# Two-Dimensional Mesoscale Finite Element Modeling of Concrete Damage and Failure
## 3 Automatic Generation of Cohesive Elements
The geometry to be discretized contains different components, e.g., mortar and aggregates, and one important aspect of the discretization process is that the finite element boundaries must coincide with different material surfaces and the continuity between their surfaces must be preserved. The free meshing technique is utilized during the discretization process to avoid restrictions on the model geometry. The geometry is meshed using three-node linear triangular elements which enable rather realistic crack propagation paths. After the discretization, the Abaqus FE mesh file is imported into the developed algorithm to generate cohesive elements. The material properties of the aggregates particles and mortar are obtained from [46] and the input parameters for different TSLs are obtained through the calibration process carried out in this paper. The material properties for mesoscale simulations of concrete material are listed in Table 2. Since the strength of aggregate particles is much higher than the other phases and the cracks normally do not propagate inside aggregates in normal concretes, the damage properties of aggregate particles are not defined. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Parameter** & **Aggregate** & **Mortar** & **TSL-ITZ** & **TSL-Mortar** & **TSL-Aggregate** \\ \hline Elastic modulus, \(E\) (GPa) & 72 & 28 & - & - & - \\ Poission’s ratio, \(v_{12}\) & 0.16 & 0.2 & - & - & - \\ Elastic stiffness, \(K_{u}\) (MPa/mm) & - & - & \(10^{6}\) & \(10^{6}\) & \(10^{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Material properties. The properties of aggregate particles and mortar matrix including Young’s modulus, Poisson’s ratio, and the \(\eta\) parameter are derived from the study conducted by [46]. The elastic stiffness of CIEs is referenced from [32]. The input parameters for the TSLs in relation to CIEs have been determined through a calibration process undertaken in the current study. Figure 4: Generation of three sets of cohesive elements at three different phases of concrete mesostructure. The cohesive elements are generated within the mortar matrix, inside the aggregate, and in the ITZ zone. ### Constitutive Behavior of CIEs

The basic idea behind the cohesive zone model as first proposed by [PERSON] [29; 47] is to study the behavior of the material in a zone that lies directly ahead of a traction-free crack tip that nowadays is called the _cohesive zone_ or the _fracture process zone_ (FPZ). [PERSON] called the FPZ the _terminal region_ and stated that since in this zone the two crack surfaces are very close to each other, they are attracted to each other with very large interatomic forces, and the degree of the forces depends upon their relative displacements. The first cohesive crack model to simulate discrete cracks in the fracture process zone (FPZ) of concrete was introduced by [PERSON] in 1976 [31]. To describe the mechanical behavior of FPZ, the _traction-separation laws_ (TSLs) are used. The TSLs represent the underlying relationship between the traction of a cohesive element and the corresponding relative displacement of its surfaces in the cohesive zone. Different types of TSLs have been developed for the description of engineering materials over time but all of them, in general, can be divided into two parts: the first part represents a functional relationship that describes the behavior of the material prior to the initiation of the damage and a so-called softening zone which represents the degradation of the material and its complete failure after the damage is initiated. The two parts are separated by the threshold stress \(\sigma_{max}\) above which the damage starts to initiate. The traction-separation law for mode-I and mode-II fractures can be represented, respectively, as follows [48]:

\[ \begin{split}\sigma_{n}&=f_{n}(\delta_{n})\\ \sigma_{t}&=f_{s}(\delta_{s}),\end{split}