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# Method of Dimensionality Reduction (MDR)

MDR is a new method of simulation of contact and frictional forces between elastic and viscoelastic solids. MDR is simple in application, easy to learn and does not need any special knowledge in contact mechanics. Numerical implementation of MDR is almost trivial and allows to integrate the direct simulation of frictional contacts in finite elements or multi-body programs.

The MDR is based on the observation that close analogies exist between certain types of three-dimensional contact problems and the simplest contacts with a one-dimensional elastic foundation (Fig.). Thereby, it is important to emphasize that this is not an approximation: The properties of one-dimensional systems coincide exactly with those of the original three-dimensional system, if the form of the bodies is modified and the elements of the foundation are defined according to the rules of the MDR.

Based on the MDR, several software-versions have been developed (FaCom, Triboflex, xTribo). The software is developed and distributed by the company xTribo GmbH (http://www.xtribo.de [1]).

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In the following, various application fields of MDR are listed and the corresponding references:

Basics of the method of dimensionality reduction
Description
Source
A short practical introduction of the application of MDR to axis-symmetrical contracts
Popov V.L., Hess M.: Method of dimensionality reduction in contact mechanics and friction: a users handbook. I. Axially-symmetric contacts, Facta Universitatis, series Mechanical Engineering, 12 (1): 1-14, 2014  (PDF, 646,2 KB) [3]
A comprehensive treatise, including all necessary proofs
Popov V.L., Hess M.: Method of dimensionality reduction in contact mechanics and friction, Springer [4] 2014
A lecture on the MDR at TU Berlin
A lecture on MDR by Prof. Popov held at the TU Berlin (130 min [5], in German)
English Translation of this lecture can be seen here [6]
A lecture on the MDR on International friction forum
A lecture on the MDR on International friction forum (115 min [7])
Two Review papers on MDR
Popov V.L.: Basic ideas and applications of the method of reduction of dimensionality in contact mechanics. - Physical Mesomechanics, 2012, v. 15, N. 5-6, 254-263.  (PDF, 203,2 KB) [8]

Popov V.L.: Method of reduction of dimensionality in contact and friction mechanics: A linkage between micro and macro scales. Friction, 2013, v. 1, N. 1, pp. 41-62.   (PDF, 1,4 MB) [9]
Early Publications on MDR
Description
Source
Sketch of general ideas for multi-scale modeling of friction including the first ideas of MDR
Popov V.L. and Psakhie S.G.: Numerical Simulation methods in tribology - Tribology International, 40, 916-923 (200/)  (PDF, 687,8 KB) [10]
Initial ideas of application of MDR to rough surfaces (later better understood and corrected)
Geike T. and Popov V.L.: Reduction of three-dimensional contact Problems to one-dimensional ones. - Tribology International, 40, 924-929 (2007)  (PDF, 607,8 KB) [11]

Geike T. and Popov V.L.: Mapping of three-dimensional contact Problems into one Dimension. - Phys. Rev. E., 76, 036710 (2007)   (PDF, 155,4 KB) [12]
Simulation of friction in the simplest model case of a pure viscous medium
Popov V.L., Filippov A.E.: Force of Friction between Fractal Rough Surface and Elastomer. - Tech. Phys. Lett., 36, pp. 525 -527 (2010)  (PDF, 80,6 KB) [13]
Verification of MDR
Description
Source
Verification of MDR for normal contact of self-affine surfaces
Pohrt R., Popov V.L., Filippov A.E.: Normal contact stiffness of elastic solids with fractal rough surfaces for one- and three-dimensional Systems. - Phys. Rev. E, 86, 026710 (2010)  (PDF, 544,8 KB) [14]

Pohrt R., Popov V.L.: Investigation of the dry normal contact between fractal rough surfaces using the reduction method, comparison to 3D simulations. - Physical Mesomechanics, 15, 275-279 (2012)  (PDF, 194,6 KB) [15]
Verification of MDR for normal contact of self-affine rough surfaces with a viscous medium
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, 87, 042802 (2013)  (PDF, 694,1 KB) [16]
Verification of MDR for contacts of rough spheres
Pohrt R. and Popov V.L.: Contact Mechanics of Rough Spheres: Crossover from Fractal to Hertzian Behavior, Advances in Tribology, 2013, 974178 (2013)  (PDF, 1,2 MB) [17]
Verification of MDR in a broad range of surfaces profiles (from "White noise" roughness to smooth Hertz`contact)
Pohrt R., Popov V.L.: Contact stiffness of randomly rough surfaces. Sci. Rep. 3, 3293 (2013); DOI: 10.1038/srep03293  (PDF, 519,1 KB) [18]
Applications of MDR
Description
Source
Shakedown in oscillating rolling contact
Wetter R., Popov V.L.: Shakedown limits for an oscillating, elastic rolling contact with Coulomb friction. International Journal of Solids and Structures, 51 930-935  (2014).  (PDF, 641,3 KB) [19]

Wetter R., Popov V.L. Influence of the alignment of load and oscillation on the frictional shakedown of an elastic rolling contact with Coulomb friction. Physical Mesomechanics, 17, 31-38 (2014).  (PDF, 200,3 KB) [20]
Fretting wear
Popov V.L.: Analytic solution for the limiting shape of profiles due to fretting wear, Sci. Rep. 4, 3749 (2014); DOI: 10.1038/srep03749.  (PDF, 484,9 KB) [21]

Dimaki A.V., Dmitriev A.I., Chai Y.S., Popov V.L.: Rapid simulation procedure for fretting wear on the basis of the method of dimensionality reduction. - International Journal of Solids and Structures, 51, 4215–4220 (2014).   (PDF, 1,1 MB) [22]

Li Q., Filippov A.E., Dimaki A.V., Chai Y.S., Popov V.L.: Simplified simulation of fretting wear using the method of dimensionality reduction. Physical Mesomechanics 17, 236-241 (2014), DOI: 10.1134/S1029959914030102.  (PDF, 179,4 KB) [23]
Simulation of stick-slip drives with frictional contacts
Teidelt E., Willert E., Filippov A.E., Popov V.L. :Modeling of the dynamic contact in stick-slip micro-drives using the method of reduction of dimensionality. - Physical Mesomechanics, 15, N.5-6, 287-292 (2012).  (PDF, 228,0 KB) [24]

Nguyen H.X., Teidelt E., Popov V.L., Fatikow S.: Dynamical tangential contact of rough surfaces in stick-slip microdrives: modeling and validation using the method of dimensionality reduction. Physical Mesomechanics, 17, 304-310 (2014). DOI:10.1134/S1029959914040079.  (PDF, 136,8 KB) [25]

Nguyen H.X., Teidelt E., Popov V.L., Fatikow S.: Modeling and waveform optimization of stick-slip micro-drives using the method of dimensionality reduction. Archive of Applied Mechanics, DOI:10.1007/s00419-014-0934-y  Springer Link [26]
Elastomer friction
Li Q., Popov M., Dimaki A., Filippov A.E., Kürschner S., Popov V.L.: Friction Between a Viscoe-lastic Body and a Rigid Surface with Random Self-Affine Roughness, Physical Review Letters, 111, 034301 (2013).  (PDF, 527,6 KB) [27]

Popov V.L., Voll L., Li Q., Chai, Y.S. & Popov M.: Generalized law of friction between elastomers and differently shaped rough bodies. Sci. Rep. 4, 3750 (2014); DOI:10.1038/srep03750.  (PDF, 1,1 MB) [28]

Li Q., Dimaki A., Popov M., Psakhie S.G. and Popov V.L. : Kinetics of the coefficient of friction of elastomers. Sci. Rep. 4, 5795 (2014); DOI:10.1038/srep05795.  (PDF, 1,1 MB) [29]
Relaxation damping in oscillating contacts
Popov M, Popov V.L. & Pohrt R., Relaxation damping in oscillating contacts, Scientific Reports, 5,16189 (1-9 pp) (2015).   (PDF, 749,0 KB) [30]
Influence of oscillations on friction
Starcevic J., Filippov A.E.: Simulation of the influence of ultrasonic in-plane oscillations on dry friction accounting for stick and creep. – Physical Mesomechanics, 15, 330-332 (2012).   (PDF, 84,4 KB) [31]

Milahin N., Starcevic J.: Influence of the normal force and contact geometry on the static force of friction of an oscillating sample. Physical Mesomechanics, 17, 228-231 (2014), 10.1134/S1029959914030084.  (PDF, 113,5 KB) [32]
Acoustic emission in rolling contacts on rough surfaces
Popov M., Benad J., Popov V.L., Heß M.: Acoustic Emission in Rolling Contacts. In: Method of Dimensionality Reduction in Contact Mechanics and Friction, Springer, 2014, 207-214.  (PDF, 149,0 KB) [33]
Impact of elastic bodies
Lyashenko, I.A. & Popov, V.L.: Impact of an elastic sphere with an elastic half space revisited: Numerical analysis based on the method of dimensionality reduction. Sci. Rep. 5, 8479; DOI:10.1038/srep08479 (2015).   (PDF, 376,4 KB) [34]
I.A. Lyashenko, E. Willert, V.L. Popov, Adhesive impact of an elastic sphere with an elastic half space: Numericalanalysis based on the method of dimensionality reduction, Mechanics of Materials, 92, 155-163
(2016), doi: 10.1016/j.mechmat.2015.09.009 (PDF, 1,6 MB) [35]
Further generalization of MDR
Description
Source
Application of MDR to heterogeneous media
Popov V.L.: Method of dimensionality reduction in contact mechanics and tribology. Heterogeneous media, Physical Mesomechanics, 17, 50-57 (2014).  (PDF, 123,0 KB) [36]
Criticism of MDR (There is a public discussion about the region of validity of MDR.)
Description
Source
A critical discussion of the MDR
Ivan Argatov: A discussion of the method of dimensionality reduction, J. Mechanical Engineering Science, 2015, DOI: 10.1177/0954406215602512 (PDF, 222,4 KB) [37]
The main critical arguments by B.N.J. Persson et. al.
Lyashenko A., Pastewka L., and Persson B.N.J.: Comment on ‘‘Friction Between a Viscoelastic Body and a Rigid Surface with Random Self-Affine Roughness’’, Phys. Rev. Lett, 111, 189401 (2013).  (PDF, 72,0 KB) [38]
Response to the criticism by Persson
Li Q., Popov M., Dimaki A., Filippov A.E., Kürschner, S., Popov V.L. and Pohrt R.: Reply to the above Comment, Phys. Rev. Lett.,111, 189402 (2013).  (PDF, 97,5 KB) [39]
Detailed Response to the crtiticism by Persson
Popov V.L.: Comment on ‘‘Contact Mechanics for Randomly Rough Surfaces: On the Validity of the Method of Reduction of Dimensionality’’ by Bo Persson in Tribology Letters , Tribol Lett, 2015.doi:10.1007/s11249-015-0608-0  (PDF, 268,8 KB) [40]