Skip to main content
SpringerLink
Log in

Buckling Analysis of Nanobeams Resting on Viscoelastic Foundation

  • Original Paper
  • Published:
Journal of Vibration Engineering & Technologies Aims and scope Submit manuscript

Abstract

Purpose

This paper uses analytical methods to analyze the buckling behavior of sandwich nanobeams composed of three layers of material, with the central layer including a hollow hole. This study is the first instance of using such analytical techniques for investigating the buckling response of these specific FGM (functionally graded material) nanobeams.

Methods

The calculation formulas have been derived using a novel third-order shear deformation theory. The equilibrium equation of the nanobeam is formulated based on the notion of virtual work. Analytical solutions have been used to get the precise answer for the ultimate load of the beam. Empirical evidence has substantiated the dependability of this particular approach.

Results

This feature complicates the response of the beam and gives rise to several fascinating phenomena, including the idea of the virtual critical load which allows for the evaluation of the loss of the critical load. This research also looks at how the buckling behavior of the beam is affected by some geometric elements, material characteristics, hollow features of the intermediate layer, and viscous resistance-related parameters.

Conclusion

Based on the findings of this investigation, several inferences can be inferred. The third-order theory employed in this investigation exhibits commendable reliability in the computation of the buckling issue pertaining to sandwich beams experiencing substantial deformations. The presence of a viscoelastic foundation has a significant impact on the critical buckling load of the beam, resulting in the inclusion of both real and fictitious components. The critical buckling load is influenced by various factors, including the core thickness, foundation parameters, material composition ratio, and the number of hollows, nonlocal characteristics, and core layer material. These factors not only impact the actual magnitude of the critical buckling load, but also affect the rate at which it is lost. The findings of this study hold considerable importance for engineers in their calculations and designs of similar structures within the field of engineering. Moreover, this serves as the foundation for other forthcoming study avenues that can serve as valuable sources of reference.

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

Access this article

Log in via an institution

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

Data Availability

Data used to support the findings of this study are included in the article.

References

  1. Xin L, Kiani Y (2023) Vibration characteristics of arbitrary thick sandwich beam with metal foam core resting on elastic medium. Structures 49:1–11. https://doi.org/10.1016/j.istruc.2023.01.108

    Article  Google Scholar 

  2. Thanikasalam N, Ramamoorthy M (2023) Free and forced vibration response of a laminated composite sandwich beam with dual honeycomb core: Numerical and experimental study. Polym Compos. https://doi.org/10.1002/pc.27485

    Article  Google Scholar 

  3. Gunasegeran M, Edwin Sudhagar P (2023) Investigation of free and forced vibration of GFRP corrugated bio-inspired sandwich beam with HSDT: Numerical and experimental study. Mech Adv Mater Struct 30(18):3734–3748

    Article  CAS  Google Scholar 

  4. Zhang J, Huang W, Yuan H, Wu X (2023) Failure behavior of a sandwich beam with GLARE face-sheets and aluminum foam core under three-point bending. Thin-Walled Struct. https://doi.org/10.1016/j.tws.2022.110438

    Article  Google Scholar 

  5. Yu Z, Liu K, Zhou X, Jing L (2023) Low-velocity impact response of aluminum alloy corrugated sandwich beams used for high-speed trains. Thin-Walled Struct. https://doi.org/10.1016/j.tws.2022.110375

    Article  Google Scholar 

  6. Hui Y et al (2023) A geometrically nonlinear analysis through hierarchical one-dimensional modelling of sandwich beam structures. Acta Mech 234(1):67–83. https://doi.org/10.1007/s00707-022-03194-7

    Article  MathSciNet  Google Scholar 

  7. Nguyen Thai D, Van Minh P, Phan Hoang C, Ta Duc T, Nguyen Thi Cam N, Nguyen Thi D (2021) Bending of symmetric sandwich fgm beams with shear connectors. Math Probl Eng. https://doi.org/10.1155/2021/7596300

    Article  MathSciNet  Google Scholar 

  8. Phung VM (2020) Static bending analysis of symmetrical three-layer fgm beam with shear connectors under static load. J Sci Tech 15(3):68–78. https://doi.org/10.56651/lqdtu.jst.v15.n03.213

    Article  Google Scholar 

  9. Dat PT, Van Thom D, Luat DT (2016) Free vibration of functionally graded sandwich plates with stiffeners based on the third-order shear deformation theory. Vietnam J Mech 38(2):103–122. https://doi.org/10.15625/0866-7136/38/2/6730

    Article  Google Scholar 

  10. Thai LM, Luat DT, Van Ke T, Van Phung M (2023) Finite-element modeling for static bending analysis of rotating two-layer FGM beams with shear connectors resting on imperfect elastic foundations. J Aeros Eng. https://doi.org/10.1061/jaeeez.aseng-4771

    Article  Google Scholar 

  11. Van Vinh P, Dung NT, Thom DV, Tho NC (2021) Modified single variable shear deformation plate theory for free vibration analysis of rectangular FGM plates. Struct 29:1435–1444. https://doi.org/10.1016/j.istruc.2020.12.027

    Article  Google Scholar 

  12. Tho NC, Thanh NT, Tho TD, Van Minh P, Hoa LK (2021) Modelling of the flexoelectric effect on rotating nanobeams with geometrical imperfection. J Brazilian Soc Mech Sci Eng. https://doi.org/10.1007/s40430-021-03189-w

    Article  Google Scholar 

  13. Van Dung N, Thai LM, Dung NT, Van Minh P 2023 Free Vibration Response of Micro FG Beams Taking the Initial Geometrical Imperfection into Consideration, p 197–203

  14. Thai LM, Hieu NT, Dung NT, Tam TD, Van Minh P 2023 On the Free Vibration Analysis of Micro FG Beams Considering the Initial Geometrical Imperfection, p 181–187

  15. Derikvand M, Farhatnia F, Hodges DH (2021) Functionally graded thick sandwich beams with porous core: Buckling analysis via differential transform method. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2021.1931309

    Article  Google Scholar 

  16. Zhang W, Li J, Wang Z, Li K, Bai C, Qin Q (2023) The influence of asymmetric faces on low-velocity impact failure of CFRP/aluminum foam composite sandwich beams. Eng Struct. https://doi.org/10.1016/j.engstruct.2023.116574

    Article  Google Scholar 

  17. Marandi SM, Karimipour I (2023) Free vibration analysis of a nanoscale FG-CNTRCs sandwich beam with flexible core: Implementing an extended high order approach. Eng Struct. https://doi.org/10.1016/j.engstruct.2022.115320

    Article  Google Scholar 

  18. Hao N, Zhu L, Wu Z, Ke L (2023) Softening-spring nonlinearity in large amplitude vibration of unsymmetric double-layer lattice truss core sandwich beams. Thin-Walled Struct. https://doi.org/10.1016/j.tws.2022.110164

    Article  Google Scholar 

  19. De Chen C, Chen PY (2023) An improved model of refined zigzag theory with equivalent spring for mode II dominant strain energy release rate of a cracked sandwich beam. Theor Appl Fract Mech. https://doi.org/10.1016/j.tafmec.2023.103874

    Article  Google Scholar 

  20. Yasin MY, Prakash B, Ur Rahman N, Alam MN, Khan AH (2023) Design, fabrication, nonlinear analysis, and experimental validation for an active sandwich beam in strong electric field and thermal environment. J Sound Vib. https://doi.org/10.1016/j.jsv.2023.117828

    Article  Google Scholar 

  21. Zhang J, Yuan L, Zhang J, Yan J, Yuan H (2023) Dynamic response of foam-filled X-type sandwich beam under low-velocity impact. Eng Struct. https://doi.org/10.1016/j.engstruct.2023.116588

    Article  Google Scholar 

  22. Farajpour A, Yazdi MRH, Rastgoo A, Mohammadi M (2016) A higher-order nonlocal strain gradient plate model for buckling of orthotropic nanoplates in thermal environment. Acta Mech 227(7):1849–1867. https://doi.org/10.1007/s00707-016-1605-6

    Article  MathSciNet  Google Scholar 

  23. Karimi M, Farajpour MR (2019) Bending and buckling analyses of BiTiO3–CoFe2O4 nanoplates based on nonlocal strain gradient and modified couple stress hypotheses: rate of surface layers variations. Appl Phys A Mater Sci Process. https://doi.org/10.1007/s00339-019-2811-6

    Article  Google Scholar 

  24. Al-Furjan MSH, Farrokhian A, Keshtegar B, Kolahchi R, Trung NT (2020) Higher order nonlocal viscoelastic strain gradient theory for dynamic buckling analysis of carbon nanocones. Aerosp Sci Technol. https://doi.org/10.1016/j.ast.2020.106259

    Article  Google Scholar 

  25. Barretta R, Fabbrocino F, Luciano R, de Sciarra FM, Ruta G (2020) Buckling loads of nano-beams in stress-driven nonlocal elasticity. Mech Adv Mater Struct 27(11):869–875. https://doi.org/10.1080/15376494.2018.1501523

    Article  Google Scholar 

  26. Alam M, Mishra SK, Kant T (2021) Scale dependent critical external pressure for buckling of spherical shell based on nonlocal strain gradient theory. Int J Struct Stab Dyn. https://doi.org/10.1142/S0219455421500036

    Article  MathSciNet  Google Scholar 

  27. Akgöz B, Civalek Ö (2011) Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int J Eng Sci 49(11):1268–1280. https://doi.org/10.1016/j.ijengsci.2010.12.009

    Article  MathSciNet  Google Scholar 

  28. Fan F, Safaei B, Sahmani S (2021) Buckling and postbuckling response of nonlocal strain gradient porous functionally graded micro/nano-plates via NURBS-based isogeometric analysis. Thin-Walled Struct. https://doi.org/10.1016/j.tws.2020.107231

    Article  Google Scholar 

  29. Rao R, Ye Z, Yang Z, Sahmani S, Safaei B (2022) Nonlinear buckling mode transition analysis of axial–thermal–electrical-loaded FG piezoelectric nanopanels incorporating nonlocal and couple stress tensors. Arch Civ Mech Eng. https://doi.org/10.1007/s43452-022-00437-1

    Article  Google Scholar 

  30. Bakhshi Khaniki H, Hosseini-Hashemi S (2017) Buckling analysis of tapered nanobeams using nonlocal strain gradient theory and a generalized differential quadrature method. Mater Res Express. https://doi.org/10.1088/2053-1591/aa7111

    Article  Google Scholar 

  31. Sahmani S, Fattahi AM (2018) Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory. Appl Math Mech (English Ed.,) 3(4):561–580

    Article  MathSciNet  Google Scholar 

  32. Thai LM, Luat DT, Phung VB, Van Minh P, Van Thom D (2022) Finite element modeling of mechanical behaviors of piezoelectric nanoplates with flexoelectric effects. Arch Appl Mech 92(1):163–182. https://doi.org/10.1007/s00419-021-02048-3

    Article  Google Scholar 

  33. Duc DH, Van Thom D, Cong PH, Van Minh P, Nguyen NX (2022) Vibration and static buckling behavior of variable thickness flexoelectric nanoplates. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2022.2088558

    Article  Google Scholar 

  34. Dung NT, Thai LM, Van Ke T, Huyen TTH, Van Minh P (2022) Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory. Comptes Rendus Mec 350:121–141. https://doi.org/10.5802/crmeca.105

    Article  ADS  Google Scholar 

  35. Tuan LT, Dung NT, Van Thom D, Van Minh P, Zenkour AM (2021) Propagation of non-stationary kinematic disturbances from a spherical cavity in the pseudo-elastic cosserat medium. Eur Phys J Plus. https://doi.org/10.1140/epjp/s13360-021-02191-4

    Article  Google Scholar 

  36. Phung VM (2022) Static bending analysis of nanoplates on discontinuous elastic foundation with flexoelectric effect. J Sci Tech 17(5):47–57. https://doi.org/10.56651/lqdtu.jst.v17.n05.529

    Article  MathSciNet  Google Scholar 

  37. Van Minh P, Van Ke T (2023) A comprehensive study on mechanical responses of non-uniform thickness piezoelectric nanoplates taking into account the flexoelectric effect. Arab J Sci Eng 48(9):11457–11482. https://doi.org/10.1007/s13369-022-07362-8

    Article  CAS  Google Scholar 

  38. Nguyen TCN (2022) Static bending analysis of variable thickness microplates using the finite element method and modified couple stress theory. J Sci Tech. https://doi.org/10.56651/lqdtu.jst.v17.n03.351

    Article  Google Scholar 

  39. Van Minh P, Thai LM, Dung NT, Tounsi A, Nhung NTC, Van Thom D (2023) An overview of the flexoelectric phenomenon, potential applications, and proposals for further research directions. Int J Mech Mater Des. https://doi.org/10.1007/s10999-023-09678-1

    Article  Google Scholar 

  40. Duc DH, Thom DV, Phuc PM (2022) Buckling analysis of variable thickness cracked nanoplatesconsiderting the flexoelectric effect. Transp Comm Sci J 73(5):470–485

    Google Scholar 

  41. Khaniki HB, Hosseini-Hashemi S, Nezamabadi A (2018) Buckling analysis of nonuniform nonlocal strain gradient beams using generalized differential quadrature method. Alexandria Eng J 57(3):1361–1368. https://doi.org/10.1016/j.aej.2017.06.001

    Article  Google Scholar 

  42. Daikh AA, Belarbi MO, Khechai A, Li L, Ahmed HM, Eltaher MA (2023) Buckling of bi-coated functionally graded porous nanoplates via a nonlocal strain gradient quasi-3D theory. Acta Mech 234(8):3397–3420. https://doi.org/10.1007/s00707-023-03548-9

    Article  MathSciNet  Google Scholar 

  43. Gui Y, Wu R (2023) Buckling analysis of embedded thermo-magneto-electro-elastic nano cylindrical shell subjected to axial load with nonlocal strain gradient theory. Mech Res Commun. https://doi.org/10.1016/j.mechrescom.2023.104043

    Article  Google Scholar 

  44. Gui Y, Li Z (2023) A nonlocal strain gradient shell model with the surface effect for buckling analysis of a magneto-electro-thermo-elastic cylindrical nanoshell subjected to axial load. Phys Chem Chem Phys. https://doi.org/10.1039/d3cp02880a

    Article  PubMed  Google Scholar 

  45. Yue XG, Sahmani S, Safaei B (2023) Nonlocal couple stress-based quasi-3D nonlinear dynamics of agglomerated CNT-reinforced micro/nano-plates before and after bifurcation phenomenon. Phys Scr. https://doi.org/10.1088/1402-4896/acb858

    Article  Google Scholar 

  46. Dhanoriya A, Alam M, Mishra SK (2023) Postcritical behavior of nonlocal strain gradient arches: formulation and differential quadrature solution. J Eng Mech. https://doi.org/10.1061/jenmdt.emeng-6727

    Article  Google Scholar 

  47. Boyina K, Piska R, Natarajan S (2023) Nonlocal strain gradient model for thermal buckling analysis of functionally graded nanobeams. Acta Mech 234(10):5053–5069. https://doi.org/10.1007/s00707-023-03637-9

    Article  MathSciNet  Google Scholar 

  48. Yang Z, Hurdoganoglu D, Sahmani S, Nuhu AA, Safaei B (2023) Nonlocal strain gradient-based nonlinear in-plane thermomechanical stability of FG multilayer micro/nanoarches. Arch Civ Mech Eng. https://doi.org/10.1007/s43452-023-00623-9

    Article  Google Scholar 

  49. Tanzadeh H, Amoushahi H (2022) Buckling analysis of orthotropic nanoplates based on nonlocal strain gradient theory using the higher-order finite strip method (H-FSM). Eur J Mech A/Solids. https://doi.org/10.1016/j.euromechsol.2022.104622

    Article  MathSciNet  Google Scholar 

  50. Tuan LT, Dung NV, Minh PV, Tan BD, Thom DV, Zenkour AM (2023) Analysis of the stress-strain state of the elastic moment medium when a spherical cavity diffracts the wave. J Vib Eng Technol. https://doi.org/10.1007/s42417-023-01155-5

    Article  Google Scholar 

  51. Duc ND, Trinh TD, Van Do T, Doan DH (2018) On the buckling behavior of multi-cracked FGM plates. Lect Notes Mech Eng. https://doi.org/10.1007/978-981-10-7149-2_3

    Article  Google Scholar 

  52. Doan DH, Zenkour AM, Van Thom D (2022) Finite element modeling of free vibration of cracked nanoplates with flexoelectric effects. Eur Phys J Plus. https://doi.org/10.1140/epjp/s13360-022-02631-9

    Article  Google Scholar 

  53. Tho NC, Ta NT, Thom DV (2019) New numerical results from simulations of beams and space frame systems with a tuned mass damper. Materials 12(8):1329. https://doi.org/10.3390/ma12081329

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  54. Do TV, Bui TQ, Yu TT, Pham DT, Nguyen CT (2017) Role of material combination and new results of mechanical behavior for FG sandwich plates in thermal environment. J Comput Sci 21:164–181. https://doi.org/10.1016/j.jocs.2017.06.015

    Article  MathSciNet  Google Scholar 

  55. Tien DM, Thom DV, Minh PV, Tho NC, Doan TN, Mai DN (2023) The application of the nonlocal theory and various shear strain theories for bending and free vibration analysis of organic nanoplates. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2023.2186893

    Article  Google Scholar 

  56. Van Tuyen B, Luu GT (2023) Static buckling analysis of FG sandwich nanobeams. J Vib Eng Technol. https://doi.org/10.1007/s42417-023-01081-6

    Article  Google Scholar 

  57. Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45(2–8):288–307. https://doi.org/10.1016/j.ijengsci.2007.04.004

    Article  CAS  Google Scholar 

  58. Aydogdu M (2009) A general nonlocal beam theory: Its application to nanobeam bending, buckling and vibration. Phys E Low-Dimensional Syst Nanostructures 41(9):1651–1655. https://doi.org/10.1016/j.physe.2009.05.014

    Article  ADS  Google Scholar 

  59. Eltaher MA, Emam SA, Mahmoud FF (2013) Static and stability analysis of nonlocal functionally graded nanobeams. Compos Struct 96:82–88. https://doi.org/10.1016/j.compstruct.2012.09.030

    Article  Google Scholar 

  60. Tossapanon P, Wattanasakulpong N (2016) Stability and free vibration of functionally graded sandwich beams resting on two-parameter elastic foundation. Compos Struct 142:215–225. https://doi.org/10.1016/j.compstruct.2016.01.085

    Article  Google Scholar 

  61. Vo TP, Thai HT, Nguyen TK, Maheri A, Lee J (2014) Finite element model for vibration and buckling of functionally graded sandwich beams based on a refined shear deformation theory. Eng Struct 64:12–22. https://doi.org/10.1016/j.engstruct.2014.01.029

    Article  Google Scholar 

  62. Emam SA (2013) A general nonlocal nonlinear model for buckling of nanobeams. Appl Math Model 37(10–11):6929–6939. https://doi.org/10.1016/j.apm.2013.01.043

    Article  MathSciNet  Google Scholar 

  63. Quang DV, Doan TN, Luat DT, Thom DV (2022) Static analysis and boundary effect of FG-CNTRC cylindrical shells with various boundary conditions using quasi-3D shear and normal deformations theory. Struct 44:828–850. https://doi.org/10.1016/j.istruc.2022.08.039

    Article  Google Scholar 

  64. Anh TT, Do TV, Tien DP, Duc ND (2019) The effects of strength models in numerical study of metal plate destruction by contact explosive charge. Mech Adv Mat Struct 26(8):661–670. https://doi.org/10.1080/15376494.2017.1410907

    Article  Google Scholar 

  65. Tien DM, Thom DV, Van NTH, Tounsi A, Minh PV, Mai DN (2023) Buckling and forced oscillation of organic nanoplates taking the structural drag coefficient into account. Comp Concr 6(32):553–565. https://doi.org/10.12989/cac.2023.32.6.553

    Article  Google Scholar 

  66. Do VT, Pham VV, Nguyen HN (2020) On the development of refined plate theory for static bending behavior of functionally graded plates. Math Probl Eng. https://doi.org/10.1155/2020/2836763

    Article  MathSciNet  Google Scholar 

  67. Van Do T, Hong Doan D, Chi Tho N, Dinh Duc N (2022) Thermal Buckling Analysis of Cracked Functionally Graded Plates. Int J Struct Stab Dyn. https://doi.org/10.1142/S0219455422500894

    Article  MathSciNet  Google Scholar 

  68. Nam VH, Vinh PV, Chinh NV, Thom DV, Hong TT (2019) A new beam model for simulation of the mechanical behaviour of variable thickness functionally graded material beams based on modified first order shear deformation theory. Materials 12(3):404. https://doi.org/10.3390/ma12030404

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  69. Tho NC, Cong PH, Zenkour AM, Doan DH, Minh PV (2023) Finite element modeling of the bending and vibration behavior of three-layer composite plates with a crack in the core layer. Compos Struct 305:116529. https://doi.org/10.1016/j.compstruct.2022.116529

    Article  Google Scholar 

  70. Luschi L, Pieri F (2016) An analytical model for the resonance frequency of square perforated Lam´ e-mode resonators. Sensor Actuator B Chem 222:1233–1239. https://doi.org/10.1016/j.snb.2015.07.085

    Article  CAS  Google Scholar 

  71. Luschi L, Pieri F (2014) An analytical model for the determination of resonance frequencies of perforated beams”. J Micromech Microeng 24:055004. https://doi.org/10.1088/0960-1317/24/5/055004

    Article  Google Scholar 

  72. Chai Q, Wang YQ (2022) Traveling wave vibration of graphene platelet reinforced porous joined conical-cylindrical shells in a spinning motion. Eng Struct 252:113718. https://doi.org/10.1016/j.engstruct.2021.113718

    Article  Google Scholar 

  73. Ye C, Wang YQ (2021) Nonlinear forced vibration of functionally graded graphene platelet-reinforced metal foam cylindrical shells: internal resonances. Nonl Dyn 104:2051–2069. https://doi.org/10.1007/s11071-021-06401-7

    Article  Google Scholar 

  74. Wang YQ, Ye C, Zu JW (2019) Nonlinear vibration of metal foam cylindrical shells reinforced with graphene platelets. Aerosp Science Techn 85:359–370. https://doi.org/10.1016/j.ast.2018.12.022

    Article  Google Scholar 

  75. Wang YQ, Ye C, Zu JW (2018) Identifying the temperature effect on the vibrations of functionally graded cylindrical shells with porosities. Appl Math Mech 39:1587–1604. https://doi.org/10.1007/s10483-018-2388-6

    Article  MathSciNet  Google Scholar 

  76. Wang YQ, Wan YH, Zhang YF (2017) Vibrations of longitudinally traveling functionally graded material plates with porosities. Eur J Mech A/Solids 66:55–68. https://doi.org/10.1016/j.euromechsol.2017.06.006

    Article  MathSciNet  CAS  Google Scholar 

  77. Wang Y, Zhang W (2022) On the thermal buckling and postbuckling responses of temperature-dependent graphene platelets reinforced porous nanocomposite beams. Comp Struct 296:115880. https://doi.org/10.1016/j.compstruct.2022.115880

    Article  CAS  Google Scholar 

  78. Wang Y, Zhou A, Xie K, Fu T, Shi C (2020) Nonlinear static behaviors of functionally graded polymer-based circular microarches reinforced by graphene oxide nanofillers. Res Phys 16:102894. https://doi.org/10.1016/j.rinp.2019.102894

    Article  Google Scholar 

  79. Zhang W, Ma H, Wang Y (2023) Stability and vibration of nanocomposite microbeams reinforced by graphene oxides using an MCST-based improved shear deformable computational framework. Acta Mech 234:1471–1488. https://doi.org/10.1007/s00707-022-03467-1

    Article  MathSciNet  Google Scholar 

  80. Wang Y, Liu H, Zhang W, Liu Y (2022) A size-dependent shear deformable computational framework for transient response of GNP-reinforced metal foam cylindrical shells subjected to localized impulsive loads. Appl Math Mod 109:578–598. https://doi.org/10.1016/j.apm.2022.05.018

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Economics - Technology for Industries, 456 Minh Khai, Hai Ba Trung, Hanoi, Vietnam

    Pham Van Lieu

  2. Laboratory of Applied Physics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam

    Gia Thien Luu

  3. Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam

    Gia Thien Luu

Authors
  1. Pham Van Lieu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gia Thien Luu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gia Thien Luu.

Ethics declarations

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Van Lieu, P., Luu, G.T. Buckling Analysis of Nanobeams Resting on Viscoelastic Foundation. J. Vib. Eng. Technol. (2024). https://doi.org/10.1007/s42417-024-01277-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s42417-024-01277-4

Keywords

Search

Navigation