Stress fluctuations during monotonic loading of dense three-dimensional granular materials

Abstract

The paper examines the sudden and irregular fluctuations of stress that are commonly observed in DEM and laboratory tests of slow monotonic loading of granular materials. The stresss fluctuations occur as stress-drops that are spaced in an irregular, random manner as the material is loaded, and occur at an average rate of about 0.05 drops per particle per one percent of strain. Each fluctuation is accompanied by a drop in the number of contacts and in the number of sliding contacts, a brief increase in the particles’ kinetic energy, a reduction in the elastic energy of the contacts, and a reduction in bulk volume. Stress-drops are shown to originate within small regions of the larger assembly and are likely the result of a multi-slip mechanism. An advanced discrete element method (DEM) is used in the study, with non-convex non-spherical particles and an exact implementation of the Hertz-like Cattaneo–Mindlin contact model. The simulations reveal differences with the avalanche phenomenon of amorphous solids, in particular the different scalings of stress-drop magnitude with assembly size and drop frequency.

DEM modeling details

Fig. 9

Sphere-cluster particle

Discrete element simulations were conducted with a cubical assembly of 10,648 particles contained within periodic boundaries. The simulations were intended to produce a modest fidelity to the bulk behavior of sands at low confining stress. Because sphere assemblies produce unrealistic rolling between particles and have a low bulk strength, we used a bumpy, non-convex cluster shape for the particles: a large central sphere with six smaller embedded outer spheres in an octahedral arrangement (Fig. 9). The use of non-spherical particles circumvents the need of artificial measures to restrain the particle rotation (for example, the use of rotational contact springs or the direct restraint of particle rotations, as in [48, 49]). Through trial and error, we chose the radii of the central and outer spheres so that the bulk behavior approximated that of Nevada Sand, a standard poorly graded sand (SP) use in laboratory and centrifuge testing programs [22, 23]. The particle size range was 0.074–0.28 mm, with $D_{50}=0.165\,\hbox {mm}$.

To create assemblies with a range of densities, we began with the particles sparsely and randomly arranged within a cubic periodic cell. With an initial low inter-particle friction coefficient ($\mu = 0.20$), the assembly was isotropically reduced in dimension until it “seized” when a sufficiently complete contact network had formed. A series of 25 progressively denser assemblies were created by repeatedly assigning random velocities to the particles of the previous assembly and then further reducing the cell’s dimensions until it, too, had seized. The 25 assemblies had void ratios ranging from 0.781 to 0.586 (solid fractions of 0.561 to 0.664). The single assembly used in the paper had void ratio 0.690 (solid fraction 0.592) and approximates the behavior of Nevada Sand at a relative density of 40%. After compaction, the assembly was allowed to quiesce with friction coefficient $\mu =0.40$, which was then raised to $\mu =0.55$ for the subsequent loading simulations.

The particles are durable (non-breaking) and interact only at their contacts. A Hertz, Cattaneo–Mindlin contact model was used in the simulations. The model is a full implementation of a Hertz–Mindlin contact between elastic–frictional bodies. We used the Jäger algorithm, which can model arbitrary sequences of normal and tangential contact movements [21]. With the Cattaneo–Mindlin contacts, the loading simulations were conducted with an inter-particle friction coefficient $\mu =0.55$, particle shear modulus $G=29\,\hbox {GPa}$, and Poisson ratio $\nu =0.15$. The contact model is rate-independent, such that the contact force depends upon the movement history and not on the particles’ velocities. A special servo-algorithm was used for precisely maintaining constant mean stress during the constant-p simulations [50]. Because the shear modulus was much greater than the mean stress of 100 kPa, the contacts were relatively stiff, and the average overlap at a contact was about 0.1% of the particle diameter.

In Fig. 2d, e, we show the kinetic and elastic energies per unit of bulk volume. The energies were derived from an energy-audit of the assembly: the increment of stress-work, $\int \sigma _{ij}\,d\varepsilon _{ij}$, during an increment of strain must equal the sum of the volume-averages of the change in elastic energy within the contacts, the frictional dissipation at the contacts, and the change in the particles’ kinetic energy. By directly computing the stress-work, the elastic energy, and the frictional dissipation, as described below, we could find the change in the kinetic energy. The kinetic energy was also independently computed from the particles’ velocities and masses, and we found that the two values almost exactly matched, thus verifying the energy balance.

The stress-work was computed in the usual manner from bulk measures of stress and strain (the former was computed from the contact forces and branch vectors by using the usual Navier–Love–Rothenburg equation; whereas, the strain was computed from the boundary displacements). Contact elastic energy was computed as the sum of normal and tangential parts. In conventional simulations, contacts are assumed to occur between spherical surfaces, but this approach leads to a bulk small-strain stiffness that increases with the means stress p as $p^{1/3}$. In our simulations, we allowed for non-spherical contours (asperities) at the contacts, as this approach yielded a small-strain modulus that was proportional to the mean stress p raised to the power of 0.5, a common observation in geotechnical practice (see [22] for details). We used a power-law type ($z=A_{\alpha }r^{\alpha }$) of contact profile as described by Jäger [51] with a profile parameter $\alpha =1.3$. The normal elastic energy $E_{\text {n}}$ in a single contact is given as

$$\begin{aligned} E_{\text {n}} = \frac{4 \alpha ^{2} G a \zeta ^{2}}{(1+2\alpha )(1+\alpha )(1-\nu )} \end{aligned}$$

(1)

where G and $\nu$ are the shear modulus and Poisson ratio of the particles, $\zeta$ is the contact indentation (one half of the overlap), a is the radius of the contact zone,

$$\begin{aligned} a = \left[ \frac{\zeta }{A_{\alpha }\sqrt{\pi }} \frac{\varGamma (\frac{\alpha +1}{2})}{\varGamma (\frac{\alpha +2}{2})} \right] ^{1/\alpha } \end{aligned}$$

(2)

and $\varGamma (\cdot )$ is the gamma function (see [51]). Note that the normal part of elastic energy can be directly computed from the accumulated indentation of the contact. The tangential response, however, is history-dependent and must be computed in an incremental manner. For each time increment $\varDelta t$ in our DEM simulations with Cattaneo–Mindlin contacts, we computed the increment of elastic (reversible) tangential displacement $\varDelta {\varvec{\xi }}^{\text {(r)}}$ and subtracted this reversible increment from the full tangential increment $\varDelta {\varvec{\xi }}$ to find the irreversible, frictional tangential increment,

$$\begin{aligned} \varDelta {\varvec{\xi }}^{\text {(i)}} = \varDelta {\varvec{\xi }} - \varDelta {\varvec{\xi }}^{\text {(r)}} \end{aligned}$$

(3)

and then computed the inner product of the irreversible increment and the tangential force to find the increment of dissipation:

$$\begin{aligned} \text {Contact dissipation} = {\mathbf {f}}^{\text {t}} \cdot \varDelta {\varvec{\xi }}^{\text {(i)}} \end{aligned}$$

(4)

To find the incremental elastic (reversible) tangential movement $\varDelta {\varvec{\xi }}^{\text {(r)}}$ in Eq. (3), which represents the response to the increment of tangential force $\varDelta {\mathbf {f}}^{t}$ in the absence of micro-slip, we used Walton’s solution for the tangential contact response of rough (zero-slip) surfaces [52]:

$$\begin{aligned} \varDelta {\varvec{\xi }}^{\text {(r)}} = \frac{2-\nu }{8G}\frac{1}{a} \varDelta {\mathbf {f}}^{\text {t}} \end{aligned}$$

(5)

In this manner, we tracked the incremental frictional dissipation of each Cattaneo–Mindlin contact so that the total contact frictional dissipation, the total contact elastic energy, the stress-work, and the kinetic energy of the assembly could be computed in a consistent manner.

In conducting the simulations, we intended to model quasi-static, rate-independent behavior. To this end, we used a slow strain rate (strain increments of $2\times 10^{-9}$ and minimal viscous damping (2% of critical damping). The inertial number $I={\dot{\varepsilon }}\sqrt{m/(pd)}$, a relative measure of loading and inertial rates, was about $1\times 10^{-11}$, signifying nearly quasi-static loading [53]. During loading, the average imbalance of force on a particle was less than 0.003% of the average contact force (parameter $\chi$ in [54, 55]). The average kinetic energy of the particles was less than $3\times 10^{-7}$ times the elastic energy in the contacts. With the very slow strain rates of the simulations, doubling the strain increment from $1\times 10^{-8}$ to $2\times 10^{-8}$ had minimal effect on the monotonic stress-strain response. Boundary movements were regulated so that any six of the stress or strain components (or any six linear combination of these components) could be controlled at desired rates. During the constant-p (constant mean stress) triaxial loading, the mean stress would remain within 0.01 Pa of the 100 kPa target. All of these measurements indicate that the behavior in the simulations was nearly quasi-static and independent of loading rate.

The DEM simulations were done with the authors’ OVAL code (see [22]) and were run with a 4th generation Intel i7 processor on a single thread. The monotonic loading in Fig. 1, in which the strain was advanced to 16%, took 60 days of compute time.