An analytical solution for static stability of multi-scale hybrid nanocomposite plates

Abstract

An analytical answer to the buckling problem of a composite plate consisted of multi-scale hybrid nanocomposites is presented here for the first time. In other words, the constituent material of the structure is made of an epoxy matrix which is reinforced by both macro- and nanosize reinforcements, namely, carbon fiber (CF) and carbon nanotube (CNT). The effective material properties such as Young’s modulus or density are derived utilizing a micromechanical scheme incorporated with the Halpin–Tsai model. To present a more realistic problem, the plate is placed on a two-parameter elastic substrate. Then, on the basis of an energy-based Hamiltonian approach, the equations of motion are derived using the classical theory of plates. Finally, the governing equations are solved analytically to obtain the critical buckling load of the system. Afterward, the normalized form of the results is presented to emphasize the impact of each parameter on the dimensionless buckling load of composite plates. It is worth mentioning that the effects of various boundary conditions are covered, too. To show the efficiency of presented modeling, the results of this article are compared to those of former attempts.

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References

  1. 1.

    Ansari R, Torabi J, Faghih Shojaei M (2017) Buckling and vibration analysis of embedded functionally graded carbon nanotube-reinforced composite annular sector plates under thermal loading. Compos B Eng 109:197–213

    Google Scholar 

  2. 2.

    Barati MR, Zenkour AM (2017) Post-buckling analysis of refined shear deformable graphene platelet reinforced beams with porosities and geometrical imperfection. Compos Struct 181:194–202

    Google Scholar 

  3. 3.

    Catapano A, Giunta G, Belouettar S et al (2011) Static analysis of laminated beams via a unified formulation. Compos Struct 94:75–83

    Google Scholar 

  4. 4.

    Chakrapani SK, Barnard DJ, Dayal V (2016) Nonlinear forced vibration of carbon fiber/epoxy prepreg composite beams: theory and experiment. Compos B Eng 91:513–521

    Google Scholar 

  5. 5.

    Dabbagh A, Rastgoo A, Ebrahimi F (2019) Finite element vibration analysis of multi-scale hybrid nanocomposite beams via a refined beam theory. Thin Walled Struct 140:304–317

    Google Scholar 

  6. 6.

    Ebrahimi F, Dabbagh A (2019) On thermo-mechanical vibration analysis of multi-scale hybrid composite beams. J Vib Control 25:933–945

    MathSciNet  Google Scholar 

  7. 7.

    Ebrahimi F, Dabbagh A (2019) Vibration analysis of graphene oxide powder-/carbon fiber-reinforced multi-scale porous nanocomposite beams: a finite-element study. Eur Phys J Plus 134:225

    Google Scholar 

  8. 8.

    Ebrahimi F, Dabbagh A (2019) Vibration analysis of multi-scale hybrid nanocomposite plates based on a Halpin-Tsai homogenization model. Compos B Eng 173:106955

    Google Scholar 

  9. 9.

    Ebrahimi F, Farazmandnia N (2017) Thermo-mechanical analysis of carbon nanotube-reinforced composite sandwich beams. Coupled Syst Mech 6:207–227

    Google Scholar 

  10. 10.

    Ebrahimi F, Habibi S (2017) Low-velocity impact response of laminated FG-CNT reinforced composite plates in thermal environment. Adv Nano Res 5:69–97

    Google Scholar 

  11. 11.

    Ebrahimi F, Habibi S (2018) Nonlinear eccentric low-velocity impact response of a polymer-carbon nanotube-fiber multiscale nanocomposite plate resting on elastic foundations in hygrothermal environments. Mech Adv Mater Struct 25:425–438

    Google Scholar 

  12. 12.

    Ebrahimi F, Jafari A, Barati MR (2017) Vibration analysis of magneto-electro-elastic heterogeneous porous material plates resting on elastic foundations. Thin Walled Struct 119:33–46

    Google Scholar 

  13. 13.

    Elliott JA, Sandler JKW, Windle AH et al (2004) Collapse of single-wall carbon nanotubes is diameter dependent. Phys Rev Lett 92:095501

    Google Scholar 

  14. 14.

    Emam S, Eltaher MA (2016) Buckling and postbuckling of composite beams in hygrothermal environments. Compos Struct 152:665–675

    Google Scholar 

  15. 15.

    Fan Y, Xiang Y, Shen H-S et al (2018) Nonlinear low-velocity impact response of FG-GRC laminated plates resting on visco-elastic foundations. Compos B Eng 144:184–194

    Google Scholar 

  16. 16.

    Fantuzzi N, Tornabene F, Bacciocchi M et al (2017) Free vibration analysis of arbitrarily shaped functionally graded carbon nanotube-reinforced plates. Compos B Eng 115:384–408

    Google Scholar 

  17. 17.

    Feng C, Kitipornchai S, Yang J (2017) Nonlinear bending of polymer nanocomposite beams reinforced with non-uniformly distributed graphene platelets (GPLs). Compos B Eng 110:132–140

    Google Scholar 

  18. 18.

    García-Macías E, Rodríguez-Tembleque L, Castro-Triguero R et al (2017) Eshelby-Mori-Tanaka approach for post-buckling analysis of axially compressed functionally graded CNT/polymer composite cylindrical panels. Compos B Eng 128:208–224

    Google Scholar 

  19. 19.

    García-Macías E, Rodríguez-Tembleque L, Sáez A (2018) Bending and free vibration analysis of functionally graded graphene vs. carbon nanotube reinforced composite plates. Compos Struct 186:123–138

    Google Scholar 

  20. 20.

    Ghorbanpour Arani A, BabaAkbar Zarei H, Eskandari M et al (2017) Vibration behavior of visco-elastically coupled sandwich beams with magnetorheological core and three-phase carbon nanotubes/fiber/polymer composite facesheets subjected to external magnetic field. J Sandw Struct Mater. https://doi.org/10.1177/1099636217743177

    Article  Google Scholar 

  21. 21.

    Giunta G, Biscani F, Belouettar S et al (2013) Free vibration analysis of composite beams via refined theories. Compos B Eng 44:540–552

    Google Scholar 

  22. 22.

    Gunda JB, Gupta RK, Ranga Janardhan G et al (2011) Large amplitude vibration analysis of composite beams: simple closed-form solutions. Compos Struct 93:870–879

    Google Scholar 

  23. 23.

    Han Y, Elliott J (2007) Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites. Comput Mater Sci 39:315–323

    Google Scholar 

  24. 24.

    He XQ, Rafiee M, Mareishi S et al (2015) Large amplitude vibration of fractionally damped viscoelastic CNTs/fiber/polymer multiscale composite beams. Compos Struct 131:1111–1123

    Google Scholar 

  25. 25.

    Heshmati M, Yas MH, Daneshmand F (2015) A comprehensive study on the vibrational behavior of CNT-reinforced composite beams. Compos Struct 125:434–448

    Google Scholar 

  26. 26.

    Jam JE, Kiani Y (2015) Low velocity impact response of functionally graded carbon nanotube reinforced composite beams in thermal environment. Compos Struct 132:35–43

    Google Scholar 

  27. 27.

    Jin Y, Yuan FG (2003) Simulation of elastic properties of single-walled carbon nanotubes. Compos Sci Technol 63:1507–1515

    Google Scholar 

  28. 28.

    Ke L-L, Yang J, Kitipornchai S (2010) Nonlinear free vibration of functionally graded carbon nanotube-reinforced composite beams. Compos Struct 92:676–683

    Google Scholar 

  29. 29.

    Kiani Y, Mirzaei M (2018) Enhancement of non-linear thermal stability of temperature dependent laminated beams with graphene reinforcements. Compos Struct 186:114–122

    Google Scholar 

  30. 30.

    Lei ZX, Zhang LW, Liew KM (2015) Free vibration analysis of laminated FG-CNT reinforced composite rectangular plates using the kp-Ritz method. Compos Struct 127:245–259

    Google Scholar 

  31. 31.

    Lei ZX, Zhang LW, Liew KM (2016) Parametric analysis of frequency of rotating laminated CNT reinforced functionally graded cylindrical panels. Compos B Eng 90:251–266

    Google Scholar 

  32. 32.

    Liu D, Kitipornchai S, Chen W et al (2018) Three-dimensional buckling and free vibration analyses of initially stressed functionally graded graphene reinforced composite cylindrical shell. Compos Struct 189:560–569

    Google Scholar 

  33. 33.

    Mareishi S, Rafiee M, He XQ et al (2014) Nonlinear free vibration, postbuckling and nonlinear static deflection of piezoelectric fiber-reinforced laminated composite beams. Compos B Eng 59:123–132

    Google Scholar 

  34. 34.

    Phung-Van P, Abdel-Wahab M, Liew KM et al (2015) Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory. Compos Struct 123:137–149

    Google Scholar 

  35. 35.

    Rafiee M, Liu XF, He XQ et al (2014) Geometrically nonlinear free vibration of shear deformable piezoelectric carbon nanotube/fiber/polymer multiscale laminated composite plates. J Sound Vib 333:3236–3251

    Google Scholar 

  36. 36.

    Rafiee M, Nitzsche F, Labrosse M (2016) Rotating nanocomposite thin-walled beams undergoing large deformation. Compos Struct 150:191–199

    Google Scholar 

  37. 37.

    Shen H-S (2009) A comparison of buckling and postbuckling behavior of FGM plates with piezoelectric fiber reinforced composite actuators. Compos Struct 91:375–384

    Google Scholar 

  38. 38.

    Shen H-S, Xiang Y (2014) Postbuckling of axially compressed nanotube-reinforced composite cylindrical panels resting on elastic foundations in thermal environments. Compos B Eng 67:50–61

    Google Scholar 

  39. 39.

    Shen H-S, Xiang Y, Fan Y et al (2018) Nonlinear bending analysis of FG-GRC laminated cylindrical panels on elastic foundations in thermal environments. Compos B Eng 141:148–157

    Google Scholar 

  40. 40.

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

    Google Scholar 

  41. 41.

    Song M, Kitipornchai S, Yang J (2017) Free and forced vibrations of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos Struct 159:579–588

    Google Scholar 

  42. 42.

    Song M, Yang J, Kitipornchai S (2018) Bending and buckling analyses of functionally graded polymer composite plates reinforced with graphene nanoplatelets. Compos B Eng 134:106–113

    Google Scholar 

  43. 43.

    Song ZG, Zhang LW, Liew KM (2016) Dynamic responses of CNT reinforced composite plates subjected to impact loading. Compos B Eng 99:154–161

    Google Scholar 

  44. 44.

    Thostenson ET, Li WZ, Wang DZ et al (2002) Carbon nanotube/carbon fiber hybrid multiscale composites. J Appl Phys 91:6034–6037

    Google Scholar 

  45. 45.

    Tornabene F, Fantuzzi N, Bacciocchi M et al (2016) Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Compos B Eng 89:187–218

    Google Scholar 

  46. 46.

    Wattanasakulpong N, Chaikittiratana A (2015) Exact solutions for static and dynamic analyses of carbon nanotube-reinforced composite plates with Pasternak elastic foundation. Appl Math Model 39:5459–5472

    MathSciNet  MATH  Google Scholar 

  47. 47.

    Yang J, Wu H, Kitipornchai S (2017) Buckling and postbuckling of functionally graded multilayer graphene platelet-reinforced composite beams. Compos Struct 161:111–118

    Google Scholar 

  48. 48.

    Zarei H, Fallah M, Bisadi H et al (2017) Multiple impact response of temperature-dependent carbon nanotube-reinforced composite (CNTRC) plates with general boundary conditions. Compos B Eng 113:206–217

    Google Scholar 

  49. 49.

    Zhang LW, Liew KM (2015) Large deflection analysis of FG-CNT reinforced composite skew plates resting on Pasternak foundations using an element-free approach. Compos Struct 132:974–983

    Google Scholar 

  50. 50.

    Zhang LW, Liew KM, Reddy JN (2016) Postbuckling analysis of bi-axially compressed laminated nanocomposite plates using the first-order shear deformation theory. Compos Struct 152:418–431

    Google Scholar 

  51. 51.

    Zhu P, Lei ZX, Liew KM (2012) Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory. Compos Struct 94:1450–1460

    Google Scholar 

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Appendix

Appendix

The corresponding kij arrays of stiffness matrix introduced in Eq. (36) can be computed as

$$\begin{aligned} k_{11} & = A_{11} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + A_{66} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & + 2A_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$
$$\begin{aligned} k_{12} & = \left( {A_{12} + A_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & + A_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } \\ & + A_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$
$$\begin{aligned} k_{13} & = - B_{11} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & - \left( {B_{12} + 2B_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & - 3B_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } \\ & - B_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$
$$\begin{aligned} k_{21} & = \left( {A_{12} + A_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } \\ & + A_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + A_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$
$$\begin{aligned} k_{22} & = A_{66} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } \\ & + 2A_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & + A_{22} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$
$$\begin{aligned} k_{23} & = - B_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & - \left( {B_{12} + 2B_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial Y_{n} (y)}}{\partial y}{\text{d}}x{\text{d}}y} } \\ & - 3B_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial X_{m} (x)}}{\partial x}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & - B_{22} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$
$$\begin{aligned} k_{31} & = B_{11} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{4} X_{m} (x)}}{{\partial x^{4} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + \left( {B_{12} + 2B_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & + 3B_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + B_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$
$$\begin{aligned} k_{32} & = B_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + \left( {B_{12} + 2B_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & + 3B_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } \\ & + B_{22} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{4} Y_{n} (y)}}{{\partial y^{4} }}{\text{d}}x{\text{d}}y} } , \\ \end{aligned}$$
$$\begin{aligned} k_{33} & = - D_{11} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{4} X_{m} (x)}}{{\partial x^{4} }}Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & - 2\left( {D_{12} + 2D_{66} } \right)\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } \\ & - 4D_{16} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)\frac{{\partial Y_{n} (y)}}{\partial y}\frac{{\partial^{3} X_{m} (x)}}{{\partial x^{3} }}Y_{n} (y){\text{d}}x{\text{d}}y} } - 4D_{26} \int\limits_{0}^{b} {\int\limits_{0}^{a} {\frac{{\partial X_{m} (x)}}{\partial x}Y_{n} (y)X_{m} (x)\frac{{\partial^{3} Y_{n} (y)}}{{\partial y^{3} }}{\text{d}}x{\text{d}}y} } \\ & - D_{22} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{4} Y_{n} (y)}}{{\partial y^{4} }}{\text{d}}x{\text{d}}y} } - k_{w} \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)Y_{n} (y){\text{d}}x{\text{d}}y} } \\ & + \left( {k_{p} - N^{b} } \right)\left( {\int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)\frac{{\partial^{2} X_{m} (x)}}{{\partial x^{2} }}Y_{n} (y){\text{d}}x{\text{d}}y} } + \int\limits_{0}^{b} {\int\limits_{0}^{a} {X_{m} (x)Y_{n} (y)X_{m} (x)\frac{{\partial^{2} Y_{n} (y)}}{{\partial y^{2} }}{\text{d}}x{\text{d}}y} } } \right). \\ \end{aligned}$$
(A1)

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Ebrahimi, F., Dabbagh, A. An analytical solution for static stability of multi-scale hybrid nanocomposite plates. Engineering with Computers 37, 545–559 (2021). https://doi.org/10.1007/s00366-019-00840-y

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Keywords

  • Static stability
  • Multi-scale hybrid nanocomposites
  • Halpin–Tsai homogenization model