Skip to main content
SpringerLink
Log in

On the nonlocal free vibration analysis of functionally graded porous doubly curved shallow nanoshells with variable nonlocal parameters

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

Abstract

In this paper, the free vibration behavior of functionally graded (FG) porous doubly curved shallow nanoshells with variable nonlocal parameters is investigated. The classical Eringen’s nonlocal elasticity theory is modified and applied to capture the small size effect of naturally discrete FG nanoshells. The effective material properties of the FG porous doubly curved shallow nanoshells including the nonlocal parameters are graded continuously through the thickness direction via the rule of mixture. A combination of the first-order shear deformation theory and the modified nonlocal elasticity theory is developed to describe the kinematic and constitutive relations of the FG doubly curved shallow nanoshells. The Hamilton’s principle is employed to establish the governing equations of motion of FG porous doubly curved shallow nanoshells and then solved analytically using the Navier’s solution. The accuracy and correctness of the proposed algorithm are demonstrated by comparing its results with those available from other researchers in the existing literature. Moreover, a comprehensive parametric study is presented and discussed in detail to show the effects of the geometric parameters, material properties, porosity, and the variation of the nonlocal parameter on the free vibration behavior of the FG porous doubly curved shallow nanoshells. Especially, the numerical results showed that the variation of the nonlocal parameters has significant effects on the free vibration behavior of the FG porous doubly curved shallow nanoshells. Some new results are also reported which will serve as a benchmark for future analysis of FG porous doubly curved shallow nanoshells.

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

Similar content being viewed by others

References

  1. Reddy JN (2000) Analysis of functionally graded plates. Int J Numer Methods Eng 47:663–684. https://doi.org/10.1002/(SICI)1097-0207(20000110/30)47:1/3%3c663::AID-NME787%3e3.0.CO;2-8

    Article  MATH  Google Scholar 

  2. Neves AMA, Ferreira AJM, Carrera E, Cinefra M, Roque CMC, Jorge RMN et al (2013) Static, free vibration and buckling analysis of isotropic and sandwich functionally graded plates using a quasi-3D higher-order shear deformation theory and a meshless technique. Compos Part B Eng 44:657–674. https://doi.org/10.1016/j.compositesb.2012.01.089

    Article  Google Scholar 

  3. Swaminathan K, Naveenkumar DT, Zenkour AM, Carrera E (2015) Stress, vibration and buckling analyses of FGM plates—a state-of-the-art review. Compos Struct 120:10–31. https://doi.org/10.1016/j.compstruct.2014.09.070

    Article  Google Scholar 

  4. Lee JW, Lee JY (2017) Free vibration analysis of functionally graded Bernoulli-Euler beams using an exact transfer matrix expression. Int J Mech Sci 122:1–17. https://doi.org/10.1016/j.ijmecsci.2017.01.011

    Article  Google Scholar 

  5. Yan K, Zhang Y, Cai H, Tahouneh V (2020) Vibrational characteristic of FG porous conical shells using Donnell’s shell theory. Steel Compos Struct 35:249–260. https://doi.org/10.12989/scs.2020.35.2.249

    Article  Google Scholar 

  6. Liang D, Wu Q, Lu X, Tahouneh V (2020) Vibration behavior of trapezoidal sandwich plate with functionally graded-porous core and graphene platelet-reinforced layers. Steel Compos Struct 36:47–62. https://doi.org/10.12989/scs.2020.36.1.047

    Article  Google Scholar 

  7. Kumar JS, Chakraverty S, Malikan M (2021) 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 37:3569–3589. https://doi.org/10.1007/s00366-020-01018-7

    Article  Google Scholar 

  8. Sahmani S, Fattahi AM, Ahmed NA (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. https://doi.org/10.1007/s00366-019-00782-5

    Article  Google Scholar 

  9. Van Vinh P (2022) Analysis of bi-directional functionally graded sandwich plates via higher-order shear deformation theory and finite element method. J Sandwich Struct Mater 24:860–899. https://doi.org/10.1177/10996362211025811

    Article  Google Scholar 

  10. Van Vinh P (2021) Formulation of a new mixed four-node quadrilateral element for static bending analysis of variable thickness functionally graded material plates. Math Probl Eng 2021:1–23. https://doi.org/10.1155/2021/6653350

    Article  MathSciNet  Google Scholar 

  11. Van Vinh P (2021) Deflections, stresses and free vibration analysis of bi-functionally graded sandwich plates resting on Pasternak’s elastic foundations via a hybrid quasi-3D theory. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2021.1894948

    Article  Google Scholar 

  12. Natarajan S, Manickam G (2012) Bending and vibration of functionally graded material sandwich plates using an accurate theory. Finite Elem Anal Des 57:32–42. https://doi.org/10.1016/j.finel.2012.03.006

    Article  Google Scholar 

  13. Neves AMA, Ferreira AJM, Carrera E, Roque CMC, Cinefra M, Jorge RMN et al (2012) A quasi-3D sinusoidal shear deformation theory for the static and free vibration analysis of functionally graded plates. Compos Part B Eng 43:711–725. https://doi.org/10.1016/j.compositesb.2011.08.009

    Article  Google Scholar 

  14. Zenkour AM (2005) A comprehensive analysis of functionally graded sandwich plates: Part 2—Buckling and free vibration. Int J Solids Struct 42:5243–5258. https://doi.org/10.1016/j.ijsolstr.2005.02.016

    Article  MATH  Google Scholar 

  15. Zenkour AM (2013) A simple four-unknown refined theory for bending analysis of functionally graded plates. Appl Math Model 37:9041–9051. https://doi.org/10.1016/j.apm.2013.04.022

    Article  MathSciNet  MATH  Google Scholar 

  16. Thai HT, Choi DH (2013) A simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates. Compos Struct 101:332–340. https://doi.org/10.1016/j.compstruct.2013.02.019

    Article  Google Scholar 

  17. Thai H-T, Kim S-E (2013) A simple higher-order shear deformation theory for bending and free vibration analysis of functionally graded plates. Compos Struct 96:165–173. https://doi.org/10.1016/j.compstruct.2012.08.025

    Article  Google Scholar 

  18. Demirhan PA, Taskin V (2017) Levy solution for bending analysis of functionally graded sandwich plates based on four variable plate theory. Compos Struct 177:80–95. https://doi.org/10.1016/j.compstruct.2017.06.048

    Article  Google Scholar 

  19. Vinh PV, Dung NT, Tho NC, Van TD, Hoa LK (2021) Modified single variable shear deformation plate theory for free vibration analysis of rectangular FGM plates. Structures 29:1435–1444. https://doi.org/10.1016/j.istruc.2020.12.027

    Article  Google Scholar 

  20. Pandey S, Pradyumna S (2018) Analysis of functionally graded sandwich plates using a higher-order layerwise theory. Compos Part B Eng 153:325–336. https://doi.org/10.1016/j.compositesb.2018.08.121

    Article  Google Scholar 

  21. Rezaei AS, Saidi AR, Abrishamdari M, Mohammadi MHP (2017) Natural frequencies of functionally graded plates with porosities via a simple four variable plate theory: An analytical approach. Thin-Wall Struct 120:366–377. https://doi.org/10.1016/j.tws.2017.08.003

    Article  Google Scholar 

  22. Akbaş ŞD (2017) Vibration and static analysis of functionally graded porous plates. J Appl Comput Mech 3:199–207. https://doi.org/10.22055/jacm.2017.21540.1107

    Article  Google Scholar 

  23. Riadh B, Ait AH, Belqassim A, Abdelouahed T, Adda BEA, A. A-OM (2019) Free vibration response of functionally graded Porous plates using a higher-order Shear and normal deformation theory. Earthq Struct 16:547–561. https://doi.org/10.12989/EAS.2019.16.5.547

  24. Zeng S, Wang BL, Wang KF (2019) Nonlinear vibration of piezoelectric sandwich nanoplates with functionally graded porous core with consideration of flexoelectric effect. Compos Struct 207:340–351. https://doi.org/10.1016/j.compstruct.2018.09.040

    Article  Google Scholar 

  25. Van Vinh P, Huy LQ (2021) Finite element analysis of functionally graded sandwich plates with porosity via a new hyperbolic shear deformation theory. Def Technol. https://doi.org/10.1016/j.dt.2021.03.006

    Article  Google Scholar 

  26. Pradhan SC, Loy CT, Lam KY, Reddy JN (2000) Vibration characteristics of functionally graded cylindrical shells under various boundary conditions. Appl Acoust 61:111–129. https://doi.org/10.1016/S0003-682X(99)00063-8

    Article  Google Scholar 

  27. Woo J, Meguid SA (2001) Nonlinear analysis of functionally graded plates and shallow shells. Int J Solids Struct 38:7409–7421. https://doi.org/10.1016/S0020-7683(01)00048-8

    Article  MATH  Google Scholar 

  28. Khare RK, Kant T, Garg AK (2004) Free vibration of composite and sandwich laminates with a higher-order facet shell element. Compos Struct 65:405–418. https://doi.org/10.1016/j.compstruct.2003.12.003

    Article  Google Scholar 

  29. Fadaee M, Atashipour SR, Hosseini-hashemi S (2013) Lévy-type functionally graded spherical shell panel using a new exact closed-form solution. Int J Mech Sci 77:227–238. https://doi.org/10.1016/j.ijmecsci.2013.10.008

    Article  Google Scholar 

  30. Amabili M (2005) Non-linear vibrations of doubly curved shallow shells. Int J Non Linear Mech 40:683–710. https://doi.org/10.1016/j.ijnonlinmec.2004.08.007

    Article  MATH  Google Scholar 

  31. Alijani F, Amabili M, Karagiozis K, Bakhtiari-Nejad F (2011) Nonlinear vibrations of functionally graded doubly curved shallow shells. J Sound Vib 330:1432–1454. https://doi.org/10.1016/j.jsv.2010.10.003

    Article  Google Scholar 

  32. Jouneghani FZ, Dimitri R, Bacciocchi M, Tornabene F (2017) Free vibration analysis of functionally graded porous doubly curved shells based on the First-order Shear Deformation Theory. Appl Sci 7:1252. https://doi.org/10.3390/app7121252

    Article  Google Scholar 

  33. Santos H, Mota Soares CM, Mota Soares CA, Reddy JN (2009) A semi-analytical finite element model for the analysis of cylindrical shells made of functionally graded materials. Compos Struct 91:427–432. https://doi.org/10.1016/j.compstruct.2009.04.008

    Article  Google Scholar 

  34. Viola E, Rossetti L, Fantuzzi N, Tornabene F (2014) Static analysis of functionally graded conical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery. Compos Struct 112:44–65. https://doi.org/10.1016/j.compstruct.2014.01.039

    Article  Google Scholar 

  35. Wattanasakulpong N, Chaikittiratana A (2015) An analytical investigation on free vibration of FGM doubly curved shallow shells with stiffeners under thermal environment. Aerosp Sci Technol 40:181–190. https://doi.org/10.1016/j.ast.2014.11.006

    Article  Google Scholar 

  36. Tornabene F, Fantuzzi N, Bacciocchi M, Viola E (2016) Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly curved shells. Compos Part B Eng 89:187–218. https://doi.org/10.1016/j.compositesb.2015.11.016

    Article  Google Scholar 

  37. Punera D, Kant T (2017) Free vibration of functionally graded open cylindrical shells based on several refined higher order displacement models. Thin-Walled Struct 119:707–726. https://doi.org/10.1016/j.tws.2017.07.016

    Article  Google Scholar 

  38. Punera D, Kant T (2017) Elastostatics of laminated and functionally graded sandwich cylindrical shells with two refined higher order models. Compos Struct 182:505–523. https://doi.org/10.1016/j.compstruct.2017.09.051

    Article  Google Scholar 

  39. Aliyari Parand A, Alibeigloo A (2017) Static and vibration analysis of sandwich cylindrical shell with functionally graded core and viscoelastic interface using DQM. Compos Part B Eng 126:1–16. https://doi.org/10.1016/j.compositesb.2017.05.071

    Article  Google Scholar 

  40. Chen H, Wang A, Hao Y, Zhang W (2017) Free vibration of FGM sandwich doubly curved shallow shell based on a new shear deformation theory with stretching effects. Compos Struct 179:50–60. https://doi.org/10.1016/j.compstruct.2017.07.032

    Article  Google Scholar 

  41. Wang A, Chen H, Hao Y, Zhang W (2018) Vibration and bending behavior of functionally graded nanocomposite doubly curved shallow shells reinforced by graphene nanoplatelets. Res Phys 9:550–559. https://doi.org/10.1016/j.rinp.2018.02.062

    Article  Google Scholar 

  42. Arefi M, Mohammad-Rezaei Bidgoli E, Civalek O (2020) Bending response of FG composite doubly curved nanoshells with thickness stretching via higher-order sinusoidal shear theory. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2020.1777157

    Article  Google Scholar 

  43. Szekrényes A (2021) Mechanics of shear and normal deformable doubly curved delaminated sandwich shells with soft core. Compos Struct 258:113196. https://doi.org/10.1016/j.compstruct.2020.113196

    Article  Google Scholar 

  44. Allahkarami F, Tohidi H, Dimitri R, Tornabene F (2020) Dynamic stability of bi-directional functionally graded porous cylindrical shells embedded in an elastic foundation. Appl Sci 10:1345. https://doi.org/10.3390/app10041345

    Article  Google Scholar 

  45. Liu B, Guo M, Liu C, Xing Y (2019) Free vibration of functionally graded sandwich shallow shells in thermal environments by a differential quadrature hierarchical finite element method. Compos Struct 225:111173. https://doi.org/10.1016/j.compstruct.2019.111173

    Article  Google Scholar 

  46. Eringen AC, Edelen DGB (1972) On nonlocal elasticity. Int J Eng Sci 10:233–248. https://doi.org/10.1016/0020-7225(72)90039-0

    Article  MathSciNet  MATH  Google Scholar 

  47. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710. https://doi.org/10.1063/1.332803

    Article  Google Scholar 

  48. Jouneghani FZ, Mohammadi Dashtaki P, Dimitri R, Bacciocchi M, Tornabene F (2018) First-order shear deformation theory for orthotropic doubly curved shells based on a modified couple stress elasticity. Aerosp Sci Technol 73:129–147. https://doi.org/10.1016/j.ast.2017.11.045

    Article  Google Scholar 

  49. Faleh M, Fenjan R (2020) Scale-dependent thermal vibration analysis of FG beams having porosities based on DQM. Adv Nano Res 8:59–68. https://doi.org/10.12989/anr.2020.8.4.283

    Article  Google Scholar 

  50. Razavi H, Babadi AF, Tadi BY (2017) Free vibration analysis of functionally graded piezoelectric cylindrical nanoshell based on consistent couple stress theory. Compos Struct 160:1299–1309. https://doi.org/10.1016/j.compstruct.2016.10.056

    Article  Google Scholar 

  51. Karami B, Shahsavari D, Janghorban M (2019) On the dynamics of porous doubly curved nanoshells. Int J Eng Sci 143:39–55. https://doi.org/10.1016/j.ijengsci.2019.06.014

    Article  MathSciNet  MATH  Google Scholar 

  52. Karami B, Janghorban M, Tounsi A (2020) Novel study on functionally graded anisotropic doubly curved nanoshells. Eur Phys J Plus 135:103. https://doi.org/10.1140/epjp/s13360-019-00079-y

    Article  Google Scholar 

  53. Karami B, Shahsavari D, Janghorban M, Dimitri R, Tornabene F (2019) Wave propagation of porous nanoshells. Nanomaterials 9:22. https://doi.org/10.3390/nano9010022

    Article  Google Scholar 

  54. Karami B, Janghorban M, Tounsi A (2018) Variational approach for wave dispersion in anisotropic doubly curved nanoshells based on a new nonlocal strain gradient higher order shell theory. Thin-Walled Struct 129:251–264. https://doi.org/10.1016/j.tws.2018.02.025

    Article  Google Scholar 

  55. 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 178:97–109. https://doi.org/10.1016/j.compstruct.2017.06.062

    Article  Google Scholar 

  56. Shariati A, Ebrahimi F, Karimiasl M, Vinyas M, Toghroli A (2020) On transient hygrothermal vibration of embedded viscoelastic flexoelectric/piezoelectric nanobeams under magnetic loading. Adv Nano Res 8:49–58. https://doi.org/10.12989/anr.2020.8.1.049

    Article  Google Scholar 

  57. Eltaher MA, Mohamed N (2020) Nonlinear stability and vibration of imperfect CNTs by Doublet mechanics. Appl Math Comput 382:125311. https://doi.org/10.1016/j.amc.2020.125311

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MATH  Google Scholar 

  59. Natarajan S, Chakraborty S, Thangavel M, Bordas S, 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

    Article  Google Scholar 

  60. Aghababaei R, Reddy JN (2009) Nonlocal third-order shear deformation plate theory with application to bending and vibration of plates. J Sound Vib 326:277–289. https://doi.org/10.1016/j.jsv.2009.04.044

    Article  Google Scholar 

  61. Aksencer T, Aydogdu M (2011) Levy type solution method for vibration and buckling of nanoplates using nonlocal elasticity theory. Phys E Low-Dimens Syst Nanostruct 43:954–959. https://doi.org/10.1016/j.physe.2010.11.024

    Article  Google Scholar 

  62. Ansari R, Arash B, Rouhi H (2011) Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity. Compos Struct 93:2419–2429. https://doi.org/10.1016/j.compstruct.2011.04.006

    Article  Google Scholar 

  63. Nazemnezhad R, Hosseini-Hashemi S (2014) Nonlocal nonlinear free vibration of functionally graded nanobeams. Compos Struct 110:192–199. https://doi.org/10.1016/j.compstruct.2013.12.006

    Article  MATH  Google Scholar 

  64. Belarbi MO, Houari MSA, Daikh AA, Garg A, Merzouki T, Chalak HD et al (2021) Nonlocal finite element model for the bending and buckling analysis of functionally graded nanobeams using a novel shear deformation theory. Compos Struct 264:113712. https://doi.org/10.1016/j.compstruct.2021.113712

    Article  Google Scholar 

  65. Ghandourah EE, Abdraboh AM (2020) Dynamic analysis of functionally graded nonlocal nanobeam with different porosity models. Steel Compos Struct 36:293–305. https://doi.org/10.12989/scs.2020.36.3.293

    Article  Google Scholar 

  66. Thai H-T, Vo TP, Nguyen T-K, Lee J (2014) A nonlocal sinusoidal plate model for micro/nanoscale plates. Proc Inst Mech Eng Part C J Mech Eng Sci 228:2652–2660. https://doi.org/10.1177/0954406214521391

    Article  Google Scholar 

  67. Hoa LK, Van VP, Duc ND, Trung NT, Son LT, Van TD (2021) Bending and free vibration analyses of functionally graded material nanoplates via a novel nonlocal single variable shear deformation plate theory. Proc Inst Mech Eng Part C J Mech Eng Sci 235:3641–3653. https://doi.org/10.1177/0954406220964522

    Article  Google Scholar 

  68. Daneshmehr A, Rajabpoor A, Hadi A (2015) Size dependent free vibration analysis of nanoplates made of functionally graded materials based on nonlocal elasticity theory with high order theories. Int J Eng Sci 95:23–35. https://doi.org/10.1016/j.ijengsci.2015.05.011

    Article  MathSciNet  MATH  Google Scholar 

  69. Anjomshoa A, Tahani M (2016) Vibration analysis of orthotropic circular and elliptical nano-plates embedded in elastic medium based on nonlocal Mindlin plate theory and using Galerkin method. J Mech Sci Technol 30:2463–2474. https://doi.org/10.1007/s12206-016-0506-x

    Article  Google Scholar 

  70. Mechab I, Mechab B, Benaissa S, Serier B, Bouiadjra BB (2016) Free vibration analysis of FGM nanoplate with porosities resting on Winkler Pasternak elastic foundations based on two-variable refined plate theories. J Brazilian Soc Mech Sci Eng 38:2193–2211. https://doi.org/10.1007/s40430-015-0482-6

    Article  MATH  Google Scholar 

  71. Sobhy M, Radwan AF (2017) A new quasi 3D nonlocal plate theory for vibration and buckling of FGM nanoplates. Int J Appl Mech. https://doi.org/10.1142/S1758825117500089

    Article  Google Scholar 

  72. Arefi M (2018) Nonlocal free vibration analysis of a doubly curved piezoelectric nano shell. Steel Compos Struct 27:479–493. https://doi.org/10.12989/scs.2018.27.4.479

    Article  MATH  Google Scholar 

  73. Arefi M, Rabczuk T (2019) A nonlocal higher order shear deformation theory for electro-elastic analysis of a piezoelectric doubly curved nano shell. Compos Part B Eng 168:496–510. https://doi.org/10.1016/j.compositesb.2019.03.065

    Article  Google Scholar 

  74. Van Vinh P, Tounsi A (2021) The role of spatial variation of the nonlocal parameter on the free vibration of functionally graded sandwich nanoplates. Eng Comput. https://doi.org/10.1007/s00366-021-01475-8

    Article  Google Scholar 

  75. Vinh PV, Huy LQ (2021) Influence of variable nonlocal parameter and porosity on the free vibration behavior of functionally graded nanoplates. Shock Vib 2021:1219429. https://doi.org/10.1155/2021/1219429

    Article  Google Scholar 

  76. Van Vinh P (2022) Nonlocal free vibration characteristics of power-law and sigmoid functionally graded nanoplates considering variable nonlocal parameter. Phys E Low-Dimens Syst Nanostruct 135:114951. https://doi.org/10.1016/j.physe.2021.114951

    Article  Google Scholar 

  77. Vinh PV, Belarbi M-O, Tounsi A (2022) Wave propagation analysis of functionally graded nanoplates using nonlocal higher-order shear deformation theory with spatial variation of the nonlocal parameters. Waves Random Complex Media. https://doi.org/10.1080/17455030.2022.2036387

    Article  Google Scholar 

  78. Salehipour H, Shahidi AR, Nahvi H (2015) Modified nonlocal elasticity theory for functionally graded materials. Int J Eng Sci 90:44–57. https://doi.org/10.1016/j.ijengsci.2015.01.005

    Article  MathSciNet  MATH  Google Scholar 

  79. Batra RC (2021) Misuse of Eringen’s nonlocal elasticity theory for functionally graded materials. Int J Eng Sci 159:103425. https://doi.org/10.1016/j.ijengsci.2020.103425

    Article  MathSciNet  MATH  Google Scholar 

Download references

Funding

This research did not receive any specific grants from funding agencies in the public, commercial or nonprofit sectors.

Author information

Authors and Affiliations

  1. Department of Solid Mechanics, Le Quy Don Technical University, 236 Hoang Quoc Viet Street, Hanoi, Vietnam

    Pham Van Vinh

  2. YFL (Yonsei Frontier Lab), Yonsei University, Seoul, South Korea

    Abdelouahed Tounsi

  3. Department of Civil and Environmental Engineering, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Eastern Province, Saudi Arabia

    Abdelouahed Tounsi

  4. Material and Hydrology Laboratory, Civil Engineering Department, Faculty of Technology, University of Sidi Bel Abbes, Sidi Bel Abbès, Algeria

    Abdelouahed Tounsi

  5. Laboratoire de Recherche en Génie Civil, LRGC, Université de Biskra, B.P. 145, R.P. 07000, Biskra, Algeria

    Mohamed-Ouejdi Belarbi

Authors
  1. Pham Van Vinh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Abdelouahed Tounsi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mohamed-Ouejdi Belarbi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pham Van Vinh.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in 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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Van Vinh, P., Tounsi, A. & Belarbi, MO. On the nonlocal free vibration analysis of functionally graded porous doubly curved shallow nanoshells with variable nonlocal parameters. Engineering with Computers 39, 835–855 (2023). https://doi.org/10.1007/s00366-022-01687-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-022-01687-6

Keywords

Search

Navigation