Torsional Statics of Circular Nanostructures: Numerical Approach
Based on the standard finite element method, a new
finite element method which is known as nonlocal finite element
method (NL-FEM) is numerically implemented in this article to
study the nonlocal effects for solving 1D nonlocal elastic problem.
An Eringen-type nonlocal elastic model is considered. In this model,
the constitutive stress-strain law is expressed interms of integral
equation which governs the nonlocal material behavior. The new
NL-FEM is adopted in such a way that the postulated nonlocal elastic
behavior of material is captured by a finite element endowed with a
set of (cross-stiffness) element itself by the other elements in mesh.
An example with their analytical solutions and the relevant numerical
findings for various load and boundary conditions are presented and
discussed in details. It is observed from the numerical solutions that
the torsional deformation angle decreases with increasing nonlocal
nanoscale parameter. It is also noted that the analytical solution fails
to capture the nonlocal effect in some cases where numerical
solutions handle those situation effectively which prove the
reliability and effectiveness of numerical techniques.
C. W. Lim, Z. R. Li, L. H. He, "Size dependent, non-uniform elastic
field inside a nano-scale spherical inclusion due to interface stress,"
International Journal of Solids and Structures, vol. 43 (17), pp. 5055-
D. G. B. Edelen, "Protoelastic bodies with large deformations," Archive
for Rational Mechanics Analysis, vol. 34 pp. 283-300, 1969.
D. G. B. Edelen, A. E. Green, N. Laws, "Nonlocal continuum
mechanics," Archive for Rationale Mechanics and Analysis, vol. 43, pp.
 A. C. Eringen, "Nonlocal polar elastic continua," International Journal
of Engineering Science, vol. 10, pp. 1-16, 1972.
 A.C. Eringen, "Linear theory of nonlocal elasticity and dispersion of
plane waves," International Journal of Engineering Science, vol. 10, pp.
 A. C. Eringen, "Nonlocal Continuum Field Theories," New York,
 A. C. Eringen, D. B. G. Edelen, "On nonlocal elasticity," International
Journal of Engineering Science. Vol. 10, pp. 233-248, 1972.
 A. C. Eringen, "On differential equations of nonlocal elasticity and
solution of screw dislocaltion and surface waves," Journal of Applied
Physics, vol. 54, pp.4703-4710, 1983.
 J. Peddieson, G.R. Buchanan, R. P. McNitt, "Application of nonlocal
continuum models to nanotechnology," International Journal of
Engineering Science, vol. 41, pp. 305-312, 2003.
 L. J. Sudak, "Column buckling of multi-walled carbon nanotubes using
nonlocal continuum mechanics," Journal of Applied Physics, vol. 94
(11), pp. 7281-7287, 2003.
 C. M. Wang, Y.Y. Zhang, S. S. Ramesh, S. Kitipornchai, "Buckling
analysis of micro- and nano-rods/tubes based on nonlocal Timoshenko
beam theory," Journal of Physics D: Applied Physics, vol. 39 (17), pp.
 Y.Y. Zhang, C.M. Wang, W.H. Duan, Y. Xiang, Z. Zong, "Assessment
of continuum mechanics models in predicting buckling of single-walled
carbon nanotubes," Nanotechnology, vol. 20 (39), pp. 395707(8pp),
 T. H. Thai, "A nonlocal beam theory for bending buckling, and vibration
of nanobeams," International Journal of Engineering Science, vol. 52,
pp. 56-64, 2012.
 C. Polizzotto, "Nonlocal elasticity and related variational principles,"
International Journal of Solids and Structures, vol. 38, pp. 7359-7380,
 A. C. Eringen, B. S. Kim, "Stress concentration at the tip of a crack,"
Mechanics Research Communications, vol. 1, pp. 233-237, 1974.
 A. C. Eringen, C. G. Speziale, B.S. Kim, "Crack-tip problem in nonlocal
elasticity," Journal of Mechanics, Physics and Silods, vol. 25, pp. 339-
 A. C. Eringen, "Theory of nonlocal elasticity and some applications,"
Res Mechaica, vol. 21, pp. 313-342, 1987.
 S. B. Altan, "Existance in nonlocal elasticity," Archive Mechanics, vol.
41, pp. 25-36, 1989.
 A. A. Pasano, A. Sofi, P. Fuschi, "Nonlocal integral elasticity: 2D finite
element based solutions," International Journal of Solids and Structures,
vol. 46, pp. 3836-3849, 2009.
D. G. B. Edelen, N. Laws, "On the thermodynamics of systems with
nonlocality," Archive for Rationale Mechanics and Analysis, vil. 43, pp.
E. Kröner, "Elasticity theory of materials with long range cohesive
forces," International Journal of Solids and Structures, vol. 3, pp. 731-
H. Gleiter, "Nanocristalline materials," Progress in Materials Science,
vol. 33, pp. 223-315, 1989.
I. A. Kunin, "The theory of elastic media with microstructures and the
theory of dislocaltion," In: Kröner E. ed. Mechanices of Geeneralized
Continua, Proceedings of IUTAM Symposium 1967. New York:
J. A. Krumhansl, "Some considerations on the relations between solid
state physics and generalized continuum mechanics," In: Kröner E. ed.
Mechanices of Geeneralized Continua. Berlin: Springer-Verlag, 1968,
M. E. Gurtin, A. Murdoch, "A continuum theory of elastic material
surfaces," Archive for Rational Mechanics Analysis, vol. 57 (4), pp.