# Two-Dimensional Mesoscale Finite Element Modeling of Concrete Damage and Failure
## 3 Automatic Generation of Cohesive Elements
The quadratic stress damage criterion is employed to determine the initiation of damage. As represented in the following equation, damage is assumed to start to initiate when the function involving nominal stress ratios reaches a value of one:

\[\left(\frac{<\sigma_{n}>}{\sigma_{n}^{0}}\right)^{2}+\left(\frac{\sigma_{s}}{ \sigma_{s}^{0}}\right)^{2}=1, \tag{18}\]

where \(\sigma_{n}^{0}\) and \(\sigma_{s}^{0}\) are the tensile and shear cohesive strengths, respectively. When the stress state in the model satisfies Equation (18), with a further increase in the applied displacement, the normal and shear stress decrease. This stiffness degradation represents the start of the softening region. To represent the stiffness degradation, a scalar damage variable \(D\) is utilized. \(D\) illustrates overall damage in the model and captures all the damage mechanisms. With the further accumulation of damage, \(D\) monotonically evolves from 0 to 1 and when a complete failure occurs, its value is equal to 1. The normal and shear stress in the softening region can be obtained by:

\[ \begin{split}\sigma_{n}&=(1-D)\,k_{n0}\langle \delta_{n}\rangle\quad\text{(normal direction)}\\ \sigma_{t}&=(1-D)\,k_{s0}\,\delta_{s}\quad\quad\quad \text{(shear direction)}\end{split}  \tag{19}\]

For the linear softening, the evolution of the damage variable \(D\) can be calculated by the expression proposed by [50] as follows:

\[D=\frac{\delta_{m}^{f}(\delta_{m}^{max}-\delta_{m}^{o})}{\delta_{m}^{max}( \delta_{m}^{f}-\delta_{m}^{o})}, \tag{20}\]

where \(\delta_{m}^{0}\) and \(\delta_{m}^{f}\) denote the _effective displacement at damage initiation_ and _effective displacement at complete failure_, respectively, and \(\delta_{m}^{max}\) represents the maximum value of effective displacement at the complete failure. ment during the loading history. To obtain the effective displacements in Equation (20), the following expression proposed by [50] is utilized as follows:

\[\delta_{m}= \sqrt{<\delta_{n}>^{2}+\delta_{s}^{2}}, \tag{21}\]

The B-K fracture criterion [51] is utilized to capture mixed-mode fracture in composite and quasi-brittle materials. According to the B-K fracture criterion, the effective displacement at complete failure can be calculated as:

\[\delta_{m,f}=\frac{2}{K\delta_{m,0}}\Big{[}G_{C}^{I}+(G_{C}^{II}-G_{C}^{I}) \Big{(}\frac{G_{s}}{G_{n}+G_{s}}\Big{)}^{\eta}\Big{]}, \tag{22}\]

where \(G_{C}^{I}\) and \(G_{C}^{II}\) denote the fracture energies of mode-I and mode-II, respectively, and \(G_{n}\) and \(G_{s}\) represent the work done by tractions and their corresponding displacements in the normal and shear directions, respectively. According to [50], the magnitude of \(\eta\) can be set as 1.2 for quasi-brittle materials. As is clear from Equation (22), depending on the relative contribution of mode-I and mode-II fracture, the mixed-mode fracture energy varies. Figure 5a shows a schematic representation of the mixed-mode fracture criteria. In this paper, the finite element simulations employ traction-separation laws (TSL) with bilinear behavior to characterize the cohesive zone model. Damage initiation is determined by the quadratic stress damage criterion, while the evolution of damage in the softening region is represented by a scalar damage variable (D). To capture mixed-mode fracture behavior in composite and quasi-brittle materials, the B-K fracture criterion is applied. The provided equations define the constitutive relations utilized for the cohesive zone model in the simulations. ## 5 Results

In the numerical simulation of concrete, it is found that specimens with the same aggregate size distribution but with different spatial distributions show different results, especially in the softening zone. Therefore, a single numerical model is insufficient to study concrete's overall behavior under different conditions. In this paper, a comprehensive investigation is conducted by simulating ten distinct finite element models with random aggregate distributions.