Nonlocal boundaries and paradoxes in thermoelastic vibrations of functionally graded Non-Uniform cantilever nanobeams and annular nanoplates

Abstract

The paradoxical behaviour of free vibrations of cantilever nanostructures using Eringen's nonlocal elasticity model is always a debatable issue for researchers in the past few years. According to this, the frequency of cantilever nanobeams increases with the increase in the value of the nonlocal parameter while it decreases for other boundary conditions. In thermoelastic vibrations, the frequency of nanobeams increases and decreases with the increase in temperature at the surfaces for clamped-free and other boundary conditions, respectively. Therefore, investigation of these structures is of critical importance. In view of this, the authors have proposed the modified boundary conditions for the well-established differential model of Eringen’s nonlocal elasticity theory to deal with the free edges of the structural components. Using these conditions, the vibration analysis of nonuniform functionally graded nanobeams and axisymmetric annular nanoplates is presented for clamped-free boundary conditions on the basis of first-order shear deformation theory and physical neutral plane. The thickness of these structural elements (nanobeams and annular nanoplates) is assumed to be varying linearly and parabolically in the axial direction. The temperature-dependent mechanical properties of the material are obtained by the power law. The nonlocal governing equations and modified boundary conditions for the structural elements are discretized using the generalized differential quadrature method and solved by MATLAB to obtain the fundamental frequency. For the present modifications, the behaviour of frequency parameter for different values of nonlocal parameter, volume fraction index, temperature difference, thickness parameter, and taper parameter is discussed.