What triggers failure in frictional granular assemblies?

We examine numerically the yielding or failure of small granular packings subjected to an increasing deviatoric stress. As the load increases, the packing softens and the number of sliding contacts rises. When the packing fails, the kinetic energy starts to rise exponentially in time. It is always possible to identify a contact status change that triggers the collapse of the packing. Furthermore, by use of the stiffness matrix, we show that this change often causes a mechanical instability or a motion with neutral stability. In some cases, the status change provokes an oscillation and a second status change following shortly thereafter introduces an instability. Failure can also be considered from the perspective of energy flux: before failure, the energy injected by the load is stored as potential energy in the contacts. When this is no longer possible, failure occurs and the injected energy is converted to kinetic energy. However, the force disequilibrium then soon becomes the dominant energy source.

Figure 1

Biaxial boundary conditions.

Figure 2

Strain along the direction of the changing external force. Failure is characterized by a sudden change in strain. When the external force is changed slowly enough, (i.e., $d{f}_{\text{ext}}/dt\le {10}^{-2}$) the time of failure depends solely on the critical value of the external force and not on its rate of change. During failure, the assemblies behavior is not quasistatic anymore, as can be seen from the different strain values attained thereafter.

Figure 3

Relative error of the stiffness matrix approach [Eq. (18)]. The figure shows the maximum error and the mean error of 26 assemblies from the beginning of the simulation at $\tau =-1$, through the failure at $\tau =0$, and after failure. Note that failure lasts from $\tau =0$ to about $\tau =0.01$. One source of error are vibrations; they are especially important after failure has happened.

Figure 4

Fraction of sliding contacts averaged over 26 different seeds. The fraction increases until the time when the assembly fails. During the deformation process, the fraction drops abruptly and the intergranular contacts stop sliding when the assembly attains a new stable state. The remaining fraction is the contacts with the smooth walls.

Figure 5

Rotation mode with zero stiffness: whenever four grains form a square and touch each other, they rotate at the same speed but in opposite directions at their points of contact. (The rotations are indicated by the segment of a circle inside each grain; the segments start at the rightmost position and point in the direction of rotation. Their lengths indicate the speed.) Grains without indicated rotation are rattlers.

Figure 6

Degree of hyperstaticity averaged over 26 different seeds. (a) Values from the beginning of the simulation until after failure. (b) Values for a small time interval around the trigger event. When a contact status change triggers the collapse, this leads on average to a decrease in the degree of hyperstaticity. Note that one time unit in the second figure is on average 1/500 of a unit in the first figure, hence the period covered by the entire second figure corresponds to less than the thickness of the vertical line in the first figure.

Figure 7

Comparison of the macroscopic and microscopic stiffnesses. The macroscopic stiffness is calculated from the response of the confining walls according to Eq. (34). The microscopic stiffness is calculated using ${\dot{\mathbf{f}}}_{\text{ext}}$ and $\mathbf{v}$ according to Eq. (35).

Figure 8

A close view of a status change. At $t\approx 246.35$, a contact becomes sliding. Both the stiffness and the kinetic energy make a steplike change accompanied by damped oscillations.

Figure 9

Mechanical instability at the origin of failure: when the contact status change occurs, the stiffness, represented by $\mathbf{v}\mathbf{k}\mathbf{v}/\mathbf{v}\mathbf{v}$, decreases rapidly and finally attains negative values. The purely structure-dependent quantity ${\mathbf{v}}_{\mathbf{t}\mathbf{h}}\mathbf{k}{\mathbf{v}}_{\mathbf{t}\mathbf{h}}$ changes instantaneously and hence shows the effect of the contact status change at the vertical line, whereas the simulation velocities $\mathbf{v}$ change on a time scale related to the particle inertia.

Figure 10

Contributions to ${\ddot{E}}_{\text{kin}}$, and $E_{\text{kin}}$, for the simulation shown in Fig. 9, where an instability triggers failure. The contributions to ${\ddot{E}}_{\text{kin}}$ are plotted on a linear scale, indicated on the left, while $E_{\text{kin}}$ is plotted on a logarithmic scale shown on the right. $\dot{\mathbf{v}}\mathbf{m}\dot{\mathbf{v}}$ is the contribution from the force imbalance.

Figure 11

Stiffness of the granular assembly $\left(\mathbf{v}\mathbf{k}\mathbf{v}/\mathbf{v}\mathbf{v}\right)$ at failure caused by the appearance of a null mode. The thick vertical line at $t=t_{\text{trigger}}$ shows the time when the null mode appears due to a contact status change.

Figure 12

The null mode that triggers failure in Fig. 11. Velocities are indicated by arrows and angular velocities are represented by a circle segment inside the corresponding grain. Grains for which no velocity is shown are rattlers. When the null mode occurs, the contact marked by the black circle starts to slide. The other sliding contacts are indicated by gray circles.

Figure 13

Fraction of the particle velocity $\mathbf{v}$ associated with the null mode ${\mathbf{v}}_0$ at failure for the simulation shown in Fig. 11. Shortly after the contact status changes, the velocity vector points in the direction of the null-mode movement, i.e., $\mathbf{v}{\mathbf{v}}_0/\left|\mathbf{v}\right|\left|{\mathbf{v}}_0\right|\approx 1$.

Figure 14

Contributions to ${\ddot{E}}_{\text{kin}}$ and $E_{\text{kin}}$, for the simulation shown in Fig. 11, where a null mode triggers failure. The contributions to ${\ddot{E}}_{\text{kin}}$ are plotted on a linear scale, indicated on the left, while $E_{\text{kin}}$ is plotted on a logarithmic scale shown on the right.

Figure 15

An example of an ambushed transition: the kinetic energy starts to rise at $t=t_{\text{trigger}}$, even though the packing is stable.

Figure 16

The most difficult example of failure to classify. The triggering event is shown by a heavy vertical line. Other contact status changes are shown by the short vertical arrows at the bottom of the graph. At certain short periods of time, a null mode exists. When this occurs, $\left({\mathbf{v}}_{\text{th}}\mathbf{k}{\mathbf{v}}_{\text{th}}\right)/\left({\mathbf{v}}_{\text{th}}{\mathbf{v}}_{\text{th}}\right)$ jumps to a value near 30.

Figure 17

Compound status change. The compound status change involves the two contacts in the upper left corner marked with the black circles when they are sliding. The other sliding contacts are marked with gray circles. The pictures are taken at $t-t_{\text{trigger}}=-0.04,-0.02,0$ (from left to right).

Figure 18

Relation of trigger mechanisms. All systems begin in the box marked “stability.” Then they can fail through three different ways indicated by the three arrows emerging from this box. The remaining arrows indicate that transitions are sometimes possible between different mechanisms. Though we did not observe the transition from the box “instability” to the “null-mode” oval, we do not want to claim it is not possible. The dotted horizontal line indicates $\mathbf{v}\mathbf{k}\mathbf{v}=0$.