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Analytical treatment on the nonlocal strain gradient vibrational response of postbuckled functionally graded porous micro-/nanoplates reinforced with GPL

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A Correction to this article was published on 21 March 2020

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Abstract

Through production of porous materials with remarkable complexity in geometry, functionally graded porous materials (FGPMs) have gained considerable attention for use in additive manufacturing in biomedical applications. In the current study, the size-dependent linear and nonlinear vibrational characteristics of axially loaded micro-/nanoplates made of FGPM reinforced with graphene platelets (GPLs) is investigated within both the prebuckling and postbuckling regimes. To this end, the nonlocal strain gradient continuum elasticity in conjunction with geometrical nonlinearity is implemented into the refined exponential shear deformation plate theory. On the basis of the closed-cell Gaussian random field scheme as well as the Halpin–Tsai micromechanical modeling, the mechanical properties of the FGPM reinforced with GPLs are achieved corresponding to the uniform and three different patterns of porosity dispersion. Via the variational approach, the differential equations of motion related to the nonlinear problem are constructed in the presence of nonlocality and strain gradient size dependency. Finally, with the aid of an improved perturbation technique together with the Galerkin method, analytical expressions in explicit form for the size-dependent linear frequency–load and deflection–nonlinear frequency responses of the FGPM micro-/nanoplates within stability and instability domains are obtained. It is displayed that within the prebuckling regime, the nonlocality causes reduction of the linear frequency of the micro-/nanoplate, while the strain gradient size dependency leads to increasing it. But within the postbuckling domain, these patterns are vice versa. Also, it is found that for a specific value of plate deflection, increasing the value of the porosity coefficient leads to increase in the frequency ratio of ωNL/ωL within both the prebuckling and postbuckling regimes.

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Change history

  • 21 March 2020

    In the original publication of the article, the first author's affiliation has been changed to ���School of Science and Technology, The University of Georgia, Tbilisi 0171, Georgia���.

References

  1. Popp A, Engstler J, Schneider JJ (2009) Porous carbon nanotube-reinforced metals and ceramics via a double templating approach. Carbon 47:3208–3214

    Google Scholar 

  2. Jun L, Ying L, Lixian L, Xuejuan Y (2012) Mechanical properties and oil content of CNT reinforced porous CuSn oil bearings. Compos B Eng 43:1681–1686

    Google Scholar 

  3. Hai C, Shirai T, Fuji M (2013) Fabrication of conductive porous alumina (CPA) structurally modified with carbon nanotubes (CNT). Adv Powder Technol 24:824–828

    Google Scholar 

  4. Chen L, Wang JX, Tang CY, Chen DZ, Law WC (2016) Shape memory effect of thermal-responsive nano-hydroxyapatite reinforced poly-d-l-lactide composites with porous structure. Compos B Eng 107:67–74

    Google Scholar 

  5. Xu H, Li Q (2017) Effect of carbon nanofiber concentration on mechanical properties of porous magnesium composites: experimental and theoretical analysis. Mater Sci Eng A 706:249–255

    Google Scholar 

  6. Chen D, Yang J, Kitipornchai S (2017) Nonlinear vibration and postbuckling of functionally graded graphene reinforced porous nanocomposite beams. Compos Sci Technol 142:235–245

    Google Scholar 

  7. Thai H-T, Vo TP (2012) A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling and vibration of nanobeams. Int J Eng Sci 54:58–66

    MathSciNet  MATH  Google Scholar 

  8. Hajmohammad MH, Kolahchi R, Zarei MS, Maleki M (2018) Earthquake induced dynamic deflection of submerged viscoelastic cylindrical shell reinforced by agglomerated CNTs considering thermal and moisture effects. Compos Struct 187:498–508

    Google Scholar 

  9. Hosseini H, Kolahchi R (2018) Seismic response of functionally graded-carbon nanotubes-reinforced submerged viscoelastic cylindrical shell in hygrothermal environment. Physica E 102:101–109

    Google Scholar 

  10. Wang L, Xu YY, Ni Q (2013) Size-dependent vibration analysis of three-dimensional cylindrical microbeams based on modified couple stress theory: a unified treatment. Int J Eng Sci 68:1–10

    MathSciNet  MATH  Google Scholar 

  11. Sarrami-Foroushani S, Azhari M (2014) On the use of bubble complex finite strip method in the nonlocal buckling and vibration analysis of single-layered graphene sheets. Int J Mech Sci 85:168–178

    Google Scholar 

  12. Nguyen N-T, Hui D, Lee L, Nguyen-Xuan H (2015) An efficient computational approach for size-dependent analysis of functionally graded nanoplates. Comput Methods Appl Mech Eng 297:191–218

    MathSciNet  MATH  Google Scholar 

  13. Li HB, Li D, Wang X, Huang X (2015) Nonlinear vibration characteristics of graphene/piezoelectric sandwich films under electric loading based on nonlocal elastic theory. J Sound Vib 358:285–300

    Google Scholar 

  14. Zafari E, Torabi K (2016) Semi-analytical solution for free transverse vibrations of Euler-Bernoulli nanobeams with manifold concentrated masses. Mech Adv Mater Struct 24:725–736

    Google Scholar 

  15. Yang WD, Wang X (2016) Nonlinear pull-in instability of carbon nanotubes reinforced nano-actuator with thermally corrected Casimir force and surface effect. Int J Mech Sci 107:34–42

    Google Scholar 

  16. Lou J, He L, Du J, Wu H (2016) Buckling and post-buckling analyses of piezoelectric hybrid microplates subject to thermo–electro-mechanical loads based on the modified couple stress theory. Compos Struct 153:332–344

    Google Scholar 

  17. Li HB, Yang FP, Wang X (2016) Nonlinear resonant frequency of graphene/elastic/piezoelectric laminated films under active electric loading. Int J Mech Sci 115:624–633

    Google Scholar 

  18. Ghadiri M, Shafiei N, Alavi H (2016) Thermo-mechanical vibration of orthotropic cantilever and propped cantilever nanoplate using generalized differential quadrature method. Mech Adv Mater Struct 24:636–646

    Google Scholar 

  19. Arani AJ, Kolahchi R (2016) Buckling analysis of embedded concrete columns armed with carbon nanotubes. Comput Concr 17:567–578

    Google Scholar 

  20. Madani H, Hosseini H, Shokravi M (2016) Differential cubature method for vibration analysis of embedded FG-CNT-reinforced piezoelectric cylindrical shells subjected to uniform and non-uniform temperature distributions. Steel Compos Struct 22:889–913

    Google Scholar 

  21. Kolahchi R, Hosseini H, Esmailpour M (2016) Differential cubature and quadrature-Bolotin methods for dynamic stability of embedded piezoelectric nanoplates based on visco-nonlocal-piezoelasticity theories. Compos Struct 157:174–186

    Google Scholar 

  22. Liu JC, Zhang YQ, Fan LF (2017) Nonlocal vibration and biaxial buckling of double-viscoelastic-FGM-nanoplate system with viscoelastic Pasternak medium in between. Phys Lett A 381:1228–1235

    MathSciNet  Google Scholar 

  23. Sahmani S, Aghdam MM (2017) Imperfection sensitivity of the size-dependent postbuckling response of pressurized FGM nanoshells in thermal environments. Arch Civ Mech Eng 17:623–638

    Google Scholar 

  24. Kolahchi R, Zarei MS, Hajmohammad MH, Oskouei AN (2017) Visco-nonlocal-refined Zigzag theories for dynamic buckling of laminated nanoplates using differential cubature-Bolotin methods. Thin-Walled Struct 113:162–169

    Google Scholar 

  25. Sahmani S, Aghdam MM (2017) Nonlocal strain gradient beam model for nonlinear vibration of prebuckled and postbuckled multilayer functionally graded GPLRC nanobeams. Compos Struct 179:77–88

    Google Scholar 

  26. Tuna M, Kirca M (2017) Bending, buckling and free vibration analysis of Euler-Bernoulli nanobeams using Eringen’s nonlocal integral model via finite element method. Compos Struct 179:269–284

    Google Scholar 

  27. Fattahi AM, Sahmani S (2017) Size dependency in the axial postbuckling behavior of nanopanels made of functionally graded material considering surface elasticity. Arab J Sci Eng 42:4617–4633

    MathSciNet  MATH  Google Scholar 

  28. Kolahchi R, Cheraghbak A (2017) Agglomeration effects on the dynamic buckling of viscoelastic microplates reinforced with SWCNTs using Bolotin method. Nonlinear Dyn 90:479–492

    MATH  Google Scholar 

  29. Sahmani S, Fattahi AM (2017) An efficient computational approach for size-dependent analysis of functionally graded nanoplates. Comput Methods Appl Mech Eng 322:187–207

    MATH  Google Scholar 

  30. Kolahchi R (2017) A comparative study on the bending, vibration and buckling of viscoelastic sandwich nano-plates based on different nonlocal theories using DC, HDQ and DQ methods. Aerosp Sci Technol 66:235–248

    Google Scholar 

  31. Sahmani S, Fattahi AM (2017) Imperfection sensitivity of the size-dependent nonlinear instability of axially loaded FGM nanopanels in thermal environments. Acta Mech 228:3789–3810

    MathSciNet  MATH  Google Scholar 

  32. Fernandez-Saez J, Zaera R (2017) Vibrations of Bernoulli–Euler beams using the two-phase nonlocal elasticity theory. Int J Eng Sci 119:232–248

    MathSciNet  MATH  Google Scholar 

  33. Kolahchi R, Zarei MS, Hajmohammad MH, Nouri A (2017) Wave propagation of embedded viscoelastic FG-CNT-reinforced sandwich plates integrated with sensor and actuator based on refined zigzag theory. Int J Mech Sci 130:534–545

    Google Scholar 

  34. Sahmani S, Aghdam MM, Bahrami M (2017) An efficient size-dependent shear deformable shell model and molecular dynamics simulation for axial instability analysis of silicon nanoshells. J Mol Graph Model 77:263–279

    Google Scholar 

  35. Shafiei N, Mirjavadi SS, Afshari BM, Rabby S, Kazemi M (2017) An efficient computational approach for size-dependent analysis of functionally graded nanoplates. Comput Methods Appl Mech Eng 322:615–632

    MATH  Google Scholar 

  36. Sahmani S, Fattahi AM (2017) Size-dependent nonlinear instability of shear deformable cylindrical nanopanels subjected to axial compression in thermal environments. Microsyst Technol 23:4717–4731

    Google Scholar 

  37. Zhang LW, Zhang Y, Liew KM (2017) Vibration analysis of quadrilateral graphene sheets subjected to an in-plane magnetic field based on nonlocal elasticity theory. Compos B Eng 118:96–103

    Google Scholar 

  38. Sahmani S, Aghdam MM (2017) Nonlinear instability of hydrostatic pressurized hybrid FGM exponential shear deformable nanoshells based on nonlocal continuum elasticity. Compos B Eng 114:404–417

    Google Scholar 

  39. Sahmani S, Aghdam MM (2018) Nonlinear size-dependent instability of hybrid FGM nanoshells. In: Nonlinear approaches in engineering applications. Springer, pp 107–143

  40. Shojaeefard MH, Saeidi Googarchin H, Ghadiri M, Mahinzare M (2017) Micro temperature-dependent FG porous plate: free vibration and thermal buckling analysis using modified couple stress theory with CPT and FSDT. Appl Math Model 50:633–655

    MathSciNet  MATH  Google Scholar 

  41. Sahmani S, Aghdam MM (2017) Size-dependent nonlinear bending of micro/nano-beams made of nanoporous biomaterials including a refined truncated cube cell. Phys Lett A 381:3818–3830

    MathSciNet  Google Scholar 

  42. Sahmani S, Aghdam MM (2018) Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells. Compos B Eng 132:258–274

    Google Scholar 

  43. Hajmohammad MH, Farrokhian A, Kolahchi R (2018) Smart control and vibration of viscoelastic actuator-multiphase nanocomposite conical shells-sensor considering hygrothermal load based on layerwise theory. Aerosp Sci Technol 78:260–270

    Google Scholar 

  44. Sahmani S, Khandan A (2018) Size dependency in nonlinear instability of smart magneto-electro-elastic cylindrical composite nanopanels based upon nonlocal strain gradient elasticity. Microsyst Technol. https://doi.org/doi.org/10.1007/s00542-018-4072-2

    Article  Google Scholar 

  45. Sahmani S, Fattahi AM, Ahmed NA (2018) Analytical mathematical solution for vibrational response of postbuckled laminated FG-GPLRC nonlocal strain gradient micro-/nanobeams. Eng Comput. https://doi.org/doi.org/10.1007/s00366-018-0657-8

    Article  Google Scholar 

  46. Wang KF, Wang BL, Xu MH, Yu AB (2018) Influences of surface and interface energies on the nonlinear vibration of laminated nanoscale plates. Compos Struct 183:423–433

    Google Scholar 

  47. Fakhar A, Kolahchi R (2018) Dynamic buckling of magnetorheological fluid integrated by visco-piezo-GPL reinforced plates. Int J Mech Sci 144:788–799

    Google Scholar 

  48. Sahmani S, Aghdam MM (2018) Small scale effects on the large amplitude nonlinear vibrations of multilayer functionally graded composite nanobeams reinforced with graphene-nanoplatelets. Int J Nanosci Nanotechnol 14:207–227

    Google Scholar 

  49. Dai HL, Ceballes S, Abdelkefi A, Hong YZ, Wang L (2018) Exact modes for post-buckling characteristics of nonlocal nanobeams in a longitudinal magnetic field. Appl Math Model 55:758–775

    MathSciNet  MATH  Google Scholar 

  50. Sahmani S, Aghdam MM (2018) Boundary layer modeling of nonlinear axial buckling behavior of functionally graded cylindrical nanoshells based on the surface elasticity theory. Iran J Sci Technol Trans Mech Eng 42:229–245

    Google Scholar 

  51. Ganapathi M, Merzouki T, Polit O (2018) Vibration study of curved nanobeams based on nonlocal higher-order shear deformation theory using finite element approach. Compos Struct 184:821–838

    Google Scholar 

  52. Sahmani S, Aghdam MM, Akbarzadeh A (2018) Surface stress effect on nonlinear instability of imperfect piezoelectric nanoshells under combination of hydrostatic pressure and lateral electric field. AUT J Mech Eng 2:177–190

    Google Scholar 

  53. Fang X-Q, Zhu C-S, Liu J-X, Liu X-L (2018) Surface energy effect on free vibration of nano-sized piezoelectric double-shell structures. Physica B 529:41–56

    Google Scholar 

  54. Sahmani S, Fotouhi M, Aghdam MM (2019) Size-dependent nonlinear secondary resonance of micro-/nano-beams made of nano-porous biomaterials including truncated cube cells. Acta Mech 230:1077–1103

    MathSciNet  MATH  Google Scholar 

  55. Sarafraz A, Sahmani S, Aghdam MM (2019) Nonlinear secondary resonance of nanobeams under subharmonic and superharmonic excitations including surface free energy effects. Appl Math Model 66:195–226

    MathSciNet  MATH  Google Scholar 

  56. Sahmani S, Fattahi AM, Ahmed NA (2019) Develop a refined truncated cubic lattice structure for nonlinear large-amplitude vibrations of micro/nano-beams made of nanoporous materials. Eng Comput. https://doi.org/doi.org/10.1007/s00366-019-00703-6

    Article  Google Scholar 

  57. Sahmani S, Aghdam MM (2019) Size-dependent nonlinear mechanics of biological nanoporous microbeams. In: Nanomaterials for advanced biological applications. Springer, pp 181–207

  58. 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 Solids 78:298–313

    MathSciNet  MATH  Google Scholar 

  59. 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

    MathSciNet  MATH  Google Scholar 

  60. Simsek M (2016) Nonlinear free vibration of a functionally graded nanobeam using nonlocal strain gradient theory and a novel Hamiltonian approach. Int J Eng Sci 105:10–21

    MathSciNet  MATH  Google Scholar 

  61. Tang Y, Liu Y, Zhao D (2016) Viscoelastic wave propagation in the viscoelastic single walled carbon nanotubes based on nonlocal strain gradient theory. Physica E 84:202–208

    Google Scholar 

  62. 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

    Google Scholar 

  63. Shahsavari D, Karami B, Mansouri S (2018) Shear buckling of single layer graphene sheets in hygrothermal environment resting on elastic foundation based on different nonlocal strain gradient theories. Eur J Mech A/Solids 67:200–214

    MathSciNet  MATH  Google Scholar 

  64. Sahmani S, Aghdam MM (2017) Nonlinear vibrations of pre- and post-buckled lipid supramolecular micro/nano-tubules via nonlocal strain gradient elasticity theory. J Biomech 65:49–60

    Google Scholar 

  65. Sahmani S, Aghdam MM (2018) Nonlocal strain gradient beam model for postbuckling and associated vibrational response of lipid supramolecular protein micro/nano-tubules. Math Biosci 295:24–35

    MathSciNet  MATH  Google Scholar 

  66. Radic N (2018) On buckling of porous double-layered FG nanoplates in the Pasternak elastic foundation based on nonlocal strain gradient elasticity. Compos B Eng 153:465–479

    Google Scholar 

  67. Li X, Li L, Hu Y, Ding Z, Deng W (2017) Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory. Compos Struct 165:250–265

    Google Scholar 

  68. Sahmani S, Aghdam MM, Rabczuk T (2018) Nonlinear bending of functionally graded porous micro/nano-beams reinforced with graphene platelets based upon nonlocal strain gradient theory. Compos Struct 186:68–78

    Google Scholar 

  69. 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 5:045048

    Google Scholar 

  70. Sahmani S, Aghdam MM, Rabczuk T (2018) Nonlocal strain gradient plate model for nonlinear large-amplitude vibrations of functionally graded porous micro/nano-plates reinforced with GPLs. Compos Struct 198:51–62

    Google Scholar 

  71. Zeighampour H, Tadi Beni Y, Dehkordi MB (2018) Wave propagation in viscoelastic thin cylindrical nanoshell resting on a visco-Pasternak foundation based on nonlocal strain gradient theory. Thin-Walled Struct 122:378–386

    Google Scholar 

  72. Imani Aria A, Biglari H (2018) Computational vibration and buckling analysis of microtubule bundles based on nonlocal strain gradient theory. Appl Math Comput 321:313–332

    MathSciNet  MATH  Google Scholar 

  73. Sahmani S, Safaei B (2019) Nonlinear free vibrations of bi-directional functionally graded micro/nano-beams including nonlocal stress and microstructural strain gradient size effects. Thin-Walled Struct 140:342–356

    Google Scholar 

  74. Halpin JC, Kardos JL (1976) The Halpin–Tsai equations: a review. Polym Eng Sci 16:344–352

    Google Scholar 

  75. Hejazi SM, Abtahi SM, Safaie F (2016) Investigation of thermal stress distribution in fiber reinforced roller compacted concrete pavements. J Ind Text 45:869–914

    Google Scholar 

  76. Roberts AP, Garboczi EJ (2001) Elastic moduli of model random three-dimensional closed-cell cellular solids. Acta Mater 49:189–197

    Google Scholar 

  77. Shen H-S, Xiang Y (2012) An efficient computational approach for size-dependent analysis of functionally graded nanoplates. Comput Methods Appl Mech Eng 213–216:196–205

    Google Scholar 

  78. Shen H-S, Xiang Y, Lin F, Hui D (2017) Buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates in thermal environments. Compos B Eng 119:67–78

    MATH  Google Scholar 

  79. Sahmani S, Shahali M, Khandan A, Saber-Samandari S, Aghdam MM (2018) Analytical and experimental analyses for mechanical and biological characteristics of novel nanoclay bio-nanocomposite scaffolds fabricated via space holder technique. Appl Clay Sci 165:112–123

    Google Scholar 

  80. Sahmani S, Saber-Samandari S, Shahali M, Yekta HJ, Aghadavoodi F, Montazeran AH, Aghdam MM, Khandan A (2018) Mechanical and biological performance of axially loaded novel bio-nanocomposite sandwich plate-type implant coated by biological polymer thin film. J Mech Behav Biomed Mater 88:238–250

    Google Scholar 

  81. Yu Y, Shen H-S, Wang H, Hui D (2018) Postbuckling of sandwich plates with graphene-reinforced composite face sheets in thermal environments. Compos B Eng 135:72–83

    Google Scholar 

  82. Sahmani S, Shahali M, Ghadiri Nejad M, Khandan A, Aghdam MM (2019) Effect of copper oxide nanoparticles on electrical conductivity and cell viability of calcium phosphate scaffolds with improved mechanical strength for bone tissue engineering. Eur Phys J Plus 134:7

    Google Scholar 

  83. Sahmani S, Saber-Samandari S, Khandan A, Aghdam MM (2019) Nonlinear resonance investigation of nanoclay based bio-nanocomposite scaffolds with enhanced properties for bone substitute applications. J Alloys Compd 773:636–653

    Google Scholar 

  84. Fan Y, Xiang Y, Shen H-S (2019) Nonlinear forced vibration of FG-GRC laminated plates resting on visco-Pasternak foundations. Compos Struct 209:443–452

    Google Scholar 

  85. Sahmani S, Saber-Samandari S, Khandan A, Aghdam MM (2019) Influence of MgO nanoparticles on the mechanical properties of coated hydroxyapatite nanocomposite scaffolds produced via space holder technique: fabrication, characterization and simulation. J Mech Behav Biomed Mater 95:76–88

    Google Scholar 

  86. Tjong SC (2013) Recent progress in the development and properties of novel metal matrix nanocomposites reinforced with carbon nanotubes and graphene nanosheets. Mater Sci Eng R Rep 74:281–350

    Google Scholar 

  87. Han W, Petyt M (1996) Linear vibration analysis of laminated rectangular plates using the hierarchical finite element method—I. Free vibration analysis. Comput Struct 61:705–712

    MATH  Google Scholar 

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Authors and Affiliations

  1. Mechanical Rotating Equipment Department, Niroo Research Institute (NRI), Tehran, 14665-517, Iran

    S. Sahmani

  2. Mechanical Engineering Science Department, Faculty of Engineering and the Built Environment, University of Johannesburg, Johannesburg, South Africa

    A. M. Fattahi & N. A. Ahmed

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  1. S. Sahmani
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  2. A. M. Fattahi
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  3. N. A. Ahmed
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Correspondence to A. M. Fattahi.

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Appendices

Appendix 1

The nonlocal strain gradient stress resultants can be expressed as:

$$\begin{aligned} & \left\{ {\begin{array}{*{20}c} {N_{xx} - \mu^{2} \left( {\frac{{\partial^{2} N_{xx} }}{{\partial x^{2} }} + \frac{{\partial^{2} N_{xx} }}{{\partial y^{2} }}} \right)} \\ {N_{yy} - \mu^{2} \left( {\frac{{\partial^{2} N_{yy} }}{{\partial x^{2} }} + \frac{{\partial^{2} N_{yy} }}{{\partial y^{2} }}} \right)} \\ {N_{xy} - \mu^{2} \left( {\frac{{\partial^{2} N_{xy} }}{{\partial x^{2} }} + \frac{{\partial^{2} N_{xy} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad = \left[ {\begin{array}{*{20}c} {A_{11}^{*} } & {A_{12}^{*} } & 0 \\ {A_{12}^{*} } & {A_{22}^{*} } & 0 \\ 0 & 0 & {A_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial y^{2} }}} \right)} \\ {\varepsilon_{yy}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial y^{2} }}} \right)} \\ {\gamma_{xy}^{0} - l^{2} \left( {\frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {B_{11}^{*} } & {B_{12}^{*} } & 0 \\ {B_{12}^{*} } & {B_{22}^{*} } & 0 \\ 0 & 0 & {B_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {F_{11}^{*} } & {F_{12}^{*} } & 0 \\ {F_{12}^{*} } & {F_{22}^{*} } & 0 \\ 0 & 0 & {F_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ \end{aligned},$$
$$\begin{aligned} & \left\{ {\begin{array}{*{20}c} {M_{xx} - \mu^{2} \left( {\frac{{\partial^{2} M_{xx} }}{{\partial x^{2} }} + \frac{{\partial^{2} M_{xx} }}{{\partial y^{2} }}} \right)} \\ {M_{yy} - \mu^{2} \left( {\frac{{\partial^{2} M_{yy} }}{{\partial x^{2} }} + \frac{{\partial^{2} M_{yy} }}{{\partial y^{2} }}} \right)} \\ {M_{xy} - \mu^{2} \left( {\frac{{\partial^{2} M_{xy} }}{{\partial x^{2} }} + \frac{{\partial^{2} M_{xy} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad = \left[ {\begin{array}{*{20}c} {B_{11}^{*} } & {B_{12}^{*} } & 0 \\ {B_{12}^{*} } & {B_{11}^{*} } & 0 \\ 0 & 0 & {B_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial y^{2} }}} \right)} \\ {\varepsilon_{yy}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial y^{2} }}} \right)} \\ {\gamma_{xy}^{0} - l^{2} \left( {\frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {D_{11}^{*} } & {D_{12}^{*} } & 0 \\ {D_{12}^{*} } & {D_{11}^{*} } & 0 \\ 0 & 0 & {D_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {G_{11}^{*} } & {G_{12}^{*} } & 0 \\ {G_{12}^{*} } & {G_{22}^{*} } & 0 \\ 0 & 0 & {G_{66}^{**} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ \end{aligned},$$
$$\begin{aligned} & \left\{ {\begin{array}{*{20}c} {R_{xx} - \mu^{2} \left( {\frac{{\partial^{2} R_{xx} }}{{\partial x^{2} }} + \frac{{\partial^{2} R_{xx} }}{{\partial y^{2} }}} \right)} \\ {R_{yy} - \mu^{2} \left( {\frac{{\partial^{2} R_{yy} }}{{\partial x^{2} }} + \frac{{\partial^{2} R_{yy} }}{{\partial y^{2} }}} \right)} \\ {R_{xy} - \mu^{2} \left( {\frac{{\partial^{2} R_{xy} }}{{\partial x^{2} }} + \frac{{\partial^{2} R_{xy} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad = \left[ {\begin{array}{*{20}c} {F_{11}^{*} } & {F_{12}^{*} } & 0 \\ {F_{12}^{*} } & {F_{22}^{*} } & 0 \\ 0 & 0 & {F_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{xx}^{0} }}{{\partial y^{2} }}} \right)} \\ {\varepsilon_{yy}^{0} - l^{2} \left( {\frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{yy}^{0} }}{{\partial y^{2} }}} \right)} \\ {\gamma_{xy}^{0} - l^{2} \left( {\frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial x^{2} }} + \frac{{\partial^{2} \gamma_{xy}^{0} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {G_{11}^{*} } & {G_{12}^{*} } & 0 \\ {G_{12}^{*} } & {G_{22}^{*} } & 0 \\ 0 & 0 & {G_{66}^{**} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 1 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 1 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ & \quad + \left[ {\begin{array}{*{20}c} {H_{11}^{*} } & {H_{12}^{*} } & 0 \\ {H_{12}^{*} } & {H_{22}^{*} } & 0 \\ 0 & 0 & {H_{66}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\kappa_{xx}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xx}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{yy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{yy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ {\kappa_{xy}^{\left( 2 \right)} - l^{2} \left( {\frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial x^{2} }} + \frac{{\partial^{2} \kappa_{xy}^{\left( 2 \right)} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} \\ \end{aligned},$$
$$\left\{ {\begin{array}{*{20}c} {Q_{x} - \mu^{2} \left( {\frac{{\partial^{2} Q_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} Q_{x} }}{{\partial y^{2} }}} \right)} \\ {Q_{y} - \mu^{2} \left( {\frac{{\partial^{2} Q_{y} }}{{\partial x^{2} }} + \frac{{\partial^{2} Q_{y} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {A_{44}^{*} } & 0 \\ 0 & {A_{55}^{*} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\psi_{x} - l^{2} \left( {\frac{{\partial^{2} \psi_{x} }}{{\partial x^{2} }} + \frac{{\partial^{2} \psi_{x} }}{{\partial y^{2} }}} \right)} \\ {\psi_{y} - l^{2} \left( {\frac{{\partial^{2} \psi_{y} }}{{\partial x^{2} }} + \frac{{\partial^{2} \psi_{y} }}{{\partial y^{2} }}} \right)} \\ \end{array} } \right\}.$$

Appendix 2

$$\varphi_{1} = \frac{{A_{11}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{2} = \frac{1}{{A_{66}^{*} }},\quad \varphi_{3} = \frac{{A_{12}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{4} = \frac{{A_{22}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},$$
$$\varphi_{5} = \frac{{A_{11}^{*} B_{12}^{*} - A_{12}^{*} B_{11}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{6} = \frac{{A_{11}^{*} B_{11}^{*} - A_{12}^{*} B_{12}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{7} = \frac{{A_{22}^{*} B_{22}^{*} - A_{12}^{*} B_{12}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{8} = \frac{{B_{66}^{*} }}{{A_{66}^{*} }},$$
$$\varphi_{9} = \frac{{A_{22}^{*} B_{12}^{*} - A_{12}^{*} B_{22}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{10} = \frac{{A_{11}^{*} F_{12}^{*} - A_{12}^{*} F_{11}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{11} = \frac{{A_{11}^{*} F_{11}^{*} - A_{12}^{*} F_{12}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{12} = \frac{{F_{66}^{*} }}{{A_{66}^{*} }},$$
$$\varphi_{13} = \frac{{A_{22}^{*} F_{12}^{*} - A_{12}^{*} F_{22}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{14} = \frac{{A_{22}^{*} F_{22}^{*} - A_{12}^{*} F_{12}^{*} }}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{15} = D_{11}^{*} - \frac{{A_{11}^{*} \left( {\left( {B_{11}^{*} } \right)^{2} + \left( {B_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},$$
$$\varphi_{16} = 2\left( {D_{12}^{*} + 2D_{66}^{*} } \right) - \frac{{A_{11}^{*} \left( {\left( {B_{11}^{*} } \right)^{2} + \left( {B_{12}^{*} } \right)^{2} } \right) + A_{22}^{*} \left( {\left( {B_{22}^{*} } \right)^{2} + \left( {B_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - 4\frac{{\left( {B_{66}^{*} } \right)^{2} }}{{A_{66}^{*} }},$$
$$\varphi_{17} = D_{22}^{*} - \frac{{A_{22}^{*} \left( {\left( {B_{22}^{*} } \right)^{2} + \left( {B_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} ,\quad \varphi_{18} = \frac{{A_{11}^{*} \left( {\left( {F_{11}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - G_{11}^{*},$$
$$\varphi_{19} = \frac{{2A_{11}^{*} F_{11}^{*} F_{12}^{*} - A_{12}^{*} \left( {\left( {F_{11}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} + 2\frac{{\left( {B_{66}^{**} } \right)^{2} }}{{A_{66}^{*} }} - \left( {G_{12}^{*} + 2G_{66}^{*} } \right),$$
$$\varphi_{20} = \frac{{A_{22}^{*} \left( {\left( {F_{22}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - G_{22}^{*},$$
$$\varphi_{21} = \frac{{2A_{22}^{*} F_{22}^{*} F_{12}^{*} - A_{12}^{*} \left( {\left( {F_{22}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} + 2\frac{{\left( {B_{66}^{**} } \right)^{2} }}{{A_{66}^{*} }} - \left( {G_{12}^{*} + 2G_{66}^{*} } \right),$$
$$\varphi_{22} = \frac{{A_{11}^{*} \left( {B_{11}^{*} F_{11}^{*} + B_{12}^{*} F_{12}^{*} } \right) - A_{12}^{*} \left( {B_{12}^{*} F_{11}^{*} + B_{11}^{*} F_{12}^{*} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - G_{11}^{*} ,$$
$$\varphi_{23} = \frac{{A_{11}^{*} \left( {B_{11}^{*} F_{12}^{*} + B_{12}^{*} F_{11}^{*} } \right) - A_{12}^{*} \left( {B_{12}^{*} F_{12}^{*} + B_{11}^{*} F_{11}^{*} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} + 2\frac{{B_{66}^{*} F_{66}^{*} }}{{A_{66}^{*} }} - \left( {G_{12}^{*} + 2G_{66}^{*} } \right),$$
$$\varphi_{24} = H_{11}^{*} - \frac{{A_{11}^{*} \left( {\left( {F_{11}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} ,\quad \varphi_{25} = H_{66}^{*} - \frac{{\left( {F_{66}^{*} } \right)^{2} }}{{A_{66}^{*} }},$$
$$\varphi_{26} = H_{12}^{*} + H_{66}^{*} + \frac{{A_{12}^{*} \left( {\left( {F_{11}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - \frac{{\left( {F_{66}^{*} } \right)^{2} }}{{A_{66}^{*} }},$$
$$\varphi_{27} = \frac{{A_{22}^{*} \left( {B_{22}^{*} F_{22}^{*} + B_{12}^{*} F_{12}^{*} } \right) - A_{12}^{*} \left( {B_{12}^{*} F_{22}^{*} + B_{22}^{*} B_{12}^{*} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - G_{22}^{*},$$
$$\varphi_{28} = \frac{{A_{22}^{*} \left( {B_{22}^{*} F_{12}^{*} + B_{12}^{*} F_{22}^{*} } \right) - A_{12}^{*} \left( {B_{12}^{*} F_{12}^{*} + B_{22}^{*} F_{22}^{*} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} + 2\frac{{B_{66}^{*} F_{66}^{*} }}{{A_{66}^{*} }} - \left( {G_{12}^{*} + 2G_{66}^{*} } \right),$$
$$\varphi_{29} = H_{22}^{*} - \frac{{A_{22}^{*} \left( {\left( {F_{22}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }},\quad \varphi_{30} = H_{12}^{*} + H_{66}^{*} + \frac{{A_{12}^{*} \left( {\left( {F_{22}^{*} } \right)^{2} + \left( {F_{12}^{*} } \right)^{2} } \right)}}{{A_{11}^{*} A_{22}^{*} - \left( {A_{12}^{*} } \right)^{2} }} - \frac{{\left( {F_{66}^{*} } \right)^{2} }}{{A_{66}^{*} }}.$$

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Sahmani, S., Fattahi, A.M. & Ahmed, N.A. Analytical treatment on the nonlocal strain gradient vibrational response of postbuckled functionally graded porous micro-/nanoplates reinforced with GPL. Engineering with Computers 36, 1559–1578 (2020). https://doi.org/10.1007/s00366-019-00782-5

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