Axisymmetric loading on nanoscale multilayered media

Abstract

Multilayered nanoscale structures are used in several applications. Because the effect of surface energy becomes nontrivial at such a small scale, a modified continuum theory is required to accurately predict their mechanical behaviors. A Gurtin–Murdoch continuum model of surface elasticity is implemented to establish a computational scheme for investigating an elastic multilayered system under axisymmetric loads with the incorporation of surface/interface energy. Each layer stiffness matrix is derived based on the general solutions of stresses and displacements obtained in the form of the Hankel integral transform. Numerical solutions to the global equation, which are formulated based on the continuity conditions of tractions and displacements across interfaces between layers, yield the displacements at each layer interface and on the top surface of the multilayered medium. The numerical solutions indicate that the elastic responses of multilayered structures are affected significantly by the surface material properties of both the top surface and interfaces, and that they become size dependent. In addition, the indentation problem of a multilayered nanoscale elastic medium under a rigid frictionless cylindrical punch is investigated to demonstrate the application of the proposed solution scheme.

Graphical abstract

Keywords

functionally graded layer / Gurtin–Murdoch surface elasticity / multilayered medium / size dependency / stiffness matrix

References

[1] LacazeP C. Nanotechnologies: Concepts, Production and Applications. London: Wiley, 2012

[2] Wong E W, Sheehan P E, Lieber C M. Nanobeam mechanics: Elasticity, strength, and toughness of nanorods and nanotubes. Science, 1997, 277(5334): 1971–1975

[3] Gurtin M E, Ian Murdoch A. A Continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis, 1975, 57(4): 291–323

[4] Gurtin M E, Ian Murdoch A. Surface stress in solids. International Journal of Solids and Structures, 1978, 14(6): 431–440

[5] Miller R E, Shenoy V B. Size-dependent elastic properties of nanosized structural elements. Nanotechnology, 2000, 11(3): 139–147

[6] Shenoy V B. Atomistic calculations of elastic properties of metallic FCC crystal surfaces. Physical Review B: Condensed Matter and Materials Physics, 2005, 71(9): 094104

[7] HeL HLimC WWuB S. A continuum model for size-dependent deformation of elastic films of nano-scale thickness. International Journal of Solids and Structures, 2004, 41(3−4): 847−857

[8] Tirapat S, Senjuntichai T, Rungamornrat J. Influence of surface energy effects on elastic fields of a layered elastic medium under surface loading. Advances in Materials Science and Engineering, 2017, 2017: 1–11

[9] Lu P, He L H, Lee H P, Lu C. Thin plate theory including surface effects. International Journal of Solids and Structures, 2006, 43(16): 4631–4647

[10] Sapsathiarn Y, Rajapakse R K N D. Finite-element modeling of circular nanoplates. Journal of Nanomechanics & Micromechanics, 2013, 3(3): 59–66

[11] Nanthakumar S S, Valizadeh N, Park H S, Rabczuk T. Surface effects on shape and topology optimization of nanostructures. Computational Mechanics, 2015, 56(1): 97–112

[12] Nguyen T B, Rungamornrat J, Senjuntichai T, Wijeyewickrema A C. FEM-SGBEM coupling for modeling of mode-I planar cracks in three-dimensional elastic media with residual surface tension effects. Engineering Analysis with Boundary Elements, 2015, 55: 40–51

[13] Intarit P, Senjuntichai T, Rungamornrat J, Rajapakse R K N D. Penny-shaped crack in elastic medium with surface energy effects. Acta Mechanica, 2017, 228(2): 617–630

[14] Tian L, Rajapakse R K N D. Analytical Solution for size-dependent elastic field of a nanoscale circular inhomogeneity. Journal of Applied Mechanics, 2007, 74(3): 568–574

[15] Pinyochotiwong Y, Rungamornrat J, Senjuntichai T. Rigid frictionless indentation on elastic half space with influence of surface stresses. International Journal of Engineering Science, 2013, 71: 15–35

[16] Intarit P, Senjuntichai T, Rungamornrat J. Elastic layer under axisymmetric indentation and surface energy effects. Zeitschrift für Angewandte Mathematik und Physik, 2018, 69: 29

[17] Tirapat S, Senjuntichai T, Rungamornrat J, Rajapakse R K N D. Indentation of a nanolayer on a substrate by a rigid cylinder in adhesive contact. Acta Mechanica, 2020, 231(8): 3235–3246

[18] Intarit P, Senjuntichai T, Rungamornrat J, Limkatanyu S. Influence of frictional contact on indentation of elastic layer under surface energy effects. Mechanics Research Communications, 2020, 110: 103622

[19] Eringen A C. Nonlocal polar elastic continua. International Journal of Engineering Science, 1972, 10(1): 1–16

[20] Eringen A C, Edelen D G B. On nonlocal elasticity. International Journal of Engineering Science, 1972, 10(3): 233–248

[21] Mindlin R D, Tiersten H F. Effects of couple-stresses in linear elasticity. Archive for Rational Mechanics and Analysis, 1962, 11(1): 415–448

[22] Toupin R A. Theory of elasticity with couple-stress. Archive for Rational Mechanics and Analysis, 1964, 17(2): 85–112

[23] Mindlin R D. Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 1964, 16(1): 51–78

[24] Wongviboonsin W, Gourgiotis P A, Van C N, Limkatanyu S, Rungamornrat J. Size effects in two-dimensional layered materials modeled by couple stress elasticity. Frontiers of Structural and Civil Engineering, 2021, 15(2): 425–443

[25] Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh V M, Guo H, Hamdia K, Zhuang X, Rabczuk T. An Energy Approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications. Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112790

[26] Anitescu C, Atroshchenko E, Alajlan N, Rabczuk T. Artificial neural network methods for the solution of second order boundary value problems. Computers, Materials & Continua, 2019, 59(1): 345–359

[27] Nguyen-Thanh V M, Anitescu C, Alajlan N, Rabczuk T, Zhuang X. Parametric deep energy approach for elasticity accounting for strain gradient effects. Computer Methods in Applied Mechanics and Engineering, 2021, 386: 114096

[28] GürdalZHaftkaR THajelaP. Design and Optimization of Laminated Composite Materials. New York: Wiley, 1999

[29] Senjuntichai T, Sornpakdee N, Teerawong J, Rajapakse R K N D. Time dependent response of an axially loaded elastic bar in a multi-layered poroelastic medium. Journal of Engineering Mechanics, 2007, 133(5): 578–587

[30] Senjuntichai T, Kaewjuea W. Dynamic response of multiple flexible strips on a multilayered poroelastic half-plane. Journal of Mechanics of Materials and Structures, 2008, 3(10): 1885–1901

[31] Ba Z N, Kang Z Q, Lee V W. Plane strain dynamic responses of a multi-layered transversely isotropic saturated half-space. International Journal of Engineering Science, 2017, 119: 55–77

[32] Senjuntichai T, Keawsawasvong S, Plangmal R. Vertical vibrations of rigid foundations of arbitrary shape in a multi-layered poroelastic medium. Computers and Geotechnics, 2018, 100: 121–134

[33] Ba Z N, Sang Q Z, Liang J W. Seismic analysis of a lined tunnel in a multi-layered TI saturated half-space due to qP1-and qSV-waves. Tunnelling and Underground Space Technology, 2022, 119: 104248

[34] ZhuX YLiuX JZongR LZengFPanF. Microstructure and mechanical properties of nanoscale Cu/Ni multilayers. Materials Science and Engineering A, 2010, 527(4−5): 1243−1248

[35] Kern W, Schuegraf K K. Handbook of Thin Film Deposition Processes and Techniques. 2nd ed. Norwich: William Andrew Publishing, 2001, 11–43

[36] Senjuntichai T, Rajapakse R K N D. Exact stiffness method for quasi-statics of a multi-layered poroelastic medium. International Journal of Solids and Structures, 1995, 32(11): 1535–1553

[37] SneddonI N. Fourier Transform. New York: McGraw-Hill, 1951

[38] Povstenko Y Z. Theoretical investigation of phenomena caused by heterogeneous surface tension in solids. Journal of the Mechanics and Physics of Solids, 1993, 41(9): 1499–1514

[39] Wang Z Q, Zhao Y P, Huang Z P. The effects of surface tension on the elastic properties of nano structures. International Journal of Engineering Science, 2010, 48(2): 140–150

[40] MeyersM AChawlaK K. Mechanical Behavior of Materials. 2nd ed. Cambridge: Cambridge University Press, 2008

[41] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[42] Hamdia K, Ghasemi H, Zhuang X, Alajlan N, Rabczuk T. Sensitivity and uncertainty analysis for flexoelectric nanostructures. Computer Methods in Applied Mechanics and Engineering, 2018, 337: 95–109

[43] PiessensRDoncker-KapengaEUberhuberC WKahanerD K. QUADPACK: A Subroutine Package for Automatic Integration. Berlin: Springer, 1983

[44] Katebi A, Selvadurai A P S. Undrained behaviour of a non-homogeneous elastic medium: The influence of variations in the elastic shear modulus with depth. Geotechnique, 2013, 63(13): 1159–1169

[45] NakayamaTYamamotoKSatohHKonnoT JClemensB MSinclairR. Structure and corrosion properties of Al/Si and Fe/Zr multilayers. Materials Science and Engineering A, 1995, 198(1−2): 19−24

[46] Sneddon I N. The relation between load and penetration in the axisymmetric Boussinesq problem for a punch of arbitrary profile. International Journal of Engineering Science, 1965, 3(1): 47–57

Acknowledgements

This study was supported by the Civil Engineering Centennial Scholarship of Chulalongkorn University, Thailand Research Fund under Grant MRG6280116, and the TRF Senior Research Scholar under Grant RTA 6280012.

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