Mathematical Problems in Engineering

Volume 2017, Article ID 3861526, 14 pages

https://doi.org/10.1155/2017/3861526

## Simplified Qualitative Discrete Numerical Model to Determine Cracking Pattern in Brittle Materials by Means of Finite Element Method

^{1}Departamento de Ingeniería Mecánica, Fundación Universidad de América, Bogotá, Colombia^{2}Departamento de Ingeniería Mecánica y Mecatrónica, Universidad Nacional de Colombia, Bogotá, Colombia^{3}Departamento de Ingeniería Civil y Agrícola, Universidad Nacional de Colombia, Bogotá, Colombia^{4}Universidad Politécnica de Cataluña, Barcelona, Spain

Correspondence should be addressed to J. Ochoa-Avendaño; oc.ude.lanu@aaohcofj

Received 16 March 2017; Accepted 31 May 2017; Published 2 July 2017

Academic Editor: Fabrizio Greco

Copyright © 2017 J. Ochoa-Avendaño 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

This paper presents the formulation, implementation, and validation of a simplified qualitative model to determine the crack path of solids considering static loads, infinitesimal strain, and plane stress condition. This model is based on finite element method with a special meshing technique, where nonlinear link elements are included between the faces of the linear triangular elements. The stiffness loss of some link elements represents the crack opening. Three experimental tests of bending beams are simulated, where the cracking pattern calculated with the proposed numerical model is similar to experimental result. The advantages of the proposed model compared to discrete crack approaches with interface elements can be the implementation simplicity, the numerical stability, and the very low computational cost. The simulation with greater values of the initial stiffness of the link elements does not affect the discontinuity path and the stability of the numerical solution. The exploded mesh procedure presented in this model avoids a complex nonlinear analysis and regenerative or adaptive meshes.

#### 1. Introduction

Fracture mechanics studies the cracking process of a solid subjected to progressive external load. Particularly, computational fracture mechanics allows representing the formation and propagation of cracks in solids of general geometry by means of numerical models [1]. The results obtained with these models enable studying new materials and reducing cost of experimental tests [2].

The fracture process in a solid can be represented according to the description of displacement and strain field, the material constitutive model, and the numerical approximation technique in the finite element method. Likewise, the models could be divided by the following three types: the models with propagating cohesive discontinuities, the softening continuum models with partial regularization, and the regularized softening continua models [3].

Models with propagating cohesive discontinuities assume that the fracture process zone preserves a linear elastic behavior during its formation. Cohesive forces are defined between the faces of the discontinuity. These forces disappear when the gap between faces reaches a certain distance.

Consequently, the kinematics singularity vanishes and the crack opening increases in a discrete form [4]. Some authors indicate that each change of the discontinuity path needs remeshing process [5–8]. Other authors propose to enhance the trial function of the finite element in order to represent a jump of the displacement field [9–11]. The concept of partition of unity enhances the shape function of standard finite elements and ensures a uniqueness result [3]. Particularly, extended finite element method (X-FEM) is based on previous concept, choosing a function that represents the discontinuity of the displacement field on the crack faces [12–14].

Softening continuum models with partial regularization represent the fracture process zone by means of the strain localization on a finite band. Although the displacement field is defined as a continua form, there are weak discontinuities in the boundaries of the band [3]. The softening in fracture zone is independent of the size finite element and is also associated with the fracture energy per area unit [15]. The band of fracture zone is embedded into finite element in order to ensure objectivity with respect to the orientation of the mesh [16].

Regularized softening continua models preserve continuity of the displacement and strain fields. The fracture process zone is represented by a material band, in which the softening strain increases from the band boundary until its center [3]. These models have generally high computational cost; however, they have some advantage with respect to the models of partial regulation. The influence of size element is substantially reduced with fine meshes. These models show good results with adaptive meshing technique [17].

Other classification establishes two types of numerical models in order to represent cracking in brittle materials: the smeared crack approach and the discrete crack approach [18].

The smeared crack approach considers that an infinite amount of parallel cracks, each with very small opening, are assigned to the finite elements. The constitutive material model in the element is modified, such that the tangent stiffness and the stress in normal direction of the crack are reduced while the strain increases [19].

The discrete crack approach indicates that the fracture process zone is concentrated at a surface characterized by a relationship between the traction versus displacement jump, which describes the cohesion loss of the material between the crack faces after fulfilling the failure criteria. This approach has been developed on different models. Originally, cohesive forces associated with the fracture energy between the faces of a crack are appended. The initial models as the cohesive crack model prescribe the location of crack [20, 21]. Subsequently, the fictitious crack model predicts the formation and propagation of the crack in anywhere of the solid [22, 23]. The location of the crack with respect to finite element mesh establishes two different numerical techniques. The first one states that the crack is traced on the finite element sides and the cohesive forces are obtained by means of nonlinear mixed boundary conditions [18] or through interface elements with vanished or zero thickness that connect the nodes to both sides of the discontinuity [24–30]. The second technique states that the crack crosses the finite element [31, 32].

Particularly in two-dimensional mechanical problems using discrete crack models with interface elements, some authors define two triangular finite elements with high aspect ratio in the interface [29, 30], which depend on a tension damage constitutive relationship and the same kinematics as the continuum strong discontinuity approach tangent stiffness factor. Other authors state zero-thickness interface elements that couple pairs of duplicate nodes, whose behavior is defined by a traction-separation law [28]. In other works [25–27], the fracture process in the interface surface is based on failure criteria of the three-parameter hyperbolic cracking surface [24], which relates the normal and tangential stress components to the corresponding relative displacements. These last approaches request additional procedures in order to preserve the numerical stability in the nonlinear analysis.

In composite materials, the analysis of crack nucleation and growth must be addressed taking into consideration the nonlinear behavior in its microstructure. Different phenomena such as void growth, microcracking, interfacial debonding, and other nonhomogeneities are closely related to failure mechanisms that produce macroscale failure; as a consequence, macroscopic fracture models may turn out to be inappropriate to estimate crack trajectories and the structural response of those kinds of specimens [33]. Different authors have been developed approaches to tackle these problems via multiscale methods, which, by means a combination of microscale phenomena and mathematical approaches, allow describing the behavior of multiphase materials. In this sense, different works must be pointed [34–37].

This paper presents the formulation, implementation, and validation of a qualitative numerical model, which describes qualitatively the cracking pattern in a brittle homogeneous material, considering static loads, plane stress condition, and infinitesimal strain. This model is based on both the discrete crack approach and the finite element method, where the overlapping faces of the triangular elements are doubled and connected to zero-length link elements. Some sides of triangular elements are part of the discontinuity path, where the nonlinear behavior is represented by the link elements. The tangent stiffness of these elements tends to infinity during the linear elastic behavior of the material and is equal to zero when the failure criteria are fulfilled. This model avoids the remeshing process, provides the implementation into finite element code, and maintains a low computational cost. However, the structural response cannot be obtained because the cohesive law in the cracking zone was not considered. The advantage of the proposed model compared to discrete crack approaches with interface elements can be the implementation simplicity, the numerical stability, and the very low computational cost.

#### 2. Formulation of the Numerical Model

The proposed model is based on continuum mechanics applied to solid with discontinuities. The latter are produced by the fracture process of the brittle material. Particularly, plane stress condition, infinitesimal strain, and static load are considered.

##### 2.1. Government Equations in the Continuum

A solid is subjected to body forces vector at the domain and surface forces at the contour with normal vector , as shown in Figure 1(a). The prescribed displacements are indicated by the vector at the contour . The state stress in each material point is expressed by the tensor . Equations (1a) to (1c) present the equilibrium and boundary conditions in the solid: