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Zemlyanova, Anna Y.; White, Lauren M.

Axisymmetric frictionless indentation of a rigid stamp into a semi-space with a surface energetic boundary. (English) Zbl 07590427

Math. Mech. Solids 27, No. 2, 334-347 (2022).
Summary: An axisymmetric problem for a frictionless contact of a rigid stamp with a semi-space in the presence of surface energy in the Steigmann-Ogden form is studied. The method of Boussinesq potentials is used to obtain integral representations of the stresses and the displacements. Using the Hankel transform, the problem is reduced to a single integral equation of the first kind on a contact interval with an additional condition. The integral equation is studied for solvability. It is shown that for the classic problem in the absence of surface effects and for the problem with the Gurtin-Murdoch surface energy without surface tension, the obtained equation represents a Cauchy singular integral equation. At the same time, for the Gurtin-Murdoch model with a non-zero surface tension and for the general Steigmann-Ogden model, the problem results in the integral equation of the first kind with a weakly singular or a continuous kernel, correspondingly. Hence, the contact problem is ill-posed in these cases. The integral equation of the first kind with an additional condition is solved approximately by using Gauss-Chebyshev quadrature for evaluation of the integrals. Numerical results for various values of the parameters are reported.

Cited in 5 Documents

MSC:

74-XX Mechanics of deformable solids

Keywords:

frictionless contact; axisymmetric; Steigmann-Ogden surface energy; Gurtin-murdoch surface energy; singular integral equation; Fredholm equation of first kind

Software:

IIPBF
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Full Text: DOI

References:

[1] Agrawal, R, Peng, B, Gdoutos, E, et al. Elasticity size effects in ZnO nanowires - a combined experimental-computational approach. Nano Lett 2008; 8(11): 3668-3674. · doi:10.1021/nl801724b
[2] Chen, C, Shi, Y, Zhang, Y, et al. Size dependence of Young’s modulus in ZnO nanowires. Phys Rev Lett 2006; 96: 075505. · doi:10.1103/PhysRevLett.96.075505
[3] Cheng, YT, Cheng, CM. Scaling, dimensional analysis, and indentation measurements. Mater Sci Eng 2004; 44: 91-149. · doi:10.1016/j.mser.2004.05.001
[4] Gao, XL. An expanding cavity model incorporating strain-hardening and indentation size effects. Int J Solids Struct 2006; 43: 6615-6629. · Zbl 1120.74655 · doi:10.1016/j.ijsolstr.2006.01.008
[5] Gao, XL. A new expanding cavity model for indentation hardness including strain-hardening and indentation size effects. J Mater Res 2006; 21: 1317-1326. · doi:10.1557/jmr.2006.0158
[6] Ma, Q, Clarke, D. Size dependent hardness of silver single crystals. J Mater Res 1995; 10: 853-863. · doi:10.1557/JMR.1995.0853
[7] Nix, W, Gao, H. Indentation size effects in crystalline materials: A law for strain gradient plasticity. J Mech Phys Solids 1998; 46(3): 411-425. · Zbl 0977.74557 · doi:10.1016/S0022-5096(97)00086-0
[8] Stan, G, Ciobanu, C, Parthangal, P, et al. Diameter-dependent radial and tangential elastic moduli of ZnO nanowires. Nano Lett 2007; 7(12): 3691-3697. · doi:10.1021/nl071986e
[9] Zhang, Y, Hong, J, Liu, B, et al. Strain effect on ferroelectric behaviors of BaTiO3 nanowires: A molecular dynamics study. Nanotechnology 2009; 21(1): 015701. · doi:10.1088/0957-4484/21/1/015701
[10] Zhao, M, Wang, Z, Mao, S. Piezoelectric characterization of individual zinc oxide nanobelt probed by piezoresponse force microscope. Nano Lett 2004; 4(4): 587-590. · doi:10.1021/nl035198a
[11] Gibbs, J. The scientific papers of J.W. Gibbs. 1st ed. London: Longmans-Green, 1906. · JFM 37.0035.06
[12] Gurtin, M, Murdoch, A. A continuum theory of elastic material surfaces. Arch Ration Mech Anal 1975; 57(4): 291-323. · Zbl 0326.73001 · doi:10.1007/BF00261375
[13] Gurtin, M, Murdoch, A. Surface stress in solids. Int J Solids Struct 1978; 14(6): 431-440. · Zbl 0377.73001 · doi:10.1016/0020-7683(78)90008-2
[14] Steigmann, D, Ogden, R. Plane deformations of elastic solids with intrinsic boundary elasticity. Proc R Soc London Ser A 1997; 453: 853-877. · Zbl 0938.74014 · doi:10.1098/rspa.1997.0047
[15] Steigmann, D, Ogden, R. Elastic surface-substrate interactions. Proc R Soc London Ser A 1999; 455: 437-474. · Zbl 0926.74016 · doi:10.1098/rspa.1999.0320
[16] Zemlyanova, A. Interaction of multiple straight fractures in the presence of the Steigmann-Ogden surface energy. SIAM J Appl Math 2020; 80(5): 2098-2119. · Zbl 1448.74094 · doi:10.1137/19M1305173
[17] Gao, X, Hao, F, Fang, D, et al. Boussinesq problem with the surface effect and its application to contact mechanics at the nanoscale. Int J Solids Struct 2013; 50: 2620-2630. · doi:10.1016/j.ijsolstr.2013.04.007
[18] Zhao, X, Rajapakse, R. Analytical solutions for a surface-loaded isotropic elastic layer with surface energy effects. Int J Eng Sci 2009; 47: 1433-1444. · Zbl 1213.74243 · doi:10.1016/j.ijengsci.2008.12.013
[19] Zhao, X, Rajapakse, R. Elastic field of a nano-film subjected to tangential surface load: Asymmetric problem. Eur J Mech A Solids 2013; 39: 69-75. · Zbl 1348.74234 · doi:10.1016/j.euromechsol.2012.11.005
[20] Zhou, S, Gao, XL. Solutions of half-space and half-plane contact problems based on surface elasticity. Z Angew Math Phys 2013; 64(1): 145-166. · Zbl 1318.74004 · doi:10.1007/s00033-012-0205-0
[21] Li, X, Mi, C. Nanoindentation hardness of a Steigmann-Ogden surface bounding an elastic half-space. Math Mech Solids 2019; 24(9): 2754-2766. · Zbl 07273338
[22] Mi, C. Elastic behavior of a half-space with a Steigmann-Ogden boundary under nanoscale frictionless patch loads. Int J Eng Sci 2018; 129: 129-144. · Zbl 1423.74674 · doi:10.1016/j.ijengsci.2018.04.009
[23] Li, X, Jiang, L, Mi, C. Flamant solution of a half-plane with surface flexural resistibility and its applications to nanocontact mechanics. Math Mech Solids 2020; 25(3): 664-681. · Zbl 1446.74180
[24] Long, J, Wang, G. Effects of surface tension on axisymmetric Hertzian contact problem. Mech Mater 2013; 56: 65-70. · doi:10.1016/j.mechmat.2012.09.003
[25] Long, J, Wang, G, Feng, X, et al. Two-dimensional Hertzian contact problem with surface tension. Int J Solids Struct 2012; 49(13): 1588-1594. · doi:10.1016/j.ijsolstr.2012.03.017
[26] Pinyochotiwong, Y, Rungamornrat, J, Senjuntichai, T. Analysis of rigid frictionless indentation on half-space with surface elasticity. Proc Eng 2011; 14: 2403-2410. · Zbl 1423.74676 · doi:10.1016/j.proeng.2011.07.302
[27] Pinyochotiwong, Y, Rungamornrat, J, Senjuntichai, T. Rigid frictionless indentation on elastic half space with influence of surface stresses. Int J Eng Sci 2013; 71: 15-35. · Zbl 1423.74676 · doi:10.1016/j.ijengsci.2013.04.005
[28] Zemlyanova, A. Frictionless contact of a rigid stamp with a semi-plane in the presence of surface elasticity in the Steigmann-Ogden form. Math Mech Solids 2018; 23(8): 1140-1155. · Zbl 1401.74225
[29] Zemlyanova, A. An adhesive contact problem for a semi-plane with a surface elasticity in the Steigmann-Ogden form. J Elasticity 2019; 136(1): 103-121. · Zbl 1415.74008 · doi:10.1007/s10659-018-9694-1
[30] Li, X, Mi, C. Nanoindentation of a half-space due to a rigid cylindrical roller based on Steigmann-Ogden surface mechanical model. Int J Mech Mater Des 2021; 17: 25-40. · doi:10.1007/s10999-020-09507-9
[31] Shen, J. Axisymmetric Boussinesq problem of a transversely isotropic half space with surface effects. Math Mech Solids 2019; 24(5): 1425-1437. · Zbl 1440.74249
[32] Wang, F, Shen, J, Li, Y. Axisymmetric indentation problem of a transversely isotropic elastic medium with surface stresses. Math Mech Solids 2020; 234(2): 609-619.
[33] Walton, J, Zemlyanova, A. A rigid stamp indentation into a semiplane with a curvature-dependent surface tension on the boundary. SIAM J Appl Math 2016; 76(2): 618-640. · Zbl 1332.74010 · doi:10.1137/15M1044096
[34] Eremeyev, V. On dynamic boundary conditions within the linear Steigmann-Ogden model of surface elasticity and strain gradient elasticity. In: Altenbach, H, Belyaev, A, Eremeyev, V, Krivtsov, A, Porubov, A (eds) Dynamical Processes in Generalized Continua and Structures Advanced Structured Materials, Vol. 103, pp. 195-207. Cham: Springer, 2019. · Zbl 1429.74018 · doi:10.1007/978-3-030-11665-1_10
[35] Eremeyev, V, Lebedev, L. Mathematical study of boundary-value problems within the framework of Steigmann-Ogden model of surface elasticity. Continuum Mech Thermodyn 2016; 28: 407-422. · Zbl 1348.35260 · doi:10.1007/s00161-015-0439-0
[36] Zemlyanova, A, Machina, A. A new B-spline collocation method for singular integro-differential equations of higher orders. J Comp Appl Math 2020; 380: 112949. · Zbl 1452.65415 · doi:10.1016/j.cam.2020.112949
[37] Ban, Y, Mi, C. On spherical inhomogeneity embedded in a half-space analyzed with Steigmann-Ogden surface and interface models. Int J Solids Struct 2021; 216: 123-135. · doi:10.1016/j.ijsolstr.2020.11.034
[38] Han, Z, Mogilevskaya, S, Schillinger, D. Local fields and overall transverse properties of unidirectional composite materials with multiple nanofibers and Steigmann-Ogden interfaces. Int J Solids Struct 2018; 147: 166-182. · doi:10.1016/j.ijsolstr.2018.05.019
[39] Mogilevskaya, S, Kushch, V, Zemlyanova, A. Displacement representations for the problems with sperical and circular material surfaces. Quart J Mech Appl Math 2019; 72(4): 449-471. · Zbl 1479.74110 · doi:10.1093/qjmam/hbz013
[40] Zemlyanova, A, Mogilevskaya, S. Circular inhomogeneity with Steigmann-Ogden interface: Local fields and Maxwell’s type approximation formula. Int J Solids Struct 2018; 135: 85-98. · doi:10.1016/j.ijsolstr.2017.11.012
[41] Zemlyanova, A, Mogilevskaya, S. On spherical inhomogeneity with Steigmann-Ogden interface. ASME J Appl Mech 2018; 85(12): 121009. · doi:10.1115/1.4041499
[42] Baranova, S, Mogilevskaya, S, Mantic, V, et al. Analysis of the antiplane problem with an embedded zero thickness layer described by the Gurtin-Murdoch model. J Elasticity 2020; 140: 171-195. · Zbl 1440.74192 · doi:10.1007/s10659-020-09764-x
[43] Mogilevskaya, S, Zemlyanova, A, Mantic, V. The use of the Gurtin-Murdoch theory for modeling mechanical processes in composites with two-dimensional reinforcements. Comp Sci Tech 2021; 210(7): 108751. · doi:10.1016/j.compscitech.2021.108751
[44] Intarit, P, Senjuntichai, T, Rungamornrat, J, et al. Penny-shaped crack in elastic medium with surface energy effects. Acta Mech 2017; 228: 617-630. · doi:10.1007/s00707-016-1728-9
[45] Zemlyanova, A. An axisymmetric problem for a penny-shaped crack under influence of the steigmann-ogden surface energy. Proc R Soc London Ser A 2021; 477(2248): 20200998.
[46] Dai, M, Schiavone, P. Edge dislocation interacting with a Steigmann-Ogden interface incorporating residual tension. Int J Eng Sci 2019; 139: 62-69. · Zbl 1425.74048 · doi:10.1016/j.ijengsci.2019.01.009
[47] Gorbushin, N, Eremeyev, V, Mishuris, G. On stress singularity near the tip of a crack with surface stresses. Int J Eng Sci 2020; 146: 103183. · Zbl 1476.74045 · doi:10.1016/j.ijengsci.2019.103183
[48] Kim, C, Ru, CQ, Schiavone, P. A clarification of the role of crack-tip conditions in linear elasticity with surface effects. Math Mech Solids 2012; 18(1): 59-66. · Zbl 1528.74005
[49] Kim, C, Schiavone, P, Ru, CQ. Analysis of a mode-III crack in the presence of surface elasticity and a prescribed non-uniform surface traction. Z Angew Math Phys 2010; 61: 555-564. · Zbl 1273.74456 · doi:10.1007/s00033-009-0021-3
[50] Kim, C, Schiavone, P, Ru, CQ. The effects of surface elasticity on an elastic solid with mode-III crack: Complete solution. J Appl Mech 2010; 77: 021011. · doi:10.1115/1.3177000
[51] Kim, C, Schiavone, P, Ru, CQ. Analysis of plane-strain crack problems (mode-I and mode-II) in the presence of surface elasticity. J Elasticity 2011; 104: 397-420. · Zbl 1300.74048 · doi:10.1007/s10659-010-9287-0
[52] Kim, C, Schiavone, P, Ru, CQ. Effect of surface elasticity on an interface crack in plane deformations. Proc R Soc London Ser A 2011; 467: 3530-3549. · Zbl 1243.74005 · doi:10.1098/rspa.2011.0311
[53] Kim, C, Schiavone, P, Ru, CQ. The effect of surface elasticity on a mode-III interface crack. Arch Mech 2011; 63: 267-286. · Zbl 1269.74195
[54] Ferguson, L, Muddamallappa, M, Walton, J. Numerical simulation of mode-III fracture incorporating interfacial mechanics. Int J Fract 2015; 192: 47-56. · doi:10.1007/s10704-014-9984-y
[55] Sendova, T, Walton, JR. A new approach to the modeling and analysis of fracture through extension of continuum mechanics to the nanoscale. Math Mech Solids 2010; 15: 368-413. · Zbl 1197.74010
[56] Walton, J. Plane-strain fracture with curvature-dependent surface tension: mixed-mode loading. J Elasticity 2014; 114(1): 127-142. · Zbl 1282.74082 · doi:10.1007/s10659-013-9430-9
[57] Zemlyanova, A. The effect of a curvature-dependent surface tension on the singularities at the tips of a straight interface crack. Quart J Mech Appl Math 2013; 66(2): 199-219. · Zbl 1291.74165 · doi:10.1093/qjmam/hbt001
[58] Zemlyanova, A. Curvilinear mode-I/mode-II interface fracture with a curvature-dependent surface tension on the boundary. IMA J Appl Math 2016; 81(6): 1112-1136. · Zbl 1408.74011 · doi:10.1093/imamat/hxw048
[59] Zemlyanova, A, Walton, J. Modeling of a curvilinear planar crack with a curvature-dependent surface tension. SIAM J Appl Math 2012; 72: 1474-1492. · Zbl 1264.74014 · doi:10.1137/110860100
[60] Li, X, Mi, C. Effects of surface tension and Steigmann-Ogden surface elasticity on Hertzian contact properties. Int J Eng Sci 2019; 145: 103165. · Zbl 1476.74116 · doi:10.1016/j.ijengsci.2019.103165
[61] Lucas, S. Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J Comp Appl Math 1995; 64(3): 269-282. · Zbl 0853.65028 · doi:10.1016/0377-0427(95)00143-3
[62] Ratnanather, J, Kim, J, Zhang, S, et al. Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans Math Softw 2014; 40(2): 1-12. · Zbl 1305.65105 · doi:10.1145/2508435
[63] Barber, J. Elasticity. 3rd ed. New York: Springer, 2010. · doi:10.1007/978-90-481-3809-8
[64] Gradshteyn, I, Ryzhik, I. Table of Integrals, Series and Products. 7th ed. New York: Academic Press, 2014. · Zbl 0918.65002
[65] Erdogan, F. Stress distribution in bonded dissimilar materials containing circular or ring-shaped cavities. J Appl Mech 1965; 32(4): 829-836. · doi:10.1115/1.3627323
[66] Gakhov, F. Boundary Value Problems. New York: Dover, 1990. · Zbl 0830.30026
[67] Green, A, Zerna, W. Theoretical Elasticity. 2nd ed. Oxford: Clarendon Press, 1968. · Zbl 0155.51801
[68] Groetsch, C. Integral equations of the first kind, inverse problems and regularization: A crash course. J Phys Conf Ser 2007; 734: 012001. · doi:10.1088/1742-6596/73/1/012001
[69] Kress, R. Linear Integral Equations. 3rd ed. New York: Springer, 2014. · Zbl 1328.45001 · doi:10.1007/978-1-4614-9593-2
[70] Chhapadia, P, Mohammadi, P, Sharma, P. Curvature-dependent surface energy and implications for nanostructures. J Mech Phys Solids 2011; 59: 2103-2115. · Zbl 1270.74018 · doi:10.1016/j.jmps.2011.06.007
[71] Chhapadia, P, Mohammadi, P, Sharma, P. Erratum to: Curvature-dependent surface energy and implications for nanostructures. J Mech Phys Solids 2012; 60: 1241-1242. · doi:10.1016/j.jmps.2012.01.004
[72] Dingreville, R, Qu, J. A semi-analytic method to compute surface elastic properties. Acta Mater 2007; 55: 141-147. · doi:10.1016/j.actamat.2006.08.007
[73] Miller, R, Shenoy, V. Size dependent elastic properties of nanosized structural elements. Nanotechnology 2000; 11: 139-147. · doi:10.1088/0957-4484/11/3/301
[74] Shenoy, V. Atomistic calculations of elastic properties of metallic fcc crystal surfaces. Phys Rev B 2005; 71: 094104. · doi:10.1103/PhysRevB.71.094104
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