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Vibrational behavior of truncated conical porous GPL-reinforced sandwich micro/nano-shells

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Abstract

In this paper, dynamic characteristics of microconical sandwich shells are investigated. The microshell is considered to consist of a porous core made of a polymer and two face sheets made of that polymer which is reinforced by graphene nanoplatelets (GPLs). The face sheets effective mechanical characteristics are estimated utilizing the rule of mixture along with the Halpin–Tsai model, and size effects are incorporated based on the modified couple stress theory. Hamilton’s principle is applied for derivation of governing equations of motion as well as boundary condition in which differential quadrature method is employed for numerical solution. The accuracy of the presented solution is examined using the benchmark results reported in other papers. Influences of different parameters on the natural frequencies in the various vibrational modes of the microshells are examined, including the wave number, micro-length-scale parameter, the thickness of the porous core, dispersion patterns of the pores and the GPLs, porosity parameter, total mass fraction of the GPLs, and also the semi-vertex angle of the cone.

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References

  1. Areias P, Rabczuk T, Msekh M (2016) Phase-field analysis of finite-strain plates and shells including element subdivision. Comput Methods Appl Mech Eng 312:322–350

    MathSciNet  MATH  Google Scholar 

  2. Areias P, Rabczuk T (2013) Finite strain fracture of plates and shells with configurational forces and edge rotations. Int J Numer Methods Eng 94:1099–1122

    MathSciNet  MATH  Google Scholar 

  3. Rabczuk T, Gracie R, Song JH, Belytschko T (2010) Immersed particle method for fluid–structure interaction. Int J Numer Methods Eng 81:48–71

    MathSciNet  MATH  Google Scholar 

  4. Sahmani S, Ansari R, Gholami R, Darvizeh A (2013) Dynamic stability analysis of functionally graded higher-order shear deformable microshells based on the modified couple stress elasticity theory. Compos B Eng 51:44–53

    MATH  Google Scholar 

  5. Zeighampour H, Tadi Beni Y (2014) Analysis of conical shells in the framework of coupled stresses theory. Int J Eng Sci 81:107–122

    MathSciNet  MATH  Google Scholar 

  6. Lou J, He L, Wu H, Du J (2016) Pre-buckling and buckling analyses of functionally graded microshells under axial and radial loads based on the modified couple stress theory. Compos Struct 142:226–237

    Google Scholar 

  7. Mou B, Bai Y (2018) Experimental investigation on shear behavior of steel beam-to-CFST column connections with irregular panel zone. Eng Struct 168:487–504

    Google Scholar 

  8. Zhang X, Tang Y, Zhang F, Lee C (2016) A novel aluminium–graphite dual-ion battery. Adv Energy Mater 6(11):1502588

    Google Scholar 

  9. Farokhi H, Ghayesh MH (2018) Nonlinear mechanical behaviour of microshells. Int J Eng Sci 127:127–144

    MathSciNet  MATH  Google Scholar 

  10. Heidari Y, Arefi M, Irani Rahaghi M (2020) Nonlocal vibration characteristics of a functionally graded porous cylindrical nanoshell integrated with arbitrary arrays of piezoelectric elements. Mech Based Des Struct Mach. https://doi.org/10.1080/15397734.2020.1830799

    Article  Google Scholar 

  11. Yuan Y, Zhao K, Zhao Y, Sahmani S, Safaei B (2020) Couple stress-based nonlinear buckling analysis of hydrostatic pressurized functionally graded composite conical microshells. Mech Mater 148:103507

    Google Scholar 

  12. Yuan Y, Zhao K, Han Y, Sahmani S, Safaei B (2020) Nonlinear oscillations of composite conical microshells with in-plane heterogeneity based upon a couple stress-based shell model. Thin Walled Struct 154:106857

    Google Scholar 

  13. Fan L, Sahmani S, Safaei B (2021) Couple stress-based dynamic stability analysis of functionally graded composite truncated conical microshells with magnetostrictive facesheets embedded within nonlinear viscoelastic foundations. Eng Comput 37:1635–1655

    Google Scholar 

  14. Arefi M, Mohammad-Rezaei Bidgoli E, Civalek O (2020) Bending response of FG composite doubly curved nanoshells with thickness stretching via higher-order sinusoidal shear theory. Mech Based Des Struct Mach 1–29. https://doi.org/10.1080/15397734.2020.1777157

  15. Yang Y, Sahmani S, Safaei B (2021) Couple stress-based nonlinear primary resonant dynamics of FGM composite truncated conical microshells integrated with magnetostrictive layers. Appl Math Mech 42:209–222

    MathSciNet  MATH  Google Scholar 

  16. Dehsaraji ML, Arefi M, Loghman A (2021) Size dependent free vibration analysis of functionally graded piezoelectric micro/nano shell based on modified couple stress theory with considering thickness stretching effect. Def Technol 17:119–134

    Google Scholar 

  17. Kiani Y, Dimitri R, Tornabene F (2018) Free vibration study of composite conical panels reinforced with FG-CNTs. Eng Struct 172:472–482

    Google Scholar 

  18. Tong X, Zhang F, Ji B, Sheng M, Tang Y (2016) Carbon-coated porous aluminum foil anode for high-rate, long-term cycling stability, and high energy density dual-ion batteries. Adv Mater (Weinh) 28(45):9979–9985

    Google Scholar 

  19. Arefi M, Bidgoli EMR, Rabczuk T (2019) Effect of various characteristics of graphene nanoplatelets on thermal buckling behavior of FGRC micro plate based on MCST. Eur J Mech A Solids 77:103802.

  20. Xie Y, Meng X, Mao D, Qin Z, Wan L, Huang Y (2021) Homogeneously dispersed graphene nanoplatelets as long-term corrosion inhibitors for aluminum matrix composites. ACS Appl Mater Interfaces 13(27):32161–32174

    Google Scholar 

  21. Affdl JH, Kardos J (1976) The Halpin–Tsai equations: a review. Polym Eng Sci 16:344–352

    Google Scholar 

  22. Toupin R (1962) Elastic materials with couple-stresses

  23. Yang F, Chong A, Lam DCC, Tong P (2002) Couple stress based strain gradient theory for elasticity. Int J Solids Struct 39:2731–2743

    MATH  Google Scholar 

  24. Farokhi H, Ghayesh MH (2019) Modified couple stress theory in orthogonal curvilinear coordinates. Acta Mech 230:851–869

    MathSciNet  MATH  Google Scholar 

  25. Wang M, Jiang C, Zhang S, Song X, Tang Y, Cheng H (2018) Reversible calcium alloying enables a practical room-temperature rechargeable calcium-ion battery with a high discharge voltage. Nat Chem 10(6):667–672

    Google Scholar 

  26. Liu C, Gao X, Chi D, He Y, Liang M, Wang H (2021) On-line chatter detection in milling using fast kurtogram and frequency band power. Eur J Mech A Solids 90:104341

    MATH  Google Scholar 

  27. Yang W, Lin Y, Chen X, Xu Y, Song X (2021) Wave mixing and high-harmonic generation enhancement by a two-color field driven dielectric metasurface. Chin Opt Lett 19(12):123202

    Google Scholar 

  28. Wang P, Wang S, Kang Y, Sun Z, Wang X, Meng Y, Xie W (2021) Cauliflower-shaped Bi2O3–ZnO heterojunction with superior sensing performance towards ethanol. J Alloys Compd 854:157152

    Google Scholar 

  29. Gao M, Liu B, Zhang X, Zhang Y, Li X, Han G (2022) Ultrathin MoS2 nanosheets anchored on carbon nanofibers as free-standing flexible anode with stable lithium storage performance. J Alloys Compd 894:162550

    Google Scholar 

  30. Ji X, Peng B, Ding H, Cui B, Nie H, Yan Y (2021) Purification, structure and biological activity of pumpkin polysaccharides: a review. Food Rev Int. https://doi.org/10.1080/87559129.2021.1904973

    Article  Google Scholar 

  31. Ji X, Hou C, Shi M, Yan Y, Liu Y (2020) An insight into the research concerning Panax ginseng C. A. Meyer polysaccharides: a review. Food Rev Int. https://doi.org/10.1080/87559129.2020.1771363

    Article  Google Scholar 

  32. Wang T, Liu W, Zhao J, Guo X, Terzija V (2020) A rough set-based bio-inspired fault diagnosis method for electrical substations. Int J Electr Power Energy Syst 119:105961

    Google Scholar 

  33. Wang T, Wei X, Wang J, Huang T, Peng H, Song X, Pérez-Jiménez MJ (2020) A weighted corrective fuzzy reasoning spiking neural P system for fault diagnosis in power systems with variable topologies. Eng Appl Artif Intell 92:103680

    Google Scholar 

  34. Samaniego E, Anitescu C, Goswami S, Nguyen-Thanh VM, Guo H, Hamdia K, Zhuang X, Rabczuk T (2020) An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications. Comput Methods Appl Mech Eng 362:112790

    MathSciNet  MATH  Google Scholar 

  35. Guo H, Zhuang X, Rabczuk T (2019) A deep collocation method for the bending analysis of Kirchhoff plate. arXiv preprint arXiv:2102.02617

  36. 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

    MathSciNet  MATH  Google Scholar 

  37. Zhuang X, Guo H, Alajlan N, Zhu H, Rabczuk T (2021) Deep autoencoder based energy method for the bending, vibration, and buckling analysis of Kirchhoff plates with transfer learning. Eur J Mech A Solids 87:104225

    MathSciNet  MATH  Google Scholar 

  38. Arefi M (2014) A complete set of equations for piezo-magnetoelastic analysis of a functionally graded thick shell of revolution. Lat Am J Solids Struct 11:2073–2092

    MathSciNet  Google Scholar 

  39. Arefi M, Firouzeh S, Bidgoli EMR, Civalek Ö (2020) Analysis of porous micro-plates reinforced with FG-GNPs based on Reddy plate theory. Compos Struct 247:112391

    Google Scholar 

  40. Arefi M, Bidgoli EMR, Rabczuk T (2019) Thermo-mechanical buckling behavior of FG GNP reinforced micro plate based on MSGT. Thin Walled Struct 142:444–459

    Google Scholar 

  41. Arefi M, Zenkour AM (2018) Size-dependent electro-elastic analysis of a sandwich microbeam based on higher-order sinusoidal shear deformation theory and strain gradient theory. J Intell Mater Syst Struct 29(7):1394–1406

    Google Scholar 

  42. Arefi M, Rabczuk T (2019) A nonlocal higher order shear deformation theory for electro-elastic analysis of a piezoelectric doubly curved nano shell. Compos Part B Eng 168(1):496–510

    Google Scholar 

  43. Arefi M, Bidgoli EMR, Dimitri R, Tornabene F (2018) Free vibrations of functionally graded polymer composite nanoplates reinforced with graphene nanoplatelets. Aerosp Sci Technol 81:108–117

    Google Scholar 

  44. Arefi M, Moghaddam SK, Bidgoli EMR, Kiani M, Civalek O (2021) Analysis of graphene nanoplatelet reinforced cylindrical shell subjected to thermo-mechanical loads. Compos Struct 255:112924

    Google Scholar 

  45. Bert CW, Malik M (1996) Differential quadrature method in computational mechanics: a review. Appl Mech Rev 49:1–28

    Google Scholar 

  46. Hamdia KM, Ghasemi H, Zhuang X, Alajlan N, Rabczuk T (2018) Sensitivity and uncertainty analysis for flexoelectric nanostructures. Comput Methods Appl Mech Eng 337:95–109

    MathSciNet  MATH  Google Scholar 

  47. 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

    Google Scholar 

  48. Liew KM, Ng TY, Zhao X (2005) Free vibration analysis of conical shells via the element-free kp-Ritz method. J Sound Vib 281:627–645

    Google Scholar 

  49. Tadi Beni Y, Mehralian F, Razavi H (2015) Free vibration analysis of size-dependent shear deformable functionally graded cylindrical shell on the basis of modified couple stress theory. Compos Struct 120:65–78

    Google Scholar 

  50. Mirzaei M, Kiani Y (2015) Thermal buckling of temperature dependent FG-CNT reinforced composite conical shells. Aerosp Sci Technol 47:42–53

    MATH  Google Scholar 

  51. Liu F, Ming P, Li J (2007) Ab initio calculation of ideal strength and phonon instability of graphene under tension. Phys Rev B 76:064120

    Google Scholar 

  52. 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 3:3884–3890

    Google Scholar 

  53. Yasmin A, Daniel IM (2004) Mechanical and thermal properties of graphite platelet/epoxy composites. Polymer 45:8211–8219

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Solid Mechanic, Faculty of Mechanical Engineering, University of Kashan, Kashan, 87317-51167, Iran

    Niloofar Adab & Mohammad Arefi

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  1. Niloofar Adab
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  2. Mohammad Arefi
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Correspondence to Mohammad Arefi.

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Appendices

Appendix 1

The demonstration of stiffness and mass matrices and displacement vector are as follows:

$$\left\{ s \right\} = \left\{ {\begin{array}{*{20}c} {\left\{ U \right\}} \\ {\left\{ V \right\}} \\ {\left\{ W \right\}} \\ {\left\{ X \right\}} \\ {\left\{ \Theta \right\}} \\ \end{array} } \right\},\quad \left[ K \right] = \left[ {\begin{array}{*{20}c} {k_{11} } & \ldots & {k_{15} } \\ \vdots & \ddots & \vdots \\ {k_{51} } & \ldots & {k_{55} } \\ \end{array} } \right],\quad \left[ M \right] = - \left[ {\begin{array}{*{20}c} {I_{0} I} & {\left[ 0 \right]} & {\left[ 0 \right]} & {I_{1} I} & {\left[ 0 \right]} \\ {\left[ 0 \right]} & {I_{0} I} & {\left[ 0 \right]} & {\left[ 0 \right]} & {I_{1} I} \\ {\left[ 0 \right]} & {\left[ 0 \right]} & {I_{0} I} & {\left[ 0 \right]} & {\left[ 0 \right]} \\ {I_{1} I} & {\left[ 0 \right]} & {\left[ 0 \right]} & {I_{2} I} & {\left[ 0 \right]} \\ {\left[ 0 \right]} & {I_{1} I} & {\left[ 0 \right]} & {\left[ 0 \right]} & {I_{2} I} \\ \end{array} } \right],$$
(47)

in which [0] and I are the zero and the identity matrix of order N and kij are defined accordingly

$$\begin{aligned} k_{11} & = \left( {A_{11} I + 0.25n^{2} a_{{m}} \left[ {a_{2} } \right]} \right)\left[ B \right] + \sin \alpha \left( {A_{11} \left[ {a_{1} } \right] - 0.25n^{2} a_{{m}} \left[ {a_{3} } \right]} \right)\left[ A \right] \\ & \quad - \left\{ {\left( {A_{22} \sin^{2} \alpha + n^{2} A_{66} } \right)\left[ {a_{2} } \right] + 0.25n^{2} a_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{12} & = 0.25na_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5na_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] + n\left\{ {\left( {A_{12} + A_{66} } \right)\left[ {a_{1} } \right] - 0.25a_{{m}} \left( {1 + n^{2} + \cos^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - n\sin \alpha \left\{ {\left( {A_{22} + A_{66} } \right)\left[ {a_{2} } \right] + 0.25a_{{m}} \left( {n^{2} - \sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{13} & = \cos \alpha \left( {A_{12} \left[ {a_{1} } \right] - 0.75n^{2} a_{{m}} \left[ {a_{3} } \right]} \right)\left[ A \right] - 0.5\sin 2\alpha \left( {A_{22} \left[ {a_{2} } \right] - 0.5n^{2} a_{{m}} \left[ {a_{4} } \right]} \right) \\ k_{14} & = \left( {B_{11} I + 0.25n^{2} b_{{m}} \left[ {a_{2} } \right]} \right)\left[ B \right] + \sin \alpha \left( {B_{11} \left[ {a_{1} } \right] - 0.25n^{2} b_{{m}} \left[ {a_{3} } \right]} \right)\left[ A \right] \\ & \quad - \left\{ {\left( {B_{22} \sin^{2} \alpha + n^{2} B_{66} } \right)\left[ {a_{2} } \right] - 0.75n^{2} a_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right.\left. { + 0.25n^{2} b_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{15} & = 0.25nb_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nb_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] + n\left\{ {\left( {B_{12} + B_{66} } \right)\left[ {a_{1} } \right] - 0.25b_{{m}} \left( {1 + n^{2} + \cos^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - n\sin \alpha \left\{ {\left( {B_{22} + B_{66} } \right)\left[ {a_{2} } \right] - 0.5a_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.25b_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ \end{aligned}$$
$$\begin{aligned} k_{21} & = - 0.25na_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5na_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] + \left\{ { - n\left( {A_{12} + A_{66} } \right)\left[ {a_{1} } \right] + 0.25na_{{m}} \left( {n^{2} + 2 - 5\sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - n\sin \alpha \left\{ {\left( {A_{22} + A_{66} } \right)\left[ {a_{2} } \right] + 0.25a_{{m}} \left( {2 - 5\sin^{2} \alpha + 3n^{2} } \right)\left[ {a_{4} } \right]} \right\} \\ k_{22} & = - 0.25a_{{m}} \left[ D \right] - 0.5a_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ C \right] + \left\{ {A_{66} I + 0.25a_{{m}} \left( {2 - n + \sin^{2} \alpha } \right)\left[ {a_{2} } \right]} \right\}\left[ B \right] \\ & \quad + \sin \alpha \left\{ {A_{66} \left[ {a_{1} } \right] - 0.25a_{{m}} \left( {2 + n^{2} + \sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - \left\{ {\left( {A_{44} \cos^{2} \alpha + A_{66} \sin^{2} \alpha + n^{2} A_{22} } \right)\left[ {a_{2} } \right] + 0.25a_{{m}} \left[ {n^{2} \left( {1 + 2\sin^{2} \alpha } \right) - 3\sin^{4} \alpha } \right]\left[ {a_{4} } \right]} \right\} \\ k_{23} & = 0.5na_{{m}} \cos \alpha \left[ {a_{2} } \right]\left[ B \right] - 0.625na_{{m}} \sin 2\alpha \left[ {a_{3} } \right]\left[ A \right] - n\cos \alpha \left\{ {\left( {A_{22} + A_{44} } \right)\left[ {a_{2} } \right]} \right. \\ &\left. { \quad + 0.25a_{{m}} \left( {n^{3} - 4\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{24} & = - 0.25nb_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nb_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] - n\left\{ {\left( {B_{12} + B_{66} } \right)\left[ {a_{1} } \right] + 0.5a_{{m}} \cos \alpha \left[ {a_{2} } \right]} \right. \\ & \quad \left. { - 0.25b_{{m}} \left( {2 + n^{2} - 5\sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - n\sin \alpha \left\{ {\left( {B_{22} + B_{66} } \right)\left[ {a_{2} } \right] - 0.25a_{{m}} \cos \alpha \left( {1 + 2\sin \alpha } \right)\left[ {a_{3} } \right]} \right. \\ &\left. { \quad + 0.25b_{{m}} \left[ {\sin \alpha \left( {4\cos^{2} \alpha - 3\sin \alpha } \right) + 3n^{2} } \right]\left[ {a_{4} } \right]} \right\} \\ k_{25} & = - 0.25b_{{m}} \left[ D \right] - 0.5b_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ C \right] + \left\{ {B_{66} + 0.25b_{{m}} \left( {2 + n^{2} + \sin^{2} \alpha } \right)\left[ {a_{2} } \right]} \right\}\left[ B \right] \\ & \quad + \sin \alpha \left[ {B_{66} \left[ {a_{1} } \right] - 0.5a_{{m}} \cos \alpha \left[ {a_{2} } \right] - 0.25b_{{m}} \left( {n^{2} + 3\sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right]\left[ A \right] + \left\{ {A_{44} \cos \alpha \left[ {a_{1} } \right]} \right. \\ &\left. { \quad - \left( {B_{66} \sin^{2} \alpha + n^{2} B_{22} } \right)\left[ {a_{2} } \right] + 0.25a_{{m}} \cos \alpha \left( {2\sin^{2} \alpha - n^{2} } \right)\left[ {a_{3} } \right] - 0.25b_{{m}} \sin^{2} \alpha \left( {4 + 3n^{2} - 7\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \end{aligned}$$
$$\begin{aligned} k_{31} & = - \cos \alpha \left( {A_{12} \left[ {a_{1} } \right] - 0.75n^{2} a_{{m}} \left[ {a_{3} } \right]} \right)\left[ A \right] - 0.5\sin 2\alpha \left( {A_{22} \left[ {a_{2} } \right] + n^{2} a_{{m}} \left[ {a_{4} } \right]} \right) \\ k_{32} & = 0.5na_{{m}} \cos \alpha \left[ {a_{2} } \right]\left[ B \right] + 0.25na_{{m}} \sin 2\alpha \left[ {a_{3} } \right]\left[ A \right] - n\cos \alpha \left\{ {\left( {A_{22} + A_{44} } \right)\left[ {a_{2} } \right] + 0.25n^{2} a_{{m}} \cos \alpha \left[ {a_{4} } \right]} \right\} \\ k_{33} & = - 0.25a_{{m}} \left[ D \right] - 0.5a_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ C \right] + \left[ {A_{55} I + 0.25a_{{m}} \left( {1 + 2n^{2} } \right)\left[ {a_{2} } \right]} \right]\left[ B \right] \\ & \quad + \sin \alpha \left\{ {A_{55} \left[ {a_{1} } \right] - 0.25a_{{m}} \left[ {\sin \alpha \left( {\sin \alpha + \cos \alpha } \right) + 2n^{2} } \right]\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - \left\{ {\left( {A_{22} \cos^{2} \alpha + n^{2} A_{44} } \right)\left[ {a_{2} } \right] + 0.25n^{2} a_{{m}} \left( {n^{2} - 4\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{34} & = 0.25a_{{m}} \left[ C \right] + 0.5a_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ B \right] + \left\{ {A_{55} I - B_{12} \cos \alpha \left[ {a_{1} } \right] - 0.25a_{{m}} \left( {1 + n^{2} } \right)\left[ {a_{2} } \right]} \right. \\ & \quad \left. { + 0.75n^{2} b_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right\}\left[ A \right] + \sin \alpha \left\{ {A_{55} \left[ {a_{1} } \right] - B_{22} \cos \alpha \left[ {a_{2} } \right] + 0.25a_{{m}} \left[ {\sin \alpha \left( {\sin \alpha + \cos \alpha } \right) - n^{2} } \right]\left[ {a_{3} } \right]} \right. \\ &\left. { \quad - n^{2} b_{{m}} \cos \alpha \left[ {a_{4} } \right]} \right\} \\ k_{35} & = 0.25n\left( {a_{{m}} \left[ {a_{1} } \right] + 3b_{{m}} \cos \alpha \left[ {a_{2} } \right]} \right)\left[ B \right] - 0.5n\sin \alpha \left( {0.5a_{{m}} \left[ {a_{2} } \right] - b_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right)\left[ A \right] \\ & \quad + n\left\{ {A_{44} \left[ {a_{1} } \right] - B_{22} \cos \alpha \left[ {a_{2} } \right] + 0.25a_{{m}} \left( {\sin^{2} \alpha - n^{2} } \right)\left[ {a_{3} } \right] - 0.5b_{{m}} \sin \alpha \sin 2\alpha \left[ {a_{4} } \right]} \right\} \\ \end{aligned}$$
$$\begin{aligned} k_{41} & = \left( {B_{11} I + 0.25n^{2} b_{{m}} \left[ {a_{2} } \right]} \right)\left[ B \right] + \sin \alpha \left( {B_{11} \left[ {a_{1} } \right] - 0.25n^{2} b_{{m}} \left[ {a_{3} } \right]} \right)\left[ A \right] - \left\{ {\left( {B_{22} \sin^{2} \alpha + n^{2} B_{66} } \right)\left[ {a_{2} } \right]} \right. \\ &\left. { \quad - 0.75n^{2} a_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.25n^{2} b_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{42} & = 0.25nb_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nb_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right]\, + n\left\{ {\left( {B_{12} + B_{66} } \right)\left[ {a_{1} } \right] + 0.5a_{{m}} \cos \alpha \left[ {a_{2} } \right]} \right. - 0.25b_{{m}} \left( {1 + n^{2} } \right. \\ & \quad \left. {\left. { + \cos^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - n\sin \alpha \left\{ {\left( {B_{22} + B_{66} } \right)\left[ {a_{2} } \right] - 0.25a_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.25b_{{m}} \left( {n^{2} - \sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{43} & = - 0.25a_{{m}} \left[ C \right] - 0.25a_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ B \right] + \left\{ { - A_{55} I + B_{12} \cos \alpha \left[ {a_{1} } \right] + 0.25a_{{m}} \left( {1 + n^{2} } \right)\left[ {a_{2} } \right]} \right. \\ & \quad \left. { - 0.75n^{2} b_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right\}\left[ A \right] - \sin \alpha \left( {B_{22} \cos \alpha \left[ {a_{2} } \right] + 0.5n^{2} a_{{m}} \left[ {a_{3} } \right] - 0.5n^{2} b_{{m}} \cos \alpha \left[ {a_{4} } \right]} \right) \\ k_{44} & = \left\{ {\left( {D_{11} + 0.25a_{{m}} } \right)I + 0.25n^{2} d_{{m}} \left[ {a_{2} } \right]} \right\}\left[ B \right] + 0.25\sin \alpha \left\{ {\left( {4D_{11} + a_{{m}} } \right)\left[ {a_{1} } \right] - n^{2} d_{{m}} \left[ {a_{3} } \right]} \right\}\left[ A \right] - \left\{ {A_{55} I} \right. \\ & \quad \left. { + \left[ {D_{22} \sin^{2} \alpha + n^{2} D_{66} + a_{{m}} \left( {0.25 + n^{2} } \right)} \right]\left[ {a_{2} } \right] - 1.5n^{2} b_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.25n^{2} d_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{45} & = 0.25nd_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nd_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] + n\left\{ {\left( {D_{12} + D_{66} - 0.75a_{{m}} } \right)\left[ {a_{1} } \right] + 0.75b_{{m}} \cos \alpha \left[ {a_{2} } \right]} \right. \\ & \quad \left. { - 0.25d_{{m}} \left( {1 + n^{2} + \cos^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - n\sin \alpha \left\{ {\left( {D_{22} + D_{66} + 1.25a_{{m}} } \right)\left[ {a_{2} } \right] - 1.25b_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right. \\ & \quad \left. { + 0.25d_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ \end{aligned}$$
$$\begin{aligned} k_{51} & = - 0.25nb_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nb_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] + n\left\{ { - \left( {B_{12} + B_{66} } \right)\left[ {a_{1} } \right]} \right. \\ & \quad \left. { + 0.25b_{{m}} \left( {2 + n^{2} - 5\sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - n\sin \alpha \left\{ {2I_{1} \Omega^{2} I + \left( {B_{22} + B_{66} } \right)\left[ {a_{2} } \right]} \right. \\ & \quad \left. { - 0.5a_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.75b_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{52} & = - 0.25b_{{m}} \left[ D \right] - 0.5b_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ C \right] + \left\{ {B_{66} I + 0.5a_{{m}} \cos \alpha \left[ {a_{1} } \right]\, + 0.25b_{{m}} \left( {2 + n^{2} + \sin^{2} \alpha } \right)\left[ {a_{2} } \right]} \right\}\left[ B \right] \\ & \quad + \sin \alpha \left\{ {B_{66} \left[ {a_{1} } \right] - 0.25b_{{m}} \left( {3 + n^{2} + \cos^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - \left\{ { - A_{44} \cos \alpha \left[ {a_{1} } \right] + \left( {B_{66} \sin^{2} \alpha + n^{2} B_{22} } \right)\left[ {a_{2} } \right]} \right. \\ & \quad \left. { + 0.25a_{{m}} \cos \alpha \left( {n^{2} - 2\sin^{2} \alpha } \right)\left[ {a_{3} } \right] + 0.75b_{{m}} \sin^{2} \alpha \left( {n^{2} - \sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{53} & = - 0.25n\left( {a_{{m}} \left[ {a_{1} } \right] + 3b_{{m}} \cos \alpha \left[ {a_{2} } \right]} \right)\left[ B \right] - n\sin \alpha \left( { - 0.25a_{{m}} \left[ {a_{2} } \right] + 2b_{{m}} \cos \alpha \left[ {a_{3} } \right]} \right)\left[ A \right] \\ & \quad - n\left\{ { - A_{44} \left[ {a_{1} } \right] + B_{22} \cos \alpha \left[ {a_{2} } \right] + 0.25n^{2} a_{{m}} \left[ {a_{3} } \right] - 0.75b_{{m}} \sin \alpha \sin 2\alpha \left[ {a_{4} } \right]} \right\} \\ k_{54} & = - 0.25nd_{{m}} \left[ {a_{1} } \right]\left[ C \right] + 0.5nd_{{m}} \sin \alpha \left[ {a_{2} } \right]\left[ B \right] - n\left\{ {\left( {D_{12} + D_{66} } \right)\left[ {a_{1} } \right] - 0.75a_{{m}} \left[ {a_{1} } \right]} \right. \\ & \quad \left. { + 0.75b_{{m}} \cos \alpha \left[ {a_{2} } \right] - 0.25d_{{m}} \left( {2 + n^{2} - 5\sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] \\ & \quad - n\sin \alpha \left\{ {\left( {D_{22} + D_{66} + 1.25a_{{m}} } \right)\left[ {a_{2} } \right] - 2b_{{m}} \cos \alpha \left[ {a_{3} } \right] + 0.75d_{{m}} \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ k_{55} & = - 0.25d_{{m}} \left[ D \right] - 0.5d_{{m}} \sin \alpha \left[ {a_{1} } \right]\left[ C \right] + \left\{ {\left( {D_{66} + a_{{m}} } \right)I + 0.25d_{{m}} \left( {2 + n^{2} + \sin^{2} \alpha } \right)\left[ {a_{2} } \right]} \right\}\left[ B \right] \\ & \quad - 0.25n^{2} d_{{m}} \sin \alpha \left[ {a_{3} } \right]\left[ A \right] + n\sin \alpha \left\{ {\left( {D_{66} + a_{{m}} } \right)\left[ {a_{1} } \right] - 0.25d_{{m}} \left( {2 + \sin^{2} \alpha } \right)\left[ {a_{3} } \right]} \right\}\left[ A \right] - \left\{ {A_{44} I} \right. \\ & \quad + \left[ {\left( {D_{66} + a_{{m}} } \right)\sin^{2} \alpha + n^{2} \left( {D_{22} + 0.25a_{{m}} } \right)} \right]\left[ {a_{2} } \right] - 0.75b_{{m}} \sin \alpha \sin 2\alpha \left[ {a_{3} } \right] \\ & \quad \left. { + 0.75d_{{m}} \sin^{2} \alpha \left( {2 + n^{2} - 3\sin^{2} \alpha } \right)\left[ {a_{4} } \right]} \right\} \\ \end{aligned}$$
(48)

and with the following definitions, [a1]–[a4] are four diagonal matrices:

$$\left[ {a_{1} } \right]_{ii} = r_{i}^{ - 1} ,\quad \left[ {a_{2} } \right]_{ii} = r_{i}^{ - 2} ,\quad \left[ {a_{3} } \right]_{ii} = r_{i}^{ - 3} ,\quad \left[ {a_{4} } \right]_{ii} = r_{i}^{ - 4} .$$
(49)

Appendix 2

The general form of matrix [P] is defined as follows:

$$\left[ P \right] = \left[ {\begin{array}{*{20}c} {p_{11} } & {p_{12} } & {p_{13} } & {p_{14} } & {p_{15} } \\ {p_{21} } & {p_{22} } & {p_{23} } & {p_{24} } & {p_{25} } \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {p_{161} } & {p_{162} } & {p_{163} } & {p_{164} } & {p_{165} } \\ \end{array} } \right],$$
(50)

in which p11–p85 are related to the conditions at x = 0 and p81–p165 are related to the conditions at x = L. As an example, for a microshell clamped (C) at x = 0 and simply supported (S) at x = L (abbreviated as CS), p11–p165 can be stated as follows:

$$\begin{aligned} &p_{11} = p_{22} = p_{33} = p_{44} = p_{55} = I_{1} ,\quad {p_{62} = p_{73} = p_{85} = A_{1} ,} \\ & p_{12} = p_{13} = p_{14} = p_{15} = p_{21} = p_{23} = p_{24} = p_{25} = p_{31} = p_{32} = p_{34} = p_{35} = p_{41} = p_{42} = p_{43} = p_{45} = p_{51} \\ & \quad = p_{52} = p_{53} = p_{54} = p_{61} = p_{63} = p_{64} = p_{65} = p_{71} = p_{72} = p_{74} = p_{75} = p_{81} = p_{82} = p_{83} = p_{84} = \left\{ 0 \right\}_{1 \times N} , \end{aligned}$$
(51)
$$\begin{aligned} & p_{91} = A_{11} A_{N} + \frac{{A_{12} \sin \alpha }}{b}I_{N} ,\quad {p_{94} = p_{121} = B_{11} A_{N} + \frac{{B_{12} \sin \alpha }}{b}I_{N} ,} \quad {p_{124} = D_{11} A_{N} + \frac{{D_{12} \sin \alpha }}{b}I_{N} ,} \\ & p_{102} = p_{113} = p_{135} = I_{N} , \\ & p_{141} = \frac{{na_{{m}} }}{b}A_{N} - \frac{{na_{{m}} \sin \alpha }}{{b^{2} }}I_{N} , \quad {p_{142} = a_{{m}} B_{N} + \frac{{a_{{m}} \sin \alpha }}{b}A_{N} ,} \quad {p_{144} = \frac{{nb_{{m}} }}{b}A_{N} - \frac{{nb_{{m}} \sin \alpha }}{{b^{2} }}I_{N} ,} \\ & p_{145} = b_{{m}} B_{N} + \frac{{b_{{m}} \sin \alpha }}{b}A_{N} , \quad {p_{153} = - a_{{m}} B_{N} + \frac{{a_{{m}} \sin \alpha }}{b}A_{N} ,} \quad {p_{154} = a_{{m}} A_{N} - \frac{{a_{{m}} \sin \alpha }}{b}I_{N} ,} \\ & p_{161} = \frac{{nb_{{m}} }}{r}A_{N} - \frac{{nb_{{m}} \sin \alpha }}{{r^{2} }}I_{N} , \quad {p_{162} = b_{{m}} B_{N} + \frac{{b_{{m}} \sin \alpha }}{b}A_{N} ,} \quad {p_{164} = \frac{{nd_{{m}} }}{b}A_{N} - \frac{{nd_{{m}} \sin \alpha }}{{b^{2} }}I_{N} ,} \\ \\ & p_{165} = d_{{m}} B_{N} + \frac{{d_{{m}} \sin \alpha }}{b}A_{N} , \\ & p_{92} = p_{93} = p_{95} = p_{101} = p_{103} = p_{104} = p_{105} = p_{111} = p_{112} = p_{114} = p_{115} = p_{122} \\ & \quad = p_{123} = p_{125} = p_{131} = p_{132} = p_{133} = p_{134} = p_{143} = p_{151} = p_{152} = p_{155} = p_{163} = \left\{ 0 \right\}_{1 \times N} , \\ \end{aligned}$$

in which the subscripts 1 and N, respectively, indicate the first and Nth row of the matrices.

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Adab, N., Arefi, M. Vibrational behavior of truncated conical porous GPL-reinforced sandwich micro/nano-shells. Engineering with Computers 39, 419–443 (2023). https://doi.org/10.1007/s00366-021-01580-8

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