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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 ([email protected])
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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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References

1. Fleck, N. A. et al., “Strain gradient plasticity: theory and experiment,” Acta Metallurgica et Materialia, 42, pp. 475–487 (1994).Google Scholar
2. Lam, D. C. C. et al., “Experiments and theory in strain gradient elasticity,” Journal of the Mechanics and Physics of Solids, 51, pp. 1477–1508 (2003).Google Scholar
3. Stölken, J. S. and Evans, A. G., “A microbend test method for measuring the plasticity length scale,” Acta Materialia, 46, pp. 5109–5115 (1998).Google Scholar
4. Ma, Q. and Clarke, D. R., “Size dependent hardness of silver single crystals,” Journal of Materials Research, 10, pp. 853–863 (1995).Google Scholar
5. Tang, C. and Alici, G., “Evaluation of length-scale effects for mechanical behaviour of micro- and nanocantilevers: I. Experimental determination of length-scale factors,” Journal of Physics D: Applied Physics, 44, pp. 335501–1-12 (2011).Google Scholar
6. Chasiotis, I. and Knauss, W. G., “The mechanical strength of polysilicon films: Part 2. Size effects associated with elliptical and circular perforations,” Journal of the Mechanics and Physics of Solids, 51, pp. 1551–1572 (2003).Google Scholar
7. Mindlin, R. D. and Tiersten, H. F., “Effects of couple-stresses in linear elasticity,” Archive for Rational Mechanics and Analysis, 11, pp. 415–448 (1962).Google Scholar
8. Toupin, R., “Elastic materials with couple-stresses,” Archive for Rational Mechanics and Analysis, 11, pp. 385–414 (1962).Google Scholar
9. Koiter, W. T., “Couple stresses in the theory of elasticity, I and II,” Philosophical Transactions of the Royal Society of London B, 67, pp. 17–29 (1964).Google Scholar
10. Yang, F. et al., “Couple stress based strain gradient theory for elasticity,” International Journal of Solids and Structures, 39, pp. 2731–2743 (2002).Google Scholar
11. Hadjesfandiari, A. R. and Dargush, G. F., “Couple stress theory for solids,” International Journal of Solids and Structures, 48, pp. 2496–2510 (2011).CrossRefGoogle Scholar
12. Mindlin, R. D., “Micro-structure in linear elasticity,” Archive for Rational Mechanics and Analysis, 16, pp. 51–78 (1964).Google Scholar
13. Aifantis, E. C., “On the role of gradients in the localization of deformation and fracture,” International Journal of Engineering Science, 30, pp. 1279–1299 (1992).Google Scholar
14. Gao, X. L. and Park, S. K., “Variational formulations of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem,” International Journal of Solids and Structures, 44, pp. 7486–7499 (2007).Google Scholar
15. Zhou, S. J. et al., “A reformulation of constitutive relations in the strain gradient elasticity theory for isotropic materials,” International Journal of Solids and Structures, 80, pp. 28–37 (2016).CrossRefGoogle Scholar
16. Li, A. Q. et al., “A size-dependent bilayered microbeam model based on strain gradient elasticity theory,” Composite Structures, 108, pp. 259–266 (2014).Google Scholar
17. Wang, B. L. et al., “A micro scale Timoshenko beam model based on strain gradient elasticity theory,” European Journal of Mechanics A, 29, pp. 591–599 (2010).Google Scholar
18. Wang, B. L. et al., “A size-dependent Kirchhoff micro-plate model based on strain gradient elasticity theory,” European Journal of Mechanics A-Solids, 30, pp. 517–524 (2011).Google Scholar
19. Movassagh, A. A. and Mahmoodi, M. J., “A micro-scale modeling of Kirchhoff plate based on modified strain-gradient elasticity theory,” European Journal of Mechanics A-Solids, 40, pp. 50–59 (2013).Google Scholar
20. Li, A. Q. et al., “A size-dependent model for bi-layered Kirchhoff micro-plate based on strain gradient elasticity theory,” Composite Structures, 113, pp. 272–280 (2014).CrossRefGoogle Scholar
21. Zhao, J. and Pedroso, D., “Strain gradient theory in orthogonal curvilinear coordinates,” International Journal of Solids and Structures, 45, pp. 3507–3520 (2008).CrossRefGoogle Scholar
22. Zhang, B. et al., “A novel size-dependent functionally graded curved microbeam model based on the strain gradient elasticity theory,” Composite Structures, 106, pp. 374–392 (2013).Google Scholar