Theoretical Research, The Force Distribution In The Two-dimensional Granular System

In this paper we will discuss the force propagation and distribution in 2D granular material systems. Firstly, considering there are some arches in the real granular packing which the classical q model can not describe correctly, we revise theq model with two probabilities ρ₀ and ρ₁ .With our calculation, we find when f �' 0, p(f) is not tend to 0, but the maximal value of the distribution ,which is well consistent with the experiments under the f≤1 and that is not obtained by theq model.Secondly, we define a peculiar " stool "model to describe the arches in the granular packing. We define some stochastic quantities such as the length, the weight and the contact points of the "stool "to describe the disorder packing of the granular material. From this model, we calculate the system without external force and find that it is approximately consistent with the q model when f ≥ 1 . When f ≤ 1 , it is wellconsistent with the experiments. In addition, we also calculate the force distribution of the system which is under the external force and find whose results is well consistentwith f^-a like q model.Thirdly, we simulate a vector model for repairing the defection of the classical q model. In this model, all the sites of the system must satisfy the force and torquebalance and all the force must great than zero. We analyze and simplify the restriction of the force which is not less than zero in the frictionless systems in detail, but to consider the friction system we also use the Nguygen 's tentative method. With all these limits, we find that the distribution of vertical force in the system is similar to the scale q model.Finally, a special model of the force distribution of each layer in the granular packing will be considered that is a 2D hexagonal packing under a vertical downward point force at the top layer. In our work, we find that the distribution in each layer ofthe system is double-peaked when the layer r < N and the height of the peak quickly decrease exponentially with the depth. In our calculation we use some approximate methods in different parts of the system to describe the properties of force and theresults is well consistent with the experiments and Srdjan's conclusion.