# Fracture Nucleation from a Compression-parallel, Finite-width Elliptical Flaw

Y. G. YUANt E. Z. LAJTAI t M. L. AYARIt

INTRODUCTION

How tensile fractures form and propagate in response to crack-parallel compression has been puzzling the rock mechanicscommunityfordecades. Thedirectapplication of engineering fracture mechanics to the problem of tensile fracture in a compressive environment appears to be invalid, because the zero-width mathematical crack model is unresponsive to the normal stress that is coaxial to the crack direction. Fractures can however be observed to propagate parallel to a compressive stress, and pre-existing compression-parallel cracks are known to extend in their own plane when subjected to increasing compressive stress [1, 2].

Fracture mechanics needs tensile stresses to nucleate tensile fractures. Theoretical models exploit the inhomogeneity of rock materials (flaws) to produce localized tensile stresses in an otherwise compressive stress environment. The crack starting flaws can be pre-existing microcracks, grain boundaries, elastic moduli mismatches, cylindrical pores, dislocation pile-ups and Herztian contacts [l, 3]. Perhaps, the best example for this approach is the sliding-crack model of Nemmat-Nasser [4, 5]. The fracture nucleating flaw is a “sliding crack”, a closed flat ellipse that is inclined to the compression direction. The tension is generated at the crack tip through the shear stress along the plane of the crack and through the normal stress that is perpendicular to it (the crack opening and the in-plane sliding modes of fracture mechanics). The stress intensity factor formulation of the mathematical crack model neglects the other normal

stresses and in particular the one that lies parallel with the major axis of the ellipse. The nucleated cracks (wing- cracks) propagate along the gently curving maximum principal stress trajectory until the crack becomes parallel with the far-field (applied) compressive stress. The fracture process stalls at this point.

Practically, all the numerical simulations that use inhomogeneities show that the extension fracture arising from an inhomogeneity is comparable in size to the inhomogeneity itself. Once, the newly generated fracture

propagates beyond the zone of influence of the inhomogeneity, tension yields to compression and the fracture process halts. The sliding crack or any other stress starter should therefore be observable under the microscope. Microscopic observations, however, report only the presence of compression parallel extension fractures, and only rarely can the actual crack starter be identified [1, 2, 3]. Since, the tensile fractures aligned with the compression direction extend in their own plane with increasing compression, they themselves must be considered as potential crack starters. The zero-width, mathematical crack of fracture mechanics is clearly unsuitable for this purpose.

Sensitivity to crack-parallel compression can be introduced by replacing the zero-width, mathematical crack with a general elliptical crack, with the major axis representing the crack length oriented along the maximum principal stress trajectory, and the minor axis representing the crack width. We call this the Finite-Width Elliptical Crack, or FIWEC for short. Using the FIWEC model in connection with some type of fracture criterion, the propagation of compression-parallel tensile cracks can be modelled as a series of crack nucleation events from an open,finite-widthellipticalflawofprogressivelyincreasing length.

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**REFERENCES**

1. KranzR.LMicrocracksinrocks:areview. Tectonophysws, I00, pp. 449-480 (1983).

2. LajtaiE.Z.,CarterB.J.andAyariM.L.Criteriaforbrittle fracture in compression. Eng. Frac. Mech.. 37, pp. 59-74 (1990).

3. Kemeny J.M. and Cook N.G.W. Crack models for the failure of rocks in compression. Prec. 2nd Int. Conf. Constitutive Lawsfor Eng. Mat., 2, pp. 879-887 (1987). Horri H. and Nemmat-Nasser S. Compression-induced microcraek growth in brittle solids: axial splitting and shear failure. J. Geophys. Res., 90, pp. 3105-3125 (1985).

5. Nemmat-Nasser S. and Horri H. Compression-induced nonplanar crack extension with application to splitting, exfoliation and rockburst. J. Geophys. Res., 87, pp. 6805- 6821 (1982).

6. Carter B.J., Lajtai E.Z. and Petukhov A. Primary and remote fracture around underground cavities. Int. J. Num. Analy. Meths. in Geomechanics., 15, pp. 21-40 (1991).

7. Lajtai E.Z. Effect of tensile stress gradient oF brittle fracture in compression. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 9, pp. 569-578 (1972).

8. Ritchie R.O., Knott J.F. and Rice J.R. On the relationship between critical tensile stress and fracture toughness in mild steel. J. Mech. Phys. Solids, 21, pp. 395-410 (1973).

9. Jaeger J.C. and Cook N.G.W. Fundamentals of Rock Mechanics, 3rd edition, Chapman and Hall, London, 593 pp. (1979).

10. Carter B . J . , Lajtai E.Z. and Yuan Y.G. Tensile fracture from circular cavities loaded in compression. Int. J. Fract., 57. pp. 221-236 (1992).

11. Carter B.J., Duncan E.J. and Lajtai E.Z. Fitting strength criteria to intact rock. Geotechnical and Geological Engineering, 9, pp. 73-81 (1991).

12. Samrnis C.G. and Ashby M.F. The failure of brittle porous solids under compressive stress states. Acta. Metall., 34, pp. 511-526 (1986).