Advances in Civil Engineering

Volume 2018, Article ID 3067120, 9 pages

https://doi.org/10.1155/2018/3067120

## Multiphysics Coupling Model of Rock Mass considering Damage and Disturbance and Its Application

^{1}School of Resources and Safety Engineering, Central South University, Changsha 410083, China^{2}Jiangxi Copper Technology Research Institute Co., Ltd, Nanchang 330000, China^{3}Fankou Lead-Zinc Mine, Shaoguan 512325, China

Correspondence should be addressed to Yaguang Qin; moc.361@gyqusc

Received 26 June 2018; Accepted 16 August 2018; Published 12 September 2018

Academic Editor: Dengke Wang

Copyright © 2018 Wei Wang 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

Aiming at the deficiency of the conventional multiphysics coupling model, the deterioration of strength parameters was considered by defining elastoplastic damage variables, and the heterogeneity of strength parameters was expressed by the Weibull distribution function. In addition, the relation between effective stress and the anisotropic permeability matrix was established, and the blast was transformed into a load boundary condition. On this basis, an improved multiphysics coupling model that considered damage and disturbance was constructed, while a corresponding finite element calculation program was developed. Taking an excavation stope as the object, the characteristics of the mining-induced stress, seepage, and failure were analyzed by an improved multiphysics coupling model and compared with actual detection data. The results show that the improved model reflects the extent and range of mining-induced failure more accurately and fits well with the actual detection. These results are compared to the conventional multiphysics coupling model and a single physics model. It is indicated that the improved multiphysics coupling model and corresponding calculation program are effective and rational.

#### 1. Introduction

Mineral resources are the material basis for developing a national economy and safeguarding national security. With the rapid industrialization and urbanization process in China, the demand for mineral resources is significant, resulting in the depletion of shallow resources, and many mines have now advanced to a deep mining stage. Deep mining is accompanied by complex mining conditions. Rock mass engineering is in a coupled system composed of a stress field, seepage field, and temperature field, and rock deformation and its failure mechanism is very complex. The conventional single field (usually stress field) analysis has major limitations, and the analysis of rock mass response and failure under multifield coupling has not only become an important subject of rock mechanics research [1], but also has important practical significance.

For most rock mass engineering applications, the variation in temperature gradient is small, and the study of multifield coupling is mainly focused on the coupling of the stress field and seepage field. Baghbanan and Jing [2] simulated the coupling process of seepage and stress under a different fracture distribution and pressure coefficient using the discrete element software known as UDEC and obtained the rule of stress variation on permeability and seepage path. Figueiredo et al. [3] established the function relationship between rock mass porosity and isotropic volume strain. Then, the quantitative relationship between rock mass deformation and permeability was determined, and the numerical analysis of a fluid-solid coupling process in a deep rock mass project was realized. Based on pore elasticity theory, Zhang and Wang [4] deduced the relationship between permeability coefficient and stress variation and determined the range of mining failure in a mine. A fully coupled mathematical model of a seepage field and stress field was proposed by Wang et al. [5], and the two developments of a corresponding numerical calculation program and the application analysis of the fluid solid response characteristics of a hydropower station were realized. According to the actual construction situation of a tunnel [6], the analysis model of tunnel excavation under the action of fluid structure interaction was established by using elastoplastic theory, which provided technical reference for the design and construction of a tunnel.

At present, multifield coupling research is usually based on the Biot consolidation theory, which assumes that the deformation of rock is linearly elastic and the seepage obeys Darcy’s law. The conventional multifield coupling analysis has good practicability, which can realize reliable analysis and solve problems in engineering, but there are also several shortcomings. On the one hand, conventional multifield coupling research does not consider the effect of damage, which means that elastic modulus, cohesion, and coupling strength parameters are constant in the calculation process and that the research will not reflect the nonlinear elastic compression or expansion of microdefects caused by deformation, an important feature of strain hardening or softening stage. The rock mass will behave as an ideal elastoplastic body, which is inconsistent with a real-world scenario. Several scholars have gradually focused on the influence of damage and built the corresponding multifield coupling model [7, 8], but the assumption is that damage occurs only in the plastic stage, which ignores the existence of elastic damage. On the other hand, the engineering disturbance (especially blasting) time is very short compared to the geological action, but the effect and influence of blasting on rock engineering is very strong. In the area of multifield coupling research, the disturbance effect is seldom considered. In addition, the conventional multifield coupling study also lacks the consideration of the important characteristics such as rock mass strength heterogeneity and seepage anisotropy.

Therefore, in this study, the definition of staged damage variables and the equivalent calculation of blasting were taken into account. Considering the heterogeneity of rock mass strength parameters and seepage anisotropy, an improved multifield coupling model that considers damage and disturbances was established, and the corresponding numerical calculation program was compiled. Through numerical analysis and comparison of the excavation response and failure characteristics of a stope, the effectiveness of the improved multifield coupling model and its calculation program was verified.

#### 2. Conventional Multifield Coupling Model

In order to simplify the expression of the formula, this study adopts the abstract notation of a tensor (black body representation). The definition of the rock stress tensor sigma, strain tensor for, based on the assumption of small deformation, equilibrium equations and geometric equations of rock mass are as follows:where is the Laplace operator, is the volume force tensor, and is the displacement tensor.

It is generally believed that the internal seepage of a rock mass is incompressible and obeys Darcy’s law:where is the seepage speed, is the source item, is the seepage pressure, is the dynamic viscosity coefficient of fluid, is the fluid density, is the gravitational acceleration, and is the permeability tensor.

The conventional multifield coupling method is mainly based on the Biot consolidation equation [9], that is, the coupling of stress and seepage through effective stress principle and volume strain source term:where is the effective stress tensor, is the elastic stiffness tensor of rock mass, is the effective stress coefficient of Biot, and is the volumetric strain of rock mass.

Compared to the single stress field analysis, the multifield coupling model considers the interaction between seepage and stress, which can reflect a real-world scenario more effectively. However, the conventional multifield coupling model does not consider the influence of damage, disturbance, rock mass heterogeneity, and other important factors, it is still not sufficient to solve the deformation and failure analysis of a rock mass under complex conditions. Therefore, it is necessary to improve the conventional multifield coupling model.

#### 3. Improved Multifield Coupling Model

##### 3.1. Damage Variable Definition

Damage is the existence and evolution of microdefects, and the deterioration of material stiffness and strength is caused by macroscopic damage. The conventional multifield coupling model does not consider the influence of damage, which will lead to the omission of nonlinear elastic rock mass deformation and the softening or hardening of plastic strain. The rock mass becomes an ideal elastoplastic body, which is inconsistent with real-world behavior. Therefore, the corresponding damage variables are defined for the elastic and plastic stages, respectively. Referring to the related research [10, 11], the damage evolution of the elastic stage is related to elastic strain, and the damage evolution of plastic stage is related to the equivalent plastic strain. The expression of damage variable, , is defined by the following stages:where is the elastic strain, is the mean value of elastic strain, is the equivalent plastic strain, and and are correction coefficients.

The experimental results of Yu et al. [12] show that the damage has obvious influence on the elastic modulus and cohesion, but has little influence on the internal friction angle. According to the Lemaitre strain equivalence principle, the relationship between the strength parameters of rock mass and the damage degradation is obtained:where and are initial modulus of elasticity and cohesion and and are modulus of elasticity and cohesion considering damage.

##### 3.2. Improved Multifield Coupling Model

The damage variable, , is introduced into the conventional multifield coupling model, and the improved multifield coupling model at the elastic stage is obtained as follows:

In the improved multifield coupling model, by considering the damage elastic stiffness tensor and permeability tensor , we can reflect the effect of stress and seepage damage. In addition, in the rock constitutive equation reflects the effective stress effect of seepage, and the bulk strain source, , reflects the rock mass effect of deformation on seepage continuity. The coupling action of stress, seepage, and damage has taken into account in the improved model. At the same time, seepage usually shows anisotropic characteristics [13], the relationship between permeability and effective stress is as follows:where is the initial permeability, is the correlation coefficient (0.35 MPa^{−1}), and is the effective principal stress.

After entering the plastic stage, the improved multifield coupling model lies in the difference of the constitutive equations of the rock mass, that is, the stress and strain are no longer subjected to one-to-one correspondence and are related to the loading-unloading path. Therefore, it is necessary to use the incremental theory to express the relationship between stress and strain. In this study, the Drucker–Prager yield model (DP model) [14, 15] which considers three principal stresses are used, and the damage variable is introduced into the DP model, while the yield function and potential function are in turn:where is the first invariant of the stress tensor, is the second invariant of the tensor of partial stress, and are internal friction angle and expansion angle of rock mass, and is the hardening function.

The plastic strain increment is calculated by the associated flow rule:where is the plastic multiplier, .

The loading-unloading criterion is described as follows:

The piecewise linear hardening function is used to approximate the nonlinear hardening function, and the expression is described as follows:where is plastic hardening or softening internal variables of a rock mass [16].

Compared to the conventional yield model, the modified yield model takes into account the influence of damage variables. In the calculation process, the strength parameters will deteriorate with the evolution of damage. Meanwhile, the piecewise linear function is used to approximate the nonlinear hardening function. Therefore, the improved model can overcome the deficiency of rock mass as an ideal elastoplastic body in conventional multifield coupling, thus reflecting the strain hardening or softening characteristics of the rock mass yielding stage.

##### 3.3. Blasting Equivalent and Expression of Strength Heterogeneity

The effect and influence of blasting on rock engineering is very strong, but it is seldom considered in a multifield coupling study. Blasting can be converted into an action load by an equivalent calculation based on the charge configuration in a given engineering application, and it is applied to the boundary of a free surface in a numerical simulation to reflect the effect of blasting on the multifield coupling process. The method of uncoupled charge is usually used in deep hole blasting in a mine, and the equivalent load of blasting is calculated as follows [17]:where is the density of explosives, is the explosive velocity, is the isentropic exponent of explosives, is the charge diameter, is the hole diameter, is the total length of charge, is the hole length, is the pressure increase coefficient, and and are the attenuation coefficients of the blasting load.

The Weibull probability density function [18] is used to characterize the heterogeneity of rock mass strength parameters:where are the rock mass strength parameters (such as elastic modulus, cohesive force, etc.), is the mean value of the strength parameter, and is the correlation coefficient of inhomogeneity.

As shown in Figure 1, when the value of the nonhomogeneous coefficient, , is higher, the value of the intensity parameter becomes more concentrated and tends to the mean value.