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A Journal of Russian Academy of Sciences | | Founded
in January 1966
Issued 6 times a year
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Online ISSN 1934-7936 |
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| I. M. Lavit, "Crack growth in the process of quasibrittle fracture under an increasing or cyclic load," Mech. Solids. 36 (2), 95-103 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
2 |
Pages |
95-103 |
| Title |
Crack growth in the process of quasibrittle fracture under an increasing or cyclic load |
| Author(s) |
I. M. Lavit (Tula) |
| Abstract |
For an elastic-plastic body in the plane strain state,
we consider the problem of growth for a rectilinear tensile surface crack of
length a. The body is subjected to
distributed surface loads p whose variation is quasistatic and
proportional to a single parameter (Fig. 1). The plastic region near the
crack tip is assumed small in comparison with the dimensions of the body
and the length of the crack. The behavior of the crack in this situation,
with the load parameter monotonically increasing,
can be described, with an acceptable degree of accuracy, by the
Irwin-Orowan [1-2] theory of quasibrittle fracture based on the
assumption that the said plastic region is moving unchanged together with
the crack tip. If the calculated value of the stress intensity
factor KI is less than the crack resistance KIC
(a material constant), then the crack does not move. The quasistatic
growth of the crack implies that its length and the load parameter are
related by KI=IIC. The crack growth resistance due to
the plastic deformation of the material as the plastic region is moving
with the crack tip is assumed constant and is taken into consideration
indirectly in terms of KIC.
Thus, the crack of length a0 begins to grow only when
the load parameter reaches a certain critical value, which agrees with
the experimental data obtained in the case of a monotonically increasing
load parameter [3]. It is known, however, that in certain situations
crack growth may be observed for smaller values of the load parameter
(subcritical fracture growth), in contrast to the theory of
quasibrittle fracture. One of the most important and well-studied
types of such a growth is the fatigue crack growth.
In spite of the fact that fracture mechanics gives an acceptable
description of the laws of such growth, it does not explain
why the fatigue crack grows, unless we introduce some additional
crack growth criteria.
An attempt to overcome this difficulty is undertaken in the present paper,
which continues the studies started in [4, 5]. Our aim is to create
a unified mathematical model of quasistatic fracture growth in
elastic-plastic media - a model adequately describing both the stable
growth of a crack under a monotonically increasing load and its fatigue
growth under the constraints ensuring quasibrittle fracture. The
creation of such a model requires that stresses and strains be
calculated in the plastic region near the crack tip, and therefore, we
have to pass from linear to nonlinear fracture mechanics. This
leads us to the problem of singular points, since the fields of stresses
and strains have singularities at the crack tip [3]. The problem of
singularities exists also in linear fracture mechanics, but in this case
it can be easily solved on the basis of reliable methods which allow us
to find the stresses and the strains near the crack tip. This is possible
because the character of the singular solution is known beforehand; thus,
all successful numerical methods reproduce this solution. The situation
is different when solving the elastic-plastic problem. The asymptotics of
the stress field near the crack tip in the framework of linear fracture
mechanics is known [3]. Under certain additional assumptions, the
asymptotics of the stress field in nonlinear fracture mechanics is also
known (the HRR-asymptotics [3]). These two types of asymptotic behavior
are quite different. But there is no method of solving plasticity
problems, like the Il'yushin method of elastic solutions, that would
allow us to construct a numerical algorithm for obtaining the
HRR-asymptotics.
The singularity of stresses and strains at the crack tip is
not a physical property (in reality, infinite stresses or
strains does not occur), but an attribute of its mathematical
description. This description can be changed so as to eliminate
singularities. Suppose that there are interaction forces between the
edges of a crack (called adhesion forces) distributed in
such a way that the crack tip ceases to be a singular point of the
stress-strain state. This assumption has been stated independently by
Leonov and Panasyuk [7], Barenblatt [8-10], and Dugdale [11], whose
arguments are based on previous investigations (see [12, 13]). Solutions of
elastic-plastic problems with adhesion forces are considered in [14-18]
and elsewhere. These publications do not even discuss the possibility of
applying the theory of adhesion forces in mathematical models of
fatigue crack growth. It turns out that if we know how to
calculate stresses and strains near the crack tip (and the introduction
of adhesion forces allows us to do this correctly - by the method
of elastic solutions [6]), we can model the motion of the crack
not only as the load parameter grows (and the crack increases),
but also as the load parameter decreases (and the crack becomes shorter).
The equations describing these processes, the numerical algorithms,
and some calculation results are given below. |
| References |
| 1. | G. R. Irwin, "Fracture dynamics," in Fracturing of Metals,
ASM, Cleveland, pp. 157-166, 1948. |
| 2. | E. Orowan, "Fundamentals of brittle behavior of metals,"
in Fatigue and Fracture of Metals, pp. 139-167, Wiley, New York, 1950. |
| 3. | K. Hellan, Introduction to Fracture Mechanics [Russian
translation], Mir, Moscow, 1988. |
| 4. | I. M. Lavit, "A mathematical model of quasistatic crack growth
in an elastic-plastic medium. 1. Basic assumptions and statement of
boundary value problems," Izv. Tulsk. Univ., Mat., Mekh., Inform.,
Vol. 3, No. 1, pp. 118-123, 1997. |
| 5. | I. M. Lavit, "A mathematical model of quasistatic crack growth
in an elastic-plastic medium. 2. Numerical algorithm and
calculation results," Izv. Tulsk. Univ., Mat., Mekh., Inform., Vol. 3,
No. 1, pp. 124-129, 1997. |
| 6. | A. A. Il'yushin, Plasticity. Part I. Elastic-Plastic Deformations
[in Russian], Gostekhizdat, Moscow, Leningrad, 1948. |
| 7. | M. Ya. Leonov and V. V. Panasyuk, "Propagation of
cracks in solid bodies," Prikl. Mekh., Vol. 5, No. 4, pp. 391-401, 1959. |
| 8. | G. I. Barenblatt, "On equilibrium cracks in brittle bodies.
General ideas and hypotheses. Axially symmetric cracks,"
PMM [Applied Mathematics and Mechanics], Vol. 23, No. 3, pp. 434-444,
1959. |
| 9. | G. I. Barenblatt, "On equilibrium cracks in brittle bodies.
Rectilinear cracks in plates,"
PMM [Applied Mathematics and Mechanics], Vol. 23, No. 4, pp. 706-721,
1959. |
| 10. | G. I. Barenblatt, "On equilibrium cracks in brittle bodies.
Stability of isolated cracks. Relation to energy theories,"
PMM [Applied Mathematics and Mechanics], Vol. 23, No. 5, pp. 893-900,
1959. |
| 11. | D. S. Dugdale, "Yielding of steel sheets containing slits,"
J. Mech. and Phys, Vol. 8, No. 2, pp. 100-104, 1960. |
| 12. | G. I. Barenblatt, "Mathematical theory of equilibrium cracks in
brittle bodies," Zh. Prikl. Mekh. i Tekhn. Fiziki, No. 4, pp. 3-56, 1961. |
| 13. | G. Goodier, "Mathematical theory of equilibrium
cracks" [Russian translation], in Fracture. Vol. 2, Mir, Moscow, 1975. |
| 14. | I. M. Lavit, "On the stable growth of a crack in an
elastic-plastic material," Problemy Prochnosti, No. 7, pp. 18-23, 1988. |
| 15. | I. M. Lavit and L. A. Tolokonnikov,
"Thermoelastic-plastic problem of fracture mechanics for a hollow cylinder
with internal cracks," in Applied problems of Strength and Plasticity.
Methods of Solution [in Russian], pp. 56-61, Izd-vo Gorkovsk. Un-ta, Gorky, 1990. |
| 16. | A. Cornec, H. Yuan, and G. Lin, "Cohesive zone model for
ductile fracture," GKSS Rept., No. E73, pp. 269-274, 1994. |
| 17. | V. Tvergaard and J. W. Hutchinson,
"The relation between crack growth resistance and fracture process
parameters in elastic-plastic solids," J. Mech. and Phys. Solids,
Vol. 40, No. 6, pp. 1377-1397, 1992. |
| 18. | I. M. Lavit and L. A. Tolokonnikov, "Investigation of crack growth
in an elastic-plastic material," in Proceedings of the 9-th Conference on
Strength and Plasticity, Vol. 1, pp. 114-119, IPM RAN, Moscow, 1996. |
| 19. | G. I. Barenblatt, "On some problems in brittle fracture
mechanics," Inzh. Zh. MTT, No. 6, pp. 153-163, 1968. |
| 20. | I. M. Strojman, Cold Welding of Metals [in Russian], Mashinostroenie,
Leningrad, 1985. |
| 21. | J. T. Hahn, B. L. Averbach, B. C. Owen, and M. Cohen,
"Initiation of microcleavage in polycrystalline iron and steel"
[Russian translation], in Atomic Mechanism of Fracture, pp. 109-134,
Metallurgizdat, Moscow, 1963. |
| 22. | A. I. Lur'e, Theory of Elasticity [in Russian], Nauka, Moscow, 1979. |
| 23. | G. Reiss, "Mathematical methods in fracture mechanics"
[Russian translation], in Fracture. Volume 2, pp. 204-335, Mir, Moscow,
1975. |
| 24. | J. Nott, "Fracture mechanics" [Russian translation],
in Atomistics of Fracture, pp. 145-176, Mir, Moscow, 1987. |
|
| Received |
18 September 1998 |
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