9 Furthermore, as pointed out in Carpinteri et. al. (2009), the statistical analysis of b-values is closely correlated with the fractal geometry approach in the damage and fracture mechanics of heterogeneous materials. Fractal geometry is the natural tool to characterize self-organized processes, emphasizing their universality and the scaling laws arising at the critical points. 3. THE TRUSS-LIKE DISCRETE ELEMENT METHOD The truss-like DEM used in this work represents the continuum by means of a periodic spacial arrangement of bars with the masses lumped at their ends. A lumped mass of ( 3 2 Lρ ) corresponds to each internal node, where ρ is the density and L the length of a cubic module. The nodes will have a lumped mass of ( 3 16 Lρ ) if they are localized in the corner, ( 3 8 Lρ ) on the edges and ( 3 4 Lρ ) on a free surface. The discretization uses a basic cubic module constructed using 20 bar elements and 9 nodes showed in Figure 1(a) and 1(b). Every node has three degrees of freedom, which are the three components of the displacement vector in the global reference system. Figure 1. DEM discretization strategy: (a) basic cubic module, (b) generation of a prismatic body. In case of an isotropic elastic material, the cross-sectional area Al of the longitudinal elements (those defining the edges of the module and those parallel to the edges connected to the node located at the centre of the module) in the equivalent discrete model is: 2 lA Lφ = (4) where L is the length of the side of the cubic module under consideration. The function ( ) ( ) 9 8 / 18 24 φ δ δ = + + , where ( ) 9 / 4 8 δ ν ν = − , accounts for the effect of the Poisson’s ratio ν. Similarly, the area Ad of the diagonal elements is: 2 2 3 d A Lδφ = (5) The coefficient 2 3 in equation (2) accounts for the difference in length between the longitudinal and the diagonal elements, this is, 2 3 d L L = ⋅ . To arrive at expression of φ it is necessary to have equivalence between the isotropic elastic coefficient matrix and a computation of the equivalent directional properties of the bars as proposed by Nayfes Heftzy (1978). y x (b) z (a) L
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