Finite Strain Singular Field Near the Tip of a Crack Terminating at a Material Interface

Abstract

The paper contains an asymptotic finite-strain analysis of the deformations and stresses near the tip of a traction-free crack terminating at an interface between two dissimilar compressible elastic half-planes under conditions of plane strain. Each of the two half-planes is assumed to be hyperelastic with a harmonic-type strain energy density introduced by John. The near-tip singular field is derived for an arbitrary meeting angle at which the crack meets the interface and the deformations and stresses around the crack tip are found to be free of oscillatory singularities arising in the linearized theory. Regardless of the loading conditions, the explicit conditions for the strict positivity of the Jacobian determinant in the vicinity of the crack tip are given in terms of the known material parameters and the meeting angle. In particular, these conditions are unconditionally satisfied for an interface crack tip. As two typical examples, the interface crack and the crack perpendicular to the interface are discussed.

Get full access to this article

View all access options for this article.

References

[1] Williams, M. L. : The stresses around a fault within the dissimilar media. Bulletin of the Seismological Society of America, 49, 199-204 (1959).

[2] England, A. E. : A crack between dissimilar media. ASME, Journal of Applied Mechanics, 32, 400-403 (1965).

[3] Bogy, D. B. : On the plane elastostatic problem of a loaded crack terminating at a material interface. ASME, Journal of Applied Mechanics, 38, 911-918, (1971).

[4] Comninou, M. : The interface crack. ASME, Journal of Applied Mechanics, 44, 631-634 (1979).

[5] Knowles, J. K. and Sternberg, E. : Large deformation near a tip of an interface-crack between two Neo-Hookean sheets, J. Elast., 13, 257-293 (1983).

[6] Herrmann, J. M. : An asymptotic analysis of finite deformation near the tip of an interface-crack. J. Elast., 21, 227-269 (1989).

[7] Geubelle, P. and Knauss, W. G. : Finite strains at the tip of a crack in a sheet of hyperelastic material: I. Homogeneous case, II. Special bimaterial cases, III. General bimaterial case. J. Elast., 35, 61-98, 99-138, 139-174 (1994).

[8] Geubelle, P. : Finite deformation effects in homogeneous and interfacial fracture. Int. J. Solids Structures, 32, 1003-1016 (1995).

[9] John, F. : Plane strain problems for a perfectly elastic material of harmonic type. Communications in Pure and Applied Mathematics, XIII, 239-290 (1960).

10.

[10] Knowles, J. K. and Sternberg, E. : On the singularity induced by certain mixed boundary conditions in linearized and nonlinear elastostatics. Int. J. Solids Structures, 11, 1173-1201 (1975).

11.

[11] Ogden, R. W. and Isherwood, D. A. : Solution of some finite plane-strain problems for compressible elastic solids. Quarterly Journal of Mechanics and Applied Mathematics, XXXI, 219-249 (1978).

12.

[12] Varley, E. and Cumberbatch, E. : Finite deformation of elastic materials surrounding cylindrical holes. J. Elast., 10, 341-405 (1980).

13.

[13] Carroll, M. M. : Finite strain solutions in compressible isotropic elasticity. J. Elast., 19, 65-92 (1987).

14.

[14] Steigmann, D. J. and Pipkin, A. C. : Stability of harmonic materials in plane strain. Quarterly of Applied Mathematics, XLVI, 559-568 (1988).

15.

[15] Cook, T. S. and Erdogan, F. : Stresses in bonded materials with a crack perpendicular to the interface. Int. J. Eng. Sci., 10, 677-697 (1972).

16.

[16] Le, K. C. : On the singular elastostatic field induced by a crack in a Hadamard material. Quarterly Journal of Applied Mathematics and Mechanics, 45, 101-117 (1992).

17.

[17] Le, K. C. , and Stumpf, H. :The singular elastostatic field due to a crack in rubberlike materials. J. Elast., 32, 183-222 (1993).