## A constitutive law for dense granular flows

#### Abstract

A continuum description of granular flows would be of considerable help in predicting natural geophysical hazards or in designing industrial processes. However, the constitutive equations for dry granular flows, which govern how the material moves under shear, are still a matter of debate 1–10. One difficulty is that grains can behave 11 like a solid (in a sand pile), a liquid (when poured from a silo) or a gas (when strongly agitated). For the two extreme regimes, constitutive equations have been proposed based on kinetic theory for collisional rapid flows 12 , and soil mechanics for slow plastic flows 13. However, the intermediate dense regime, where the granular material flows like a liquid, still lacks a unified view and has motivated many studies over the past decade 14. The main characteristics of granular liquids are: a yield criterion (a critical shear stress below which flow is not possible) and a complex dependence on shear rate when flowing. In this sense, granular matter shares similarities with classical visco-plastic fluids such as Bingham fluids. Here we propose a new constitutive relation for dense granular flows, inspired by this analogy and recent numerical 15,16 and experimental work 17–19. We then test our three-dimensional (3D) model through experiments on granular flows on a pile between rough sidewalls, in which a complex 3D flow pattern develops. We show that, without any fitting parameter , the model gives quantitative predictions for the flow shape and velocity profiles. Our results support the idea that a simple visco-plastic approach can quantitatively capture granular flow properties, and could serve as a basic tool for modelling more complex flows in geophysical or industrial applications. Advances in our understanding of dense granular flows have recently been made by comparing different flow configurations 14. The simplest configuration from a rheological point of view is the one sketched in the inset to Fig. 1. A granular material confined under a normal stress P in between two rough planes is sheared at a given shear rate _ g by applying a shear stress t. In refs 15 and 16, for stiff particles the shear stress is shown, using dimensional arguments and numerical simulations, to be proportional to the normal stress, with a coefficient of proportionality that is a function of a single dimensionless number, called the inertial number I: t ¼ mðIÞP with I ¼ _ gd=ðP=r s Þ 0:5 ð1Þ where r(I) is the friction coefficient, d is the particle diameter and m s is the particle density. They found that the volume fraction F of the sample is also a function of I but varies only slightly in the dense regime. The inertial number, which is the square root of the Savage number 20 or of the Coulomb number 21 introduced previously in the literature, can be interpreted as the ratio between two timescales, a macroscopic deformation timescale (1/ _ g) and an inertial timescale (d 2 r s /P) 0.5. By confronting results from the simple shear test with experimental measurements of granular flows on rough inclined planes 17,22 , it can be shown that the friction coefficient m(I) has the shape given in Fig. 1. It starts from a critical value of m s at zero shear rate and converges to a limiting value of m 2 at high I. The following friction law can then be proposed, compatible with the experiments 19 : mðIÞ ¼ m s þ ðm 2 2 m s Þ=ðI 0 =I þ 1Þ ð 2Þ where I 0 is a constant. Very recently, this simple description of granular flows has been successful in predicting two-dimensional configurations , capturing velocity profiles on inclined planes 14,23 and important features of flows on a pile 19. However, the simple scalar law (equation (1)) cannot be applied in more complex flows where shear in different LETTERS Figure 1 | Friction coefficient m as a function of the dimensionless parameter I (m s 5 tan(20.9), m 2 5 tan(32.76) and I 0 5 0.279). Inset, definition of the pressure P, the shear stress t, and the shear rate _ g in the simple plane shear configuration. Figure 2 | Experimental setup of granular flows on a pile between rough sidewalls. The channel is partially closed at the bottom end to create a static pile on top of which the grains flow 19. The sidewalls are made rough by gluing one layer of beads on them. The channel is 120 cm long and the width W varies from 0.9 cm (16.5d) up to 28.9 cm (546d).