 | | 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. V. Georgievskii and A. S. Kirillov, "Acceleration and deceleration of a heavy viscoplastic layer (glacier) along an inclined plane," Mech. Solids. 38 (3), 88-94 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
3 |
Pages |
88-94 |
| Title |
Acceleration and deceleration of a heavy viscoplastic layer (glacier) along an inclined plane |
| Author(s) |
D. V. Georgievskii (Moscow)
A. S. Kirillov (Moscow) |
| Abstract |
The problem of deformation of a heavy viscoplastic layer
with high viscosity on an inclined plane models the behavior of
geophysical structures on an inclined bed in the gravity field.
Such structures include snow-dust avalanches formed by soft snow [1],
yielding parts of the upper layers of the earth crust [2, 3],
rock salt in a limit state [4], rain-induced landslide [5], and
glacier formations [6]. The latter are characterized by a massive
non-deformable crust occupying about 95% of the glacier thickness and
a thin shear layer at its base. The fact that ice has a viscoplastic nature
was noted in [7, pp. 166]: "Since no ultimate long-term strength
has been observed in ice, we introduce the notion of its "virtually
ultimate state" in which the time-evolution of its deformation is so
insignificant that further deformation may be assumed absent... one
can define long-term strength, which is of great importance for
characterizing the time-evolution of the bearing capacity of ice."
This long-term strength is in fact the shear yield stress.
Experimentally determined values of material functions for a wide range of
ice structures and snow flows of avalanche type can be found in the review [8].
Steady-state plane-parallel motion of a viscoplastic layer on an inclined
plane is one of the classical problems studied long ago [9]. In this
motion, the region next to the free surface of the layer is occupied by
a rigid zone, which is always present and may fill up the entire
layer, provided that the gravity force is insufficient for flow
initiation. The stability of this flow
with respect to two-dimensional perturbations was studied in [10]. There
exist numerous generalizations of this problem for viscoplastic fluids with
nonlinear viscosity or elastic compressibility and for double-layer or
stratified flows [11].
On the other hand, there are many investigations
of the motion of phase boundaries in materials subjected to nonstationary
shear. In such problems, boundary conditions are prescribed on an unknown
boundary whose motion is found in the process of solving the problem
(problems of Stefan type). Monograph [1] contains a fairly complete review
of the studies regarding unsteady deformations,
with the motion of rigid zones taken into account.
In the present paper, we determine the parameters of the nonstationary one-dimensional
shear of a viscoplastic layer on an inclined plane in the gravity field. We
determine the time-variation of the thickness of the rigid zone near the free boundary,
as well as some other characteristics of the flow.
The problem with unknown boundary is reduced to a problem with a boundary
fixed in time. A method is proposed for solving problems similar to the one just mentioned.
We study the asymptotic behavior of solutions as t→∞ in the case of
the Reynolds number being much smaller than the Froude number. |
| References |
| 1. | M. E. Eglit, "Dynamics of snow avalanches,"
Trudy. Matem. In-ta AN SSSR, Vol. 186, pp. 162-167, 1989. |
| 2. | Kh. Ramberg, The Gravity Force and Deformations of the Earth Crust
[in Russian], Nedra, Moscow, 1985. |
| 3. | W. K. Nowacki, "On the dynamics description of the rock failure
process," Arch. Mech. Stosow., Vol. 38, No. 1-2, pp. 25-37, 1986. |
| 4. | A. K. Chernikov,
"Variational methods for problems of viscoplastic salt rock flow,"
Izv. Vuzov. Gorny Zh., No. 10, pp. 29-33, 1985. |
| 5. | A. Ya. Sagomonyan, "Rain erosion of soil on hill slopes,"
Vestnik MGU [Bulletin of Moscow State University], Ser. 1, Mat. Mekh.,
No. 4, pp. 28-34, 2000. |
| 6. | B. A. Savel'ev and D. A. Latalin,
"Artificial ice platforms," in Advances in Science and Technology.
Ser. Oceanology. Volume 7 [in Russian], pp. 3-193, VINITI, Moscow, 1986. |
| 7. | B. A. Savel'ev, Glaciology [in Russian], Izd-vo MGU, Moscow, 1991. |
| 8. | A. N. Nazarov,
"Principles of mathematical modeling of friction and entrainment
in flows of avalanche type," Vestnik MGU [Bulletin of Moscow State
University], Ser. Mat. Mekh., No. 4, pp. 79-85, 1995. |
| 9. | M. P. Volarovich,
"A study of rheological properties of dispersed systems,"
Kolloidn. Zh., Vol. 16, No. 3, pp. 227-240, 1954. |
| 10. | D. V. Georgievskii,
Stability of Deformation Processes in Viscoplastic Bodies [in Russian],
Izd-vo URSS, Moscow, 1998. |
| 11. | A. T. Listrov and A. D. Chernyshev,
"On steady-state flow of a viscoplastic medium with nonlinear viscosity,"
Doklady AN. SSSR, Vol. 158, No. 4, pp. 805-807, 1964. |
| 12. | P. M. Ogibalov and A. Kh. Mirzadzhanzade,
Nonstationary Motion of Viscoplastic Media [in Russian], Nauka, Moscow, 1977. |
| 13. | A. A. Il'yushin,
"Deformation of viscoplastic bodies," Uchen. Zapiski MGU,
Mekh., Vol. 39, pp. 3-81, 1940. |
| 14. | G. I. Barenblatt and A. Yu. Ishlinskii,
"On the impact of a viscoplastic beam against a rigid obstacle,"
PMM [Applied Mathematics and Mechanics], Vol. 26, No. 3, pp. 497-502, 1962. |
| 15. | S. N. Kruzhkov, "On some problems with unknown boundary for
the heat equation," PMM [Applied Mathematics and Mechanics], Vol. 31, No. 6, pp. 1009-1020, 1967. |
| 16. | A. V. Gnoevoi, D. M. Klimov, A. G. Petrov, and V. M. Chesnokov,
"Flow of a viscoplastic medium between parallel circular plates moving
toward or away from one another," Izv. AN. MZhG [Fluid Dynamics], No. 1,
pp. 9-17, 1996. |
|
| Received |
15 January 2003 |
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