Imperfections can significantly reduce the load-carrying capacity of structures, especially in thin shells. Such imperfections can stem from inaccurate fabrication and erection and they should be taken into account in the analysis and design. For the first time, post-buckling behavior of functionally graded triply periodic minimal surface (FG-TPMS) shells under axial compression is investigated in this paper. The proposed formulation considers both geometric imperfection and porosity which can be considered as material imperfection. The two types of porosity in this study are the even and uneven porosity distributions. The nonlinear responses of FG-TPMS shells with six density distribution patterns along the thickness are investigated. The mechanical properties of the FG-TPMS materials were calculated using a fitting technique. The present formulation is based on isogeometric analysis (IGA) and first-order shear deformation shell theory (FSDT). Non-uniform rational B-Spline (NURBS) basis functions are utilized to model exact geometries and to approximate displacements. The non-linearity of shells is formulated based on the von Karman assumption and the total Lagrangian approach. A modified Riks method is employed to solve the discrete nonlinear equation system iteratively. The high reliability of the present formulation is confirmed by solving several problems. Effects of the density distribution pattern, geometrical imperfection, curvature, porosity volume fraction, and porosity distribution on post-buckling strength of FG-TPMS panel are thoroughly studied. Moreover, numerous new load-deflection paths of FG-TPMS shells subjected to compression and considering both geometric imperfection and porosity are proposed.

An improved meshfree moving-Kriging (MK) formulation for free vibration analysis of functionally graded material-functionally graded carbon nanotube-reinforced composite (FGM-FGCNTRC) sandwich shells is first proposed in this article. The proposed sandwich structure consists of skins of FGM layers and an FGCNTRC core. This structure possesses all the advantages of FGM and FGCNTRC, including high electrical or thermal insulating properties, high fatigue resistance, good corrosion resistance, high stiffness, low density, high strength, and high aspect ratios. Such sandwich structures can be used to replace conventional FGM structures. The present formulation has been established by using an improved meshfree MK method and the first-order shear deformation shell theory (FSDT). The effective material characteristics of the FGM-skin layers and the FGCNTRC core were calculated using the rule of mixture. Key parameters and factors such as the thickness-to-radius ratio, the length-to-radius ratio, layer-thickness ratios, CNT distributions, the volume fraction of CNTs, the power-law index, and various boundary conditions were rigorously investigated. A nonlinear CNT distribution that we term FG-nX is first proposed in this work, and many new results of FGM-FGCNTRC sandwich shells have been provided.