Postbuckling analysis of piezoelectric multiscale sandwich composite doubly curved porous shallow shells via Homotopy Perturbation Method

Abstract

In this study postbuckling behaviors of multiscale composite sandwich doubly curved piezoelectric shell with a flexible core and MR layers by employing Homotopy Perturbation Method in hygrothermal environment has been investigated. By using Reddy third shear deformable theory the face sheets and third-order polynomial theory of the flexible core the strains and stresses are obtained. A mathematical model for the multiscale composite layered shell with a flexible core and magnetorheological layer (MR) that incorporates the nonlinearity of the in-plane and the vertical displacements of the core is assumed. Three-phase composite shells with polymer/Carbon nanotube/fiber and polymer/Graphene platelet/fiber either uniformly or non-uniformly based on different patterns according to Halpin–Tsai model have been considered. The governing equations of multiscale shell have been derived by implementing Hamilton’s principle. Meanwhile, simply supported boundary conditions are employed to the shell. For investigating correctness and accuracy, this paper is validated by other previous researches. Finally, different parameters such as temperature rise, various distribution patterns, magnetic fields and curvature ratio are considered in this article. It is found these parameters have significant effect on the frequency–amplitude curves.

Appendix

Transformed shell principle coordinate can be expressed by the following:

$$ \begin{aligned} & \bar{Q}_{11}^{n} = Q_{11}^{n} \cos^{4} \theta + 2\left( {Q_{12}^{n} + 2Q_{66}^{n} } \right)\sin^{2} \theta \cos^{2} \theta + Q_{22}^{n} \sin^{4} \theta \\ & \bar{Q}_{12}^{n} = (Q_{11}^{n} + Q_{22}^{n} - 4Q_{66}^{n} )\sin^{2} \theta \cos^{2} \theta + Q_{12}^{n} \left( {\sin^{4} \theta + \cos^{4} \theta } \right) \\ & \bar{Q}_{22}^{n} = Q_{11}^{n} \sin^{4} \theta + 2\left( {Q_{12}^{n} + 2Q_{66}^{n} } \right)\sin^{2} \theta \cos^{2} \theta + Q_{22}^{n} \cos^{4} \theta \\ & \bar{Q}_{66}^{n} = \left( {Q_{11}^{n} + Q_{22}^{n} - 2Q_{12}^{n} - 2Q_{66}^{n} } \right)\sin^{2} \theta \cos^{2} \theta + Q_{66}^{n} \left( {\sin^{4} \theta + \cos^{4} \theta } \right) \\ & \bar{Q}_{44}^{n} = Q_{44}^{n} \cos^{2} \theta + Q_{55}^{n} \sin^{2} \theta \\ & \bar{Q}_{55}^{n} = Q_{55}^{n} \cos^{2} \theta + Q_{44}^{n} \sin^{2} \theta , \\ \end{aligned} $$

(67)

where $\bar{Q}_{ij} \left( {i,j = 1,2,3,4,5,6} \right)$ presented the transformed reduced stiffness modulus.