Skip to main content
Springer Nature Link
Log in

Porosity, mass and geometric imperfection sensitivity in coupled vibration characteristics of CNT-strengthened beams with different boundary conditions

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

Structures face different types of imperfections and defects during the fabrication process, installation and working environment. In this paper, the imperfection effects in the coupled vibration behaviour of axially functionally graded carbon nanotube (CNT)-strengthened beam structures with different boundary conditions are analysed considering porosity as well as geometric and mass imperfections in the structure. Porosity is modelled using different types of formulations for simple-cell, open-cell and closed-cell porous structures. The porosity is assumed to be either uniform or by varying through the thickness of the hollow beam using different functions. Mass imperfection effect is added to the system by considering a concentrated mass in the system affecting the mass homogeneity of the structure. Geometry imperfection is also considered by having an initial deformation in the structure which could be caused by an improper fabrication process. Coupled axial and transverse equations of motion are obtained using Hamilton’s principle and the von Kármán geometrical nonlinearity. Governing equations are solved for different types of boundary conditions using a semi-analytical modal decomposition technique. It is shown that strengthening the base matrix with CNT fibres can improve the vibration behaviour of imperfect structures and the influence of CNT volume fraction and distribution through the length of the beam is discussed. The results provided in this paper may be used as a benchmark to validate future experimental results to prevent imperfection, delamination and stress singularities in the structures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Matsunawa A, Mizutani M, Katayama S, Seto N (2003) Porosity formation mechanism and its prevention in laser welding. Weld Int 17(6):431–437

    Google Scholar 

  2. Zhang B, Liu S, Shin YC (2019) In-Process monitoring of porosity during laser additive manufacturing process. Addit Manuf 28:497–505

    Google Scholar 

  3. Zhou J, Tsai H-L (2007) Porosity formation and prevention in pulsed laser welding. J Heat Transf 129(8):1014–1024

    Google Scholar 

  4. Malikan M, Eremeyev VA (2020) A new hyperbolic-polynomial higher-order elasticity theory for mechanics of thick FGM beams with imperfection in the material composition. Compos Struct 249:112486

    Google Scholar 

  5. Dastjerdi S, Tadi Beni Y, Malikan M (2020) A comprehensive study on nonlinear hygro-thermo-mechanical analysis of thick functionally graded porous rotating disk based on two quasi-three-dimensional theories. Mech Based Design Struct Mach. https://doi.org/10.1080/15397734.2020.1814812

    Article  Google Scholar 

  6. Akgöz B, Civalek Ö (2013a) Buckling analysis of functionally graded microbeams based on the strain gradient theory. Acta Mech 224(9):2185–2201

    MathSciNet  MATH  Google Scholar 

  7. Sayyad A, Ghumare S (2019) A new quasi-3D model for functionally graded plates. J Appl Comput Mech 5(2):367–380

    Google Scholar 

  8. Akgöz B, Civalek Ö (2013b) Free vibration analysis of axially functionally graded tapered Bernoulli–Euler microbeams based on the modified couple stress theory. Compos Struct 98:314–322

    Google Scholar 

  9. Ghayesh MH (2019a) Viscoelastic dynamics of axially FG microbeams. Int J Eng Sci 135:75–85

    MathSciNet  MATH  Google Scholar 

  10. Ghayesh MH (2019b) Nonlinear oscillations of FG cantilevers. Appl Acoust 145:393–398

    Google Scholar 

  11. Khaniki HB (2019) On vibrations of FG nanobeams. Int J Eng Sci 135:23–36

    MathSciNet  MATH  Google Scholar 

  12. Jena SK, Chakraverty S, Malikan M (2020) Application of shifted Chebyshev polynomial-based Rayleigh–Ritz method and Navier’s technique for vibration analysis of a functionally graded porous beam embedded in Kerr foundation. Eng Comput. https://doi.org/10.1007/s00366-020-01018-7

    Article  Google Scholar 

  13. Xie B, Sahmani S, Safaei B, Xu B (2020) Nonlinear secondary resonance of FG porous silicon nanobeams under periodic hard excitations based on surface elasticity theory. Eng Comput. https://doi.org/10.1007/s00366-019-00931-w

    Article  Google Scholar 

  14. Liu Z, Yang C, Gao W, Wu D, Li G (2019) Nonlinear behaviour and stability of functionally graded porous arches with graphene platelets reinforcements. Int J Eng Sci 137:37–56

    MathSciNet  MATH  Google Scholar 

  15. Akbaş ŞD (2018) Forced vibration analysis of functionally graded porous deep beams. Compos Struct 186:293–302

    Google Scholar 

  16. Wu D, Liu A, Huang Y, Huang Y, Pi Y, Gao W (2018) Dynamic analysis of functionally graded porous structures through finite element analysis. Eng Struct 165:287–301

    Google Scholar 

  17. Gao K, Huang Q, Kitipornchai S, Yang J (2019) Nonlinear dynamic buckling of functionally graded porous beams. Mech Adv Mat Struct. https://doi.org/10.1080/15376494.2019.1567888

    Article  Google Scholar 

  18. Fattahi A, Sahmani S, Ahmed N (2019) Nonlocal strain gradient beam model for nonlinear secondary resonance analysis of functionally graded porous micro/nano-beams under periodic hard excitations. Mech Based Design Struct Mach 48(4):403–432. https://doi.org/10.1080/15397734.2019.1624176

    Google Scholar 

  19. Ebrahimi F, Farazmandnia N, Kokaba MR, Mahesh V (2019) Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory. Eng Comput. https://doi.org/10.1007/s00366-019-00864-4

    Article  Google Scholar 

  20. Ebrahimi F, Dabbagh A, Taheri M (2020) Vibration analysis of porous metal foam plates rested on viscoelastic substrate. Eng Comput. https://doi.org/10.1007/s00366-020-01031-w

    Article  Google Scholar 

  21. Sahmani S, Fattahi A, Ahmed N (2020) Analytical treatment on the nonlocal strain gradient vibrational response of postbuckled functionally graded porous micro-/nanoplates reinforced with GPL. Eng Comput 36:1559–1578

    Google Scholar 

  22. Rahmani M, Mohammadi Y, Kakavand F, Raeisifard H (2020) Vibration analysis of different types of porous FG conical sandwich shells in various thermal surroundings. J Appl Comput Mech 6(3):416–432

    Google Scholar 

  23. Jena SK, Chakraverty S, Malikan M, Sedighi H (2020) Implementation of Hermite–Ritz method and Navier’s technique for vibration of functionally graded porous nanobeam embedded in Winkler–Pasternak elastic foundation using bi-Helmholtz nonlocal elasticity. J Mech Mat Struct 15(3):405–434

    MathSciNet  Google Scholar 

  24. Malikan M, Tornabene F, Dimitri R (2018) Nonlocal three-dimensional theory of elasticity for buckling behavior of functionally graded porous nanoplates using volume integrals. Mat Res Express 5(9):095006

    Google Scholar 

  25. Dastjerdi S, Malikan M, Dimitri R, Tornabene F (2021) Nonlocal elasticity analysis of moderately thick porous functionally graded plates in a hygro-thermal environment. Compos Struct 255:112925

    Google Scholar 

  26. Akbaş Ş, Fageehi Y, Assie A, Eltaher M (2020) Dynamic analysis of viscoelastic functionally graded porous thick beams under pulse load. Eng Comput 55:1–13

    Google Scholar 

  27. Alambeigi K, Mohammadimehr M, Bamdad M, Rabczuk T (2020) Free and forced vibration analysis of a sandwich beam considering porous core and SMA hybrid composite face layers on Vlasov’s foundation. Acta Mech 231:3199–3218

    MathSciNet  Google Scholar 

  28. Dat ND, Quan TQ, Mahesh V, Duc ND (2020) Analytical solutions for nonlinear magneto-electro-elastic vibration of smart sandwich plate with carbon nanotube reinforced nanocomposite core in hygrothermal environment. Int J Mech Sci 186:105906

    Google Scholar 

  29. Chen D, Kitipornchai S, Yang J (2016) Nonlinear free vibration of shear deformable sandwich beam with a functionally graded porous core. Thin-Walled Struct 107:39–48

    Google Scholar 

  30. Fazzolari FA (2018) Generalized exponential, polynomial and trigonometric theories for vibration and stability analysis of porous FG sandwich beams resting on elastic foundations. Compos B Eng 136:254–271

    Google Scholar 

  31. Liu Y, Su S, Huang H, Liang Y (2019) Thermal-mechanical coupling buckling analysis of porous functionally graded sandwich beams based on physical neutral plane. Compos B Eng 168:236–242

    Google Scholar 

  32. Rostami R, Mohammadimehr M (2020) Vibration control of rotating sandwich cylindrical shell-reinforced nanocomposite face sheet and porous core integrated with functionally graded magneto-electro-elastic layers. Eng Comput. https://doi.org/10.1007/s00366-020-01052-5

    Article  Google Scholar 

  33. Hamed M, Abo-bakr R, Mohamed S, Eltaher M (2020) Influence of axial load function and optimization on static stability of sandwich functionally graded beams with porous core. Eng Comput 36:1929–1946

    Google Scholar 

  34. Karimiasl M, Ebrahimi F, Mahesh V (2019) Postbuckling analysis of piezoelectric multiscale sandwich composite doubly curved porous shallow shells via Homotopy Perturbation Method. Eng Comput. https://doi.org/10.1007/s00366-019-00841-x

    Article  Google Scholar 

  35. Duong TM, Vu TTA, Pham DN, Nguyen DD (2020) Nonlinear post-buckling of CNTs reinforced sandwich-structured composite annular spherical shells. Intern J Struct Stab Dyn 20(02):2050018

    MathSciNet  Google Scholar 

  36. Do QC, Pham DN, Vu DQ, Vu TTA, Nguyen DD (2019) Nonlinear buckling and post-buckling of functionally graded CNTs reinforced composite truncated conical shells subjected to axial load. Steel Compos Struct 31(3):243–259

    Google Scholar 

  37. Ghayesh MH (2018a) Nonlinear vibration analysis of axially functionally graded shear-deformable tapered beams. Appl Math Model 59:583–596

    MathSciNet  MATH  Google Scholar 

  38. Ghayesh MH (2019c) Mechanics of viscoelastic functionally graded microcantilevers. Eur J Mech A/Solids 73:492–499

    MathSciNet  MATH  Google Scholar 

  39. Nguyen DD, Pham DN (2017) The dynamic response and vibration of functionally graded carbon nanotubes reinforced composite (FG-CNTRC) truncated conical shells resting on elastic foundation. Materials 10(10):1194

    Google Scholar 

  40. Dat ND, Khoa ND, Nguyen PD, Duc ND (2020) An analytical solution for nonlinear dynamic response and vibration of FG-CNT reinforced nanocomposite elliptical cylindrical shells resting on elastic foundations. ZAMM J Appl Math Mech 100(1):e201800238

    MathSciNet  Google Scholar 

  41. Thanh NV, Khoa ND, Tuan ND, Tran P, Duc ND (2017) Nonlinear dynamic response and vibration of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shear deformable plates with temperature-dependent material properties and surrounded on elastic foundations. J Therm Stresses 40(10):1254–1274

    Google Scholar 

  42. Nguyen DD (2018) Nonlinear thermo-electro-mechanical dynamic response of shear deformable piezoelectric sigmoid functionally graded sandwich circular cylindrical shells on elastic foundations. J Sandwich Struct Mater 20(3):351–378

    Google Scholar 

  43. Nguyen DD, Tran QQ, Nguyen DK (2017) New approach to investigate nonlinear dynamic response and vibration of imperfect functionally graded carbon nanotube reinforced composite double curved shallow shells subjected to blast load and temperature. Aerosp Sci Technol 71:360–372

    Google Scholar 

  44. Dat ND, Quan TQ, Duc ND (2019) Nonlinear thermal vibration of carbon nanotube polymer composite elliptical cylindrical shells. Intern J Mech Mat Design 1–20

  45. Van Thanh N, Dinh Quang V, Dinh Khoa N, Seung-Eock K, Dinh Duc N (2019) Nonlinear dynamic response and vibration of FG CNTRC shear deformable circular cylindrical shell with temperature-dependent material properties and surrounded on elastic foundations. J Sandwich Struct Mater 21(7):2456–2483

    Google Scholar 

  46. Duc ND, Nguyen PD, Cuong NH, Van Sy N, Khoa ND (2019) An analytical approach on nonlinear mechanical and thermal post-buckling of nanocomposite double-curved shallow shells reinforced by carbon nanotubes. Proc Inst Mech Eng Part C J Mech Eng Sci 233(11):3888–3903

    Google Scholar 

  47. Shafiei N, Mirjavadi SS, MohaselAfshari B, Rabby S, Kazemi M (2017) Vibration of two-dimensional imperfect functionally graded (2D-FG) porous nano-/micro-beams. Comput Methods Appl Mech Eng 322:615–632

    MathSciNet  MATH  Google Scholar 

  48. Khaniki HB, Ghayesh MH (2020a) A review on the mechanics of carbon nanotube strengthened deformable structures. Eng Struct 220:110711

    Google Scholar 

  49. Konsta-Gdoutos MS, Metaxa ZS, Shah SP (2010) Highly dispersed carbon nanotube reinforced cement based materials. Cem Concr Res 40(7):1052–1059

    Google Scholar 

  50. Liew K, Kai M, Zhang L (2016) Carbon nanotube reinforced cementitious composites: an overview. Compos A Appl Sci Manuf 91:301–323

    Google Scholar 

  51. Yanase K, Moriyama S, Ju J (2013) Effects of CNT waviness on the effective elastic responses of CNT-reinforced polymer composites. Acta Mech 224(7):1351–1364

    Google Scholar 

  52. Ghayesh MH (2018b) Vibration analysis of shear-deformable AFG imperfect beams. Compos Struct 200:910–920

    Google Scholar 

  53. Mirjavadi SS, Forsat M, Badnava S, Barati MR (2020) Analyzing nonlocal nonlinear vibrations of two-phase geometrically imperfect piezo-magnetic beams considering piezoelectric reinforcement scheme. J Strain Anal Eng Design 55(7–8):258–270. https://doi.org/10.1177/0309324720917285

    Article  Google Scholar 

  54. Wu H, Liu H (2020) Nonlinear thermo-mechanical response of temperature-dependent FG sandwich nanobeams with geometric imperfection. Eng Comput. https://doi.org/10.1007/s00366-020-01005-y

    Article  Google Scholar 

  55. Malikan M, Eremeyev VA, Sedighi HM (2020) Buckling analysis of a non-concentric double-walled carbon nanotube. Acta Mech 231:5007–5020. https://doi.org/10.1007/s00707-020-02784-7

    Article  MathSciNet  Google Scholar 

  56. Malikan M (2020) On the plastic buckling of curved carbon nanotubes. Theor Appl Mech Lett 10(1):46–56

    Google Scholar 

  57. Ghayesh MH (2019d) Viscoelastic mechanics of Timoshenko functionally graded imperfect microbeams. Compos Struct 225:110974

    Google Scholar 

  58. Duc ND, Hadavinia H, Quan TQ, Khoa ND (2019) Free vibration and nonlinear dynamic response of imperfect nanocomposite FG-CNTRC double curved shallow shells in thermal environment. Eur J Mech A/Solids 75:355–366

    MathSciNet  MATH  Google Scholar 

  59. Khaniki HB, Ghayesh MH (2020b) On the dynamics of axially functionally graded CNT strengthened deformable beams. Eur Phys J Plus 135(6):415

    Google Scholar 

  60. Wattanasakulpong N, Chaikittiratana A (2015) Flexural vibration of imperfect functionally graded beams based on Timoshenko beam theory: Chebyshev collocation method. Meccanica 50(5):1331–1342

    MathSciNet  MATH  Google Scholar 

  61. Gao K, Li R, Yang J (2019) Dynamic characteristics of functionally graded porous beams with interval material properties. Eng Struct 197:109441

    Google Scholar 

  62. Chen D, Yang J, Kitipornchai S (2015) Elastic buckling and static bending of shear deformable functionally graded porous beam. Compos Struct 133:54–61

    Google Scholar 

  63. Gibson I, Ashby MF (1982) The mechanics of three-dimensional cellular materials. Proc R Soc Lond Mathe Phys Sci 382(1782):43–59

    Google Scholar 

  64. Roberts AP, Garboczi EJ (2001) Elastic moduli of model random three-dimensional closed-cell cellular solids. Acta Mater 49(2):189–197

    Google Scholar 

  65. Kitipornchai S, Chen D, Yang J (2017) Free vibration and elastic buckling of functionally graded porous beams reinforced by graphene platelets. Mater Des 116:656–665

    Google Scholar 

  66. Lin F, Xiang Y (2014) Vibration of carbon nanotube reinforced composite beams based on the first and third order beam theories. Appl Math Model 38(15–16):3741–3754

    MathSciNet  MATH  Google Scholar 

  67. Khaniki HB (2018a) On vibrations of nanobeam systems. Int J Eng Sci 124:85–103

    MathSciNet  MATH  Google Scholar 

  68. Khaniki HB (2018b) Vibration analysis of rotating nanobeam systems using Eringen’s two-phase local/nonlocal model. Physica E 99:310–319

    Google Scholar 

  69. Kutz JN, Brunton SL, Brunton BW, Proctor JL (2016) Dynamic mode decomposition: data-driven modeling of complex systems. SIAM, USA

    MATH  Google Scholar 

  70. Vega JM, Le Clainche S (2020) Higher order dynamic mode decomposition and its applications. Academic Press, USA

    MATH  Google Scholar 

  71. ANSYS® MultiphysicsTM, Workbench 19.2, Workbench User's Guide, ANSYS Workbench Systems, Analysis Systems, Modal

Download references

Author information

Authors and Affiliations

  1. School of Mechanical Engineering, University of Adelaide, Adelaide, South Australia, 5005, Australia

    Hossein B. Khaniki & Mergen H. Ghayesh

  2. Faculty of Science and Technology, University of Canberra, Canberra, Australia

    Shahid Hussain

  3. Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Canada

    Marco Amabili

Authors
  1. Hossein B. Khaniki
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Mergen H. Ghayesh
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Shahid Hussain
    View author publications

    You can also search for this author inPubMed Google Scholar

  4. Marco Amabili
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding authors

Correspondence to Hossein B. Khaniki or Mergen H. Ghayesh.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Khaniki, H.B., Ghayesh, M.H., Hussain, S. et al. Porosity, mass and geometric imperfection sensitivity in coupled vibration characteristics of CNT-strengthened beams with different boundary conditions. Engineering with Computers 38, 2313–2339 (2022). https://doi.org/10.1007/s00366-020-01208-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-020-01208-3

Keywords