Advances in Materials Science and Engineering

Volume 2015 (2015), Article ID 540621, 6 pages

http://dx.doi.org/10.1155/2015/540621

## Micromechanism Underlying Nonlinear Stress-Dependent of Clays at a Wide Range of Pressures

^{1}State Key Laboratory for Geomechanics & Deep Underground Engineering, China University of Mining and Technology, Xuzhou 221116, China^{2}School of Mechanics & Civil Engineering, China University of Mining and Technology, Xuzhou 221116, China

Received 7 April 2015; Revised 18 May 2015; Accepted 21 May 2015

Academic Editor: Ana S. Guimarães

Copyright © 2015 Xiang-yu Shang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to investigate the mechanism underlying the reported nonlinear at-rest coefficient of earth pressure, of clays at high pressure, a particle-scale model which can be used to calculate vertical and horizontal repulsion between clay particles has been proposed. This model has two initial states which represent the clays at low pressure and high pressure, and the particles in this model can undergo rotation and vertical translation. The computation shows that the majority of particles in a clay sample at high pressure state would experience rotation during one-dimensional compression. In addition, rotation of particles which tends to form a parallel structure causes an increase of the horizontal interparticle force, while vertical translation leads to a decrease in it. Finally, the link between interparticle force, microstructure, and macroscopic is analyzed and it can be used to interpret well the nonlinear changes in with both vertical consolidation stress and height-diameter ratio.

#### 1. Introduction

The coefficient of earth pressure at rest, , defined as the ratio of horizontal effective stress to vertical effective stress under the condition of zero horizontal deformation representing the in situ stress state of the ground, is a fundamental parameter in the analysis and design of geotechnical structures.

Numerous studies in the past have indicated that of a given soil is a constant depending on its strength parameter [1]. However, accumulated evidence over recent two decades demonstrated that is not necessarily a constant but generally a function of void ratio, stress level, and critical state friction angle even for given normally consolidated clay [2–5]. In particular, of clay increases nonlinearly with consolidation stresses over a wide range of pressures [6–9].

In addition, previous investigation has indicated that, during one-dimensional compression, the distance between clay particles decreases continuously and the orientations of clay particles tend to be parallel to each other with their normal line pointing to the vertical direction [3, 10, 11].

It is well known that there exist noncontact forces such as repulsion between clay particles due to the electric charge on the surface of clay particle, and these interparticle forces usually dominate the mechanical behaviors of clayey material. In addition, interparticle forces, which are balanced with macroscopic stress in the clays, determine the arrangements and orientations (i.e., microscopic structure) of clay particles. The macroscopic stress and deformation can be readily measured during mechanical tests on the clay specimens, but the information related to microscopic structure such as pore size, arrangement, and orientation of clay particles only can be analyzed after stopping the mechanical tests. Since the evolution of microscopic structure of clay during tests is usually unknown, it is difficult to establish the relation between macroscopic mechanical behavior and microscopic structure. Moreover, it is almost impossible to measure the interaction forces between clay particles during macroscopic tests. Therefore, the studies on the links between microscopic structure, interparticle forces, and macroscopic behavior, which is important for the thorough understanding of the intrinsic mechanism of macroscopic mechanical properties of clayey material, are seldom reported. It is not surprising that the micromechanism relating above nonlinear of clay to its microscopic structure during high pressure one-dimensional compression has not been exploited.

This study aims to reveal the links between microscopic structure, interparticle forces, and macroscopic using numerical method and get insight into the micromechanism underlying the reported nonlinear of clay at high pressure.

#### 2. Numerical Study on the Links between Interparticle Forces and Microscopic Structure

##### 2.1. Repulsive Forces between Clay Particles

Since it has been revealed that the calculated relations between void ratio and vertical pressure, based on the double layer repulsive forces between clay particles, agree well with the measured results of both low and high pressure one-dimensional compression tests [12, 13], the noncontact forces between clay particles in this study are limited to the repulsive forces. The repulsion consists of osmotic pressure and electrical stress . and at arbitrary positon in a system of clay-electrolyte are given as follows [14]:where is the position vector, is the concentration of th ion in the electrolyte, J/(mol·K) is the universal gas constant, is the absolute temperature, is the permittivity of the electrolyte, and is electric field intensity. The repulsive pressure between two clay particles iswhere is placed at any position between two particles and is an arbitrary position outside the region between these two particles.

Both in (1) and in (2) are related to the electric potential:in which is the concentration of background electrolyte, is the electronic unit charge, is the valence of ion in the electrolyte, J/K is Boltzmann’s constant, denotes the gradient operator, and is the electric potential. It should be noted that (4) only applies to binary monovalent electrolytes which is the common case in the study of saturated clay.

In order to calculate the repulsion between clay particles, it is necessary to obtain the potential distribution in the clay-electrolyte system. Dimensionless potential can be determined by solving the following well-known Poisson-Boltzmann equation:whereis the reciprocal of double-layer thickness in which ’s number.

Combined with the following boundary conditions, the particular solution of (7) in a given domain can be determined. Considerin which is the boundary enclosing the domain and , , , and are constant parameters used to determine the boundary conditions.

There are few analytical solutions of (6) due to its strong nonlinearity, and it is necessary to solve it using numerical method. Using a partial differential equation solver FreeFem++ which is open finite element software [14], (6) can be solved numerically. The first step to solve (6) in this software is to derive its weak form as follows: in which is an arbitrary weight function which is forced to be zero on the boundary . Equation (12) is nonlinear and can be solved using iterative method. Assuming that there is only Dirichlet boundary condition, the following linear iterative equation based on Newton method can be obtained:where the superscript denotes the number of iteration steps.

Figure 1 presents the finite element mesh used for computation of potential around two inclined charged plates with finite length and the corresponding potential nephogram. Substituting the calculated potential into (4), the repulsion between clay particles can be solved.