13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- 5. Conclusions Particulate materials such as rocks with granular microstructure and concrete permit relative rotations of the grains independent on their displacements. Therefore the criteria of crack propagation in such materials should include the grain (particle) rotations and the associated moment stresses. This is achieved by considering bending of the cement links/bonds between neighbouring particles caused by their relative rotation. The bending produces tensile stress on one side of the link, which eventually initiates a flexure crack (microcrack as seen from the scale of the propagating fracture). The initiation of the flexure crack leads to the link breakage and ultimately effects the particle detachment from the bulk of the material. The fracture mechanism based on link breakage and particle detachment is independent of the sign and source of the moment stress; only the side of the link where the flexural crack starts and the orientation/position of the link that is fractured first are affected. It is assumed that after the first link is broken the resistance to particle rotation is diminished sufficiently to permit breakage of other links and allow for the complete particle detachment. This criterion can explain the in-plane growth of Mode I and Mode II cracks as well as the anti-cracks (compaction bands). Modelling of fracture propagation that involves moment stress requires the use of Cosserat (micropolar) continuum, which includes rotational degrees of freedom on top of the conventional translation ones. The Cosserat continuum possesses characteristic lengths that in the case of particulate materials with cement bonds/links between the particles are of the order of the particle size. Since the resolution of a continuum cannot be better than the microstructural length (the particle size in our case) the stress singularity at the crack tip only refers to the distances from the crack tip larger than the particle size and therefore larger than the Cosserat characteristic lengths. This leads to the concept of small-scale Cosserat continuum that is an asymptotics of small Cosserat lengths. This asymptotics formally leads to the Cosserat continuum with constrained rotations (the couple stress continuum). Modelling dislocations and Mode I and II cracks in such a continuum allows further simplification whereby the stress and moment stress distribution can directly be obtained from the displacement field produced by conventional dislocations and cracks by applying the relations of the couple stress theory. It was found that while the stresses have conventional square root singularity, the moment stresses have singularity of the power 3/2. The actual fracture criterion is based on the stress and moment stress computed at a distance from the crack tip equal to the particle size (the Cosserat length). The tensile stress produced in the link/bond between particles by the moment stress is an order of magnitude higher than the one associated with the classical stress singularity. This suggests that the rotational mechanism of crack growth can actually supersede the traditionally perceived mechanism based on the tensile stress concentration only. The flexure cracks formed in the process of particle rotations are seen at the scale of the crack as microcracks which are either coplanar to the main crack in the cases of Mode I crack or anti-crack or form en-echelon in the case of Mode II crack. The actual crack propagation is however caused by detachment (separation) of the particles from the bulk of the material and hence the appearance of en-echelon cracks is essentially a secondary effect accompanying the rotational mechanism of crack propagation. Acknowledgements The work has been supported by the Deep Exploration Technologies Cooperative Research Centre
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