13th International Conference on Fracture June 16–21, 2013, Beijing, China -7- µ+mµm ( ) 5 µ2 4µ+3 ( ) = µm 3 1+m ( ) 5 4µm+3 1−c ( ) 5 . (11) The point of instability is reached when µ=µ1. This happens at the concentration of negative stiffness inclusions ccr =1− µ1 +mµm 1+m 4µm+3 µm 3 µ 1 2 4µ 1 +3 ( ) " # $ $ % & ' ' 1 5 . (12) The dependencies (11) and (12) are shown in Figs. 3 and 4 respectively for different values of m and µm. It is seen from Fig. 3 that the effective shear modulus can both increase and decrease with concentration of the negative stiffness inclusions depending upon the values of parameters m and µm. The plot of critical concentration, Fig. 4a, shows that there exist combinations of parameters m and µm at which the critical concentration is zero. That means that at the instance when the particle rotations start and make the corresponding shear modulus negative, the particulate material loses stability. Dependence of the value of negative shear modulus of inclusions vs. the shear modulus of the matrix is shown in Fig 4b. It is seen that the dependence is relatively weak; the value of the negative shear modulus that delivers zero critical concentration is of the order of the shear modulus of the matrix. 0 0.2 0.4 0.6 0.8 0 2 4 6 c/ccr m=0.2, µm=0.2 m=0.2, µm=1 m=0.2, µm=5 m=1, µm=1 m=5, µm=1 m=5, µm=5 m=5, µm=0.2 µ µm Figure 3. Effective shear modulus µ vs. volumetric fraction (concentration) of negative stiffness inclusions c. Three pairs of parameters m and µm on the left side of the picture refer to nearly indistinguishable dependencies in the same order from top to bottom.
RkJQdWJsaXNoZXIy MjM0NDE=