Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Small scale effect on the vibration of non-uniform nanoplates

  • Chakraverty, S. (Department of Mathematics, National Institute of Technology Rourkela) ;
  • Behera, Laxmi (Department of Mathematics, National Institute of Technology Rourkela)
  • Received : 2015.02.20
  • Accepted : 2015.05.03
  • Published : 2015.08.10
⟨ Previous Next ⟩

Abstract

Free vibration of non-uniform embedded nanoplates based on classical (Kirchhoff's) plate theory in conjunction with nonlocal elasticity theory has been studied. The nanoplate is assumed to be rested on two-parameter Winkler-Pasternak elastic foundation. Non-uniform material properties of nanoplates have been considered by taking linear as well as quadratic variations of Young's modulus and density along the space coordinates. Detailed analysis has been reported for all possible casesof such variations. Trial functions denoting transverse deflection of the plate are expressed in simple algebraic polynomial forms. Application of the present method converts the problem into generalised eigen value problem. The study aims to investigate the effects of non-uniform parameter, elastic foundation, nonlocal parameter, boundary condition, aspect ratio and length of nanoplates on the frequency parameters. Three-dimensional mode shapes for some of the boundary conditions have also been illustrated. One may note that present method is easier to handle any sets of boundary conditions at the edges.

Keywords

References

  1. Adali, S. (2012), "Variational principles for nonlocal continuum model of orthotropic graphene sheets embedded in an elastic medium", Acta Mathematica Scientia, 32, 325-338. https://doi.org/10.1016/S0252-9602(12)60020-4
  2. Aghababaei, R. and Reddy, J.N. (2009), "Nonlocal third-order shear deformation plate theory with application to bending andv ibration of plates", J. Sound Vib., 326, 277289.
  3. Aksencer, T. and Aydogdu, M. (2011), "Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory", Physica E, 43, 954-959. https://doi.org/10.1016/j.physe.2010.11.024
  4. Anjomshoa, A. (2013), "Application of ritz functions in buckling analysis of embedded orthotropic circular and elliptical micro/nano-plates based on nonlocal elasticity theory", Meccanica, 48, 1337-1353. https://doi.org/10.1007/s11012-012-9670-y
  5. Ansari, R., Ashrafi, M.A., Pourashraf, T. and Sahmani, S. (2015), "Vibration and buckling characteristics of functionally graded nanoplates subjected to thermal loading based on surface elasticity theory", Acta Astronautica, 109, 42-51. https://doi.org/10.1016/j.actaastro.2014.12.015
  6. Behera, L. and Chakraverty, S. (2013), "Free vibration of nonhomogeneous Timoshenko nanobeams", Meccanica, 49(1), 51-67. https://doi.org/10.1007/s11012-013-9771-2
  7. Belkorissat I., Houari M.S.A., Tounsi, A., Bedia, E.A. and Mahmoud, S.R.(2015), "On vibration properties of functionally graded nano-plate using a new nonlocal refined four variable mode", Steel Compos. Struct., 18, 1063-1081. https://doi.org/10.12989/scs.2015.18.4.1063
  8. Beni, A.A. and Malekzadeh, P. (2012), "Nonlocal free vibration of orthotropic nonprismatic skew nanoplates", Compos. Struct., 94, 3215-3222. https://doi.org/10.1016/j.compstruct.2012.04.013
  9. Bhat, R.B. (1985), "Plate deflections using orthogonal polynomials", J. Eng. Mech., 111, 1301-1309. https://doi.org/10.1061/(ASCE)0733-9399(1985)111:11(1301)
  10. Bhat, R.B. (1991), "Vibration of rectangular plates on point and line supports using characteristic orthogonal polynomials in the Rayleigh-Ritz method", J. Sound Vib., 149, 170-172. https://doi.org/10.1016/0022-460X(91)90923-8
  11. Chakraverty, S. and Behera, L. (2014), "Free vibration of rectangular nanoplates using Rayleigh-Ritz method", Physica E, 56, 357-363. https://doi.org/10.1016/j.physe.2013.08.014
  12. Chakraverty, S., Jindal, R. and Agarwal, V.K. (2007), "Effect of non-homogeneity on natural frequencies of vibration of elliptic plates", Meccanica, 42, 585-599. https://doi.org/10.1007/s11012-007-9077-3
  13. Dubey, A., Sharma, G., Mavroidis, C., Tomassone, M.S., Nikitczuk, K. and Yarmush, M.L. (2004), "Computational Studies of Viral Protein Nano-Actuators", J. Comput. Theor. Nanosci., 1, 18-28. https://doi.org/10.1166/jctn.2003.003
  14. Eringen, A.C. (1972), "Nonlocal polar elastic continua", Int. J. Eng. Sci., 10, 1-16. https://doi.org/10.1016/0020-7225(72)90070-5
  15. Eringen, A.C. (1983), "On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves", J. Appl. Phys., 54, 4703-4710.15. https://doi.org/10.1063/1.332803
  16. Farajpour, A., Danesh, M. and Mohammadi, M. (2011), "Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics", Physica E, 44, 719-727. https://doi.org/10.1016/j.physe.2011.11.022
  17. Jomehzadeh, E. and Saidi, A.R. (2012), "Study of small scale effect on nonlinear vibration of nano-plates", J. Comput. Theor. Nanosci., 9, 864-871. https://doi.org/10.1166/jctn.2012.2108
  18. Kiani, K. (2011), "Small-scaleeffect on the vibration of thin nanoplates subjected to amoving nanoparticle via nonlocal continuum theory", J. Sound Vib., 330, 4896-4914. https://doi.org/10.1016/j.jsv.2011.03.033
  19. Kiani, K. (2011a), "Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle, Part I: theoretical formulations", Physica E: Low-dimen. Syst. Nanostruct., 44, 229248.
  20. Kiani, K. (2011b), "Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle, Part II: parametric studies", Physica E: Low-dimen. Syst. Nanostruct., 44, 249269.
  21. Kiani, K. (2014), "Free vibration of conducting nanoplates exposed to unidirectional in-plane magnetic fields using nonlocal shear deformable plate theories", Physica E: Low-dimen. Syst. Nanostruct., 57C, 179-192.
  22. Liang, Y.J. and Han, Q. (2012), "Prediction of nonlocal scale parameter for carbon nanotubes", Sci. China Phys. Mech. Astron., 55, 1670-1678. https://doi.org/10.1007/s11433-012-4826-2
  23. Liang, Y.J. and Han, Q. (2014), "Prediction of the nonlocal scaling parameter for graphene sheet," Eur. J. Mech. A Solid., 45, 153-160. https://doi.org/10.1016/j.euromechsol.2013.12.009
  24. Liu, C., Ke, L.,Wang, Y.S., Yang, J. and Kitipornchai, S. (2013), "Thermo-electro-mechanical vibration of piezoelectric nanoplates based on the nonlocal theory", Compos. Struct., 106, 167-174, https://doi.org/10.1016/j.compstruct.2013.05.031
  25. Malekzadeh, P., Setoodeh, A. and Alibeygi Beni, A. (2011), "Small scale effect on the free vibration of orthotropic arbitrary straight-sided quadrilateral nanoplates", Compos. Struct., 93, 1631-1639. https://doi.org/10.1016/j.compstruct.2011.01.008
  26. Malekzadeh, P. and Shojaee, M. (2013), "Free vibration of nanoplates based on a nonlocal two-variable refined plate theory", Compos. Struct., 95, 443-452. https://doi.org/10.1016/j.compstruct.2012.07.006
  27. Murmu, T. and Pradhan, S.C. (2009), "Small-scale effect on the vibration of nonuniform nanocantilever based on nonlocal elasticity theory", Physica E, 41, 1451.
  28. Murmu, T. and Pradhan, S.C. (2009), "Vibration analysis of nanoplates under uniaxial prestressed conditions via nonlocal elasticity", J. Appl. Phys., 106, 104301. https://doi.org/10.1063/1.3233914
  29. 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
  30. Narendar, S. (2011), "Buckling analysis of micro-/nano-scale plates based on two-variable refined plate theory incorporating nonlocal scale effects", Compos. Struct., 93. 3093-3103. https://doi.org/10.1016/j.compstruct.2011.06.028
  31. Narendar, S., Roy Mahapatra, D. and Gopalakrishnan, S. (2011), "Prediction of nonlocal scaling parameter for armchair and zigzag single-walled carbon nanotubes based on molecular structural mechanics, nonlocal elasticity and wave propagation", Int. J. Eng. Sci., 49, 509-522. https://doi.org/10.1016/j.ijengsci.2011.01.002
  32. Natarajan S., Chakraborty, S., Thangavel, M., Bordas, S. and Rabczuk, T. (2012), "Size dependent free flexural vibration behavior of functionally graded nanoplates", Comput. Mater. Sci., 65, 74-80. https://doi.org/10.1016/j.commatsci.2012.06.031
  33. Peng, H.B., Chang, C.W., Aloni, S., Yuzvinsky, T.D. and Zettl, A. (2006), "Ultrahigh frequency nanotube resonators", Phys. Rev. Lett., 97, 087203.
  34. Phadikar, J.K. and Pradhan, S.C. (2010), "Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates", Comput. Mater. Sci., 49, 492-499. https://doi.org/10.1016/j.commatsci.2010.05.040
  35. Pradhan, S. and Phadikar J. (2009), "Nonlocal elasticity theory for vibration of nanoplates", J. Sound Vib., 325, 206-223. https://doi.org/10.1016/j.jsv.2009.03.007
  36. Pradhan, S.C. and Murmu, T. (2010), "Application of nonlocal elasticity and DQM in the flapwise bending vibration of a rotating nanocantilever", Physica E, 42, 1944-1949. https://doi.org/10.1016/j.physe.2010.03.004
  37. Rajalingham, C., Bhat, R.B. and Xistris, G.D. (1996), "Vibration of rectangular plates using plate characteristic functions as shape functions in the Rayleigh-Ritz method", J. Sound Vib., 193, 585-599.
  38. Ravari, M.K. and Shahidi, A. (2013), "Axisymmetric buckling of the circular annular nanoplates using finite difference method", Meccanica, 48, 135-144. https://doi.org/10.1007/s11012-012-9589-3
  39. Ruud, J., Jervis, T. and Spaepen, F. (1994), "Nanoindentation of ag/ni multilayered thin films", J. Appl. Phys., 75, 4969-4974. https://doi.org/10.1063/1.355787
  40. Salehipour, H., Nahvi, H. and Shahidi, A.R. (2015), "Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity", Physica E: Low-dimen. Syst. Nanostruct., 66, 350-358. https://doi.org/10.1016/j.physe.2014.10.001
  41. Singh, B. ans Chakraverty, S. (1994a), "Boundary characteristic orthogonal polynomials in numerical approximation", Commun. Numer. Meth. Eng., 10, 1027-1043. https://doi.org/10.1002/cnm.1640101209
  42. Wang, K. and Wang, B. (2011), "Vibration of nanoscale plates with surface energy via nonlocal elasticity", Physica E: Low-dimen. Syst. Nanostruct., 44, 448-453. https://doi.org/10.1016/j.physe.2011.09.019

Cited by

  1. Dynamic buckling of FGM viscoelastic nano-plates resting on orthotropic elastic medium based on sinusoidal shear deformation theory vol.60, pp.3, 2016, https://doi.org/10.12989/sem.2016.60.3.489
  2. Thermal stability of functionally graded sandwich plates using a simple shear deformation theory vol.58, pp.3, 2016, https://doi.org/10.12989/sem.2016.58.3.397
  3. The computation of bending eigenfrequencies of single-walled carbon nanotubes based on the nonlocal theory vol.9, pp.2, 2018, https://doi.org/10.5194/ms-9-349-2018
  4. Electro-mechanical vibration of nanoshells using consistent size-dependent piezoelectric theory vol.22, pp.6, 2015, https://doi.org/10.12989/scs.2016.22.6.1301
  5. Vibration analysis of FG nanoplates with nanovoids on viscoelastic substrate under hygro-thermo-mechanical loading using nonlocal strain gradient theory vol.64, pp.6, 2015, https://doi.org/10.12989/sem.2017.64.6.683
  6. Influence of shear preload on wave propagation in small-scale plates with nanofibers vol.70, pp.4, 2015, https://doi.org/10.12989/sem.2019.70.4.407