Mathematical Problems in Engineering

Volume 2016, Article ID 9794605, 9 pages

http://dx.doi.org/10.1155/2016/9794605

## The Application of Nonordinary, State-Based Peridynamic Theory on the Damage Process of the Rock-Like Materials

^{1}School of Civil Engineering, Sichuan University of Science & Engineering, Sichuan, China^{2}School of Architecture and Civil Engineering, Chengdu University, Chengdu, China

Received 30 May 2016; Revised 3 August 2016; Accepted 16 August 2016

Academic Editor: Ninshu Ma

Copyright © 2016 X. B. Gu and Q. H. Wu. 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

Peridynamics has a great advantage over modeling the damage process of rock-like materials, which is assumed to be in a continuum interaction with each other across a finite distance. In the paper, an approach to incorporate classical elastic damage model in the nonordinary, state-based peridynamics is introduced. This method can model the dynamic damage process and stress change of rock-like materials. Then two instances about three-point bend experiment are simulated in the rock-like materials. Finally the conclusions are drawn that numerical results are close to the experimental results. So the method has a great predictable value in the geotechnical engineering.

#### 1. Introduction

The rock-like material is a quasibrittle material; it is widely applied to the geotechnical engineering, for example, tunnels and lots of other underground buildings. Especially for the mountain area, the research on the damage process of rock-like materials is essential, so the prediction for the damage process of rock-like materials becomes a focus problem gradually. But it is difficult to model because the mechanical character of rock-like materials is very complex and the precise damage prediction of rock-like materials is elusive. In the geotechnical engineering, the rock-like mass includes lots of tiny cracks, even before the load is applied. The damage process of the rock-like materials is often caused by tension, comparison, the change of the temperature, and so on, and the stress and displacement fields are influenced by the propagation and coalescence of the crack. Even through the damage process of rock-like materials has been investigated for many years, the damage mechanism and prediction of rock-like material are still not understood. In fact, the damage process of rock-like materials includes the initiation, propagation, and coalescence of the crack, which will lead to a sudden collapse due to brittle damage. How to model the damage character of the rock-like materials has brought great challenge.

Over the past decades, many methods are put forward to model the damage process of rock-like materials. In the finite element-based method, singular crack-tip elements are frequently encountered [1]. Because of the crack-tip stress singularity, an external fracture criterion must be introduced to determine propagation and bifurcation of the cracks, and the nucleation question of the crack is still not solved [2]. In order to overcome the above difficulties, the extended finite element theory [3] is proposed to simulate the propagation of cracks. Although many crack questions are solved by virtue of the extended finite element theory, external and bifurcation criterion must still be introduced when displacement is discontinuous and when interaction and bifurcation of multiple cracks are involved. Besides, a series of difficulties are encountered for the problem of the three-dimensional cracks by the XFEM. In order to solve the problems of the three-dimensional cracks, such as interactions among cracks and branching phenomenon of multiple cracks, meshless methods are developed [4]. The propagation and coalescence process of cracks can be simulated by Smooth Particle Hydrodynamics (SPH); however, the tensile instability problems are still encountered in method of SPH [5]. In order to avoid the aforementioned lacks, peridynamic theory, which is a numerical method based on the nonlocal thoughts, is introduced to model propagation and bifurcation process of cracks.

The peridynamic theory is a nonlocal meshless method; it is put forward by Silling [6], at Sandia National Laboratory. It is assumed that particles in a continuum interact with each other across a finite distance, and it formulates problems in terms of intergral equations rather than partial differential equations [7]. Therefore, the peridynamic method can be applied to model the problems of continuous or discontinuous displacements [8].

After this theory is put forward, it has been widely applied to model the damage process of different materials. Firstly, the theory is applied to model the fracture process of composite material; for example, the fracture processes in laminated composites subjected to low-velocity impact and in woven composites subject to static indentation are predicted by Askari et al. [9] and Colavito et al. [10, 11] and the fatigue crack growth analysis is done in layered heterogeneous material system using peridynamic method by Jung and Soek [12]. In addition, the notched laminated composite under biaxial loads is considered by Xu et al. [13]. Then it is applied to model the fracture process of metal material; for example, Foster et al. [14] use peridynamic viscoplastic theory to model the collapse process of metal and Wu et al. [15, 16] analyze the ductile fracture of metal materials using nonordinary state-based peridynamics. Meanwhile, Sun and Sundararaghavan [17] model the crystal material using the peridynamic plasticity theory. Immediately, it is extended to model the damage of concrete; for example, Gerstle et al. [18] model the damage process of concrete by introducing “micropolar peridynamic model.” The damage results of these materials are rather good by using the peridynamics, but for the geotechnical engineering, rock-like materials are an important material, the peridynamic theory is seldom used to model the damage process of rock-like materials, and only Ha et al. [19] use peridynamics to model the fracturing patterns of rock-like materials in compression, but the stress field description is not considered. In the paper, the application of nonordinary, state-based peridynamics in the damage process of rock-like material will be investigated. Not only the damage process of rock-like materials is described by using the method, but also the change of stress field in the damage process is depicted, so it provides a new idea to model the damage process of rock-like materials by using the peridynamics.

The paper is organized as follows. In Section 2, the state-based peridynamic theory is introduced at first, and the implementation on this method for the specific damage model is discussed. In Section 3, the state-based peridynamic numerical discretization is described. In Section 4, we represent numerical results consisting of the numerical simulation about the three-point bend test of the beam and the numerical simulation about the three-point bend test of Brazilian disk. In Section 5, conclusions are drawn.

#### 2. Model Description

For completeness, nonordinary state-based peridynamics is reviewed briefly; it includes a summary of basic peridynamic equation, the idea of constitutive relation, and its property.

##### 2.1. Basic Theory

Peridynamic theory is put forward by Silling [6] at Sandia laboratory in the United States; its basic equation of motion is shown as follows:where is the density of material point , is a neighborhood (Figure 1), and is a prescribed body force density field of material point at the instance , which represents the external force per unit reference volume square. and are the force density vector of material points and , respectively.