Abstract
The dynamic behaviour of micro- and nano-beams is investigated by the nonlocal continuum mechanics, a computationally convenient approach with respect to atomistic strategies. Specifically, size effects are modelled by expressing elastic curvatures in terms of the integral mixture of stress-driven local and nonlocal phases, which leads to well-posed structural problems. Relevant nonlocal equations of the motion of slender beams are formulated and integrated by an analytical approach. The presented strategy is applied to simple case-problems of nanotechnological interest. Validation of the proposed nonlocal methodology is provided by comparing natural frequencies with the ones obtained by the classical strain gradient model of elasticity. The obtained outcomes can be useful for the design and optimisation of micro- and nano-electro-mechanical systems (M/NEMS).
Highlights
The modelling and design of advanced small-scale structures is a topic of major interest in nanoengineering
Analysis of micro and nanostructures has to be carried out by adequately modelling the effect of molecular interactions and inter-atomic forces which are technically significant. These long-range interactions result in size effects which cannot be overlooked
Denoting with v : [0, L] 7→ < the transverse displacement field of beam axis, the kinematic hypothesis of Bernoulli–Euler theory prescribes that the linearised geometric curvature field χt : [0, L] 7→ < at a time t is related to displacements as follows χt = v(2) = χel + χnel, (1)
Summary
The modelling and design of advanced small-scale structures is a topic of major interest in nanoengineering. Eringen formulated a nonlocal model of elasticity based on a strain-driven integral convolution efficiently applied to screw dislocation and wave propagation problems involving unbounded domains [12,13]. Since local/nonlocal mixtures are able to model both stiffening and softening behaviors of small-scale structures [18], they can be conveniently adopted to solve applicative problems of nanoengineering. The motivation of the present research is to develop a well-posed stress-driven two-phase methodology to model the size-dependent dynamic behaviour of small-scale elastic beams, generalising the treatment contributed in [28] confined to stress-driven purely nonlocal nanomaterials. Extension of the stress-driven elasticity mixture to size-dependent buckling and dynamic problems of advanced materials and 2-D structures, such as graphene nanoribbons [31], will be contributed in a forthcoming paper.
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