The steady-state dynamic contact of a viscoelastic FGM-coated half-plane under a rigid flat punch

doi:10.1007/s10483-026-3332-7

Abstract

Abstract:

This study explores the dynamic contact response of a viscoelastic functionally graded material (FGM)-coated half-plane under a rigid flat punch subjected to a time-harmonic vertical force. The elastic modulus and mass density of the FGM coating vary exponentially along the thickness direction. The FGM coating and the homogeneous half-plane possess viscoelastic properties, which are described by a linearly hysteretic damping model. By applying the asymptotic method and the Fourier integral transform technique, the contact problem is converted into a Cauchy singular integral equation. The effects of excitation frequency, gradient index, damping factor ratio, and punch width on the vertical impedance and dynamic contact stress are analyzed. The results indicate that adjusting the gradient index of the FGM coating can significantly affect the contact stress and vertical impedance.

Key words: dynamic contact, functionally graded material (FGM) coating, hysteretic damping, flat punch

2010 MSC Number:

O343

Xiaomin WANG, Liaoliang KE, Jie SU, Junhong GUO. The steady-state dynamic contact of a viscoelastic FGM-coated half-plane under a rigid flat punch. Applied Mathematics and Mechanics (English Edition), 2026, 47(1): 61-76.

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Figures/Tables 10

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References 47

[1]	SURESH, S. and MORTENSEN, A. Fundamentals of Functionally Graded Materials: Processing and Thermomechanical Behavior of Graded Metals and Metal-ceramic Composites, IOM Communications Ltd., London (1998)
[2]	SATHISH, M., RADHIKA, N., and SALEH, B. A critical review on functionally graded coatings: methods, properties, and challenges. Composites Part B: Engineering, 225, 109278 (2021)
[3]	NAEBE, M. and SHIRVANIMOGHADDAM, K. Functionally graded materials: a review of fabrication and properties. Applied Materials Today, 5, 223–245 (2016)
[4]	SALEH, B., JIANG, J. H., FATHI, R., AL-HABABI, T., XU, Q., WANG, L. S., SONG, D., and MA, A. B. 30 Years of functionally graded materials: an overview of manufacturing methods, applications and future challenges. Composites Part B: Engineering, 201, 108376 (2020)
[5]	SINGH, B., MULIK, R. S., and HARSHA, S. P. Dynamic response analysis of functionally graded gears. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials-Design and Applications, 237(1), 52–69 (2023)
[6]	SURESH, S. Graded materials for resistance to contact deformation and damage. Science, 292, 2447–2451 (2001)
[7]	GULER, M. A. and ERDOGAN, F. Contact mechanics of graded coatings. International Journal of Solids and Structures, 41, 3865–3889 (2004)
[8]	GULER, M. A. and ERDOGAN, F. Contact mechanics of two deformable elastic solids with graded coatings. Mechanics of Materials, 38, 633–647 (2006)
[9]	GULER, M. A. and ERDOGAN, F. The frictional sliding contact problems of rigid parabolic and cylindrical stamps on graded coatings. International Journal of Mechanical Sciences, 49, 161–182 (2007)
[10]	RHIMI, M., EL-BORGI, S., SAÏD, W. B., and JEMAA, F. B. A receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate. International Journal of Solids and Structures, 46, 3633–3642 (2009)
[11]	EL-BORGI, S., USMAN, S., and GÜLER, M. A. A frictional receding contact plane problem between a functionally graded layer and a homogeneous substrate. International Journal of Solids and Structures, 51, 4462–4476 (2014)
[12]	EL-BORGI, S. and ÇÖMEZ, I. A receding frictional contact problem between a graded layer and a homogeneous substrate pressed by a rigid punch. Mechanics of Materials, 114, 201–214 (2017)
[13]	ÇÖMEZ, I., EL-BORGI, S., and YILDIRIM, B. Frictional receding contact problem of a functionally graded layer resting on a homogeneous coated half-plane. Archive of Applied Mechanics, 90, 2113–2131 (2020)
[14]	RHIMI, M., EL-BORGI, S., and LAJNEF, N. A double receding contact axisymmetric problem between a functionally graded layer and a homogeneous substrate. Mechanics of Materials, 43, 787–798 (2011)
[15]	CHOI, H. J. and PAULINO, G. H. Thermoelastic contact mechanics for a flat punch sliding over a graded coating/substrate system with frictional heat generation. Journal of the Mechanics and Physics of Solids, 56, 1673–1692 (2008)
[16]	CHEN, P. J. and CHEN, S. H. Partial slip contact between a rigid punch with an arbitrary tip-shape and an elastic graded solid with a finite thickness. Mechanics of Materials, 59, 24–35 (2013)
[17]	TIAN, X. J., ZHOU, Y. T., WANG, L. H., and DING, S. H. Surface contact behavior of functionally graded thermoelectric materials indented by a conducting punch. Applied Mathematics and Mechanics (English Edition), 42(5), 649–664 (2021) https://doi.org/10.1007/s10483-021-2732-8
[18]	PENG, J., LI, D. K., HUANG, Z. X., PENG, G. J., CHEN, P. J., and CHEN, S. H. Interfacial behavior of a thermoelectric film bonded to a graded substrate. Applied Mathematics and Mechanics (English Edition), 44(11), 1853–1870 (2023) https://doi.org/10.1007/s10483-023-3045-8
[19]	KE, L. L. and WANG, Y. S. Two-dimensional contact mechanics of functionally graded materials with arbitrary spatial variations of material properties. International Journal of Solids and Structures, 43, 5779–5798 (2006)
[20]	KE, L. L. and WANG, Y. S. Two-dimensional sliding frictional contact of functionally graded materials. European Journal of Mechanics-A/Solids, 26, 171–188 (2007)
[21]	LIU, T. J., WANG, Y. S., and ZHANG, C. Z. Axisymmetric frictionless contact of functionally graded materials. Archive of Applied Mechanics, 78, 267–282 (2008)
[22]	LIU, T. J., WANG, Y. S., and XING, Y. M. The axisymmetric partial slip contact problem of a graded coating. Meccanica, 47, 1673–1693 (2012)
[23]	XIAO, S. and YUE, Z. Q. Complete solutions for elastic fields induced by point load vector in functionally graded material model with transverse isotropy. Applied Mathematics and Mechanics (English Edition), 44(3), 411–430 (2023) https://doi.org/10.1007/s10483-023-2958-8
[24]	LIU, Y. W., XU, Y. Q., LI, J. P., and XU, H. Three-dimensional semiperiodic line-contact mixed lubrication analysis of functionally graded coated gear pairs. International Journal of Applied Mechanics, 17(5), 2550033 (2025)
[25]	MUNIRAJ, D., VIGNESH, S., and SREEHARI, V. M. Mechanical characterization and impact damage assessment of Al/SiC functionally graded coating under elevated temperatures. Scientific Reports, 14(1), 29134 (2024)
[26]	SHEN, M. Y., WANG, Q. S., and WANG, R. H. Investigation on the vibration mechanisms of a rotating FG-GPLRC shaft-disk-shell combined system. Structures, 56, 105049 (2023)
[27]	CHEN, Y. K., YE, T. G., JIN, G. Y., ZHONG, S. F., LV, W. B., and MAO, Q. Z. Thermoelastic vibration analysis of rotational pre-twisted and curved blades covered with functionally graded thermal barrier coatings. Mechanical Systems and Signal Processing, 222, 111721 (2025)
[28]	TOKTAS, S. E. and DAG, S. Stresses in multi-layer coatings in Hertzian contact with a moving circular punch. Tribology International, 171, 107565 (2022)
[29]	ÇÖMEZ, I. Contact problem for a functionally graded layer indented by a moving punch. International Journal of Mechanical Sciences, 100, 339–344 (2015)
[30]	ÇÖMEZ, I. Frictional moving contact problem between a functionally graded monoclinic layer and a rigid punch of an arbitrary profile. Acta Mechanica, 233, 1435–1453 (2022)
[31]	KARABULUT, P. M., ÇETINKAYA, I. T., OĞUZ, H., and ÇÖMEZ, I. Moving contact problem of a functionally graded orthotropic coated half plane. Acta Mechanica, 235(6), 3989–4004 (2024)
[32]	LYU, X., KE, L. L., TIAN, J. Y., and SU, J. Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch. Applied Mathematics and Mechanics (English Edition), 43, 1187–1202 (2022) https://doi.org/10.1007/s10483-022-2885-7
[33]	ÇÖMEZ, I. Analysis of stress and deformation of an exponentially graded viscoelastic coated half plane under indentation by a rigid flat punch indenter tip. Mechanics of Time-Dependent Materials, 28(3), 1271–1289 (2024)
[34]	WANG, X. M., KE, L. L., and WANG, Y. S. The dynamic contact response of the FGM coated half-plane with hysteretic damping under time harmonic force. Structural Engineering and Mechanics, 87(1), 95–106 (2023)
[35]	MYKLESTAD, N. O. The concept of complex damping. Journal of Applied Mechanics, 19(3), 20–30 (1952)
[36]	ERDOGAN, F. and GUPTA, G. D. On the numerical solution of singular integral equations. Quarterly of Applied Mathematics, 29(4), 525–534 (1972)
[37]	VOROVICH, I. I., ALEKSANDROV, V. M., and BABESHKO, V. A. Non-Classical Mixed Problems of the Theory of Elasticity, Nauka, Moscow (1974)
[38]	ALEKSANDROV, V. M. and ZELENTSOV, V. B. Asymptotic methods of solving two-dimensional dynamic problems of a viscoelastic layer with mixed boundary conditions. Journal of Applied Mathematics and Mechanics, 45(2), 238–243 (1981)
[39]	ALEKSANDROV, V. M. and POZHARSKII, D. A. An asymptotic method in contact problems. Journal of Applied Mathematics and Mechanics, 63(2), 283–290 (1999)
[40]	WANG, X. M., KE, L. L., and WANG, Y. S. The dynamic contact of a viscoelastic coated half-plane under a rigid flat punch. Mechanics Based Design of Structures and Machines, 51(10), 5925–5940 (2023)
[41]	LUCO, J. E. and WESTMANN, R. A. Dynamic response of a rigid footing bonded to an elastic half space. Journal of Applied Mechanics, 39(2), 527–534 (1972)
[42]	HUANG, Y. M., HOU, S. S., YANG, L. J., TIAN, G., YONG, Z., and LIU, S. Y. Effect of arc dynamic behavior on deposition quality of additive manufactured aluminum alloys. Journal of Materials Processing Technology, 295, 117172 (2021)
[43]	HUANG, Y. M., ZHANG, F., YUAN, J., JIA, C. P., REN, X. K., and YANG, L. J. Investigation on surface morphology and microstructure of double-wire+ arc additive manufactured aluminum alloys based on spectral analysis. Journal of Manufacturing Processes, 84, 639–651 (2022)
[44]	BAI, X. R., XIE, H. N., ZHANG, X., ZHAO, D. D., RONG, X. D., JIN, S. B., LIU, E. Z., ZHAO, N. Q., and HE, C. N. Heat-resistant super-dispersed oxide strengthened aluminium alloys. Nature Materials, 23, 747–754 (2024)
[45]	HRYNIEWICZ, Z. Dynamic response of a rigid strip on an elastic half-space. Computer Methods in Applied Mechanics and Engineering, 25(3), 355–364 (1981)
[46]	KARASUDHI, P., KEER, L. M., and LEE, S. L. Vibratory motion of a body on an elastic half plane. Journal of Applied Mechanics, 35(4), 697–705 (1968)
[47]	TIAN, J. Y. and XIE, Z. M. Dynamic contact stiffness of vibrating rigid sphere contacting semi-infinite transversely isotropic viscoelastic solid. Acta Mechanica Solida Sinica, 21, 580–588 (2008)