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Nonlinear analysis of size-dependent frequencies in porous FG curved nanotubes based on nonlocal strain gradient theory

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Abstract

A nonlocal strain gradient model is developed in this research to analyse the nonlinear frequencies of functionally graded porous curved nanotubes. It is assumed that the curved nanotube is in contact with a two-parameter nonlinear elastic foundation and is also subjected to the uniform temperature rise. The non-classical theory presented for curved nanotubes contains a nonlocal parameter and a material length scale parameter which can capture the size effect. A power law distribution function is used to describe the graded properties through the thickness direction of curved nanotubes. The even dispersion pattern is used to model the porosities distribution. The high-order shear deformation theory and the von Kármán type of geometric non-linearity are utilized to obtain the nonlinear governing equations of the structure. The size-dependent equations of motion for the large amplitude vibrations of curved nanotubes are obtained by employing Hamilton’s principle. The analytical solutions are extracted for the curved nanotube with immovable hinged-hinged boundary conditions. Size-dependent frequencies of the curved nanotube exposed to thermal field are obtained using the two-step perturbation technique and Galerkin procedure. The effects of important parameters such as nonlocal and length scale parameters, temperature field, elastic foundation, porosity, power law index and geometrical parameters are studied in detail.

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References

  1. Huang Y, Li XF (2010) Buckling of functionally graded circular columns including shear deformation. Mater Des 31:3159–3166

    Google Scholar 

  2. Huang Y, Li XF (2010) Bending and vibration of circular cylindrical beams with arbitrary radial nonhomogeneity. Int J Mech Sci 52:595–601

    Google Scholar 

  3. Zhang P, Fu Y (2013) A higher-order beam model for tubes. Eur J Mech A Solids 38:12–19

    MathSciNet  MATH  Google Scholar 

  4. Fu Y, Zhong J, Shao X, Chen Y (2015) Thermal postbuckling analysis of functionally graded tubes based on a refined beam model. Int J Mech Sci 96:58–64

    Google Scholar 

  5. Chen Y, Fu Y, Zhong J, Li Y (2017) Nonlinear dynamic responses of functionally graded tubes subjected to moving load based on a refined beam model. Nonlinear Dyn 88:1441–1452

    Google Scholar 

  6. Wang Y, Xie K, Fu T (2018) Vibration analysis of functionally graded porous shear deformable tubes excited by moving distributed load. Acta Astronaut 151:603–613

    Google Scholar 

  7. Dehrouyeh-Semnani AM, Dehdashti E, Hairi Yazdi MR, Nikkhah-Bahrami M (2019) Nonlinear thermo-resonant behavior of fluid-conveying FG pipes. Int J Eng Sci 144:103141

    MathSciNet  MATH  Google Scholar 

  8. Dai J, Liu Y, Liu H, Miao C, Tong G (2019) A parametric study on thermo-mechanical vibration of axially functionally graded material pipe conveying fluid. Int J Mech Mater Des 15:715–726

    Google Scholar 

  9. Babaei H, Kiani Y, Eslami MR (2020) Large amplitude free vibrations of FGM shallow curved tubes in thermal environment. Smart Struct Syst 25:693–705

    Google Scholar 

  10. Sofiyev AH, Schnack E (2004) The stability of functionally graded cylindrical shells under linearly increasing dynamic torsional loading. Eng Struct 26:1321–1331

    Google Scholar 

  11. Sofiyev AH (2009) The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure. Compos Struct 89:356–366

    Google Scholar 

  12. Shen HS (2012) Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium. Compos Struct 94:1144–1154

    Google Scholar 

  13. Sofiyev AH (2016) Large amplitude vibration of FGM orthotropic cylindrical shells interacting with the nonlinear Winkler elastic foundation. Compos Part B 98:141–150

    Google Scholar 

  14. Sofiyev AH, Karaca Z, Zerin Z (2017) Non-linear vibration of composite orthotropic cylindrical shells on the non-linear elastic foundations within the shear deformation theory. Compos Struct 159:53–62

    Google Scholar 

  15. Sofiyev AH (2019) Review of research on the vibration and buckling of the FGM conical shells. Compos Struct 211:301–317

    Google Scholar 

  16. Avey M, Yusufoglu E (2020) On the solution of large-amplitude vibration of carbon nanotube-based double-curved shallow shells. Math Methods Appl Sci. https://doi.org/10.1002/mma.6820

    Article  Google Scholar 

  17. Marin M (2010) Some estimates on vibrations in thermoelasticity of dipolar bodies. J Vib Control 16:33–47

    MathSciNet  MATH  Google Scholar 

  18. Abd-Elaziz EM, Marin M, Othman MIA (2019) On the effect of Thomson and initial stress in a thermo-Porous elastic solid under GN electromagnetic theory. Symmetry 11(3):413. https://doi.org/10.3390/sym11030413

    Article  MATH  Google Scholar 

  19. Vlase S, Marin M, Ochsner A, Scutaru ML (2019) Motion equation for a flexible one-dimensional element used in the dynamical analysis of a multibody system. Continuum Mech Thermodyn 31:715–724

    MathSciNet  MATH  Google Scholar 

  20. Bhatti MM, Marin M, Zeeshan A, Ellahi R, Abdelsalam SI (2020) Swimming of motile gyrotactic microorganisms and nanoparticles in blood flow through anisotropically tapered arteries. Front Phys 8:95. https://doi.org/10.3389/fphy.2020.00095

    Article  Google Scholar 

  21. Civalek O, Demir C, Akgöz B (2010) Free vibration and bending analyses of cantilever microtubules based on nonlocal continuum model. Math Comput Appl 15:289–98

    MathSciNet  MATH  Google Scholar 

  22. Setoodeh AR, Afrahim S (2014) Nonlinear dynamic analysis of FG micro-pipes conveying fluid based on strain gradient theory. Compos Struct 116:128–135

    Google Scholar 

  23. Tang M, Ni Q, Wang L, Luo Y, Wang Y (2014) Nonlinear modeling and size-dependent vibration analysis of curved microtubes conveying fluid based on modified couple stress theory. Int J Eng Sci 84:1–10

    MathSciNet  MATH  Google Scholar 

  24. Civalek O, Demir C (2016) A simple mathematical model of microtubules surrounded by an elastic matrix by nonlocal finite element method. Appl Math Comput 289:335–52

    MathSciNet  MATH  Google Scholar 

  25. Mashrouteh S, Sadri M, Younesian D, Esmailzadeh E (2016) Nonlinear vibration analysis of fluid-conveying microtubes. Nonlinear Dyn 85:1007–1021

    MathSciNet  Google Scholar 

  26. Dehrouyeh-Semnani AM, Nikkhah-Bahrami M, Hairi Yazdi MR (2017) On nonlinear vibrations of micropipes conveying fluid. Int J Eng Sci 117:20–33

    MathSciNet  MATH  Google Scholar 

  27. Ghazavi MR, Molki H, Alibeigloo A (2017) Nonlinear vibration and stability analysis of the curved microtube conveying fluid as a model of the micro coriolis flowmeters based on strain gradient theory. Appl Math Model 45:1020–1030

    MathSciNet  MATH  Google Scholar 

  28. Zhu X, Li L (2017) Closed form solution for a nonlocal strain gradient rod in tension. Int J Eng Sci 119:16–28

    MathSciNet  MATH  Google Scholar 

  29. Numanoglu HM, Akgöz B, Civalek O (2018) On dynamic analysis of nanorods. Int J Eng Sci 130:33–50

    Google Scholar 

  30. Sharifi N, She GL (2018) On vibration of functionally graded nanotubes in thermal environment. Int J Eng Sci 133:84–98

    Google Scholar 

  31. She GL, Yuan FG, Ren YR, Liua HB, Xiaoa WS (2018) Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory. Compos Struct 203:614–623

    Google Scholar 

  32. Li L, Hu Y (2019) Torsional statics of two-dimensionally functionally graded microtubes. Mech Adv Mater Struct 26:430–442

    Google Scholar 

  33. She GL, Yuan FG, Karami B, Ren YR, Xiaoa WS (2019) On nonlinear bending behavior of FG porous curved nanotubes. Int J Eng Sci 135:58–74

    MathSciNet  MATH  Google Scholar 

  34. Gao Y, Xiao WS, Zhu H (2019) Nonlinear vibration of functionally graded nano-tubes using nonlocal strain gradient theory and a two-steps perturbation method. Struct Eng Mech 69:205–219

    Google Scholar 

  35. Gao Y, Xiao WS, Zhu H (2019) Nonlinear vibration of different types of functionally graded nanotubes using nonlocal strain gradient theory. Eur Phys J Plus 134:345–365

    Google Scholar 

  36. Karami B, Janghorban M (2019) On the dynamics of porous nanotubes with variable material properties and variable thickness. Int J Eng Sci 136:53–66

    MathSciNet  MATH  Google Scholar 

  37. Liu H, Lv Z, Tang H (2019) Nonlinear vibration and instability of FG nanopipes with initial imperfection conveying fluid. Appl Math Model 76:133–150

    MathSciNet  MATH  Google Scholar 

  38. Eltaher MA, Mohamed N, Mohamed S, Seddek LF (2019) Postbuckling of curved carbon nanotubes using energy equivalent model. J Nano Res 57:136–157

    Google Scholar 

  39. Jena SK, Chakraverty S, Malikan M, Tornabene F (2019) Stability analysis of single-walled carbon nanotubes embedded in Winkler foundation placed in a thermal environment considering the surface effect using a new refined beam theory. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2019.1698437

    Article  Google Scholar 

  40. Babaei H, Eslami MR (2020) Limit load analysis and imperfection sensitivity of porous FG micro-tubes surrounded by a nonlinear softening elastic medium. Acta Mech 231:4563–4583

    MathSciNet  MATH  Google Scholar 

  41. Babaei H, Eslami MR (2020) On nonlinear vibration and snap-through stability of porous FG curved micro-tubes using two-step perturbation technique. Compos Struct 247:112447

    Google Scholar 

  42. Babaei H, Eslami MR (2020) Nonlinear bending analysis of size-dependent FG porous microtubes in thermal environment based on modified couple stress theory. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2020.1784202

    Article  Google Scholar 

  43. Jamali S, Hosseini-Hashemi S, Nazemnezhad R (2020) Torsional vibrations investigation of nonlinear nonlocal behavior in terms of functionally graded nanotubes. Int J Nonlinear Mech 124:103513

    Google Scholar 

  44. Xiao WS, Dai P (2020) Static analysis of a circular nanotube made of functionally graded bi-semi-tubes using nonlocal strain gradient theory and a refined shear model. Eur J Mech A Solid 82:103979

    MathSciNet  MATH  Google Scholar 

  45. Tong G, Liu Y, Cengh Q, Dai J (2020) Stability analysis of cantilever functionally graded material nanotube under thermo-magnetic coupling effect. Eur J Mech A Solid 80:103929

    MathSciNet  MATH  Google Scholar 

  46. Mohammad Sedighi H, Ouakad H, Dimitri R, Tornabene F (2020) Stress-driven nonlocal elasticity for instability analysis of fluid-conveying C-BN hybrid-nanotube in magnetic and thermal environment. Phys Scr 95:065204

    Google Scholar 

  47. Zhu X, Lu Z, Wang Z, Xue L, Ebramimi-Mamaghani A (2020) Vibration of spinning functionally graded nanotubes conveying fluid. Eng Comput. https://doi.org/10.1007/s00366-020-01123-7

    Article  Google Scholar 

  48. Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solid 78:298–313

    MathSciNet  MATH  Google Scholar 

  49. Babaei H, Eslami MR (2021) On nonlinear vibration and snap-through buckling of long FG porous cylindrical panels using nonlocal strain gradient theory. Compos Struct 256:113125

    Google Scholar 

  50. Babaei H, Eslami MR (2021) Study on nonlinear vibrations of temperature- and size-dependent FG porous arches on elastic foundation using nonlocal strain gradient theory. Eur Phys J Plus 136:24. https://doi.org/10.1140/epjp/s13360-020-00959-8

    Article  Google Scholar 

  51. Babaei H, Kiani Y, Eslami MR (2019) Thermally induced large deflection analysis of shear deformable FGM shallow curved tubes using perturbation method. ZAMM J Appl Math Mech 99:e201800148

    MathSciNet  Google Scholar 

  52. Babaei H, Eslami MR (2020) Size-dependent vibrations of thermally pre/post-buckled FG porous micro-tubes based on modified couple stress theory. Int J Mech Sci 180:105694

    Google Scholar 

  53. Reddy JN (2003) Mechanics of laminated composite plates and shells. CRC Press, Boca Raton

    Google Scholar 

  54. Eslami MR (2018) Buckling and postbuckling of beams, plates, and shells. Springer, Cham

    MATH  Google Scholar 

  55. Shen HS (2013) A two-step perturbation method in nonlinear analysis of beams. Plates and shells. Wiley, Singapore

    MATH  Google Scholar 

  56. Shen HS, Wang ZX (2014) Nonlinear analysis of shear deformable FGM beams resting on elastic foundations in thermal environments. Int J Mech Sci 81:195–206

    Google Scholar 

  57. Babaei H, Eslami MR (2019) Thermally induced large deflection of FGM shallow micro-arches with integrated surface piezoelectric layers based on modified couple stress theory. Acta Mech 230:2363–2384

    MathSciNet  MATH  Google Scholar 

  58. Shen HS (2009) Functionally graded materials nonlinear analysis of plates and shells. CRC Press, Boca Raton

    Google Scholar 

  59. She GL, Ren YR, Yuan FG, Xiao WS (2018) On vibrations of porous nanotubes. Int J Eng Sci 125:23–35

    MathSciNet  MATH  Google Scholar 

  60. Zhong J, Fu Y, Wan D, Li Y (2016) Nonlinear bending and vibration of functionally graded tubes resting on elastic foundations in thermal environment based on a refined beam model. Appl Math Model 40:1–14

    MathSciNet  MATH  Google Scholar 

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  1. Mechanical Engineering Department, South Tehran Branch, Islamic Azad University, Tehran, Iran

    Hadi Babaei

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Babaei, H. Nonlinear analysis of size-dependent frequencies in porous FG curved nanotubes based on nonlocal strain gradient theory. Engineering with Computers 38 (Suppl 3), 1717–1734 (2022). https://doi.org/10.1007/s00366-021-01317-7

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