Geofluids

Volume 2018, Article ID 3269423, 10 pages

https://doi.org/10.1155/2018/3269423

## A Statistical Constitutive Model considering Deterioration for Brittle Rocks under a Coupled Thermal-Mechanical Condition

^{1}State Key Laboratory of Geohazard Prevention and Geoenvironment Protection, Chengdu University of Technology, Chengdu, Sichuan 610059, China^{2}College of Environment and Civil Engineering, Chengdu University of Technology, Chengdu, Sichuan 610059, China

Correspondence should be addressed to Tianbin Li; nc.ude.tudc@btl

Received 2 March 2018; Revised 16 April 2018; Accepted 30 April 2018; Published 30 July 2018

Academic Editor: Ming Zhang

Copyright © 2018 Meiben Gao 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

Due to active actions of groundwater and geothermal, the stability of underground engineering is important during geological structure active area. The damage mechanical theory and statistical mesoscopic strength theory based on Weibull distribution are widely used to discuss constitutive behaviors of rocks. In these theories, a statistical method is used to capture mesoscopic properties of rocks in order to generate a realistic behavior at a macroscopic scale. Based on the above theories, this paper aims at establishing a constitutive relation of brittle rocks under thermal-mechanical coupling conditions. First, a statistical damage constitutive model was established by considering the thermal effects and crack initiation strength. Subsequently, the parameters of the model were determined and expressed according to the characteristics of stress-strain curve. Third, the model was verified by conventional triaxial experiments under thermal-mechanical actions, and the experimental data and theoretical results were compared and analyzed in the case study. Finally, the physical meaning of the parameters and their effects on the model performance were discussed.

#### 1. Introduction

With this longer, larger, and deeper underground engineering’s construction, the rock deterioration caused by groundwater and geothermal has become increasingly prominent. Constitutive modeling of rocks, as a key in the theory development and numerical analysis of rock mechanics, has become a topic of everlasting interest to researchers and practitioners working on geosciences and geotechnics. In addition, many factors affect rock behaviors, such as water-rock interactions and temperature, which would lead to its deterioration. With the deterioration of rocks and soil, the state of the geoenvironment would be changed resulting in landslide [1, 2], rock avalanche [3, 4], and rock burst [5]. For brittle rocks, the deterioration process in laboratory samples is a consequence of microscale and macroscale fracturing that occurs in several stages. Recent research suggests that crack initiation stress can be used as an estimate for in situ spalling strength, which is commonly observed in brittle rocks around underground openings [6–10]. Additionally, under high-temperature and high-pressure conditions, the mechanical characteristics of deep rocks exhibit different behaviors compared to those at more mild temperatures at lower depths. A reasonable constitutive model considering thermal action and crack initiation is the key to accurate prediction and judgment of the reliability and stability of those works such as the exploitation of deep mining resources and geothermal resources. Therefore, the constitutive behaviors of rocks considering thermal effects and crack initiation strength are significant to be studied and explored in order to better solve rock engineering problems in thermal-mechanical coupling conditions.

During the preceding four decades, various rock constitutive models have been established from theoretical, experimental approaches (e.g., [11–15]). Based on the traditional continuum mechanics and damage theory, thermoplastic and thermoelastic brittle models were proposed [16, 17]. However, these models failed to reflect the damage features of brittle rocks. For rock materials, numerous microscopic cracks, ranging from 0.01 to 1.0 mm in length, are statistically distributed, which have significant influences on the damage processes and failure characteristics of rocks. The nucleation and growth of microcracks would lead to a concentration of these microcracks into a narrow zone and produce a visible macroscopic fissure wider than 1.0 mm [18], so the process from damage to fracture could be studied on a mesoscopic scale. Since the statistical damage-based approach has been used successfully to address rock constitutive behaviors [19–21], as a quite attractive tool for investigating deformation processes and failure mechanisms in the mechanics of geomaterial community, it has been especially favored by many researchers. However, these previous statistical models did not reflect the residual strength of rocks. By introducing a coefficient into the damage variable, the statistical damage-based model will be able to describe the residual strength [22, 23]. For thermal effects, Zhang et al. [24] proposed a three-parameter Weibull distribution to express the rock constitutive behavior under a uniaxial compression test, which considers the thermal-mechanical coupling conditions.

In the present study, a mesoscopic element is considered to be isotropically elastic and its properties are defined by Young’s modulus or Poisson’s ratio. The stress-strain relationship is linearly elastic until given damage threshold (crack initiation strength) is attained [17]. Thus, the macroscopic strength and the properties of rock depend on the statistical mechanical properties of individual mesoscopic elements, which could be described by a phenomenological model through a statistical method. A previous research work showed that continuum damage models can effectively simulate the elastic degradation caused by preexisting microcracks in rocks. Although rocks always exhibit anisotropy after macroscopic fissures occur, an isotropic damage model is still an effective method to estimate the gross damage of rocks subjected to external loading.

This paper aims at exploring a statistical damage constitutive model considering crack initiation strength, deterioration, and thermal effects. Based on a statistical damage approach, the statistical constitutive model is established by introducing a three-parameter Weibull distribution, which describes a brittle constitutive behavior of rocks under thermal-mechanical coupling conditions.

#### 2. Constitutive Model

##### 2.1. Thermal-Mechanical-Damage Evolution Equations

###### 2.1.1. Thermal Damage

Under the thermal-mechanical coupling effect, numerous microcracks and considerable damage occur in the rock, which changes the mechanical properties. Macroscopically, the elastic modulus is usually chosen as the damage variable. According to Liu et al*.* [17], the thermal damage can be given by
where and are the elastic moduli at and room temperature, respectively.

###### 2.1.2. Damage Evolution Equation considering the Thermal Effect

The randomly distributed preexisting microcracks in rocks are the main factors leading to the damage of the rock.

Suppose that a microunit within the rock is sufficiently large to contain numerous cracks and defects and adequately small in dimension compared with the whole rock structure. Obviously, the strengths of these microunits vary randomly according to a Weibull distribution. The rock damage level can be expressed by the ratio of the number of damaged microunits to total units. On the microscopic level, the rock damage evolution process includes three stages, that is, crack initiation, propagation, and coalescence. Based on this, the coupled thermal-mechanical constitutive model of the complete failure process of the rock is proposed from a damage statistical point.

Next, before the damage evolution equation presented, some assumptions are given in the following [22]: (a)Rock is an isotropic homogenous geomaterial on a macroscale.(b)Microunits conform to Hooke’s law, prior to failure.(c)Rock damage occurs continuously and is the gradual accumulation of failures on the mesoscopic level.(d)Heat diffusion in a rock is only in the form of heat transfer, without considering convection and radiation.(e)Microunit strength follows a Weibull distribution, where the three-parameter density and distribution functions are given by the following: where is the microunit strength and , , and are the mean uniformity, peak strength, and damage evolution threshold, respectively, which represent the shape, scale, dimension, and position (as shown in Figure 1).