Some New Approximate Solutions in Closed-Form to Problems of Nanobars

Year 2021, , 161 - 167, 20.12.2021

Uğurcan Eroğlu

https://doi.org/10.26701/ems.773106

Abstract

Following recent technological advancements, a great attention has been paid to mechanical behaviour of structural elements of nanosize. In this study, some solutions to mechanical problems of bars of nanosize is examined using Eringen’s two-phase nonlocal elasticity. Assuming the fraction coefficient of nonlocal part of the material is small, a perturbation expansion with respect to it is performed. With this procedure, the original nonlocal problem is broken into a set of local elasticity problems. Solutions to some example problems of nanobars are provided in closed-form for the first time, and commented on. The new solutions provided herein may well serve for benchmark studies, as well as identification of material parameters of nano-sized structural elements, such as carbon nanotubes.

Keywords

nanobars, nonlocal elasticity, nanomechanics, closed-form solutions, approximate methods

References

• Navier, C.-L.-M.-H. (1827) . Mémoire sur le lois de l’équilibre et du mouvement des corps solides élastiques (1821), Mémoires de l’Academie des Sciences de l’Institut de France, s. II, 7: 375–393.
• Cauchy, A.-L. (1828) . Sur l’équilibre et le mouvement d’un système de points matériels sollicités par des forces d’attraction ou de répulsion mutuelle, Exercices de Mathématiques, 3, 188–213, 1822, 1827; Oeuvres, 2(8): 227–252 .
• Poisson, S.D. (1829). Mémoire sur l’équilibre et le mouvement des corps élastiques 1828. Mémoires de l’Académie des Sciences de l’Institut de France, s. II, 8: 357–380.
• Trovalusci, P., Capecchi, D., Ruta, G. (2009). Genesis of the multiscale approach for materials with microstructure. Archive of Applied Mechanics 79: 981-997.
• Mindlin, R. D. (1964). Micro-structure in linear elasticity. Archive for Rational Mechanics and Analysis, 16(1): 51–78,.
• Kunin, I. A. (1968). The theory of elastic media with microstructure and the theory of dislocation. Kröner, E. (Eds.), Mechanics of Generalized Continua, Springer, Berlin Heidelberg, p. 321.
• Capriz, G. (1989). Continua with Microstructure. Springer Tracts in Natural Philosophy. Springer-Verlag.
• Maugin, G.A. (1993). Material Inhomogeneities in Elasticity. Applied Mathematics. Taylor & Francis.
• Eringen, A. C. (1999). Microcontinuum Field Theory. Springer.
• Eringen, A. C. (2002). Nonlocal Continuum Field Theories. Springer-Verlag.
• Rapaport, D.C. (1995). The Art of Molecular Dynamics Simulation. Cambridge University Press, Cambridge.
• Baggio, C., Trovalusci, P. (1998). Limit analysis for no-tension and friction three dimensional discrete systems. Mechanics of Structures and Mach., 26:287 – 304,.
• Trovalusci, P. (2014). Molecular approaches for multifield continua: origins and current developments. Tomasz Sadowski and Patrizia Trovalusci (Eds.), Multiscale Modeling of Complex Materials: Phenomenological, Theoretical and Computational Aspects. Springer Vienna, p. 211–278.
• Kunin, I. A. (1984). On foundations of the theory of elastic media with microstructure. International Journal of Engineering Sciences, 22(8):969 – 978.
• Tuna, M., Leonetti, L., Trovalusci, P., Kirca, M. (2020). ‘Explicit’ and ‘implicit’ non-local continuous descriptions for a plate with circular inclusion in tension. Meccanica 55: 927–944.
• Tuna, M., Trovalusci, P. (2020). Scale dependent continuum approaches for discontinuous assemblies: ’explicit’ and ’implicit’ non-local models. Mech. Res. Commun., 103:103461, 6 pages.
• Abdollahi, R., Boroomand, B. (2013) Benchmarks in nonlocal elasticity defined by Eringen’s integral model. International Journal of Solids and Structures 50(18): 2758-2771.
• Benvenuti, E., Simone, A. (2013). One-dimensional nonlocal and gradient elasticity: Closed-form solution and size effect. Mechanics Research Communications, 48: 46-51.
• Zaera R., Serrano, Ó., Fernández-Sáez, J. (2019). On the consistency of the nonlocal strain gradient elasticity. International Journal of Engineering Sciences 138:65-81.
• Eroglu, U. (2020). Perturbation approach to Eringen’s local/non-local constitutive equation with applications to 1-D structures. Meccanica, 55: 1119-1134.
• Romano, G., Barretta, R., Diaco, M., de Sciarra, F.M. (2017). Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams. International Journal of Mechanical Sciences 121:151-156.
• Polyanin, P., Manzhirov, A. (2008). Handbook of integral equations. Chapman and Hall/CRC, London.