Free Vibrations of Bernoulli-Euler Nanobeams with Point Mass Interacting with Heavy Fluid Using Nonlocal Elasticity.

Eigenfrequencies of a nanobeam with a point mass interacting with a heavy fluid are calculated using Bernoulli-Euler kinematics and consistent nonlocal elasticity model. The proposed approach is applicable to a variety of nanotechnology materials and structures, especially mass nanosensors. Eigenfrequencies are compared with those of local theory and conclusions are drawn. Influence of nonlocal effects, heavy fluid interaction and added point mass on dynamic responses is analyzed and the results are documented. Size phenomena are noted and discussed.

Dynamics of Stress-Driven Two-Phase Elastic Beams.

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).

Elastostatics of Bernoulli-Euler Beams Resting on Displacement-Driven Nonlocal Foundation.

The simplest elasticity model of the foundation underlying a slender beam under flexure was conceived by Winkler, requiring local proportionality between soil reactions and beam deflection. Such an approach leads to well-posed elastostatic and elastodynamic problems, but as highlighted by Wieghardt, it provides elastic responses that are not technically significant for a wide variety of engineering applications. Thus, Winkler's model was replaced by Wieghardt himself by assuming that the beam deflection is the convolution integral between soil reaction field and an averaging kernel. Due to conflict between constitutive and kinematic compatibility requirements, the corresponding elastic problem of an inflected beam resting on a Wieghardt foundation is ill-posed. Modifications of the original Wieghardt model were proposed by introducing fictitious boundary concentrated forces of constitutive type, which are physically questionable, being significantly influenced on prescribed kinematic boundary conditions. Inherent difficulties and issues are overcome in the present research using a displacement-driven nonlocal integral strategy obtained by swapping the input and output fields involved in Wieghardt's original formulation. That is, nonlocal soil reaction fields are the output of integral convolutions of beam deflection fields with an averaging kernel. Equipping the displacement-driven nonlocal integral law with the bi-exponential averaging kernel, an equivalent nonlocal differential problem, supplemented with non-standard constitutive boundary conditions involving nonlocal soil reactions, is established. As a key implication, the integrodifferential equations governing the elastostatic problem of an inflected elastic slender beam resting on a displacement-driven nonlocal integral foundation are replaced with much simpler differential equations supplemented with kinematic, static, and new constitutive boundary conditions. The proposed nonlocal approach is illustrated by examining and analytically solving exemplar problems of structural engineering. Benchmark solutions for numerical analyses are also detected.