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Steel and Composite Structures Volume 35, Number 3, May10 2020 , pages 449-462 DOI: https://doi.org/10.12989/scs.2020.35.3.449 |
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Free axial vibration of cracked axially functionally graded nanoscale rods incorporating surface effect |
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Reza Nazemnezhad and Hassan Shokrollahi
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| Abstract | ||
| This work aims to study effects of the crack and the surface energy on the free longitudinal vibration of axially functionally graded nanorods. The surface energy parameters considered are the surface stress, the surface density, and the surface Lamé constants. The cracked nanorod is modelled by dividing it into two parts connected by a linear spring in which its stiffness is related to the crack severity. The surface and bulk material properties are considered to vary in the length direction according to the power law distribution. Hamilton\'s principle is implemented to derive the governing equation of motion and boundary conditions. Considering the surface stress causes that the derived governing equation of motion becomes non-homogeneous while this was not the case in works that only the surface density and the surface Lamé constants were considered. To extract the frequencies of nanorod, firstly the non-homogeneous governing equation is converted to a homogeneous one using an appropriate change of variable, and then for clamped-clamped and clamped-free boundary conditions the governing equation is solved using the harmonic differential quadrature method. Since the present work considers effects of all the surface energy parameters, it can be claimed that this is a comprehensive work in this regard. | ||
| Key Words | ||
| functionally graded materials; free axial vibration; cracked nanorod; surface energy; harmonic differential quadrature method | ||
| Address | ||
| Reza Nazemnezhad: School of Engineering, Damghan University, Damghan, Iran Hassan Shokrollahi: Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran | ||