry friction is always accompanied by wear, which plays an important role in practically all tribological application as e.g. engine components, gears, cams and composite materials [1-3]. Many new coating techniques and advanced materials have been developed to reduce the energy loss caused by friction and wear, such as use of nanotechnology [4, 5]. Another widely used way for reduction of friction is application of ultrasonic vibration, for example in metal forming [6]. The pioneer work on the effect of vibration on friction was the experimental study of Godfrey [7] and Lenkiewicz [8] who found that the sliding friction was re-
duced by vibration. Later the effect of load, sliding velocity, frequency and vibration directions including along and perpendicular to the sliding direction were intensively studied and theoretical models were also developed for interpretation of experimental findings [9, 10]. However, the wear in oscillating contacts has not been studied so far. In this paper we shortly present an experimental investigation on the influence of ultrasonic oscillation on the coefficient of sliding friction, although it has been studied in [9] and [10], then we focus on the wear of the indenter and provide interpretation of experimental findings through a theoretical analysis and numerical simulation using the method of dimensionality reduction.
© Milahin N., Li Q., 2015
Milahin N., Li Q. / &u3unecKan Me30MexaHum 18 4 (2015) 38-41
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2. Experimental measurement and results
The experiment was carried out on the ultrasonic pin-on-disk tribometer [10]. The specimen is made of a hard steel ball (much harder than the other contact body-rotating disc) and embedded into a steel element with build-in piezo elements whose natural frequency is about Q = = 30 kHz. A sketch of sliding contact is shown in Fig. 1. Under a normal load Fn the specimen is pressed against a rotating disc. During the sliding movement it is forced to perform oscillations perpendicular to the sliding direction (so-called out-of-plane-mode). The variable amplitude of the oscillations is measured by the high-precision laser vibrometer, and frictional and normal forces are measured by a force sensor.
The influence of sliding velocity, normal load, and amplitude of oscillation on the coefficient of sliding friction for this contact configuration has been presented in paper [9, 10], therefore, here we show only one example with the following parameters: normal load Fn = 4N, oscillation amplitudes Auz = 0.13 and 0.27 ^m and 16 sliding velocities v ranging from 0.001 to 0.009 m/s.
In Fig. 2, it is clearly seen that the sliding friction in the contact without oscillation effect is almost constant and is only slightly dependent of sliding velocity. When the ultrasonic oscillation is applied, the friction becomes essentially smaller. For larger oscillation amplitude, the reduction of friction is more pronounced. Focusing on a curve for certain value of oscillation amplitude, it is found that the sliding friction is also related to the sliding velocity: it increases generally with velocity.
Similar dependences as the one shown in Fig. 2 can be also found in paper [9, 10]. In this paper we focus on the wear of the indenter. The results presented below have been obtained as follows: We started each series of experiments with a new specimen, which mean the steel ball before experiment was completely round. Each sliding case with a certain constant velocity lasts one minute, after that the radius of worn cap of steel ball was measured with a digital microscope. This measurement was conducted for both conditions without oscillation and with oscillations (amplitude Auz = 0.13 ^m).
The measured contact radii as function of sliding distance are shown in Fig. 3. For a visual impression, four
Fig. 2. Dependence of the coefficient of sliding friction on the sliding velocity and amplitude of oscillation
photos in the case of "with oscillation" are selected and also presented. It can be seen that the radius of the worn part increases with the sliding distance, and the oscillation has a slight influence on the wear. The dependence in Fig. 3 can be approximated by a power function
a = cxa, (1)
where a is the contact radius, x is the sliding distance. The constant c and the power a providing the best fitting are equal to c = 155 and a = 0.247 for the case without oscillations and c = 150.8 and a = 0.233 in the presence of oscillations. In the following we give a theoretical analysis and numerical simulation of the wear process.
3. Theoretical analysis and numerical simulation
3.1. Theoretical analysis
The classic Reye-Archard-Khrushchov law of wear states that the wear volume dV is proportional to the nor-
Fig. 1. Sketch of experiment for sliding contact between a steel ball and a friction disc
Fig. 3. Measurements and approximations of contact radius of indenter after the sliding movement. Four selected photos of spherical indenter are presented. Experiment without oscillation
(A) and with oscillation (•); fitting without oscillation (----)
and with oscillation (-)
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Milahin N., Li Q. / &u3uuecKan Me30MexanuKa 18 4 (2015) 38-41
mal force Fn, the relative tangential displacement dx of the contacting bodies and inversely proportional to the hard-
ness a
0'
dV = kFndx.
(2)
The wear equation was suggested already in 1860 by Reye [11], and was later derived and experimentally justified by Khrushchov for abrasive [12] and by Archard for adhesive wear [13]. For an indenter with spherical profile, we can rewrite this equation as 2 F
na dd = k—dx, (3)
a0
where a is the contact radius of the worn cap with an indentation depth d. Using the geometrical relation a2 = 2 Rd, where R is the radius of the sphere, gives
dd = a da. (4)
R
Substituting Eq. (4) into Eq. (3), leads to the equation
—da = kFndx. (5)
R a0
Integrating Eq. (5), we obtain the dependence of contact radius on the sliding distance
a =
4 kFn R TCGn
1/4
For a constant load, it gives
a = cx1/4
(6)
(7)
with the constant factor c = ((4kFn R)/(na 0))14.
According to Eq. (6), the contact radius of the worn cap for a spherical profile is proportional to the sliding distance to the power 1/4 when the normal load is kept constant. Let us note that the radius a is not dependent on the sliding velocity. With the same parameter as conducted in experiment, we compare this analytical solution with the results from measurement. The bold line as seen in Fig. 4, shows this theoretical dependence of the contact radius on the slid-
Fig. 4. Comparison among experimental measurement, theoretical analysis with Eq. (6) and numerical simulation using the method of dimensionality reduction (MDR)
ing distance, where the typical values of steel indenter hardness and wear coefficient are used, a0 = 2.5 GPa and k = 2.2 -10-4.
3.2. Numerical simulation using the method of dimensionality reduction
The method of dimensionality reduction is a fast numerical tool for solving contact problems of elastic and vis-coelastic materials [14]. It provides exact solutions for contact problems of rotationally symmetric profiles. Recently it was applied to problem of fretting wear [15-18]. It was found that the calculation of wear problem using the method of dimensionality reduction drastically reduces the simulation time compared with conventional method like as finite element simulations. With the rules of the method of dimensionality reduction we simulate the wear of a spherical indenter in a sliding contact for the case without oscillation.
Firstly, a three-dimensional profile z = f (r), should be transformed into an equivalent one-dimensional profile by
I x |'i1 -&L dr. (8)
g ( x)
x
0 V x - r
The back transformation is given by the integral
f (r ) = - J-
g ( x )
rdx.
(9)
n Wr 2 - x-
According to Eq. (8), the one-dimensional form of a parabolic indenter f (r) = r 2/(2 R) is expressed as g (x) = x 2/R . Under a normal load Fn the profile (8) is pressed into an elastic foundation consisting of independent springs with discrete spacing Ax whose normal stiffness is given by
kz = E * Ax, (10)
where E * is the effective elastic modulus
J_
E
1 -v
i +i-Vi
(11)
M 2
E1 and E2 are the Young's moduli of contacting bodies, v1 and v 2 are their Poisson ratios.
The relation between the resulting vertical displacements of springs and indentation depth d is calculated by
uz (x) = d - g(x) (12)
and the normal force can be written in an integral form if the spacing is enough small
Fn = J kzuz (x)dx = 2 J E*[d - g(x)]dx.
(13)
With a given normal force Fn, the indentation depth d is determined by Eq. (13), then the contact radius a is calculated by the condition
g(a) = d. (14)
From Eq. (13), the linear force density is calculated as
* *
q(x) = E uz (x) = E (d - g(x)). (15)
According to the method of dimensionality reduction, the distribution of normal pressurep in the initial three-dimen-
Milahin N., Li Q. / Физическая мезомеханика 18 4 (2015) 38-41
41
sional problem can be calculated using the following integral transformation
П rjx2 -
= dx = t. J^XL dx.
2 п rVX^7
(16)
-r r y/x - r
According to the Reye-Archard-Khrushchov wear law (2), the wear rate of the three-dimensional profile will be df (r) kp(r)
du
(17)
where u is tangential displacement of the indenter. By determining an incremental change of the profile z = f (r) according to equation
df (r ) = » du,
(18)
a new profile f(r) is obtained after a tangential displacement du. Then we go back to the Eq. (8) and calculate the new one-dimensional profile. Repeating these iterative procedures, the worn profile of the indenter can be obtained, including the contact radius changing with tangential displacement.
In the simulation the same parameters are used as described in the experiment. The dependence of contact radius on the tangential displacement is shown in Fig. 4 as indicated with stars. If we approximate it with a power function a ~ ua, the exponent a is equal to a = 0.2526, which is very close to the value obtained in the simple theoretical estimation.
4. Conclusion
We measured the sliding friction and wear of a spherical indenter under the influence of out-of-plane ultrasonic oscillations. As in paper [9, 10] described, the friction is reduced by the oscillation, and this reduction effect is related to the oscillation amplitude and sliding velocity. The measurement of radius of the worn spherical cap shows that the contact radius increases with the sliding distance, however the influence of oscillation is quite small. A theoretical analysis of this wear problem and simple numerical simulation using the method of dimensionality reduction show a good agreement with the experimental results. It is found that the contact radius of the worn cap of spherical indenter is approximately proportional to the sliding distance to the power 1/4. The future work on this wear problem can be carried out for different profiles of indenter, normal loads and oscillation amplitudes, and a further model in the frame
of the method of dimensionality reduction including oscillation and consideration of partial slip will be developed.
5. Acknowledgement
The authors acknowledge many valuable discussions with V.L. Popov. This work was supported by Deutsche Forschungsgemeinschaft (DFG).
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Поступила в редакцию 26.05.2015 г.
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Сведения об авторах
Natalie Milahin, Dipl.-Ing., Berlin University of Technology, Germany, natalie.sabelfeld@mailbox.tu-berlin.de Qiang Li, MSc., Dr.-Ing., Researcher, Berlin University of Technology, Germany, qiang.li@tu-berlin.de
