Structural modelling of nanorods and nanobeams using doublet mechanics theory
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Structural modelling of nanorods and nanobeams using doublet mechanics theory Ufuk Gul . Metin Aydogdu
Received: 7 February 2017 / Accepted: 7 April 2017 Springer Science+Business Media Dordrecht 2017
Abstract In this study, statics and dynamics of nanorods and nanobeams are investigated by using doublet mechanics. Classical rod theory and Euler– Bernoulli beam theory is used in the formulation. After deriving governing equations static deformation, buckling, vibration and wave propagation problems in nanorods and nanobeams are investigated in detail. The results obtained by using of doublet mechanics are compared to that of the classical elasticity theory. The importance of the size dependent mechanical behavior at the nano scale is shown in the considered problems. In doublet mechanics, bond length of atoms of the considered solid are used as an intrinsic length scale. Keywords Bending Buckling Doublet mechanics Vibration Wave propagation Carbon nanotubes
1 Introduction In his famous talk (There’s Plenty of Room at the Bottom) Feynman (1960) stated the possibility of direct manipulation of individual atoms in order to design nanoscale devices and structures. In recent U. Gul M. Aydogdu (&) Department of Mechanical Engineering, Trakya University, 22030 Edirne, Turkey e-mail: [email protected]
years, nanostructures have taken great interest due to their extraordinary properties. They are used in composites as a fiber, in atomic force microscope as a probe tip and in tissue engineering scaffolding for bone growth. Moreover, they are planned to be used in solar cells, hydrogen storage, actuator and nanomotors like nanoscale applications. In order to design nanoscale structures, different size dependent continuum models have been used such as stress and strain type gradient models, modified couple stress theory and peridynamics. Couple stress theory has been proposed by Cosserat brothers (1909) is the first generalized continuum theory. Then, Mindlin (1964) proposed the strain gradient theory. Eringen (1976, 1983) proposed a stress gradient nonlocal elasticity (NL) theory. Aydogdu (2009a, b) investigated the statics and dynamics of nanobeams by using a general nonlocal elasticity theory. The main purpose of these theories was the consideration of the internal length scale when modelling their mechanical responses. Stress gradient nonlocal model has been used for static and dynamic analysis of nanotubes (Peddieson et al. 2003; Sudak 2003; Ece and Aydogdu 2007; Reddy 2007; Aydogdu 2009a, b) and this model has been reviewed by Arash and Wang (2012). The strain gradient model has been applied to many problems during last three decades (Aifantis 1992; Askes et al. 2002; Tsepoura et al. 2002; Beskou et al. 2003a). Considering the positive or negative sign of the gradient term softening and hardening behaviors occur with respect to classical
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elasticity in the materials. Strain gradient elasticity model with negative sign is derived from a positive energy functional whereas posi
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