Size-dependent axisymmetric buckling and free vibration of FGP-microplate using well-posed nonlocal integral polar models

Abstract

Softening and toughening size-dependent axisymmetric elastic buckling and free vibration of functionally graded porous (FGP) Kirchhoff microplates with two different porous distribution patterns are investigated through strain-driven ($𝜀$D) and stress-driven ($\sigma$D) two-phase local/nonlocal integral polar models (TPNIPM), respectively. The Hamilton’s principle is used to derive the differential governing equation and boundary conditions. A few nominal variables are introduced to simplify the differential governing equation and boundary conditions, and equivalent differential constitutive relations and constitutive constraints are expressed in united nominal forms. The general differential quadrature method is applied to discretize differential governing equation and constitutive relations as well as boundary conditions and constitutive constraints. L’Hospital’s rule is applied to deal with the boundary conditions and constitutive constraints at center for circular microplate. A general eigenvalue problem is obtained in matrix form, from which one can determine buckling loads and vibration frequency for different boundary conditions. The effects of nonlocal parameters, FGP distribution patterns, geometrical dimensions and buckling/vibration order on the buckling load and vibration frequency are investigated numerically for different boundary conditions, and consistent size-effects are obtained for $𝜀$D- and $\sigma$D-TPNIPMs TPNIPMs, respectively.