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.
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 [
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 [
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 [
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 [
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 [
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 [
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 [
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 [
Particularly in twodimensional mechanical problems using discrete crack models with interface elements, some authors define two triangular finite elements with high aspect ratio in the interface [
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 [
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 zerolength 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.
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.
A solid is subjected to body forces vector
Solid with discontinuity subjected to body and surface forces: (a) general sketch and (b) detailed traction vector inside of the discontinuity.
In Figure
The proposed model uses Rankine’s failure criteria restricted by tensile stress states. These criteria establish that the failure of material takes place when the positive maximum principal stress
The fracture process in mode I appears with the failure of material, where the crack is normal to the direction of the positive maximum principal stress
The cohesion between crack faces is lost inside the cracking zone, while elastic unloading is shown at neighborhood of the cracking zone.
The mechanical problem raised in the previous section is implemented in finite element method. The nonlinear analysis procedure, the meshing technique with potential discontinuities, the description of used type elements, and the evaluation of its tangent stiffness are presented as follows.
A solid in plane stress condition can be represented with a mesh of
Connecting an overlapping side of two triangular elements by means of two link elements: (a) sketch of two triangular elements in real scale, (b) sketch of two triangular elements and two link elements out of scale, and (c) detailed link element and its orientation with regard to neighboring triangular elements in a virtual gap.
Each link element connects two overlapping nodes; therefore, its length is null and its longitudinal axis is perpendicular to the side of associated triangular element, as shown in the virtual gap of Figure
The discontinuity path on a contour
In each triangular finite element
The external virtual work of the solid, expressed in (
The cohesive virtual work in the discontinuity of the solid, expressed in (
Equations (
According to NewtonRaphson method, the residual force vector
Nonlinear numerical solution method searches the trial incremental nodal displacement
The tangent stiffness matrices
A linear elastic material is assumed in the domain of the triangular elements; therefore
In contrast, each link element
This numerical model assumes that the crack path does not substantially depend on the cohesive law of the brittle material. Likewise, the single goal of this work is to predict the crack path, without representing the structural response of the solid. Consequently, this model proposes a simplification of the cohesive law, in which the tangent stiffness factor
First, a conventional mesh of linear triangular elements and the boundary conditions are generated. Next, a new mesh of linear triangular and link elements is produced with the information of the conventional mesh. The new mesh is called exploded mesh and is made only once during the simulation. Figure
Finite element meshes: (a) conventional mesh and (b) exploded mesh out of scale.
The triangular finite elements preserve the linear elastic behavior during the whole loading process and depend on the mechanical elastic properties of the material. Particularly, in the first loading step, the tendency to infinity of the tangent stiffness factor of all link elements is kept, which ensures the displacement compatibility between nodes.
In the implemented numerical procedure, the numbering of triangular finite elements of the exploded and conventional meshes are the same. However, the numbering of each set of overlapping nodes of the exploded mesh is associated with the node number at the same location of the conventional mesh.
The nodal stress components
If the positive maximum principal stress
Outscale sketch of the exploded mesh around node 6 of the conventional mesh: (a) exploded mesh in node 6 of the conventional mesh, (b) directional vector of the link elements near node 6 and comparative direction, and (c) angle between the comparative direction and the directional vector of a link element.
Since there is no perfect alignment between the comparative direction
The angle of the comparative direction
The side of triangular element normal to vector
The crack path is considered normal to direction of positive maximum principal stress at each material point of solid. However, the discontinuity path is traced on the sides of triangular finite elements in the numerical model. In order to reduce the difference caused by the mesh alignment, the comparative direction
Detailed discontinuity path in a mesh of triangular finite elements: (a) with correction and (b) without correction.
The three experimental tests developed by other authors are simulated by means of proposed numerical model. These tests correspond to notched beams with and without holes, subjected to transversal load in one or two points. The deformed shape of the finite mesh in the last loading step exhibits a path where the relative displacement between nodes is greater. This path represents the discontinuity in the solid, which is compared with the cracking pattern of the experimental test.
A simply supported beam of polymethylmethacrylate (PMMA) is subjected to load
Meshes of Case I of Example
Number of elements  Number of nodes  

Mesh 1  1413  765 
Mesh 2  4412  2310 
Mesh 3  9432  4869 
Sketch of the three points beam with nonconcentric notch. The measurements are given in millimeters.
Figure
Comparison between the discontinuity path in three finite element meshes and the crack path of the experimental test in Case I of Example 1.
Mesh 1
Mesh 2
Mesh 3
The simulation with mesh
In Case II,
Numerical and experimental crack path in Case II of Example 1: (a) comparison between the discontinuity path in numerical model and the crack path of experimental test [
Example 2 has the same material and external geometry of the previous example but includes three internal holes which are located between the notch and the load, as shown in Figure
Finite element meshes of Example
Case I  Case II  

Number of elements  Number of nodes  Number of elements  Number of nodes  
Mesh 1  2630  1398  2191  1172 
Mesh 2  10520  5428  10284  5311 
Mesh 3  23200  11846  16296  8360 
Sketch of the threepoint beam with notch and three holes. The measurements are given in millimeters.
A depth
Comparison between the discontinuity path in numerical model and the crack path of experimental test in Case I of Example 2.
Mesh 1
Mesh 2
Mesh 3
Other approaches as the element free Galerkin method [
Figure
Evolution of the discontinuity path in the numerical simulation of the finest mesh in Case I of Example 2.
10th step
30th step
Final step
Comparison between the discontinuity path in numerical model and the crack path of experimental test in Case II of Example 2.
Mesh 1
Mesh 2
Mesh 3
A concrete beam is subjected to the loads
Sketch of the beam with notch and two loading points. The measures are given in millimeters.
The notorious relative displacement between nodes of the deformed shape of the numerical model is shown in Figure
Numerical and experimental crack path in Example 3: (a) comparison between the discontinuity path in numerical model and the crack path of experimental test [
The works of Rots with smeared crack models [
The initial stiffness factor of link element
Results of the sensibility analysis of the discontinuity path to the initial stiffness of link elements.





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Example 
2  Y  2  Y  2  Y  0  N  Figure 
Example 
2  Y  2  Y  2  Y  0  N  Figure 
Example 
2  Y  2  Y  2  Y  0  N  Figure 
Example 
2  Y  2  Y  2  Y  0  N  Figure 
Example 
2  Y  2  Y  2  Y  2  Y  Figure 
Low values of initial stiffness of the link elements were not assigned to the simulations because these do not represent the displacement compatibility in the overlapping nodes before the formation of the crack.
Convergence criteria of residual forces were used for these nonlinear simulations with a tolerance of
The main conclusions of this work are as follows.
The formulation, implementation, and validation of a discrete numerical model which predicts the cracking pattern of a solid, considering infinitesimal strain, static loads, and plane stress condition, are presented in this paper. A nonlinear analysis with the finite element method is implemented.
This model assumes that the crack is produced between the sides of the linear triangular elements, connected by the link elements. The latter element type defines the activation of the fracture process in accordance with the magnitude and direction of the positive maximum principal stress. A link element with high stiffness represents the noncracking zone and a link element with null stiffness establishes the crack path.
The exploded mesh is generated from the conventional mesh one time in the simulation. This procedure separates the sides of the triangular finite elements and joins the adjacent nodes with the link elements, without modifying the original geometry.
Linear elastic behavior of the triangular elements and nonlinear behavior of the link elements in the exploded mesh are taken into consideration. Initially, the tangent stiffness of all link elements tends to infinity. In the following loading steps of the nonlinear analysis, the tangent stiffness of several link elements is reduced to zero in order to represent the crack path. The simplicity and stability of this procedure sustain a good approximation of the crack path, reduce the computational cost, and allow implementing it on a standard finite element code with few modifications. However, the structural response cannot be obtained because the cohesive law in the cracking zone was not considered.
A simple procedure of correction of the discontinuity path caused by the mesh alignment is presented. Here, the orientation of the discontinuity path is defined by the link elements with stiffness null. A link element loses the stiffness when its orientation is nearest to the comparative direction and the tensile strength is reached. The comparative direction is defined by the direction of the positive maximum principal stress. The difference between the link element orientation and the comparative direction is used in order to correct the comparative direction in the next set of overlapping nodes.
This procedure presents successful results of the discontinuity path of several examples of threepoint beam with nonconcentric notch, subjected to the simultaneous action of normal and shear stresses. The topology of the numerical discontinuity path depends on the size and orientation of the finite elements. However, the finest meshes of Examples 1 and 2 show that the difference with respect to experimental test is negligible. Example 3 shows lesser accuracy because of the strong curvature of the real crack. Generally, the numerical model exhibits a good approximation of the cracking pattern, using meshes of about 400 triangular elements at height of the beam.
The discontinuity paths of Examples 2 and 3 obtained with the finest mesh of the proposed model are similar to the ones computed by other authors using different numerical models. 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. In the proposed model, once the conventional mesh is modified, a nonlinear problem is solved with elastic triangle elements and inelastic link elements which are normally included on finite element software. 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. This value is limited by the precision of the computer. The exploded mesh procedure presented in this model avoids the regeneration or adaptation of the mesh during the formation of crack and consequently the computational cost is low.
The tests of structural members of brittle material reinforced with ductile material, for example, the reinforced concrete beams, exhibit several cracks. These tests could be simulated using the proposed numerical model. In future works, no models and others failure criteria of the fracture process could be implemented in order to describe the structural response.
The authors declare that they have no conflicts of interest.