The density-density correlation functions were evaluated with the fractal networks of prepared in the box . The averages and standard deviations were taken over ten samples. The red line indicates the slope corresponding to .
The three-point bending test for a linear aggregate consists of the three steps i–iii. When a smaller external force ( ) is applied, the elastic deformation is seen (a), and when it is slightly larger ( ), the bond breakups cause the plastic deformation (b).
The y dependence of packing fractions is shown for every second compression step ( , i = 1, 3, 5…). The solid line indicates the initial configuration.
The compressive yield stress is evaluated by the simulations of stepwise compression. Its averages and standard deviations taken over ten runs are shown. The red squares represent the result of the bond 1 simulations, and the blue circles of the bond 2 simulations. The dotted curve shows the fitting function of Eq. (11) , and (solid and dashed) straight lines show the power-law functions (12) . The arrows indicate ranges of the three regimes: (I) Elastic-dominant regime, (II) single-mode plastic regime, and (III) multimode plastic regime. The average exponents λ are written on the figure.
(a) The mean contact number per particles increases as the compression proceeds. The standard deviations are taken over the averages of ten simulations. (b) The successive increments of mean bond stored and mean dissipated energy rates are shown, where . The rates are obtained by normalizing with the compressive strain . The averages and standard deviations are taken over ten runs of the bond 1 simulation.
Bond rupture rates are compared for two types of bond: (a) Bond 1 and (b) bond 2. The bond ruptures are recorded by the causing stress. The rates of the separation due to normal forces are plotted by squares, the ones of the regeneration due to sliding forces or rolling moments by triangles and circles, respectively.
Snapshots of the equilibrium configurations under uniaxial compression. The initial configuration (a) is a prepared sample in Sec. II A , whose packing fraction is . The configurations (b) and (c) show the beginning of the single-mode plastic regime ( ) and the multimode plastic regime ( ), respectively and (d) shows . The red circles express the correlation lengths; their diameters are , which will be evaluated in the later section (Sec. III F ).
The density-density correlation functions C(r) of compressed networks are shown. The solid line shows the initial configuration and the dashed or dotted lines the every second compression step (i = 1, 3, 5,…). They are averaged values over ten runs of simulations.
(a) The fractal dimension profiles evaluated by Eq. (13) are plotted. The solid line shows the initial configuration , and the dashed or dotted solid lines the every second compression step (i = 1, 3, 5,…). The numerical values indicate the packing fractions. (b) The -dependence of the correlation length (see the definition in the text).
Parameters for the contact model. C is the parameter in Eq. (1) to avoid significant overlap by compression.
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