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The Strain Gradient Elasticity Theory in Orthogonal Curvilinear Coordinates and its Applications

Published online by Cambridge University Press:  13 December 2016

X. Ji ,
A. Q. Li  and
S. J. Zhou
X. Ji
Affiliation:
School of Mechanical & Automotive EngineeringQilu University of TechnologyJinan, China School of Mechanical EngineeringShandong UniversityJinan, China
A. Q. Li*
Affiliation:
School of Mechanical & Automotive EngineeringQilu University of TechnologyJinan, China School of Mechanical EngineeringShandong UniversityJinan, China
S. J. Zhou
Affiliation:
School of Mechanical EngineeringKey Laboratory of High Efficiency and Clean Mechanical ManufactureShandong UniversityJinan, China
*
*Corresponding author (akin@qlu.edu.cn)
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Abstract

The strain gradient elasticity theory including only three independent material length scale parameters has been proposed by Zhou et al. to explain the size effect phenomena in micro scales. In this paper, the general formulations of strain gradient elasticity theory in orthogonal curvilinear coordinates are derived, and then are specified for the cylindrical and spherical coordinates for the convenience of applications in cases where orthogonal curvilinear coordinates are suitable. Two basic problems, one is the twist of a cylindrical bar and the other is the radial deformation of a solid sphere, are analyzed under the cylindrical and spherical coordinates, respectively. The results reveal that only the material length scale parameter l2 enters the torsion problem, while completely disappears in the problem of radial deformation of a sphere. The size effect of radial deformation of a solid sphere is controlled by the material length scale parameters l1 and l2. In addition, for the incompressible solid sphere especially, only the material length scale parameter l1 enters this radial deformation problem by neglecting the strain gradient terms associated with hydrostatic strains. Predictably, the present paper offers an alternative avenue for measuring the three independent material length scale parameters from bar twisting and sphere expansion tests.

Keywords

Strain gradient elasticity theoryOrthogonal curvilinear coordinatesCylindrical coordinatesSpherical coordinates
Type
Research Article
Information
Journal of Mechanics , Volume 34 , Issue 3 , June 2018 , pp. 311 - 323
Copyright
Copyright © The Society of Theoretical and Applied Mechanics 2018 

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