Well-posed nonlocal elasticity model for finite domains and its application to the mechanical behavior of nanorods
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O R I G I NA L PA P E R
Mohammad A. Maneshi · Esmaeal Ghavanloo Fazelzadeh
· S. Ahmad
Well-posed nonlocal elasticity model for finite domains and its application to the mechanical behavior of nanorods
Received: 22 January 2020 / Revised: 10 May 2020 © Springer-Verlag GmbH Austria, part of Springer Nature 2020
Abstract Eringen’s nonlocal elasticity theory is one of the most attractive approaches to investigate the intrinsic scale effect of nanoscopic structures. Eringen proposed both integral and differential nonlocal models which are equivalent to each other over unbounded continuous domains. Although the Eringen nonlocal models can be used as very useful tools for modeling the mechanical characteristics of nanoscopic structures, however, several researchers have reported some paradoxical results when they used the nonlocal differential model. In this paper, we develop a well-posed nonlocal differential model for finite domains, and its applicability to predict the static and dynamic behavior of a nanorod is investigated. It is shown that the proposed integral and differential nonlocal models are equivalent to each other over bounded continuous domains, and the corresponding elastic problems are well-posed and consistent. In addition, some paradigmatic static problems are solved the and we show that the paradoxical results disappear by using the present model.
1 Introduction Nonlocal continuum theory is one of the most efficient approaches to model the mechanical characteristics of nanoscopic structures. The initial concepts of this theory were proposed by Kröner [1]. A systematic representation of the nonlocal continuum field theories was developed by Eringen [2]. Eringen’s nonlocal elasticity theory is one of most attractive fields among various topics of the nonlocal theories. In this theory, the stress is calculated from an integral of strain with a weight function over the whole body. This weight function is the so-called averaging kernel. The kernel is assumed to fulfill some mathematical properties including symmetry, limit impulsivity, and normalization conditions [3]. Eringen [4] proposed the differential type of the nonlocal elasticity theory by using the Green’s function approach. The nonlocal differential type is equivalent to the nonlocal integral type over unbounded continuous domains, and the corresponding elastic problems are well-posed. In the year 2003, the first attempt to show the applicability of the nonlocal differential model for predicting the mechanical behavior of nano-beams was carried out by Peddieson et al. [5]. They indicated that the differential type of the nonlocal elasticity theory could be used in nanotechnology applications. They also found that the nonlocal transverse deflections of a cantilever nano-beam subjected to concentrated forces are identical to the local ones, and so the nonlocality does not produce any changes of the results. In other words, the differential nonlocal model is not able by its own nature to predict size effects in the cantilever beam. It is the fi
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