A new displacement-based framework for non-local Timoshenko beams

Abstract

In this paper, a new theoretical framework is presented for modeling non-locality in shear deformable beams. The driving idea is to represent non-local effects as long-range volume forces and moments, exchanged by non-adjacent beam segments as a result of their relative motion described in terms of pure deformation modes of the beam. The use of these generalized measures of relative motion allows constructing an equivalent mechanical model of non-local effects. Specifically, long-range volume forces and moments are associated with three spring-like connections acting in parallel between couples of nonadjacent beam segments, and separately accounting for pure axial, pure bending and pure shear deformation modes. The variational consistency of the proposed non-local beam model is demonstrated by minimization of an appropriate total potential energy functional. Numerical results concerning the static behavior for different boundary and loading conditions are presented. It is shown that the proposed nonlocal beam model is able to capture experimental data on the static deflection of micro-beams, available in the literature.

Random vibrations of stress-driven nonlocal beams with external damping

“…Paradoxical results and unacceptable nonlocal responses are obtained [36,37] if CBCs are ignored. Several nonlocal theories have been adopted to overcome the aforementioned difficulties, such as: two-phase models [38,39], strain and stress gradient theories [40][41][42][43][44], nonlocal gradient techniques [45][46][47], strain-difference approaches [48][49][50], displacement-based nonlocal models [51][52][53][54], stress-driven formulation of nonlocality [55,56]. Advantageously, the stress-driven approach has been shown to be able to effectively model the nonlocal behaviour of small-scale structures and provides exact solutions for problems of applicative interest in Nano-Engineering [57,58].…”

Stochastic Analysis of a Nonlocal Fractional Viscoelastic Bar Forced by Gaussian White Noise

“…These results are compared with the stationary variances obtained by numerical integration of the PSD functions of the Eq. (29) and depicted in dotted line in Figure 4 for the EV case and in Figure 6 for the VE case.…”

“…As for the non local damping effects at macroscale, clear examples are the application on beams of external damping patches, adhesive joints, surface treatments or the presence of fibers in composites that produce a coupling between nonadjacent points in terms of viscoelastic forces [12,13]. In the last years the authors have proposed non-local EB and TM beam models which treats non-local effects as longrange interactions depending on the relative motion of nonadjacent volume elements [27][28][29][30]. In particular non-local forces are considered as volume forces resulting from a relative motion of non-adjacent beam segments; the relative motion is measured by the pure deformation modes of the beams [31,32] and weighted by a spatial attenuation function.…”

On the nonlocal bending problem with fractional hereditariness

“…However, some mathematical issues and physical paradoxes occur when the model is applied in a bounded domain [41,42]. Nevertheless, there are other various nonlocal approaches to overcome these difficulties, e.g., two-phase models [43,44], strain-difference approaches [45,46], strain and stress gradient theory [47][48][49][50][51][52][53], displacement-based nonlocal model [54][55][56][57], stress-driven integral formulation [58,59]. Among these well-posed nonlocal formulations the stressdriven approach is used in this paper to perform the study of the nonlocal interactions in bending problem.…”