Research Article
A Note on Torsion of Nonlocal Composite Nanobeams
Luciano Feo and Rosa Penna
Department of Civil Engineering, University of Salerno, Via Ponte don Melillo, 84084 Fisciano, Italy
Correspondence should be addressed to Luciano Feo; l.feo@unisa.it
Received 4 December 2015; Accepted 5 October 2016
Academic Editor: Teodoros C. Rousakis
Copyright © 2016 L. Feo and R. Penna. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Te Eringen elastic constitutive relation is used in this paper in order to assess small-scale efects in nanobeams. Structural behavior
is studied for functionally graded materials in the cross-sectional plane and torsional loading conditions. Te governing boundary
value problem has been formulated in a mixed framework. Torsional rotations and equilibrated moments are evaluated by solving
a frst-order diferential equation of elastic equilibrium with boundary conditions of kinematic-type. Benchmarks examples are
briefy discussed, enlightening thus efectiveness of the proposed methodology.
1. Introduction
Assessments of stress and displacement felds in continuous
media are a subject of special interest in the theory of
structures. Numerous case studies have been examined in
the current literature with reference to beams [1–6], half-
spaces [7, 8], thin plates [9, 10], compressible cubes [11],
and concrete [12, 13]. Several methodologies of analysis have
been developed in the research feld of geometric continuum
mechanics [14, 15], limit analysis [16–19], homogenization
[20], elastodynamics [21–25], thermal problems [26–28],
random composites [29–32], and nonlocal and gradient
formulations [33–38]. A comprehensive analysis of classical
and generalized models of elastic structures, with special
emphasis on rods, can be found in the interesting book by
Ies ¸an [39]. In particular, Ies ¸an [40–42] formulated a method
for the solution of Saint-Venant problems in micropolar
beams with arbitrary cross-section. Detailed solution of the
torsion problem for an isotropic micropolar beam with
circular cross-section is given in [43, 44]. Experimental
investigations are required for the evaluation of the behavior
of composite structures [45].
In the context of the present research, particular attention
is devoted to the investigation of scale efects in nanostruc-
tures; see, for example, [46–55] and the reviews [56, 57].
Recent contributions on functionally graded materials have
been developed for nanobeams under fexure [58, 59] and
torsion [60].
Unlike previous treatments on torsion of gradient elastic
bars (see, e.g., [61]) in which higher-order boundary condi-
tions have to be enforced, this paper is concerned with the
analysis of composite nanobeams with nonlocal constitutive
behavior conceived by Eringen in [62]. Basic equations
governing the Eringen model are preliminarily recalled in
Section 2. Te corresponding elastic equilibrium problem of
torsion of an Eringen circular nanobeam is then formulated
in Section 3. It is worth noting that only classical boundary
conditions are involved in the present study. Small-scale
efects are detected in Section 4 for two static schemes of
applicative interest. Some concluding remarks are delineated
in Section 5.
2. Eringen Nonlocal Elastic Model
Before formulating the elastostatic problem of a nonlocal
nanobeam subjected to torsion, we shortly recall in the
sequel some notions of nonlocal elasticity. To this end, let us
consider a body B made of a material, possibly composite,
characterized by the following integral relation between the
stress t
at a point x and the elastic strain feld e
in B [62]:
t
(x)=∫
B
(
x
− x
,) E
ℎ
(x
) e
ℎ
(x
) . (1)
Te fourth-order tensor E
ℎ
(x
), symmetric and positive
defnite, describes the material elastic stifness at the point
x
∈ B.
Hindawi Publishing Corporation
Modelling and Simulation in Engineering
Volume 2016, Article ID 5934814, 5 pages
http://dx.doi.org/10.1155/2016/5934814