Euler-Bernoulli small-scale beams in bending: exact solutions by a nonlocal elastic approach

The recent and fast development of nanotechnology has stimulated the interest of Researchers in the fields of advanced materials and, in general, in a number of well know engineering problems, such as Euler-Bernoulli beam theory, that has to be rephrased with reference to the so called size effects arising at small-scale. Structural elements such as beams, sheets and plates, in micro- or nano-scale, are indeed used as components in many micro- or nano-electromechanical systems devices, so that the analysis of simple structures at small scales has become relevant in both theoretical and computational mechanics. It is well known that size effects at small scales cannot be captured by classical continuum models which are free from any information coming from the micro-structure of the examined element. A tool, among others, to overcome such limit is given by some enhanced continuum models such as the nonlocal ones. The latter, keeping a continuum approach, are able to deal with the above effects introducing in the constitutive description some internal length material parameters. To this concern, a conceptual framework for the study of size effects exhibited by beam structures at micro- and nano-scale is the nonlocal elasticity theory advanced by Eringen in the early eighties (Eringen 1983) and then taken up by a large number of researchers. The Eringen theory is characterized by a stress-strain relation of integral type, in which the stress measured at a point is expressed as a mean weighted value in terms of the strain measured at all points within the domain occupied by the material. There exists a huge literature in which the Eringen nonlocal theory is applied to beams and plates, but, twenty years later from its publication Peddieson et al. (2003) found a “paradoxal” result in applying that theory to simple beams in bending. Namely, they found that this theory predicts no-size effects in a cantilever beam subjected to point load(s), gives rise to stiffening size effects under uniform load and, in general, predicts softening size effects for beams with different constraint conditions. Size-effects appear so related to load and constraint conditions. Such paradoxes envisaged by Peddieson et al. seems to be confirmed by recent laboratory experiments performed to measure size effects by Abazari et al. (2015). These measurements show, in fact, that stiffness in general increases with size decreasing in accord with the smaller is stronger phenomenon. A lively discussion is taking place today. The present contribution can be inserted in the above discussion. The main purpose of the study is to show that, for the analysis of size effects in small-scale beams in bending, it can be usefully applied the so-called strain-difference based nonlocal elasticity model proposed by the present authors (Polizzotto et al., 2006). Starting from this model, exact solutions for the Euler-Bernoulli beam problem, under several loading and boundary conditions, will be given. As a result, it will be shown that the strain-difference based nonlocal model always predicts stiffening size effects in small-scale beams, even in the case of the (paradoxal) cantilever beam subjected to point loads. References [1] Eringen A.C., 1983. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys. 54, 4703–4710. [2] Peddieson J., Buchanan G.R., McNitt R.P., 2003. Application of nonlocal continuous models to nanotechnology, Int. J. Eng. Sci. 41, 305–312. [3] Abazari A.M., Safavi S.M., Rezazadeh G., Villanueva L.G., 2015. Modelling the size effects on the mechanical properties of micro/nano structures. Sensors 15, 28543–28562. [4] Polizzotto C., Fuschi P., Pisano A.A., 2006. A nonhomogeneous nonlocal elasticity model. Eur. J. Mech. Sci. A/Solids 25, 308–333.