13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- r>>lm. This translates into the notion that the meaningful Cosserat solutions are those that correspond to the asymptotics r lm →∞. We call such a continuum the small-scale Cosserat continuum. In this approach the relevant stress singularities at the crack tip are given by the intermediate asymptotics, which refers to the distances infinitesimal with respect to the crack dimensions but infinitely large compared to the Cosserat lengths. It is interesting that the technique of obtaining the intermediate asymptotics simplifies the analysis since the derivation of the asymptotics is less involved than the orthodox calculation of the stress singularities in Cosserat continuum. In this paper we use the concept of the small-scale Cosserat continuum and propose a universal criterion of in-plane growth of shear cracks and compaction bands in compression based on the concentration of moment stresses. In this criterion the actual failure is produced by tensile fracturing of the cement layers between the grains caused by the bending moments associated with the concentration of moment stresses. This mechanism is independent of the sense of the moment stress; the latter only controls the location and direction of the induced tensile microcracks. 2. Crack propagation caused by moment stress concentration The classical fracture criteria are based on the notion of crack propagation by separating two surfaces by applied stresses and expressing the conditions of the separation either in terms of critical forces or critical energy. In particulate materials consisting of particles (grains) and connected by cement bonds another micro failure mechanism can be at work: bending the cement layer by mutual rotation of adjacent particles and its cracking by flexure cracks, Fig. 1. Essentially what this mechanism is doing is translating the moment stress acting at scale H into microscopic tensile stresses at scale lm. Since scale lm is beyond the resolution of the (Cosserat) continuum that models the particulate material at scale H>>lm, the failure criterion should be formulated in terms of moment stress µij. Given that both stress and moment stress have singularity at the crack tip, we will use the approach proposed in [33] in which the crack propagation criterion is based on comparing the stress at a certain distance from the crack tip with the local material strength. It is natural to use lm in place of such a length (see also [29-31]). Hence, for the simple case of fracture criterion controlled by a single moment stress component one has µij lm ( ) = µc . (1) Here µij is the moment stress component controlling local failure and the absolute value sign indicates that the instance of local failure is independent of the sense of the moment stress; the sign only controls the side of the link between the neighbouring particles from which it starts breaking. The critical value of moment stress µc represents the particular mechanism of bond breakage and microscopic properties of the material of the bond. From the symmetry analysis it can be found [29-31] that the moment stresses invoke the bond bending and fracturing shown in Fig. 2. It is seen that while the flexure cracks in Mode I crack are roughly coplanar with the main crack, the flexure cracks in Mode II crack form en-echelon of microcracks normal to the main crack. Yet, it is the bond (cement) breakage associated with relative particle rotations that separate the particles from the matrix and thus effect the crack propagation. The formation of en-echelon cracks is thus an
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