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Nonlocal strain gradient analysis of FG GPLRC nanoscale plates based on isogeometric approach

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Abstract

In this paper, a nonlocal strain gradient isogeometric model based on the higher order shear deformation theory for free vibration analysis of functionally graded graphene platelet-reinforced composites (FG GPLRC) plates is performed. Various distributed patterns of graphene platelets (GPLs) in the polymer matrix including uniform and non-uniform are considered. To capture size dependence of nanostructures, the nonlocal strain gradient theory including both nonlocal and strain gradient effects is used. Based on the modified Halpin–Tsai model, the effective Young’s modulus of the nanocomposites is expressed, while the Poisson’s ratio and density are established using the rule of mixtures. Natural frequencies of FG GPLRC nanoplates is determined using isogeometric analysis. The effects played by strain gradient parameter, distributions of GPLs, thickness-to-length ratio, and nonlocal parameter are examined, and results illustrate the interesting dynamic phenomenon. Several results are investigated and considered as benchmark results for further studies on the FG GPLRC nanoplates.

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References

  1. Rafiee MA, Rafiee J, Wang Z, Song H, Yu Z-Z, Koratkar N (2009) Enhanced mechanical properties of nanocomposites at low graphene content. ACS Nano 3(12):3884–3890

    Article  Google Scholar 

  2. Haile JM (1992) Molecular dynamics simulation: elementary methods. Wiley, New York

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  4. Xu X-J, Wang X-C, Zheng M-L, Ma Z (2017) Bending and buckling of nonlocal strain gradient elastic beams. Compos Struct 160:366–377

    Article  Google Scholar 

  5. Li L, Hu Y (2016) Nonlinear bending and free vibration analyses of nonlocal strain gradient beams made of functionally graded material. Int J Eng Sci 107:77–97

    Article  MathSciNet  MATH  Google Scholar 

  6. Farajpour A, Yazdi M, 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

    Article  MathSciNet  MATH  Google Scholar 

  7. Nematollahi MS, Mohammadi H, Nematollahi MA (2017) Thermal vibration analysis of nanoplates based on the higher-order nonlocal strain gradient theory by an analytical approach. Superlattices Microstruct 111:944–959

    Article  Google Scholar 

  8. Arefi M, Kiani M, Rabczuk T (2019) Application of nonlocal strain gradient theory to size dependent bending analysis of a sandwich porous nanoplate integrated with piezomagnetic face-sheets. Compos B Eng 168:320–333

    Article  Google Scholar 

  9. Lu L, Guo X, Zhao J (2017) A unified nonlocal strain gradient model for nanobeams and the importance of higher order terms. Int J Eng Sci 119:265–277

    Article  Google Scholar 

  10. Nematollahi MS, Mohammadi H (2019) Geometrically nonlinear vibration analysis of sandwich nanoplates based on higher-order nonlocal strain gradient theory. Int J Mech Sci 156:31–45

    Article  Google Scholar 

  11. Jalaei MH, Thai H-T (2019) Dynamic stability of viscoelastic porous FG nanoplate under longitudinal magnetic field via a nonlocal strain gradient quasi-3D theory. Compos B Eng 175:107164

    Article  Google Scholar 

  12. T. Rabczuk, H. Ren and X. Zhuang (2019) A nonlocal operator method for partial differential equations with application to electromagnetic waveguide problem. Computers, Materials & Continua, vol. 59, no.1, pp. 31–55.

  13. Ren H, Zhuang X, Rabczuk T (2020) A nonlocal operator method for solving partial differential equations. Comput Methods Appl Mech Eng 358:112621

    Article  MathSciNet  MATH  Google Scholar 

  14. Nguyen-Thanh VM, Anitescu C, Alajlan N, Rabczuk T, Zhuang X (2021) Parametric deep energy approach for elasticity accounting for strain gradient effects. Comput Methods Appl Mech Eng 386:114096

    Article  MathSciNet  MATH  Google Scholar 

  15. Anitescu C, Atroshchenko E, Alajlan N, Rabczuk T (2019) Artificial neural network methods for the solution of second order boundary value problems. Comput Mater Continua 59(1):345–359

    Article  Google Scholar 

  16. Sahmani S, Aghdam M (2017) Axial postbuckling analysis of multilayer functionally graded composite nanoplates reinforced with GPLs based on nonlocal strain gradient theory. Eur Phys J Plus 132(11):1–17

    Article  Google Scholar 

  17. Sahmani S, Aghdam MM, Rabczuk T (2018) A unified nonlocal strain gradient plate model for nonlinear axial instability of functionally graded porous micro/nano-plates reinforced with graphene platelets. Mater Res Express 5(4):045048

    Article  Google Scholar 

  18. Sahmani S, Fattahi AM, Ahmed N (2019) Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. Eng Comput 35(4):1173–1189

    Article  Google Scholar 

  19. Sahmani S, Aghdam M (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

    Article  Google Scholar 

  20. Karami B, Shahsavari D, Janghorban M, Tounsi A (2019) Resonance behavior of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets. Int J Mech Sci 156:94–105

    Article  Google Scholar 

  21. Thai CH, Ferreira A, Phung-Van P (2019) Size dependent free vibration analysis of multilayer functionally graded GPLRC microplates based on modified strain gradient theory. Compos B Eng 169:174–188

    Article  Google Scholar 

  22. Karami B, Shahsavari D, Ordookhani A, Gheisari P, Li L, Eyvazian A (2020) Dynamics of graphene-nanoplatelets reinforced composite nanoplates including different boundary conditions. Steel Compos Struct Int J 36(6):689–702

    Google Scholar 

  23. Phung-Van P, Lieu QX, Ferreira A, Thai CH (2021) A refined nonlocal isogeometric model for multilayer functionally graded graphene platelet-reinforced composite nanoplates. Thin-Wall Struct 164:107862

    Article  Google Scholar 

  24. Thai CH, Ferreira A, Phung-Van P (2020) A nonlocal strain gradient isogeometric model for free vibration and bending analyses of functionally graded plates. Compos Struct 251:112634

    Article  Google Scholar 

  25. Phung-Van P, Thai CH (2021) A novel size-dependent nonlocal strain gradient isogeometric model for functionally graded carbon nanotube-reinforced composite nanoplates. Eng Comput 1–14. https://doi.org/10.1007/s00366-021-01353-3

  26. Phung-Van P, Ferreira A, Nguyen-Xuan H, Thai CH (2021) A nonlocal strain gradient isogeometric nonlinear analysis of nanoporous metal foam plates. Eng Anal Bound Elem 130:58–68

    Article  MathSciNet  MATH  Google Scholar 

  27. Phung-Van P, Ferreira A, Nguyen-Xuan H, Thai CH (2021) Scale-dependent nonlocal strain gradient isogeometric analysis of metal foam nanoscale plates with various porosity distributions. Compos Struct 268:113949

    Article  Google Scholar 

  28. Thai CH, Ferreira A, Nguyen-Xuan H, Nguyen LB, Phung-Van P (2021) A nonlocal strain gradient analysis of laminated composites and sandwich nanoplates using meshfree approach. Eng Comput 1–17. https://doi.org/10.1007/s00366-021-01501-9

  29. Ghasemi H, Park HS, Alajlan N, Rabczuk T (2020) A computational framework for design and optimization of flexoelectric materials. Int J Comput Methods 17(01):1850097

    Article  MathSciNet  MATH  Google Scholar 

  30. Ghasemi H, Brighenti R, Zhuang X, Muthu J, Rabczuk T (2015) Optimal fiber content and distribution in fiber-reinforced solids using a reliability and NURBS based sequential optimization approach. Struct Multidiscipl Optim 51(1):99–112

    Article  MathSciNet  Google Scholar 

  31. Ghasemi H, Brighenti R, Zhuang X, Muthu J, Rabczuk T (2014) Optimization of fiber distribution in fiber reinforced composite by using NURBS functions. Comput Mater Sci 83:463–473

    Article  Google Scholar 

  32. Ghasemi H (2021) A multiscale material model for heterogeneous liquid droplets in solid soft composites. Front Struct Civ Eng 15(5):1292–1299

    Article  Google Scholar 

  33. Ghasemi H, Rafiee R, Zhuang X, Muthu J, Rabczuk T (2014) Uncertainties propagation in metamodel-based probabilistic optimization of CNT/polymer composite structure using stochastic multi-scale modeling. Comput Mater Sci 85:295–305

    Article  Google Scholar 

  34. Phung-Van P, Thai CH, Abdel-Wahab M, Nguyen-Xuan H (2020) Optimal design of FG sandwich nanoplates using size-dependent isogeometric analysis. Mech Mater 142:103277

    Article  Google Scholar 

  35. Thai CH, Tran T, Phung-Van P (2020) A size-dependent moving Kriging meshfree model for deformation and free vibration analysis of functionally graded carbon nanotube-reinforced composite nanoplates. Eng Anal Bound Elem 115:52–63

    Article  MathSciNet  MATH  Google Scholar 

  36. Hughes TJ, Cottrell JA, Bazilevs Y (2005) Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39–41):4135–4195

    Article  MathSciNet  MATH  Google Scholar 

  37. Thai CH, Ferreira A, Tran T, Phung-Van P (2020) A size-dependent quasi-3D isogeometric model for functionally graded graphene platelet-reinforced composite microplates based on the modified couple stress theory. Compos Struct 234:111695

    Article  Google Scholar 

  38. Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T (2016) A software framework for probabilistic sensitivity analysis for computationally expensive models. Adv Eng Softw 100:19–31

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculty of Civil Engineering, HUTECH University, Ho Chi Minh City, Vietnam

    P. Phung-Van

  2. CIRTech Institute, HUTECH University, Ho Chi Minh City, Vietnam

    H. Nguyen-Xuan

  3. Division of Computational Mechanics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Chien H. Thai

  4. Faculty of Civil Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam

    Chien H. Thai

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  1. P. Phung-Van
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  2. H. Nguyen-Xuan
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  3. Chien H. Thai
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Correspondence to Chien H. Thai.

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Phung-Van, P., Nguyen-Xuan, H. & Thai, C.H. Nonlocal strain gradient analysis of FG GPLRC nanoscale plates based on isogeometric approach. Engineering with Computers 39, 857–866 (2023). https://doi.org/10.1007/s00366-022-01689-4

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  • DOI: https://doi.org/10.1007/s00366-022-01689-4

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