Steel and Composite Structures

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Nonlocal strain gradient 3D elasticity theory for anisotropic spherical nanoparticles

  • Karami, Behrouz (Department of Mechanical Engineering, Marvdasht Branch, Islamic Azad University) ;
  • Janghorban, Maziar (Department of Mechanical Engineering, Marvdasht Branch, Islamic Azad University) ;
  • Tounsi, Abdelouahed (Material and Hydrology Laboratory, University of Sidi Bel Abbes, Faculty of Technology, Civil Engineering Department)
  • Received : 2017.09.02
  • Accepted : 2018.03.02
  • Published : 2018.04.25
⟨ Previous Next ⟩

Abstract

In this paper, three-dimensional (3D) elasticity theory in conjunction with nonlocal strain gradient theory (NSGT) is developed for mechanical analysis of anisotropic nanoparticles. The present model incorporates two scale coefficients to examine the mechanical characteristics much accurately. All the elastic constants are considered and assumed to be the functions of (r, ${\theta}$, ${\varphi}$), so all kind of anisotropic structures can be modeled. Moreover, all types of functionally graded spherical structures can be investigated. To justify our model, our results for the radial vibration of spherical nanoparticles are compared with experimental results available in the literature and great agreement is achieved. Next, several examples of the radial vibration and wave propagation in spherical nanoparticles including nonlocal strain gradient parameters are presented for more than 10 different anisotropic nanoparticles. From the best knowledge of authors, it is the first time that 3D elasticity theory and NSGT are used together with no approximation to derive the governing equations in the spherical coordinate. Moreover, up to now, the NSGT has not been used for spherical anisotropic nanoparticles. It is also the first time that all the 36 elastic constants as functions of (r, ${\theta}$, ${\varphi}$) are considered for anisotropic and functionally graded nanostructures including size effects. According to the lack of any common approximations in the displacement field or in elastic constant, present theory can be assumed as a benchmark for future works.

Keywords

References

  1. Aifantis, E.C. (1999), "Strain gradient interpretation of size effects", Int. J. Fract., 95(1-4), 299. https://doi.org/10.1023/A:1018625006804
  2. Askes, H. and Aifantis, E.C. (2009), "Gradient elasticity and flexural wave dispersion in carbon nanotubes", Phys. Rev. B, 80(19), 195412. https://doi.org/10.1103/PhysRevB.80.195412
  3. Askes, H. and Aifantis, E.C. (2011), "Gradient elasticity in statics and dynamics: an overview of formulations, length scale identification procedures, finite element implementations and new results", Int. J. Solids Struct., 48(13), 1962-1990. https://doi.org/10.1016/j.ijsolstr.2011.03.006
  4. Bagdatli, S.M. (2015), "Non-linear transverse vibrations of tensioned nanobeams using nonlocal beam theory", Struct. Eng. Mech., Int. J., 55(2), 281-298. https://doi.org/10.12989/sem.2015.55.2.281
  5. Benahmed, A., Houari, M.S.A., Benyoucef, S., Belakhdar, K. and Tounsi, A. (2017), "A novel quasi-3D hyperbolic shear deformation theory for functionally graded thick rectangular plates on elastic foundation", Geomech. Eng., Int. J., 12(1), 9-34. https://doi.org/10.12989/gae.2017.12.1.009
  6. Bourada, F., Amara, K. and Tounsi, A. (2016), "Buckling analysis of isotropic and orthotropic plates using a novel four variable refined plate theory", Steel Compos. Struct., Int. J., 21(6), 1287-1306. https://doi.org/10.12989/scs.2016.21.6.1287
  7. Carrera, E. (2002), "Theories and finite elements for multilayered, anisotropic, composite plates and shells", Arch. Computat. Method. Eng., 9(2), 87-140. https://doi.org/10.1007/BF02736649
  8. Chaht, F.L., Kaci, A., Houari, M.S.A., Tounsi, A., Beg, O.A. and Mahmoud, S. (2015), "Bending and buckling analyses of functionally graded material (FGM) size-dependent nanoscale beams including the thickness stretching effect", Steel Compos. Struct., Int. J., 18(2), 425-442. https://doi.org/10.12989/scs.2015.18.2.425
  9. Chronopoulos, D. (2017), "Wave steering effects in anisotropic composite structures: Direct calculation of the energy skew angle through a finite element scheme", Ultrasonics, 73, 43-48. https://doi.org/10.1016/j.ultras.2016.08.020
  10. Combe, N., Huntzinger, J.R. and Mlayah, A. (2007), "Vibrations of quantum dots and light scattering properties: Atomistic versus continuous models", Phys. Rev. B, 76(20), 205425. https://doi.org/10.1103/PhysRevB.76.205425
  11. Ebrahimi, F. and Barati, M.R. (2017a), "Hygrothermal effects on vibration characteristics of viscoelastic FG nanobeams based on nonlocal strain gradient theory", Compo s. Struct., 159, 433-444. https://doi.org/10.1016/j.compstruct.2016.09.092
  12. Ebrahimi, F. and Barati, M.R. (2017b), "Vibration analysis of embedded size dependent FG nanobeams based on third-order shear deformation beam theory", Struct. Eng. Mech., Int. J., 61(6), 721-736. https://doi.org/10.12989/sem.2017.61.6.721
  13. Eringen, A.C. (1967), "Theory of micropolar plates", Zeitschrift fur angewandte Mathematik und Physik ZAMP, 18(1), 12-30. https://doi.org/10.1007/BF01593891
  14. Eringen, A.C. and Edelen, D. (1972), "On nonlocal elasticity", Int. J. Eng. Sci., 10(3), 233-248. https://doi.org/10.1016/0020-7225(72)90039-0
  15. Farajpour, A., Yazdi, M.H., Rastgoo, A. and Mohammadi, M. (2016), "A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment", Acta Mechanica, 227(7), 1849-1867. https://doi.org/10.1007/s00707-016-1605-6
  16. Ghavanloo, E. and Fazelzadeh, S.A. (2013), "Radial vibration of free anisotropic nanoparticles based on nonlocal continuum mechanics", Nanotechnology, 24(7), 075702. https://doi.org/10.1088/0957-4484/24/7/075702
  17. Grundmann, M., Sturm, C., Kranert, C., Richter, S., Schmidt-Grund, R., Deparis, C. and Zuniga-Perez, J. (2016), "Optically anisotropic media: New approaches to the dielectric function, singular axes, microcavity modes and Raman scattering intensities", Physica Status Solidi (RRL)-Rapid Research Letters, 11(1).
  18. Gupta, S.K., Sahoo, S., Jha, P.K., Arora, A. and Azhniuk, Y. (2009), "Observation of torsional mode in CdS1- xSex nanoparticles in a borosilicate glass", J. Appl. Phys., 106(2), 024307. https://doi.org/10.1063/1.3171925
  19. Gurtin, M., Weissmuller, J. and Larche, F. (1998), "A general theory of curved deformable interfaces in solids at equilibrium", Philosophical Magazine A, 78(5), 1093-1109. https://doi.org/10.1080/01418619808239977
  20. Hamidi, A., Houari, M.S.A., Mahmoud, S. and Tounsi, A. (2015), "A sinusoidal plate theory with 5-unknowns and stretching effect for thermomechanical bending of functionally graded sandwich plates", Steel Compos. Struct., Int. J., 18(1), 235-253. https://doi.org/10.12989/scs.2015.18.1.235
  21. Houari, M.S.A., Tounsi, A., Bessaim, A. and Mahmoud, S. (2016), "A new simple three-unknown sinusoidal shear deformation theory for functionally graded plates", Steel Compos. Struct., Int. J., 22(2), 257-276. https://doi.org/10.12989/scs.2016.22.2.257
  22. Jandaghian, A.A. and Rahmani, O. (2017), "Vibration analysis of FG nanobeams based on third-order shear deformation theory under various boundary conditions", Steel Compos. Struct., Int. J., 25(1), 67-78.
  23. Karami, B. and Janghorban, M. (2016), "Effect of magnetic field on the wave propagation in nanoplates based on strain gradient theory with one parameter and two-variable refined plate theory", Modern Physics Letters B, 30(36), 1650421. https://doi.org/10.1142/S0217984916504212
  24. Karami, B., Janghorban, M. and Tounsi, A. (2017a), "Effects of triaxial magnetic field on the anisotropic nanoplates", Steel Compos. Struct., Int. J., 25(3), 361-374.
  25. Karami, B., Shahsavari, D. and Janghorban, M. (2017b), "Wave propagation analysis in functionally graded (FG) nanoplates under in-plane magnetic field based on nonlocal strain gradient theory and four variable refined plate theory", Mech. Adv. Mater. Struct., 1-11.
  26. Karami, B., Janghorban, M. and Li, L. (2018a), "On guided wave propagation in fully clamped porous functionally graded nanoplates", Acta Astronautica, 143, 380-390. https://doi.org/10.1016/j.actaastro.2017.12.011
  27. Karami, B., Shahsavari, D., Janghorban, M. and Li, L. (2018b), "Wave dispersion of mounted graphene with initial stress", Thin-Wall. Struct., 122, 102-111. https://doi.org/10.1016/j.tws.2017.10.004
  28. Karami, B., Shahsavari, D. and Li, L. (2018c), "Hygrothermal wave propagation in viscoelastic graphene under in-plane magnetic field based on nonlocal strain gradient theory", Physica E; Low-dimens. Syst. Nanostruct., 97, 317-327. https://doi.org/10.1016/j.physe.2017.11.020
  29. Karami, B., Shahsavari, D. and Li, L. (2018d), "Temperaturedependent flexural wave propagation in nanoplate-type porous heterogenous material subjected to in-plane magnetic field", J. Therm. Stress., 41(4), 483-499. https://doi.org/10.1080/01495739.2017.1393781
  30. Karami, B., Shahsavari, D., Li, L., Karami, M. and Janghorban, M. (2018e), "Thermal buckling of embedded sandwich piezoelectric nanoplates with functionally graded core by a nonlocal second-order shear deformation theory", Proceedings of the Institution of Mechanical Engineers, Part C; Journal of Mechanical Engineering Science.
  31. Li, L. and Hu, Y. (2017), "Post-buckling analysis of functionally graded nanobeams incorporating nonlocal stress and microstructure-dependent strain gradient effects", Int. J. Mech. Sci., 120, 159-170. https://doi.org/10.1016/j.ijmecsci.2016.11.025
  32. Li, L., Hu, Y. and Ling, L. (2015), "Flexural wave propagation in small-scaled functionally graded beams via a nonlocal strain gradient theory", Compos. Struct., 133, 1079-1092. https://doi.org/10.1016/j.compstruct.2015.08.014
  33. Li, L., Hu, Y. and Ling, L. (2016a), "Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory", Physica E; Low-dimens. Syst. Nanostruct., 75, 118-124. https://doi.org/10.1016/j.physe.2015.09.028
  34. Li, L., Li, X. and Hu, Y. (2016b), "Free vibration analysis of nonlocal strain gradient beams made of functionally graded material", Int. J. Eng. Sci., 102, 77-92. https://doi.org/10.1016/j.ijengsci.2016.02.010
  35. Li, X., Li, L., Hu, Y., Ding, Z. and Deng, W. (2017), "Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory", Compos. Struct., 165, 250-265. https://doi.org/10.1016/j.compstruct.2017.01.032
  36. Lim, C., Zhang, G. and Reddy, J. (2015), "A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation", J. Mech. Phys. Solids, 78, 298-313. https://doi.org/10.1016/j.jmps.2015.02.001
  37. Mankad, V., Mishra, K., Gupta, S.K., Ravindran, T. and Jha, P.K. (2012), "Low frequency Raman scattering from confined acoustic phonons in freestanding silver nanoparticles", Vibrational Spectroscopy, 61, 183-187. https://doi.org/10.1016/j.vibspec.2012.02.004
  38. Mehralian, F., Beni, Y.T. and Zeverdejani, M.K. (2017), "Calibration of nonlocal strain gradient shell model for buckling analysis of nanotubes using molecular dynamics simulations", Physica B; Condensed Matter.
  39. Mock, A., Korlacki, R., Knight, S. and Schubert, M. (2017), "Anisotropy, phonon modes, and lattice anharmonicity from dielectric function tensor analysis of monoclinic cadmium tungstate", arXiv preprint arXiv;1701.00813. https://doi.org/10.1103/PhysRevB.95.165202
  40. Mousavi, S., Reddy, J. and Romanoff, J. (2016), "Analysis of anisotropic gradient elastic shear deformable plates", Acta Mechanica, 227(12), 3639-3656. https://doi.org/10.1007/s00707-016-1689-z
  41. Nami, M.R. and Janghorban, M. (2014), "Resonance behavior of FG rectangular micro/nano plate based on nonlocal elasticity theory and strain gradient theory with one gradient constant", Compos. Struct., 111, 349-353. https://doi.org/10.1016/j.compstruct.2014.01.012
  42. Ng, M.-Y. and Chang, Y.-C. (2011), "Laser-induced breathing modes in metallic nanoparticles: a symmetric molecular dynamics study", J. Chem. Phys., 134(9), 094116. https://doi.org/10.1063/1.3563803
  43. Portales, H., Saviot, L., Duval, E., Fujii, M., Hayashi, S., Del Fatti, N. and Vallee, F. (2001), "Resonant Raman scattering by breathing modes of metal nanoparticles", J. Chem. Phys., 115(8), 3444-3447. https://doi.org/10.1063/1.1396817
  44. Ruijgrok, P.V., Zijlstra, P., Tchebotareva, A.L. and Orrit, M. (2012), "Damping of acoustic vibrations of single gold nanoparticles optically trapped in water", Nano Letters, 12(2), 1063-1069. https://doi.org/10.1021/nl204311q
  45. Sadd, M.H. (2009), Elasticity; Theory, Applications, and Numerics, Academic Press.
  46. Sahmani, S. and Aghdam, M. (2017), "Nonlinear instability of axially loaded functionally graded multilayer graphene plateletreinforced nanoshells based on nonlocal strain gradient elasticity theory", Int. J. Mech. Sci., 131, 95-106.
  47. Saviot, L., Murray, D.B. and De Lucas, M.D.C.M. (2004), "Vibrations of free and embedded anisotropic elastic spheres: Application to low-frequency Raman scattering of silicon nanoparticles in silica", Phys. Rev. B, 69(11), 113402. https://doi.org/10.1103/PhysRevB.69.113402
  48. Shahsavari, D. and Janghorban, M. (2017), "Bending and shearing responses for dynamic analysis of single-layer graphene sheets under moving load", J. Brazil. Soc. Mech. Sci. Eng., 39(10), 3849-3861. https://doi.org/10.1007/s40430-017-0863-0
  49. Shahsavari, D., Karami, B., Janghorban, M. and Li, L. (2017), "Dynamic characteristics of viscoelastic nanoplates under moving load embedded within visco-Pasternak substrate and hygrothermal environment", Mater. Res. Express, 4(8), 085013. https://doi.org/10.1088/2053-1591/aa7d89
  50. Shahsavari, D., Karami, B. and Mansouri, S. (2018a), "Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories", Eur. J. Mech.-A/Solids, 67, 200-214. https://doi.org/10.1016/j.euromechsol.2017.09.004
  51. Shahsavari, D., Shahsavari, M., Li, L. and Karami, B. (2018b), "A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation", Aerosp. Sci. Technol., 72, 134-149. https://doi.org/10.1016/j.ast.2017.11.004
  52. Shen, J., Li, C., Fan, X. and Jung, C. (2017), "Dynamics of silicon nanobeams with axial motion subjected to transverse and longitudinal loads considering nonlocal and surface effects", Smart Struct. Syst., Int. J., 19(1), 105-113. https://doi.org/10.12989/sss.2017.19.1.105
  53. Shukla, A. and Kumar, V. (2011), "Low-frequency Raman scattering from silicon nanostructures", J. Appl. Phys., 110(6), 064317. https://doi.org/10.1063/1.3633235
  54. Simsek, M. (2016), "Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach", Int. J. Eng. Sci., 105, 12-27. https://doi.org/10.1016/j.ijengsci.2016.04.013
  55. Sobhy, M. (2017), "Hygro-thermo-mechanical vibration and buckling of exponentially graded nanoplates resting on elastic foundations via nonlocal elasticity theory", Struct. Eng. Mech., Int. J., 63(3), 401-415.
  56. Srinivas, S. and Rao, A. (1970), "Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates", Int. J. Solids Struct.res, 6(11), 1463-1481. https://doi.org/10.1016/0020-7683(70)90076-4
  57. Teodosiu, C. (1982), The Elastic Field of Point Defects, Springer.
  58. Voisin, C., Del Fatti, N., Christofilos, D. and Vallee, F. (2000), "Time-resolved investigation of the vibrational dynamics of metal nanoparticles", Appl. Surf. Sci., 164(1), 131-139. https://doi.org/10.1016/S0169-4332(00)00347-0
  59. Yang, F., Chong, A., Lam, D.C. and Tong, P. (2002), "Couple stress based strain gradient theory for elasticity", Int. J. Solids Struct., 39(10), 2731-2743. https://doi.org/10.1016/S0020-7683(02)00152-X
  60. Zbib, H. and Aifantis, E. (2003), "Size effects and length scales in gradient plasticity and dislocation dynamics", Scripta Materialia, 48(2), 155-160. https://doi.org/10.1016/S1359-6462(02)00342-1
  61. Zenkour, A.M. and Abouelregal, A.E. (2015), "Thermoelastic interaction in functionally graded nanobeams subjected to timedependent heat flux", Steel Compos. Struct., Int. J., 18(4), 909-924. https://doi.org/10.12989/scs.2015.18.4.909
  62. Zhu, X. and Li, L. (2017a), "Closed form solution for a nonlocal strain gradient rod in tension", Int. J. Eng. Sci., 119, 16-28. https://doi.org/10.1016/j.ijengsci.2017.06.019
  63. Zhu, X. and Li, L. (2017b), "Longitudinal and torsional vibrations of size-dependent rods via nonlocal integral elasticity", Int. J. Mech. Sci., 133, 639-650. https://doi.org/10.1016/j.ijmecsci.2017.09.030
  64. Zhu, X. and Li, L. (2017c), "On longitudinal dynamics of nanorods", Int. J. Eng. Sci., 120, 129-145. https://doi.org/10.1016/j.ijengsci.2017.08.003
  65. Zhu, X. and Li, L. (2017d), "Twisting statics of functionally graded nanotubes using Eringen's nonlocal integral model", Compos. Struct., 178, 87-96. https://doi.org/10.1016/j.compstruct.2017.06.067
  66. Ziane, N., Meftah, S.A., Belhadj, H.A. and Tounsi, A. (2013), "Free vibration analysis of thin and thick-walled FGM box beams", Int. J. Mech. Sci., 66, 273-282. https://doi.org/10.1016/j.ijmecsci.2012.12.001

Cited by

  1. Nonlocal three-dimensional theory of elasticity for buckling behavior of functionally graded porous nanoplates using volume integrals vol.5, pp.9, 2018, https://doi.org/10.1088/2053-1591/aad4c3
  2. A novel approach for nonlinear bending response of macro- and nanoplates with irregular variable thickness under nonuniform loading in thermal environment pp.1539-7742, 2019, https://doi.org/10.1080/15397734.2018.1557529
  3. Nonlocal Thermal and Mechanical Buckling of Nonlinear Orthotropic Viscoelastic Nanoplates Embedded in a Visco-Pasternak Medium vol.10, pp.8, 2018, https://doi.org/10.1142/s1758825118500862
  4. On nonlinear bending behavior of FG porous curved nanotubes vol.135, pp.None, 2018, https://doi.org/10.1016/j.ijengsci.2018.11.005
  5. Influence of shear preload on wave propagation in small-scale plates with nanofibers vol.70, pp.4, 2018, https://doi.org/10.12989/sem.2019.70.4.407
  6. A simple quasi-3D HSDT for the dynamics analysis of FG thick plate on elastic foundation vol.31, pp.5, 2018, https://doi.org/10.12989/scs.2019.31.5.503
  7. Wave dispersion properties in imperfect sigmoid plates using various HSDTs vol.33, pp.5, 2018, https://doi.org/10.12989/scs.2019.33.5.699
  8. Using IGA and trimming approaches for vibrational analysis of L-shape graphene sheets via nonlocal elasticity theory vol.33, pp.5, 2019, https://doi.org/10.12989/scs.2019.33.5.717
  9. A new higher-order shear and normal deformation theory for the buckling analysis of new type of FGM sandwich plates vol.72, pp.5, 2019, https://doi.org/10.12989/sem.2019.72.5.653
  10. On the modeling of dynamic behavior of composite plates using a simple nth-HSDT vol.29, pp.6, 2018, https://doi.org/10.12989/was.2019.29.6.371
  11. Dynamic Stress around a Cylindrical Nano-Inclusion with an Interface in a Right-Angle Plane under SH-Wave vol.2020, pp.None, 2018, https://doi.org/10.1155/2020/9717386
  12. Influence of vacancy defects on vibration analysis of graphene sheets applying isogeometric method: Molecular and continuum approaches vol.34, pp.2, 2020, https://doi.org/10.12989/scs.2020.34.2.261
  13. Free vibration analysis of sandwich FGM shells using isogeometric B-spline finite strip method vol.34, pp.3, 2020, https://doi.org/10.12989/scs.2020.34.3.361
  14. A numerical method for dynamic characteristics of nonlocal porous metal-ceramic plates under periodic dynamic loads vol.7, pp.1, 2020, https://doi.org/10.12989/smm.2020.7.1.027
  15. Study of transversely isotropic nonlocal thermoelastic thin nano-beam resonators with multi-dual-phase-lag theory vol.91, pp.1, 2018, https://doi.org/10.1007/s00419-020-01771-7
  16. Flow of casson nanofluid along permeable exponentially stretching cylinder: Variation of mass concentration profile vol.38, pp.1, 2018, https://doi.org/10.12989/scs.2021.38.1.033
  17. Stoneley wave propagation in nonlocal isotropic magneto-thermoelastic solid with multi-dual-phase lag heat transfer vol.38, pp.2, 2018, https://doi.org/10.12989/scs.2021.38.2.141
  18. On the mechanics of nanocomposites reinforced by wavy/defected/aggregated nanotubes vol.38, pp.5, 2018, https://doi.org/10.12989/scs.2021.38.5.533
  19. Thermal stress effects on microtubules based on orthotropic model: Vibrational analysis vol.11, pp.3, 2018, https://doi.org/10.12989/acc.2021.11.3.255
  20. Effect of suction on flow of dusty fluid along exponentially stretching cylinder vol.10, pp.3, 2018, https://doi.org/10.12989/anr.2021.10.3.263
  21. Dispersion of waves characteristics of laminated composite nanoplate vol.40, pp.3, 2018, https://doi.org/10.12989/scs.2021.40.3.355
  22. On static buckling of multilayered carbon nanotubes reinforced composite nanobeams supported on non-linear elastic foundations vol.40, pp.3, 2021, https://doi.org/10.12989/scs.2021.40.3.389