Various beam theories are formulated in literature using the nonlocal differential constitutive relation proposed by Eringen. A new variational framework is derived in the present paper by following a consistent thermodynamic approach based on a nonlocal constitutive law of gradient-type. Contrary to the results obtained by Eringen, the new model exhibits the nonlocality effect also for constant axial load distributions. The treatment can be adopted to get new benchmarks for numerical analyses. 1. Introduction Carbon nanotubes (CNTs) are a topic of major interest both from theoretical and applicative points of view. This subject is widely investigated in literature to describe small-scale effects [1–4], vibration and buckling [5–13], and nonlocal finite element analysis [14–18]. A comprehensive review on applications of nonlocal elastic models for CNTs is reported in  and therein references. Buckling of triple-walled CNTs under temperature fields is dealt with in . An alternative methodology is based on an atomistic-based approach  which predicts the positions of atoms in terms of interactive forces and boundary conditions. The standard approach to analyze CNTs under axial loads consists in solving an inhomogeneous second-order ordinary differential equation providing the axial displacement field, see, for example, . The known term of the differential equation is the sum of two contributions. The former describes the local effects linearly depending on the axial load. The latter characterizes the small-scale effects depending linearly on the second derivative along the rod axis of the axial load. This model is thus not able to evaluate small-scale effects due to, constant axial loads per unit length. This approach, commonly adopted in literature, is based on the following nonlocal linearly elastic constitutive law proposed by Eringen : where is a material constant, is the internal length, is the Young modulus, is the normal stress, the apex is second derivative along the rod axis, and is the axial elongation. Indeed, integrating on the rod cross section domain and imposing that the axial force is equal to the resultant of normal stress field we get the differential equation where , with being first derivative along the rod axis of the axial displacement field , where is the rod length and denotes the cross section area. Since the equilibrium prescribes that the first derivative of is opposite to the axial load , we infer the well-known differential equation (see, e.g., ) as follows: Note that the nonlocal contribution vanishes for
M. A. De Rosa and C. Franciosi, “A simple approach to detect the nonlocal effects in the static analysis of Euler-Bernoulli and Timoshenko beams,” Mechanics Research Communications, vol. 48, pp. 66–69, 2013.
M. A. Kazemi-Lari, S. A. Fazelzadeh, and E. Ghavanloo, “Non-conservative instability of cantilever carbon nanotubes resting on viscoelastic foundation,” Physica E, vol. 44, no. 7-8, pp. 1623–1630, 2012.
H.-T. Thai and T. P. Vo, “A nonlocal sinusoidal shear deformation beam theory with application to bending, buckling, and vibration of nanobeams,” International Journal of Engineering Science, vol. 54, pp. 58–66, 2012.
B. Fang, Y.-X. Zhen, C.-P. Zhang, and Y. Tang, “Nonlinear vibration analysis of double-walled carbon nanotubes based on nonlocal elasticity theory,” Applied Mathematical Modelling, vol. 37, no. 3, pp. 1096–1107, 2013.
M. ？im？ek and H. H. Yurtcu, “Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory,” Composite Structures, vol. 97, pp. 378–386, 2013.
J. K. Phadikar and S. C. Pradhan, “Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates,” Computational Materials Science, vol. 49, no. 3, pp. 492–499, 2010.
C. M. C. Roque, A. J. M. Ferreira, and J. N. Reddy, “Analysis of Timoshenko nanobeams with a nonlocal formulation and meshless method,” International Journal of Engineering Science, vol. 49, no. 9, pp. 976–984, 2011.
F. F. Mahmoud, M. A. Eltaher, A. E. Alshorbagy, and E. I. Meletis, “Static analysis of nanobeams including surface effects by nonlocal finite elements,” Journal of Mechanical Science and Technology, vol. 26, no. 11, pp. 3555–3563, 2012.
M. A. Eltaher, A. E. Alshorbagy, and F. F. Mahmoud, “Vibration analysis of Euler-Bernoulli nanobeams by using finite element method,” Applied Mathematical Modelling, vol. 37, no. 7, pp. 4787–4797, 2013.
Y. Yan, W. Q. Wang, and L. X. Zhang, “Nonlocal effect on axially compressed buckling of triple-walled carbon nanotubes under temperature field,” Applied Mathematical Modelling, vol. 34, no. 11, pp. 3422–3429, 2010.
F. Marotti de Sciarra, “Variational formulations, convergence and stability properties in nonlocal elastoplasticity,” International Journal of Solids and Structures, vol. 45, no. 7-8, pp. 2322–2354, 2008.