Mathematical Problems in Engineering

Volume 2015, Article ID 136971, 10 pages

http://dx.doi.org/10.1155/2015/136971

## A Compressive Damage Constitutive Model for Rock Mass with a Set of Nonpersistently Closed Joints under Biaxial Conditions

^{1}College of Engineering & Technology, China University of Geosciences (Beijing), Beijing 100083, China^{2}School of Engineering, Tibet University, Lhasa, Xizang 850000, China^{3}Laboratoire de Géologie, CNRS, Ecole Normale Supérieure, 24 rue Lhomond, 75005 Paris, France

Received 6 October 2014; Revised 9 March 2015; Accepted 11 March 2015

Academic Editor: Chaudry Masood Khalique

Copyright © 2015 Hongyan Liu and Xiaoping Yuan. 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

One of the most problems faced by the practical rock engineering is to evaluate the rock mass strength. Now the existing theoretical evaluation of the mechanical property of jointed rock mass has no satisfactory answer yet because of the great number of variables involved. One of them is the nonpersistent joints which inherently affect the rock mass mechanical behavior. The previous models for rock mass can only reflect the effect of joint geometrical property on its mechanical behavior. Accordingly, this paper presents a new theoretical model to evaluate the mechanical behavior of the rock mass with a set of nonpersistently closed joints under biaxial conditions, which can reflect the effect of both the joint geometrical and mechanical property on the mechanical behavior of the rock mass under biaxial conditions at the same time. A series of calculation examples validate that the proposed model is capable of presenting the effect of joint geometrical and mechanical properties and the confining pressure on rock mass strength at the same time.

#### 1. Introduction

Existing in many rock mass projects, such as rock slopes, open pits, and underground caverns, joints have significant effects on the rock mass mechanical behaviors, such as its strength, deformability, failure mode, and stability, which will directly influence the design and construction of the rock mass engineering [1–3]. The existence of the nonpersistent joints and their interaction under the far-field stresses often lead to high stress concentration at the tips of the joint and become the source of weakening and failure of the rock mass [4, 5]. Jennings [6] proposed to compute the combined strength of joint and rock bridges from the simple linear weighing of the strength contributed by each fraction of material:where and represent the cohesion and friction angle of the joint and of the intact rock, respectively, and is the joint continuity factor given bywhere and are the length of the joint and of the rock bridge, respectively.

Equation (1) disregards the influence of the joints on the stress distribution, and assumes simultaneous failure of the intact material and the joints.

Different procedures have been used to study the strength of rock mass with nonpersistent joints: field observations, numerical studies or laboratory tests, and analytical solutions. All these methods have their own advantages and disadvantages. Among them, the analytical solution develops quickly in recent years with the widespread application of mechanical and mathematical tools in the study of the rock mass mechanical behavior. Now the following three analytical approaches are often used to study the constitutive relation of a jointed rock mass. The first is the phenomenological approach based on continuum damage mechanics [7, 8], in which the effects of microscopic damage mechanisms on property of the rock mass are reflected by scalar, vector, or tensor damage variables. For example, Kawamoto et al. [9] and Swoboda et al. [10] both adopted the second-order tensor to describe the damage anisotropy in the rock mass caused by the joints. Kawamoto et al. [9] presented a damage model for the jointed rock mass with one set of joints:where is the damage tensor caused by the joints to the rock mass, is the size of the jointed area, is the unit normal vector to the joint surface, is the average spacing of the joints, and is the volume of the rock mass.

For -set of joints, the damage tensor is given as the sum over all the related damage tensor obtained from (3):

It is a common method to define the damage variable in rock mass geometrical damage theory [9–11]. But its deficiency is obvious, in which only the geometrical property of the joint such as its length and dip angle is included, while the mechanical property of the joint such as its shear strength, namely, its cohesion and internal friction angle, is not included. That is to say, the above damage variable definition method thinks that the damage cannot transfer the stress, which is nearly true for the rock mass under tension, but not for the rock mass under compression. This is because the joint will close and then slide along its surface under compression, in which the joint will transfer part of the compressive and shear stresses. And the transferring coefficients are much related to the mechanical property of the joint such as its internal friction angle and cohesion [10]. Therefore, many scholars began to adopt different methods to improve the above model. When the rock mass is subjected to compression, Kawamoto et al. [9] and Yuan et al. [12] introduced the joint transferring coefficients of the compressive and shear stresses to revise the above model according to the fact that the joint can transfer part of the compressive and shear stresses. However, how to accurately obtain these two coefficients also becomes a new problem. While Swoboda and Yang [13] introduced the material constant to consider the effect of joint closure on the stress transferring, now it is obtained mainly by experience.

The second approach is based on mesomechanical damage mechanics, which leads to an improved understanding of the underlying physical process. In this approach, the nucleation, growth, and coalescence of microcracks are studied and their influences on mechanical properties are reflected in the constitutive relation in certain ways [14]. To study the mechanical behaviors of a joint-weakened rock mass by the micromechanical approach, several micromechanics-based joint models, such as the cylindrical pore model [15], dislocation pile-up model [16], and the frictional sliding crack model, have been proposed. Among these models, the frictional sliding crack model is widely accepted [17–20].

The third is the phenomenological approach based on mesomechanical mechanics, developed in recent years, which lies between the above two approaches and combines the advantages of them. It has a certain application in rock mechanics study [21].

In sum, it can be seen from the existing studies that the above three approaches all have been used in the study of the damage constitutive model for the jointed rock mass. And it is widely accepted that the geometrical property of the joint such as its length, dip angle, and number and the mechanical property such as its internal friction angle and cohesion should be both included in the rock mass constitutive model, only by which the effect of the joint on rock mass mechanical behavior is objectively reflected. But these two kinds of properties of the joint such as the geometrical ones and mechanical ones are separately considered in the existing study. Namely, the geometrical property of the joint is firstly adopted to define the damage tensor, and then its mechanical property such as its shear strength are adopted to revise the above calculation result, which not only causes inconvenience in application of this model but also is prone to lead to much error in the calculation result because of the arbitrariness in selecting these parameters. Can a damage tensor which includes both geometrical and mechanical property of the joint be proposed? This kind of damage tensor is not only in good agreement with the failure mechanism of the jointed rock mass but also applicable to use, which can avoid the error caused by the selection of the parameters to a large extent. So some research works have been done on this subject. For instance, Li et al. [22] obtained the calculation method of the mesodamage tensor of the rock mass with nonpersistently closed joints based on the strain energy theory, which perfectly considers the joint geometrical and mechanical property at the same time. It provides a good idea for studying the damage mechanics of the rock mass with nonpersistent joints. However, they only study the jointed rock mass under uniaxial condition, while in practical engineering, the rock mass is always in the complicated stress conditions such as biaxial or triaxial conditions. Arora [23] made triaxial compressive tests on specimens of plaster of Paris containing a single joint, and the result showed that increase in the lateral pressure leads to a more isotropic behavior. The study done by Prudencio and van Sint Jan [24] also illustrated that the confining stress resulted in different failure modes and higher peak strength of the jointed specimens. Therefore, it is very important and necessary to study the mechanical behavior of the rock mass under the complicated stress conditions.

Overall, the existing models provided a basis for estimating the mechanical behavior of the rock mass. Yet, since only a part is involved in the existing models, further key factors should be taken into account for the sophisticated mechanical behavior of the rock mass as follows:(1)the major property of the joints, such as its shear strength, can be further considered in the calculation of the damage tensor;(2)the effect of the complicated stress conditions such as the biaxial stress condition on the rock mass mechanical behavior should be studied in order to perfectly solve the practical engineering problem.

As such, the present work aims to establish a mathematical model to consider the above-mentioned factors.

The establishment of the proposed model is presented. Next, the calculation examples are done to check its validity and to further demonstrate the characteristics of the proposed model. The proposed model considers the major factors of joints to provide a mean in determining the mechanical behavior of the rock mass with a set of nonpersistently closed joints under biaxial conditions.

#### 2. Establishment of the Damage Model for the Rock Mass with Nonpersistently Closed Joints

According to fracture mechanics, for a planar problem under biaxial conditions, the increment of the additional strain energy because of the existence of the joints is as follows (the third stress intensity factor in the planar problem):where is the energy release rate,* A* is the joint area, and and are the first and second stress intensity factors of the joint tip, respectively. For a planar strain problem, , and for a planar stress problem , where and are Young’s elastic modulus and Poisson’s ratio of the corresponding intact rock, respectively. Because the planar stress problem is studied in this paper, is adopted.

For a single joint, (unilateral joint) or (central joint). For many joints, (unilateral joint) or (central joint), where is the joint number, is the joint depth, and is the joint half length, as shown in Figure 1.