On scale-dependent vibration of circular cylindrical nanoporous metal foam shells

  • Yan Qing Wang
  • Yun Fei Liu
  • Jean W. Zu
Technical Paper


In this paper, free vibrations of cylindrical nanoshells made of nanoporous metal foam are investigated for the first time. Based on the modified couple stress theory and Love’s thin shell theory, the governing equations of the present system are derived by using Hamilton’s principle. Two types of nanoporosity distribution are considered in the construction of the nanoporous shells. Then, the Navier method and Galerkin method are utilized to solve natural frequencies of the nanoporous shells under different boundary conditions. Afterwards, a detailed parametric study is conducted. Results show that the nanoporosity type, the material length scale parameter, the porosity coefficient, the length-to-radius ratio, and the radius-to-thickness ratio play important role on the free vibrations of nanoporous shells. To check the validity of the present analysis, the results are compared with those in previous studies for the special cases.



This research was supported by the National Natural Science Foundation of China (Grant no. 11672071) and the Fundamental Research Funds for the Central Universities (Grant no. N170504023).


  1. Alibeigloo A, Shaban M (2013) Free vibration analysis of carbon nanotubes by using three-dimensional theory of elasticity. Acta Mech 224:1415–1427MathSciNetCrossRefGoogle Scholar
  2. Ansari R, Gholami R, Norouzzadeh A, Sahmani S (2015) Size-dependent vibration and instability of fluid-conveying functionally graded microshells based on the modified couple stress theory. Microfluid Nanofluidics 19:509–522CrossRefGoogle Scholar
  3. Ansari R, Pourashraf T, Gholami R, Rouhi H (2016) Analytical solution approach for nonlinear buckling and postbuckling analysis of cylindrical nanoshells based on surface elasticity theory. Appl Math Mech 37:903–918MathSciNetCrossRefGoogle Scholar
  4. Asghari M, Rahaeifard M, Kahrobaiyan MH, Ahmadian MT (2011) The modified couple stress functionally graded Timoshenko beam formulation. Mater Des 32:1435–1443CrossRefGoogle Scholar
  5. Barati MR (2018) A general nonlocal stress–strain gradient theory for forced vibration analysis of heterogeneous porous nanoplates. Eur J Mech A/Solids 67:215–230MathSciNetCrossRefGoogle Scholar
  6. Barati MR, Zenkour AM (2017) Investigating post-buckling of geometrically imperfect metal foam nanobeams with symmetric and asymmetric porosity distributions. Compos Struct 182:91–98CrossRefGoogle Scholar
  7. Biener J, Wittstock A, Zepeda-Ruiz LA et al (2009) Surface-chemistry-driven actuation in nanoporous gold. Nat Mater 8:47CrossRefGoogle Scholar
  8. Bringa EM, Monk JD, Caro A et al (2011) Are nanoporous materials radiation resistant? Nano Lett 12:3351–3355CrossRefGoogle Scholar
  9. Chen D, Yang J, Kitipornchai S (2015) Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos Struct 133:54–61CrossRefGoogle Scholar
  10. Chen D, Kitipornchai S, Yang J (2016) Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core. Thin-Walled Struct 107:39–48CrossRefGoogle Scholar
  11. Cheng IC, Hodge AM (2013) Strength scale behavior of nanoporous Ag, Pd and Cu foams. Scr Mater 69:295–298CrossRefGoogle Scholar
  12. Detsi E, Punzhin S, Rao J et al (2012) Enhanced strain in functional nanoporous gold with a dual microscopic length scale structure. ACS Nano 6:3734–3744CrossRefGoogle Scholar
  13. Ebrahimi N, Beni YT (2016) Electro-mechanical vibration of nanoshells using consistent size-dependent piezoelectric theory. Steel Compos Struct 22:1301–1336CrossRefGoogle Scholar
  14. Eringen AC (1980) Mechanics of continua. Robert E Krieger Publ Co, Huntington, p 606Google Scholar
  15. Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248MathSciNetCrossRefGoogle Scholar
  16. Ghadiri M, SafarPour H (2017) Free vibration analysis of size-dependent functionally graded porous cylindrical microshells in thermal environment. J Therm Stress 40:55–71CrossRefGoogle Scholar
  17. Gibson LJ, Ashby MF (1982) The mechanics of three-dimensional cellular materials. Proc R Soc A Math Phys Eng Sci 382:43–59CrossRefGoogle Scholar
  18. Gurtin ME, Ian Murdoch A (1978) Surface stress in solids. Int J Solids Struct 14:431–440CrossRefGoogle Scholar
  19. Heydari H, Moosavifard SE, Shahraki M, Elyasi S (2017) Facile synthesis of nanoporous CuS nanospheres for high-performance supercapacitor electrodes. J Energy Chem 26:762–767CrossRefGoogle Scholar
  20. Jabbari M, Mojahedin A, Khorshidvand AR, Eslami MR (2014) Buckling analysis of a functionally graded thin circular plate made of saturated porous materials. J Eng Mech 140:287–295CrossRefGoogle Scholar
  21. Jin H-J, Wang X-L, Parida S et al (2009) Nanoporous Au–Pt alloys as large strain electrochemical actuators. Nano Lett 10:187–194CrossRefGoogle Scholar
  22. Ke LL, Wang Y (2011) Size effect on dynamic stability of functionally graded microbeams based on a modified couple stress theory. Compos Struct 93:342–350CrossRefGoogle Scholar
  23. Ke LL, Wang YS, Reddy JN (2014a) Thermo-electro-mechanical vibration of size-dependent piezoelectric cylindrical nanoshells under various boundary conditions. Compos Struct 116:626–636CrossRefGoogle Scholar
  24. Ke LL, Wang YS, Yang J, Kitipornchai S (2014b) The size-dependent vibration of embedded magneto-electro-elastic cylindrical nanoshells. Smart Mater Struct 23:125036CrossRefGoogle Scholar
  25. Kheibari F, Beni YT (2017) Size dependent electro-mechanical vibration of single-walled piezoelectric nanotubes using thin shell model. Mater Des 114:572–583CrossRefGoogle Scholar
  26. Lam DCC, Yang F, Chong ACM et al (2003) Experiments and theory in strain gradient elasticity. J Mech Phys Solids 51:1477–1508CrossRefGoogle Scholar
  27. Li J, Wang S, Xiao T et al (2017) Controllable preparation of nanoporous Ni3S2 films by sulfuration of nickel foam as promising asymmetric supercapacitor electrodes. Appl Surf Sci 420:919–926CrossRefGoogle Scholar
  28. Loy CT, Lam KY (1997) Vibration of cylindrical shells with ring support. Int J Mech Sci 39:455–471CrossRefGoogle Scholar
  29. Magnucka-Blandzi E (2010) Non-Linear analysis of dynamic stability of metal foam circular plate. J Theor Appl Mech 48:207–217Google Scholar
  30. Mehralian F, Beni YT (2018) Vibration analysis of size-dependent bimorph functionally graded piezoelectric cylindrical shell based on nonlocal strain gradient theory. J Braz Soc Mech Sci Eng 40:27CrossRefGoogle Scholar
  31. Mehralian F, Tadi Beni Y, Ansari R (2016) Size dependent buckling analysis of functionally graded piezoelectric cylindrical nanoshell. Compos Struct 152:45–61CrossRefGoogle Scholar
  32. Mohammadi K, Mahinzare M, Ghorbani K, Ghadiri M (2017) Cylindrical functionally graded shell model based on the first order shear deformation nonlocal strain gradient elasticity theory. Microsyst Technol 24:1133–1146CrossRefGoogle Scholar
  33. Nieman GW, Weertman JR, Siegel RW (1992) Mechanical behavior of nanocrystalline metals. Nanostructured Mater 1:185–190CrossRefGoogle Scholar
  34. Park SK, Gao XL (2006) Bernoulli–Euler beam model based on a modified couple stress theory. J Micromechanics Microengineering 16:2355–2359CrossRefGoogle Scholar
  35. Park SK, Gao XL (2008) Variational formulation of a modified couple stress theory and its application to a simple shear problem. Zeitschrift fur Angew Math und Phys 59:904–917MathSciNetCrossRefGoogle Scholar
  36. Park H, Ahn C, Jo H et al (2014) Large-area metal foams with highly ordered sub-micrometer-scale pores for potential applications in energy areas. Mater Lett 129:174–177CrossRefGoogle Scholar
  37. Pia G, Delogu F (2013) On the elastic deformation behavior of nanoporous metal foams. Scr Mater 69:781–784CrossRefGoogle Scholar
  38. Razavi H, Babadi AF, Tadi Beni Y (2017) Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory. Compos Struct 160:1299–1309CrossRefGoogle Scholar
  39. Rouhi H, Ansari R, Darvizeh M (2016) Size-dependent free vibration analysis of nanoshells based on the surface stress elasticity. Appl Math Model 40:3128–3140MathSciNetCrossRefGoogle Scholar
  40. Sahmani S, Aghdam MM (2017) A nonlocal strain gradient hyperbolic shear deformable shell model for radial postbuckling analysis of functionally graded multilayer GPLRC nanoshells. Compos Struct 178:97–109CrossRefGoogle Scholar
  41. Sahmani S, Aghdam MM, Bahrami M (2016) Size-dependent axial buckling and postbuckling characteristics of cylindrical nanoshells in different temperatures. Int J Mech Sci 107:170–179CrossRefGoogle Scholar
  42. Sahmani S, Aghdam MM, Bahrami M (2017) Surface free energy effects on the postbuckling behavior of cylindrical shear deformable nanoshells under combined axial and radial compressions. Meccanica 52:1329–1352MathSciNetCrossRefGoogle Scholar
  43. Schiøtz J, Vegge T, Di Tolla FD, Jacobsen KW (1999) Atomic-scale simulations of the mechanical deformation of nanocrystalline metals. Phys Rev B Condens Matter Mater Phys 60:11971–11983CrossRefGoogle Scholar
  44. Shin H, Liu M (2005) Three-dimensional porous Copper–Tin alloy electrodes for rechargeable lithium batteries. Adv Funct Mater 15:582–586CrossRefGoogle Scholar
  45. Şimşek M, Reddy JN (2013) Bending and vibration of functionally graded microbeams using a new higher order beam theory and the modified couple stress theory. Int J Eng Sci 64:37–53MathSciNetCrossRefGoogle Scholar
  46. Soedel W (2004) Vibrations of shells and plates. CRC Press, Boca RatonzbMATHGoogle Scholar
  47. Soleimani I, Beni YT (2018) Vibration analysis of nanotubes based on two-node size dependent axisymmetric shell element. Arch Civ Mech Eng 18:1345–1358CrossRefGoogle Scholar
  48. Tadi Beni Y, Mehralian F, Razavi H (2015) Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory. Compos Struct 120:65–78CrossRefGoogle Scholar
  49. Tadi Beni Y, Mehralian F, Zeighampour H (2016) The modified couple stress functionally graded cylindrical thin shell formulation. Mech Adv Mater Struct 23:791–801CrossRefGoogle Scholar
  50. Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11:385–414MathSciNetCrossRefGoogle Scholar
  51. Van Vliet KJ, Li J, Zhu T et al (2003) Quantifying the early stages of plasticity through nanoscale experiments and simulations. Phys Rev B 67:104105CrossRefGoogle Scholar
  52. Wang YQ (2018) Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state. Acta Astronaut 143:263–271CrossRefGoogle Scholar
  53. Wang Q, Varadan VK (2007) Application of nonlocal elastic shell theory in wave propagation analysis of carbon nanotubes. Smart Mater Struct 16:178–190CrossRefGoogle Scholar
  54. Wang YQ, Zu JW (2017a) Analytical analysis for vibration of longitudinally moving plate submerged in infinite liquid domain. Appl Math Mech 38:625–646MathSciNetCrossRefGoogle Scholar
  55. Wang YQ, Zu JW (2017b) Nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid. Compos Struct 164:130–144CrossRefGoogle Scholar
  56. Wang YQ, Zu JW (2017c) Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment. Aerosp Sci Technol 69:550–562CrossRefGoogle Scholar
  57. Wang YQ, Zu JW (2017d) Instability of viscoelastic plates with longitudinally variable speed and immersed in ideal liquid. Int J Appl Mech 9:1750005CrossRefGoogle Scholar
  58. Wang YQ, Zu JW (2018) Nonlinear dynamics of a translational FGM plate with strong mode interaction. Int J Struct Stab Dyn 18:1850031MathSciNetCrossRefGoogle Scholar
  59. Wang R, Wang C, Cai W, Ding Y (2010) Ultralow-platinum-loading high-performance nanoporous electrocatalysts with nanoengineered surface structures. Adv Mater 22:1845–1848CrossRefGoogle Scholar
  60. Wang YQ, Huang XB, Li J (2016) Hydroelastic dynamic analysis of axially moving plates in continuous hot-dip galvanizing process. Int J Mech Sci 110:201–216CrossRefGoogle Scholar
  61. Wang YQ, Ye C, Zu JW (2018a) Identifying the temperature effect on the vibrations of functionally graded cylindrical shells with porosities. Appl Math Mech 39:1587–1604CrossRefGoogle Scholar
  62. Wang YQ, Zhao HL, Ye C, Zu JW (2018b) A porous microbeam model for bending and vibration analysis based on the sinusoidal beam theory and modified strain gradient theory. Int J Appl Mech 10:1850059CrossRefGoogle Scholar
  63. Wittstock A, Zielasek V, Biener J et al (2010) Nanoporous gold catalysts for selective gas-phase oxidative coupling of methanol at low temperature. Science 327:319–322CrossRefGoogle Scholar
  64. Yamakov V, Wolf D, Phillpot SR et al (2004) Deformation-mechanism map for nanocrystalline metals by molecular-dynamics simulation. Nat Mater 3:43–47CrossRefGoogle Scholar
  65. Yang F, Chong ACM, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743CrossRefGoogle Scholar
  66. Zeighampour H, Beni YT (2014a) Size-dependent vibration of fluid-conveying double-walled carbon nanotubes using couple stress shell theory. Phys E Low-Dimens Syst Nanostructures 61:28–39CrossRefGoogle Scholar
  67. Zeighampour H, Beni YT (2014b) Analysis of conical shells in the framework of coupled stresses theory. Int J Eng Sci 81:107–122MathSciNetCrossRefGoogle Scholar
  68. Zeighampour H, Beni YT (2014c) Cylindrical thin-shell model based on modified strain gradient theory. Int J Eng Sci 78:27–47MathSciNetCrossRefGoogle Scholar
  69. Zeighampour H, Beni YT (2015) A shear deformable cylindrical shell model based on couple stress theory. Arch Appl Mech 85:539–553CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanics, College of SciencesNortheastern UniversityShenyangChina
  2. 2.Key Laboratory of Ministry of Education on Safe Mining of Deep Metal MinesNortheastern UniversityShenyangChina
  3. 3.Schaefer School of Engineering and ScienceStevens Institute of TechnologyHobokenUSA

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