 | | 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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| 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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