Justus Benad

DOI Number
First page
Last page


In the present paper a numerical implementation technique for the transformations of the Method of Dimensionality Reduction (MDR) is described. The MDR has become, in the past few years, a standard tool in contact mechanics for solving axially-symmetric contacts. The numerical implementation of the integral transformations of the MDR can be performed in several different ways. In this study, the focus is on a simple and robust algorithm on the uniform grid using integration by parts, a central difference scheme to obtain the derivatives, and a trapezoidal rule to perform the summation. The results are compared to the analytical solutions for the contact of a cone and the Hertzian contact. For the tested examples, the proposed method gives more accurate results with the same number of discretization points than other tested numerical techniques. The implementation method is further tested in a wear simulation of a heterogeneous cylinder composed of rings of different material having the same elastic properties but different wear coefficients. These discontinuous transitions in the material properties are handled well with the proposed method.


Normal Contact, Method of Dimensionality Reduction, Stress, Wear

Full Text:



Popov, V.L., Psakhie, S., 2007, Numerical simulation methods in tribology, Tribology International, 40(6), pp. 916-923.

Popov, V.L., Heß, M., 2016, Method of dimensionality reduction in contact mechanics and friction, Springer, Berlin.

Popov, M., Benad, J., Popov, V.L., Heß, M., 2015, Acoustic Emission in Rolling Contacts, Method of Dimensionality Reduction in Contact Mechanics and Friction, Springer, Berlin, pp. 207-214.

Dimaki, A., Dmitriev, A., Chai, Y., Popov, V.L., 2014, Rapid simulation procedure for fretting wear on the basis of the method of dimensionality reduction, International Journal of Solids and Structures, 51(25-26), pp. 4215-4220.

Dimaki, A., Dmitriev, A., Menga, N., Papangelo, A., Ciavarella, M., Popov, V.L., 2016, Fast high-resolution simulation of the gross slip wear of axially symmetric contacts, Tribology Transactions, 59(1), pp. 189-194.

Li, Q., Forsbach, F., Schuster, M., Pielsticker, D., Popov, V.L., 2018, Wear Analysis of a Heterogeneous Annular Cylinder, Lubricants, 6(1), 28.

Murio, D., Hinestroza, D., Mejía, C., 1992, New stable numerical inversion of Abel's integral equation, Computers & Mathematics with Applications, 23(11), pp. 3-11.

Hansen, E., Law, P., 1985, Recursive methods for computing the Abel transform and its inverse, Journal of the Optical Society of America A, 2(4), pp. 510-520.

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

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.

Popov, V.L., Heß, M., Willert, E., 2017, Handbuch der Kontaktmechanik, Springer, Berlin.

Archard, J., Hirst, W., 1956, The wear of metals under unlubricated conditions, Proc. R. Soc. Lond. A, 236(1206), pp. 397-410.

Pohrt, R., Li, Q., 2014, Complete Boundary Element Formulation for Normal and Tangential Contact Problems, Physical Mesomechanics, 17(4), pp. 334-340.



  • There are currently no refbacks.

ISSN: 0354-2025 (Print)

ISSN: 2335-0164 (Online)

COBISS.SR-ID 98732551

ZDB-ID: 2766459-4