13th International Conference on Fracture June 16–21, 2013, Beijing, China -8- 0 1 2 3 4 5 0 0.2 0.4 ccr m µm=0.1 µm=10 µm=1 Ccr=0 (a) 0 2 4 6 8 10 0.6 0.8 1 1.2 mc µm Ccr=0 (b) Figure 4. The critical concentration of negative stiffness inclusions ccr: (a) its dependence upon the value m of negative shear modulus of inclusions; (b) the values mcr of negative shear modulus of inclusions delivering zero critical concentration. From here it is seen that the magnitude of compressive stress that produces the global instability of the particulate material is pcr κm =mc sin 3 ϕ+sin ϕcos2 ϕ. (13) 4. Conclusions and outlook The ability of partially detached non-spherical particles to roll or rotate leads to the effect of apparent negative stiffness (negative shear modulus), whose value depends on the magnitude of the applied compressive stress. This is a property of particle non-sphericity: rotation of spherical (or circular in 2D) particles does not produce negative stiffness. Rotation of non-spherical particles also produces elbowing which results in dilation of the surrounding material. Depending of the initial packing, dilation can lead to the reduction of the value of negative shear modulus such that the magnitude of compressive stress needed to effect negative stiffness is of the order of the bulk modulus. Therefore the effect of negative stiffness is only relevant to the particulate materials loaded in compression up to the peak when the damage created in the course of loading has considerably weakened the material and made the moduli sufficiently low. The global instability of the particulate material with rolling or rotating particles is reached when the effective shear modulus is no longer positive. This happens when the concentration (volumetric fraction) of negative stiffness areas reaches a certain critical value that depends upon the value of negative shear modulus and the shear modulus of the surrounding material. There exist a combination of these parameters which makes the critical concentration zero, meaning that the first rolling particle results in global instability. The theory proposed casts light on the mechanics of compressive failure of particular materials such as rock and concrete as well as on the mechanism instability of granular materials. Another possible application of this theory is in the design of a special class of hybrid materials based on specially
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