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Abstract:
Free flexural vibrations of nanobeams with non-rigid edge supports are studied by means of the stress-driven nonlocal elasticity model and Euler-Bernoulli kinematics. The elastic deformations of the supports are modelled by transversal and flexural springs, so that, in the limit conditions when the springs stiffnesses tend to zero or infinity, the classical free, pinned, and clamped boundary conditions may be recovered. An analytical procedure is used to derive the closed form solution of the spatial differential equation. The problem of finding the natural frequencies is then reduced to find the roots of the determinant of a matrix, whose elements are explicitly given. The proposed technique, then, avoids the numerical instabilities usually arising when the numerical techniques are used to obtain the solution. The effects of both non-rigid supports elastic deformations and nonlocal parameter on the natural frequencies are studied also for higher vibrations modes. The comparison between the solutions of the proposed model and those available in the literature shows an excellent agreement, and new insightful results and discussions are presented.

Keywords:
elastically constrained beam, nanostructures, natural frequency, size effects, well-posed nonlocal formulation

Abstract:
In this work, variational formulations are proposed for solving numerically the problem of bending and buckling of Timoshenko nano-beams. The present work belongs to research branch in which the non-local theory of elasticity has been used for analysis of beam-like elements in smart materials, micro-electro-mechanical (MEMS) or nano-electro-mechanical systems (NEMS). In fact, the local beam theory is not adequate to describe the behavior of beam-like elements of smart materials at the nano-scale, so that different non-local models have been proposed in last decades for nano-beams. The nano-beam model considered in this work is a convex combination (mixture) of local and non-local phases. In the non-local phase, the kinematic entities in a point of the nano-beam are expressed as integral convolutions between internal forces and an exponential kernel. The aim is to construct a functional whose stationary condition provides the solution of the problem. Two different functionals are defined: one for the pure non-local model, where the local fraction of the mixture is absent, and the other for the mixture with both local and non-local phases. The Euler equations of the two functionals are derived; then, attention focuses on the mixture model. The functional of the mixture depends on unknown Lagrange multipliers and the Euler equations of the functional provide not only the governing equations of the problem but also the relationships between these Lagrange multipliers and the other variables on which the functional depends. In fact, approximations of the variables of the functional can not be chosen arbitrarily in numerical analyzes but have to satisfy suitable conditions. The Euler equations involving the Lagrange multipliers are essential in the numerical analyzes and suggest the correct approximations that have to be adopted for Lagrange multipliers and the other unknown variables of the functional. The proposed method is verified by comparing numerical solutions with exact solutions in bending problem. Finally, the method is used to determine the buckling load of Timoshenko nano-beams with mixture of phases.

Keywords:
non-local elasticity, variational methods, Timoshenko beam, buckling load, smart materials