 | | 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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| D. I. Chernyavskii, "Elastoplastic impact of two deformable solids at medium impact velocities," Mech. Solids. 38 (2), 71-80 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
2 |
Pages |
71-80 |
| Title |
Elastoplastic impact of two deformable solids at medium impact velocities |
| Author(s) |
D. I. Chernyavskii (Moscow) |
| Abstract |
An impact interaction of two deformable solid bodies is accompanied
by various physical processes depending on the shape and material
of these bodies, the impact velocity, and a number of other parameters.
The major difference between a dynamic impact interaction and a static
loading is that for the impact interaction,
the forces at the point of contact of the bodies have an extremely
short duration, measured by tenth and hundredth of microsecond.
For that reason, an impact interaction leads to the formation
of shock stress waves that cover the entire system of bodies
and, in addition, the interacting bodies penetrate one another,
except for cases rarely encountered in practice.
As is known, the mechanical behavior of materials subject to an impact
is classified into elastic, plastic, viscous, or combined. The
main criterion for this classification is the relationship between
stresses, strains, and rates of change of these quantities.
In practice, impact velocities are divided into low, medium, and
high ones. Low velocities lead only to elastic strains. For high
impact velocities, the interacting bodies fail or are completely
disintegrated. For fairly high impact velocities, at which the pressure
attains the modulus of elasticity, phase transformations
can occur in the materials of the impacting bodies.
In practice, the most frequent case is an impact with medium
velocities at which the stress level exceeds the yield stress
at most by two orders of magnitude and the velocity of motion
of particles of the bodies is lower than the sound speed for
the respective material. In this velocity range, a residual hollow appears
in the contact zone, and the plastic deformation is confined
only to a neighborhood of this zone and does not spread over the
entire cross section of the bodies. In this case, we have the inequality
Pm <σTSi, where Pm is the maximum force, σT
is the yield stress, and Si is the cross-sectional area behind
the contact region of the impactor (i=1) or the second body (i=2).
For simplicity, such processes are regarded as isothermal and, hence,
temperature and other thermodynamic effects are not taken into account.
Impacts of such a type are frequently encountered when solving
topical problems of applied mechanics related to the behavior
of various structures subject to an impulsive loading.
Processes of propagation of waves in solids, irrespective
of their source, have been fairly well studied theoretically.
The results of these studies are presented in [1-9]. The governing
equations involve differential equations of motion and an equation
of state. The solution of these equations depends on the geometric
properties and anticipated behavior of the colliding bodies.
A great number of solutions have been obtained for homogeneous
isotropic elastic materials. The simplest differential
equations of motion are those of propagation of waves in an unbounded
medium or in an elastic half-space. Exact solutions of equations
governing wave processes in bodies of finite size have been obtained
for plates (Rayleigh and Lamb), cylindrical rods
(Pokhammer and Cree), shells (Mirsky and Hermann),
and some other structures.
Complications in applications of strict solutions of elasticity to
bodies of finite size have given rise to the appearance of various
simplified approximate solutions for rods, beams, plates, curved rods,
and other simple structures subject to a great variety of boundary
conditions.
The classification of substances distinguishes, apart from
elastic materials, anisotropically elastic, multiphase loose,
elastoplastic, viscoelastic, and some other types of materials.
The differential equations of motion for such materials are
substantially more complex than the equations of motion
for elastic materials. Of most importance for practice is
the description of dynamic interaction between elastoplastic
bodies. This is for that reason that metals, having been
currently basic structural materials, demonstrate elastoplastic
properties when being subjected to dynamic interaction at medium
impact velocities. |
| References |
| 1. | E. V. Aleksandrov and V. B. Sokolinskii,
Applied Theory and Calculations of Impact Systems [in Russian],
Mauka, Moscow, 1969. |
| 2. | G. S. Batuev, Yu. V. Golubkov, A. K. Efremov, and A. A. Fedosov,
Engineering methods of analysis of impact processes [in Russian],
Mashinostroenie, Moscow, 1969. |
| 3. | W. Goldsmith, "Impact and contact phenomena under
medium velocities," in Physics of Transient Processes. Volume 2 [in
Russian], pp. 153-203, Mir, Moscow, 1971. |
| 4. | W. Goldsmith, Impact: Theory and Physical Properties
of Colliding Bodies [Russian translation], Stroiizdat, Moscow, 1965. |
| 5. | A. Yu. Ishlinskii, "Axially symmetric problem of plasticity
and Brinnel test," PMM [Applied Mathematics and Mechanics],
Vol. 8, No. 3, pp. 201-224, 1944. |
| 6. | Yu. V. Kolesnikov and E. V. Morozov, Mechanics of Contact Fracture
[in Russian], Nauka, Moscow, 1989. |
| 7. | L. D. Landau and E. M. Lifshits, Theoretical Physics. Volume 7.
Theory of Elasticity [in Russian], Nauka, Moscow, 1965. |
| 8. | I. Ya. Shtaerman, Contact Problem of Elasticity [in Russian],
Gostekhizdat, Moscow, 1949. |
| 9. | D. Tabor, "A simple theory of static and dynamic hardness,"
Proc. Roy. Soc. London, Ser. A, Vol. 192, No. 1029, pp. 247-274, 1948. |
|
| Received |
10 May 2001 |
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