The interaction observed between two surfaces in contact with one another is part of a number of physical processes, such as wear. In this paper, we present a numerical study of the asperities between two surfaces in contact with each other. The real contact area between two surfaces varies due to the multiple roughness scales caused by the stochastic nature of asperities. In our research, we employ a tribological system comprising two partitions: C1 is the contact set (CS), where the two surfaces are in direct contact with each other, and C2 is the noncontact set, where the two surfaces are not in contact with each other. Here, we have developed a new numerical model to describe the CS using ε-entropy to prove the existence of a minimum value for entropy in sliding contact scenarios. In this system, the lower and upper bounds of entropy are determined through the Kolmogorov approach using the aforementioned model. Using this model, we conclude that the ε-entropy value is bound between ln 2 and 2·ln 2 for a tribological system comprising two partitions. Additionally, we conclude that a correlation between the stochastic tribological contact behavior and the rate of entropy change is the key parameter in thermal nonequilibrium scenarios.

References

References
1.
Lorenz
,
B.
,
Oh
,
Y. R.
,
Nam
,
S. K.
,
Jeon
,
S. H.
, and
Persson
,
B. N. J.
,
2015
, “
Rubber Friction on Road Surfaces: Experiment and Theory for Low Sliding Speeds
,”
J. Chem. Phys.
,
142
(
19
), p.
194701
.
2.
Rowe
,
K. G.
,
Bennet
,
A. I.
,
Krick
,
B. A.
, and
Sawyer
,
W. G.
,
2013
, “
In Situ Thermal Measurements of Sliding Contacts
,”
Tribol. Int.
,
62
, pp.
208
214
.
3.
Ciavarella
,
M.
, “
A Comment on “Meeting the Contact-Mechanics Challenge” by Muser et al. [1]
,”
Tribol. Lett.
,
66
, pp.
37
39
.
4.
Shannon
,
С. E.
,
1948
, “
A Mathematical Theory of Communication
,”
Bell Syst. Tech. J.
,
27
(
4
), pp.
379
423
; 623–656.
5.
You
,
Y.
,
2017
, “
Random Dynamics of Stochastic Reaction-Diffusion Systems With Additive Noise
,”
J. Dyn. Differ. Eq.
,
29
(
1
), pp.
83
90
.
6.
Weedbrook
,
C.
,
Pirandola
,
S.
,
García-Patrón
,
R.
,
Cerf
,
N. J.
,
Ralph
,
T.
,
Shapiro
,
J.
, and
Lloyd
,
S.
,
2012
, “
Gaussian Quantum Information
,”
Rev. Mod. Phys.
,
84
(
2
), pp.
621
627
.
7.
Anisimov
,
M. A.
,
2004
, “
Thermodynamics at the Meso- and Nanoscale
,”
Dekker Encyclopedia of Nanoscience and Nanotechnology
, J. A. Schwarz, C. Contescu, and K. Putyera, eds., Marcel Dekker, New York, pp.
3893
3904
.
8.
Bol'shanin
,
A. A.
,
Slobodyan
,
S. M.
,
Yakovlev
,
A. R.
, and
Vasil'eva
,
L. A.
,
1987
, “
Two-Channel Optical Transducer for an Industrial Inspection System
,”
Meas. Tech.
,
30
(
10
), pp.
954
956
.
9.
Qi
,
G.
,
2017
, “
Energy Cycle of Brushless DC Motor Chaotic System
,”
Appl. Math. Mod.
,
51
, pp.
686
688
.
10.
Deeva
,
V. S.
,
Slobodyan
,
S. M.
, and
Teterin
,
V. S.
,
2016
, “
Optimization of Oil Particles Separation Disperser Parameters
,”
Mater. Sci. Forum
,
870
, pp.
677
682
.
11.
Mo
,
F.
,
Shen
,
C.
,
Zhou
,
J.
, and
Khonsari
,
M.
,
2017
, “
Statistical Analysis of Surface Texture Performance With Provisions With Uncertainty in Texture Dimensions
,”
IEEE Access
,
5
, pp.
5388
5398
.
12.
Zhou
,
Y.
,
Bosman
,
R.
, and
Lugt
,
P. M.
,
2018
, “
A Model for Shear Degradation of Lithium Soap Grease at Ambient Temperature
,”
Tribol. Trans.
,
61
(
1
), pp.
61
70
.
13.
Slobodyan
,
M. S.
,
2011
, “
The Probability Factor of Contact Measurements
,”
Meas. Tech.
,
54
(
1
), pp.
68
73
.
14.
Su
,
J.
,
Ke
,
L.
,
Wang
,
Y.
, and
Xiang
,
Y.
,
2017
, “
Axisymmetric Torsional Fretting Contact Between a Spherical Punch and an FGPM Coating
,”
Appl. Math. Modell.
,
52
, pp.
576
581
.
15.
Gallavotti
,
G.
,
2004
, “
Entropy Production and Thermodynamics of Nonequilibrium Stationary States: A Point of View
,”
Chaos
,
14
(
3
), pp.
680
690
.
16.
Feng
,
Z.
,
2017
, “
Magnetic Entropy Change of Layered Perovskites La2 − 2xSr1 + 2xMn2O7
,”
J. Appl. Phys.
,
97
(
10
), p.
103906
.
17.
Ciavarella
,
M.
, and
Papangelo
,
A.
,
2018
, “
The “Sport” of Rough Contacts and the Fractal Paradox in Wear Laws
,”
Facta Univ.: Mech. Eng.
,
16
(
1
), pp.
65
75
.
18.
Deeva
,
V.
,
Slobodyan
,
S.
, and
Martikyan
,
M.
,
2016
, “
Physical Model of the Sliding Contact of Conductors of the Alloy Cu-Zr and Cu-Re Under High Current Density
,”
Mater. Today: Proc.
,
3
(
9
), pp.
3114
3120
.
19.
Zhang
,
Y.
,
Kovalev
,
A.
,
Hayashi
,
N.
,
Nishiura
,
K.
, and
Meng
,
Y.
,
2018
, “
Numerical Prediction of Surface Wear and Roughness Parameters During Running-In for Line Contacts Under Mixed Lubrication
,”
ASME J. Tribol.
,
140
(
6
), p.
061501
.
20.
Imanian
,
A.
, and
Modarres
,
M.
,
2016
, “
Development of a Generalized Entropic Framework for Damage Assessment
,”
Fracture, Fatigue, Failure and Damage Evolution
(Conference Proceedings of the Society for Experimental Mechanics Series, Vol. 8),
A.
Beese
,
A.
Zehnder
, and
S.
Xia
, eds., Springer, Cham, Switzerland, pp.
73
78
.
21.
Bryant
,
M. D.
,
2009
, “
Entropy and Dissipative Processes of Friction and Wear
,”
FME Trans.
,
37
, pp.
55
59
.
22.
Aghdam
,
A. B.
, and
Khonsari
,
M. M.
,
2011
, “
On the Correlation Between Wear and Entropy in Dry Sliding Contact
,”
Wear
,
270
(
11–12
), pp.
781
783
.
23.
Kumar
,
N.
,
Singh
,
T.
,
Rajoria
,
R. S.
, and
Patnaik
,
A.
,
2016
, “
Optimum Design of Brake Friction Material Using Hybrid Entropy-GRA Approach
,”
MATEC Web Conf.
,
57
, p. 03002.
24.
Romanishina
,
T.
,
Romanishina
,
S.
,
Deeva
,
V.
, and
Slobodyan
,
S.
,
2017
, “
Numerical Modeling of Synovial Fluid Layer
,”
IEEE International Young Scientists Forum on Applied Physics and Engineering
(
YSF
), Lviv, Ukraine, Oct. 17–20, pp.
143
146
.
25.
Tufano
,
D.
, and
Sotoudeh
,
Z.
,
2017
, “
Exploring the Entropy Concept for Coupled Oscillators
,”
Int. J. Eng. Sci.
,
112
, pp.
18
31
.
26.
Banjac
,
M.
,
Vencl
,
A.
, and
Otović
,
S.
,
2014
, “
Friction and Wear Processes–Thermodynamic Approach
,”
Tribol. Ind.
,
36
(4), pp.
341
343
.https://www.researchgate.net/publication/269989594_Friction_and_Wear_Processes_-_Thermodynamic_Approach
27.
Volkov
,
V. F.
,
Peshel'
,
A. K.
,
Slobodyan
,
S. M.
, and
Tyryshkin
,
I. S.
,
1981
, “
Registration of a Pulsed Laser Beam by a Matrix of Charge-Coupled Devices
,”
Instrum. Exp. Tech.
,
24
(6 pt. 2), pp.
1522
1524
.
28.
Carpick
,
L. R. W.
,
2018
, “
The Contact Sport of Rough Surfaces
,”
Science
,
359
(
6371
), pp.
38
38
.
29.
Deeva
,
V.
, and
Slobodyan
,
S.
,
2017
, “
Influence of Gravity and Thermodynamics on the Sliding Electrical Contact
,”
Tribol. Int.
,
105
, pp.
299
303
.
30.
Machado
,
J. A.
,
2010
, “
Entropy Analysis of Integer and Fractional Dynamical Systems
,”
Nonlinear Dyn.
,
62
(1–2), pp.
371
378
.
31.
Majcherczak
,
D.
,
Dufrenoy
,
P.
,
Berthier
,
Y.
, and
Nait-Abdelaziz
,
M.
,
2006
, “
Experimental Thermal Study of Contact With Third Body
,”
Wear
,
261
(
5–6
), pp.
467
470
.
32.
Collet
,
P.
, and
Eckmann
,
J.
,
2006
,
Concepts and Results in Chaotic Dynamics: A Short Course
,
Heidelberg
,
Berlin
.
33.
Wright
,
S.
,
Scott
,
D.
,
Haddow
,
J.
, and
Rosen
,
M.
,
2001
, “
On the Entropy of Radiative Heat Transfer in Engineering Thermodynamics
,”
Int. J. Eng. Sci.
,
39
(
15
), pp.
1691
1693
.
34.
Kanazawa
,
Y.
,
Sayles
,
R.
, and
Kadiric
,
A.
,
2017
, “
Film Formation and Friction in Grease Lubricated Rolling-Sliding Non-Conformal Contacts
,”
Tribol. Int.
,
109
, pp.
505
510
.
35.
Lieb
,
E. H.
, and
Yngvason
,
J.
,
2014
, “
Entropy Meters and the Entropy of Non-Extensive Systems
,”
Proc. R. Soc. A
,
470
(2167), p.
0192
.
36.
Kolmogorov
,
A. N.
,
1958
, “
On the Entropy per Unit of Time as the Metric Invariant of the Automorphism
,”
Dokl. Akad. Nauk SSSR
,
124
, pp.
754
755
(in Russian).
37.
Sinai
,
Y. G.
,
1959
, “
On the Concept of the Entropy for a Dynamic System
,”
Dokl. Akad. Nauk SSSR
,
125
, pp.
768
771
(in Russian).
38.
Sinai
,
Y. G.
,
1988
, “
About A.N. Kolmogorov's Work on the Entropy of Dynamical Systems
,”
Ergodic Theory Dyn. Syst.
,
8
, pp.
501
505
(in Russian).
39.
Deeva
,
V. S.
, and
Slobodyan
,
S. M.
,
2017
,
Entropy Estimation of a Dynamical System Via a Contact Interaction
,
CRC Press/Safety and Reliability–Theory and Application: ESREL/Taylor & Francis
, Portoroz, Slovenia, p.
373
.
40.
Carcaterra
,
A.
,
2014
, “
Thermodynamic Temperature in Linear and Nonlinear Hamiltonian Systems
,”
Int. J. Eng. Sci.
,
80
, pp.
189
195
.
41.
Frigg
,
R.
,
2006
, “
Chaos and Randomness: An Equivalence Proof of a Generalized Version of the Shannon Entropy and the Kolmogorov–Sinai Entropy for Hamiltonian Dynamical Systems
,” Chaos
Solitons Fractals
,
28
(
1
), pp.
26
34
.
42.
Keller
,
K.
,
Mangold
,
T.
,
Stolz
,
I.
, and
Werner
,
J.
,
2017
, “
Permutation Entropy: New Ideas and Challenges
,”
Entropy
,
19
(3), pp.
134
150
.
43.
Ahmed
,
M. U.
, and
Mandic
,
D. P.
,
2012
, “
Multivariate Multiscale Entropy Analysis
,”
IEEE Signal Proc. Lett.
,
19
(
2
), pp.
91
96
.
44.
Ford
,
I.
,
2013
,
Statistical Physics: An Entropic Approach
,
Wiley
,
New York
.
45.
La
,
H. P.
,
Sarkar
,
S.
, and
Gupta
,
S.
,
2017
, “
Stochastic Model Order Reduction in Randomly Parametered Linear Dynamical Systems
,”
Appl. Math. Mod.
,
51
, pp.
744
763
.
46.
Prajapati
,
D. K.
, and
Tiwari
,
M.
,
2017
, “
Topography Analysis of Random Anisotropic Gaussian Rough Surfaces
,”
ASME J. Tribol.
,
139
(
4
), p.
041402
.
47.
Deng
,
C. Y.
,
Zhang
,
H. B.
,
Yin
,
J.
,
Xiong
,
X.
,
Wang
,
P.
, and
Sun
,
M.
,
2017
, “
Carbon Fiber/Copper Mesh Reinforced Carbon Composite for Sliding Contact Material
,”
Mater. Res. Express
,
4
(
2
), p.
025602
.
48.
Morris
,
N.
,
Mohammadpour
,
M.
,
Rahmani
,
R.
,
Johns-Rahnejat
,
P. M.
,
Rahnejat
,
H.
, and
Dowson
,
D.
,
2018
, “
Effect of Cylinder Deactivation on Tribological Performance of Piston Compression Ring and Connecting Rod Bearing
,”
Tribol. Int.
,
120
, pp.
243
254
.
49.
Gaspard
,
P.
, and
Wang
,
X.
,
1993
, “
Noise, Chaos, and (ε, τ)-Entropy per Unit Time
,”
Phys. Rep.
,
235
(
6
), pp.
291
343
.
50.
Amiri
,
M.
, and
Khonsari
,
M. M.
,
2010
, “
On the Thermodynamics of Friction and Wear
,”
Entropy
,
12
(
5
), pp.
1021
1049
.
51.
Nosonovsky
,
M.
,
2010
, “
Entropy in Tribology: In the Search for Applications
,”
Entropy
,
12
(
6
), pp.
1345
1390
.
52.
Bershadski
,
L. I.
,
1992
, “
On the Self-Organization and Concepts of Wear-Resistance in Tribosystems
,”
Trenie I Iznos (Russian Friction and Wear)
,
13
, pp.
1077
1094
.
53.
Nosonovsky
,
M.
, and
Mortazavi
,
V.
,
2018
,
Friction-Induced Vibrations and Self-Organization: Mechanics and Non-Equilibrium Thermodynamics of Sliding Contact
,
CRC Press
,
Boca Raton, FL
.
54.
Gershman
,
I. S.
,
Mironov
,
A.
,
Fox-Rabinovich
,
G. S.
, and
Veldhuis
,
S. C.
,
2015
, “
Self-Organization During Friction of Slide Bearing Antifriction Materials
,”
Entropy
,
17
(
12
), pp.
7967
7978
.
55.
Fleurquin
,
P.
,
Fort
,
H.
,
Kornbluth
,
M.
,
Sandler
,
R.
,
Segall
,
M.
, and
Zypman
,
F.
,
2010
, “
Negentropy Generation and Fractality in Dry Friction of Polished Surfaces
,”
Entropy
,
12
(
3
), pp.
480
489
.
56.
Kheifets
,
M. L.
,
2016
, “
Self-Organization of Structure Formation Processes in Intense Treatment and Operation of Materials
,”
Adv. Mater. Technol.
,
3
, pp.
14
20
.
57.
Creeger
,
P.
, and
Zypman
,
F.
,
2014
, “
Entropy Content During Nanometric Stick-Slip Motion
,”
Entropy
,
16
(
6
), pp.
3062
3073
.
58.
Barszcz
,
M.
,
Paszeczko
,
M.
, and
Lenik
,
K.
,
2015
, “
Self-Organization of Friction Surface of Fe-Mn-C-B Coating With Increased Resistance to Abrasion
,”
Arch. Metall. Mater.
,
60
(
4
), pp.
2651
2656
.
59.
Klameski
,
B. E.
,
1984
, “
An Entropy Based Model of Plastic Deformation Energy Dissipation in Sliding
,”
Wear
,
96
, pp.
319
329
.
60.
Obozov, A. A.
,
Serpik, I. N.
,
Mihalchenko, G. S.
, and
Fedyaeva, G. A.
,
2017
, “
Theoretical Aspects of the Patterns Recognition Statistical Theory Used for Developing the Diagnosis Algorithms for Complicated Technical Systems
,”
J. Phys.: Conf. Ser.
,
803
, p.
012109
.
61.
Myshkin
,
N.
,
1991
, “
Tribological Problems in Electrical Contacts
,”
Tribol. Int.
,
24
(
1
), pp.
45
49
.
62.
Leighton
,
M.
,
Morris
,
N.
,
Core
,
M.
,
Rahmani
,
R.
,
Rahnejat
,
H.
, and
King
,
P. D.
,
2016
, “
Boundary Interactions of Rough Non-Gaussian Surfaces
,”
Proc. Inst. Mech. Eng., Part J
,
230
, pp.
1359
1370
.
63.
Maciąg
,
M.
,
2015
, “
Specific Heat of Tribological Wear Debris Material
,”
ASME J. Tribol.
,
137
(
3
), p.
031601
.
64.
Klimontovich
,
Y. L.
,
2007
, “
Entropy and Information of Open Systems
,”
Phys.-Usp.
,
42
(
4
), pp.
375
399
(in Russian).
65.
Slobodyan
,
S. M.
,
2006
, “
Optimizing Phase-Space Scanning for a Dynamic System Monitoring Chaotic Media
,”
Meas. Tech.
,
49
(
1
), pp.
1
6
.
66.
Van Trees
,
H. L.
,
1968
,
Detection, Estimation and Modulation Theory: Part I, II, and III
,
Wiley
,
New York
.
67.
Posner
,
E. C.
,
Rodemish
,
E. R.
, and
Rumsey
,
H. G.
,
1967
, “
ε-Entropy of Stochastic Processes
,”
Ann. Math. Statist.
,
38
(
4
), pp.
1000
1020
.
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