×

Modelling tangential contact problem with surface stress. (English) Zbl 1507.74250

Summary: In this paper, we investigate the effects of surface stress on the spherical contact problem under combined normal and lateral loads. The boundary element method based on fast Fourier transform and conjugate gradient algorithm is employed to perform the contact analysis with partial slip. It is shown that as the size of contact reduces to the characteristic scale that surface stress dominates, the contact behavior will significantly deviate from the classical model. Under the same loading condition, the tangential traction within contact region tends to distribute more evenly and the width of slip zone would be relatively smaller due to the presence of surface stress. In addition, it is found that the variations of the total tangential force and the lateral contact stiffness with respect to the lateral displacement depend strongly on the contact scale. This work provides a theoretical basis for the study of contact and friction of some soft materials and biological systems.

MSC:

74M15 Contact in solid mechanics
74S15 Boundary element methods applied to problems in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bico, J.; Reyssat, É.; Roman, B., Elastocapillarity: when surface tension deforms elastic solids, Annu. Rev. Fluid Mech., 50, 629-659 (2018) · Zbl 1384.76015
[2] Cattaneo, C. Sul contatto di due corpi elastici: distribuzione locale degli sforzi. Rendiconti dell’Accademia nazionale dei Lincei, 27, 342-348, 434-436, 474-478. · JFM 64.0838.02
[3] Cheng, Z.; Shurer, C. R.; Schmidt, S.; Gupta, V. K.; Chuang, G.; Su, J., The surface stress of biomedical silicones is a stimulant of cellular response, Sci. Adv., 6, Article eaay0076 pp. (2020)
[4] Ciavarella, M., The generalized Cattaneo partial slip plane contact problem, I—Theory. Int. J. Solids Struct., 35, 18, 2349-2362 (1998) · Zbl 0934.74054
[5] Gao, X.; Hao, F.; Fang, D.; Huang, Z., Boussinesq problem with the surface effect and its application to contact mechanics at the nanoscale, Int. J. Solid Struct., 50, 16-17, 2620-2630 (2013)
[6] Gurtin, M. E.; Murdoch, A. I., A continuum theory of elastic material surfaces, Arch. Ration. Mech. Anal., 57, 4, 291-323 (1975) · Zbl 0326.73001
[7] Gurtin, M. E.; Weissmüller, J.; Larche, F., A general theory of curved deformable interfaces in solids at equilibrium, Philos. Mag. A, 78, 5, 1093-1109 (1998)
[8] Hajji, M. A., Indentation of a membrane on an elastic half space, ASME J. Appl. Mech., 45, 2, 320-324 (1978)
[9] Hayashi, T.; Koguchi, H.; Nishi, N., Contact analysis for anisotropic elastic materials considering surface stress and surface elasticity, J. Mech. Phys. Solid., 61, 8, 1753-1767 (2013)
[10] He, L. H.; Lim, C. W., Surface Green function for a soft elastic half-space: influence of surface stress, Int. J. Solid Struct., 43, 1, 132-143 (2006) · Zbl 1119.74311
[11] Hui, C. Y.; Liu, T.; Salez, T.; Raphael, E.; Jagota, A., Indentation of a rigid sphere into an elastic substrate with surface tension and adhesion, Proc. Math. Phys. Eng. Sci., 471, 2175, 20140727 (2015) · Zbl 1371.74053
[12] Jäger, J., A new principle in contact mechanics, ASME J. Tribol., 120, 4, 677-684 (1998)
[13] Jin, F.; Wan, Q.; Guo, X., Plane contact and partial slip behaviors of elastic layers with randomly rough surfaces, ASME J. Appl. Mech., 82, 9, Article 091006 pp. (2015)
[14] Johnson, K. L., Contact Mechanics (1985), Cambridge University Press: Cambridge University Press London · Zbl 0599.73108
[15] Johnson, K. L., Adhesion and friction between a smooth elastic spherical asperity and a plane surface, Proc. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci., 453, 1956, 163-179 (1997)
[16] Kim, J. H.; Gouldstone, A., Spherical indentation of a membrane on an elastic half-space, J. Mater. Res., 23, 8, 2212-2220 (2008)
[17] Koguchi, H., Surface Green function with surface stresses and surface elasticity using Stroh’s formalism, ASME J. Appl. Mech., 75, 6 (2008)
[18] Liu, S.; Wang, Q.; Liu, G., A versatile method of discrete convolution and FFT (DC-FFT) for contact analyses, Wear, 243, 1-2, 101-111 (2000)
[19] Long, J.; Ding, Y.; Yuan, W.; Chen, W.; Wang, G., General relations of indentations on solids with surface tension, ASME J. Appl. Mech., 84, 5 (2017)
[20] Long, J.; Yuan, W.; Chen, W.; Wang, G., Analytic relations for two-dimensional indentations with surface tension, Mech. Mater., 119, 34-41 (2018)
[21] Lu, P.; Yang, L.; Wang, G., Studies of low-loading micro-slip contacts on rough surfaces with GW model, Int. J. Appl. Mech., 9, 4, 1750049 (2017)
[22] McMeeking, R. M.; Ciavarella, M.; Cricrì, G.; Kim, K. S., The interaction of frictional slip and adhesion for a stiff sphere on a compliant substrate, ASME J. Appl. Mech., 87, 3 (2020)
[23] Mindlin, R. D., Compliance of elastic bodies in contact, ASME J. Appl. Mech., 16, 259-268 (1949) · Zbl 0034.26602
[24] Mora, S.; Phou, T.; Fromental, J. M.; Pismen, L. M.; Pomeau, Y., Capillarity driven instability of a soft solid, Phys. Rev. Lett., 105, 21, 214301 (2010)
[25] Paggi, M.; Pohrt, R.; Popov, V. L., Partial-slip frictional response of rough surfaces, Sci. Rep., 4, 5178 (2014)
[26] Paretkar, D.; Xu, X.; Hui, C. Y.; Jagota, A., Flattening of a patterned compliant solid by surface stress, Soft Matter, 10, 23, 4084-4090 (2014)
[27] Peng, B.; Li, Q.; Feng, X. Q.; Gao, H., Effect of shear stress on adhesive contact with a generalized Maugis-Dugdale cohesive zone model, J. Mech. Phys. Solid., 148, 104275 (2021)
[28] Pohrt, R.; Li, Q., Complete boundary element formulation for normal and tangential contact problems, Phys. Mesomech., 17, 4, 334-340 (2014)
[29] Polonsky, I. A.; Keer, L. M., A numerical method for solving rough contact problems based on the multi-level multi-summation and conjugate gradient techniques, Wear, 231, 2, 206-219 (1999)
[30] Savkoor, A. R.; Briggs, G. A.D., The effect of tangential force on the contact of elastic solids in adhesion, Proc. R. Soc. Lond. A. Math. Phys. Sci., 356, 1684, 103-114 (1977) · Zbl 0364.73095
[31] Style, R. W.; Hyland, C.; Boltyanskiy, R.; Wettlaufer, J. S.; Dufresne, E. R., Surface tension and contact with soft elastic solids, Nat. Commun., 4, 2728 (2013)
[32] Wang, G. F.; Feng, X. Q., Effects of surface stresses on contact problems at nanoscale, J. Appl. Phys., 101, 1, Article 013510 pp. (2007)
[33] Xia, K.; Liu, C. F.; Leong, W. H.; Kwok, M. H.; Yang, Z. Y.; Feng, X., Nanometer-precision non-local deformation reconstruction using nanodiamond sensing, Nat. Commun., 10, 3259 (2019)
[34] Yuan, W.; Wang, G., Boundary element calculations for normal contact of soft materials with tensed surface membrane, Front. Mech. Eng., 6, 57 (2020)
[35] Zhou, S.; Gao, X. L., Solutions of half-space and half-plane contact problems based on surface elasticity, Z. Angew. Math. Phys., 64, 1, 145-166 (2013) · Zbl 1318.74004
[36] Zhou, S.; Gao, X. L., Solutions of the generalized half-plane and half-space Cerruti problems with surface effects, Z. Angew. Math. Phys., 66, 1, 1125-1142 (2015) · Zbl 1317.74007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.