Mathematical Problems in Engineering

Volume 2018, Article ID 5264319, 9 pages

https://doi.org/10.1155/2018/5264319

## Particle Equilibrium Method for Crack Propagation Simulation

College of Hydraulic and Environmental Engineering, Three Gorges University, Yichang, China

Correspondence should be addressed to Xiao-chun Lu; moc.361@4101nuhcoaixul

Received 18 December 2017; Revised 24 April 2018; Accepted 10 May 2018; Published 11 June 2018

Academic Editor: George S. Dulikravich

Copyright © 2018 Bo-Bo Xiong 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

Crack propagation simulation is constantly of great significance. This study presents the particle equilibrium method (PEM) to achieve this goal. PEM is based on the idealization of the problem domain as an assemblage of distinct particles, which release interaction forces to their surrounding particles. Once the distance between two particles exceeds the extreme distance, a permanent crack will emerge between them. The function of interaction force describes the mechanical response of elastic and plastic materials. The calculated structure attains its final state to the external load when all the particles reach the equilibrium condition of force. A calculation program is developed based on the proposed method’s theory and is applied to three numerical examples with reliable calculation results.

#### 1. Introduction

Crack propagation is one of the most common phenomena in structures, and this remarkable feature is consistently associated with the damage or destruction of structures. Thus, crack propagation simulation is of great significance, and several methods have been applied to address this problem. The finite element method (FEM) is a well-known numerical analysis method that has been extensively used in predicting crack paths [1]. However, FEM requires several remeshing processes of the finite element model to represent arbitrary and complex crack paths, which is slightly difficult; moreover, automatic remeshing can result in highly distorted elements, which worsens the performance of the FEM [1, 2]. To improve the FEM in terms of fracture simulation, extended finite element method (XFEM) [3], fractal finite element method [4], node-based smoothed extended finite element method (NS-XFEM) [5], and edge-based smoothed finite element method (ESFEM) [6] were proposed. However, these FEM versions are mesh-based approximation methods and have certain inherent limitations [7].

Thus, several meshless methods have been developed. Meshless methods discretize the problem domain only with a set of unstructured points or nodes without using a predefined element mesh. In contrast to the “element” concept of the FEM, nodal connectivity in meshless methods is enforced with the “influence-domain” concept [1]. This feature makes the meshless method suitable for modeling fracture propagation. Meshless methods that commonly use the weak form of Galerkin can be classified as approximation meshless methods and interpolation meshless methods [1]. Approximation meshless methods were proposed first, including smooth particle hydrodynamics (SPH) [8–12], diffuse element method (DEM) [13], and element-free Galerkin method (EFG) [14, 15]. Although these meshless methods have been widely used, they have difficulty in imposing essential and natural boundary conditions. Consequently, interpolation meshless methods, such as nature element method (NEM) [16], point interpolation method (PIM) [7], radial PIM (RPIM) [17], and nature neighbor RPIM [18–20], have been developed. These methods have improved the solution for the problem.

The present study proposes an approach for modeling structural responses to external loads, especially for simulating the crack growth phenomenon in brittle materials. This method begins by discretizing the problem domain as a set of particles, which interact solely with their surrounding particles through an interaction force. The concept of “element” is used in this method to record the relative position relationship between particles and their surrounding particles. This interaction force appears as a repulsion when two particles are relatively near each other and as attraction when the distance between two particles is far. In this study, the interaction force is expressed by a function, and different functions are established to describe the mechanical properties of elastic and plastic materials. If the distance between two particles exceeds the extreme distance, then a crack is generated between them and their interaction force would be interrupted. Thus, the crack propagation of the material can be naturally simulated and the damage rate of the material from the cracks can be obtained simultaneously. When all particles reach the equilibrium condition of forces, the system attains a stable state, and this is the final response to the external load. On the basis of the principle, a calculation program is developed and applied on three examples.

In the following sections, the basic principle of the proposed method is introduced. Second, the development of the numerical calculation program of the proposed method is presented. Finally, the performance of the proposed method is evaluated using three numerical examples.

#### 2. Basic Principle of the Particle Equilibrium Method (PEM)

Suppose that a target object occupies a reference configuration in region Ω and this object consists of many particles that are not mutually in contact. Any particle may undergo the effect of various loads, such as the interaction force among the surrounding particles, as well as the body (gravity or seepage force) and boundary loads, if this particle is in boundary . The particles within the target object may develop displacement under the applied loads. If all particles reach balance, then this displacement is the final response of the object to the external load.

##### 2.1. Discrete Particle Systems

The target object should be divided into many nontouching particles , where is the total number of particles that constitute the target object, to determine the response of the object under an external load. The discretization procedure is the same as the conventional FEM. Thus, PEM can use the data of the finite element mesh, which is easily acceptable to numerous users. The divided particles split the object volume; however, the volume concentrates on the particle. As shown in Figure 1, the volume of the particles is obtained as follows:where* m* is the total number of elements that contain particle ; is the volume of elements containing ; and is the weight of node to , which is determined by the geometry of the grid.