13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- more complex since they can rotate out of plane still being parallel to the direction of maximum compression. Another challenge is to understand the mechanism of in-plane propagation of compaction bands and anti-wing cracks. For instance, the criterion of compaction band propagation cannot be based on simple substitution of rock failure in tension with rock failure in compression, as proposed e.g., in [15]. Indeed, the failure in compression is preceded by the formation of multitude of parallel tensile cracks in the direction of maximum compression (e.g., wing cracks). Such wing cracks are not observed in the compaction bands. On the contrary observations of thin sections suggest random orientations of cracks and grains within the compaction bands (e.g. [6, 16, 17]). Multiple microcracks are even observed in the directions normal to the direction of compression (see for instance the thin sections presented in [16]). Obviously, such microcracks cannot be produced by compressive stresses but it is conceivable that they are produced as a result of mutual rotations of the grains. We hypothesise that both the compaction band and anti-wing crack propagation involves mutual rotation of the grains followed by fragmentation of the cement connecting the grains and subsequent rearrangement and compaction of the grains. (We note that the anti-wing cracks were observed in rocks with grain structure such as granodiorite, but never in homogeneous materials such as PMMA [10].) Given that grain rotations are directly observed in the shear bands in granular materials [18-20], we assume that grain rotations can also form the mechanism of in-plane shear band propagation. Continuum modelling of rocks with grain rotations requires the use of Cosserat or micropolar continuum (e.g. [21-23]), which additionally considers rotational degrees of freedom and introduces the moment stress. Another difference from the classical continuum is that the Cosserat continuum possesses characteristic lengths (Cosserat characteristic lengths). Extensive research was devoted to cracks in such continua based on considering stress singularities at the crack tip, e.g. [24-27]. These singularities reflect the asymptotics of stress concentration when the distance to the crack tip tends to zero. In other words the traditional approach considers the distances smaller than the Cosserat characteristic lengths. It was however pointed out in [29-31] that when a Cosserat continuum is used to model a particulate material such as rocks with grain microstructure, the asymptotics of small distances to the crack tip is beyond the resolution of the continuum. Indeed, a continuum description of a heterogeneous material is based on the introduction of representative volume elements whose size, H, is naturally much larger than the characteristic microstructural length, lm, which is the scale at which the material can no longer be considered smooth. Then the equivalent continuum is introduced by averaging the relevant physical fields over these volume elements (e.g. [28]). While the equivalent continuum can formally address any distances, including infinitesimal, the interpretation of the calculated physical fields in terms of the original material (needed for instance to formulate the fracture criteria) is only possible in terms of distances larger than lm. This concept poses no restrictions on classical continuum modelling since the classical continua are scale independent. The situation however is different for higher order continua, such a Cosserat continuum, since they possess internal length scales. It was shown in [29-32] that for particulate materials in which particles are cemented to each other (e.g. rocks with granular structure or concrete), the Cosserat lengths are of the order of the grain size, lm. Therefore the only asymptotics in the Cosserat continuum that are relevant to the particulate materials are the asymptotics that concern distances
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