 | | 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 |
Archive of Issues
| Total articles in the database: | | 12804 |
| In Russian (Èçâ. ÐÀÍ. ÌÒÒ): | | 8044
|
| In English (Mech. Solids): | | 4760 |
|
| << Previous article | Volume 36, Issue 1 / 2001 | Next article >> |
| R. V. Goldstein and M. N. Perelmuter, "An interface crack with bonds between the surfaces," Mech. Solids. 36 (1), 77-92 (2001) |
| Year |
2001 |
Volume |
36 |
Number |
1 |
Pages |
77-92 |
| Title |
An interface crack with bonds between the surfaces |
| Author(s) |
R. V. Goldstein (Moscow)
M. N. Perelmuter (Moscow) |
| Abstract |
A crack located at the interface between different materials is
modeled taking into account regions in which the surfaces of the
crack interact with each other. It is assumed that these regions
are adjacent to the crack tip (in what follows, we refer to these
regions as the end regions). The size of the end regions can be
comparable with the size of the crack. The interaction between the
surfaces of the crack in the end regions is modeled by bonds
between these surfaces with a given law of deformation. The
physical nature of these bonds and size of the end regions depend
on the type of the materials and scale of the crack. The mechanisms
of the interaction between the crack surfaces change depending on the
distance from the crack tip. High stress concentration
near the crack tip leads to the softening of the material
surrounding the crack, causing the formation of microcracks, pores,
and zones of plastic flow. The analysis of the conditions of
equilibrium and development of the crack taking into account the
interaction between its surfaces and formation of the
softening zones leads to the model of a crack with an end region.
In this region,
the adhesive forces (bonds) act, which resist the crack opening.
Such models were suggested in [1, 2] for brittle materials.
The models of [3, 4] assumed the plastic flow in the end regions
under the action of constant stress.
In the case where the zones of action of adhesive forces are introduced,
the total stress intensity factor (SIF), defined as the difference of the
SIF due to the tensile forces and the SIF due to the compressive (adhesive)
forces applied at the crack end region, is equal to zero.
The size of the adhesion zone was assumed to be constant as the
crack grows. G. I. Barenblatt [1, 2] assumed also that the size of
this zone is much smaller than the size of the crack. Problems of
the fracture mechanics were solved analytically and numerically
in [5-7] on the basis of such models. In composite materials and
structures consisting of various materials (for example, polymer
and ceramics, polymer and metal), the deformation and
fracture must be considered on various scale levels, beginning with
the analysis of the local fracture at the interface between two
components [8]. In such materials, the local fracture near a
crack tip is accounted for by different mechanisms in different zones of the
end region of the crack (multi-scale fracture). In the general case,
the size of the end region changes as the crack grows and is not
small in comparison with the characteristic size of the crack. For
such materials, the models of a crack with the end region assume
that the loads which resist the crack opening lead to the decrease in
the SIF at the crack tip, but the total SIF is not zero [9, 10]. In
accordance with these models, the problem of the limit equilibrium
is solved by prescribing the law of deformation of the bonds in the
end region of the crack and analyzing the stress state near the
crack using an appropriate criterion of fracture. We
assume that the law of deformation of the bonds is given (equations
of deformation of the bonds are considered in [11-13] for various
materials). The stress state in the end region of a crack was
analyzed in the two-dimensional formulation in [14-17] for
isotropic materials, in [18] for transversely isotropic materials,
in [19] for orthotropic materials, and in [20] for isotropic
axisymmetric domains with a circular crack. The three-dimensional
problem for an isotropic body with a plane crack and linear
elastic bonds was considered in [21]. The development of a crack at
the interface between materials attended with the softening in the
end region was simulated numerically in [22, 23] for the case where
one of the materials undergoes the plastic deformation and in [24]
for a crack between a viscoelastic and a rigid bodies.
In the present study, the equilibrium of a crack at the
interface between materials is considered. The crack is subjected to
external tensile loads and stresses in the bonds which resist
the crack opening. This problem is reduced to the system of nonlinear singular
integro-differential equations with the kernel of the Cauchy type.
By solving this system, we determine the normal and tangential
stresses in the bonds. The SIF at the crack tip are calculated
taking into account the bonds in the end region of the crack, the
influence of which
compensates for the external loads. We also consider the energy
characteristics of a crack with the end region, such as the release rate of
the strain energy and rate of absorption of the energy by bonds. The
condition of the limit equilibrium of a crack with the end region is stated
taking into account the criterion of limit stretching of bonds. The
parametric analysis of the force and energy characteristics of the crack as
functions of the size of the end region and physical and mechanical
parameters of the bonds is carried out. |
| References |
| 1. | G. I. Barenblatt, "On the equilibrium cracks formed during
the brittle fracture," PMM [Applied Mathematics and Mechanics],
Vol. 23, No. 3, pp. 434-444; No. 4, pp. 706-721; No. 5,
pp. 893-900, 1959. |
| 2. | G. I. Barenblatt, "Mathematical theory of equilibrium
cracks formed during the brittle fracture," Zh. Prikl. Mekh. i Tekhn. Fiziki,
No. 4, pp. 3-53, 1961. |
| 3. | M. Ya. Leonov and V. V. Panasyuk, "Development of finest
cracks in a solid," Prikl. Mekhanika, Vol. 5, No. 4, pp. 391-401,
1959. |
| 4. | D. S. Dugdale, "Yielding of steel sheets containing
slits," J. Mech. and Phys. Solids, Vol. 8, No. 2, pp. 100-108,
1960. |
| 5. | V. M. Alexandrov, B. I. Smetanin, and B. V. Sobol', Thin
Concentrators of Stresses in Elastic Bodies [in Russian], Nauka,
Moscow, 1993. |
| 6. | N. C. Huang, "On the size of the cohesive zone at the
crack tip," Trans. ASME J. Appl. Mech., Vol. 52, No. 2,
pp. 490-492, 1985. |
| 7. | T. Ungsuwarungsri and W. G. Knauss, "A nonlinear analysis of an
equilibrium craze. Part I and II," Trans. ASME. J. Appl. Mech.,
Vol. 55, No. 1, pp. 44-58, 1988. |
| 8. | V. V. Bolotin and Yu. N. Novichkov, Mechanics of
Multilayered Structures [in Russian], Mashinostroenie, Moscow,
1980. |
| 9. | L. N. McCartney, "Mechanics of matrix cracking in
brittle-matrix fiber-reinforced composites," Proc. Royal Soc.,
No. A409, pp. 329-350, 1987. |
| 10. | B. N. Cox and D. B. Marshall, "Concepts for bridged
cracks in fracture and fatigue," Acta Met. Mater., Vol. 42, No. 2,
pp. 341-363, 1994. |
| 11. | B. Budiansky, A. G. Evans, and J. W. Hutchinson,
"Fiber-matrix debonding effects on cracking in aligned fiber
ceramic composites," Int. J. Solid structures, Vol. 32, No. 3-4,
pp. 315-328, 1995. |
| 12. | H. Ji and P. G. de Gennes, "Adhesion via connector
molecules: the many-stitch problem," Macromolecules, Vol. 26,
pp. 520-525, 1993. |
| 13. | R. V. Goldstein, V. F. Barikov, and M. N. Perelmuter,
"Modeling of the adhesion strength and fracture kinetics of the
microelectronic package polymer-polymer joints," in Proc. In-ta
Phys. Technol. RAS. Volume 13. Modeling and Simulation
of Submicron Technology and Devices, pp. 115-125, Moscow, 1997. |
| 14. | V. M. Entov and R. L. Salganik, "To the Prandtl model of
brittle fracture," Izv. AN SSSR. MTT [Mechanics of Solid], No. 6,
pp. 87-99, 1968. |
| 15. | Y. Weitsman, "Nonlinear analysis of crazes," Trans.
ASME. J. Appl. Mech., Vol. 53, pp. 97-102, 1986. |
| 16. | L. R. F. Rose, "Crack reinforcement by distributed
springs," J. Mech. Phys. Solids, Vol. 35, No. 4, pp. 383-405,
1987. |
| 17. | B. Budiansky, J. c. Amazigo, and A. G. Evans,
"Small-scale crack bridging and the fracture toughness of
particulate-reinforced ceramics," J. Mech. Phys. Solids, Vol. 36,
No. 2, pp. 167-187, 1998. |
| 18. | S. Nemat-Nasser and M. Hori, "Toughening by partial or full
bridging of cracks in ceramics and fiber reinforced composites," Mech.
Mater., Vol. 6, No. 3, pp. 245-269, 1987. |
| 19. | N. Morozov, M. Paukshto, and N. Ponikarov, "On the
problem of equilibrium length of bridged crack," Trans. ASME. J.
Appl. Mech., Vol. 64, pp. 427-430, 1997. |
| 20. | N. V. Movchan and J. R. Willis, "Penny shaped crack
bridged by fibers," Quart. Appl. Math., No. 2, pp. `327-340,
1998. |
| 21. | E. I. Shifrin, "A plane crack of normal fracture the
surfaces of which interact in accordance with a linear law," Izv.
AN SSSR. MTT [Mechanics of Solid], No. 5,
pp. 94-100, 1988. |
| 22. | V. Tvergaard and J. W. Hutchinson, "The influence of
plasticity on mixed mode interface toughness," J. Mech. Phys.
Solids, Vol. 41, No. 6, pp. 1119-1135, 1993. |
| 23. | Y. Wei and J. W. Hutchinson, "Models of interface
separation accompanied by plastic dissipation at multiple scales,"
Int. J. Fracture, Vol. 95, No. 1, pp. 1-17, 1999. |
| 24. | A. Needleman, "An analysis of decohesion in an imperfect
interface," Int. J. Fracture, Vol. 42, No. 1, pp. 21-40, 1990. |
| 25. | L. I. Slepyan, Mechanics of Cracks [in Russian],
Sudostroenie, Leningrad, 1981. |
| 26. | J. R. Rice and G. C. Sih, "Plane problems of cracks in
dissimilar media," Trans. ASME. J. Appl. Mech., Vol. 32,
pp. 218-224, 1965. |
| 27. | R. V. Goldstein and M. N. Perelmuter,
A Crack with Bonds at an Interface Between Materials
[in Russian], Preprint No. 568, IPM RAN, Moscow,
1996. |
| 28. | R. V. Goldstein and M. N. Perelmuter, "Modeling of
bonding at the interface crack," Int. J. Fracture, Vol. 99,
No. 1-2, pp. 53-79, 1999. |
| 29. | J. R. Rice, "Elastic fracture mechanics concepts for
interface cracks," Trans. ASME. J. Appl. Mech., Vol. 55,
pp. 98-103, 1988. |
| 30. | R. L. Salganik, "On the elastic fracture of glued
bodies," PMM [Applied Mathematics and Mechanics], Vol. 27, No. 5,
pp. 957-962, 1963. |
| 31. | J. W. Hutchinson and Z. Suo, "Mixed mode cracking in
layered materials," in J. W. Hutchinson and T. Y. Wu (Editors), Adv.
Appl. Mech., Vol. 29, pp. 63-191, 1992. |
| 32. | A. A. Il'yusin, Plasticity [in Russian], GITTL, Moscow,
Leningrad, 1948. |
| 33. | I. A. Birger, "Design of structures taking into account
plasticity and creep," Izv. AN SSSR. Mekhanika, No. 2,
pp. 113-119, 1965. |
| 34. | V. Z. Parton and P. I. Perlin, Integral Equations of
Elasticity [in Russian], Nauka, Moscow, 1975. |
| 35. | R. V. Goldstein and M. N. Perelmuter,
An Interface Crack With Nonlinear Interaction Between the Surfaces
[in Russian], Preprint No. 619, IPM RAN, Moscow,
1998. |
| 36. | M. N. Perelmuter, "Polymer interface crack and adhesion
resistance," in Proc. of Workshop. Mechanical Reliability of
Polymeric Materials and Plastic Packages of IC Devices, pp. 231-236, Paris,
France, 1998. |
|
| Received |
12 October 2000 |
| << Previous article | Volume 36, Issue 1 / 2001 | Next article >> |
|
 If you find a misprint on a webpage, please help us correct it promptly - just highlight and press Ctrl+Enter
|
|
Russian 
English