 | | Mechanics of Solids
A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
Print ISSN 0025-6544
Online ISSN 1934-7936 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
|
| In English (Mech. Solids): | | 4760 |
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| S.A. Lurie, E.L. Kuznetsova, L.N. Rabinskii, and E.I. Popova, "Refined Gradient Theory of Scale-Dependent Superthin Rods," Mech. Solids. 50 (2), 135-146 (2015) |
| Year |
2015 |
Volume |
50 |
Number |
2 |
Pages |
135-146 |
| DOI |
10.3103/S002565441502003X |
| Title |
Refined Gradient Theory of Scale-Dependent Superthin Rods |
| Author(s) |
S.A. Lurie (Dorodnicyn Computing Center, Russian Academy of Sciences, ul. Vavilova 40, Moscow, 119333 Russia; Ishlinsky Institute for Problems in Mechanics, Russian Academy of Sciences, pr. Vernadskogo 101, str. 1, Moscow, 119526 Russia, lurie@ccas.ru)
E.L. Kuznetsova (Moscow Aviation Institute (State University of Aerospace Technologies), Volokolamskoe sh. 4, Moscow, 125993 Russia, vida_ku@mail.ru)
L.N. Rabinskii (Moscow Aviation Institute (State University of Aerospace Technologies), Volokolamskoe sh. 4, Moscow, 125993 Russia, f9_dec@mai.ru)
E.I. Popova (Moscow Aviation Institute (State University of Aerospace Technologies), Volokolamskoe sh. 4, Moscow, 125993 Russia) |
| Abstract |
A version of the refined nonclassical theory of thin beams whose thickness is comparable with the scale characteristic of the material structure is constructed on the basis of the gradient theory of elasticity which, in contrast to the classical theory, contains some additional physical characteristics depending on the structure scale parameters and is therefore most appropriate for modeling the strains of scale-dependent systems. The fundamental conditions for the well-posedness of the gradient theories are obtained for the first time, and it is shown that some of the known applied gradient theories do not generally satisfy the well-posedness criterion. A version of the well-posed gradient strain theory which satisfies the symmetry condition is proposed. The well-posed gradient theory is then used to implement the method of kinematic hypotheses for constructing a refined theory of scale-dependent beams. The equilibrium equations of the refined theory of scale-dependent Timoshenko and Bernoulli beams are obtained. It is shown that the scale effects are localized near the beam ends, and therefore, taking the scale effects into account does not give any correction to the bending rigidity of long beams as noted in the previously published papers dealing with the scale-dependent beams. |
| Keywords |
gradient theories of elasticity, tensor of gradient moduli, well-posedness condition, kinematic relations, variational method, scale-dependent beam, refined theory |
| References |
| 1. | S. Kakunai, J. Masaki, R. Kuroda, et al.,
"Measurement of Apparent Young's Modulus in the Bending of Cantilever Beam
by Heterodyne Holographic Interferometry,"
Exp. Mech.
25 (4), 408-412 (1985). |
| 2. | D. C. C. Lam, F. Yang, A. C. M. Chong, et al.,
"Experiments and Theory in Strain Gradient Elasticity,"
J. Mech. Phys. Solids
51, 1477-1508 (2003). |
| 3. | A. W. McFarland and J. S. Colton,
"Role of Material Microstructure in Plate Stiffness with Relevance to Microcantilever Sensors,"
J. Micromech. Microeng.
15 (5), 1060-1067 (2005). |
| 4. | F. Yang, A. C. M. Chong, D. C. C. Lam, and P. Tong,
"Couple Stress Based Strain Gradient Theory for Elasticity,"
Int. J. Solids Struct.
39, 2731-2743 (2002). |
| 5. | C. M. Wang, Y. Y. Zhang, and X. Q. He,
"Vibration of Nonlocal Timoshenko Beams,"
Nanotech.
18, 105401 (2007). |
| 6. | H. M. Ma, X.-L. Gao, and J. N. Reddy,
"A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory,"
J. Mech. Phys. Solids
56 (12), 3379-3391 (2008). |
| 7. | H. M. Ma, X.-L. Gao, and J. N. Reddy,
"A Non-Classical Plate Model Based on a Modified Couple Stress Theory,"
Acta Mech.
220 (1-4), 217-235 (2011). |
| 8. | R. D. Mindlin,
"Micro-Structure in Linear Elasticity,"
Arch. Rat. Mech. Anal.
16 (1), 51-77 (1964). |
| 9. | R. D. Mindlin and N. N. Eshel,
"On First Strain-Gradient Theories in Linear Elasticity,"
Int. J. Solids Struct.
4 (1), 109-124 (1968). |
| 10. | X.-L. Gao and S. K. Park,
"Variational Formulation of a Simplified Strain Gradient Elasticity Theory
and Its Application to a Pressurized Thick-Walled Cylinder Problem,"
Int. J. Solids Struct.
44, 7486-7499 (2007). |
| 11. | S. Lurie, D. Volkov-Bogorodskii, A. Leontiev, and E. Aifantis,
"Eshelby's Inclusion Problem in the Gradient Theory of Elasticity.
Applications to Composite Materials,"
Int. J. Engng Sci.
49, 1517-1525 (2011). |
| 12. | D. B. Volkov-Bogorodskii and S. A. Lurie,
"Eshelby Integral Formulas in Gradient Elasticity,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 4, 182-192 (2010)
[Mech. Solids (Engl. Transl.)
45 (4), 648-656 (2010)]. |
| 13. | V. V. Vasiliev and S. A. Lurie,
"To the Problem of Construction of Nonclassical Theories of Plates,"
Izv. Akad. Nauk SSSR. Mekh. Tverd. Tela,
No. 2, 158-167 (1990)
[Mech. Solids (Engl. Transl.)]. |
| 14. | V. V. Vasiliev and S. A. Lurie,
"On Refined Theories of Beams, Plates, and Shells,"
J. Compos. Mater.
26 (4), 546-557 (1992). |
| 15. | P. A. Belov and S. A. Lurie,
"Theory of Ideal Adhesion Interactions,"
Mekh. Komp. Mater. Konstruktsii
13 (4), 45-56 (2007)
[J. Comp. Mech. Design (Engl. Transl.)]. |
| 16. | S. A. Lurie and N. P. Tuchkova,
"Continual Models of Adhesion for Deformable Solids and Media with Nanostructures,"
Kompoz. Nanostrukt.
2 (2), 25-43 (2009). |
| 17. | S. A. Lurie, P. A. Belov, and Yu. O. Solyaev,
"Adhesion Interactions in Continuum Mechanics,"
Mat. Model. Sist. Prots.
16, 75-85 (2008). |
| 18. | S. A. Lurie, D. B. Volkov-Bogorodskii, V. I.Zubov, and N. P. Tuchkova,
"Advanced Theoretical and Numerical Multiscale Modeling of Cohesion/Adhesion Interactions
in Continuum Mechanics and Its Applications
for Filled Nanocomposites,"
Int. J. Comp. Mater.
45 (3), 709-714 (2009). |
|
| Received |
03 December 2014 |
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