Abstract
Static and sliding friction are phenomenas whose understanding is necessary for the design of safe and energy-saving products. Knowledge of the exact laws of friction is of interest for numerous applications, for example, compression, braking, fatigue, using bushings and bearings, internal combustion engines, hinges, gaskets, foundries, machine building, welding, electrical contacts, and many others (Persson in Sliding Friction: Physical Principles and Applications. Springer, p 462, 1999 [1], Pashayev and Janahmadov in Fractal Approach to Fracture Mechanics, p 440, 2015 [2]). The phenomenon of friction interested people hundreds and even thousands of years ago and remains to date one of the main problems in the production of new products and the improvement of technology.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Literature
Persson B.N.J. Sliding Friction: Physical Principles and Applications – Springer, 1999 – 462 p.
Pashayev A.M., Janahmadov A.Kh. Fractal Approach to Fracture Mechanics. Baku: “APOSTROFF”, 2015 – 440 p.
Amontons G. De la resistance cause’e dans les machines (About resistance and force in machines) // Mem l’Acedemic R.A. – 1699 –P.257–282.
Coulomb C.A. Theoric des machines simple (Theory of Simple Machines) – Paris: Bachelier, 1821 – 368 p.
Popov V.L. Contact Mechanics and Friction: Physical Principles and Applications. – Berlin: Springer – Verlag, 2010 – 362 p.
Bowden F.P., Tabor D. The Friction and Lubrication of Solids. – Oxford: Clarendon Press, 1986. -374 p.
He G., Müser M.H., Robbins M.O. Adsorbed layers and the origin of static friction // Science – 1999. – V.284.- P.1650–1652.
Barber J.R. Multiscale surfaces and Amontons’ law of friction // Tribol. Left – 2013. – V. 49. P.539–543.
Otsuki M., Matsukawa H. Systematic breakdown of Amontons’ law of friction for an elastic object locally obeying Amontons’ law // Sci. Rep.- 2013. – V.3. –P.1586.
Rubinstein S.M., Cohen G., Fineberg J. Detachment fronts and onset of dynamic friction // Nature. -2004.-V.430.-P.1005–1009.
Li Q., Popov M., Dimaki A., Filippov A.E., Kürschner S., Popov V.L. Friction between a viscoelastic body and a rigid surface with random self – affine roughness // Phys. Rev. Lett. – 2013. – V.111.-P.034301.
De Wit C.C., Olsson H., Astrom K., Lischinsky P. A new model for control of systems with friction // IEEE Trans. Autom. Control. -1995. –V.40. –P. 419–425.
Peng J.Y., Chen D.B. Modeling of piezoelectric – driven stick – slip actuators // IEEE/ASME Trans. Mechatron – 2010-V.99. –P.1–4.
Dupont P., Armstrong D., Hayward V. Elasto-Plastic Friction Model: Contact Compliance and Stiction // Proc. Am. Control Conf. 2. – Chicago, 2000. –P.1072–1077.
Milahin N., Starcevic S. Influence of the normal force and contact geometry on the static force of friction of an oscillating sample// Phys. Mesomech. – 2014. –V.17. - №3. P.228–231.
Nguyen H.X., Teidelt E., Popov V.L., Fatikov S. Modeling and waveform optimization of stick - slip micro – drives using the method of dimensionality reduction // Arch. Appl. Mech.- doi 10.1007/s 00419-014-0934-y.
Dieterich J.H. Time dependent friction and the mechanics of stick-slip // Pure Appl. Geophys. – 1978 – V. 116. –P. 790–806.
Dieterich J.H. Modeling of rock friction: 1. Experiment results and constitutive equations // J.Geophys. Res. Solid Earth. – 1979. V.84. – P.2161–2168.
Rice J.R., Ruina A.L. Stability of steady frictional slipping // J. Appl. Mech. – 1983. V.50. –P.343–349.
Grosch K.A. The relation between the friction and visco - elastic properties of rubber // Proc. R. Soc. Lond. A-1963. –V.274.-P.21–39.
Le Gal A., Yang X., Klüppel M. Evaluation of sliding friction and contact mechanics of elastomers based on dynamic – mechanical analysis // J. Chem. Phys. -2005. –V.123.- P.014704.
Popov V.L., Voll L., Li Q., Chai Y.S., Popov M. Generalized law of friction between elastomer and differently shaped rough bodies // Sci. Rep. -2014.-V4-P.3750- https://doi.org/10.1038/srep 03750.
Popov V.L., Dimaki A., Psakhie S., Popov M. On the role of scales in contact mechanics and friction between elastomers and randomly rough self –affine surfaces // Sci. Rep. -2015. –V.5. – P.1139.
Lee E.H. Stress analysis in viscoelastic bodies //Quart Appl. Math. – 1955. V.13. –P.183–190.
Radok J.R.M. Visco-elastic stress analysis // Quart Appl. Math. – 1957.- V.15. –P.198–202.
Kürschner S., Popov V.L. Penetration of self-affine fractal rough rigid bodies into a model elastomer having a linear viscous rheology // Phys. Rev. E. -2013. V.87.-P.042802.
Norburg A.L., Samuel T. The recovery and sinking-in or piling-up of material in the Brinell test // J. Iron Steel Inst. – 1928 – V. 117.- P.673.
Popov V.L., Heβ M. Method of Dimensionality Reduction in Contact Mechanics and Friction. – Berlin: Springer, 2014. – 265 p.
Argatov I.I., Sabina F.J. Spherical indentation of a transversely isotropic elastic half-space reinforced with a thin layer // Int. J. Eng. Sci – 2012. –V.50.-P.132–143.
Gao H.J., Chiu C.H., Lee J. Elastic contact versus indentation modeling of multi-layered materials// Int. J. Solid Struct. – 1992. –V.29.-P.2471–2492.
Popov V.L. Method of dimensionality reduction in contact mechanics and tribology. Heterogeneous media // Phys. Mesomech. -2014. – V. 17. №1-P.50–57.
Teidelt E. Oscillating Contacts: Friction Induced Motion and Control of Friction: Dissertation – Berlin: Berlin University of Technology, 2015. – 131 p.-https:opus4. kobv.de/opus4-tuberlin/files/6108/ teidelt_elena.pdf.
Starcevic J., Filippov A.E. Simulation of the influence of ultrasonic in-plane oscillations on dry friction accounting for stick and creep//Phys. Mesomech. – 2012.- V.15.-№5-6.-P.330–332.
Milanin N., Starcevic J. Influence of the normal force and contact geometry on the static force of friction of an oscillating sample//Phys. Mesomech. – 2014. –V.17.-№3.-P.228–231.
Milamin N., Li Q., Starčević J. Influence of the normal force on the sliding friction under ultrasonic oscillations// Facta Univ. Ser. Mech. Eng. – 2015.-V.13(11). –P.27–32.
Popov V.L. Kontaktmechanik and Reibung. Von der Nanotribologie bis zur Erdbebendynamik. – Berlin: Springer, 2015. – 398 p.
Paggi M., Pohrt R., Popov V.L. Partial – slip frictional response of rough surfaces // Sci. Rep. – 2014. – V.4.-P.5178 – https://doi.org/10.1038/srep05178.
Grzemba B., Pohrt R., Teidelt E., Popov V.L. Maximum micro-slip in tangential contact of randomly rough self-affine surfaces // Wear. -2014.-V.309(1).-P.256–258.
Archard J.F. Elastik deformation and the law of friction // Proc. R. Soc. A. – 1957. – V.243 – P.190–205.
Persson B.N.J., Albohr O., Tartaglino U., Volokin A.I., Tosatti E. On the nature of surface roughness with application to contact mechanics, sealing, rubber friction and adhesion // J.Phys. : Condens. Matter. R. -2005.- V.17. – P.1–62.
Sivebaek I.M., Samoiliv V.N., Persson B.N.J. Effective viscosity of confined hydrocarbons // Phys. Rev. Lett. – 20012. – V.108 P.036102.
Hyun S., Pei L., Molinari J.-F., Robbins M.O. Finite-element analysis of contact between elastic self-affine surfaces // Phys. Rev. E. – 2004. – V.70. –P.02117.
Pohrt R., Popov V.L. Normal contact stiffness of elastic solids with fractal rough surfaces // Phys. Rev. Lett. – 2012. – V.108 P.104301.
Pohrt R. Normal stiffness of multiscale rough surfaces in elastic contact: Thesis.- Berlin: Berlin University of Technology, 2013.
Greenwood J.A., Williamson J.B.P. Contact of nominally flat surfaces // Proc. R. Soc. A.- 1966. V.295. – P.300.
Popov V.L. What does friction really depend on? Robust governing parameters in contact mechanics and friction // Физ. мeзoмex. 18 4 (2015) 5–11.
Dimaki A.V., Popov V.L. Coefficient of friction between a rigid conical indenter and a model elastomer: Influence of local frictional heating // Phys. Mesomech. – 2015. – V.18- №1.- P.75–80.
Li Q., Dimaki A., Popov M., Psakhie S.G., Popov V.L. Kinetics of the coefficient of friction of elastomers // Sci. Rep. -2014. – V.4. – P.5795.
Psakhie S.G., Popov V.L. Mesoscopic nature of friction and numerical simulation methods in tribology // Физ. мeзoмex. 15 4 (2012) 5–7.
Persson B.N.J. Contact mechanics for randomly rough surfaces // Surf. Sci. Rep. – 2006. – V.61. –P.201–227.
Prandtl L. Ein Gedankenmodel zur kinetischen Theorie der festen Körper // ZAMM. 1928- V.8-P.85–106.
Janahmadov A.Kh., Javadov M.Y. Synergetic and Fractals in Tribology.- Berlin: Springer – Verlag, 2016 – 382 p.
Panin V.E. Overview on mesomechanics of plastic deformation and fracture of solids // Teor. Appl. Fract. Mech.- 1998 – V.30. – P.1–11.
Popov V.L., Psakhie S.G. Numerical simulation methods in tribology // Tribology Int. – 2007. – V40. P.916–923.
Geike T., Popov V.L. Mapping of three – dimensional contact problem into one dimension //Phys. Rev. E.- 2007.- V76. – P.036710.
Kürschner S., Filippov A.E. Normal contact between a rigid surface and a viscous body: Verification of the method of reduction of dimensionality for viscous media // Физ. мeзoмex. 15 4 (2012) 25–30.
Pohrt R., Popov V.L. Investigation of the dry normal contact between fractal rough surfaces using the reduction method, comparison to 3D simulations // Физ. мeзoмex. 15 4 (2012) 31–35.
Li Q. Dependence of the kinetic force of friction between a randomly rough surface and simple elastomer on the normal force // Физ. мeзoмex. 15 4 (2012) 63–65.
Heβ M. Über die exakte Abbildung ausgewählter dreidimensionaler Kontakte auf Systems mit niedrigerer räumlicher Dimension. – Göttingen: Cuvillier – Verlag, 2011.- 172 p.
Psakhie S.G., Smolin A.Y., Stefanov Y.P. et al. Modeling the behavior of complex media by jointly using discrete and continuum approaches // Tech. Phys. Lett.-2004. –V.30. –P.712–714.
Filippov A.E., Popov V.L. Fractal Tomlinson model for mesoscopic friction: From microscopic velocity – dependent damping to macroscopic Coulomb friction // Phys. Rev. E.-2007.-V.75. – P.027103.
Popov V.L. Basic ideas and applications of the method of reduction of dimensionality in contact mechanics //// Физ. мeзoмex. 15 4 (2012) 9–18.
Sneddon I.N The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile // Int. J. Eng. Sci. – 1965. – V.3. –P.47–57.
Johnson K.L. Contact Mechanics.- Cambridge: Cambridge University Press, 1987. – 468 p.
Johnson K.L., Kendall K., Roberts A.D. Surface energy and the contact of elastic solid // Proc. Roy. Soc. Lond. A. Math. – 1971.-V. 324.- P.301–313.
Heβ M. On the reduction method of dimensionality: The exact mapping of axisymmetric contact problems with and without adhesion //// Физ. мeзoмex. -2012 T.15.-№4.- C.19–24.
Landau L.D., Lifschitz E.M. Lehrbuch der Theoretischen Physik. Band 7. Elastizitätstheorie. – Berlin: Akademic – Verlag. 1965 –183 p.
Landau L.D., Lifschitz E.M. Lehrbuch der Theoretischen Physik. Band 6. Hydrodynamic. – Berlin: Akademic – Verlag. 1991 –683 p.
Barbe J.R. Bounds on the electrical resistance between contacting elastic rough bodies // Proc. Roy. Soc. Lond. A. – 2003. V. 495. – P.53–66.
Savkoor A.R. On the friction of rubber // Wear, 1965, v. 8.
Kummer H.W. United theory of rubber and tire friction // Eng. Res. Bulletin B-9, USA, 1966.
Janahmadov A.Kh. Mechanics of elastomers in oil and gas equipment. Baku: Çaşıoglu. 2002. – 308 p.
Myp Д. Tpeниe и cмaзкa элacтoмepoв. M.:Xимия, 1977.
Griffith A.A. The phenomena of rapture and flow in solid // Philos. T. Roy. Soc. A. – 1921. – v.221. – P.163–198.
Prandtl L. Ein Gedankenmodell für den Zerreiβ vorgang spröder Körper // J.Appl. Math. Mech. – 1933. –V.13. – P.129–133.
Maugis D. Contact, Adhesion and Rupture of Elastic Solids – Berlin: Springer-Verlag, 2000. -414p.
Hyun S., Robbins M.O. Elastic contact between rough surfaces: Effect of roughness at large and small wavelengths // Tribol. Int. – 2007. –V.40. – P.1413–1422.
Campana C., Müser M.H. Practical Green’s function approach to the simulation of elastic, semi-infinite solids//Phys. Rev.B. 2006. –V.74. – P.075420.
Akarapu S., Sharp T., Robbins M.O, Stiffness of contacts between rough surfaces // Phys. Rev.Lett. 2011. –V.106. – P.204301.
Campana C., Persson B.N.J., Müser M.H. Transverse and normal interfacial stiffness of solids with randomly rough surfaces// J. Phys. Condens. Matt.- 2011. –V.23. – P.085001.
Popov V.L., Filippov A.E. Force of friction between fractal rough surface and elastomer // Tech. Phys. Lett.- 2010.- V.36.-P.525–527.
Popov V.L., Dimaki A.V. Using hierarchial memory to calculate friction force between fractal rough solid surface and elastomer with arbitrary linear rheological properties // Tech. Phys. Lett. -2011.-V.37.-P8–11.
Popov V.L. A theory of the transition from static to kinetic friction in boundary lubrication layers // Solid State Commun. -2000.-V.115.-P.369–373.
Meyer E., Overney R.M., Dransfeld K., Gyalog T. Nanoscience: Friction and Rheology of Nanometer Scale-Singapore: World Scientific, 1998–392p.
Hertz H. Über die Berührung fester elastischer Körper // J.für die reine und angewandte Mathematik – 1882. –V.92-. P.156–171.
Boussinesq J. Application des Potentiels a L’equilibre et du Mouvement des Solides Elastiques- Paris: Gauthier-Villars, 1885.
Sneddon I.N. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile // Int. J.Eng. Sci. -1965. –V.3.-P.47–57.
Oliver W.C., Pharr G.M. Measurement of hardness and elastic modulus leg instrumented indentation: Advances in understanding and refinements to methodology // J.Mater. Res.-2004. – V. 19. №1. –P.3–20.
Maugis D., Barquins M. Fracture mechanics and the adherence of viscoelastic bodies // J.Phys. D. – 1978-V.11- P.1989–2023.
Barquins M., Maugis D. Adhesive contact of axisymmetric punches on an elastic half – space. The modified Hertz – Huber’s stress tensor for contacting spheres // J. Mec. Theor. Appl. – 1982. -V.1. P.331–357.
Mauqis D., Barquins M. Adhesive contact of sectionally smooth-ended punches on elastic half-spaces: Theory and experiment // J. Phys. D. – 1983. –V. 16.- P.1843–1874.
Yao H., Gao H. Optimal shapes for adhesive binding between two elastic bodies // J. Colloid Interf. Sci.-2006.-V.298 №2. P.564–572.
Popova E., Popov V.L. The research works of Coulomb and Amontons and generalized laws of friction // Friction. -2015. –V.3(2). –P.183–190.
Steuermann E. To Hertz’s theory of local deformations in compressed elastic bodies // Dokl. AS URSS. -1939. –V.25. –P.359–361.
Segedin C.M. The relation between load and penetration for a spherical punch // Mathematika. – 1957.- V.4. –P.156–161.
Pharr G.M., Oliver W.C., Brotzen F.R. On the generality of the relationship among contact stiffness, contact area, and elastic modulus during indentation // Mater. Res.-1992.-V.7.№3 – P.613–617.
Gibson R.E. Some results concerning displacements and stresses in a non-homogeneous elastic half –space // Geotechnique . – 1967. –V.17. №1.- P.58–67.
Irwin G.R. Fracture // Handbook of Physics. – Berlin: Springer – Verlag, 1958 – V.6. – P.551–590.
Geike T., Popov V.L. Reduction of three – dimensional contact problem to one-dimensional ones // Tribology Int. – 2007.-V40.-P.924–929.
Popov V.L., Filippov A.E. Applicability of a reduced model to description of real contacts between rough surfaces with different Hurst exponents // Tech. Phys. Lett. – 2008 –V.34. –P.722–724.
Popov V. L., Filippov A.E. Statistics of contacts and the dependence of their total length on the normal force for fractal surfaces with different Hurst exponents // Tech. Phys. Lett.- 2008 –V.34. – P.792–794.
Димaки A.B., Пoпoв B.Л. Meтoд peдyкции paзмepнocти иeгo пpимeнeниe для мoдeлиpoвaния тpeния элacтoмepoв в ycлoвияx cлoжныx динaмичecкиx нaгpyзoк // Физ. мeзoмex. 15 4 (2012) 81–86.
Ruina A.L. Slip instability and state variable friction laws // J. Geophys. Res. – 1983.- V.88-P.10359–10370.
Heslot F., Baumberger T., Perrin B. et al. Creep, stick-slip and dry friction dynamics: Experiment and heuristic model // Phys. Rev. E. – 1994. –V.49. –P.4973–4988.
Popov V.L., Grzemba B., Starcevic J., Popov M. Rate and state dependent friction laws and the prediction of earthquakes: What can we learn from laboratory models ? // Tectonophysics. -2012. –V. 532–535. –P.291-300.
Eaves A., Smith A., Waterhouse W., Sansome D. Review of the application of ultrasonic vibrations to deforming metals // Ultrasonics . – 1975. –V. 13. - №4. –P.162–170.
Siegert K., Ulmer J. Superimposing ultrasonic waves on the dies in tube and wire drawing // J. Eng. Mat. Tech. – 2001. – V.125.- №4. –P.517–523.
Блexмaн И.И.. Джaнeлидзe Г.Ю. Bибpaциoннoe пepeмeщeниe –M.: Hayкa, 1964. -410 c.
Hess D., Soom A., Kim C. Normal vibrations and friction at a Hertzian contact under random excitation: Theory and experiments // J. Sound Vibration – 1992. V. 153. №3.- P.491–508.
Tolstoy M. Significance of the normal degree of freedom and natural normal vibrations in contact friction // Wear. – 1967. –V. 10. №3. –P.199–213.
Popov V.L., Starcevic J., Filippov A.E. Influence of ultrasonic in – plane oscillations on static and sliding friction and intrinsic length scale of dry friction // Trib. Lett. -2010. –V.39. – p. 25–30.
Пoпoв B.Л., Cтapкeвич Я., Taйдeльт E. Bлияниe yльтpaзвyкoвыx кoлeбaний в плocкocти cкoльжeния и пepпeндикyляpнo к нeй нa cилy тpeния пoкoя и cкoльжeния // Tpeниe и cмaзкa в мaшинax и мexaнизмax -2011. №2. c.3–9.
Teidelt E., Popov V.L., Starcevic J. Influence of in – plane and out-of-plane ultrasonic oscillations on sliding friction // SAE Int. J. Passeng. Cars-Mech. Syst.-2011.-V.4(3). –P.1387–1393.
Ben-David O., Fineberg J. Static friction coefficient is not a material constant //Phys. Rev. Lett. -2011. –V.106.- P.254301.
Pohrt R., Popov V.L.// Private communication -2012.
Popov V.L., Filippov A.E. Adhesive properties of contacts between elastic bodies with randomly rough self-affine surfaces: A simulation with the method of reduction of dimensionality // Физ. мeзoмex. 15 4 (2012) 87–92.
Schargott M., Popov V.L., Gorle. Spring model of biological attachment pads // J.Theor. Biology. -2006- V.243. –P.48–53.
Yao H., Gao H. Mechanics of robust and releasable adhesion in biology: Bottom-up designed hierarchical structures of gecko // J. Mech. Phys. Solids.-2006.-V.54.-P.1120–1146.
Heise R., Popov V.L. Adhesive contribution to the coefficient of friction between rough surfaces //Tribol. Lett .-2010.-V.39.-P.245–250.
Geike T., Popov V.L. Reduction of three-dimensional contact problems to one-dimensional ones // Tribol. Int .-2007.-V.40.-P.924–929.
Fuller K.N.G., Tabor D. The effect of surface roughness on the adhesion of elastic solids // Proc. R. Soc. Lond. A- 1975.-V.345.-P.327–342.
Persson B.N.J. Elastoplastic contact between randomly rough surfaces // Phys. Rev. Lett .-2001.-V.87.-№11ю-P.116101.
Leidner M. Kontakt physikalische Simulation von Schichtsystemen: PhD Thesis. – Technische Universität Darmstadt, 2009.
Venner C.H, Lubrecht A.A. Multilevel Methods in Lubricatio. – Amsterdam: Elsevier, 2000.-379 p.
Geike T. Theoretische Grundlagen eines schnellen Berechnungsver - fahrens für den Kontakt rauer Oberflächen. –Berlin: TU Berlin, 2007.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2019 Springer International Publishing AG
About this chapter
Cite this chapter
Janahmadov, A.K., Javadov, M. (2019). Dimension Reduction as Modeling Method for Elastomers Under Complex Dynamic Loading. In: Fractal Approach to Tribology of Elastomers. Materials Forming, Machining and Tribology. Springer, Cham. https://doi.org/10.1007/978-3-319-93861-5_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-93861-5_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-93860-8
Online ISBN: 978-3-319-93861-5
eBook Packages: EngineeringEngineering (R0)