## 2 Multiscale Analysis of Fiber-Reinforced Concrete Structures
### Structural Modeling of Fiber-Reinforced Concrete Components: Concrete Damage Plasticity Model  
We use the concrete damage plasticity (CDP) model by [PERSON] et al. [25] to analyze the fracture behavior of keyed dry joints under shear. This material model, including the modifications by [PERSON] and [PERSON] [34], is available in the commercial software Abaqus [26] and has been used successfully to simulate damage in reinforced concrete structures [35; 36; 37].  
The CDP model combines isotropic elastic damage theory with isotropic tensile and compressive plasticity to model the inelastic behavior of concrete. This model is suitable for materials with quasi-brittle behavior, as well as for all kinds of elements (trusses, beams, plates, and solids). Furthermore, it is suitable for modeling for both plain and reinforced concrete and allows static, cyclic, and dynamic loads. Two main fracture mechanisms are considered in the CDP model: tensile  
Figure 2: Virtual testing of SFRC specimens: (**a**) double-edge-notched (DEN) uniaxial tension test; (**b**) uniaxial compression test. From [27].  
cracking and compressive crushing. The evolution of the failure surface is controlled by two hardening variables: tensile and compressive equivalent plastic strains, \(\overline{\varepsilon}_{c}^{pl}\) and \(\overline{\varepsilon}_{t}^{pl}\) respectively. The continuum damage formulation in elasticity [38] can be expressed as:  
\[\mathbf{\sigma}=(1-d)\mathbf{D}_{0}^{d}:\left(\mathbf{\epsilon}-\mathbf{\epsilon}^{in}\right) \tag{5}\]  
where \(\mathbf{\sigma}\) is the Cauchy stress tensor, \(\mathbf{d}\) is the damage variable (which can be defined both for tension and compression, \(d_{t}\) and \(d_{c}\), respectively), \(\mathbf{D}_{0}^{\text{el}}\) is the undamaged elastic stiffness tensor, \(\mathbf{\epsilon}\) is the total strain tensor and \(\mathbf{\epsilon}^{\text{in}}\) the inelastic strain tensor, which takes into account the irreversible nature of concrete cracking. For instance, in the case of uniaxial tension, the equivalent inelastic strain \(\overline{\varepsilon}_{t}^{in}\) is defined as:  
\[\overline{\varepsilon}_{t}^{in}=\overline{\varepsilon}_{t}^{ck}-\left(\frac{ d_{t}}{1-d_{t}}\right)\frac{f_{t}}{E_{0}} \tag{6}\]  
where \(f_{t}\) is the tensile strength, \(E_{0}\) the undamaged elastic modulus, and \(\overline{\varepsilon}_{t}^{ck}\) the cracking strain defined as:  
\[\overline{\varepsilon}_{t}^{ck}=\varepsilon_{t}-\frac{\sigma_{t}}{E_{0}} \tag{7}\]  
Although the application plotted in this work presents some important zones under compression (mainly due to prestressing), these are far from crushing whereas tensile cracking is the principal fracture mechanism. Such tensile behavior can be directly described by means of stress-cracking strain curves, for instance. In this work, we use a linear softening approximation defined in terms of the tensile strength (\(f_{t}\)) and fracture energy (\(G_{F}\)), which also reduces mesh-dependency [10].  
Finally, the yield surface of the model depends on four parameters: dilation angles (\(\psi\)), eccentricity (\(\epsilon\)), yield surface parameter (\(K\)), and compressive biaxial-to-uniaxial strength ratio (\(f\)).