 | | 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
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| In English (Mech. Solids): | | 4811 |
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| I. M. Lavit, "On the asymptotics of the stress and strain fields near a crack tip," Mech. Solids. 44 (3), 389-399 (2009) |
| Year |
2009 |
Volume |
44 |
Number |
3 |
Pages |
389-399 |
| DOI |
10.3103/S002565440903008X |
| Title |
On the asymptotics of the stress and strain fields near a crack tip |
| Author(s) |
I. M. Lavit (Tula State University, pr-t Lenina 92, Tula, 300600 Russia, IgorLavit@yandex.ru) |
| Abstract |
The problem is solved under the plane strain conditions for a crack of general form, which in general is neither a mode I nor a mode II crack. We assume that the strains are small and the material is nonlinearly elastic. The mathematical statement of the problem is reduced to the eigenvalue problem for a system of ordinary nonlinear differential equations. Its solution is obtained numerically. We show that, for an incompressible material with power-law relations between the stress and strain deviators, the solution (the well-known HRR-asymptotics [1, 2]) exists only for mode I and II cracks. In the general case, we can only speak of approximate solutions. A similar conclusion can be made for different-modulus materials. We analyze the results of the preceding papers [1-7], where specific cases of the problem were considered. |
| References |
| 1. | J.R. Rice and G.F. Rosengren,
"Plane Strain Deformation near a Crack Tip in a Power-Law Hardening Material,"
J. Mech. Phys. Solids 16 (1), 1-12 (1968). |
| 2. | J.W. Hutchinson,
"Singular Behavior at the End of a Tensile Crack in a Hardening Material,"
J. Mech. Phys. Solids 16 (1), 13-31 (1968). |
| 3. | V.I. Astaf'ev and V.N. Krutov,
"Stress Distribution near the Tip of an Inclined Crack
in the Nonlinear Fracture Mechanics,"
Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 5, 125-133 (2001)
[Mech. Solids (Engl. Transl.) 36 (5), 101-108 (2001)]. |
| 4. | T.A. Belyakova and E.V. Lomakin,
"A Tensile Crack in an Elastic Medium with Variable Properties Subject to Plane Deformation,"
Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 97-105 (1999)
[Mech. Solids (Engl. Transl.) 34 (3), 82-89 (1999)]. |
| 5. | A.V. Berezin,
Effect of Damage on Strain Characteristics and Strength of Solids
(Nauka, Moscow, 1990) [in Russian]. |
| 6. | A.V. Berezin and P.L. Ponomarev,
"Transverse and Longitudinal Shear Cracks in Dilating Media with Different Moduli,"
Izv. Akad. Nauk. Mekh. Tverd. Tela, No. 3, 127-135 (2002)
[Mech. Solids (Engl. Transl.) 37 (3), 110-116 (2002)]. |
| 7. | J.R. Rice,
"Mathematical Methods in Fracture Mechanics,"
in Fracture, Vol. 2
(Mir, Moscow, 1975), pp. 204-335 [in Russian]. |
| 8. | A.I. Lurie,
The Theory of Elasticity
(Nauka, Moscow, 1970) [in Russian]. |
| 9. | G.C. Sih and H. Liebowitz,
"Mathematical Theory of Brittle Failure,"
in Fracture, Vol. 2
(Mir, Moscow, 1975), pp. 83-203 [in Russian]. |
| 10. | L.M. Kachanov,
Foundations of the Theory of Plasticity
(Nauka, Moscow, 1969; North-Holland, Amsterdam, 1971). |
| 11. | Ph.E. Gill, W. Murrey, and M. Wright,
Practical Optimization
(Academic Press, London, 1981; Mir, Moscow, 1985). |
| 12. | G.I. Barenblatt,
"Mathematical Theory of Equilibrium Cracks Formed under Brittle Fracture,"
Zh. Prikl. Mekh. Tekhn. Fiz., 2 (4), 3-56 (1961). |
| 13. | I.M. Lavit,
"Stable Crack Growth in an Elastoplastic Material,"
Probl. Prochn., No. 7, 18-23 (1988)
[Strength of Materials (Engl. Transl.) 20 (7), 854-860 (1988)]. |
| 14. | V. Tvergaad and J.W. Hutchinson,
"The Relation between Crack Growth Resistance
and Fracture Process Parameters in Elastic-Plastic Solids,"
J. Mech. Phys. Solids 40 (6), 1377-1397 (1992). |
|
| Received |
27 September 2006 |
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