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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.** **The following book is recommended as a practical guide to the MDR.

Description | Source |
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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 |

A comprehensive treatise, including all necessary proofs | Popov V.L., Hess M.: Method of dimensionality reduction in contact mechanics and friction, Springer 2014 |

A lecture on the MDR at TU Berlin | A lecture on MDR by Prof. Popov held at the TU Berlin (130 min, in German) English Translation of this lecture can be seen here |

A lecture on the MDR on International friction forum | A lecture on the MDR on International friction forum (115 min) |

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. 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. |

Description | Source |
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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). 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). |

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. 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). 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. |

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). 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. 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 |

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). 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. 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. |

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). |

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). 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. |

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. |

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). |

Adhesive impact of elastic bodies | 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 |

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