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Size-dependent longitudinal and torsional vibrations of nano-beams are examined by two-phase mixture integral elasticity. A new and efficient elastodynamic model is conceived by convexly combining the local phase with strain- and stress-driven purely nonlocal phases. The proposed stress-driven nonlocal integral mixture leads to well-posed structural problems for any value of the scale parameter. Effectiveness of stress-driven mixture is illustrated by analyzing axial and torsional free vibrations of cantilever and doubly clamped nano-beams. The local/nonlocal integral mixture is conveniently replaced with an equivalent differential law equipped with higher-order constitutive boundary conditions. Exact solutions of fundamental natural frequencies associated with strain- and stress-driven mixtures are evaluated and compared with counterpart results obtained by strain gradient elasticity theory. The provided new numerical benchmarks can be effectively employed for modelling and design of Nano-Electro-Mechanical-Systems (NEMS).

A consistent nonlocal viscoelastic beam model is proposed in this paper. Specifically, a Timoshenko bending problem, where size- and time-dependent effects cannot be neglected, is investigated. In order to inspect scale phenomena, a stress-driven nonlocal formulation is used, whereas to simulate time-dependent effects, fractional linear viscoelasticity is considered. These two approaches are adopted to develop a new Timoshenko bending model. Analytical solutions and application samples of the proposed formulation are presented. Moreover, in order to show influences of viscoelastic and size effects on mechanical response, parametric analyses are provided. The contributed results can be useful for the design and optimization of small-scale devices exhibiting flexural behaviour.

The bending behaviour of systems of straight elastic beams at different scales is investigated by the well-posed stress-driven nonlocal continuum mechanics. An effective computational methodology, based on nonlocal two-noded finite elements, is developed in order to take accurately into account long-range interactions present in the whole structural domain. The idea consists in partitioning the beam in subdomains and in observing that the nonlocal stress-driven convolution integral, equipped with the Helmholtz averaging kernel, can be equivalently formulated by expressing nonlocal bending interaction fields in terms of zero-th and second-order derivatives of elastic curvature fields which have to fulfil appropriate non-classical constitutive boundary and interface conditions. Relevant mesh-dependent shape functions governing the FEM technique are analytically detected. Each element is characterized by shape functions whose number is equal to four times the number of elements of the considered mesh. A simple analytical strategy to obtain nonlocal stiffness matrices and equivalent nodal forces of a finite element is exposed. The global nonlocal stiffness matrix is got by assembling the nonlocal element stiffness matrices accounting for long-range interactions among the elements. The proposed numerical approach is examined by exactly solving exemplar nonlocal case-problems of current interest in nano-engineering. The presented nonlocal strategy extends previous contributions on the matter and offers designers a consistent computational tool.

In this research, the size-dependent static behaviour of elastic curved stubby beams is investigated by Timoshenko kinematics. Stress-driven two-phase integral elasticity is adopted to model size effects which soften or stiffen classical local responses. The corresponding governing equations of nonlocal elasticity are established and discussed, non-classical boundary conditions are detected and an effective coordinate-free solution procedure is proposed. The presented mixture approach is elucidated by solving simple curved small-scale beams of current interest in Nanotechnology. The contributed results could be useful for design and optimization of modern sensors and actuators.

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).

The size-dependent bending behavior of nano-beams is investigated by the modified nonlocal strain gradient elasticity theory. According to this model, the bending moment is expressed by integral convolutions of elastic flexural curvature and of its derivative with the special bi-exponential averaging kernel. It has been recently proven that such a relation is equivalent to a differential equation, involving bending moment and flexural curvature fields, equipped with natural higher-order boundary conditions of constitutive type. The associated elastostatic problem of a Bernoulli-Euler functionally graded nanobeam is formulated and solved for simple statical schemes of technical interest. An effective analytical approach is presented and exploited to establish exact expressions of nonlocal strain gradient transverse displacements of doubly clamped, cantilever, clamped-simply supported and simply supported nano-beams, detecting thus also new benchmarks for numerical analyses. Comparisons with results of literature, corresponding to selected higher-order boundary conditions are provided and discussed. The considered nonlocal strain gradient model can be advantageously adopted to characterize scale phenomena in nano-engineering problems.

Numerous contributions can be found in the recent literature exploiting the nonlocal strain gradient model, introduced in consequence of unification of the differential relation (consequent but not equivalent to Eringen nonlocal integral law) and strain gradient elasticity. In the present paper, Eringen nonlocal integral and Lam modified strain gradient theories are coupled to formulate a nonlocal Lam strain gradient model of elasticity. Three scale parameters, describ-ing nonlocality, dilatation, and stretch gradient, are utilized to significantly estimate size-dependent responses of 1D nanocontinua. The governing constitutive law is established via a variationally consistent approach, based on suitably selected test fields, projected for formulating well-posed static and dynamic problems of engineering interest. The non-local Lam strain gradient model, developed for nanorods, provides axial force fields in terms of integral convolutions involving elastic axial strain fields. The integral law, equivalent to an expedient set of constitutive differential and boundary conditions, is exploited for studying static and free vibration behaviors of simple nanostructural schemes. Exact analytical solutions are gotten in terms of nonlocal and gradient characteristic parameters. Validation of the proposed strategy is carried out by comparing the contributed results with those obtained by the modified nonlocal strain gradient theory.

In this paper, the mechanical behavior of multilayered small-scale beams in nonisothermal environment is investigated. Scale phenomena are modeled by means of the mathematically well-posed and experimentally consistent stress-driven integral formulation of elasticity. The present research extends the treatment in Barretta, Canađija, Luciano, and de Sciarra (2018b) confined to elastically homogeneous nano-scopic structures. It is shown that the non-locality leads to a complex coupling between axial and transverse elastic displacements. Such a size-dependent phenomenon makes the solution of the relevant nonlocal thermoelastostatic problem, governed by a system of two ordinary differential equations with ten standard boundary conditions and non-classical constitutive boundary conditions, significantly more involved with respect to treatments in literature. Thus, a novel solution methodology, based on Laplace transforms, is proposed and illustrated by examining simple structural schemes of current applicative interest in Nanomechanics and Nanotechnology.

Size-dependent buckling of compressed Bernoulli-Euler nano-beams is investigated by stress-driven nonlocal continuum mechanics. The nonlocal elastic strain is obtained by convoluting the stress fie...

An effective nonlocal integral formulation for functionally graded Bernoulli-Euler beams in nonisothermal environment is developed. Both thermal and mechanical loadings are accounted for. The proposed model, of stress-driven integral type, is shown to be governed by a thermodynamically consistent differential problem with proper constitutive boundary conditions. The new thermoelastic strategy is illustrated by investigating a set of examples. It is demonstrated that in nonisothermal statically indeterminate problems rather complex structural behaviours can appear and that both the shift of the neutral surface and nonlocality have a dominating influence at small-scales.