# Components of the Fracture Response of Alkali-Activated Slag Composites with Steel Microfibers
## Materials and Methods
### Identification of Material Parameters  
After the previously mentioned adjustment, the _F-d_ diagrams were utilized as input data for parameter identification with the aim of transferring the fracture test response data to the desired material parameters. The FiCubS application [40] developed by co-author [PERSON] was used for this purpose. The FiCubS application is used to simulate the performance of a fracture test in the three-point bending configuration on a fiber-reinforced composite prism with a notch.  
Using symmetry, half of the specimen is modeled as an elastic body connected to the plane of symmetry by boundary conditions, which can be released and replaced by cohesive forces. The release of a particular condition occurs after the tensile strength of the material is overstepped. Cohesive forces are dependent on displacements of the released ligament area and on the identified cohesive function. The used cohesive function consists of two components: cohesion provided by the matrix and by fiber resistance. The matrix cohesive function is modeled by the Hordijk function [41]:  
\[\sigma=f_{t}\left[\left(1+\left(c_{1}\frac{w}{w_{c}}\right)^{3}\right)e^{ \left(-c_{2}\frac{w}{w_{c}}\right)}-\frac{w}{w_{c}}\left(1+{c_{1}}^{3}\right) e^{-c_{2}}\right] \tag{1}\]  
where \(\sigma\) is cohesive stress, \(f_{t}\) is tensile strength, \(w\) is crack opening displacement, \(w_{c}\) is critical crack opening displacement, and \(c_{1}\) and \(c_{2}\) are material constants.  
The fiber resistance is modeled by the proposed function:  
\[\sigma=\sigma_{\text{max}}\left(1-\frac{1}{w_{cf}}\left(we^{p\left(w-w_{cf} \right)}+\left(w_{cf}-w\right)e^{-kw}\right)\right) \tag{2}\]  
where \(\sigma\) is the cohesive stress applied for displacement \(w\), \(\sigma_{\text{max}}\) is the approximate maximum cohesive stress, \(w_{cf}\) is the displacement limit where cohesion disappears and \(k,p\) are parameters determining the initial and finite slope. Initial and finite slopes are given by:  
\[\frac{\partial}{\partial w}\sigma\left|\right|^{w=0}=\frac{\sigma_{\text{max }}}{w_{cf}}\left(1+kw_{cf}-e^{-pw_{cf}}\right) \tag{3}\]  
Figure 4: _F-CMOD_ diagrams of alkali-activated slag (AAS) composites with various amounts of microfibers.  
\[\frac{\partial}{\partial w}\sigma\Big{|}^{w=w_{cf}}=\frac{\sigma_{\max}}{w_{cf}} \Big{(}1-\Big{(}pw_{cf}+1\Big{)}e^{w_{cf}}\Big{)}e^{-kw_{cf}} \tag{4}\]  
The total fracture energy \(G\) represented by this function is given by the relation:  
\[G=\sigma_{\max}\Bigg{(}\frac{1}{w_{cf}}\Bigg{(}\frac{1}{k^{2}}\Big{(}1-e^{-w_{ cf}k}\Big{)}+\frac{1}{p^{2}}\Big{(}1-e^{-w_{cf}p}\Big{)}\Bigg{)}-\frac{1}{k}- \frac{1}{p}+w_{cf}\Bigg{)} \tag{5}\]  
From the identified parameters of the model (seven independent values) it was necessary to determine the effective modulus of elasticity of the composite, the tensile strength of the composite, the fracture energy and the coefficient of transverse contraction.  
To illustrate, Figure 5 shows \(F\)-\(d\) diagrams for selected specimens of AAS composite with various amounts of steel microfibers. The following diagrams are plotted in the graphs: a measured diagram of composite reinforced by microfibers (COMP); a simulated diagram corresponding to the plain AAS matrix (MTX); an identified diagram of AAS composite reinforced by microfibers (ID).  
Figure 5: \(F\)-\(d\) diagrams for selected specimens of individual AAS composites with various amounts of steel microfibers: COMP—a measured diagram of composite reinforced by microfibers; MTX—a simulated diagram corresponding to the plain AAS matrix; ID—an identified diagram of composite reinforced by microfibers.