A model for decoding the life cycle of granular avalanches in a rotating drum


Granular materials can behave as harmless sand dunes or as devastating landslides. A granular avalanche marks the transition between these distinct solid-like and fluid-like states. The solid-like state is typically described using plasticity models from critical state theory. In the fluid regime, granular flow is commonly captured using a visco-plastic model. However, due to our limited understanding of the mechanism governing the solid–fluid-like transition, characterizing the material behavior throughout the life cycle of an avalanche remains an open challenge. Here, we employ laboratory experiments of transient avalanches spontaneously generated by a rotating drum. We report measurements of dilatancy and grain kinematics before, during, and after each avalanche. Those measurements are directly incorporated into a rate-dependent plasticity model that quantitatively predicts the granular flow measured in experiments. Furthermore, we find that dilatancy in the solid-like state controls the triggering of granular avalanches and therefore plays a key role in the solid–fluid-like transition. With the proposed approach, we demonstrate that the life cycle of a laboratory avalanche, from triggering to run out, can be fully explained. Our results represent an important step toward a unified understanding of the physical phenomena associated with transitional behavior in granular media.

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  1. 1.

    Andrade JE, Chen Q, Le PH, Avila CF, Evans TM (2012) On the rheology of dilative granular media: bridging solid and fluid-like behavior. J Mech Phys Solids 60:1122–1136

    MathSciNet  Article  Google Scholar 

  2. 2.

    Courrech du Pont S, Fischer R, Gondret P, Perrin B, Rabaud M (2005) Instantaneous velocity profiles during granular avalanches. Phys Rev Lett 94:048003

    Article  Google Scholar 

  3. 3.

    Daerr A, Douady S (1999) Two types of avalanche behaviour in granular media. Nature 399:241–243

    Article  Google Scholar 

  4. 4.

    Evesque P (1991) Analysis of the statistics of sandpile avalanches using soil-mechanics results and concepts. Phys Rev A 43:2720–2740

    Article  Google Scholar 

  5. 5.

    Fischer R, Gondret P, Perrin B, Rabaud M (2008) Dynamics of dry granular avalanches. Phys Rev E 78:021302

    Article  Google Scholar 

  6. 6.

    Forterre Y, Pouliquen O (2008) Flows of dense granular media. Annu Rev Fluid Mech 40:1–24

    MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    George DL, Iverson RM (2014) A depth-averaged debris-flow model that includes the effects of evolving dilatancy II Numerical predictions and experimental tests. Proc R Soc A 470:20130820

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Gray JMNT (2001) Granular flow in partially filled slowly rotating drums. J Fluid Mech 441:1–29

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Iverson RM, George DL (2014) A depth-averaged debris-flow model that includes the effects of evolving dilatancy. I physical basis. Proc R Soc A 470:20130819

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Jaeger HM, Nagel SR (1992) Physics of granular state. Science 255:1523–1531

    Article  Google Scholar 

  11. 11.

    Jaeger HM, Nagel SR, Behringer RP (1996) Granular solids, liquid and gases. Rev Mod Phys 68:1259

    Article  Google Scholar 

  12. 12.

    Jop P, Forterre Y, Pouliquen O (2005) Crucial role of side walls for granular surface flows: consequences for the rheology. J Fluid Mech 541:167–192

    Article  MATH  Google Scholar 

  13. 13.

    Jop P, Forterre Y, Pouliquen O (2006) A constitutive law for dense granular flows. Nature 441:727–730

    Article  Google Scholar 

  14. 14.

    Kabla AJ, Senden TJ (2009) Dilatancy in slow granular flow. Phys Rev Lett 102:28301

    Article  Google Scholar 

  15. 15.

    Lowe DR (1976) Grain flow and grain flow deposits. J Sediment Petrol 46:188–199

    Google Scholar 

  16. 16.

    Midi GDR (2004) On dense granular flow. Eur Phys J E 14:341–365

    Article  Google Scholar 

  17. 17.

    Orpe A, Khakhar DV (2001) Scaling relations for granular flow in quasi-two-dimensional rotating cylinders. Phys Rev E 64:031302

    Article  Google Scholar 

  18. 18.

    Pailha M, Pouliquen O (2009) A two-phase flow description of the initiation of underwater granular avalanches. J Fluid Mech 633:115–136

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Pailha M, Nicolas M, Pouliquen O (2008) Initiation of underwater granular avalanches: influence of the initial volume fraction. Phys Fluids 20:11701

    Article  MATH  Google Scholar 

  20. 20.

    Pan B, Qian K, Xie H, Asundi A (2009) Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review. Meas Sci Technol 20:062001

    Article  Google Scholar 

  21. 21.

    Peng C, Guo X, Wu W, Wang Y (2016) Unified modelling of granular media with smoothed particle hydrodynamics. Acta Geotech 11:1231

    Article  Google Scholar 

  22. 22.

    Pouliquen O, Renaut N (1996) Onset of granular flows on an inclined rough surface: dilatancy effects. J Phys II Fr 6:923–935

    Google Scholar 

  23. 23.

    Prime N, Dufour F, Darve F (2014) Solid-fluid transition modelling in geomaterials and application to a mudflow interacting with an obstacle. Int J Numer Anal Methods Geomech 38:1341–1361

    Article  Google Scholar 

  24. 24.

    Rajchenbach J (1990) Flow in powders: from discrete avalanches to continuous regime. Phys Rev Lett 65:2221–2224

    Article  Google Scholar 

  25. 25.

    Reinsch CH (1967) Smoothing by spline functions. Numer Math 10:177–183

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Reynolds O (1885) On the dilatancy of media composed of rigid particle in contact. Philos Mag Ser 5(20):469–481

    Article  Google Scholar 

  27. 27.

    Rowe PW (1962) The stress-dilatancy relation for static equilibrium of an assembly of particles in contact. Proc R Soc Lond A 269:500–527

    Article  Google Scholar 

  28. 28.

    Schofield AN, Wroth P (1968) Critical state soil mechanics. McGraw-Hill, New York City

    Google Scholar 

  29. 29.

    Sutton MA, Orteu JJ, Schreier H (2009) Image correlation for shape, motion and deformation measurements: basic concepts theory and applications. Springer, Berlin

    Google Scholar 

  30. 30.

    Wood DM (1991) Soil behaviour and critical state soil mechanics. Cambridge University Press, Cambridge

    Google Scholar 

Download references


The author would like to thank Ryan Hurley for his fruitful comments. This work has been partially funded by Keck Institute for Space Studies (KISS) and the California Institute of Technology; this support is gratefully acknowledged.

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Correspondence to José E. Andrade.

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Appendix 1: Determination of the governing equation of the flow

In this appendix, we show how the governing equation of the granular flow (5) is derived from the linear momentum balance equation:

$$\frac{a}{g} = \cos \theta \left( {\tan \theta - \mu } \right).$$

Let us consider a granular flow along an inclined infinite slope. The position in the flow direction and in the flow depth direction is labeled x and y, respectively. The free surface of the flow makes an angle θ with the horizontal.

The local statement of linear momentum balance expressed in full gives:

$$\frac{{\partial \sigma_{xx} }}{\partial x} + \frac{{\partial \sigma_{xy} }}{\partial y} + \rho g\sin \theta = \rho a_{x} ,$$
$$\frac{{\partial \sigma_{yx} }}{\partial x} + \frac{{\partial \sigma_{yy} }}{\partial y} + \rho g \cos \theta = \rho a_{y} .$$

Exploiting the infinite character of the problem, we assume that the stress components do not vary in the x-direction, such that: \(\frac{{\partial \sigma_{xx} }}{\partial x} = \frac{{\partial \sigma_{yx} }}{\partial x} = 0\). Hence, by writing \(\sigma_{yy} = \sigma\) and \(\sigma_{xy} = \tau ,\) we obtain:

$$\frac{\partial \tau }{\partial y} + \rho g\sin \theta = \rho a_{x} ,$$
$$\frac{\partial \sigma }{\partial y} + \rho g\cos \theta = \rho a_{y} .$$

The definition of dilatancy β is given by:

$$\beta = \frac{{{\text{d}}\varepsilon_{yy} }}{{{\text{d}}\varepsilon_{xy} }} = \frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{v_{y} }}{{v_{x} }} \Rightarrow v_{y} = \beta v_{x} .$$

We know that in the fluid-like state the granular material reaches critical state so that β = 0. Therefore, we must have v y  = 0 and a y  = 0. Equation (7b) can be rewritten as:

$$\frac{\partial \sigma }{\partial y} = - \rho g\cos \theta .$$

Recalling Coulomb friction law and using the previous expression, we have:

$$\frac{\partial \tau }{\partial y} = \mu \frac{\partial \sigma }{\partial y} = - \mu \rho g\cos \theta .$$

Combining Eq. (7a) with Eq. (10) further gives:

$$- \mu g\cos \theta + g\sin \theta = a_{x} .$$

By denoting the surface acceleration in the flow direction a x by a, we finally obtain the governing equation of the granular flow from (11):

$$\frac{a}{g} = \cos \theta \left( {\tan \theta - \mu } \right).$$

Appendix 2: Determination of the difference between the angle of avalanche θ A and angle of repose θ R

Knowing that in the solid-like state the surface acceleration a = 0 and the mobilized friction \(\mu = \beta + \mu_{1}\), Eq. (1) can be expressed as:

$$\sin \theta = \beta \cos \theta + \mu_{l} \cos \theta .$$

Multiplying each side of Eq. (13) by cos θ R, we have:

$$\sin \theta \cos \theta_{R} = \beta \cos \theta \cos \theta_{R} + \mu_{l} \cos \theta \cos \theta_{R} .$$

Using the formula sin (x − y) = sin x cos y − cos x sin y, we obtain:

$$\sin \left( {\theta - \theta_{R} } \right) + \cos \theta \sin \theta_{R} = \beta \cos \theta \cos \theta_{R} + \mu_{l} \cos \theta \cos \theta_{R} .$$

Moreover, we defined the parameter μ l to be \(\mu_{\text{l}} = \tan \theta_{R} ,\) such that:

$$\sin \left( {\theta - \theta_{R} } \right) + \cos \theta \sin \theta_{R} = \beta \cos \theta \cos \theta_{R} + \cos \theta \sin \theta_{R}$$
$$\Leftrightarrow \sin \left( {\theta - \theta_{R} } \right) = \beta \cos \theta \cos \theta_{R}$$
$$\Leftrightarrow \tan \left( {\theta - \theta_{R} } \right) = \beta \frac{{\cos \theta \cos \theta_{R} }}{{\cos \left( {\theta - \theta_{R} } \right)}}.$$

Once the inclination angle is equal to the angle of avalanche θ A, the dilatancy reaches its peak value β*. Therefore, the previous equation can be rewritten as:

$$\tan \Delta \theta = \beta^{*} \frac{{\cos \theta_{A} \cos \theta_{R} }}{\cos \Delta \theta }.$$

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Marteau, E., Andrade, J.E. A model for decoding the life cycle of granular avalanches in a rotating drum. Acta Geotech. 13, 549–555 (2018). https://doi.org/10.1007/s11440-017-0609-2

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  • Avalanches
  • Dilatancy
  • Granular materials
  • Plasticity
  • Rate dependent
  • Solid/fluid transition