Mathematical Problems in Engineering

Volume 2018 (2018), Article ID 3080173, 11 pages

https://doi.org/10.1155/2018/3080173

## A New Equivalent Statistical Damage Constitutive Model on Rock Block Mixed Up with Fluid Inclusions

State Key Laboratory of Disaster Prevention & Mitigation of Explosion & Impact, National Defense Engineering College, Army Engineering University of PLA, Nanjing 210000, China

Correspondence should be addressed to Hongfa Xu; moc.anis@1afgnohux

Received 31 October 2017; Revised 3 January 2018; Accepted 9 January 2018; Published 18 March 2018

Academic Editor: Michele Brun

Copyright © 2018 Xiao Chen 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

So far, there are few studies concerning the effect of closed “fluid inclusions” on the macroscopic constitutive relation of deep rock. Fluid-matrix element (FME) is defined based on rock element in statistical damage model. The properties of FME are related to the size of inclusions, fluid properties, and pore pressure. Using FME, the equivalent elastic modulus of rock block containing fluid inclusions is obtained with Eshelby inclusion theory and the double M-T homogenization method. The new statistical damage model of rock is established on the equivalent elastic modulus. Besides, the porosity and confining pressure are important influencing factors of the model. The model reflects the initial damage (void and fluid inclusion) and the macroscopic deformation law of rock, which is an improvement of the traditional statistical damage model. Additionally, the model can not only be consistent with the rock damage experiment date and three-axis compression experiment date of rock containing pore water but also describe the locked-in stress experiment in rock-like material. It is a new fundamental study of the constitutive relation of locked-in stress in deep rock mass.

#### 1. Introduction

As the research of rock mechanics gradually develops to deep and complicated geological conditions, the traditional theory of rock mechanics has been continuously improved and perfected. The application of CT technology in rock mechanics helps people have a new understanding of microscopic pore structure of rock, thus establishing a microscopic system of rock [1–4].

Pores are classified into connected pores and closed pores. Actually, the theoretical and experimental research on the connected pores in rock is relatively mature [5–8], and the theory about it has developed to fluid interaction with all forms of subsurface materials, whether the materials are unconsolidated or crystalline [9]. From the perspective of engineering applications, Gassmann-Biot equation describes the relationship between rock physical properties and pore fluid characteristics under the conditions of low frequency [10]. Besides, it is an important theoretical basis of rock fluid replacement or seismic wave detection in oil and gas engineering [11–15]. Compared with the connected pore problems, theoretical and experimental investigations about closed pores are relatively few. Closed pore is an important carrier of locked-in stress. In 1979, the concept of “locked-in” stress was first proposed by Zongji Chen. Based on the microscopic rheological analysis of rock, Chen found that the tectonic loading and thermal loading lead to nonuniform stress field, and some of them are retained in the form of locked-in stress. Additionally, locked-in stress is considered as an important cause of engineering disaster [16]. Barrows considered that the internal shape of viscoelastic Earth material possesses an internal equilibrium pressure over extended periods of geologic time. Internal equilibrium pressure should be regarded as an intrinsic physical property of Earth materials [17]. For a long period of time, the locked-in stress hypothesis has only a few quantitative studies and developments, but the situation has changed in recent years. Wang analyzed the complex environment of diagenesis and the influence of geological processes on the inhomogeneity and discontinuity of rock based on the geological characteristics of rocks [18]. It is also an important source of locked-in stress. Using the hypothesis of locked-in stress, Yue proposed and demonstrated that high-pressure fluid inclusions are concrete, measurable, and computational stress inclusions. Besides, they are prevalent in geological rocks and minerals. The effect of fluid inclusions on surrounding rock is calculated. Yue believed that small, confined, and compacted fluid inclusions are common volume stress of rock burst, surrounding rock rupture, and large deformation of roadways in deep rock excavation chambers [19–21].

In general, the research on the constitutive relations of rock fluid inclusions under deep high pressure is very few. Therefore, this study attempts to establish a concise mathematical and physical model that reflects the constitutive relationship between locked-in stress and rock block. The relationship between microcosmic characteristics and macroscopic rules has good applicability, which also lays a good foundation for the mechanical problems of deep rock mass containing closed fluid inclusions.

#### 2. Related Work

The statistical damage model is a good mathematical tool for establishing the constitutive relationship between the microscopic and macroscopic properties of rocks. “Rock element” is the basic element of the rock statistical damage model and the “rock element” satisfies the D-P yield criterion. Rock elements gradually yield under external stress, and their strength intensity conforms to Weibull distributions. Moreover, the process can well reflect the stress-strain law of the rock under external stress.

According to the Lemaitre strain equivalence hypothesis, the elastic modulus of rock after damage is as follows [22, 23]:where is modulus of elasticity before rock damage, is modulus of elasticity after rock damage, and is damage variable.

The stress-strain curve is consistent with the law of linear elasticity before material damage. The macroscopic constitutive relation considering cumulative damage of rock can be expressed as

The solution to is as follows.

According to the study of Wengui Cao, the yield judgment of rock element is appropriate with Drucker-Prager criterion, and the strength intensity of rock elements conforms to Weibull distribution [24, 25]where is the parameter related to cohesion and internal friction angle ; is the spherical tensor invariant; is the deviator tensor invariant; , .

Assuming that the strength of rock element obeys the Weibull distribution, its probability density function is where and in (4) are Weibull distribution parameters:

According to (2), the constitutive relation of rock is as follows:where is confining pressure of rock and is the elasticity modulus of rock matrix. and can be obtained with value method of multivariate function. It uses the mathematical nature that the derivative of the highest point of the curve is 0:whereThe parameters , in (7) need to be gotten from fitting methods by experiment date and conform to the linear rule.

However, the damage defined in the traditional statistical model essentially reflects the damage caused by the yielding of the nondestructive matrix due to the external stress. Besides, the initial damage of the “fluid inclusions,” cracks, and holes existing in the rock before the external loading are not reflected in the model. Obviously, these are important factors influencing the initial elastic modulus of the rock. To establish the relationship between fluid inclusions and macroscopic constitutive relations of rocks, the initial factors of these rocks must be clearly described.

To solve this problem, it is necessary to establish the “rock element” which can reflect the constitutive relation of the rock part representatively, so fluid inclusion-matrix element (FME) is defined [26]. Theoretical basis of FME is RVE, which originates from rock element. Representative volume element (RVE) is an effective method to study such problems, which has been widely used in theoretical and numerical simulation of voids, fractured composites, and so on [27–31]. FME differs from rock element because a typical element of the same size contains a fluid inclusion, and the size is 3 to 5 times of the diameter around the inclusion. There are two types of pores in FME. One is connected pore, which indicates that the pore is connected with each other or connected with the cracks in the matrix. This kind of problem includes the pore which has fluid exchange with the outside, whether it has obvious boundary with the matrix or not. It covers the scope of fluid solid coupling, fluid replacement, and other engineering problems. Another kind is the closed pore, which has obvious boundary with matrix or no external exchange of fluid (gas) under certain confining pressure. This kind of FME contains rock pressure existing in the formation of geology, and it is an important carrier of the locked-in stress. Special statement is that pores mixed up with fluid inside FME are collectively referred to as “fluid inclusions” in this study.

#### 3. Model

FME is mixed up with a fluid inclusion and rock matrix around a finite range, as shown in Figure 1 (0 is matrix; 1 is inclusion).