 | | 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: | | 12854 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
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| In English (Mech. Solids): | | 4810 |
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| I.E. Keller, "Prandtl-Meyer Solutions of Viscoplasticity Equations with Negative Strain Rate Sensitivity," Mech. Solids. 49 (1), 40-48 (2014) |
| Year |
2014 |
Volume |
49 |
Number |
1 |
Pages |
40-48 |
| DOI |
10.3103/S0025654414010051 |
| Title |
Prandtl-Meyer Solutions of Viscoplasticity Equations with Negative Strain Rate Sensitivity |
| Author(s) |
I.E. Keller (Perm National Research Polytechnic University, Komsomolsky pr-t 29, Perm, 614990 Russia, kie@icmm.ru) |
| Abstract |
We present a further study of a viscoplasticity model with nonmonotonic strain rate dependence ensuring the complete integrability of the two-dimensional equilibrium and consistency equations. The considered nonlinear equations change their type from hyperbolic to elliptic at a certain critical value of the strain rate intensity; the type change is accompanied by the formation of an interphase in the solid. This model is of interest for describing spatial autowave processes in active continua, and the integrability of equations allows one to construct efficient methods for the numerical solution of boundary value problems and ensures the existence of closed-form solutions.
The present paper shows that the considered material function satisfies a criterion for the separation of the system of these equations into two noninteracting subsystems. We derive kinematic equations on the characteristics. We obtain and analyze centered self-similar solutions (Prandtl-Meyer solutions) in the domain of hyperbolicity of the equations, which describe flows in convergent and divergent channels. |
| Keywords |
viscoplasticity, negative strain rate sensitivity, complete integrability, kinematic relation on characteristics, Prandtl-Meyer solution |
| References |
| 1. | I. E. Keller,
"Integrability of the Equilibrium and Compatibility Equations for a
Viscoplastic Medium with Negative Strain Rate Sensitivity,"
Dokl. Ross. Akad. Nauk
451 (6), 643-646 (2013)
[Dokl. Phys. (Engl. Transl.)
58 (8), 362-365 (2013)]. |
| 2. | M. Lebyodkin, L. Dunin-Barkowskii, Y. Bréchet, et al.,
"Spatio-Temporal Dynamics of the Portevin-Le Chatelier Effect:
Experiment and Modelling,"
Acta Mater.
48 (10), 2529-2541 (2000). |
| 3. | A. I. Rudskoi and Ya. I. Rudaev,
Mechanics of Dynamic Superplasticity of Aluminum Alloys
(Nauka, St.Petersburg, 2009)
[in Russian]. |
| 4. | S. L. Bazhenov and E. P. Kovalchuk,
"Self-Oscillating Plastic Deformation of Polymers,"
Dokl. Ross. Akad. Nauk
417 (3), 353-356 (2007)
[Dokl. Phys. Chem. (Engl. Transl.)
417 (1), 308-301 (2007)]. |
| 5. | J. H. Dieterich,
"Modeling of Rock Friction 1. Experimental Results and Constitutive Equations,"
J. Geophys. Res.
84, 2161-2168 (1979). |
| 6. | A. A. Il'yushin,
"Deformation of Viscoplastic Bodies,"
Ucheb. Zap. MGU. Mekh.
No. 39, 3-81 (1940). |
| 7. | A. M. Freudental and H. Geiringer,
The Mathematical Theories of Inelastic Continuum
(Springer-Verlag, Berlin-Goettingen-Heidelberg, 1958; Fizmatgiz, Moscow, 1962) |
| 8. | P. K. Rozhdestvenskii and N. N. Yanenko,
Systems of Quasilinear Equations and Their Applications to Gas Dynamics
(Nauka, Moscow, 1978)
[in Russian]. |
| 9. | P. K. Rashevskii,
Geometric Theory of Partial Differential Equations
(Gostekhizdat, Moscow-Leningrad, 1947)
[in Russian]. |
| 10. | L. V. Ovsyannikov,
Lectures in Foundations of Gas Dynamics
(IKI, Moscow-Izhevsk, 2003)
[in Russian]. |
| 11. | O. I. Bogoyavlenskij,
"Decoupling Problem for Systems of Quasi-linear PDE's,"
Commun. Math. Phys.,
No. 269, 545-556 (2007). |
| 12. | A. G. Kulikovskii, N. V. Pogorelov, and A. Yu. Semenov,
Mathematical Problems of Numerical Solution of Hyperbolic Systems of Equations
(Fizmatlit, Moscow, 2001)
[in Russian]. |
|
| Received |
03 September 2013 |
| Link to Fulltext |
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