Ivan Argatov

DOI Number
First page
Last page


In recent years, the method of dimensionality reduction (MDR) has started to figure as a very convenient tool for dealing with a wide class of elastic contact problems. The MDR modeling framework introduces an equivalent punch profile and a one-dimensional Winkler-type elastic foundation, called henceforth Popov’s foundation. While the former mainly accounts for the geometry of contact configuration, the Popov foundation inherits the main characteristics of both the contact interface (like friction and adhesion) and the contacting elastic bodies (e.g., anisotropy, viscoelasticity or inhomogeneity). The discussion is illustrated with an example of the Kendall-type adhesive contact for an isotropic elastic half-space.


Elastic Contact, Winkler Foundation, Method of Dimensionality Reduction, Contact Stiffness, Adhesion Strength

Full Text:



Johnson, K.L., 1985, Contact Mechanics, Cambridge University Press, Cambridge.

Popov, V.L., 2010, Contact Mechanics and Friction: Physical Principles and Applications, Springer, Heidelberg, New York.

Kalker, J.J., 1990, Three-Dimensional Elastic Bodies in Rolling Contact, Kluwer Academic Publications, Dordrecht.

Goryacheva, I.G., 1998, Contact Mechanics in Tribology, Kluwer Academic Publishers, Dordrecht.

Argatov, I., Mishuris, G., 2015, Contact Mechanics of Articular Cartilage Layers: Asymptotic Models, Springer, Cham.

Wriggers, P., 2006, Computational Contact Mechanics, Springer, Berlin.

Yastrebov, V.A., 2013, Numerical Methods in Contact Mechanics, John Wiley & Sons, Hoboken.

Barber, J.R., 2018, Contact Mechanics, Springer, Cham.

Argatov, I., Mishuris, G., 2018, Indentation Testing of Biological Materials, Cham, Springer.

Hertz, H., 1882, Über die Berührung fester elastischer Körper, Journal für die reine und angewandte Mathematik, 92, pp. 156–171 (in German).

Popov, V.L., Heß, M., 2014, Method of Dimensionality Reduction in Contact Mechanics and Friction, Springer, Berlin.

Argatov, I., 2016, A discussion of the method of dimensionality reduction, Proceedings of the Institution of Mechanical Engineers, Pt. C: Journal of Mechanical Engineering Science 230(9), pp. 1424–1431.

Johnson, K.L., Kendall, K., Roberts, A.D., 1971, Surface energy and the contact of elastic solids, Proceedings of the Royal Society A, 324, pp. 301–313.

Frýba, L., 1995, History of Winkler foundation, Vehicle System Dynamics Supplement 24, pp. 7–12.

Argatov, I., Hess, M., Pohrt, R., Popov, V.L., 2016, The extension of the method of dimensionality reduction to non‐compact and non‐axisymmetric contacts, Journal of Applied Mathematics and Mechanics (ZAMM), 96(10), pp. 1144–1155.

Fabrikant, V.I., 1986, Flat punch of arbitrary shape on an elastic half-space, International Journal of Engineering Science 24, 1731–1740.

Galin, L.A., 2008, Contact Problems: The Legacy of L.A. Galin., In: Gladwell, G.M.L. (ed.). Springer, Dordrecht.

Sevostianov, I., Kachanov, M., 2004, Connection between elastic and conductive properties of microstructures with Hertzian contacts, Proceedings of the Royal Society A 460, pp. 1529–1534.

Holm, R., 1929, Uber metallische Kontaktwiderstände, Wissenschaftliche Veröffentlichungen aus den Siemens-Werken, 7, pp. 217–258 (in German).

Greenwood, J.A., 1966, Constriction resistance and the real area of contact, British Journal of Applied Physics, 17, pp. 1621–1622.

Argatov, I., Sevostianov, I., 2009, On relations between geometries of microcontact clusters and their overall properties, International Journal of Engineering Science, 47, pp. 959–973.

Popov, V.L., Psakhie, S.G., 2007, Numerical simulation methods in tribology, Tribology International, 40, pp. 916–923.

Geike, T., Popov, V.L., 2007, Mapping of three-dimensional contact problems into one dimension, Physical Review E, 76(3), 036710.

Popov, V.L., Heß, M., 2014, Method of dimensionality reduction in contact mechanics and friction: A user’s handbook. I. Axially-symmetric contacts, Facta Universitatis-Series Mechanical Engineering, 12(1), pp. 1–14.

Heß, M., Popov, V.L., 2016, Method of dimensionality reduction in contact mechanics and friction: A user’s handbook. II. Power-law graded materials, Facta Universitatis-Series Mechanical Engineering, 14(3), pp. 251–268.

Popov, V.L., Willert, E., Heß, M., 2018, Method of dimenionality reduction in contact mechanics and friction: A user’s handbook. III. Viscoelastic contacts. Facta Universitatis, Series Mechanical Engineering, 16(2), pp. 99–113.

Heß, M., 2011, Über die exakte Abbildung ausgewählter dreidimensionaler Kontakte auf Systeme mit niedrigerer räumlicher Dimension, Göttingen: Cuvillier Verlag (in German).

Aleksandrov, V.M., 1962, On the approximate solution of a certain type of integral equation, Journal of Applied Mathematics and Mechanics (PMM), 26, 1410–1424.

Argatov, I.I., Mishuris, G.S., Popov, V.L., 2016, Asymptotic modelling of the JKR adhsion contact for thin elastic layer, Quarterly Journal of Mechanics and Applied Mathematics, 69(2), pp. 161–179.

Maugis, D., 1995, Extension of the Johnson–Kendall–Roberts theory of the elastic contact of spheres to large contact radii, Langmuir, 11, pp. 679–682.

Johnson, K.L., Greenwood, J.A., 2005, An approximate JKR theory for elliptical contacts, Journal of Physics D: Applied Physics, 38, pp. 1042–1046.

Kendall, K., 1971, The adhesion and surface energy of elastic solids, Journal of Physics D: Applied Physics, 4, pp. 1186–1195.

Pólya, G., Szegö, G., 1951, Isoperimetric Inequalities in Mathematical Physics, Princeton Univ. Press, Princeton, NJ.

Li, Q., Argatov, I.I., Popov, V.L., 2018, Onset of detachment in adhesive contact of an elastic half-space and flat-ended punches with noncircular shape: Analytic estimations and comparison with numeric analysis, Journal of Physics D: Applied Physics, 51(14), 145601.

Popov, V.L., Pohrt, R., Li, Q., 2017, Strength of adhesive contacts: influence of contact geometry and material gradients, Friction, 5, pp. 308–325.

Kerr, A.D., 1964, Elastic and viscoelastic foundation models, Journal of Applied Mechanics, 31(3), pp. 491–498.

Dillard, D.A., Mukherjee, B., Karnal, P., Batra, R.C., Frechette, J., 2018, A review of Winkler’s foundation and its profound influence on adhesion and soft matter applications, Soft matter, 14(19), pp. 3669–3683.

Willert, E., Popov, V.L., 2017, Exact one‐dimensional mapping of axially symmetric elastic contacts with superimposed normal and torsional loading, Journal of Applied Mathematics and Mechanics (ZAMM), 97(2), pp. 173–182.

Argatov, I.I., Popov, V.L., 2016, Rebound indentation problem for a viscoelastic half-space and axisymmetric indenter — Solution by the method of dimensionality reduction, Journal of Applied Mathematics and Mechanics (ZAMM), 96(8), pp. 956–967.

Hess, M., 2016, A simple method for solving adhesive and non-adhesive axisymmetric contact problems of elastically graded materials, International Journal of Engineering Science, 104, pp. 20–33.

Argatov, I., Hess, M., Popov, V.L., 2018, The extension of the method of dimensionality reduction to layered elastic media, Journal of Applied Mathematics and Mechanics (ZAMM), 98(4), pp. 622–634.



  • There are currently no refbacks.

ISSN: 0354-2025 (Print)

ISSN: 2335-0164 (Online)

COBISS.SR-ID 98732551

ZDB-ID: 2766459-4