Dilatancy plays a key role in mixtures of grains and fluid but is poorly investigated in dry granular flows. These flows may however dilate by more than 10% in granular column collapses. We investigate here dilatancy effects in dry flows with a shallow depth-averaged model designed to be further applied to simulate natural landslides. We use a compressible $\mu (I)$, ${\varphi }_{eq}(I)$ rheology with a dilatancy law, where ${\varphi }_{eq}(I)$ is the volume fraction at the equilibrium (i.e. critical) state and I the so-called inertial number. This law is obtained by simply removing the fluid phase in the solid/fluid model of our previous work [J. Fluid Mech, 801, 166-221 (2016)] and derived from critical state theory. A numerical method is proposed to solve the equations, that have however singularities that are rather difficult to handle.

Simulations of granular collapses on horizontal and sloping beds show that the maximum height of the deposits changes as a function of the initial volume fraction with higher (lower) deposits for initially denser (looser) granular masses, as observed with Discrete Element simulations. The front position and the deposit shape behind it are on the contrary poorly affected by the initial volume fraction, as if the flow had almost forgotten its initial state. However subtle effects can be observed with the occurrence of low velocity regimes on steep slopes that strongly depend on the initial volume fraction. Simulations show complex compression/dilation effects during the flow, in particular with front dilation (compression) during the acceleration (deceleration) phases. These effects may dramatically change the effective friction that is observed to decrease at the front in some situations, while the $\mu (I)$ rheology without dilatancy would have predicted an increasing friction. The model predicts an increasing dilation of the mass for increasing slopes by up to 10% in the studied configurations, in agreement with laboratory experiments. Our results suggest that this compressible model contains key features to describe granular dilatancy.

膨胀率在谷物和流体的混合物中起关键作用，但在干燥的颗粒流中研究很少。但是，在颗粒状柱塌陷时，这些流量可能会膨胀10％以上。我们在这里使用浅深度平均模型研究干流中的剪胀效应，该模型被设计为进一步应用于模拟自然滑坡。我们使用可压缩的$\mu （\mathrm{一世}）$， ${\varphi }_{Ëq}（\mathrm{一世}）$ 流变学与膨胀定律，其中 ${\varphi }_{Ëq}（\mathrm{一世}）$是平衡（即临界）状态下的体积分数，I是所谓的惯性数。通过简单地从我们先前的工作[ J. Fluid Mech，801，166-221（2016）]的固体/流体模型中去除流体相，即可得出该定律，并且可以从临界状态理论中得出。提出了一种数值方法来求解方程，但是这些奇点很难处理。

对水平和倾斜床的颗粒塌陷进行的模拟表明，对于离散的模拟，观察到的最大沉积高度随初始体积分数的变化而变化，对于较高密度（较小）的颗粒，初始密度较高（较小）的颗粒质量有所变化。相反，前部位置和其后部的沉积物形状受初始体积分数的影响很小，就好像流动几乎忘记了其初始状态一样。但是，在陡峭的斜坡上发生低速运动时，可以观察到细微的影响，而陡峭的斜坡很大程度上取决于初始体积分数。仿真显示了流动期间的复杂压缩/膨胀效果，尤其是在加速（减速）阶段的前部膨胀（压缩）时。$\mu （\mathrm{一世}）$没有膨胀性的流变学预示着摩擦会增加。与实验室实验一致，该模型预测了在所研究的结构中，坡度增加了10％时，质量的扩张程度将增加。我们的结果表明，该可压缩模型包含描述粒度扩张的关键特征。