Evolution of internal granular structure at the flow-arrest transition

Abstract

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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References

  1. 1.

    Schofield, A., Wroth, P.: Critical State Soil Mechanics, vol. 310. McGraw-Hill, London (1968)

    Google Scholar 

  2. 2.

    Jop, P., Forterre, Y., Pouliquen, O.: A constitutive law for dense granular flows. Nature 441, 727 (2006)

    ADS  Article  Google Scholar 

  3. 3.

    Salerno, K.M., Bolintineanu, D.S., Grest, G.S., Lechman, J.B., Plimpton, S.J., Srivastava, I., Silbert, L.E.: Effect of shape and friction on the packing and flow of granular materials. Phys. Rev. E 98, 050901 (2018)

    ADS  Article  Google Scholar 

  4. 4.

    Singh, A., Magnanimo, V., Luding, S.: Effect of friction and cohesion on anisotropy in quasi-static granular materials under shear. In: AIP Conference Proceedings, vol. 1542 (2013)

  5. 5.

    Srivastava, I., Silbert, L.E., Grest, G.S., Lechman, J.B.: Flow-arrest transitions in frictional granular matter. Phys. Rev. Lett. 122, 048003 (2019)

    ADS  Article  Google Scholar 

  6. 6.

    Liu, A.J., Nagel, S.R.: The jamming transition and the marginally jammed solid. Annu. Rev. Condens. Matter Phys. 1, 347 (2010)

    ADS  Article  Google Scholar 

  7. 7.

    O’Hern, C.S., Silbert, L.E., Liu, A.J., Nagel, S.R.: Jamming at zero temperature and zero applied stress: the epitome of disorder. Phys. Rev. E 68, 011306 (2003)

    ADS  Article  Google Scholar 

  8. 8.

    Bi, D., Zhang, J., Chakraborty, B., Behringer, R.P.: Jamming by shear. Nature 480, 355 (2011)

    ADS  Article  Google Scholar 

  9. 9.

    Vinutha, H., Sastry, S.: Disentangling the role of structure and friction in shear jamming. Nat. Phys. 12, 578 (2016)

    Article  Google Scholar 

  10. 10.

    Chen, S., Bertrand, T., Jin, W., Shattuck, M.D., O’Hern, C.S.: Stress anisotropy in shear-jammed packings of frictionless disks. Phys. Rev. E 98, 042906 (2018)

    ADS  Article  Google Scholar 

  11. 11.

    Clark, A.H., Thompson, J.D., Shattuck, M.D., Ouellette, N.T., O’Hern, C.S.: Critical scaling near the yielding transition in granular media. Phys. Rev. E 97, 062901 (2018)

    ADS  Article  Google Scholar 

  12. 12.

    Majmudar, T.S., Behringer, R.P.: Contact force measurements and stress-induced anisotropy in granular materials. Nature 435, 1079 (2005)

    ADS  Article  Google Scholar 

  13. 13.

    Radjaï, F., Wolf, D.E., Jean, M., Moreau, J.J.: Bimodal character of stress transmission in granular packings. Phys. Rev. Lett. 80, 61 (1998)

    ADS  Article  Google Scholar 

  14. 14.

    Radjai, F., Delenne, J.Y., Azéma, E., Roux, S.: Fabric evolution and accessible geometrical states in granular materials. Granul. Matter 14, 259 (2012)

    Article  Google Scholar 

  15. 15.

    Rothenburg, L., Bathurst, R.J.: Analytical study of induced anisotropy in idealized granular materials. Géotechnique 39, 601 (1989)

    Article  Google Scholar 

  16. 16.

    Bathurst, R.J., Rothenburg, L.: Observations on stress-force-fabric relationships in idealized granular materials. Mech. Mat. 9, 65 (1990)

    Article  Google Scholar 

  17. 17.

    Sun, J., Sundaresan, S.: A constitutive model with microstructure evolution for flow of rate-independent granular materials. J. Fluid Mech. 682, 590 (2011)

    ADS  MathSciNet  Article  Google Scholar 

  18. 18.

    Guo, N., Zhao, J.: The signature of shear-induced anisotropy in granular media. Comput. Geotech. 47, 1 (2013)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Azéma, E., Radjaï, F.: Internal structure of inertial granular flows. Phys. Rev. Lett. 112, 93 (2014)

    Article  Google Scholar 

  20. 20.

    Silbert, L.E., Ertas, D., Grest, G.S., Halsey, T.C., Levine, D., Plimpton, S.J.: Granular flow down an inclined plane: Bagnold scaling and rheology. Phys. Rev. E 64(5), 051302 (2001)

    ADS  Article  Google Scholar 

  21. 21.

    Shinoda, W., Shiga, M., Mikami, M.: Rapid estimation of elastic constants by molecular dynamics simulation under constant stress. Phys. Rev. B 69, 134103 (2004)

    ADS  Article  Google Scholar 

  22. 22.

    Parrinello, M., Rahman, A.: Polymorphic transitions in single crystals: a new molecular dynamics method. J. Appl. Phys. 52, 7182 (1981)

    ADS  Article  Google Scholar 

  23. 23.

    Plimpton, S.: Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117, 1 (1995)

    ADS  Article  Google Scholar 

  24. 24.

    Silbert, L.E., Grest, G.S., Plimpton, S.J., Levine, D.: Boundary effects and self-organization in dense granular flows. Phys. Fluids 14, 2637 (2002)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Schuhmacher, P., Radjai, F., Roux, S.: Wall roughness and nonlinear velocity profiles in granular shear flows. In: EPJ Web of Conferences, vol. 140, p. 03090 (2017)

    Article  Google Scholar 

  26. 26.

    Thompson, A.P., Plimpton, S.J., Mattson, W.: General formulation of pressure and stress tensor for arbitrary many-body interaction potentials under periodic boundary conditions. J Chem. Phys. 131, 154107 (2009)

    ADS  Article  Google Scholar 

  27. 27.

    Silbert, L.E.: Jamming of frictional spheres and random loose packing. Soft Matter 6, 2918 (2010)

    ADS  Article  Google Scholar 

  28. 28.

    Otsuki, M., Hayakawa, H.: Critical scaling near jamming transition for frictional granular particles. Phys. Rev. E 83, 051301 (2011)

    ADS  Article  Google Scholar 

  29. 29.

    Goodrich, C.P., Liu, A.J., Sethna, J.P.: Scaling ansatz for the jamming transition. Proc. Natl. Acad. Sci. U.S.A. 113, 9745 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  30. 30.

    Kanatani, K.I.: Distribution of directional data and fabric tensors. Int. J. Eng. Sci. 22, 149 (1984)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Oda, M.: Fabric tensor for discontinuous geological materials. Soils Found. 22, 96 (1982)

    Article  Google Scholar 

  32. 32.

    Kanatani, K.I.: A theory of contact force distribution in granular materials. Powder Technol. 28, 167 (1981)

    Article  Google Scholar 

  33. 33.

    Sarkar, S., Bi, D., Zhang, J., Behringer, R.P., Chakraborty, B.: Origin of rigidity in dry granular solids. Phys. Rev. Lett. 111, 068301 (2013)

    ADS  Article  Google Scholar 

  34. 34.

    Seto, R., Singh, A., Chakraborty, B., Denn, M.M., Morris, J.F.: Shear jamming and fragility in dense suspensions. Granul. Matter 21, 82 (2019)

    Article  Google Scholar 

  35. 35.

    Radjaï, F., Roux, J.N., Daouadji, A.: Modeling granular materials: century-long research across scales. J. Eng. Mech. 143, 04017002 (2017)

    Article  Google Scholar 

  36. 36.

    Zhao, J., Guo, N.: Unique critical state characteristics in granular media considering fabric anisotropy. Géotechnique 63, 695 (2013)

    Article  Google Scholar 

  37. 37.

    Li, X.S., Dafalias, Y.F.: Anisotropic critical state theory: role of fabric. J. Eng. Mech. 138, 263 (2012)

    Article  Google Scholar 

  38. 38.

    Li, X.S., Dafalias, Y.F.: Dissipation consistent fabric tensor definition from DEM to continuum for granular media. J. Mech. Phys. Solids 78, 141 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  39. 39.

    Dafalias, Y.F.: Must critical state theory be revisited to include fabric effects? Acta Geotech. 11, 479 (2016)

    Article  Google Scholar 

  40. 40.

    Wang, R., Fu, P., Zhang, J.M., Dafalias, Y.F.: Evolution of various fabric tensors for granular media toward the critical state. J. Eng. Mech. 143, 04017117 (2017)

    Article  Google Scholar 

  41. 41.

    Souza, I., Martins, J.L.: Metric tensor as the dynamical variable for variable-cell-shape molecular dynamics. Phys. Rev. B 55, 8733 (1997)

    ADS  Article  Google Scholar 

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Acknowledgements

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.

Appendix

Appendix

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:

$$\begin{aligned} \dot{{\varvec{r}}}_{i}= & {} \frac{{\varvec{p}}_{i}}{m_{i}}+\frac{{\varvec{P}}_{g}}{W_{g}}{\varvec{r}}_{i}, \end{aligned}$$
(8)
$$\begin{aligned} \dot{{\varvec{p}}}_{i}&= {\varvec{F}}_{i}-\frac{{\varvec{P}}_{g}}{W_{g}}{\varvec{p}}_{i}-\frac{1}{3N}\frac{\mathrm {Tr}\left[ {\varvec{P}}_{g}\right] }{W_{g}}{\varvec{p}}_{i}, \end{aligned}$$
(9)
$$\begin{aligned} \dot{{\varvec{H}}}= & {} \frac{{\varvec{P}}_{g}}{W_{g}}{\varvec{H}}, \end{aligned}$$
(10)
$$\begin{aligned} \dot{{\varvec{P}}_{g}}= & {} V\left( {\varvec{\sigma }}-{\varvec{I}}p_{a}\right) -{\varvec{H}}{\varvec{\varSigma }}{\varvec{H}}^{T}+\left( \frac{1}{3N}\sum _{i=1}^{N}\frac{{\varvec{p}}_{i}^{2}}{m_{i}}\right) {\varvec{I}}, \end{aligned}$$
(11)

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:

$$\begin{aligned} {\varvec{\varSigma }}={\varvec{H}}_{0}^{-1}\left( {\varvec{\sigma }}_{a}-{\varvec{I}}p_{a}\right) {\varvec{H}}_{0}^{T-1}, \end{aligned}$$
(12)

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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Keywords

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