## 4 Numerical Simulation
### Damage Criterion  
Since there is no material model for ECC composites in ABAQUS, the CDP and DP material models were tested as concrete material models to capture the ductile behavior of ECC. Also, the Damage Initiation Criterion and Element Deletion options [35] were utilized in these models for defining the material's tensile failure criterion [43]. The uniaxial compressive behavior of ECC is characterized by the stress-strain relationship presented by [PERSON] [44] as follows:  
\[\sigma_{c}=\frac{E_{0}\epsilon_{c}}{1+\left(\frac{E_{0}\epsilon_{p}}{\sigma_{p }}-2\right)\left(\frac{\epsilon_{c}}{\epsilon_{p}}\right)+(\frac{\epsilon_{c} }{\epsilon_{p}})^{2}} \tag{8}\]  
where \(\sigma_{c}\) and \(\epsilon_{c}\) are the compressive stress and strain, respectively. \(\sigma_{p}\) and \(\epsilon_{p}\) are the maximum stress and its corresponding strain, which were evaluated experimentally in this study and considered to be the compressive strength \(fc\)' (MPa) and 0.002, respectively. Furthermore, the elastic modulus (\(E_{0}\)) and Poisson's ratio (\(\
u\)) of the ECC material were estimated experimentally, as shown in Figure 4a,b.  
On the other hand, the problem in the case of uniaxial tension involves tensile cracking. The tension stiffening is defined according to [PERSON] et al. [45], with linear uniaxial tensile stress for the ECC up to the reaching of the tensile strength assumed to be 0.1 of \(f_{c0}\). The softening response is then defined by means of fracture energy \(G_{F}\) depending on the uniaxial compressive strength\(f_{c0}\) and the maximum coarse aggregate size \(d_{0}\). The fracture energy is defined as the energy required to create a unit area of the stress-free crack surface, which is size independent, instead of the descending branch of the  
Figure 12: Finite element modeling of drop weight impact test.  
stress-strain curve, which is size dependent [46]. Accordingly, the stress-crack opening displacement relationship proposed by [PERSON] [47] is utilized:  
\[\frac{\sigma_{t}}{f_{t}}=1+(c_{1}\frac{w_{t}}{w_{cr}})^{3}e^{(-c_{2}\frac{w_{t}} {w_{cr}})}-\frac{w_{t}}{w_{cr}}\Big{(}1+c_{1}^{3}\Big{)}e^{(-c_{2})} \tag{9}\]  
\[w_{cr}=5.14\frac{G_{F}}{f_{t}} \tag{10}\]  
where \(w_{t}\) is the crack opening width, \(w_{cr}\) is the crack opening width at tensile stress failure, \(\sigma_{t}\) is the tensile stress normal to the crack direction, \(f_{t}\) is the uniaxial tensile strength, and \(c_{1}\) and \(c_{2}\) are constants determined from tensile tests of ECC. The CEB-FIB [48] model is used in this paper to estimate \(f_{t}\) and \(G_{F}\):  
\[f_{t}=1.4(\frac{f_{c}^{\prime}-8}{10})^{2/3} \tag{11}\]  
\[G_{F}=(0.0469d_{a}^{2}-0.5d_{a}+26)(\frac{f_{c}^{\prime}}{10})^{0.7} \tag{12}\]  
where \(d_{a}\) is the maximum aggregate size. \(d_{a}\) is assumed to be zero in this study since no coarse aggregate was utilized. Once the stress-crack opening displacement relationship is known, the stress-strain relationship can be determined for each element. Pertinent ECC material parameters, including compressive strength, density, and tensile behavior, were implemented in the models using actual corresponding experimental values shown in Table 3. Also, the constitutive parameters of the CDP and DP models, such as dilation angle \(\Psi\) (ranges between 36\({}^{\circ}\) and 40\({}^{\circ}\)), plastic potential eccentricity \(e\) (recommended value \(e\) = 0.1), stress ratio \(f_{bo}\)/\(f_{co}\) (ratio between the compressive strength in bi-and uni-compression stress field, respectively), shape of the loading surface \(K_{c}\), and viscosity parameter \(V\), were estimated based on the recommended range of values by the ABAQUS manual [35] and Kmiecik and Kaminski [49].