Fragmentation and shear band formation by slow compression of brittle porous media

Localized fragmentation is an important phenomenon associated with the formation of shear bands and faults in granular media. It can be studied by empirical observation, by laboratory experiment, or by numerical simulation. Here we investigate the spatial structure and statistics of fragmentation using discrete element simulations of the strain-controlled uniaxial compression of cylindrical samples of different finite size. As the system approaches failure, damage localizes in a narrow shear band or synthetic fault “gouge” containing a large number of poorly sorted noncohesive fragments on a broad bandwidth of scales, with properties similar to those of natural and experimental faults. We determine the position and orientation of the central fault plane, the width of the shear band, and the spatial and mass distribution of fragments. The relative width of the shear band decreases as a power law of the system size, and the probability distribution of the angle of the central fault plane converges to around 30 degrees, representing an internal coefficient of friction of 0.7 or so. The mass of fragments is power law distributed, with an exponent that does not depend on scale, and is near that inferred for experimental and natural fault gouges. The fragments are in general angular, with a clear self-affine geometry. The consistency of this model with experimental and field results confirms the critical roles of preexisting heterogeneity, elastic interactions, and finite system size to grain size ratio on the development of shear bands and faults in porous media.

Figure 1

(a) A cylindrical sample of ${10}^5$ particles. The clamped particle layers at the bottom and top of the sample are highlighted by yellow color. (b) Final reassembled configuration where particles are colored according to the size of fragments they belong to. The stripe of contiguous small sized fragments with a large number of different colors indicates the position of the shear band.

Figure 2

Plane placed in the middle of the shear band. The black dots represent the center of mass position of fragments including those that are noncontiguous with the shear band. The figure also illustrates the coordinate systems used in the calculations. The origin of the $x,y,z$ coordinate system is the center of the bottom circle of the cylinder and the $z$ axis is directed along the cylinder axis. The $(x^{\prime },y^{\prime },z^{\prime })$ system is centered at the support point of the plane and the $z^{\prime }$ axis is aligned with the normal vector ${\vec{n}}_d$ of the best fitting central fault plane.

Figure 3

Probability distribution $p\left(\mathrm{\Theta }\right)$ of the angle $\mathrm{\Theta }$ of the shear band with respect to the load direction (vertical direction in Fig. 1) for all system sizes $N$ considered. Increasing the system size the distributions get more and more peaked and their maximum gradually shifts to lower values.

Figure 4

Finite size scaling of the probability distribution $p\left(\mathrm{\Theta }\right)$. Rescaling the two axes of Fig. 3 according to Eq. (2) a good quality data collapse is obtained.

Figure 5

Probability distribution of the angle $\varphi$ of the shear band measured from the $x$ axis. The horizontal straight line represents the value $p\left(\varphi \right)=1/2\pi$ of the uniform distribution over the $-\pi \le \varphi \le +\pi$ interval. The legend is the same as in Fig. 4.

Figure 6

Probability distribution of the coordinate $z^{\prime }$ of the center of mass of fragments for all system sizes $N$ considered.

Figure 7

Scaling plot of the probability distributions $p(z^{\prime },N)$ obtained at different system sizes $N$. The value $\gamma =0.25$ of the scaling exponent provides the best data collapse. Deviations from the master curve can only be observed for the smallest system.

Figure 8

Fragments comprising more than one particle in a sample of $N={10}^5$ particles seen from the directions of the cylinder axes. The four largest fragments are not shown so that all the fragments of the figure fall inside the shear band. The color of fragments is randomly selected.

Figure 9

Mass distribution of fragments for all sample sizes $N$. The power law regime is well separated from the fine powder and large residues. The bold line represents a fit with Eq. (5). The fragment mass is made dimensionless by dividing it with the average mass of single particles $m_s$.

Figure 10

Average mass of fragments as a function of the radius of gyration $R_g$ normalized by the average radius of single particles $R_s$. Results of the different system sizes fall on the top of each other. The two straight lines represent power laws of exponents 3 and 2.7.