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

Abstract

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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Acknowledgements

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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Appendices

Appendices

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:

$$\frac{a}{g} = \cos \theta \left( {\tan \theta - \mu } \right).$$
(5)

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:

$$\frac{{\partial \sigma_{xx} }}{\partial x} + \frac{{\partial \sigma_{xy} }}{\partial y} + \rho g\sin \theta = \rho a_{x} ,$$
(6a)
$$\frac{{\partial \sigma_{yx} }}{\partial x} + \frac{{\partial \sigma_{yy} }}{\partial y} + \rho g \cos \theta = \rho a_{y} .$$
(6b)

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:

$$\frac{\partial \tau }{\partial y} + \rho g\sin \theta = \rho a_{x} ,$$
(7a)
$$\frac{\partial \sigma }{\partial y} + \rho g\cos \theta = \rho a_{y} .$$
(7b)

The definition of dilatancy β is given by:

$$\beta = \frac{{{\text{d}}\varepsilon_{yy} }}{{{\text{d}}\varepsilon_{xy} }} = \frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{{v_{y} }}{{v_{x} }} \Rightarrow v_{y} = \beta v_{x} .$$
(8)

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:

$$\frac{\partial \sigma }{\partial y} = - \rho g\cos \theta .$$
(9)

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

$$\frac{\partial \tau }{\partial y} = \mu \frac{\partial \sigma }{\partial y} = - \mu \rho g\cos \theta .$$
(10)

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

$$- \mu g\cos \theta + g\sin \theta = a_{x} .$$
(11)

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):

$$\frac{a}{g} = \cos \theta \left( {\tan \theta - \mu } \right).$$
(12)

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:

$$\sin \theta = \beta \cos \theta + \mu_{l} \cos \theta .$$
(13)

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

$$\sin \theta \cos \theta_{R} = \beta \cos \theta \cos \theta_{R} + \mu_{l} \cos \theta \cos \theta_{R} .$$
(14)

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

$$\sin \left( {\theta - \theta_{R} } \right) + \cos \theta \sin \theta_{R} = \beta \cos \theta \cos \theta_{R} + \mu_{l} \cos \theta \cos \theta_{R} .$$
(15)

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

$$\sin \left( {\theta - \theta_{R} } \right) + \cos \theta \sin \theta_{R} = \beta \cos \theta \cos \theta_{R} + \cos \theta \sin \theta_{R}$$
(16)
$$\Leftrightarrow \sin \left( {\theta - \theta_{R} } \right) = \beta \cos \theta \cos \theta_{R}$$
(17)
$$\Leftrightarrow \tan \left( {\theta - \theta_{R} } \right) = \beta \frac{{\cos \theta \cos \theta_{R} }}{{\cos \left( {\theta - \theta_{R} } \right)}}.$$
(18)

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:

$$\tan \Delta \theta = \beta^{*} \frac{{\cos \theta_{A} \cos \theta_{R} }}{\cos \Delta \theta }.$$
(19)

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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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Keywords

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