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УДК 621.891, 532.135
Influence of stress dependence of lubricant shear modulus on self-similar behavior of stress time series
N.N. Manko1, I.A. Lyashenko12
1 Sumy State University, Sumy, 40007, Ukraine 2 Technische Universität Berlin, Berlin, 10623, Germany
Here we consider melting of an ultrathin lubricant layer between two atomically smooth solid surfaces taking into account the stress dependence of the lubricant shear modulus and its decrease with increasing stress (strain). In the adiabatic approximation with the stress relaxation time far longer than strain and temperature relaxation times, a Langevin equation is written and its respective Fokker-Planck equation is derived using the Stratonovich calculus. Phase diagrams for the steady case are presented illustrating the effect of the system parameters on the lubricant behavior. A joint numerical and analytical analysis demonstrates a very close match between probability distributions at different parameters. It is shown that in a limited stress range, a self-similar mode of dry friction is established showing up in self-similar behavior of stress time series.
Keywords: Lorentz system, shear stress and strain, boundary friction, uniform distribution, Langevin and Fokker-Planck equations
Влияние деформационного дефекта модуля сдвига граничной смазки на характер самоподобного поведения временных рядов напряжений
H.H. Манько1, Я.А. Ляшенко12
1 Сумский государственный университет, Сумы, 40007, Украина 2 Берлинский технический университет, Берлин, 10623, Германия
В работе исследуется плавление ультратонкого смазочного слоя, зажатого между двумя твердыми атомарно-гладкими поверхностями, при их взаимном перемещении, с учетом влияния деформационного дефекта модуля сдвига смазочного материала, что позволяет описать уменьшение величины модуля сдвига с ростом напряжений (деформаций). В рамках адиабатического приближения, в котором время релаксации напряжений намного превышает времена релаксации деформации и температуры, записано уравнение Ланжевена, для которого с использованием исчисления Стратоновича найдено соответствующее уравнение Фоккера-Планка. В стационарном случае построены фазовые диаграммы, иллюстрирующие влияние параметров системы на характер поведения смазочного материала. При численном и аналитическом анализе кривые распределения вероятности при различных параметрах совпадают с высокой точностью. Показано, что в ограниченном диапазоне напряжений устанавливается самоподобный режим поведения твердоподобной смазки, который выражается в самоподобном характере поведения временных рядов напряжений.
Ключевые слова: система Лоренца, сдвиговые напряжения и деформации, граничное трение, однородное распределение, уравнения Ланжевена и Фоккера-Планка
1. Introduction
Boundary friction in nanosized tribological systems remains an urgent research problem. Such systems are used in microelectronics, aerospace industry, computer technologies, advanced power engineering, etc. Research in boundary friction is also significant for biomedical applications, e.g., for operation of natural and artificial joints. In particular, stick-slip friction of articular joints [1] can result in
their inflammation and early wear. One of the intriguing modes is friction of atomically smooth solid surfaces separated by an ultrathin lubricant layer. Such systems hold promise for practical application and are of interest as displaying properties distinct from those of bulk lubricants. Moreover, this mode occurs in most of the friction units when contact surface asperities are engaged in sliding. By now, many theoretical and experimental papers are avail-
© Manko N.N., Lyashenko I.A., 2017
able on the behavior of an ultrathin lubricant layer between atomically smooth solid surfaces [2-7]. As has been shown, ultrathin lubricants differ from bulk ones in melting and solidification temperatures, have nonmonotonic dependences of friction force on velocity, provide stick-slip, melt during surface motion, reveal memory effects, etc. [6, 8]. One of the approaches to the description of boundary friction [9-11] is based on the Landau theory of second-order phase transitions. The approach assumes that as the elastic stress or the temperature of a lubricant in a tribological system exceeds a critical value, the lubricant melts and provides liquid friction in the system. In particular, a study based on this approach [12] demonstrates melting of a lubricant from the boundary of an axially symmetric contact toward its center. Another much used approach to such microscopic systems is based on molecular and dynamic models [13, 14]. However, because of insufficient computer speeds, phenomenological methods and theories still remain very valuable. A large role in studies of contact phenomena for medical and biological applications belong to contact interaction mechanics [15], which allows one to describe a wide range of properties of contact materials, including their friction, surface roughness, adhesion, gradient states like those of articular joints, and many others.
Among the models that analyze the specific behavior of an ultrathin lubricant between two mutually displaced solids is a synergetic boundary friction model resting on a system of three differential equations for stress, strain, and temperature and allowing research in various boundary effects [5, 16]. For example, the model suggests that periodic and chaotic stick-slip is realizable in deterministic cases [17]. A review of numerous related results, including hysteresis phenomena, decrease in shear modulus with increasing stress, additive fluctuations, self-similar behavior, etc. can be found elsewhere [5]. A recent study of the spacetime evolution of stress, strain, and temperature in the plane of a lubricant layer [18] shows that these parameters tend to steady values which correspond to a homogeneous lubricant state; that is, the lubricant in its plane is assumed inhomogeneous but stochastic effects are ignored [18]. Such a situation is valid for thin lubricants which can be taken homogenous throughout the thickness with due regard to fluctuations of the above parameters. It is also valid, e.g., for monomolecular lubricants in which such fluctuations are critical because the number of lubricant particles (molecules or atoms) is limited making useless the use of average values. Note that nanosized systems are largely affected by thermal noise fluctuations, which are present in all friction experiments. In this context, it is worth investigating the influence of noise and accidental impurities on the behavior of lubricants during boundary friction [19]. Previously, we studied the effect of stress-induced variations in shear modulus on kinetic boundary friction modes with additive fluctuations of stress, strain, and temperature [20].
Here we continue the research and consider a specific case in which the time series of stress display self-similar properties. We also provide a joint analysis covering the influence of stress-induced variations in shear modulus on hysteresis and of fluctuations of stress, strain, and temperature on kinetic friction modes, which extends our model to a wider range of tribological systems.
2. Basic equations and phase diagrams
Previously [5, 16], using a rheological model of heat-conducting viscoelastic media, we derived a system of three first-order differential equations for shear stress a, strain e, and temperature T in an ultrathin lubricant layer between atomically smooth solid surfaces. The equations are dimen-sionless, and the units of a, e, and T are defined as
as =
iPcvnoTc ^
G0
(1)
where p is the lubricant density, cv is the specific heat capacity, Tc is the critical temperature, n0 =n(T = 2Tc) is the characteristic shear viscosity, tt = ph2 cv/k is the temperature relaxation time, h is the lubricant thickness, k is the heat conductivity, Te is the strain relaxation time,
G0 =no/Te •
The system of basic equations is written as [5, 16]: xca = -a + g (CT)e + 7£5i(t), (2)
xee = -e + (T -1)a+VU2(t), (3)
TtT = (Te - T) -ae + a2 +7^3 (t), (4)
where Te is the friction surface temperature, Ta is the stress relaxation time. Equations (2)-(4) take into account the fluctuations of three parameters through 8 being correlated stochastic Gaussian sources (t) with intensities Ia, Ie, and IT [5, 21]. The moments of the functions (t) are determined as
<iv(t)> = 0, <iv(t)lj(t')> = 28j8(t -1'). (5)
The effect of noise in Eqs. (2)-(4) goes beyond thermal fluctuations because higher intensities are further selected, which does not fit the fluctuation-dissipation theorem. In the case considered, the noise represents some imperfection of experimental equipment and possible irregularities of the lubricant and friction surfaces. In addition, there can be noise from external sources [22]. However, we consider the case where the intensity of thermal noise is many times greater than the intensities of the other two fluctuation sources (see expression (26)). It should be noted that this condition can be provided as well by purely thermal fluctuations at Ia = Ie = 0. In Eq. (2), g(a) = G(a)/G0 is introduced [5, 20], with the lubricant shear modulus G(a) being a function of dimensional stress:
G(a) = ®+,.G-e|P- (6)
l+ja/.
aP
The function g(a) in dimensionless form is written as
g (a) = ;
1 + -
0-1 -1
(7)
1+|a/a|'
where we have dimensionless constants g0 =0/G0 < 1, 0 = 0/G <1, and a = /p/as. For a = 0 with no noise, Eq. (2) gives the dependence
a = g(a)8 (8)
representing a loading curve [23] whose form is determined by steady friction. The dependence /(8) in (8) have two portions. First comes Hooke's portion whose slope is specified by the dimensionless shear modulus G/G0 = g0/0. When the stress a exceeds its critical value, the loading curve reveals a flatten portion (let term it a plastic portion1) whose slope is specified by the dimensionless hardening coefficient 0/G0 = g0 < g0/0. Relations (6) and (7) contain phenomenological constants a (ap in dimensional form) and p. The parameter a specifies the critical stress at which Hooke's portion gives way to the plastic portion, and the parameter P specifies the general form of the loading curve /(8) [5, 22-24]. For our further analysis, we also need the derivative of (7):
g, . M/) = -g 0pl/lP (0-1 -1>. (9)
da /ap(1+|/a|p)
In (6)-(9), | a/a | means that the stress a is taken in absolute value because a > 0. For even values of P, this condition is met automatically. Figure 1, a shows diagrams of g(a> and g'(a) at fixed model parameters. It is seen that the function g(a> identically describes the singularity in the positive and negative stress ranges, which is a key point because the stress is proportional to the shear velocity and its sign points to one of two possible equivalent motion directions of the upper friction surface. The derivative g'(a) corresponds to g(a> and is equal to zero at | a|<< 1 and |a|>> 1, where g(a> = const. This is because the function g(a> at low stress describes Hooke's portion of loading curve (8) with a constant slope g0/0 (see Fig. 1, b), and as the stress is increased, the curve becomes flatter and its characteristic slope is g0.
For our further analysis, we use a method described in detail elsewhere [21]. In the adiabatic approximation
Ta >> t8 , tt , we can put T8è ~ 0,
ttT
<0 in Eqs. (3), (4).
Then, (2) takes the form of the Langevin equation (stepwise transform can be found elsewhere [5, 21]):
T// = f (a) + 4l(a>^(t >, (10)
1 Althouhg we use the terms of elastic and plastic deformation, the actual situation is for a lubricant with pseudoelastic properties such that its quasi-static evolution is described by a = g(a)8 both under loading and unloading, i.e., no hysteresis occurs under unloading [23]. The same refers to the dynamic case in which the stress and strain are specified by Eqs. (2)-(4).
where the generalized force f (a) and the effective noise intensity I(a) are specified by the expressions [5]:
2 - T
f (a) = -a+ag (a)
1
1 + a2
I (a) = I a +
g (a) (1 + a2)2
( Ie+ It a)•
(11)
(12)
In the general case, many forms of the Fokker-Planck equation can correspond to (10). Here we use the Stratonovich calculus due to the possibility to automatically account for correlations on small time scales typical of all physical systems. Thus, the Fokker-Planck equation is written as [25]: dP(a, t) d
dt
da
f (a ) P (a, t )
da
4I(â)—4I(â)P (a, t ) da
(13)
With time, the distribution of solutions (10) becomes steady, and its explicit form can be found from (13) at dP (a, t )/dt = 0:
P (a) = Z -1 exp {-U (a)}, (14)
where Z is the normalization constant, and the effective potential U(a) is specified by the expression [5]:
U (a) = 1ln I (a)
a f (a')
"J
da'.
(15)
0 I (a')
The extremum points of distribution (14) (potential (15)) are determined by the condition d^/ da = 0, which leads
Fig. 1. Diagrams of g(a) from (7) as a solid line and gr(a) from (9) as a dash line (a), and loading curve a(8) from (8) at g0 = 0.5, 0 = 0.4, a = 0.2, p = 5 (b)
to the equation dl/da - 2 f = 0. Let us introduce the notation x = 1 + a2 and write the explicit extremum condition [20]:
[1 - g(a)]x3 + g(a)[2 - Te]x2 -g '( a)
■g ( a)
g (a) It
-( Ie + It a2)
x +
+ 2g (a)[It - Ie ] = 0, (16)
where the functions g(a) and gr(a) are defined by equalities (7) and (9). For a = 0, condition (16) gives the relation
Te = 1 + g0-1e + g9e-1(It - 21£ ) + + le gea-ppop-2(e-1 -1), (17)
which specifies the boundary for the existence of a zero stationary solution a0 = 0. In view of (17), the solution a 0 = 0 does not exist at P < 2. At P = 2, expression (17) takes the form
Te = 1 + ge-1e + gee-1(It - 21, ) +
+ 21e ge«-2(e-1 -1), (18)
and at P > 2, we have
Te = 1 + g-1e + gee-1(lT - 2Ie ). (19)
The influence of P on the behavior of the system is depicted in Fig. 2. At P = 1, the diagram reveals liquid friction (SF) and a combination of metastable and stable liquid friction (MSF + SF). The absence of dry friction (DF) is explained by the absence of the zero stationary solution a0 = 0 according to (17). Therefore, distribution function (14) have no zero maximum. In the SF region, the distribution function P(a) has a single nonzero maximum, and the MSF + SF region is characterized by the coexistence of two nonzero maxima of P(a), which fits stick-slip motion with attendant transitions between metastable and stable liquid friction.
The phase diagram at P = 2 (Fig. 2, b) reveals straight line A specified by Eq. (18), which gives the boundary for the existence of a0 = 0. Above this straight line, we have the same modes as in Fig. 2, a; that is, P(a) shows no maximum at a = 0. Below line A, the zero stationary solution a 0 = 0 always exists. In the DF region, the phase diagram reveals one maximum of the distribution function at a 0 = 0. The two-phase region corresponds to stick-slip (SS) with coexistent maxima of P(a) at zero and nonzero stress and to transitions between dry and liquid friction.
At integer values P = 3^10, the phase diagrams show only slight quantitative differences and qualitatively replicate the diagram at P = 3 in Fig. 2, c. In this figure, like in Fig. 2, b, there is straight line A which, however, corresponds to Eq. (19) and is therefore more leftward than line A in Fig. 2, b. Above this straight line, as opposed to P = 2, we have one SF region with one nonzero maximum of P(a). In this case, no MSF + SF region is found. At the same time, below line A, we have three different regions in which
the distribution function P(a) has a zero maximum: a region of dry friction DF, a region of stick-slip SS, and a more complex region of stick-slip and liquid friction SS + SF with three (one zero and two nonzero) maxima of the distribution function, which fits specific stick-slip motion with transitions between stable liquid, metastable liquid, and dry friction modes.
Figures 3, a and b present the steady-state stress a0 versus the noise intensity Ie at different intensities IT, allowing a detailed analysis of the friction modes in Figs. 2, a and b. Dash and solid portions correspond to unstable and stable steady states, respectively. In Fig. 3, a, all curves issue out of the point of origin, showing the absence of DF on the phase diagram in Fig. 2, a. Curve 1 corresponds to IT = 10. Here, at any Ie, a0 > 0 (liquid friction) which corresponds to the SF region on the phase diagram in Fig. 2, a. The value IT = 30 fits curve 2 on which there are two critical noise intensities Ie. Before the
16"
12"
SF MSF + SF ^^
\ ss
DF
20
40
60 IT
16
12"
4-
y/
SF A/ 'SS + SF/'
1 / o // // 2 SS
A, O DF
0
0
20
40
60 IT
Fig. 2. Phase diagrams at ge = 0.5, a = 0.2, e = 0.95, Te = 1.7 with regions SF, DF, SS, MSF + SF, and SS + SF: P = 1 (a), 2 (b), 3(c)
first critical value and after the second one, curve 2 demonstrates the existence of a single nonzero maximum of the distribution function P(a), as is for curve 1, which corresponds to the SF region on the phase diagram. Between the first and second critical values, there are two nonzero maxima (solid portions) separated by a minimum (dash portion) which corresponds to a combination of metastable and stable liquid friction MSF + SF. Curve 3 (IT = 60) has one critical value of Ie before which the distribution function P(a) has two maxima (MSF + SF on the phase diagram in Fig. 2, a). When this critical value is exceeded, P(a) reveals a single maximum (solid line, SF).
In Fig. 3, b, all curves at small values of Ie demonstrate the existence of a zero maximum of P(a) because the dependences, unlike those in Fig. 3, a, issue not out of the point of origin. For curves 1 and 2 at IT equal to 10 and 30, respectively, dry friction is observed up to a certain critical value, which corresponds to DF on the phase diagram in Fig. 2, b. Above this critical value, their behavior is similar that of curves 1 and 2 in Fig. 3, a. Curve 3 fits the value IT = 60 and also has an additional critical value of Ie at a0 = 0 following which its behavior coincides with that in Fig. 3, a. Before this critical value is attained, two (zero and nonzero) maxima of P(a) are observed, corresponding to stick-slip motion. Note that in both figures, curve 3 passes through the negative, nonphysical range of the noise intensity Ie, which provides the presence of a nonzero maximum ofP(a) at zero Ie. If the curve in Fig. 3, b
was only in the positive range of I8, we would first observe dry friction, and only then, stick-slip would occur.
Figure 4 shows probability distribution (14) for the parameters of Fig. 2, c. Curve 1 is for the case of one nonzero maximum of P(a) in Fig. 2, c, which corresponds to liquid friction SF. Curve 2 is for stick-slip and liquid friction SS + SF, i.e., for three maxima of P(a) with one being zero. Curve 3 is for SS with two (zero and nonzero) maxima of P(a) between which transitions occur. Curve 4 is for dry friction DF with one zero maximum of P (a). The dependences in Fig. 4 fully agree with the regions observed on the phase diagrams.
3. Numerical analysis
Let us use analytical representations in parallel with numerical analysis and show that these approaches are equivalent. Multiplication of (10) by dt gives the Langevin differential relation:
Ta da = f (a)dt + VI(0)dW (t), (20)
where dW(t) = W(t + dt) - W(t) is the Wiener process having the properties of white noise [25]:
<dW(t)) = 0, <(dW(t))2 ) = 2dt.
(21)
For numerically solving equation (20), it is sufficient to use the Euler method such that for (20), in view of (21), we have Ito SDE in the form [25]:
Ta d a:
f ( a) + V7(â) -VI(â)
d t +
+ VI(a)dW (t). (22)
Taking into account the discrete analog of random force dW (t) = yf^tWn, we obtain the iterative procedure for the stress time series:
Fig. 3. Steady stress a0 versus noise intensity I8 at IT = 10 (1), 30 (2), and 60 (3) for phase diagrams in Fig. 2, a and b, respectively
Fig. 4. Probability distributions P(a) from (14) for parameters and points at Ia = 1 in Fig. 2, c: I8 = 8, IT = 5 (1); I8 = 10, it = 30 (2); l = 4, IT = 10 (3); A = 4, IT = 20 (4)
Jn+1
= a„ +
f ( a n ) +
g К [ It (1 -a„2) - 2Ie ]
(1 + °2n )3
g ( an ) g e ( 1)ß|a„ |p ( I e+ It К ) an aß (1 + a П )2(1+|a J a|ß )2
At
(23)
The equation is solved on a time interval t e [0, !"]. For a given number of iterations N (number of time series points), the time increment is defined as At = T'/ N. The force Wt has the properties
Wn> = 0, <Wn22.
(24)
For adequate representation of random force (24) with white noise properties, we can use the Box-Muller model [26]:
Wn = >/|%/-2 ln r1 cos (2nr2), r e (0,1], (25)
where |2 = 2 is the dispersion, r1 and r2 are pseudorandom numbers with uniform distribution.
Figure 5 shows dependences obtained by procedure (23) for the DF and SF regions of the phase diagram at P = 3 in Fig. 2, c. The stress in Fig. 5 is presented in absolute values |g| because Eq. (23) describes the motion with a symmetric potential [5]. Although their positive and negative values are equiprobable, we analyze the absolute values reasoning that the surfaces, in our case, move in one direction and the stresses are proportional to the shear velocity. The same explains the probabilities for a > 0 in Fig. 4.
The dependences in Fig. 5 are similar to time dependences of shear stresses obtained by molecular dynamics methods for boundary friction of smooth mica surfaces with
a lubricant layer composed of dodecane molecules [27]. They also qualitatively agree with experimental dependences [28] and with time dependences of friction force presented elsewhere [29]. In all these cases, the time dependences of stresses and friction forces reveals a clearly defined stochastic component.
The probabilities P(a) were also calculated numerically in four parallel computational processes, each generating a stress time series |a|(t) on an interval t e [0, 7x 106] with a time step At = 10-4 by procedure (23). Each time series contained 7x 1010 points. The abscissa in the chosen range a e [105, 5] was divided into 105 segments with an interval Aa = 5 x 105, and the number of time series points in each segment was determined. Upon completion, the results of four processes were added up and the equivalent number of time series points was 2.8 x 1011. The number of time series points which fell on each interval Aa represented the value of the nonnormalized probability density P(a) for a in the middle of a segment. The computations were ended with normalization which gave the final form of P(a). The obtained numerical dependences P(a) coincide with the analytical dependences in Fig. 4. If instead of |a|, we analyze only positive values in computations, discarding negative, we will have curves accurately the same as in Fig. 4. However, to obtain equally smooth dependences P(a), the computation time (number of time series points) should be doubled. Thus, we can see that the analytical and numerical approaches provide fully coincident results for the probability density distribution function P(a). Note that visually it is rather difficult to distinguish between the DF and SF modes in Fig. 5 because both dependences |a|(t) are for the same temperature Te. If we increase Te we will easily distinguish between them [22].
4. Self-similar mode
For the condition
it >> i a'i8 ,
distribution (14) can be written as1
P (a) = a"1n (a), where n(a) is defined as n (a) = Z "V/2 g (a)-1 (1 + a2) x
x exp
- 1a 1 - g(a)[1 - (2- T.)a+a2>-'] da'
ag (a)2 (1 + a2)-2
(26)
(27)
(28)
Self-similar systems fit uniform distribution functions. Distribution (27) is uniform if n(a) is constant. As has been shown [30], expression (28) little contributes to (27) at a < 0.8, and when a exceeds a certain value, distribution (27) due to n(a) becomes exponentially decreasing. Thus,
Fig. 5. Time dependences of absolute stress |a| for parameters at Ia = 1 in Fig. 2, c: upper plot—DF, IT = 20, I8 = 4; lower plot— SF, IT = 5, I8 = 8
1 In the boundary case IT ^ 0, Ia = Ie = 0 , relations (14) and (27) are
identical.
a
Fig. 6. Distributions P(g) for parameters at Ia = IE = 10"10 in Fig. 2: IT = 20 (1) and 80 (2)
the power series distribution characteristic of self-similar behavior is realized in a limited stress range, and above the critical stress, the properties of self-similarity are lost. Remind that the Stratonovich calculus gives the index -1 in (27), whereas the Ito calculus gives -2. The choice of calculus influences the analysis in Sect. 2 only quantitatively, leading to renormalization of the noise intensities IT and I e •
Let us consider at length the dependences P(a) in Fig. 6 for condition (26) and phase diagrams in Fig. 2. The group of curves 1 (one upon the other) consists of three dependences P(a) at IT = 20 and P = 1, 2, 3. The group of curves 2 also consists of three superimposed dependences P(a) but at IT = 80 and P = 1, 2, 3. The dependences in Fig. 6, a were obtained directly from formulae (14), (15), and the curves in Fig. 6, b from the numerical experiment described in detail at the end of Sect. 3.
As can be seen, all dependences on a double log scale at a < 0.1 reveal a linear portion which fits a uniform form of P(a). The slope of this linear portion is not constant and depends on the model parameters. However, the linear portions of curves 2 are much closer to form (27) with an index of -1. This is because curves 2 were plotted for higher IT, and the distribution function is uniform subject to (26). Thus, condition (26) provides self-similar behavior of stress
time series with no characteristic stress scale at a<< 1. In this case, multifractal behavior of the stress time series shown in Fig. 5 is established [30, 31].
Note that curves 1 for P = 2, 3 in Fig. 6 correspond to the parameters of DF with one zero maximum of P(a), and curves 2 to those of SS with zero and nonzero maxima of P(a). However, in both cases, a self-similar mode of dry fraction is established because the linear portion lies in the range a<< 1 close to the zero maximum of P(a). At P = 1, the situation is somewhat different. Although the curves are superimposed on the respective ones at P = 2, 3, no zero maximum of P(a) at P = 1 exists according to the phase diagram in Fig. 2. In this case, curve 1 corresponds to single-phase SF with one nonzero maximum of P(a) in the range a<< 1 (not shown in the figure) for the parameters chosen. Curve 2 at P = 1 has two maxima, of which one can be seen in the figure and the other lies in the range a<< 1. Because it is the time series domain at a << 1 which is responsible for self-similar properties in boundary case (26), one should expect that the series at P = 1 will differ in statistical properties of those at P > 2. This feature can be studied in detail by multifractal analysis [30, 31] but it is beyond the scope of our paper.
Let us perform a correlation analysis to verify the linearity of the curves at a<< 1 in Fig. 6. Using the least square
Table 1
Coefficients A, B (29) and R2 for all curves in Fig. 6, a
Curves ß = 1 ß = 2 ß = 3
A -1.108881 -1.109690 -1.110726
1 B -1.230103 -1.232633 -1.234874
R2 0.999994 0.999991 0.999994
A -1.022208 -1.023911 -1.025154
2 B -1.181199 -1.185552 -1.188216
R2 0.999987 0.999986 0.999991
Table 2
Coefficients A, B (29) and R2 for all curves in Fig. 6, b
Curves ß = 1 ß = 2 ß = 3
A -1.128942 -1.129036 -1.130398
1 B -1.278614 -1.279978 -1.283051
R2 0.999982 0.999976 0.999980
A -1.025394 -1.026780 -1.028140
2 B -1.189762 -1.193522 -1.196533
R2 0.999985 0.999984 0.999990
method, we can derive the regression equation for the range 10-4 < a < 0.1 where the dependences are visually straight:
logP(a) = Aloga + B. (29)
The coefficients A and B for all curves shown in Fig. 6, a and 6, b are presented in Tables 1 and 2, respectively. It is seen from the tables that the obtained values diverge by no greater 4%, suggesting adequacy of the numerical methods.
The value A ~ -1 at IT = 80 agrees with power law (27). Note that at IT = 20, we have A ~ -1.1, which is also traceable in Fig. 6. The correlation coefficient R2 in Table 1 is determined in a standard way:
£ (y - y)2
R 2 = 1-^-, (30)
n -
£ (y - y)2
i=1
where y = log P(a), y is the value for (29),
n
y = £ yJn i-1
is the average and yi is the current value for Fig. 6. In the range 10-4 < a < 0.1, we have 1000 values (n = 1000) for Fig. 6, a and 2000 values (n = 2000) for Fig. 6, b. The values of R2 demonstrate high correlation of regression equation (29) and linear portions of the dependences in Fig. 6; that is, we do have a power series distribution.
5. Conclusion
Thus, our model allows one to analyze the behavior of an ultrathin lubricant layer between two atomically smooth solid surfaces in boundary friction taking into account additive fluctuations of stress, strain, and temperature. The analytical solution of the Fokker-Planck equation using the Stratonovich calculus shows that the system can be involved in liquid friction (SF), dry friction (DF), stick-slip (SS), metastable and stable liquid friction (MSF + SF), and stickslip with liquid friction (SS + SF). Our research in the influence of the system parameters on the lubricant behavior using phase diagrams demonstrates the possibility to select parameters that provide optimum friction conditions: only SF or MSF + SF. The joint numerical and analytical analysis of the Langevin equation shows a very close match between the probability distributions at different parameter. The time dependences of stress plotted for the parameters of phase diagrams suggest that when the intensity of thermal noise is many times greater than those of stress and strain, a self-similar mode of dry friction is established. In this case, the stress dependence of the probability distribution represents a uniform function. The data allows the conclusion that the stress time series possess multifractal characteristics. However, the fact requires additional research for verification, e.g., using the method of multifractal fluctuation analysis [30, 31].
Acknowledgments
The work was supported by the Ministry of Education and Science of Ukraine (project No. 0116U006818).
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Поступила в редакцию 07.04.2017 г.
Сведения об авторах
Natalia N. Manko, Cand. Sci. (Phys.-Math.), Assist., Sumy State University, Ukraine, mtashan@rambler.ru Iakov A. Lyashenko, Dr. Sci. (Phys.-Math.), Assoc. Prof., Sumy State University, Ukraine, nabla04@ukr.net
