Buckling analysis of coupled DLGSs systems resting on elastic medium using sinusoidal shear deformation orthotropic plate theory

Abstract

This study proposes a solution for the analysis of critical buckling load of a coupled double-layer graphene sheets (DLGSs) system resting on elastic medium. In the upper and lower parts of the system, two graphene layers, which are under van der Waals force and related to each other through an elastic medium on the basis of Pasternak and Winkler models, are utilized. The performed model is considered to be based on Eringen’s nonlocal elasticity theory and sinusoidal nonlocal shear deformation theory. In addition, the properties of orthotropic plate are applied in the model. Constitutive relations and equations together with boundary conditions are derived on the basis of Hamilton’s principle. Furthermore, the effect of surface stress according to Gurtin–Murdoch theory is considered. Moreover, critical buckling load for in-phase, out-of-phase, and one side fixed states subjected to in-plane forces is obtained. The results of the study reveal the minimum value of critical buckling load for in-phase state and the maximum value of that for out-of-phase state, and that the value of one side fixed state critical buckling load exists between these two states.

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References

  1. 1.

    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 228:1–9. doi:10.1177/0954406214521391

    Article  Google Scholar 

  2. 2.

    Jomehzadeh E, Saidi AR (2011) A study on large amplitude vibration of multilayered graphene sheets. Comput Mater Sci 50:1043–1051

    Article  Google Scholar 

  3. 3.

    Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Eringen AC (2002) Nonlocal continuum field theories. Springer, New York

    MATH  Google Scholar 

  5. 5.

    Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710

    Article  Google Scholar 

  6. 6.

    Ouakad HM, Sedighi HM (2016) Rippling effect on the structural response of electrostatically actuated single-walled carbon nanotube based NEMS actuators. Int J Non Linear Mech 87:97–108

    Article  Google Scholar 

  7. 7.

    Hashemi SH, Samaei AT (2011) Buckling analysis of micro/nanoscale plates via nonlocal elasticity theory. Phys E Low-Dimens Syst Nanostructures 43:1400–1404

    Article  Google Scholar 

  8. 8.

    Touratier M (1991) An efficient standard plate theory. Int J Eng Sci 29:901–916

    Article  MATH  Google Scholar 

  9. 9.

    Pradhan SC (2009) Buckling of single layer graphene sheet based on nonlocal elasticity and higher order shear deformation theory. Phys Lett A 373:4182–4188

    Article  MATH  Google Scholar 

  10. 10.

    Zhang Y, Zhang LW, Liew KM, Yu JL (2016) Buckling analysis of graphene sheets embedded in an elastic medium based on the kp-Ritz method and non-local elasticity theory. Eng Anal Bound Elem 70:31–39

    MathSciNet  Article  Google Scholar 

  11. 11.

    Radić N, Jeremić D (2016) Thermal buckling of double-layered graphene sheets embedded in an elastic medium with various boundary conditions using a nonlocal new first-order shear deformation theory. Compos Part B Eng 97:201–215

    Article  Google Scholar 

  12. 12.

    Arani AG, Kolahchi R, Vossough H (2012) Buckling analysis and smart control of SLGS using elastically coupled PVDF nanoplate based on the nonlocal Mindlin plate theory. Phys B Condens Matter 407:4458–4465

    Article  Google Scholar 

  13. 13.

    Shokrani MH, Karimi M, Tehrani MS, Mirdamadi HR (2015) Buckling analysis of double-orthotropic nanoplates embedded in elastic media based on non-local two-variable refined plate theory using the GDQ method. J Braz Soc Mech Sci Eng. doi:10.1007/s40430-015-0370-0

    Google Scholar 

  14. 14.

    Murmu T, Pradhan SC (2009) Buckling of biaxially compressed orthotropic plates at small scales. Mech Res Commun 36:933–938

    Article  MATH  Google Scholar 

  15. 15.

    Soleimani A, Naei MH, Mashhadi MM (2016) Buckling analysis of graphene sheets using nonlocal isogeometric finite element method for NEMS applications. Microsyst Technol. doi:10.1007/s00542-016-3098-6

    Google Scholar 

  16. 16.

    Arani AG, Jalaei MH (2016) Transient behavior of an orthotropic graphene sheet resting on orthotropic visco-Pasternak foundation. Int J Eng Sci 103:97–113

    MathSciNet  Article  Google Scholar 

  17. 17.

    Ansari R, Sahmani S (2013) Prediction of biaxial buckling behavior of single-layered graphene sheets based on nonlocal plate models and molecular dynamics simulations. Appl Math Model 37:7338–7351

    MathSciNet  Article  Google Scholar 

  18. 18.

    Arani AG, Shiravand A, Rahi M, Kolahchi R (2012) Nonlocal vibration of coupled DLGS systems embedded on visco-Pasternak foundation. Phys B Condens Matter 407:4123–4131

    Article  Google Scholar 

  19. 19.

    Natsuki T (2015) Theoretical analysis of vibration frequency of graphene sheets used as nanomechanical mass sensor. Electronics 4:723–738

    Article  Google Scholar 

  20. 20.

    Farajpour A, Solghar AA, Shahidi A (2013) Postbuckling analysis of multi-layered graphene sheets under non-uniform biaxial compression. Phys E Low-Dimens Syst Nanostructures 47:197–206

    Article  Google Scholar 

  21. 21.

    Arani AG, Kolahchi R, Barzoki AAM, Mozdianfard MR, Farahani SMN (2012) Elastic foundation effect on nonlinear thermo-vibration of embedded double layered orthotropic graphene sheets using differential quadrature method. Proc Inst Mech Eng Part C J Mech Eng Sci 227:862–879

    Article  Google Scholar 

  22. 22.

    Pradhan SC, Kumar A (2011) Vibration analysis of orthotropic graphene sheets using nonlocal elasticity theory and differential quadrature method. Compos Struct 93:774–779

    Article  Google Scholar 

  23. 23.

    Sedighi HM (2015) Modeling of surface stress effects on the dynamic behavior of actuated non-classical nano-bridges. Trans Can Soc Mech Eng 39:137–151

    Google Scholar 

  24. 24.

    Ansari R, Sahmani S (2011) Surface stress effects on the free vibration behavior of nanoplates. Int J Eng Sci 49:1204–1215

    MathSciNet  Article  Google Scholar 

  25. 25.

    Ansari R, Mohammadi V, Shojaei MF, Gholami R, Darabi MA (2014) A geometrically non-linear plate model including surface stress effect for the pull-in instability analysis of rectangular nanoplates under hydrostatic and electrostatic actuations. Int J Non Linear Mech 67:16–26

    Article  Google Scholar 

  26. 26.

    Mohammadimehr M, Najafabadi MMM, Nasiri H, Navi BR (2014) Surface stress effects on the free vibration and bending analysis of the nonlocal single-layer graphene sheet embedded in an elastic medium using energy method. Proc Inst Mech Eng Part N J Nanoeng Nanosyst 230:148–160

    Google Scholar 

  27. 27.

    Farajpour A, Dehghany M, Shahidi AR (2013) Surface and nonlocal effects on the axisymmetric buckling of circular graphene sheets in thermal environment. Compos Part B Eng 50:333–343

    Article  Google Scholar 

  28. 28.

    Gurtin ME, Murdoch AI (1978) Surface stress in solids. Int J Solids Struct 14:431–440

    Article  MATH  Google Scholar 

  29. 29.

    Gurtin ME, Murdoch AI (1975) A continuum theory of elastic material surfaces. Arch Ration Mech Anal 57:291–323

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

    Thai H-T, Vo TP (2013) A new sinusoidal shear deformation theory for bending, buckling, and vibration of functionally graded plates. Appl Math Model 37:3269–3281

    MathSciNet  Article  MATH  Google Scholar 

  31. 31.

    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

    Article  Google Scholar 

  32. 32.

    Murmu T, Sienz J, Adhikari S, Arnold C (2013) Nonlocal buckling of double-nanoplate-systems under biaxial compression. Compos Part B 44:84–94

    Article  Google Scholar 

  33. 33.

    Kim S-E, Thai H-T, Lee J (2009) A two variable refined plate theory for laminated composite plates. Compos Struct 89:197–205

    Article  Google Scholar 

  34. 34.

    Kheirikhah MM, Babaghasabha V (2016) Bending and buckling analysis of corrugated composite sandwich plates. J Braz Soc Mech Sci Eng. doi:10.1007/s40430-016-0498-6

    Google Scholar 

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Correspondence to Mehdi Khajehdehi Kavanroodi.

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Technical Editor: Eduardo Alberto Fancello.

Appendix

Appendix

The elements of matrix A are obtained by deriving \(u_{i} , v_{i} , w_{i} , \phi_{x i} , \phi_{y i}\) coefficients from the following relations, regarding the states of in-phase, out-of-phase and one side fixed.

$$\delta u_{i} : \left[ {\left[ {h\left( {C_{11} - \frac{{C_{13}^{2} }}{{C_{33} }}} \right) + 2C^{S}_{11} } \right]\left( {\frac{m}{l}} \right)^{2} + \left[ {C_{66} h + 2C^{S}_{66} } \right]\left( {\frac{n}{b}} \right)^{2} } \right]u_{imn} + \left[ {h\left( {C_{12} - \frac{{C_{13} C_{32} }}{{C_{33} }}} \right) + 2C^{S}_{12} + C_{66} h + 2C^{S}_{66} } \right]v_{imn} \left( {\frac{mn}{lb}} \right) = 0$$
(24)
$$\delta v_{i}:\left[ {\left[ {C_{66} h + 2C_{66}^{\text{S}} } \right]\left( {\frac{m}{l}} \right)^{2} + \left[ {h\left( {C_{22} - \frac{{C_{23}^{2} }}{{C_{33} }}} \right) + 2C_{66}^{\text{S}} } \right]\left( {\frac{n}{b}} \right)^{2} } \right]v_{imn} + \left[ {C_{66} h + 2C_{66}^{\text{S}} + h\left( {C_{21} - \frac{{C_{23} C_{31} }}{{C_{33} }}} \right) + 2C_{21}^{\text{S}} } \right]u_{imn} \left( {\frac{mn}{lb}} \right) = 0$$
(25)
$$\delta {w_1}: \left[ { - {{{h^3}} \over {12}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{{{C_{13}}} \over {{C_{33}}}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}{C^{\rm{S}}}_{11} - \mu {\gamma _1}{N_{cr}}} \right]{w_{1mn}}{\left( {{{m\pi } \over l}} \right)^4} + \left[ { - {{{h^3}} \over {12}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{{{C_{23}}} \over {{C_{33}}}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}C_{22}^{\rm{S}} - \mu {\gamma _2}{N_{cr}}} \right]{w_{1mn}}{\left( {{{n\pi } \over b}} \right)^4} + \cdots \left[ { - {{{h^3}} \over 6}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}\left( {{{{C_{13}}} \over {{C_{33}}}} + {{{C_{23}}} \over {{C_{33}}}}} \right){\tau ^{\rm{s}}} - {h^2}C_{12}^{\rm{S}} - {{{h^3}{C_{66}}} \over 3} - 2{h^2}C_{66}^{\rm{S}} - \mu {\gamma _1}{N_{cr}} - \mu {\gamma _2}{N_{cr}}} \right]{w_{1mn}}\left( {{{{m^2}{n^2}{\pi ^4}} \over {{l^2}{b^2}}}} \right) + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{11}^{\rm{S}}} \right]{\phi _{1xmn}}{\left( {{{m\pi } \over l}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{22}^{\rm{S}}} \right]{\phi _{1ymn}}{\left( {{{n\pi } \over b}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{12}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{1ymn}}\left( {{{{m^2}n{\pi ^3}} \over {{l^2}b}}} \right) + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{21}} - {{{C_{23}}{C_{31}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{21}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{1xmn}}\left( {{{m{n^2}{\pi ^3}} \over {l{b^2}}}} \right) - \left[ {{\gamma _1}{N_{cr}} - {c_{12}}\mu } \right]{w_{1mn}}{\left( {{{m\pi } \over l}} \right)^2} - \left[ {{\gamma _2}{N_{cr}} - {c_{12}}\mu } \right]{w_{1mn}}{\left( {{{n\pi } \over b}} \right)^2} - {c_{12}}\mu {w_{2mn}}{\left( {{{m\pi } \over l}} \right)^2} - {c_{12}}\mu {w_{2mn}}{\left( {{{n\pi } \over b}} \right)^2} + {c_{12}}\left( {{w_{1mn}} - {w_{2mn}}} \right) = 0$$
(26)
$$\delta w_{2}: \left[ { - \frac{{h^{3} }}{12}\left( {C_{11} - \frac{{C_{13}^{2} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{6} \frac{{C_{13} }}{{C_{33} }} \tau^{\text{s}} - \frac{{h^{2} }}{2}C_{11}^{\text{S}} - G_{23} \mu - \mu \gamma_{1} N_{cr} } \right]w_{2mn} \left( {\frac{m\pi }{l}} \right)^{4} + \left[ { - \frac{{h^{3} }}{12}\left( {C_{22} - \frac{{C_{23}^{2} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{6} \frac{{C_{23} }}{{C_{33} }}\tau^{\text{s}} - \frac{{h^{2} }}{2}C_{22}^{\text{S}} - G_{23} \mu - \mu \gamma_{2} N_{cr} } \right]w_{2mn} \left( {\frac{n\pi }{b}} \right)^{4} + \left[ { - \frac{{h^{3} }}{6}\left( {C_{12} - \frac{{C_{13} C_{32} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{6}\left( { \frac{{C_{13} }}{{C_{33} }} + \frac{{C_{23} }}{{C_{33} }}} \right)\tau^{\text{s}} - h^{2} C_{12}^{\text{S}} - \frac{{h^{3} C_{66} }}{3} - 2h^{2} C_{66}^{\text{S}} } \right] - 2G_{23} \mu - \mu \gamma_{1} N_{cr} - \mu \gamma_{2} N_{cr} w_{2mn} \left( {\frac{{m^{2} n^{2} \pi^{4} }}{{l^{2} b^{2} }}} \right) + \left[ {\frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{11} - \frac{{C_{13}^{2} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{\pi }C_{11}^{\text{S}} } \right]\phi_{2xmn} \left( {\frac{m\pi }{l}} \right)^{3} + \left[ {\frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{22} - \frac{{C_{23}^{2} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{\pi }C_{22}^{\text{S}} } \right]\phi_{2ymn} \left( {\frac{n\pi }{b}} \right)^{3} + \left[ {\frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{12} - \frac{{C_{13} C_{32} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{\pi }C_{12}^{\text{S}} + \frac{{4h^{3} C_{66} }}{{\pi^{3} }} + \frac{{2h^{2} C_{66}^{\text{S}} }}{\pi }} \right]\phi_{2ymn} \left( {\frac{{m^{2} n\pi^{3} }}{{l^{2} b}}} \right) + \left[ {\frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{21} - \frac{{C_{23} C_{31} }}{{C_{33} }}} \right) + \frac{{h^{2} }}{\pi }C_{21}^{\text{S}} + \frac{{4h^{3} C_{66} }}{{\pi^{3} }} + \frac{{2h^{2} C_{66}^{\text{S}} }}{\pi }} \right]\phi_{2xmn} \left( {\frac{{mn^{2} \pi^{3} }}{{lb^{2} }}} \right) - c_{21} \mu w_{1mn} \left( {\frac{m\pi }{l}} \right)^{2} - \left[ {\gamma_{1} N_{cr} + G_{23} - c_{21} \mu + K_{23} \mu } \right]w_{2mn} \left( {\frac{m\pi }{l}} \right)^{2} + \left[ {K_{23} \mu + G_{23} } \right]w_{3mn} \left( {\frac{m\pi }{l}} \right)^{2} - c_{21} \mu w_{1mn} \left( {\frac{n\pi }{b}} \right)^{2} - \left[ {\gamma_{2} N_{cr} + G_{23} - c_{21} \mu + K_{23} \mu } \right]w_{2mn} \left( {\frac{n\pi }{b}} \right)^{2} + \left[ {K_{23} \mu + G_{23} } \right]w_{3mn} \left( {\frac{n\pi }{l}} \right)^{2} + G_{23} \mu w_{3mn} \left( {\frac{m\pi }{l}} \right)^{4} + 2G_{23} \mu w_{3mn} \left( {\frac{{m^{2} n^{2} \pi^{4} }}{{l^{2} b^{2} }}} \right) + G_{23} \mu w_{3mn} \left( {\frac{n\pi }{b}} \right)^{4} + c_{21} \left( {w_{2mn} - w_{1mn} } \right) - K_{23} \left( {w_{2mn} - w_{3mn} } \right) = 0$$
(27)
$$\delta {w_3}: \left[ { - {{{h^3}} \over {12}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{\rm{}}{{{C_{13}}} \over {{C_{33}}}}{\rm{}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}C_{11}^{\rm{S}} - {G_{32}}\mu - \mu {\gamma _1}{N_{cr}}} \right]{w_{3mn}}{\left( {{{m\pi } \over l}} \right)^4} + \left[ { - {{{h^3}} \over {12}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{\rm{}}{{{C_{23}}} \over {{C_{33}}}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}C_{22}^{\rm{S}} - {G_{32}}\mu - \mu {\gamma _2}{N_{cr}}} \right]{w_{3mn}}{\left( {{{n\pi } \over b}} \right)^4} + \left[ { - {{{h^3}} \over 6}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}\left( {{\rm{}}{{{C_{13}}} \over {{C_{33}}}} + {{{C_{23}}} \over {{C_{33}}}}} \right){\tau ^{\rm{s}}} - {h^2}C_{12}^{\rm{S}} - {{{h^3}{C_{66}}} \over 3} - 2{h^2}C_{66}^{\rm{S}} - 2{G_{32}}\mu - \mu {\gamma _1}{N_{cr}} - \mu {\gamma _2}{N_{cr}}} \right]{w_{3mn}}\left( {{{{m^2}{n^2}{\pi ^4}} \over {{l^2}{b^2}}}} \right) + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{11}^{\rm{S}}} \right]{\phi _{3xmn}}{\left( {{{m\pi } \over l}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{22}^{\rm{S}}} \right]{\phi _{3ymn}}{\left( {{{n\pi } \over b}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{12}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{3ymn}}\left( {{{{m^2}n{\pi ^3}} \over {{l^2}b}}} \right) + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{21}} - {{{C_{23}}{C_{31}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{21}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{3xmn}}\left( {{{m{n^2}{\pi ^3}} \over {l{b^2}}}} \right) - {c_{32}}\mu {\rm{}}{w_{4mn}}{\left( {{{m\pi } \over l}} \right)^2} - \left[ {{\gamma _1}{N_{cr}} + {G_{32}} - {c_{32}}\mu + {K_{32}}\mu } \right]{w_{3mn}}{\left( {{{m\pi } \over l}} \right)^2} + \left[ {{K_{32}}\mu + {G_{32}}} \right]{w_{2mn}}{\left( {{{m\pi } \over l}} \right)^2} - {c_{32}}\mu {w_{4mn}}{\left( {{{n\pi } \over b}} \right)^2} - \left[ {{\gamma _2}{N_{cr}} + {G_{32}} - {c_{32}}\mu + {K_{32}}\mu } \right]{w_{3mn}}{\left( {{{n\pi } \over b}} \right)^2} + \left[ {{K_{32}}\mu + {G_{32}}} \right]{w_{2mn}}{\left( {{{n\pi } \over b}} \right)^2} + {G_{32}}\mu {w_{2mn}}{\left( {{{m\pi } \over l}} \right)^4} + 2{G_{32}}\mu {w_{2mn}}\left( {{{{m^2}{n^2}{\pi ^4}} \over {{l^2}{b^2}}}} \right) + {G_{32}}\mu {w_{2mn}}{\left( {{{n\pi } \over b}} \right)^4} + {c_{34}}\left( {{w_{3mn}} - {w_{4mn}}} \right){\rm{}} - {K_{32}}\left( {{w_{3mn}} - {w_{2mn}}} \right){\rm{}} = 0$$
(28)
$$\delta {w_4}:\left[ { - {{{h^3}} \over {12}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{\rm{}}{{{C_{13}}} \over {{C_{33}}}}{\rm{}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}C_{11}^{\rm{S}} - \mu {\gamma _1}{N_{cr}}} \right]{w_{4mn}}{\left( {{{m\pi } \over l}} \right)^4} + \left[ { - {{{h^3}} \over {12}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}{\rm{}}{{{C_{23}}} \over {{C_{33}}}}{\tau ^{\rm{s}}} - {{{h^2}} \over 2}C_{22}^{\rm{S}} - \mu {\gamma _2}{N_{cr}}} \right]{w_{4mn}}{\left( {{{n\pi } \over b}} \right)^4} + \left[ { - {{{h^3}} \over 6}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over 6}\left( {{\rm{}}{{{C_{13}}} \over {{C_{33}}}} + {{{C_{23}}} \over {{C_{33}}}}} \right){\tau ^{\rm{s}}} - {h^2}C_{12}^{\rm{S}} - {{{h^3}{C_{66}}} \over 3} - 2{h^2}C_{66}^{\rm{S}} - \mu {\gamma _1}{N_{cr}} - \mu {\gamma _2}{N_{cr}}} \right]{w_{4mn}}\left( {{{{m^2}{n^2}{\pi ^4}} \over {{l^2}{b^2}}}} \right) + \cdots \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{11}} - {{{C_{13}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{11}^{\rm{S}}} \right]{\phi _{4xmn}}{\left( {{{m\pi } \over l}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{22}} - {{{C_{23}}^2} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{22}^{\rm{S}}} \right]{\phi _{4ymn}}{\left( {{{n\pi } \over b}} \right)^3} + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{12}} - {{{C_{13}}{C_{32}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{12}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{4ymn}}\left( {{{{m^2}n{\pi ^3}} \over {{l^2}b}}} \right) + \left[ {{{2{h^3}} \over {{\pi ^3}}}\left( {{C_{21}} - {{{C_{23}}{C_{31}}} \over {{C_{33}}}}} \right) + {{{h^2}} \over \pi }C_{21}^{\rm{S}} + {{4{h^3}{C_{66}}} \over {{\pi ^3}}} + {{2{h^2}C_{66}^{\rm{S}}} \over \pi }} \right]{\phi _{4xmn}}\left( {{{m{n^2}{\pi ^3}} \over {l{b^2}}}} \right) - \left[ {{\gamma _1}{N_{cr}} - {c_{43}}\mu } \right]{w_{4mn}}{\left( {{{m\pi } \over l}} \right)^2} - \left[ {{\gamma _2}{N_{cr}} - {c_{43}}\mu } \right]{w_{4mn}}{\left( {{{n\pi } \over b}} \right)^2} - {c_{43}}\mu {w_{3mn}}{\left( {{{m\pi } \over l}} \right)^2} - {c_{43}}\mu {w_{3mn}}{\left( {{{n\pi } \over b}} \right)^2} + {c_{43}}\left( {{w_{4mn}} - {w_{3mn}}} \right) = 0.$$
(29)
$$\delta \phi_{x i}: \left[ {- \frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{11} - \frac{{C_{13}^{2}}}{{C_{33} }}} \right) + \frac{{4h^{2} }}{{\pi^{3} }} \frac{{C_{13}}}{{C_{33} }}\tau^{\text{s}} - \frac{{h^{2} }}{\pi}C_{11}^{\text{S}} } \right]w_{imn} \left( {\frac{m\pi }{l}}\right)^{3} + \left[ { - \frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{12}- \frac{{C_{13} C_{32} }}{{C_{33} }}} \right) + \frac{{4h^{2}}}{{\pi^{3} }} \frac{{C_{13} }}{{C_{33} }} \tau^{\text{s}} -\frac{{h^{2} }}{\pi }C_{12}^{\text{S}} - \frac{{4h^{3} C_{66}}}{{\pi^{3} }}} \right] - \frac{{2h^{2} C_{66}^{\text{S}} }}{\pi}w_{imn} \left( {\frac{{mn^{2} \pi^{3} }}{{lb^{2} }}} \right) +\left[ {\frac{{h^{3} }}{{2\pi^{2} }}\left( {C_{11} -\frac{{C_{13}^{2} }}{{C_{33} }}} \right) + \frac{{2h^{2} }}{{\pi^{2}}}C_{11}^{\text{S}} } \right]\phi_{ixmn} \left( {\frac{m\pi }{l}}\right)^{2} + \left[ {\frac{{h^{3} }}{{2\pi^{2} }}\left( {C_{12} -\frac{{C_{13} C_{32} }}{{C_{33} }}} \right) + \frac{{2h^{2}}}{{\pi^{2} }}C_{12}^{\text{S}} + \frac{{h^{3} C_{66} }}{{2\pi^{2}}} + \frac{{2h^{2} C_{66}^{\text{S}} }}{{\pi^{2} }}}\right]\phi_{iymn} \left( {\frac{{mn\pi^{2} }}{lb}} \right) + \left[{\frac{{h^{3} C_{66} }}{{2\pi^{2} }} + \frac{{2h^{2}C_{66}^{\text{S}} }}{{\pi^{2} }}} \right]\phi_{ixmn} \left({\frac{n\pi }{b}} \right)^{2} + \frac{{C_{55} h}}{2} \phi_{ixmn} = 0$$
(30)
$$\delta \phi_{y i}: \left[ { - \frac{{2h^{3}}}{{\pi^{3} }}\left( {C_{22} - \frac{{C_{23}^{2} }}{{C_{33} }}}\right) + \frac{{4h^{2} }}{{\pi^{3} }} \frac{{C_{23} }}{{C_{33} }}\tau^{\text{s}} - \frac{{h^{2} }}{\pi }C_{22}^{\text{S}} }\right]w_{imn} \left( {\frac{n\pi }{b}} \right)^{3} + \left[ { -\frac{{2h^{3} }}{{\pi^{3} }}\left( {C_{21} - \frac{{C_{23} C_{31}}}{{C_{33} }}} \right) + \frac{{4h^{2} }}{{\pi^{3} }} \frac{{C_{23}}}{{C_{33} }} \tau^{\text{s}} - \frac{{h^{2} }}{\pi}C_{21}^{\text{S}} - \frac{{4h^{3} C_{66} }}{{\pi^{3} }}} \right] -\frac{{2h^{2} C_{66}^{\text{S}} }}{\pi }w_{imn} \left( {\frac{{m^{2}n\pi^{3} }}{{l^{2} b}}} \right) + \left[ {\frac{{h^{3} C_{66}}}{{2\pi^{2} }} + \frac{{2h^{2} C_{66}^{\text{S}} }}{{\pi^{2} }}}\right]\phi_{iymn} \left( {\frac{m\pi }{l}} \right)^{2} + \left[{\frac{{h^{3} }}{{2\pi^{2} }}\left( {C_{21} - \frac{{C_{23} C_{31}}}{{C_{33} }}} \right) + \frac{{2h^{2} }}{{\pi^{2}}}C_{21}^{\text{S}} + \frac{{h^{3} C_{66} }}{{2\pi^{2} }} +\frac{{2h^{2} C_{66}^{\text{S}} }}{{\pi^{2} }}} \right]\phi_{ixmn}\left( {\frac{{mn\pi^{2} }}{lb}} \right) + \left[ {\frac{{h^{3}}}{{2\pi^{2} }}\left( {C_{22} - \frac{{C_{23}^{2} }}{{C_{33} }}}\right) + \frac{{2h^{2} }}{{\pi^{2} }}C_{22}^{\text{S}} }\right]\phi_{iymn} \left( {\frac{n\pi }{b}} \right)^{2} +\frac{{C_{44} h}}{2} \phi_{iymn} = 0.$$
(31)

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Khajehdehi Kavanroodi, M., Fereidoon, A. & Mirafzal, A.R. Buckling analysis of coupled DLGSs systems resting on elastic medium using sinusoidal shear deformation orthotropic plate theory. J Braz. Soc. Mech. Sci. Eng. 39, 2817–2829 (2017). https://doi.org/10.1007/s40430-017-0784-y

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Keywords

  • Critical buckling load
  • Double-layer graphene sheets
  • Sinusoidal shear deformation theory
  • Orthotropic plate
  • Surface stress effect