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IssuesArchive of Issues2009-3pp.389-399

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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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