Micro structures of granular materials with various grain size distributions
Graphical abstract
A series of packing simulation on 9 types of disk assemblies with various compositions were conducted using two dimensional discrete element method (DEM). A general relationship between the number of contact points and the particle size was found. The distribution of the coordination number was predicted from the composition using the general relationship and validated by DEM.
Highlights
► 9 types of disk assemblies with various compositions were packed using DEM. ► Coordination number generally decreased with increasing grain size dispersion. ► A general relationship between contact points' number and particle size was found. ► Coordination number was predicted only from composition using the relationship.
Introduction
Numerical analyses based on the continuum mechanics such as FEM have been commonly used to simulate the mechanical behaviors of ground in the field of geotechnical engineering. Meanwhile “From Micro to Macro Approach,” in which the macro behavior of granular materials is simulated on the basis of the micro granular mechanics, has attracted attention mainly from academic interests [1], [2]. For example, Hassanpour et al. [3] studied the relationship between the single particle crushing strength and bulk compression behavior by experiment and computer simulation. Moreover, it is also a very useful tool in the practical geotechnical engineering for estimating mechanical properties of very rare materials such as lunar soils [e.g. 4]. In this approach it is essential to understand the micro structure of granular materials; the coordination number, defined as the number of contact points per particle, is one of the most important parameters characterizing the micro structure of granular materials.
Smith et al. [5] measured experimentally the mean coordination numbers of poorly-graded spheres in various packing states in 1929. Goodling [6] measured experimentally the frequency distribution of the coordination numbers, and Iwata et al. theoretically predicted it [7]. These researches indicate that the coordination number of poorly-graded granular materials of spherical shape simply depends on their packing state.
The coordination number of well-graded granular materials, however, is dependent not only on their packing state but also on their composition. Oda [8] studied experimentally the effects of the composition on the coordination number using spherical binary packing, and pointed out that the frequency distribution of the coordination number plays an important role in the shear strength behavior. However, this type of experimental measurement is not applicable to well-graded granular materials including countless numbers of fine particles. In 1980s a simple geometrical consideration of the particle configuration yielded models predicting the mean coordination number of well-graded spheres [[9], [10], [11]]. These models, however, were not able to predict the frequency distribution of the coordination number which is also considered to affect the shear strength of granular materials [8].
As to well-sorted granular materials whose particle size dispersion is considerably small, i.e. as small as the order of several percent of the mean diameter, only a few works are currently available for the coordination number even though their flow behavior has been discussed to some extent [12].
The present situation of researches as mentioned above indicates the necessity of systematic studies on the effect of the grain size distribution, which is expressed by the difference in dispersion, on the packing parameters such as the coordination number and the void ratio. Especially, it is essential to establish a model predicting the coordination number from basic parameters of granular material because of the difficulty and uncertainness of experimental measurement. From this point of view the present study aims to analyze the coordination number and the void ratio of granular materials with a variety of grain size distributions using a two dimensional Discrete Element Method (DEM) [13], and build a model predicting the frequency distribution of the coordination number from the grain size distribution.
Section snippets
Granular assembly prepared for simulation
A series of simulations was conducted on nine types of granular assemblies listed in Table 1; Fig. 1 shows grain size distribution curves of these assemblies. Each assembly had a grain size distribution of log-normal type, with the same mean diameter (D50 = 10 mm) but a different value of standard deviation (σ), defined by the following formula,where D and f(D) are the diameter and the frequency of D, respectively. In the preparation of granular assemblies, circular
Simulation results
Fig. 3 shows the configuration of granular assemblies under densely packed conditions in which particles forming the hexagonal closest packing (HCP) are colored gray; a particle was designated as HCP when they had 6 contact points and each angle between two adjacent contact points with respect to the particle center was between 60 ± 1°. It is to be noted that the number of particles forming HCP was large in the sample consisting only of equi-diameter particles (S000), and rapidly decreased with
A prediction model for coordination number distribution
As mentioned in the previous chapter, the change in the mean coordination number can be divided into two parts according to the standard deviation σ of the particle size: when σ is so small as to be regarded as mono-sized assemblies, the number of particles forming HCP is a principal factor governing the coordination number; when σ is sufficiently large, the dispersion of particle size becomes a principal factor instead. The tendency observed in the former part is unique only for mono-sized
Conclusion
Using a two-dimensional Discrete Element Method, nine types of disk assemblies having different grain size distributions were packed densely or loosely and their coordination numbers and void ratios were measured. The results obtained from the simulations are summarized as follows:
- 1)
The void ratio of densely packed granular materials sharply increased with increasing value of the standard deviation of the grain size distribution σ when σ was less than 0.0737 and then moderately decreased with
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