13th International Conference on Fracture June 16–21, 2013, Beijing, China -2- 2. Apparent negative stiffness caused by rotating particles. The effect of elbowing Consider a particulate material loaded in compression to near the peak load. We assume that in the process of loading considerable amount of defects have been accumulated, mainly on the bonds (cement) between the particles such that some particles are now partially detached from the matrix, Fig. 1. The balance of moments of shear and normal forces shown in Fig. 1 about point O reads Tsin ϕ+Pcos ϕ=0, π 2≤ ϕ≤ π. (1) Here l is the corresponding particle diameter, T and P are the magnitudes of the shear and normal forces and angle ϕ is related to the position of the particle at the moment of detachment. Obviously, the moment equilibrium is only possible for the range of angles ϕ indicated in (1) and in Fig. 1a; when ϕ<π/2, Fig. 1b, the particle becomes unstable. It is reasonable to assume that the initial packing of particles was a stable one similar to the configuration shown in Fig. 1a. The analysis below is based in infinitesimal deformations and hence all movements considered will leave the initially stable configuration in its stable state. For that reason we will disregard the unstable configurations shown in Fig. 1b. x y ϕ" T P O A x y ϕ" T P O A (a) (b) Figure 1. A moment balance of rotating (rolling) partially detached particle: (a) stable configuration, (b) unstable configuration. Suppose the particle underwent an infinitesimal rotation dϕ. This will change the coordinates (x, y) of point A, Fig. 1 by dx=-lsinϕ dϕ and dy=lcos ϕ dϕ. The moment equilibrium (1) imposes the following force increments:
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