Understanding the response of granular matter to intrusion of solid objects is key to modelling many aspects of behaviour of granular matter, including plastic flow. Here we report a general model for such a quasistatic process. Using a range of experiments, we first show that the relation between the penetration depth and the force resisting it, transiently nonlinear and then linear, is scalable to a universal form. We show that the gradient of the steady-state part, K _ϕ , depends only on the medium's internal friction angle, ϕ, and that it is nonlinear in μ = tan ϕ, in contrast to an existing conjecture. We further show that the intrusion of any convex solid shape satisfies a modified Archimedes' law and use this to: relate the zero-depth intercept of the linear part to K _ϕ and the intruder's cross-section; explain the curve's nonlinear part in terms of the stagnant zone's development.

中文翻译：

---

阿基米德定律解释了固体渗透到颗粒介质中的过程。

理解粒状物质对固体物体侵入的响应是建模粒状物质行为的许多方面（包括塑性流动）的关键。在这里，我们报告这种准静态过程的一般模型。通过一系列实验，我们首先显示出穿透深度和抵抗穿透力之间的关系（瞬态非线性然后线性）可以扩展为通用形式。我们表明，稳态部分的梯度，K _φ，仅取决于介质的内摩擦角，φ，并且它是在μ=非线性黄褐色φ，相对于现有的猜想。我们进一步表明，任何凸形实体的入侵都满足修改后的阿基米德定律，并将其用于：将线性部分的零深度截距与K _{relate相关联}以及入侵者的横截面；用停滞区的发展来解释曲线的非线性部分。