 | | 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 |
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| Total articles in the database: | | 12854 |
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| In English (Mech. Solids): | | 4810 |
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| A. A. Burenin and V. E. Ragozina, "Construction of approximate solutions of boundary value impact strain problems," Mech. Solids. 43 (2), 254-260 (2008) |
| Year |
2008 |
Volume |
43 |
Number |
2 |
Pages |
254-260 |
| DOI |
10.3103/S0025654408020118 |
| Title |
Construction of approximate solutions of boundary value impact strain problems |
| Author(s) |
A. A. Burenin (Institute for Automation and Control Processes, Far East Branch of Russian Academy of Sciences, Radio 5, Vladivostok, 690041, Russia, Byrenin@iacp.dvo.ru)
V. E. Ragozina (Institute for Automation and Control Processes, Far East Branch of Russian Academy of Sciences, Radio 5, Vladivostok, 690041, Russia, razogina@vlc.ru) |
| Abstract |
The method for constructing approximate solutions of boundary value problems of impact strain dynamics in the form of ray expansions behind the strain discontinuity fronts is generalized to the case of curvilinear and diverging rays. This proposed generalization is illustrated by an example of dynamics of an antiplane motion of an elastic medium. The ray method is one of the methods for constructing approximate solutions of nonstationary boundary value problems of strain dynamics. It was proposed in [1, 2] and then widely used in nonstationary problems of mathematical physics involving surfaces on which the desired function or its derivatives have discontinuities [3-7]. A complete, qualified survey of papers in this direction can be found in [8]. This method is based on the expansion of the solution in a Taylor-type series behind the moving discontinuity surface rather than in a neighborhood of a stationary point. The coefficients of this series are the jumps of the derivatives of the unknown functions, for which, as a consequence of the compatibility conditions, one can obtain ordinary differential equations, i.e., discontinuity damping equations. In the case where the problem with velocity discontinuity surfaces is considered in a nonlinear medium, this method cannot be used directly, because one cannot obtain the damping equation. A modification of this method for the purpose of using it to solve problems of that type was proposed in [9-11], where, as an example, the solutions of several one-dimensional problems were considered. In the present paper, we show how this method can be transferred to the case of multidimensional impact strain problems in which the geometry of the ray is not known in advance and the rays become curvilinear and diverging. By way of example, we consider a simple problem on the antiplane motion of a nonlinearly elastic incompressible medium. |
| References |
| 1. | J. D. Achenback and D. P. Reddy,
"Note of Wave Propagation in Linearly Viscoelastic Media,"
ZAMP
18 (1), 141-144 (1967). |
| 2. | L. A. Babicheva, G. I. Bykovtsev, and N. D. Verveiko,
"Ray Method of Solving Dynamic Problems in Elastic-Viscoplastic Media,"
Prikl. Mat. Mekh.
37 (1), 145-155 (1973)
[J. Appl. Math. Mech. (Engl. Transl.)
37 (1), 132-141 (1973)]. |
| 3. | G. I. Bykovtsev and I. A. Vlasova,
"A Ray Method for Solving Spatial Problems of the Theory of Ideal Plasticity,"
in Mechanics of Deformable Rigid Bodies
(Nauka, Novosibirsk, 1979),
pp. 31-36
[in Russian]. |
| 4. | Yu. A. Rossikhin,
"Impact of a Rigid Sphere onto Elastic Half Space,"
Prikl. Mekh.
22 (5), 15-21 (1986)
[Int. Appl. Mech. (Engl. Transl.)
22 (5), 403-409 (1986)]. |
| 5. | A. G. Shatalov,
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in Mechanics of Deformable Media
(Izd. Kuibyshev Univ., Kuibyshev, 1979),
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[in Russian]. |
| 6. | Yu. N. Podil'chuk and Yu. K. Rubtsov,
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Problems of the Dynamic Theory of Elasticity,"
Prikl. Mekh.
22 (3), 3-9 (1986)
[Int. Appl. Mech. (Engl. Transl.)
22 (3), 201-207 (1986)]. |
| 7. | Yu. A. Rossikhin and M. V. Shitikova,
"The Impact of Elastic Bodies upon Beams and Plates with
Consideration for the Transverse Deformations and Extension
of a Middle Surface,"
ZAMM
76 (5), 433-434 (1996). |
| 8. | Yu. A. Rossikhin and M. V. Shitikova,
"Ray Method for Solving Dynamic Problems Connected with the Propagation
of Wave Surfaces of Strong and Weak Discontinuities,"
Appl. Mech. Reviews
48 (1), 1-39 (1995). |
| 9. | A. A. Burenin and Yu. A. Rossikhin, "A Ray Method for Solving
One-Dimensional Problems of Nonlinear Dynamic Theory of Elasticity
with Plane Surfaces of Strong Discontinuities," in Applied
Problems of Mechanics of Deformable Media (Izd. DVO AN SSSR,
Vladivostok, 1991), pp. 129-137 [in Russian]. |
| 10. | A. A. Burenin, Yu. A. Rossikhin, and M. V. Shitikova, "A Ray
Method for Solving Boundary Value Problems Connected with the
Propagation of Finite Amplitude Shock Waves," in Proc.
1993 Int. Symp. on Nonlinear Theory and Its Applications. Hawaii,
1993 (Sheraton Waikiki, Hawaii, 1993), Vol. 3, pp. 1085-1088. |
| 11. | A. A. Burenin,
"On a Possibility of Constructing Approximate Solutions of Nonstationary
Problems of Dynamics of Elastic Media under Impact Actions,"
Dal'nevost. Mat. Sb.,
No. 8, 49-72 (1999). |
| 12. | G. I. Bykovtsev and D. D. Ivlev,
Theory of Plasticity
(Dal'nauka, Vladivostok, 1998)
[in Russian]. |
| 13. | A. A. Burenin and P. V. Zinov'ev,
"To the Problem of Distinguishing the Discontinuity Surfaces
in Numerical Methods of Dynamics of Deformable Media,"
in Problems of Mechanics. Collection of Papers Dedicated to
A. Yu. Ishlinskii on the Occasion of his 90th Birthday
Ed. by D. M. Klimov
(Fizmatlit, Moscow, 2003),
pp. 146-155
[in Russian]. |
| 14. | A. A. Burenin, P. V. Zinov'ev, and V. E. Ragozina, "Distinguishing
the Discontinuity Surfaces by a Ray Method in Problems of Dynamics
of Elastic Media," in Int. Sci. Conf. "Fundamental and
Applied Problems of Mechanics" (Izd. Khaborovsk. Gos. Tekhn.
Univ., Khabarovsk, 2003), pp. 62-64 [in Russian]. |
|
| Received |
12 May 2005 |
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