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Li Q., Popov V.L. / Физическая мезомеханика 20 5 (2017) 91-95
91
УДК 539.612, 544.722.54
On the possibility of frictional damping with reduced wear: A note on the applicability of Archard's law of adhesive wear under conditions of fretting
Q. Li1, V.L. Popov123
1 Technische Universität Berlin, Berlin, 10623, Germany 2 National Research Tomsk Polytechnic University, Tomsk, 634050, Russia 3 National Research Tomsk State University, Tomsk, 634050, Russia
Based on the recent proof of the existence of a critical length scale controlling mechanisms of adhesive wear, we discuss conditions under which the energy dissipation rate and the wear rate in oscillating contacts may become uncoupled. This potentially opens up a possibility of using frictional damping with low wear rate.
Keywords: wear, fretting, frictional damping, Archard's law
Возможность фрикционного затухания c низким уровнем износа: о применимости закона Арчарда адгезионного износа в условиях фреттинга
Q. Li1, V.L. Popov1-2-3
1 Берлинский технический университет, Берлин, 10623, Германия
2 Национальный исследовательский Томский политехнический университет, Томск, 634050, Россия
3 Национальный исследовательский Томский государственный университет, Томск, 634050, Россия
На основе недавно полученного доказательства существования критической длины, определяющей механизм адгезивного износа, мы обсуждаем условия, при которых интенсивность диссипации энергии и интенсивность износа в контактах, подверженных вибрационному воздействию, могут управляться независимо. Это потенциально открывает возможность реализации фрикционного затухания c низким уровнем износа.
Ключевые слова: износ, фреттинг, фрикционное затухание, закон Арчарда
1. Introduction
It is well-known that joints significantly contribute to structural damping [1, 2]. This plays a major role in structural mechanics [3, 4], tribology [3] and materials science [4, 5]. This damping is usually attributed to relative movement in the contact (microslip). This was first analyzed by Cattaneo [6] and later independently Mindlin [7]. Mindlin et al. also obtained the energy dissipation due to the microslip in a contact under oscillating tangential loading [8]. More recently, it was also noted that superposition of tangential and normal oscillations can significantly influence frictional damping [9, 10]. In the most pure form, this has been shown in [11], where the limiting case of infinite co-
efficient of friction was considered. In this limiting case, Mindlin's dissipation vanishes, but the dissipation due to superimposed tangential and normal oscillations remains finite and depends only on the elastic properties of the contact. The authors of this paper called this effect "relaxation damping".
The interesting feature of relaxation damping is that there is no slip in the system at any time. Thus, the question may arise whether this kind of damping is connected with any wear, which, according to Archard's law of wear [12], requires relative slip of the contacting surfaces. However, a more detailed analysis of the process of energy dissipation in a system with high but finite coefficient of friction
© Li Q., Popov V.L., 2017
[13] shows that the dissipation is still due to the relative sliding in a very narrow slip zone at the boundary of the contact area. Assuming the validity of Coulomb's law of friction and Archard's law of wear in the local form (i.e. independently on the space scale), one can easily see that the energy dissipation rate and the wear rate are strongly coupled. Indeed, the amount of locally dissipated energy is proportional to the local tangential force multiplied with the relative displacement of the contacting surfaces, while the amount of wear is proportional to the local normal force multiplied with the same relative displacement. Because the tangential force during slip is equal to the normal force multiplied with the coefficient of friction, both local and global energy and material dissipation rates will be, for the given materials, rigorously proportional to each other, independently of the particular loading history. Of course, the relative magnitudes of the coefficient of friction and wear coefficient can be vastly different for different materials.
On the other hand, there exists experimental evidence that wear and frictional dissipation can become uncoupled
[14] under superimposed tangential and normal oscillations, which suggests the violation of either Coulomb's or Archard's laws. In the present note we inspect the conditions of validity of Archard's law of wear and, consequently, the conditions under which it is violated. We restrict ourselves to a very general analysis of characteristic parameters governing the process of wear. Final conclusions can only be drawn based on extensive numerical analysis and experimental investigations.
2. Characteristic length governing adhesive wear and conditions for the applicability of Archard's law
One of the main mechanisms of wear—adhesive wear— occurs in sliding contacts between materials of comparable hardness. It is governed essentially by the surface energy of the frictional partners (or more exactly by the work of adhesion) [15]. Based on an energy balance analysis very similar to that of Griffith [16], Rabinowicz [17, 18] calculated conditions for the detachment of wear particles. One of the main findings was the existence of a characteristic length:
, 30EAy
(1)
where E is the Young modulus, Ay is the work of adhesion and ct0 the yield strength of the material. If the size of an asperity is larger than this critical value, the elastic energy stored in this "potential wear particle" is sufficient for detachment. Otherwise the asperities are only plastically deformed and wear occurs by removal of single atoms as suggested by Holm [19]. However, the model of Rabinowicz was only a very rough estimation that left many open questions. Only recently, Aghababaei et al. [20] managed to prove by large-scale numerical simulations that the length
scale introduced by Rabinowitz [17] is indeed the main governing parameter determining the character of adhesive wear: If the size of the asperities is less than the critical value, roughness is plastically smoothed as predicted in a later paper of Rabinowicz [21]. Otherwise, adhesive particles are produced and the surface is roughened. For over-critical size of asperities, the overall amount of wear is much higher than in the case of undercritical size.
If we follow closely the standard derivation of Archard's law [22] (see also [4]), we see that the validity of the derivation is conditional on the assumption that the size of microcontacts is overcritical, so that detachment of particles can take place. If this characteristic length becomes smaller than the critical length introduced by Rabinowicz, we can expect the breakdown of Archard's law of adhesive wear. The wear rate would then be reduced drastically.
In the case of fretting wear, there are several characteristic length scales which depend on the amplitude of tangential oscillation, e.g. the width of the slip zone of the contact. One therefore can generally await that decreasing of the amplitude of oscillation will finally lead the characteristic contact length to falling below the critical value (1) and thus changing the regime of wear from strong adhesive wear to the "least wear" [21]. Experiments confirm the existence of critical oscillation amplitude under which the fretting wear practically vanishes [23]. However, there are not systematic investigations of the physical nature of this threshold, so that it is not clear whether it is related to the critical length of Rabinowicz or not.
3. Width of the slip area by bimodal loading
There exists no experimental work on wear under the conditions of bimodal loading (while exactly this kind of loading is of interest for structural damping.) Let us assume that the width of the sliding zone is essential for the determining character of the adhesive wear. Then it is important to determine the parameters governing this width. To estimate the width of the slip zone let us consider the following experiment: A conical indenter z = f(r) = rtan6 (r is the polar radius in the contact plane) is first indented to the depth d0, then moved tangentially by uX' and finally raised. The simplest way to calculate the radii of the contact and of the slip region is to consider this problem in the "MDR space" [24, 25]. In the MDR, the original three-dimensional profile must be replaced by the MDR-trans-formed profile
g(x) = |x|f f}r\ dr = n | x| tan6. (2)
0V x2 - r 2 The contact radius after the initial indentation is given by equation
g(a0)= d0. (3) If the bodies are displaced tangentially by then the central part of the contact with radius c0 remains in stick, while the outer region between c0 and a0 slips. The ra-
dius of the stick zone, c0, is determined by the equations [24]
GV^ = M-E*(do - g(co)) = ^E* (g(ao)-g(ft,)) = = n2 ^E*(flo -Co), (4)
where |x is the coefficient of friction, and the effective moduli in (4) are defined according to [24]:
1 1 -v,2 1 -v22 1 2-v, 2-v2
—r =-— +- , —»■ =-1 +--• (5)
E E E2 G 4Gj 4G2
If now the indenter is raised, than the condition (4) will take the form
G*uf ) =nl2 ViE*(a -c), (6)
where a and c are current values of contact radius and the radius of the stick zone. The width of the slip zone, (a - c), thus remains constant during the raising phase of a conical indenter.
Summarizing this result, one can state that the width of the slip zone depends only on the amplitude of the hori-
zontal oscillation but not on the character of the vertical displacement. This means that by keeping the horizontal amplitude below the critical value, it might be possible to achieve relative large damping by introducing large normal oscillation amplitude.
4. Generalized Rabinowicz length
We used so far two different views of the process on two different scales. While discussing the stick and slip, we consider the contact as Cattaneo-Mindlin problem between smooth surfaces, while when discussing the wear we consider the contact as consisting of separate "asperities". To combine both views and thus to get a deeper insight into the possible mechanism of adhesive wear, we conducted the following numerical simulation.
A rough sphere with roughness having the Hurst exponent 0.7 was generated according to the rules described in [26] (Fig. 1, a). The generated indenter was pressed into
Z 0.0
-0.2
-0.4
-0.6
V
-1 -1
Tangential stress t/t(
0
Fig. 1. Simulation of tangential contact of an elastic half-space and a rough sphere: the surface shape of the rough indenter (a); contact area as some given indentation depth. Black color shows the stick regions and gray color the slip regions (b); tangential stress distribution in the contact (c); tangential displacement of elastic half-space. Black areas show a rigid-body translation and gray areas the slip regions (d )
the elastic half-space and then moved tangentially by a u^0 much smaller than the displacement corresponding to the complete sliding [27]. Both the normal indentation and tangential loading were simulated using the boundary element method as described in [28]. Other than in [28], we assumed that any two points of bodies are in the stick state as long as the local stress is smaller than a fixed critical value Tc. After beginning of slip, the stress remained constant and equal Tc thus mimicking a sort of elastic-ideally plastic behaviour in the contact interface. In this way we tried to follow the concept of friction introduced by Bowden and Tabor [29] with the exception that the normal contact is considered as being purely elastic. The tangential contact comes into plastic state by overcoming by the tangential stress the critical value Tc which in this context plays the role of the "yield stress" used by Rabinowicz in Eq. (1).
Considering the plots in Fig. 1, we recognize readily the overall Cattaneo-Mindlin picture saying us that the bodies are in the stick state in the inner parts of the contact and in the sliding state in the outer parts. Note that contrary to the macroscopic Cattaneo-Mindlin picture, where tangential stresses in the slip zone are very small, here they achieve its maximum value exactly in the outer slip zone. At the same time, the relative displacement does exist only in this zone (Fig. 1, d). This makes clear why the wear is practically completely concentrated in the slip zone.
Another important observation is that it is very difficult to define what an "asperity" is and what the size of asperity is. Instead of separated asperities, we see more or less continuous clusters of contact areas. It is interesting to understand how in this case the criterion of Rabinowicz (1), has to be applied. We would like to suggest a "modified Rabi-nowicz criterion" which is largely independent on the definition of an asperity.
First, we assume that some tangential displacement is required for generating wear. Thus, we confine ourselves with consideration of the slip area. Let as consider in the slip area any region with characteristic size of D in any direction, e.g. a circular region with diameter D. The macroscopic tangential stress in this area is equal to £tc, where £ is the "filling factor" defining the part of the real contact area in this circle to the area of the circle ~D2. The elastic energy which would be released if a wear particle with the characteristic volume ~D3 would detach, has the order of magnitude (£tc)2/(2G) D3. If it is not enough for creating the free surface of the order of 4D2, the detaching cannot go on. Thus the criterion for the possibility of detaching awear particle with the size D is (£tc)2/(2G) D3 > > 4D2Ay or 8GAy
D>
(7)
(^c)2'
Note that in our consideration there is no need to define what an asperity is. However, the modified criterion de-
pends explicitly on the filling factor £ which clearly becomes smaller when approaching the boundary of the contact.
Finally, let us estimate the order of magnitude of the energy dissipation in a contact oscillating with an amplitude of uf ~ D-6 m, tan9 = 0.01, and G* ~ 1011 Pa. Using equation
32 G*
W = -
3n tan 6
uf2|uf|sm2 %
(8)
for the energy dissipated in one cycle [11], we obtain W~ ~3 x 10-5 J/cycle. This energy dissipation is still in the range providing effective damping—at least for acoustic purposes.
5. Conclusion
We have analyzed the characteristic length appearing in the problem of oscillating frictional contact, which may be relevant to the process of adhesive wear. One of the candidates for such length is the width of the slip zone. The order of magnitude of this length is determined solely by the amplitude of tangential oscillations, while the energy dissipation is determined by both amplitudes. Thus, if the tangential amplitude is kept in the "wearless" range, the energy dissipation can still be (independently) controlled through the amplitude of normal oscillations. This may open the possibility for designing fhctional damping systems with reduced wear. Another important aspect of the interrelation of frictional losses and wear is that the dissipation of energy can be due to radiation losses due to sudden release of the stuck region as suggested in [11] and discussed in detail in [30]. It would be very interesting to investigate this interrelation experimentally.
The authors thank M. Popov for a very intensive and controversial discussion of the paper and the Deutsche Forschungsgemeinschaft for financial support.
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Поступила в редакцию 04.06.2017 г.
Сведения об авторах
Qiang Li, Dr.-Ing., Technische Universität Berlin, Germany, qiang.li@tu-berlin.de
Valentin L. Popov, Prof., Technische Universität Berlin; Prof., TPU; Prof., TSU, v.popov@tu-berlin.de
