New trends in nano-engineering

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A Nonlocal Model for Carbon Nanotubes under Axial Loads

Various beam theories are formulated in literature using the nonlocal differential constitutive relation proposed by Eringen. A new variational framework is derived in the present paper by following a consistent thermodynamic approach based on a nonlocal constitutive law of gradient-type. Contrary to the results obtained by Eringen, the new model exhibits the nonlocality effect also for constant axial load distributions. The treatment can be adopted to get new benchmarks for numerical analyses

ON MAUPERTUIS PRINCIPLE IN DYNAMICS

A new statement of Maupertuis principle of extremal action is contributed on the basis of a constrained action principle in the velocity phase-space in which the condition of energy conservation is imposed on virtual velocities. Dynamical systems governed by time-dependent Lagrangians on nonlinear configuration manifolds and subject to the action of time-dependent forces are considered. In time-independent systems, and in particular in conservative systems, the constrained action principle specializes to a formulation of the original Maupertuis least action principle in which however conservation of energy along the trajectory is a natural consequence of the variational principle and not an a priori assumption as in classical statements

A Fully Gradient Model for Euler-Bernoulli Nanobeams

A fully gradient elasticity model for bending of nanobeams is proposed by using a nonlocal thermodynamic approach. As a basic theoretical novelty, the proposed constitutive law is assumed to depend on the axial strain gradient, while existing gradient elasticity formulations for nanobeams contemplate only the derivative of the axial strain with respect to the axis of the structure. Variational equations governing the elastic equilibrium problem of bending of a fully gradient nanobeam and the corresponding differential and boundary conditions are thus provided. Analytical solutions for a nanocantilever are given and the results are compared with those predicted by other theories. As a relevant implication of applicative interest in the research field of nanobeams used in nanoelectromechanical systems (NEMS), it is shown that displacements obtained by the present model are quite different from those predicted by the known gradient elasticity treatments

On the regularity of curvature fields in stress-driven nonlocal elastic beams

AbstractElastostatic problems of Bernoulli–Euler nanobeams, involving internal kinematic constraints and discontinuous and/or concentrated force systems, are investigated by the stress-driven nonlocal elasticity model. The field of elastic curvature is output by the convolution integral with a special averaging kernel and a piecewise smooth source field of elastic curvature, pointwise generated by the bending interaction. The total curvature is got by adding nonelastic curvatures due to thermal and/or electromagnetic effects and similar ones. It is shown that fields of elastic curvature, associated with piecewise smooth source fields and bi-exponential kernel, are continuously differentiable in the whole domain. The nonlocal elastic stress-driven integral law is then equivalent to a constitutive differential problem equipped with boundary and interface constitutive conditions expressing continuity of elastic curvature and its derivative. Effectiveness of the interface conditions is evidenced by the solution of an exemplar assemblage of beams subjected to discontinuous and concentrated loadings and to thermal curvatures, nonlocally associated with discontinuous thermal gradients. Analytical solutions of structural problems and their nonlocal-to-local limits are evaluated and commented upon

Modulated linear dynamics of nanobeams accounting for higher gradient effects

We present some numerical results for the linear dynamics of nanobeams modulated by an axial force, basing on a recent proposal of literature that encompasses both the standard nonlocal elasticity, according to Eringen, and second-order strain elasticity. Three different possibilities for the elastic potential energy provide different responses that highlight the contributions of nonlocality and strain gradient, plus their combination. An axial force affects the linear stationary dynamics of such nanobeams, inducing suitable variation of the natural angular frequencies for benchmark cases, until static buckling occurs when the natural angular frequency vanishes. Effects of the various elastic potentials on this modulation are investigated and thoroughly commented

Nano-beams under torsion: a stress-driven nonlocal approach

Purpose This study aims to model scale effects in nano-beams under torsion. Design/methodology/approach The elastostatic problem of a nano-beam is formulated by a novel stress-driven nonlocal approach. Findings Unlike the standard strain-driven nonlocal methodology, the proposed stress-driven nonlocal model is mathematically and mechanically consistent. The contributed results are useful for the design of modern devices at nanoscale. Originality/value The innovative stress-driven integral nonlocal model, recently proposed in literature for inflected nano-beams, is formulated in the present submission to study size-dependent torsional behavior of nano-beams