Coupled effects of surface elasticity, couple stresses, and adhesion in nanocontact mechanics

doi:10.1007/s10483-025-3267-6

摘要/Abstract

Abstract:

This paper investigates the adhesive nanocontact behavior of an elastic half-plane indented by a rigid cylindrical indenter, incorporating the simultaneous effects of surface elasticity, couple stresses, and adhesion. The free surface of the half-plane is modeled by the Steigmann-Ogden surface elasticity theory, while the bulk material behavior is described by the classical couple-stress elasticity theory. The adhesion at the contact interface is characterized by the Maugis-Dugdale (MD) adhesive contact model. Building on the fundamental nonclassical Flamant solution, the governing equations and boundary conditions of the nanocontact problem are reformulated into a system of triple integral equations. These equations are solved numerically by the Gauss-Chebyshev quadratures in combination with an iterative algorithm. The validation against the existing literature confirms the accuracy and robustness of the proposed solution methodology. Comprehensive parametric studies are performed to elucidate the critical roles of surface elasticity and couple stresses in adhesive nanocontact. The numerical results provide insights into the complex interactions among surface, couple-stress, and adhesive effects. Specifically, the interplay between the surface and adhesive effects is predominantly competitive, while the interaction between the couple stresses and adhesion exhibits an intricate nature. The findings highlight the necessity of simultaneously considering surface elasticity, couple stresses, and adhesion in nanoindentation analyses to achieve accurate predictions of material responses.

Key words: adhesive nanocontact, Maugis-Dugdale (MD) adhesion, Steigmann-Ogden surface effect, classical couple-stress theory, Gauss-Chebyshev quadrature

中图分类号:

O343.3

. [J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(7): 1237-1260.

Youxue BAN, Xinyao YANG, Q. X. LI, Changwen MI. Coupled effects of surface elasticity, couple stresses, and adhesion in nanocontact mechanics[J]. Applied Mathematics and Mechanics (English Edition), 2025, 46(7): 1237-1260.

图/表 11

Comparative pair	Tabor's parameter	$P ¯ = 1$	$P ¯ = 4$
Reductions due to surface effects	$μ MD = 0$	38.79%	22.55%
$Δ$ vs. $○$	$μ MD = 4$	25.29%	19.35%
Reductions due to couple stresses	$μ MD = 0$	31.80%	30.28%
$⋄$ vs. $○$	$μ MD = 4$	25.31%	26.28%
Reductions due to both of them	$μ MD = 0$	60.93%	47.15%
$∗$ vs. $○$	$μ MD = 4$	44.87%	41.51%

参考文献 84

[1]	HU, J., SUN, W., JIANG, Z., ZHANG, W., LU, J., HUO, W., ZHANG, Y., and ZHANG, P. Indentation size effect on hardness in the body-centered cubic coarse grained and nanocrystalline tantalum. Materials Science and Engineering A: Structural Materials Properties Microstructure and Processing, 686, 19–25 (2017)
[2]	LI, X. D. and BHUSHAN, B. A review of nanoindentation continuous stiffness measurement technique and its applications. Materials Characterization, 48(1), 11–36 (2002)
[3]	ZHOU, Y., FILLON, A., LAILLE, D., and GLORIANT, T. Probing grain size effect in the superelastic Ti-20Zr-3Mo-3Sn alloy using spherical nanoindentation. Materials Characterization, 184, 111691 (2022)
[4]	ZAN, X. D., GUO, X., WENG, G. J., and CHEN, G. Nanoindentation study of δ-phase zirconium hydride using the crystal plasticity model. International Journal of Plasticity, 167, 103675 (2023)
[5]	GERBERICH, W., TYMIAK, N., GRUNLAN, J., HORSTEMEYER, M., and BASKES, M. Interpretations of indentation size effects. Journal of Applied Mechanics-Transactions of the ASME, 69(4), 433–442 (2002)
[6]	HAN, C. and NIKOLOV, S. Indentation size effects in polymers and related rotation gradients. Journal of Materials Research, 22(6), 1662–1672 (2007)
[7]	MAUGIS, D. Adhesion of spheres: the JKR-DMT transition using a Dugdale model. Journal of Colloid and Interface Science, 150(1), 243–269 (1992)
[8]	TSAI, P., JENG, Y., MAO, C., WU, K., and HONG, F. Effects of surface morphology, size effect and wettability on interfacial adhesion of carbon nanotube arrays. Thin Solid Films, 545, 401–407 (2013)
[9]	JOHNSON, K. L., KENDALL, K., and ROBERTS, A. D. Surface energy and the contact of elastic solids. Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences, 324(1558), 301–313 (1971)
[10]	MULLER, V. M., YUSHCHENKO, V. S., and DERJAGUIN, B. V. On the influence of molecular forces on the deformation of an elastic sphere and its sticking to a rigid plane. Journal of Colloid and Interface Science, 77(1), 91–101 (1980)
[11]	STYLE, R. W., HYLAND, C., BOLTYANSKIY, R., WETTLAUFER, J. S., and DUFRESNE, E. R. Surface tension and contact with soft elastic solids. Nature Communications, 4, 2728 (2013)
[12]	MILLER, R. E. and SHENOY, V. B. Size-dependent elastic properties of nanosized structural elements. Nanotechnology, 11(3), 139–147 (2000)
[13]	GURTIN, M. E. and MURDOCH, A. I. A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis, 57(4), 291–323 (1975)
[14]	GURTIN, M. E. and MURDOCH, A. I. Surface stress in solids. International Journal of Solids and Structures, 14(6), 431–440 (1978)
[15]	STEIGMANN, D. J. and OGDEN, R. W. Plane deformations of elastic solids with intrinsic boundary elasticity. Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 453(1959), 853–877 (1997)
[16]	STEIGMANN, D. J. and OGDEN, R. W. Elastic surface-substrate interactions. Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences, 455(1982), 437–474 (1999)
[17]	CHHAPADIA, P., MOHAMMADI, P., and SHARMA, P. Curvature-dependent surface energy and implications for nanostructures. Journal of the Mechanics and Physics of Solids, 59(10), 2103–2115 (2011)
[18]	DAI, M., GHARAHI, A., and SCHIAVONE, P. Note on the deformation-induced change in the curvature of a material surface in plane deformations. Mechanics Research Communications, 94, 88–90 (2018)
[19]	LAWONGKERD, J., LE, T. M., KEAWSAWASVONG, S., INTARIT, P. I., LIMKATANYU, S., and RUNGAMORNRAT, J. Elastic solutions of axisymmetrically loaded half-space with surface and couple stress effects. Mechanics of Advanced Materials and Structures, 30(4), 835–855 (2023)
[20]	MINDLIN, R. D. and TIERSTEN, H. F. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis, 11(5), 415–448 (1963)
[21]	KOITER, W. T. Couple-stresses in the theory of elasticity: I and II. Koninklijke Nederlandse Akademie Van Weteschappen: Series B, 67, 17–44 (1964)
[22]	MINDLIN, R. D. Second gradient of strain and surface-tension in linear elasticity. International Journal of Solids and Structures, 1(4), 417–438 (1965)
[23]	LAM, D. C. C., YANG, F., CHONG, A. C. M., WANG, J., and TONG, P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51(8), 1477–1508 (2003)
[24]	ERINGEN, A. C. Linear theory of micropolar elasticity. Journal of Mathematics and Mechanics, 15(6), 909–923 (1966)
[25]	LIU, Y. and WEI, Y. Effect of surface energy on the indentation response of hard nanofilm/soft substrate composite structure. International Journal of Mechanical Sciences, 185, 105759 (2020)
[26]	LI, Y., ZHANG, H., LI, X., SHI, P., FENG, X., and DING, S. Surface effects on the indentation of a soft layer on a rigid substrate with an elliptical cylinder indenter. Acta Mechanica Sinica, 38(9), 422098 (2022)
[27]	XIAO, S., PENG, Z., WU, H., YAO, Y., and CHEN, S. Surface effect in nano-scale fretting contact problems. Journal of Applied Mechanics, 90(11), 111005 (2023)
[28]	XIAO, S., WU, H., PENG, Z., YAO, Y., and CHEN, S. Surface effect on the partial-slip contact of a nano-sized flat indenter. Mechanics of Materials, 195, 105057 (2024)
[29]	INTARIT, P. I., SENJUNTICHAI, T., RUNGAMORNRAT, J., and LIMKATANYU, S. Influence of frictional contact on indentation of elastic layer under surface energy effects. Mechanics Research Communications, 110, 103622 (2020)
[30]	WANG, F. F., SHEN, J. J., and LI, Y. G. Axisymmetric indentation problem of a transversely isotropic elastic medium with surface stresses. Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science, 234(2), 609–619 (2020)
[31]	ZHOU, W. and YANG, F. Effects of surface stress on the indentation response of an elastic half-space. International Journal of Mechanical Sciences, 229, 107512 (2022)
[32]	YUAN, W., ZHENG, Y., and WANG, G. Modelling tangential contact problem with surface stress. European Journal of Mechanics-A/Solids, 91, 104381 (2022)
[33]	LI, X. B. and MI, C. W. Effects of surface tension and Steigmann-Ogden surface elasticity on Hertzian contact properties. International Journal of Engineering Science, 145, 103165 (2019)
[34]	LI, X. B. and MI, C. W. Nanoindentation of a half-space due to a rigid cylindrical roller based on Steigmann-Ogden surface mechanical model. International Journal of Mechanics and Materials in Design, 17, 25–40 (2021)
[35]	ZEMLYANOVA, A. Y. and WHITE, L. M. Axisymmetric frictionless indentation of a rigid stamp into a semi-space with a surface energetic boundary. Mathematics and Mechanics of Solids, 27(2), 334–347 (2022)
[36]	ZEMLYANOVA, A. Y. and WHITE, L. M. An axisymmetric problem for a patch loaded material surface attached to the boundary of an elastic semi-space. SIAM Journal on Applied Mathematics, 83(2), 603–624 (2023)
[37]	CAO, R., YAN, J., and MI, C. On the sliding frictional nanocontact of an exponentially graded layer/substrate structure. International Journal of Mechanics and Materials in Design, 19(1), 95–119 (2023)
[38]	BAN, Y., LI, Z., and MI, C. On the tractive rolling nanocontact of an exponentially graded coating-substrate structure. Mechanics of Materials, 198, 105127 (2024)
[39]	BAN, Y. and MI, C. On the axisymmetric nanoindentation of an exponentially graded coating substrate structure with both surface and interface effects. International Journal of Solids and Structures, 310, 113211 (2025)
[40]	COSSERAT, E. and COSSERAT, F. Théorie des corps déformables. nature, 81, 67 (1909)
[41]	MUKI, R. and STERNBERG, E. The influence of couple-stresses on singular stress concentrations in elastic solids. Zeitschrift für Angewandte Mathematik und Physik, 16(5), 611–648 (1965)
[42]	ZISIS, T., GOURGIOTIS, P., BAXEVANAKIS, K., and GEORGIADIS, H. Some basic contact problems in couple stress elasticity. International Journal of Solids and Structures, 51, 2084–2095 (2014)
[43]	NIKOLOPOULOS, S., GOURGIOTIS, P. A., and ZISIS, T. Analysis of the tilted shallow wedge problem in couple-stress elasticity. Journal of Elasticity, 144(2), 205–221 (2021)
[44]	ZISIS, T., GOURGIOTIS, P., and DAL CORSO, F. A contact problem in couple stress thermoelasticity: the indentation by a hot flat punch. International Journal of Solids and Structures, 63, 226–239 (2015)
[45]	SONG, H., KE, L., and WANG, Y. Sliding frictional contact analysis of an elastic solid with couple stresses. International Journal of Mechanical Sciences, 133, 804–816 (2017)
[46]	WANG, Y., SHEN, H., ZHANG, X., ZHANG, B., LIU, J., and LI, X. Semi-analytical study of microscopic two-dimensional partial slip contact problem within the framework of couple stress elasticity: cylindrical indenter. International Journal of Solids and Structures, 138, 76–86 (2018)
[47]	WONGVIBOONSIN, W., GOURGIOTIS, P., VAN, C., LIMKATANYU, S., and RUNGAMORNRAT, J. Size effects in two-dimensional layered materials modeled by couple stress elasticity. Frontiers of Structural and Civil Engineering, 15, 425–443 (2021)
[48]	WONGVIBOONSIN, W., LE, T. M., LAWONGKERD, J. A., GOURGIOTIS, P., and RUNGAMORNRAT, J. Microstructural effects on the response of a multi-layered elastic substrate. International Journal of Solids and Structures, 241, 111394 (2022)
[49]	ÇÖMEZ, I. and EL-BORGI, S. Sliding frictional contact problem of a layer indented by a rigid punch in couple stress elasticity. Mathematics and Mechanics of Solids, 28, 730–747 (2023)
[50]	GOURGIOTIS, P., ZISIS, T., GIANNAKOPOULOS, A., and GEORGIADIS, H. The Hertz contact problem in couple-stress elasticity. International Journal of Solids and Structures, 168, 228–237 (2019)
[51]	LI, P. and LIU, T. Axisymmetric adhesive contact of multi-layer couple-stress elastic structures involving graded nanostructured materials. Applied Mathematical Modelling, 111, 501–520 (2022)
[52]	WANG, Y., ZHANG, X., SHEN, H., LIU, J., ZHANG, B., and XU, S. Three-dimensional contact analysis with couple stress elasticity. International Journal of Mechanical Sciences, 153, 369–379 (2019)
[53]	WANG, Y., ZHANG, X., SHEN, H., LIU, J., and ZHANG, B. Couple stress-based 3D contact of elastic films. International Journal of Solids and Structures, 191, 449–463 (2020)
[54]	WANG, Y., SHEN, H., LI, J., WANG, L., LIU, J., WANG, J., and LIU, H. Three-dimensional frictional contact within the framework of couple stress elasticity. Applied Mathematical Modelling, 131, 288–305 (2024)
[55]	CAO, Z., STEVENS, M. J., and DOBRYNIN, A. V. Adhesion and wetting of nanoparticles on soft surfaces. Macromolecules, 47(9), 3203–3209 (2014)
[56]	JENSEN, K. E., STYLE, R. W., XU, Q., and DUFRESNE, E. R. Strain-dependent solid surface stress and the stiffness of soft contacts. Physical Review X, 7(4), 041031 (2017)
[57]	GAO, X., HAO, F., HUANG, Z., and FANG, D. Mechanics of adhesive contact at the nanoscale: the effect of surface stress. International Journal of Solids and Structures, 51(3-4), 566–574 (2014)
[58]	ZHANG, L. and RU, C. Q. A refined JKR model for adhesion of a rigid sphere on a soft elastic substrate. Journal of Applied Mechanics-Transactions of the ASME, 86(5), 051004 (2019)
[59]	YUAN, W., NIU, X., and WANG, G. Axisymmetric indentations of an elastic half-space with tensed surface/membrane in the Johnson-Kendall-Roberts adhesive approximation. Journal of Applied Mechanics-Transactions of the ASME, 90(6), 061010 (2023)
[60]	ZHU, X. and XU, W. Effect of surface tension on the behavior of adhesive contact based on Lennard-Jones potential law. Journal of the Mechanics and Physics of Solids, 111, 170–183 (2018)
[61]	JIA, N., YAO, Y., PENG, Z., and CHEN, S. Surface effect in nanoscale adhesive contact. Journal of Adhesion, 97(4), 380–398 (2021)
[62]	ZHU, X., XU, W., and QIN, Y. Effect of surface tension on the behavior of adhesive contact based on Maugis-Dugdale model. European Journal of Mechanics-A/Solids, 81, 103930 (2020)
[63]	ZEMLYANOVA, A. Y. An Adhesive contact problem for a semi-plane with a surface elasticity in the Steigmann-Ogden form. Journal of Elasticity, 136(1), 103–121 (2019)
[64]	BAN, Y. and MI, C. On the competition between adhesive and surface effects in the nanocontact properties of an exponentially graded coating. Applied Mathematics and Computation, 432, 127364 (2022)
[65]	BAN, Y. and MI, C. On the adhesive nanocontact of a graded coating. European Journal of Mechanics-A/Solids, 97, 104840 (2023)
[66]	BAN, Y., YAN, J., LI, Z., and MI, C. Understanding adhesive sliding nanocontact mechanics in an exponentially graded coating substrate structure. European Journal of Mechanics-A/Solids, 109, 105482 (2025)
[67]	LI, P. and LIU, T. The axisymmetric contact in couple-stress elasticity taking into account adhesion. Journal of Adhesion Science and Technology, 37(1), 50–71 (2023)
[68]	LI, P. and LIU, T. The size effect in adhesive contact on gradient nanostructured coating. Nanotechnology, 32(23), 235704 (2021)
[69]	YUE, Y. M., XU, K. Y., TAN, Z. Q., WANG, W. J., and WANG, D. The influence of surface stress and surface-induced internal residual stresses on the size-dependent behaviors of Kirchhoff microplate. Archive of Applied Mechanics, 89(7), 1301–1315 (2019)
[70]	YIN, S., DENG, Y., YU, T., GU, S., and ZHANG, G. Isogeometric analysis for non-classical Bernoulli-Euler beam model incorporating microstructure and surface energy effects. Applied Mathematical Modelling, 89, 470–485 (2021)
[71]	MORADWEYSI, P., ANSARI, R., GHOLAMI, R., BAZDID-VAHDATI, M., and ROUHI, H. Half-space contact problem considering strain gradient and surface effects: an analytical approach. Zeitschrift für Angewandte Mathematik und Mechanik, 99(6), e201700190 (2019)
[72]	YUAN, W., ZHANG, J., NIU, X., and WANG, G. Axisymmetric Hertzian contact problem accounting for surface tension and strain gradient elasticity. Applied Mathematical Modelling, 137, 115698 (2025)
[73]	LE, T. M., LAWONGKERD, J., TINH QUOC, B., LIMKATANYU, S., and RUNGAMORNRAT, J. Elastic response of surface-loaded half plane with influence of surface and couple stresses. Applied Mathematical Modelling, 91, 892–912 (2021)
[74]	LE, T. M., WONGVIBOONSIN, W., LAWONGKERD, J., BUI, T. Q., and RUNGAMORNRAT, J. Influence of surface and couple stresses on response of elastic substrate under tilted flat indenter. Applied Mathematical Modelling, 104, 644–665 (2022)
[75]	FATHABADI, S. A. A. and ALINIA, Y. A nano-scale frictional contact problem incorporating the size dependency and the surface effects. Applied Mathematical Modelling, 83, 107–121 (2020)
[76]	EREMEYEV, V. A. and LEBEDEV, L. P. Mathematical study of boundary-value problems within the framework of Steigmann-Ogden model of surface elasticity. Continuum Mechanics and Thermodynamics, 28(1-2), 407–422 (2016)
[77]	ZEMLYANOVA, A. Y. and MOGILEVSKAYA, S. G. Circular inhomogeneity with Steigmann-Ogden interface: local fields, neutrality, and Maxwell's type approximation formula. International Journal of Solids and Structures, 135, 85–98 (2018)
[78]	MI, C. Elastic behavior of a half-space with a Steigmann-Ogden boundary under nanoscale frictionless patch loads. International Journal of Engineering Science, 129, 129–144 (2018)
[79]	BAN, Y., YANG, X., and MI, C. On the nanoindentation of a couple-stress half-plane with Steigmann-Ogden surface effects. Mechanics of Advanced Materials and Structures (2025) https://doi.org/10.1080/15376494.2025.2476782
[80]	ERDOGAN, F. and GUPTA, G. D. On the numerical solution of singular integral equations. Quarterly of Applied Mathematics, 29(4), 525–534 (1972)
[81]	SHENOY, V. B. Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Physical Review B, 71, 094104 (2005)
[82]	BARQUINS, M. Adherence and rolling kinetics of a rigid cylinder in contact with a natural rubber surface. Journal of Adhesion, 26(1), 1–12 (1988)
[83]	TABOR, D. Surface forces and surface interactions. Journal of Colloid and Interface Science, 58(1), 2–13 (1977)
[84]	LONG, J. M., WANG, G. F., FENG, X. Q., and YU, S. W. Two-dimensional Hertzian contact problem with surface tension. International Journal of Solids and Structures, 49(13), 1588–1594 (2012)