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IssuesArchive of Issues2001-5pp.67-74

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V. M. Sadovskii, "On the theory of shock waves in compressible plastic media," Mech. Solids. 36 (5), 67-74 (2001)
Year 2001 Volume 36 Number 5 Pages 67-74
Title On the theory of shock waves in compressible plastic media
Author(s) V. M. Sadovskii (Krasnoyarsk)
Abstract Within the framework of the plastic flow model for compressible elastic media subjected to a finite deformation, we investigate a class of discontinuous solutions with shock waves. The constitutive equations of the model are reduced to a variational inequality. This inequality, written in a generalized integral form allows us to obtain a complete system of relations describing strong discontinuities. On the basis of the analysis of this system, we prove that von Mises and Tresca-Saint-Venant plasticity conditions, allow longitudinal shock waves, which propagate along the principal stress axes, and also transverse and quasi-transverse waves may be observed with the direction vectors lying in the principal planes.

The problem of the construction of generalized solutions with shock waves in the theory of elastoplastic flow is a much less studied topic than the same problem in the mechanics of ideal (thermodynamically reversible) media [1]. It turns out that the method of direct integral generalization of this theory does not yield a complete system of equations on the discontinuity surface, since the associated friction law, which describes the dissipative mechanism of deformation, cannot be reduced to an equivalent system of integral conservation laws [2, 3].

A complete system of relations for a strong discontinuity, within the model of small deformations of an elastoplastic medium, was initially obtained with the help of an auxiliary hypothesis, namely, that energy dissipation power is maximal on the wavefront [4]. This hypotheses was justified by an alternative way to construct relations on the basis of an integral generalization of the variational inequality connected with the principle of maximum power of energy dissipation. Such an investigation of geometrically linear models of materials with hardening was conducted in [6, 7]. Of course, the expressions for the velocities of the waves and the equations relating the jumps of the solution obtained in [6, 7] are applicable only to the analysis of shock waves of small amplitude.

Another unified method for the construction of a system of relations on the discontinuity surface is connected with the approximation of the discontinuous solution by a sequence of solutions obtained within the model of a viscous heat conducting medium with smoothed shock waves, as the viscosity and the heat conductivity tend to zero. This method was applied to the analysis of particular models in the theory of elastoplastic flow with finite deformations, namely, the models describing plane waves of uniaxial deformation [8-10]. If the state near the wave front has three-dimensional nature, the realization of this method encounters insuperable technical difficulties.

In the present paper, shock waves of finite amplitudes are studied on the basis of a simplified thermomechanical model that describes dynamical deformation of compressible media with elastic volume change and plastic shape forming.
References
1.  L. I. Sedov, Continuum Mechanics. Volume 2 [in Russian], Nauka, Moscow, 1976.
2.  J. Mandel, "Ondes plastiques dans un milieu á treis dimensions," Mekhanika, No. 5, pp. 119-141, 1963.
3.  V. N. Kukudzhanov, "An investigation of the dynamical equations of elastoplastic media with finite deformations," in Nonlinear Deformation Waves. Volume 2 [in Russian], pp. 102-105, Tallin, 1977.
4.  G. I. Bykovtsev and L. D. Kretova, "On shock wave propagation in elastoplastic media," PMM [Applied Mathematics and Mechanics], Vol. 36, No. 1, pp. 106-116, 1972.
5.  V. M. Sadovskii, "On the dynamical consistency of the theory of elastic-ideally-plastic flow," in Dynamics of Continuum, No. 63, pp. 147-151, In-t Gidrodinamiki SO RAN, Novosibirsk, 1983.
6.  A. A. Burenin, G. I. Bykovtsev, and V. A. Rychkov, "Discontinuity surfaces for velocities in the dynamics of irreversibly compressible media," in Problems in Continuum Mechanics [in Russian], pp. 116-127, IAPU DVO RAN, Vladivostok, 1966.
7.  V. M. Sadovskii, Discontinuous Solutions of Dynamic Problems for Elastoplastic Media [in Russian], Nauka, Moscow, 1997.
8.  Ya. A. Pachepskii, "On the structure of shock waves in elastoplastic media," PMM [Applied Mathematics and Mechanics], Vol. 37, No. 2, pp. 300-305, 1973.
9.  B. A. Druyanov and E. A. Svyatova, "A problem of discontinuity structure for elastoplastic media with hardening," PMM [Applied Mathematics and Mechanics], Vol. 51, No. 6, pp. 1047-1049, 1987.
10.  V. N. Kukudjanov, "Investigation of shock wave structure in elasto-visco-plastic bars using asymptotic method," Arch. Mech., Vol. 33, No. 5, pp. 739-751, 1981.
11.  G. B. Ivanov, "On the equations of elastoplastic deformation with arbitrary rotations and strains," Zh. Prikl. Mekhaniki i Tekhn. Fiziki, No. 3, pp. 130-135, 1978.
Received 02 November 2000
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