13th International Conference on Fracture June 16–21, 2013, Beijing, China -6- dκeff dc = κincl −κeff 1−c κeff +κ ∗ κincl +κ ∗ , κ ∗ = 4 3 µeff dµeff dc = µincl − µeff 1−c µeff + µ ∗ µincl + µ ∗ , µ ∗ = µeff 6 9κeff +8µeff κeff +2µeff κeff c=0 = κm, µeff c=0 = µm $ % & & & & ' & & & & . (7) where c is the volumetric fraction (concentration) of inclusions. In our case, κincl = κm and therefore κeff = κm is an obvious solution of the first equation in (7). Due to the uniqueness of the solution of this system of differential equations, there are no other solutions. We now normalise the moduli with κm by formally assuming that κm =1. After introducing the notations µeff = µ, µincl =−mµm, m= p sin3 ϕ − cos2 ϕ sin2 ϕ . (8) system (7) is reduced to dµ dc = −5µ 1−c ⋅ (4µ+3)(mµm+ µ) 8µ2 −3(4mµ m −3) µ−6mµm µ c=0 = µm # $ % & % . (9) Derivative dµ/dc is discontinuous when the denominator in the right hand site of (9) vanishes. The discontinuities correspond to points µ1 and µ2: µ1,2 = 3 16 (4mµm −3) 3± 48µm 2m2 −8mµ m+27 " #$ % &' . (10) It can be shown that µ2<0<µ1. Since the initial condition in (9) is µ(0)=µm>0, the solution of (9) can only reach point µ1, after which the effective shear modulus drops to a certain negative value determined by the global loading device which applies the load to the particulate material [23, 26]. We therefore treat the modulus µ1 as a point of global intrinsic instability of the particulate material. Solution of (9) can be obtained in the following implicit form
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