ORIGINAL ARTICLE Analytical treatment on the nonlocal strain gradient vibrational response of postbuckled functionally graded porous micro-/nanoplates reinforced with GPL S. Sahmani 1 • A. M. Fattahi 2 • N. A. Ahmed 2 Received: 24 September 2018 / Accepted: 17 May 2019 Ó Springer-Verlag London Ltd., part of Springer Nature 2019 Abstract Through production of porous materials with remarkable complexity in geometry, functionally graded porous materials (FGPMs) have gained considerable attention for use in additive manufacturing in biomedical applications. In the current study, the size-dependent linear and nonlinear vibrational characteristics of axially loaded micro-/nanoplates made of FGPM reinforced with graphene platelets (GPLs) is investigated within both the prebuckling and postbuckling regimes. To this end, the nonlocal strain gradient continuum elasticity in conjunction with geometrical nonlinearity is implemented into the refined exponential shear deformation plate theory. On the basis of the closed-cell Gaussian random field scheme as well as the Halpin–Tsai micromechanical modeling, the mechanical properties of the FGPM reinforced with GPLs are achieved corresponding to the uniform and three different patterns of porosity dispersion. Via the variational approach, the differential equations of motion related to the nonlinear problem are constructed in the presence of nonlocality and strain gradient size dependency. Finally, with the aid of an improved perturbation technique together with the Galerkin method, analytical expressions in explicit form for the size-dependent linear frequency–load and deflection–nonlinear frequency responses of the FGPM micro-/nanoplates within stability and instability domains are obtained. It is displayed that within the prebuckling regime, the nonlocality causes reduction of the linear frequency of the micro-/nanoplate, while the strain gradient size dependency leads to increasing it. But within the postbuckling domain, these patterns are vice versa. Also, it is found that for a specific value of plate deflection, increasing the value of the porosity coefficient leads to increase in the frequency ratio of x NL /x L within both the prebuckling and postbuckling regimes. Keywords Perturbation technique  Porous materials  Nanocomposites  Size dependency  Nonlinear dynamics 1 Introduction With the aid of advancements in material science, a large number of inorganic porous materials have been manu- factured in recent years to be utilized in various applica- tions such as insulation, biomedication, impact protection and membranes. The characteristics of a porous material vary depending on the size, arrangement and shape of the pores. However, through addition of nanofillers, it is pos- sible to improve the mechanical properties of porous materials. Pop et al. [1] fabricated monolithic porous car- bon nanotube-reinforced metals and ceramics by non-cat- alytic chemical vapor deposition using a double templating approach. Jun et al. [2] reported the mechanical properties of carbon nanotube-reinforced porous CuSn oil bearings using the powder metallurgy method. Hai et al. [3] pro- duced a novel binary conductive porous alumina including carbon nanotubes. Chen et al. [4] studied the influence of the inclusion of thermal-responsive nano-hydroxyapatite and the porous structure of polylactide composites. Xu and Li [5] examined the effect of carbon nanofiber reinforce- ments on the mechanical properties of porous magnesium- matrix composites. Chen et al. [6] predicted the nonlinear & A. M. Fattahi afattahi@uj.ac.za 1 Mechanical Rotating Equipment Department, Niroo Research Institute (NRI), Tehran 14665-517, Iran 2 Mechanical Engineering Science Department, Faculty of Engineering and the Built Environment, University of Johannesburg, Johannesburg, South Africa 123 Engineering with Computers https://doi.org/10.1007/s00366-019-00782-5