Modified nonlocal elasticity theory for functionally graded materials

In this paper, it will be shown that the nonlocal theory of Eringen is not generally suitable for analysis of functionally graded (FG) materials at micro/nano scale and should be modified. In the current work, an imaginary nonlocal strain tensor is introduced and used to directly obtain the nonlocal stress tensor. Similar to the stress tensor in Eringen’s nonlocal theory, the imaginary nonlocal strain tensor at a point is assumed to be a function of local strain tensor at all neighbor points. To compare the new modified nonlocal theory with Eringen’s theory, free vibration of FG rectangular micro/nanoplates with simply supported boundary conditions are investigated based on the first-order plate theory and three-dimensional (3-D) elasticity theory. The material properties are assumed to be functionally graded only along the plate thickness. The effects of nonlocal parameter and material gradient index on the natural frequencies of FG micro/nano plates are discussed. The present developed nonlocal theory can be used in conjunction with different analytical and numerical methods to analyze mechanical response of micro/nano structures made of FG materials.