A Size-Dependent Meshfree Approach for Free Vibration Analysis of Functionally Graded Microplates Using the Modified Strain Gradient Elasticity Theory

Abstract

In this work, we present a size-dependent numerical approach for free vibration analysis of functionally graded (FG) microplates based on the modified strain gradient theory (MSGT), simple first-order shear deformation theory (sFSDT) and moving Kriging meshfree method. The present approach decreases one variable compared with the original first-order shear deformation theory (FSDT). Moreover, it only uses three material length scale parameters to capture the size effects. The effective material properties as Young’s modulus, Poisson’s ratio and density mass are homogenized by a rule of mixture. Thanks to the principle of virtual work, the discrete system equations solved by the moving Kriging meshfree method, are derived. In addition, due to satisfying a Kronecker delta function property of the moving Kriging integration shape function, the essential boundary conditions are easily enforced similar to the standard finite element method. Rectangular and circular FG microplates with different boundary conditions, material length scale parameter and volume fraction are exampled to evaluate natural frequencies.