Bending and free vibration analysis of nanobeams by differential and integral forms of nonlocal strain gradient with Rayleigh–Ritz method. (13th December 2017) - Fakher, Mahmood; Hosseini-Hashemi, Shahrokh
Abstract: The static bending and free vibration behavior of Euler nanobeams are studied via nonlocal strain gradient elasticity. Three different approaches to nonlocal strain gradient are investigated including differential form of nonlocal strain gradient, integral nonlocal strain gradient with satisfying higher order boundary conditions and basic form of integral nonlocal strain gradient without satisfying higher order boundary conditions. Two different types of Rayleigh–Ritz method are adopted; in one of them, polynomial is chosen as admissible function and in the other, combination of polynomial and trigonometric are regarded to obviate ill-conditioning and achieve more accurate results in integral nonlocal elasticity in which high number of terms is needed to extract the results with desired accuracy. Next, the bending deflections and natural frequencies of nanobeams with different boundary conditions are obtained. The results show that using integral form of nonlocal strain gradient obviates the paradoxes of differential form. In addition, significant differences are observed between the results. Moreover, comparison between two different integral forms indicates that there are considerable discrepancies, especially in the cases of high material characteristic parameter with low nonlocal parameter. Advantage of using the basic form of nonlocal strain gradient is that it does not need to definition and satisfying the higher order boundary conditions.