 | | 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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| In English (Mech. Solids): | | 4810 |
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| A. V. Andreev, R. V. Goldstein, and Yu. V. Zhitnikov, "Evolution of equilibrium of smooth curvilinear cracks with frictional interaction between the edges in the process of loading," Mech. Solids. 38 (1), 111-121 (2003) |
| Year |
2003 |
Volume |
38 |
Number |
1 |
Pages |
111-121 |
| Title |
Evolution of equilibrium of smooth curvilinear cracks with frictional interaction between the edges in the process of loading |
| Author(s) |
A. V. Andreev (Moscow)
R. V. Goldstein (Moscow)
Yu. V. Zhitnikov (Moscow) |
| Abstract |
Analysis of equilibrium of elastic systems with friction shows that,
in general, the stress-strain state is not determined by final values of
applied loads but substantially depends on the loading history [1].
This is connected with the fact that in contact regions
there appear slip zones and adhesion zones as a result of interaction
between surfaces in accordance with the Coulomb law of dry friction. Thus,
in the case of quasistatic variation of applied loads for some values of
the parameters involved, slip may cease on part of the crack or on its
entire surface. Then, two types of adhesion zones become possible, with
zero [2-5] or a nonzero [1, 6, 7] jump in displacements. Therefore, for an
adequate investigation of the limiting equilibrium of cracks with contact
between the surfaces it is necessary to develop methods for solving
problems of evolution of cavities and cut-type cracks during the loading
and to find the laws of that evolution [1, 6-9].
In the present paper, in the framework of two-dimensional linear
elasticity, we study the evolution of equilibrium of an arbitrary smooth
curvilinear crack whose surfaces are interacting with friction during the
process of complex loading. We develop an incremental method which allows
us to study a wide class of problems concerning the limiting equilibrium of
cracks with complex geometry and various types of loading. Moreover, the
approach proposed here can be naturally extended to problems of crack
growth for cracks of arbitrary shape in the process of loading. As an
illustration, we obtain solutions of limiting equilibrium problems for
curvilinear cut-type cracks and flattened cavities in the case of biaxial
tension/compression and various types of loading.
Previously, spatial problems of equilibrium of flat
cut-type cracks and flattened cavities were studied in [1, 7]
on the basis of the asymptotic analysis of solutions near
contact/opening and adhesion/slip regions. In these works,
asymptotic analysis allowed one to trace the evolution
of the tangential displacement jump in the contact region, to obtain
conditions for the beginning of slip of the crack surfaces and coming to
the adhesion state, as well as to establish the
uniqueness of a solution for a given loading trajectory.
The aforementioned conditions can be verified prior to solving the problem
and have a convenient geometric interpretation. The analysis of slip of
crack surfaces performed in [9] was based on passing from the primary
nonlinear problem to a problem for increments; moreover, it was possible
to obtain a problem which is linear with respect to the increment of the
slip angle and to describe some properties of the solution of this
problem. Some problems for which it is possible to describe slip of
crack surfaces with various geometry, for specific trajectories of complex
loading, were considered in [8, 10]. In the framework of the plane
problem, the evolution of equilibrium was studied for rectilinear cracks
of finite length in [6, 11] and for some types of kinked cracks in [12].
In [5], a solution was obtained for the problem of an arbitrary
curvilinear crack whose surfaces interact in accordance with the Coulomb
dry friction law. The adhesion zones considered in [5] admit no
displacement jumps, i.e., the solution does not take into account the
loading history. |
| References |
| 1. | R. V. Goldstein and Yu. V. Zhitnikov,
Equilibrium of Cracks and Cavities Under Complex Loading, with
of Contact, Free Surface, Slip, and Adhesion Zones Being Taken into
Account. Preprint No. 276 [in Russian], In-t Problem Mekhaniki AN SSSR,
Moscow, 1986. |
| 2. | F.-K. Chang and M. Comninou,
"Effects of partial closure and friction on a radial crack emanating from a
circular hole," Intern. J. Fracture, Vol. 28, pp. 29-36, 1985. |
| 3. | Yu. V. Zhitnikov and B. M. Tulinov,
"Equilibrium of a circular cut subject to complex stress state,"
PMM [Applied Mathematics and Mechanics], Vol. 46, No. 3, pp. 521-524, 1982. |
| 4. | B. V. Zhitnikov and B. M. Tupinov,
"Equilibrium of a circular cut with
nonhomogeneous interaction between its edges,"
PMM [Applied Mathematics and Mechanics], Vol. 47, No. 5, pp. 874-880, 1983. |
| 5. | A. V. Andreev, R. V. Goldstein, and Yu. V. Zhitnikov,
"Equilibrium of Curvilinear Cuts, with Formation of Contact,
Slip, and Adhesion Zones Being Taken into Account. Preprint No. 676 [in
Russian], In-t Problem Mekhaniki RAN, Moscow, 2001. |
| 6. | Yu. V. Zhitnikov and B. M. Tupinov,
"Interaction between cut edges subject to complex stress state,"
Izv. AN SSSR. MTT [Mechanics of Solids], No. 4, pp. 168-172, 1982. |
| 7. | R. V. Goldstein and Yu. V. Zhitnikov,
"An investigation of mixed spatial problems with unknown boundaries
for an elastic medium subject to complex loading," in
Plasticity and Fracture of Solids [in Russian], pp. 57-73, Nauka, Moscow, 1988. |
| 8. | R. V. Goldstein and Yu. V. Zhitnikov,
"A numerical-analytical method for the solution of mixed spatial
elasticity problems with unknown boundaries for cavities and cracks.
Part 2," Izv. AN SSSR. MTT [Mechanics of Solids], No. 5,
pp. 65-78, 1988. |
| 9. | R. V. Goldstein and Yu. V. Zhitnikov,
"Analysis of the process of slip of crack surfaces
with friction under complex loading,"
Izv. AN SSSR. MTT [Mechanics of Solids],
No. 1, pp. 139-148, 1991. |
| 10. | R. V. Goldstein and Yu. V. Zhitnikov,
"Stress state of an elastic material weakened by an
elliptic crack with interacting surfaces under complex loading,"
Izv. AN SSSR. MTT [Mechanics of Solids],
No. 1, pp. 126-132, 1988. |
| 11. | P. E. Berkovich, V. I. Mossakovskii, and V. M. Rybka,
"The effect of the external loading history on the stress-strain state
of a cracked material with friction,"
Izv. AN SSSR. MTT [Mechanics of Solids], No. 4, pp. 137-142, 1977. |
| 12. | W. Zang and P. Gudmundson,
"Frictional contact problems of kinked crack modelled by a
boundary integral method. Report 121. Sweden. TRITA-HFL-0121,"
Dept. of Solid Mechanics. Royal Institute of Technology,
Stockholm, 1989. |
| 13. | N. I. Muskhelishvili,
Some Basic Problems of the Mathematical Theory of Elasticity [in Russian], Nauka,
Moscow, 1966. |
| 14. | M. P. Svaruk,
Two-Dimensional Elasticity Problems for Bodies with Cracks [in Russian],
Naukova Dumka, Kiev, 1981. |
| 15. | N. I. Muskhelishvili,
Singular Integral Equations [in Russian], Nauka, Moscow, 1968. |
| 16. | V. V. Panasyuk, M. P. Savruk, and A. P. Datsyshin,
Stress Distribution near Cracks in Plates and Shells [in Russian], Naukova Dumka,
Kiev, 1976. |
| 17. | A. M. Lin'kov,
Complex Method of Boundary Integral Equations of Elasticity [in Russian],
Nauka, St. Petersburg, 1999. |
| 18. | M. M. Chawla and T. R. Tamacrishnan,
"Modified Gauss-Jacobi quadrature formulas for the numerical evaluation
of the Cauchy type singular integrals," BIT, Vol. 14, No. 1,
pp. 14-21, 1974. |
| 19. | F. E. Erdogan, G. D. Gupta, and T. S. Cook,
"The numerical solutions of singular integral equations," in
Mechanics of Fracture. Volume 1. Methods of Analysis and Solutions
of Crack Problems, pp. 368-425, Noordhoff Intern. Publ., Leyden, 1973. |
|
| Received |
23 July 2000 |
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