# A Three Dimensional Discrete Constitutive Model for Over Coarse Grained Soil

## Abstract

Over Coarse Grained Soil is widely used as filling material for subgrade in mountainous highway, and the settlement and stabilization of high embankment settlement is critical to operation safety of mountainous highway. The constitutive relationship of over coarse grained soil is theoretical basis for settlement and stability analysis. So a rigid contact model for over coarse grained soil particles is established combined with the distribution of contact force and normal of contact force based on the research of contact characteristics of particles and ignoring the deformation of particle. The local constitutive model is acquired after analyzing the relationship between contact force of particles and local stress in over coarse grained soil. Furthermore, a three dimensional discrete constitutive model is built up. It is proved through discrete constitutive model that the fabric will change during the deformation of over coarse grained soil which results the change of physical and mechanical characteristics. The change of fabric affects the macro mechanical responses characteristics of over coarse grained soil.

## Keywords

Embankment subgrade Over course grained soils Discrete granular soil particle Fabric Constitutive model## 1 Introduction

Over coarse grained soil such as gravel soil, soil-rock aggregate mixture, boulder and rockfill is a typical granular media composed of discrete particles within two or three orders of magnitude in dimension. And it is widely applied in embankment in mountainous highway. The physical and mechanical characteristics is quite different from fine grained soil. Over coarse grained soil is a loose and discrete media in substance, and the contact between particles is discontinuous point-contact. From the paper of Luan and Ugai (1999), for over coarse grained soils and other loose granular media, the research of physical constitutive model of deformation and strength characteristics to overcome inherent limitation of mechanical methods of continuous medium with basis of micro mechanics is quite potential.

Fabric is an important concept in mechanics of granular media which can describe dimensional arrangement feature and interaction of granular particles’ (Arthur et al. 1977; Mahmood and Mitchell 1974; Oda 1977). Micro fabric of over coarse grained soil is closely related to particle size, shape, rigidity, gradation, void ratio, contact number between particles and stacking manner. Granular media is compose of large number particles, so fabric has statistical significance.

_{1}to P

_{5}, and the contact points are C

_{1}to C

_{5}, contact normal are n

^{(1)}to n

^{(5).}The relation between Particle Pi and its neighbor particles includes contact point number and contact normal, as shown is Fig. 1.

Fabric ellipse is introduced to describe contact normal. Fabric ellipse is a second-order tensor. Probability density function of contact normal \( E(\alpha ,\beta ) \) is closely related to stress ellipse and has specific physical meaning. Probability density function \( E(\alpha ,\beta ) \) changes in loading process so as to change the strength of over coarse grained soil which can result strain-hardening or strain-softening (Oda 1972a, b, c).

It is proved by experiment that probability density function \( E(\alpha ,\beta ) \) can represent the three-dimensional distribution of contact normal (Oda 1972a, b, c). Therefore, particle characteristic of over coarse grained soils, interaction and essential fabric of spatial uniform distribution between particles can be descripted objectively. Based on mathematical function of statistical distribution of the reflected fabric parameters, the relationship could be set up between inherent evolution of microstructure fabric parameters over coarse grained soils and macroscopic mechanics responses.

Granular media mechanics is assumed that over coarse grained soil is composed of solid particles contacted with each other, and interaction between particles obeys the laws of probability.

Granular media mechanics is applied in study the mechanical phenomena on the contact points of particles, and describes the phenomena according to the formula of mathematical statistics. Some mechanical models have been built by different research. However, those models can’t interpret the relationship between fabric change and mechanical response of over coarse grained soils under load very well.

## 2 Rigid Particle Contact Model of Over Coarse Grained Soil

There is a certain gradation within over coarse grained soil particles where large particles act as the skeleton and fine particles fill in the void between the large particles. Both the interaction between particles and between the filled surrounding medium should be belong to solid contact mechanics in terms of the mechanical characteristics (Stake 1983). It is supposed that the number of over coarse grained soil particles is enormous amount, so the macro mechanical parameters have statistical significance. The uneven stress of microcosmic can be described by average stress (Chang 1988a, b). When the over coarse grained soil particle number is infinity, and over coarse grained soil is continuous in the certain space, the summation in the calculation could be changed into integral (Zhong and Yuan 1992).

The virtual work for average stress is \( W = \bar{\sigma }_{ij} \varepsilon_{ij} \).

*N*

_{k}particles. Assuming that all the over coarse grained soil particle size is \( \bar{d} \), then

*i*,

*j*= 1, 2, 3.

## 3 Three Dimensional Discrete Constitutive Model

Over coarse grained soil particles are randomly stacked, so it can be described by fabric density distribution function. According to findings of Rothenburb and Bathurst (1988) and Bathurs and Rothenburb (1988), the density distribution function of over coarse grained soil particle fabric content could be described approximately by three function in the two dimensional case.

The relationship between strain and stress of over coarse grained soil can’t be derived directly, and the strain should be linked to contact force. Over coarse grained soil relationships between stress and strain can be derived when the local constitutive relation is confirmed.

There are only two independent coefficient in 12 coefficient, it indicates that the stress-strain characteristics of over coarse grained soil is isotropous when its fabric is isotropous.

It is proved that the bulk modulus and shear modulus of over coarse grained soil is related to void ratio and coordination number.

## 4 Conclusions

The physical and mechanical properties of over coarse grained soil are closely related to particle spatial stacking mode, void ratio and its’ spatial distribution. The relationship between average force and contact point is set up by analyzing the force between particles and normal direction of contact based on the fabric characteristics of over coarse grained soil. Due to fabric change during the process of deformation, relationship between force and strain is nonlinear as same as relationship between stress and contact force, the relationship between stress and contact force is affected by the fabric. The mechanical response of over coarse grained soil in the large scale proved that the changes of fabric contents play an important role on the deformation characteristics of materials.

The rigid contact model of over coarse grained soil is established based on the microstructure of over coarse grained soil particles. Local constitutive relation of over coarse grained soil is acquired by the connection between contact forces of local stress between particles. Based on the local constitutive relation of over coarse grained soil, the constitutive relation of three dimensional granular of over coarse grained soil has been set up. These studies provide foundations for the further research.

Particle flow code (PFC 3D) method is a feasible method for experimental verification of three Dimensional Discrete Constitutive Model. The author will write other paper to discuss.

The application of basic theories of discrete mechanics and particle mechanics is a promising method to study stress-strain characteristics of over coarse grained soil. And the contact force of over coarse grained soil particles and contiguous normal distribution need further research.

## Notes

### Acknowledgements

The authors gratefully acknowledge the financial supports from the National Science and Technology Support Plan under Grant No. 2015BAK09B01, from the National key research and development plan under Grant No. 2016YFC0802203, from the Natural Science Foundation Project of Chongqing Science & Technology Commission of China under Grant No. CSTC2013jcyj A300081, respectively.

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