 | | 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 |
|
| << Previous article | Volume 46, Issue 6 / 2011 | Next article >> |
| V.S. Shorkin, "Nonlinear Dispersion Properties of High-Frequency Waves in the Gradient Theory of Elasticity," Mech. Solids. 46 (6), 898-912 (2011) |
| Year |
2011 |
Volume |
46 |
Number |
6 |
Pages |
898-912 |
| DOI |
10.3103/S0025654411060094 |
| Title |
Nonlinear Dispersion Properties of High-Frequency Waves in the Gradient Theory of Elasticity |
| Author(s) |
V.S. Shorkin (Orel State Technical University, Naugorskoe sh. 29, Orel, 302020, Russia, vshorkin@yandex.ru) |
| Abstract |
The dispersion law ceases to be linear already at ultrasonic frequencies of elastic vibrations of particles as mechanical perturbation waves propagate through the medium. A variant of the continuum model of an elastic medium is proposed which is based on the assumption of pair and triplet potential interaction between infinitely small particles; this allows one to represent the dispersion law with any required accuracy. The corresponding wave equation, which is still linear, can have an arbitrarily large order of partial derivatives with respect to the coordinates. It is suggested that the results of comparing the representations of the dispersion law from the elasticity and solid-state physics viewpoints should be used to determine nonclassical characteristics of the elastic state of the medium. The theoretical conclusions are illustrated with calculations performed for plane waves propagating through aluminum. |
| Keywords |
dispersion law, elastic medium models, theory of elasticity, interaction potential |
| References |
| 1. | S. S. Gorelik and M. Ya. Dashevskii,
Semiconductor Materials and Metallography
Materials Science of Semiconductors and Insulators
(Metallurgiya, Moscow, 1973)
[in Russian]. |
| 2. | I. V. Vitcovsky, F. N. Konev, V. S. Shorkin, et al.,
"Adhesion Energy Estimation of Some Composite Materials,"
Plasma Dev. Oper.
11 (2), 81-87 (2003). |
| 3. | E. M. Lifshitz,
Theory of Molecular Attraction Forces between Condensed Bodies,
in Collected Works of E. M. Lifshitz,
Ed. by L. P. Pitaevskii and Yu. G. Rudnyi
(FIZMATLIT, Moscow, 2004)
[in Russian]. |
| 4. | M. B. Partenskii,
"Self-Consistent Electron Theory of a Metallic Surface,"
Uspekhi Fiz. Nauk
128 (1), 69-106 (1979)
[Sov. Phys. Uspekhi (Engl. Transl.)
22 (5), 330-Ö351 (1979)]. |
| 5. | M. V. Mamonova, R. V. Poterin, and V. V. Prudnikov,
"Calculations of the Adhesion Characteristics of Metals in the Model
of Generalized Heine-Abarenkov Potential,"
Vestnik Omsk Gos. Univ.,
No. 2, 44-46 (1996). |
| 6. | M. N. Magomedov,
"Dependence of the Surface Energy on the Size and Shape of a Nanocrystal,"
Fiz. Tverd. Tela
46 (5), 924-937 (2004)
[Phys. Solid State (Engl. Transl.)
46 (5), 954-968 (2004)]. |
| 7. | S. A. Lurie and P. A. Belov,
"Theory of Media with Conserved Dislocations. Particular Cases:
The Cosserat and Aero-Kuvshinskii Continua, Porous Media, and
Media with `Twinning',"
in Contemporary Problems of Mechanics of Heterogeneous Media, Vol. 1
(Inst. Problem Mekhaniki RAN, Moscow, 2005),
pp. 100-132
[in Russian]. |
| 8. | I. V. Vitkovsky, A. N. Konev, and V. S. Shorkin,
"Predicting Adhesive Properties of Liquid-Metal Materials for Fusion Reactor Blankets,"
Zh. Tekhn. Fiz.
79 (2), 11-16 (2009)
[Tech. Phys. (Engl. Transl.)
54 (2), 170-175 (2009)]. |
| 9. | C. Kittel,
Introduction to Solid-State Physics
(Wiley, New York, 1971; Nauka, Moscow, 1978). |
| 10. | V. I. Erofeev,
Wave Processes in Solids with Microstructure
(Izd-vo MGU, Moscow, 1999)
[in Russian]. |
| 11. | V. I. Erofeev and V. M. Rodyushkin,
"Observation of Elastic Wave Dispersion in Grained Composite and
a Mathematical Model for Its Description,"
Akust. Zh.
38 (6), 1116-1117 (1992)
[Sov. Phys. Acoust. (Engl. Transl.)
38, 611 (1992)]. |
| 12. | H. Askes, T. Bennett, and E. C. Aifantis,
"A New Formulation and CO-Implementation of Dynamically Consistent Gradient Elasticity,"
Int. J. Numer. Engng
72 (1), 111-126 (2007). |
| 13. | A. C. Eringen,
"Vistas of Nonlocal Continuum Physics,"
Int. J. Engng Sci.
30 (10), 1551-1565 (1992). |
| 14. | A. C. Eringen,
Nonlocal Continuum Field Theories
(Springer, New York, 2002). |
| 15. | N. F. Morozov and M. V. Paukshto,
Discrete and Hybrid Models of Fracture Mechanics
(Izdat. St. Petersburg Gos. Univ., St. Petersburg, 1995)
[in Russian]. |
| 16. | A. M. Krivtsov and V. P. Myasnikov,
"Modeling of the Change of the Internal Structure and Stress
State in a Material Subjected to High Thermal Loads on the Basis
of the Particle Dynamics Method,"
Izv. Akad. Nauk. Mekh. Tverd. Tela,
No. 1, 88-103 (2005)
[Mech. Solids (Engl. Transl.)
40 (1), 72-85 (2005)]. |
| 17. | A. D. B. Woods, W. Cochran, and B. N. Brockhouse,
"Lattice Dynamics of Alkali Halide Crystals,"
Phys. Rev. Ser. 2
119 (3), 980-999 (1960). |
| 18. | A. C. Eringen,
"Linear Theory of Nonlocal Elasticity and Dispersion of Plane Waves,"
Int. J. Engng Sci
10 (5), 425-435 (1972). |
| 19. | A. A. Il'yushin and V. A. Lomakin,
"Moment Theories in Mechanics of Solids,"
in Strength and Plasticity
(Nauka, Moscow, 1971),
pp. 54-61
[in Russian]. |
| 20. | Yu. Z. Povstenko,
"Nonlocal Gradient Theory of Elasticity and Their Application to
the Description of Defects in Rigid Bodies,"
Mat. Metody Fiz.-Mekh. Polya
46 (2), 136-146 (2003). |
| 21. | V. A. Gordon and V. S. Shorkin,
"Effect Connected with Passing of High Frequency Longitudinal Wave
through Film-Mount System in the Direction Normal to the Contact Plane,"
in Proc. 27th Intern. Symp. on Acoust. Imaging, Saarbrucken, Germany, 2003
(2004),
pp. 333-341. |
| 22. | L. I. Sedov,
Continuum Mechanics, Vol. 2
(Nauka, Moscow, 1970)
[in Russian]. |
| 23. | D. V. Sivukhin,
The General Course of Physics, Vol. 1: Mechanics
(Nauka, Moscow, 1989)
[in Russian]. |
| 24. | A. A. Il'yushin,
"Nonsymmetry of Strain and Stress Tensors in Continuum Mechanics,"
Vestnik Moskov. Univ. Ser. I. Mat. Mekh.,
No. 5, 6-14 (1996)
[Moscow Univ. Math. Bull. (Engl. Transl.)]. |
| 25. | V. S. Shorkin,
"Stress State of Materials under Nonclassical Actions,"
in Problems of Nonlinear Mechanics,
Collection of Papers on the Occasion of L. A. Tolokonnikov's Eighties Birthday
(Tula Gos. Univ., Tula, 2003),
pp. 325-331
[in Russian]. |
| 26. | W. Rudin,
Principles of Mathematical Analysis
(New York, 1964; Mir, Moscow, 1966). |
| 27. | B. A. Dubrovin, S. P. Novikov, and A. T. Fomenko,
Modern Geometry
(Nauka, Moscow, 1979)
[in Russian]. |
| 28. | R. A. Tupin,
"Theories of elasticity with couple-stress,"
Arch. Rat. Mech. Anal.
17, 85-112 (1964)
[in Mechanics, Periodic Collection of Translations of Foreign Papers (Russ. Transl.),
No. 3, 113-140 (1965)]. |
| 29. | V. L. Berdichevskii,
Variational Principles of Continuum Mechanics
(Nauka, Moscow, 1983)
[in Russian]. |
| 30. | W. Nowacki,
Theory of Elasticity
(PWN, Warsaw, 1970; Mir, Moscow, 1975). |
| 31. | V. M. Kontorovich,
"Dynamic Equations of the Theory of Elasticity of Metals,"
Uspekhi Fiz. Nauk
142 (2), 265-307 (1984)
[Sov. Phys. Uspekhi (Engl. Transl.)
27 (2), 134Ö-158 (1984)]. |
| 32. | L. A. Girifalco,
Statistical Physics of Materials, 1973
Statistical Mechanics of Solids
(Mir, Moscow, 1975)
[in Russian]. |
| 33. | J. E. Mayer and M. Geppert-Mayer,
Statistical Mechanics
(Wiley, New York, 1977; Mir, Moscow, 1980). |
| 34. | A. M. Krivtsov and N. V. Krivtsova,
"Method of Particles and Its Application to Mechanics of Solids,"
Dal'nevost. Mat. Zh.
3 (2), 254-276 (2002). |
| 35. | D. Ruelle,
Statistical Mechanics: Rigorous Results
(Benjamin, New York, 1969; Mir, Moscow, 1971). |
| 36. | A. S. Azarov and V. S. Shorkin,
"Consideration of the Influence of Three-Particle Interaction
in Elastic Continuum on Its Mechanical Characteristics,"
in Proc. 47th Intern. Conf. "Actual Problems of Strength"
(Nizhnii Novgorod, 2008),
pp. 163-165
[in Russian]. |
| 37. | A. S. Azarov and V. S. Shorkin,
"A Version of Taking Account of the Triple Potential Interaction
in Many-Particle System,"
Electronic Journal "Investigated in Russia"
8, 65-71 (2009),
http://zhurnal.ape.relarn.ru/articles/2009/008.pdf. |
| 38. | M. Born and H. Kun,
Dynamical Theory of Crystal Lattices
(Clarendon Press, Oxford, 1954; Izd-vo Inostr. Liter., Moscow, 1958). |
| 39. | G. Leibfried,
Microscopic Theory of Mechanical and Thermal Properties of Crystals
(Fizmatgiz, Moscow-Leningrad, 1963)
[in Russian]. |
|
| Received |
23 January 2007 |
| Link to Fulltext |
|
| << Previous article | Volume 46, Issue 6 / 2011 | Next article >> |
|
 If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|
Russian 
English