A Note on Torsion of Nonlocal Composite Nanobeams

TheEringen elastic constitutive relation is used in this paper in order to assess small-scale effects in nanobeams. Structural behavior is studied for functionally graded materials in the cross-sectional plane and torsional loading conditions. The governing boundary value problem has been formulated in a mixed framework. Torsional rotations and equilibrated moments are evaluated by solving a first-order differential equation of elastic equilibrium with boundary conditions of kinematic-type. Benchmarks examples are briefly discussed, enlightening thus effectiveness of the proposed methodology.

Unlike previous treatments on torsion of gradient elastic bars (see, e.g., [61]) in which higher-order boundary conditions 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. The 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 effects are detected in Section 4 for two static schemes of applicative interest.Some concluding remarks are delineated in Section 5.

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 field e  in B [62]: The fourth-order tensor E ℎ (x  ), symmetric and positive definite, describes the material elastic stiffness at the point The attenuation function  depends on the Euclidean distance |x  − x| and on a nonlocal dimensionless parameter defined by where  0 is a material constant,  is an internal characteristic length, and  stands for external characteristic length.Assuming a suitable expression of the nonlocal modulus  in terms of a variant of the Bessel function, we get the inverse differential relationship of (1) between the nonlocal stress and the elastic deformation with ∇ 2 the Laplacian.Note that (3) can be conveniently used in order to describe the law between the nonlocal shear stress field   on the cross-section of a nanobeam and the elastic shear strain   as follows: where  is the axial direction and  is the shear modulus.

Torsion of Nonlocal Circular Nanobeams
Let Ω be the cross-section of a circular nanobeam, of length , subjected to the following loading conditions depicted in Figure 1: distributed couples per unit length in the interval [0, ] , M  , concentrated couples at the end cross-sections {0, } . ( The triplet (, , ) describes a set of Cartesian axes originating at the left cross-section centre O. Equilibrium equations are expressed by where   is the twisting moment.
Components of the displacement field, up to a rigid body motion, of a circular nanobeam under torsion write as   (, ) = 0,   (, ) = − () ,   (, ) =  () , (7) where () is the torsional rotation of the cross-section at the abscissa .Shear strains, compatible with the displacement field equation (7), are given by where is the twisting curvature.The shear modulus  is assumed to be functionally graded only in the cross-sectional plane (, ).
The elastic twisting stiffness is provided by where the symbol ⋅ is the inner product between vectors.
The differential equation of nonlocal elastic equilibrium of a nanobeam under torsion is formulated as follows.Let us preliminarily multiply (4) by Rr fl {−, } and integrate on (11) with the vector  fl {  ,   } given by (8).Enforcing (6a) and imposing the static equivalence condition we get the relation This equation can be interpreted as decomposition formula of the twisting curvature   into elastic (  ) el and inelastic (  ) in parts with Accordingly, the scale effect exhibited by the torsional rotation function of a nonlocal nanobeam can be evaluated by solving a corresponding linearly elastic beam subjected to the twisting curvature distortion (  ) in (15b).

Benchmark Examples
Let us consider a nanocantilever and a fully clamped nanobeam of length  subjected to the following quadratic distribution of couples per unit length: The cross-sectional torsional rotation is evaluated by following the methodology illustrated in the previous section.The nonlocality effect is assessed by prescribing the twisting curvature distortion equation (15b)

Concluding Remarks
The basic outcomes contributed in the present paper are listed as follows: (1) Size-effects in nanobeams under torsion have been evaluated by resorting to the nonlocal theory of elasticity.
(2) Exact torsional rotations solutions of cross-sections of functionally graded nanobeams have been established for nanocantilevers and fully clamped nanobeams under a quadratic distribution of couples per unit length.
(3) It has been observed that the stiffness of a nanobeam under torsional loadings is affected by the scale parameter and depends on the boundary kinematic constraints.Indeed, as shown in Figures 2 and 3, contrary to the nanocantilever structural behavior, the fully clamped nanobeam becomes stiffer for increasing values of the nonlocal parameter.
) on corresponding (cantilever and fully clamped) local nanobeams.Let us set  = / and  * () =   /( 2 )().Torsional rotations  * versus  of both the nanobeams are displayed in Figures 2 and 3 for selected values of the nonlocal parameter  fl  0 /.