Skip to main content
SpringerLink
Log in

An isogeometric approach to free vibration analysis of bi-directional functionally graded porous doubly-curved shallow microshells with variable length-scale parameters

  • Research Article
  • Published:
Frontiers of Structural and Civil Engineering Aims and scope Submit manuscript

Abstract

This study uses iso-geometric investigation, which is based on the non-uniform rational B-splines (NURBS) basis function, to investigate natural oscillation of bi-directional functionally graded porous (BFGP) doubly-curved shallow microshells placed on Pasternak foundations with any boundary conditions. The characteristics of the present material vary in both thickness and axial directions along the x-axis. To be more specific, a material length-scale coefficient of the microshell varies in both thickness and length directions as the material’s mechanical properties. One is able to develop a differential equation system with varying coefficients that regulate the motion of BFGP double-curved shallow microshells by using Hamilton principle, Kirchhoff–Love hypothesis, and modified couple stress theory. The numerical findings are reported for thin microshells that are spherical, cylindrical, and hyperbolic paraboloidal, with a variety of planforms, including rectangles and circles. The validity and effectiveness of the established model are shown by comparing the numerical results given by the proposed formulations with previously published findings in many specific circumstances. In addition, influences of length scale parameters, power-law indexes, thickness-to-side ratio, and radius ratio on natural oscillation responses of BFGP microshells are investigated in detail.

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

Similar content being viewed by others

References

  1. Calladine C R. Theory of Shell Structures. Cambridge: Cambridge University, 1983

    Book  Google Scholar 

  2. Pietraszkiewicz W, Konopinska V. Junctions in shell structures: A review. Thin-walled Structures, 2015, 95: 310–334

    Article  Google Scholar 

  3. Nguyen V P, Anitescu C, Bordas S P A, Rabczuk T. Isogeometric analysis: An overview and computer implementation aspects. Mathematics and Computers in Simulation, 2015, 117: 89–116

    Article  MathSciNet  Google Scholar 

  4. Jüttler B, Langer U, Mantzaflaris A, Moore S E, Zulehner W. Geometry + simulation modules: implementing isogeometric analysis. Proceedings in Applied Mathematics and Mechanics, 2014, 14(1): 961–962

    Article  Google Scholar 

  5. Marussig B, Hughes T J R. A review of trimming in isogeometric analysis: Challenges, data exchange and simulation aspects. Archives of Computational Methods in Engineering, 2018, 25(4): 1059–1127

    Article  MathSciNet  PubMed  Google Scholar 

  6. Hughes T J R, Cottrell J A, Bazilevs Y. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 2005, 194(39–41): 4135–4195

    Article  MathSciNet  ADS  Google Scholar 

  7. Hughes T J R, Reali A, Sangalli G. Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: Comparison of p-method finite elements with k-method NURBS. Computer Methods in Applied Mechanics and Engineering, 2008, 197(49–50): 4104–4124

    Article  MathSciNet  ADS  Google Scholar 

  8. Cottrell J A, Reali A, Bazilevs Y, Hughes T J R. Isogeometric analysis of structural vibrations. Computer Methods in Applied Mechanics and Engineering, 2006, 195(41–43): 5257–5296

    Article  MathSciNet  ADS  Google Scholar 

  9. Hughes T J R, Reali A, Sangalli G. Efficient quadrature for NURBS-based isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2010, 199(5–8): 301–313

    Article  MathSciNet  ADS  Google Scholar 

  10. Dörfel M R, Jüttler B, Simeon B. Adaptive isogeometric analysis by local h-refinement with T-splines. Computer Methods in Applied Mechanics and Engineering, 2010, 199(5–8): 264–275

    Article  MathSciNet  ADS  Google Scholar 

  11. Buffa A, Sangalli G, Vazquez R. Isogeometric analysis in electromagnetics: B-splines approximation. Computer Methods in Applied Mechanics and Engineering, 2010, 199(17–20): 1143–1152

    Article  MathSciNet  ADS  Google Scholar 

  12. Bazilevs Y, Akkerman I. Large eddy simulation of turbulent Taylor–Couette flow using isogeometric analysis and the residualbased variational multiscale method. Journal of Computational Physics, 2010, 229(9): 3402–3414

    Article  MathSciNet  CAS  ADS  Google Scholar 

  13. Zhuang X, Guo H, Alajlan N, Zhu H, Rabczuk T. Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning. European Journal of Mechanics. A, Solids, 2021, 87: 104225

    Article  MathSciNet  ADS  Google Scholar 

  14. Guo H, Rabczuk T, Zhuang X. A deep collocation method for the bending analysis of Kirchhoff plate. Computers, Materials & Continua, 2019, 59(2): 433–456

    Article  Google Scholar 

  15. Guo H, Zheng H, Zhuang X. Numerical manifold method for vibration analysis of Kirchhoff’s plates of arbitrary geometry. Applied Mathematical Modelling, 2019, 66: 695–727

    Article  MathSciNet  Google Scholar 

  16. Guo H, Zheng H. The linear analysis of thin shell problems using the numerical manifold method. Thin-walled Structures, 2018, 124: 366–383

    Article  Google Scholar 

  17. Shaaban A M, Anitescu C, Atroshchenko E, Rabczuk T. Isogeometric indirect BEM solution based on virtual continuous sources placed directly on the boundary of 2D Helmholtz acoustic problems. Engineering Analysis with Boundary Elements, 2023, 148: 243–255

    Article  MathSciNet  Google Scholar 

  18. Cohen E, Martin T, Kirby R M, Lyche T, Riesenfeld R F. Analysis-aware modeling: Understanding quality considerations in modeling for isogeometric analysis. Computer Methods in Applied Mechanics and Engineering, 2010, 199(5–8): 334–356

    Article  MathSciNet  ADS  Google Scholar 

  19. Valizadeh N, Natarajan S, Gonzalez-Estrada O A, Rabczuk T, Bui T Q, Bordas S P A. NURBS-based finite element analysis of functionally graded plates: Static bending, vibration, buckling and flutter. Composite Structures, 2013, 99: 309–326

    Article  Google Scholar 

  20. Dsouza S M, Varghese T M, Budarapu P R, Natarajan S. A non-intrusive stochastic isogeometric analysis of functionally graded plates with material uncertainty. Axioms, 2020, 9(3): 92

    Article  Google Scholar 

  21. Hu Q, Xia Y, Natarajan S, Zilian A, Hu P, Bordas S P A. Isogeometric analysis of thin Reissner–Mindlin shells: Locking phenomena and B-bar method. Computational Mechanics, 2020, 65(5): 1323–1341

    Article  MathSciNet  ADS  Google Scholar 

  22. Ha S H, Choi K K, Cho S. Numerical method for shape optimization using T-spline based isogeometric method. Structural and Multidisciplinary Optimization, 2010, 42(3): 417–428

    Article  Google Scholar 

  23. Bazilevs Y, Calo V, Zhang Y, Hughes T. Isogeometric fluid–structure interaction analysis with applications to arterial blood flow. Computational Mechanics, 2006, 38(4–5): 310–322

    Article  MathSciNet  ADS  Google Scholar 

  24. Schmidt R, Kiendl J, Bletzinger K U, Wüchner R. Realization of an integrated structural design process: Analysis-suitable geometric modelling and isogeometric analysis. Computing and Visualization in Science, 2010, 13(7): 315–330

    Article  Google Scholar 

  25. Bazilevs J. Isogeometric Analysis of Turbulence and Fluid-Structure Interaction. Austin: The University of Texas, 2006

    Google Scholar 

  26. Wang Y, Wang Z, Xia Z, Poh L H. Structural design optimization using isogeometric analysis: A comprehensive review. Computer Modeling in Engineering & Sciences, 2018, 117(3): 455–507

    Article  ADS  Google Scholar 

  27. Bontinck Z, Corno J, de Gersem H, Kurz S, Pels A, Schöps S, Wolf F, de Falco C, Dölz J, Vazquez R, Römer U. Recent advances of isogeometric analysis in computational electromagnetics. Technical Article, Cornell University, 2017

  28. Pham Q H, Nguyen P C, Tran V K, Nguyen-Thoi T. Isogeometric analysis for free vibration of bidirectional functionally graded plates in the fluid medium. Defence Technology, 2022, 18(8): 1311–1329

    Article  Google Scholar 

  29. Pham Q H, Nguyen P C, Tran V K, Lieu Q X, Tran T T. Modified nonlocal couple stress isogeometric approach for bending and free vibration analysis of functionally graded nanoplates. Engineering with Computers, 2022, 39(1): 993–1018

    Article  Google Scholar 

  30. Ansari R, Norouzzadeh A. Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates: An isogeometric analysis. Physica E, Low-Dimensional Systems and Nanostructures, 2016, 84: 84–97

    Article  CAS  ADS  Google Scholar 

  31. Fan F, Lei B, Sahmani S, Safaei B. On the surface elastic-based shear buckling characteristics of functionally graded composite skew nanoplates. Thin-walled Structures, 2020, 154: 106841

    Article  Google Scholar 

  32. Norouzzadeh A, Ansari R. Isogeometric vibration analysis of functionally graded nanoplates with the consideration of nonlocal and surface effects. Thin-walled Structures, 2018, 127: 354–372

    Article  Google Scholar 

  33. Fan F, Xu Y, Sahmani S, Safaei B. Modified couple stress-based geometrically nonlinear oscillations of porous functionally graded microplates using NURBS-based isogeometric approach. Computer Methods in Applied Mechanics and Engineering, 2020, 372: 113400

    Article  MathSciNet  ADS  Google Scholar 

  34. Luat D T, Van Thom D, Thanh T T, Van Minh P, Van Ke T, Van Vinh P. Mechanical analysis of bi-functionally graded sandwich nanobeams. Advances in Nano Research, 2021, 11(1): 55–71

    Google Scholar 

  35. Nguyen T C N. Static bending analysis of variable thickness microplates using the finite element method and modified couple stress theory. Journal of Science and Technology, 2022, 17(3)

  36. Dung N T, Thai L M, Van Ke T, Huyen T T H, Van Minh P. Nonlinear static bending analysis of microplates resting on imperfect two-parameter elastic foundations using modified couple stress theory. Comptes Rendus. Mécanique, 2022, 350(G1): 121–141 (in French)

    Article  ADS  Google Scholar 

  37. Qiu J, Sahmani S, Safaei B. On the NURBS-based isogeometric analysis for couple stress-based nonlinear instability of PFGM microplates. Mechanics Based Design of Structures and Machines, 2020, 51(2): 816–840

    Article  Google Scholar 

  38. Rahmouni F, Elajrami M, Madani K, Campilho R D S G. Isogeometric analysis based on non-uniform rational B-splines technology of stress and failure strength in inter-ply hybrid laminated composite. Journal of Composite Materials, 2022, 56(18): 2921–2932

    Article  ADS  Google Scholar 

  39. Tho N C, Cong P H, Zenkour A M, Doan D H, Minh P V. Finite element modeling of the bending and vibration behavior of three-layer composite plates with a crack in the core layer. Composite Structures, 2023, 305: 116529

    Article  Google Scholar 

  40. Tuan L T, Dung N T, Van Thom D, Van Minh P, Zenkour A M. Propagation of non-stationary kinematic disturbances from a spherical cavity in the pseudo-elastic cosserat medium. European Physical Journal Plus, 2021, 136(12): 1199

    Article  ADS  Google Scholar 

  41. Phung V M. Static bending analysis of symmetrical three-layer FG beam with shear connectors under static load. Journal of Science and Technology, 2020, 15(3): 68–78

    Google Scholar 

  42. Van Minh P, Van Ke T. A comprehensive study on mechanical responses of non-uniform thickness piezoelectric nanoplates taking into sccount the flexoelectric effect. Arabian Journal for Science and Engineering, 2022, 48: 11457–11482

    Article  Google Scholar 

  43. Phung V M. Static bending analysis of nanoplates on discontinuous elastic foundation with flexoelectric effect. Journal of Science and Technology, 2022, 17(5): 47–57

    MathSciNet  Google Scholar 

  44. Thai L M, Luat D T, Van Ke T, Phung Van M. Finite-element modeling for static bending analysis of rotating two-layer FGM beams with shear connectors resting on imperfect elastic foundations. Journal of Aerospace Engineering, 2023, 36(3): 04023013

    Article  Google Scholar 

  45. Tien D M, Van Thom D, Van Minh P, Tho N C, Doan T N, Mai D N. The application of the nonlocal theory and various shear strain theories for bending and free vibration analysis of organic nanoplates. Mechanics Based Design of Structures and Machines, 2023, 1–23

  46. Tran V K, Tran T T, Van Phung M, Pham Q H, Nguyen-Thoi T. A finite element formulation and nonlocal theory for the static and free vibration analysis of the sandwich functionally graded nanoplates resting on elastic foundation. Journal of Nanomaterials, 2020, 2020: 1–20

    Article  Google Scholar 

  47. Van Phung M, Nguyen D T, Doan L T, Van Nguyen D, Van Duong T. Numerical investigation on static bending and free vibration responses of two-layer variable thickness plates with shear connectors. Iranian Journal of Science and Technology. Transaction of Mechanical Engineering, 2022, 46(4): 1047–1065

    Article  Google Scholar 

  48. Tran T T, Tran V K, Le P B, Phung V M, Do V T, Nguyen H N. Forced vibration analysis of laminated composite shells reinforced with graphene nanoplatelets using finite element method. Advances in Civil Engineering, 2020, 2020: 1–17

    Article  Google Scholar 

  49. Nguyen Thai D, Van Minh P, Phan Hoang C, Ta Duc T, Nguyen Thi Cam N, Nguyen Thi D. Bending of symmetric sandwich FGM beams with shear connectors. Mathematical Problems in Engineering, 2021, 2021: 1–15

    Article  MathSciNet  Google Scholar 

  50. Tho N C, Thanh N T, Tho T D, Van Minh P, Hoa L K. Modelling of the flexoelectric effect on rotating nanobeams with geometrical imperfection. Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2021, 43(11): 510

    Article  Google Scholar 

  51. Shahsavari D, Karami B, Fahham H R, Li L. On the shear buckling of porous nanoplates using a new size-dependent quasi-3D shear deformation theory. Acta Mechanica, 2018, 229(11): 4549–4573

    Article  MathSciNet  Google Scholar 

  52. Karamanli A, Vo T P. Size dependent bending analysis of two directional functionally graded microbeams via a quasi-3D theory and finite element method. Composites. Part B, Engineering, 2018, 144: 171–183

    Article  Google Scholar 

  53. Ghayesh M H, Farokhi H. Nonlinear mechanics of doubly curved shallow microshells. International Journal of Engineering Science, 2017, 119: 288–304

    Article  MathSciNet  Google Scholar 

  54. Yang F, Chong A C M, Lam D C C, Tong P. Couple stress based strain gradient theory for elasticity. International Journal of Solids and Structures, 2002, 39(10): 2731–2743

    Article  Google Scholar 

  55. Tran V K, Pham Q H, Nguyen-Thoi T. A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations. Engineering with Computers, 2022, 38(2): 1465–1490

    Article  Google Scholar 

  56. Dehrouyeh-Semnani A M, Mostafaei H. Vibration analysis of scale-dependent thin shallow microshells with arbitrary planform and boundary conditions. International Journal of Engineering Science, 2021, 158: 103413

    Article  MathSciNet  Google Scholar 

  57. Nguyen H X, Nguyen T N, Abdel-Wahab M, Bordas S P A, Nguyen-Xuan H, Vo T P. A refined quasi-3D isogeometric analysis for functionally graded microplates based on the modified couple stress theory. Computer Methods in Applied Mechanics and Engineering, 2017, 313: 904–940

    Article  MathSciNet  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering, Hanoi University of Industry, Hanoi, 100000, Vietnam

    Khuat Duc Duong

  2. Institute of Mechanics, Vietnam Academy of Science and Technology, Hanoi, 100000, Vietnam

    Dao Nhu Mai

  3. Faculty of Mechanical Engineering, Le Quy Don Technical University, Hanoi, 100000, Vietnam

    Phung Van Minh & Tran Van Ke

Authors
  1. Khuat Duc Duong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dao Nhu Mai
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Phung Van Minh
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Tran Van Ke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tran Van Ke.

Ethics declarations

Conflict of Interests The authors declare that they have no conflict of interest.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Duong, K.D., Mai, D.N., Minh, P.V. et al. An isogeometric approach to free vibration analysis of bi-directional functionally graded porous doubly-curved shallow microshells with variable length-scale parameters. Front. Struct. Civ. Eng. 17, 1871–1894 (2023). https://doi.org/10.1007/s11709-023-0021-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11709-023-0021-y

Keywords

Search

Navigation