Raffaele Barretta

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 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 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 continuum dynamics

The theory of continuous dynamical systems is developed with an intrinsic geometric approach based on the action principle formulated in the velocity-time manifold. By endowing the finite dimensional Riemannian ambient manifold with a connection, an induced connection is naturally defined in the infinite dimensional configuration manifold of maps. The motion is shown to be governed, in the Banach configuration manifold, by a generalized Lagrange law and, in the ambient manifold, by a generalized Euler law which is independent of the Banach topology of the configuration manifold. Extended versions of Euler–Poincare law, Euler classical laws and d’Alembert law are also derived as special cases. Stress fields in the body are introduced as Lagrange’s multipliers of the rigidity constraint on virtual velocities, dual to the Lie derivative of the metric. No special assumptions are made so that any constitutive behaviors can be modeled.

Stress-driven modeling of nonlocal thermoelastic behavior of nanobeams

Abstract A consistent stress-driven nonlocal integral model for nonisothermal structural analysis of elastic nano- and microbeams is proposed. Most nonlocal models of literature are strain-driven and it was shown that such approaches can lead toward a number of difficulties. Following recent contributions within the isothermal setting, the developed model abandons the classical strain-driven methodology in favour of the modern stress-driven elasticity theory by G. Romano and R. Barretta. This effectively circumvents issues associated with strain-driven formulations. The new thermoelastic nonlocal integral model is proven to be equivalent to an adequate set of differential equations, accompanied by higher-order constitutive boundary conditions, when the special Helmholtz averaging kernel is adopted in the convolution. The example section provides several applications, thus enabling insight into performance of the formulation. Exact nonlocal solutions are established, detecting also new benchmarks for thermoelastic numerical analyses.

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.

Modulated Linear Dynamics of Functionally Graded Nanobeams With Nonlocal and Gradient Elasticity

Abstract In this contribution some numerical results are presented for the linear dynamics of nanobeams modulated by an axial force. The beam model was recently proposed and encompasses both Eringens usual nonlocal elasticity and second-gradient strain elasticity. Three different combinations of such aspects provide elastic potential energies that give responses highlighting the various levels of contributions of nonlocality and strain gradient. We suppose that these elastic properties are functionally graded on the beam cross-section according to a power law, thus turning some interesting aspects on. A traction 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. The effects of the functional grading and of the different elastic potentials on this modulation are investigated and thoroughly commented.

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.

Exact solutions for a coupled nonlocal model of nanobeams

Bernoulli-Euler nanobeams under concentrated forces/couples with the nonlocal constitutive behavior proposed by Eringen do not exhibit small-scale effects. A new model obtained by coupling the Eringen and gradient models is formulated in the present note. A variational treatment is developed by imposing suitable thermodynamic restrictions for nonlocal models and the ensuing differential and boundary conditions of elastic equilibrium are provided. The nonlocal elastostatic problem is solved in a closed-form for nanocantilever and clamped nanobeams.