## 4 Numerical Modeling: Justification of the Model of Fiber-Reinforced Concrete  
General Information about the Model of Plastic Behavior of Concrete with Damage Accumulation (Concrete Damage Plasticity Model)  
The concrete damage plasticity model is designed to describe the mechanical behavior of concrete under uniaxial, biaxial and volumetric stress conditions with insignificant values of lateral compression [20; 21; 22]. The mechanical behavior of concrete in the model is based on isotropic elastic damage in combination with isotropic plastic behavior for compression and tension that allows describing irreversible deformations in concrete. This approach is suitable because it is assumed that macrofibers in concrete has a chaotic orientation, and due to its sufficiently large content, it is possible for the models to assume the same hardening of concrete in all possible directions. Two main mechanisms of concrete destruction, namely, the formation of separation cracks and plastic destruction under compression, are considered in the model. This approach allows using the model under simple or cyclic static loads or dynamic loads. The model takes into account concrete reinforcement with individual rods, meshes or dispersed reinforcement. The dispersed reinforcement of concrete in the model is set in terms of the amount of energy required for the formation and full opening of a crack.  
Parameters used to set the properties of the concrete damage plasticity model with accumulated damages:  
* Initial (undamaged) Young's modulus \(E\);
* Poisson's ratio \(v\);
* Ultimate uniaxial compression strength \(f_{cm}\);
* Ultimate uniaxial tension strength \(f_{cm}\);
* Diagram of concrete deformations in the axes \"force versus CMOD\" from bending test (see Figures 2 and 3);
* Dilation angle \(\psi\) measured in the p-q plane at high confining pressure. The value should be between 0 and 30 degrees;
* The ratio of initial equibiaxial compressive yield stress to initial uniaxial compressive yield stress \(\alpha_{f}\). Recommended default value: 1.16 [23];
* Flow potential eccentricity \(Ie\). The eccentricity is a small positive number that defines the rate at which the hyperbolic flow potential approaches its asymptote. Recommended default value: 0.1 [23];  
Figure 3: Force versus CMOD from the bending test conducted on the specimen in a form of a beam reinforced by different macrofiber mass amount.  
* The ratio of the second stress invariant on the tensile meridian to that on the compressive meridian coefficient of the shape of plastic surface flow \(K_{c}\). Recommended default value: 0.667 [23]. The values of these parameters were determined based on tests (see Section 3), requirements of regulatory documents, numerical modeling [24; 25; 26; 27] as well as recommendations set out in the scientific and technical documents [28; 29; 30; 31; 32]. The parameters of the model of plastic behavior of concrete with damage accumulation are summarized in Table 2.  
Damage is usually characterized by the degradation of stiffness. An isotropic scaled damage model from the continuum damage mechanics is introduced in Abaqus to describe the stiffness degradation, which can be represented by Equation (1) under uniaxial loading [23]:  
\[\sigma\ =\ (1-d)\,E\Big{(}\epsilon-\epsilon^{pl}\Big{)}\,, \tag{1}\]  
where \(\sigma\) represents the stress, \(\epsilon\) and \(\epsilon^{pl}\) represent, respectively, the total and plastic deformation, \(E\) represents the initial (undamaged) Young's modulus and d represents the damage factor, which characterizes the degradation of the elastic stiffness and has values in the range between 0 (undamaged) and 1 (fully damaged).  
The diagrams in the axes \"tensile damage parameter versus CMOD\" based on experimental data of CMOD tests are shown in Figure 4. These diagrams are directly used by the concrete damaged plasticity model as additional parameters to correctly simulate the accumulation of damage in concrete.