### FFT-BASED IMPLEMENTATION OF THE MDR TRANSFORMATIONS FOR HOMOGENEOUS AND POWER-LAW GRADED MATERIALS

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#### Abstract

^{2}for the standard matrix-vector-multiplication (MVM) algorithm to the order of N. Convergence and performance of the proposed method are studied in detail. As an illustrative example a fretting wear simulation based on the new implementation is shown, the results of which are compared to the standard MVM implementation.

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Popov, V.L., Heß, M., Willert, E., 2019, Handbook of contact mechanics: exact solutions of axisymmetric contact problems, Springer-Verlag, Berlin Heidelberg, 347 p.

Barquins, M., Maugis, D., 1982, Adhesive contact of axisymmetric punches on an elastic half-space: the modified Hertz-Huber’s stress tensor for contacting spheres, Journal de Mécanique Théorique et Appliquée, 1(2), pp. 331-357.

Jäger, J., 1998, A New Principle in Contact Mechanics, Journal of Tribology, 120(4), pp. 677-684.

Ciavarella, M., 1998, Tangential Loading of General Three-Dimensional Contacts, Journal of Applied Mechanics, 65(4), pp. 998-1003.

Lee, E.H., Radok, J.R.M., 1960, The Contact Problem for Viscoelastic Bodies, Journal of Applied Mechanics, 27(3), pp. 438-444.

Schubert, G., 1942, Zur Frage der Druckverteilung unter elastisch gelagerten Tragwerken, Ingenieur-Archiv, 13(3), pp. 132-147.

Popov, V.L., Heß, M., 2015, Method of Dimensionality Reduction in Contact Mechanics and Friction, Springer-Verlag, Berlin Heidelberg, 265 p.

Benad, J., 2018, Fast numerical implementation of the MDR transformations, Facta Universitatis-Series Mechanical Engineering, 16(2), pp. 127-138.

Dimaki, A.V., Dmitriev, A.I., Chai, Y.S., 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.

Liu, Z., Meyers, M.A., Zhang, Z., Ritchie, R.O., 2017, Functional gradients and heterogeneities in biological materials: Design principles, functions, and bioinspired applications, Progress in Materials Science, 88, pp. 467-498.

Selvadurai, A.P., 2007, The Analytical Method in Geomechanics, Applied Mechanics Reviews, 60, 3, pp. 87-106.

Giannakopoulos, A.E., Suresh, S., 1997, Indentation of solids with gradients in elastic properties: part II. axisymmetric indentors, International Journal of Solids and Structures, 34(19), pp. 2393-2428.

Jin, F., Guo, X., Zhang, W., 2013, A unified treatment of axisymmetric adhesive contact on a power-law graded elastic half-space, Journal of Applied Mechanics, 80(6), 061024.

Chen, S., Yan, C., Zhang, P., Gao, H., 2009, Mechanics of adhesive contact on a power-law graded elastic half-space, Journal of the Mechanics and Physics of Solids, 57(9), pp. 1437-1448.

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.

Heß, M., 2016, Normal, tangential and adhesive contacts between power-law graded materials, Presentation at the Workshop on Tribology and Contact Mechanics in Biological and Medical Applications, Technische Universität Berlin, November 14-17, Berlin.

Paulino, G.H., Jin, Z.H., 2001, Correspondence principle in viscoelastic functionally graded materials, Journal of Applied Mechanics, 68(1), pp. 129-132.

Willert, E., 2020, Ratio of loss and storage moduli determines restitution coefficient in low-velocity viscoelastic impacts, Frontiers in Mechanical Engineering, 6, 3.

Willert, E., Dmitriev, A.I., Psakhie, S.G., Popov, V.L., 2019, Effect of elastic grading on fretting wear, Scientific Reports, 9, 7791.

Bracewell, R.N., 2000, The Fourier Transform and Its Applications, 3rd Edition, New York: McGraw-Hill Book Company, 636 p.

Willert, E., 2020, Stoßprobleme in Physik, Technik und Medizin: Grundlagen und Anwendungen, Springer Vieweg, Berlin, 241 p.

Willert, E., 2021, A simple semi-analytic contact mechanical model for tangential and torsional fretting wear of axisymmetric contacts, Symmetry, submitted manuscript.

Popov, V.L., 2014, Analytic solution for the limiting shape of profiles due to fretting wear, Scientific Reports, 4, 3749.

Hickstein, D.D., Gibson, S.T., Yurchak, R., Das, D.D., Ryazanov, M., 2019, A direct comparison of high-speed methods for the numerical Abel transform, Review of Scientific Instruments, 90, 065115.

DOI: https://doi.org/10.22190/FUME210415057W

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