Mixed elastohydrodynamic lubrication contact of piezoelectric materials with different Gaussian rough surfaces

doi:10.1007/s10483-026-3380-6

Abstract

Abstract:

The mixed elastohydrodynamic lubrication (EHL) behavior of a smooth, rigid, insulating cylindrical indenter in contact with transversely isotropic piezoelectric half-planes possessing Gaussian-distributed surface roughness is analyzed. Three distinct surface topographies are considered: longitudinally oriented, isotropic, and transversely oriented. The lubricant is assumed to exhibit non-Newtonian flow characteristics, and its density and viscosity are modeled to be pressure-dependent. A modified Reynolds equation, incorporating both pressure and shear flow factors, is utilized to compute the hydrodynamic pressure distribution within the lubricating film. An iterative computational scheme is developed for the coupled resolution of the modified Reynolds equation, flow rheology equations, asperity contact pressure equation, load balance equation, and film thickness equation. Parametric investigations are conducted to examine the influence of the total normal load, entrainment velocity, hydrodynamic roughness parameter, slide-to-roll ratio, contact roughness parameter, and surface pattern parameter on the film thickness, asperity contact pressure, and fluid hydrodynamic pressure. The results obtained may provide valuable insights for mitigating surface degradation at piezoelectric contact interfaces and enhancing the operational reliability of associated electromechanical systems.

Key words: contact mechanics, mixed elastohydrodynamic lubrication (EHL), Gaussian rough surface, piezoelectric material, iterative method

2010 MSC Number:

O373

Zhengzhe XIE, S. EL-BORGI, Jie SU, Liaoliang KE. Mixed elastohydrodynamic lubrication contact of piezoelectric materials with different Gaussian rough surfaces. Applied Mathematics and Mechanics (English Edition), 2026, 47(5): 965-984.

TrendMD

Add to citation manager EndNote|Reference Manager|ProCite|BibTeX|RefWorks

URL: https://www.amm.shu.edu.cn/EN/10.1007/s10483-026-3380-6

https://www.amm.shu.edu.cn/EN/Y2026/V47/I5/965

Figures/Tables 16

Fig. 1

Fig. 2

Fig. 3

Table 1

Coefficients of Eq. (5) for ϕx"

γ	C	r	$h / σ$
1/6	1.38	0.42	$> 1$
1	0.90	0.56	$> 0.5$
6	0.52	1.50	$> 0.5$

Table 1

Table 2

Coefficients of Eq. (6) for ϕs"

γ	A₁	$α 1$	$α 2$	$α 3$	A₂
1/6	1.962	1.08	0.77	0.03	1.754
1	1.899	0.98	0.92	0.05	1.126
6	1.290	0.62	1.09	0.08	0.388

Table 2

Fig. 4

Table 3

Electro-mechanical properties of piezoceramic materials"

Parameter	PZT-4	PZT-5H	PZT-5A	BaTiO₃
$c 11$ /GPa	139	126	121	166
$c 13$ /GPa	74.3	53	75.2	77.5
$c 33$ /GPa	115	117	111	162
$c 44$ /GPa	25.6	35.3	21.1	42.9
$e 31 / (C ⋅ m − 2)$	$− 5.2$	$− 6.5$	$− 5.4$	$− 4.4$
$e 33 / (C ⋅ m − 2)$	15.1	23.3	15.8	18.6
$e 15 / (C ⋅ m − 2)$	12.7	17	12.3	11.6
$ε 11$ /(C·V^-1·m^-1)	$64.61 × 10 − 10$	$151 × 10 − 10$	$81.07 × 10 − 10$	$111.51 × 10 − 10$
$ε 33$ /(C·V^-1·m^-1)	$56.2 × 10 − 10$	$130 × 10 − 10$	$73.46 × 10 − 10$	$125.67 × 10 − 10$

Table 3

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

References 58

[1]	LIANG, Y. W., SATAKE, U., and ENOMOTO, T. Elastohydrodynamic lubrication in rotational atherectomy: fluid film thickness characteristics and reduction of vascular damage risk. Tribology International, 210, 110798 (2025)
[2]	DOWSON, D. and HIGGINSON, G. R. A numerical solution to the elasto-hydrodynamic problem. Journal of Mechanical Engineering Science, 1, 6–15 (1959)
[3]	WEI, J., ZHANG, A. Q., and GAO, P. A study of spur gear pitting under EHL conditions: theoretical analysis and experiments. Tribology International, 94, 146–154 (2016)
[4]	OGURA, T., OKUDA, A., IMANISHI, M., MIYANO, A., AMANO, M., NISHIOKA, N., KAMIYAMA, R., and HIGUCHI, K. Electrohydraulic lithotripsy for pancreatic duct stones under digital single-operator pancreatoscopy (with video). Digestive Diseases and Sciences, 64, 1377–1382 (2019)
[5]	GRUBIN, A. N. Fundamentals of the Hydrodynamic Theory of Lubrication of Heavily Loaded Cylindrical Surfaces, Central Scientific Research Institute for Technology and Mechanical Engineering, Moscow (1949)
[6]	HABCHI, W. A Schur-complement model-order-reduction technique for the finite element solution of transient elastohydrodynamic lubrication problems. Advances in Engineering Software, 127, 28–37 (2019)
[7]	XIAO, H. F., SUN, Y. Y., and XU, J. W. Investigation into the normal contact stiffness of rough surface in line contact mixed elastohydrodynamic lubrication. Tribology Transactions, 61, 742–753 (2018)
[8]	KUMAR, R., AZAM, M. S., and GHOSH, S. K. Influence of stochastic roughness on performance of a Rayleigh step bearing operating under thermo-elastohydrodynamic lubrication considering shear flow factor. Tribology International, 134, 264–280 (2019)
[9]	EBNER, M., ZIEGLTRUM, A., LOHNER, T., MICHAELIS, K., and STAHL, K. Measurement of EHL temperature by thin film sensors — thermal insulation effects. Tribology International, 149, 105515 (2020)
[10]	SINGH, A., WOLF, M., JACOBS, G., and KÖNIG, F. Machine learning based surrogate modelling for the prediction of maximum contact temperature in EHL line contacts. Tribology International, 179, 108166 (2023)
[11]	LIU, H. C., ZHANG, B. B., SCHNEIDER, V., VENNER, C. H., and POLL, G. Two-dimensional generalized non-Newtonian EHL lubrication: shear rate-based solution versus shear stress-based solution. Proceedings of the Institution of Mechanical Engineers, Part J: Journal of Engineering Tribology, 235, 2626–2639 (2021)
[12]	GUO, C., ZHAO, E. H., QUAN, L., and WANG, C. W. Impact of speed on oil film interface slip and elastohydrodynamic lubrication characteristics. Surface Technology, 54, 62–70, 117 (2025)
[13]	KALIN, M. and KUS, M. New strategy for reducing the EHL friction in steel contacts using additive-formed oleophobic boundary films. Friction, 9, 1346–1360 (2021)
[14]	GUO, L., LI, G., and GAN, Z. L. Effects of surface roughness on CMAS corrosion behavior for thermal barrier coating applications. Journal of Advanced Ceramics, 10, 472–481 (2021)
[15]	WANG, S. H., YUAN, W. K., LIANG, X. M., and WANG, G. F. A new analytical model for the flattening of Gaussian rough surfaces. European Journal of Mechanics: A, 94, 104578 (2022)
[16]	REN, J. and YUAN, H. Contact analysis and friction prediction of non-Gaussian random surfaces. Applied Sciences, 12, 11237 (2022)
[17]	GUO, X., MA, B. B., and ZHU, Y. C. A magnification-based multi-asperity (MBMA) model of rough contact without adhesion. Journal of the Mechanics and Physics of Solids, 133, 103724 (2019)
[18]	LI, L., WANG, J. J., SHI, X. H., MA, S. L., and CAI, A. J. Contact stiffness model of joint surface considering continuous smooth characteristics and asperity interaction. Tribology Letters, 69, 43 (2021)
[19]	YU, X., SUN, Y. Y., and WU, S. J. Multi-stage contact model between fractal rough surfaces based on multi-scale asperity deformation. Applied Mathematical Modelling, 109, 229–250 (2022)
[20]	GREENWOOD, J. A. and WILLIAMSON, J. B. P. Contact of nominally flat surfaces. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 295, 300–319 (1966)
[21]	MAJUMDAR, A. and BHUSHAN, B. Fractal model of elastic-plastic contact between rough surfaces. Journal of Tribology, 113, 1–11 (1991)
[22]	PERSSON, B. N. J. Elastoplastic contact between randomly rough surfaces. Physical Review Letters, 87, 116101 (2001)
[23]	GAO, Z. Q., ZHANG, Y., WEI, X., PENG, L. X., FU, W. P., WANG, W., and HU, W. P. Study on normal contact stiffness characteristics in non-Gaussian rough surfaces under mixed lubrication. Acta Mechanica, 235, 3535–3555 (2024)
[24]	WANG, G. F., LIANG, X. M., and YAN, D. An incremental equivalent circular contact model for rough surfaces. Journal of Tribology, 143, 081503 (2021)
[25]	AUTIERO, M., PAOLI, G., CIRELLI, M., and VALENTINI, P. P. Influence of profile modifications on spur gear sliding power losses: an integrated approach with advanced mesh stiffness and partial EHL. Mechanism and Machine Theory, 214, 106118 (2025)
[26]	FENG, Y., SHI, X. J., LU, X. Q., SUN, W., LIU, K. P., and FEI, Y. F. Predictions of friction and wear in ball bearings based on a 3D point contact mixed EHL model. Surface and Coatings Technology, 502, 131939 (2025)
[27]	GU, C. X., SHENG, X. L., ZHANG, D., and MENG, X. H. Thermal mixed elastohydrodynamic lubrication modeling and analysis of the lubricated non-conformal contacts with non-Gaussian surface roughness and coating. Tribology International, 194, 109541 (2024)
[28]	SHUKLA, S. and DEHERI, G. Combine influence of surface roughness and deformation on the performance of elastohydrodynamic lubrication. AIP Conference Proceedings, 3149, 090008 (2024)
[29]	TIAN, H. X., ZHAO, X. J., HUANG, W. K., and MA, H. A stiffness model for EHL contact on smooth/rough surfaces and its application in mesh stiffness calculation of the planetary gear set. Tribology International, 196, 109720 (2024)
[30]	YOU, S. Y., TANG, J. Y., and WANG, Q. A new 3D plastoelastohydrodynamic lubrication model for rough surfaces. Friction, 12, 1176–1193 (2024)
[31]	WANG, D. S. and LIN, J. F. Effect of surface roughness on elastohydrodynamic lubrication of line contacts. Tribology International, 24, 51–62 (1991)
[32]	MAJUMDAR, B. C. and HAMROCK, B. J. Effect of surface roughness on elastohydrodynamic line contact. Journal of Lubrication Technology, 104, 401–407 (1982)
[33]	PEI, J. X., HAN, X., TAO, Y. R., and FENG, S. Z. Mixed elastohydrodynamic lubrication analysis of line contact with non-Gaussian surface roughness. Tribology International, 151, 106449 (2020)
[34]	CHTIBA, M. O., CHOURA, S., NAYFEH, A. H., and EL-BORGI, S. Vibration confinement and energy harvesting in flexible structures using collocated absorbers and piezoelectric devices. Journal of Sound and Vibration, 329, 261–276 (2010)
[35]	JIN, H. N., GAO, X. Y., REN, K. L., LIU, J. F., QIAO, L., LIU, M. Z., CHEN, W., HE, Y. H., DONG, S. X., XU, Z., and LI, F. Review on piezoelectric actuators based on high-performance piezoelectric materials. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 69, 3057–3069 (2022)
[36]	SMITH, M. and KAR-NARAYAN, S. Piezoelectric polymers: theory, challenges and opportunities. International Materials Reviews, 67, 65–88 (2022)
[37]	SEKHAR, M. C., VEENA, E., KUMAR, N. S., NAIDU, K. C. B., MALLIKARJUNA, A., and BASHA, D. B. A review on piezoelectric materials and their applications. Crystal Research and Technology, 58, 2200130 (2023)
[38]	ZASZCZYŃSKA, A., ZABIELSKI, K., GRADYS, A., KOWALCZYK, T., and SAJKIEWICZ, P. Piezoelectric scaffolds as smart materials for bone tissue engineering. Polymers, 16, 2797 (2024)
[39]	ÇÖMEZ, İ. Frictional moving contact problem between a conducting rigid cylindrical punch and a functionally graded piezoelectric layered half plane. Meccanica, 56, 3039–3058 (2021)
[40]	SUN, Y. Y., SU, J., SONG, H. X., and KE, L. L. The size-dependent elastohydrodynamic lubrication contact of piezoelectric materials. Acta Mechanica, 237, 849–869 (2026)
[41]	PENG, D. M. and LI, X. Y. Fractal contact analysis for transversely isotropic piezoelectric materials: theoretical and numerical predictions. Tribology International, 181, 108323 (2023)
[42]	ZHANG, Q., PIAO, S. C., and CHEN, H. J. A theoretical model of the intermittent contact of piezoelectric actuator based on Greenwood-Williamson theory. Ultrasonics, 114, 106428 (2021)
[43]	WANG, S. W., SUN, Y. Y., SU, J., and KE, L. L. Optimal solution for size-dependent sliding frictional contact of piezoelectric materials. International Journal of Mechanical Sciences, 302, 110557 (2025)
[44]	PATIR, N. and CHENG, H. S. An average flow model for determining effects of three-dimensional roughness on partial hydrodynamic lubrication. ASME Journal of Lubrication Technology, 100, 12–17 (1978)
[45]	YANG, P. R. and WEN, S. Z. A generalized Reynolds equation for non-Newtonian thermal elastohydrodynamic lubrication. Journal of Tribology, 112, 631–636 (1990)
[46]	PATIR, N. and CHENG, H. S. Application of average flow model to lubrication between rough sliding surfaces. ASME Journal of Lubrication Technology, 101, 220–229 (1979)
[47]	DOWSON, D. and HIGGINSON, G. R. Elasto-Hydrodynamic Lubrication, the Fundamentals of Roller and Gear Lubrication, Pergamon Press, Oxford (1966)
[48]	ROELANDS, C. Correlation Aspects of the Viscosity-Temperature-Pressure Relationship of Lubrication Oils, Technische Hogeschool Delf, The Netherlands (1966)
[49]	HOUPERT, L. New results of traction force calculations in elastohydrodynamic contacts. Journal of Tribology, 107, 241–245 (1985)
[50]	SU, J., KE, L. L., and WANG, Y. S. Two-dimensional fretting contact of piezoelectric materials under a rigid conducting cylindrical punch. Journal of Mechanics of Materials and Structures, 11, 535–558 (2016)
[51]	GRADSHTEYN, I. S. and RYZHIK, I. M. Table of Integrals, Series, and Products, Academic Press, New York (2000)
[52]	JOHSON, K. L. Contact Mechanics, Cambridge University Press, Cambridge (1985)
[53]	YANG, P. R. and WEN, S. Z. A forward iterative numerical method for steady-state elastohydrodynamically lubricated line contacts. Tribology International, 23, 17–22 (1990)
[54]	GREENWOOD, J. A. and TRIPP, J. H. The contact of two nominally flat rough surfaces. Proceedings of the Institution of Mechanical Engineers, 185, 625–633 (1970)
[55]	SU, J., KE, L. L., and WANG, Y. S. Axisymmetric frictionless contact of a functionally graded piezoelectric layered half-space under a conducting punch. International Journal of Solids and Structures, 90, 45–59 (2016)
[56]	SACKFIELD, A., HILLS, D. A., and NOWELL, D. Mechanics of Elastic Contacts, Elsevier (2013)
[57]	DOWSON, D. and HIGGINSON, G. R. A numerical solution to the elasto-hydrodynamic problem. Journal of Mechanical Engineering Science, 1, 6–15 (1959)
[58]	SU, J., KE, L. L., and WANG, Y. S. Elastohydrodynamic lubrication line contact of piezoelectric materials. International Journal of Mechanical Sciences, 163, 105145 (2019)