Evolution of internal granular structure at the flow-arrest transition


The evolution of the internal granular structure in shear-arrested and shear-flowing states of granular materials is characterized using fabric tensors as descriptors of the internal contact and force networks. When a dilute system of frictional grains is subjected to a constant pressure and shear stress, the bulk stress ratio is well-predicted from the anisotropy of its contact and force networks during transient flow prior to steady shear flow or shear arrest. Although the onset of shear arrest is a stochastic process, the fabric tensors upon arrest are distributed around nearly equal contributions of force and contact network anisotropy to the bulk stress ratio. The distribution becomes seemingly narrower with increasing system size. The anisotropy of the contact network in shear-arrested states is reminiscent of the fabric anisotropy observed in shear-jammed packings.

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This work was performed, in part, at the Center for Integrated Nanotechnologies, an Office of Science User Facility operated for the U.S. Department of Energy (DOE) Office of Science. Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. DOE’s National Nuclear Security Administration under contract DE-NA-0003525. The views expressed in the article do not necessarily represent the views of the U.S. DOE or the United States Government.

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Correspondence to Leonardo E. Silbert.

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This article is part of the Topical Collection: In Memoriam of Robert P. Behringer.



In this “Appendix”, we describe the equations of motion that govern the dynamics of the periodic boundaries that are subjected to an external stress. A modularly-invariant adaptation [21] of the Parrinello–Rahman method [22] of molecular dynamics is utilized to simulate the evolution of a granular system—consisting of N particles with positions and momenta \(\{{\varvec{r}}_{i},{\varvec{p}}_{i}\}\) contained within a triclinic periodic box \({\varvec{H}}\) and its associated momentum \({\varvec{P}}_{g}\)—under external applied stress \({\varvec{\sigma }}_{a}\). The triclinic periodic box is described by an upper-triangular matrix \(H_{ij}=\mathbf{e} _{i} \cdot \mathbf{a} _{j}\), where the three lattice vectors \(\mathbf{a} _{j}\) define the periodicity of the triclinic box, and \(\mathbf{e} _{i}\) are the three orthonormal vectors that define the Cartesian coordinate system in the laboratory frame. The equations of motion are given by:

$$\begin{aligned} \dot{{\varvec{r}}}_{i}= & {} \frac{{\varvec{p}}_{i}}{m_{i}}+\frac{{\varvec{P}}_{g}}{W_{g}}{\varvec{r}}_{i}, \end{aligned}$$
$$\begin{aligned} \dot{{\varvec{p}}}_{i}&= {\varvec{F}}_{i}-\frac{{\varvec{P}}_{g}}{W_{g}}{\varvec{p}}_{i}-\frac{1}{3N}\frac{\mathrm {Tr}\left[ {\varvec{P}}_{g}\right] }{W_{g}}{\varvec{p}}_{i}, \end{aligned}$$
$$\begin{aligned} \dot{{\varvec{H}}}= & {} \frac{{\varvec{P}}_{g}}{W_{g}}{\varvec{H}}, \end{aligned}$$
$$\begin{aligned} \dot{{\varvec{P}}_{g}}= & {} V\left( {\varvec{\sigma }}-{\varvec{I}}p_{a}\right) -{\varvec{H}}{\varvec{\varSigma }}{\varvec{H}}^{T}+\left( \frac{1}{3N}\sum _{i=1}^{N}\frac{{\varvec{p}}_{i}^{2}}{m_{i}}\right) {\varvec{I}}, \end{aligned}$$

where \({\varvec{F}}_{i}\) is the net force on a particle i, V is the variable volume of the periodic box, \({\varvec{I}}\) is the identity tensor, and \({\varvec{\sigma }}\) is the internal Cauchy stress that includes contributions from interparticle contact forces and particle momentum (kinetic stress). A ‘fictitious’ mass \(W_{g}\) associated with the inertia of the periodic box is set as \(W_{g}=Nk_{n}d^{2}/\omega _{g}^{2}\), where \(\omega _{g}\) is the frequency of oscillation associated with periodic box fluctuations. The choice of \(\omega _{g}\) controls strain rate fluctuations during granular flow. We set \(\omega _{g}=0.1\omega _{p}\), where is the \(\omega _{p}=\sqrt{m/k_{n}}\) is the frequency associated with the harmonic contact spring between two particles. An additional linear damping is applied to the motion of the periodic box for numerical stability, and its magnitude does not affect the results described here.

The first two terms on the right side of last equation denote the imbalance between internal Cauchy stress and external applied stress that drive the dynamics of the periodic box. The tensor \({\varvec{\varSigma }}\) is defined as:

$$\begin{aligned} {\varvec{\varSigma }}={\varvec{H}}_{0}^{-1}\left( {\varvec{\sigma }}_{a}-{\varvec{I}}p_{a}\right) {\varvec{H}}_{0}^{T-1}, \end{aligned}$$

where \({\varvec{H}}_{0}\) is the triclinic periodic box matrix at \(t=0\), and \(J^{-1}{\varvec{H}}{\varvec{\varSigma }}{\varvec{H}}^{T}\) represents the ‘true’ stress measure of the external deviatoric stress, which is defined with respect to the reference state. Here the Jacobian is \(J=\mathrm {det}\left[ {\varvec{F}}\right] \), and the deformation gradient is defined as \({\varvec{F}}={\varvec{H}}{\varvec{H}}_{0}^{-1}\). As a result, this implementation of a constant external stress on the granular system conserves the second Piola-Kirchoff measure of the external stress (or equivalently, the thermodynamic tension [41]). In the present simulations, the reference state is updated to the current state at the end of every time step of integration of the equations of motion, in order to minimize the deviation of internal strain energy from work done by the external stress.

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Srivastava, I., Lechman, J.B., Grest, G.S. et al. Evolution of internal granular structure at the flow-arrest transition. Granular Matter 22, 41 (2020). https://doi.org/10.1007/s10035-020-1003-6

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  • Shear jamming
  • Fabric tensor
  • Force network
  • Granular friction
  • Critical state