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IssuesArchive of Issues2008-2pp.254-260

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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, "Discontinuous Solutions in Constraint Problem of Thermoelasticity," in Mechanics of Deformable Media (Izd. Kuibyshev Univ., Kuibyshev, 1979), No. 4, pp. 85-90 [in Russian].
6.  Yu. N. Podil'chuk and Yu. K. Rubtsov, "Application of the Method of Ray Series to the Investigation of Axisymmetric Nonstationary 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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