13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- p> pmin( ϕ) = κmsin ϕcos 2 ϕ. (6) Dependence (6) and its interpretation are shown in Fig. 2. It is seen that the dense packing produces negative stiffness at lower pressures that the loose one. Two interim conclusions could be made here. Firstly, the compressive stress required to produce the effect of apparent negative stiffness could be of the order of the bulk modulus of the surrounding rock. That is only possible when the bulk modulus is sufficiently reduced by the damage accumulated in the preceding loading, namely near the peak load. Another conclusion is that the phenomenon of apparent negative stiffness depends upon the density of the initial packing. pmin/ κm Figure 2. Dependence (6) and its interpretation: the smallest values of the minimal compressive stress magnitude pmin needed to ensure the negative stiffness are achieved at dense packing (angles close to π /2 and π), while the loose packing (angles close to 3π /4) requires higher compressive stresses to show negative stiffness. 3. Effect of rotating particles on global stability The presence of negative stiffness means the loss of positive definiteness of the tensor of elastic
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