## 2 Numerical Models for Concrete Cracking in Statics and Dynamics  
### Cracking Model in Statics  
The model proposed is based on two material parameters related to the concrete behavior in uniaxial tension: the tensile strength and the post-cracking dissipated energy. The tensile strength (\(f_{t}\)) indicates the initiation of the macrocrack propagation (macrocrack localization) and the post-cracking energy (\(G_{IC}\)) is directly related to the macrocrack propagation. This mode I critical fracture energy comes from the Linear Elastic Fracture Mechanics (LEFM) theory. The macrocrack propagates only when \(G_{IC}\) is fully dissipated in a volume element. The use of a damage parameter, in the model, has no physical meaning and its sake objective is to dissipate \(G_{IC}\) in the simplest way. So, this model is not a classical damage model. Indeed, in damage models, the zones that are not fully damaged are considered physically relevant and are used to determine some crack propagation and opening. Finally, \(f_{t}\) and \(G_{IC}\) are considered to be random parameters. These values depend on the size of the volume elements. They have been determined in the framework of previous experimental and numerical works [4; 5; 6; 7; 8; 9; 10; 11]. They are linked to the degree of heterogeneity of the mesh element, \(r_{e}\), defined by the following relation:  
\[r_{e}=\frac{V_{e}}{V_{a}} \tag{1}\]  
where \(V_{e}\) is the volume of the mesh element and \(V_{a}\) is the volume of the greatest aggregate size of concrete. The probabilistic properties of \(f_{t}\) and \(G_{IC}\) depend on its mean value \(\mu\) and the standard deviation \(\frac{\sigma}{\mu}\). For tensile strength, the mean value, \(\mu_{f_{t}}\), is written as  
\[\mu_{f_{t}}=a.(r_{e})^{-y} \tag{2}\]  
where \(a=6.5\) MPa. The parameter \(y\) is given by the following equation  
\[y=0.25-3.6\times 10^{-3}\bigg{(}\frac{f_{c}}{\alpha}\bigg{)}+1.3\times 10^{-5} \bigg{(}\frac{f_{c}}{\alpha}\bigg{)}^{2} \tag{3}\]  
where \(f_{c}\) represents the concrete compressive strength in MPa and \(\alpha=1\) MPa. The compressive strength is determined by performing a standard test on a 160 \(\times\) 320 mm cylinder specimen. The standard deviation, \(\frac{\sigma_{f_{t}}}{\mu_{f_{t}}}\), is given by  
\[\frac{\sigma_{f_{t}}}{\mu_{f_{t}}}=c^{-d} \tag{4}\]  
where \(c=0.35\) and \(d\) is evaluated as  
\[d=4.5\times 10^{-2}+4.5\times 10^{-3}\bigg{(}\frac{f_{c}}{\alpha}\bigg{)}-1.8 \times 10^{-5}\bigg{(}\frac{f_{c}}{\alpha}\bigg{)}^{2} \tag{5}\]  
In the model, \(\mu_{G_{IC}}\) does not depend on mesh refinement, but \(\sigma_{G_{IC}}\) does [4; 5; 6]. They are written as [4; 11]:  
\[\mu_{G_{IC}}=0.4f_{c}+110 \tag{6}\]  
\[\frac{\sigma_{G_{IC}}}{\mu_{G_{IC}}}=A\bigg{[}\frac{1}{d}\ln\bigg{(}c\frac{ \mu_{f_{t}}}{\sigma_{f_{t}}}\bigg{)}+\frac{B}{A}\bigg{]} \tag{7}\]where \(A=-8.538\) and \(B=70.88\) Finally, it is important to point out that it has been demonstrated that the use of the 3D probabilistic semi-explicit cracking model leads to mesh-independent results in statics [4; 5; 6]. Equations (2)-(7) have been validated for \(f_{\rm c}\leq 130\) MPa and a maximum aggregate size \(\geq 10\) mm. The general aspects of the model are schematized in Figure 1.  
As shown in Figure 1, the introduction of a damage parameter in the theoretical approach is arbitrary and serves solely as a method to dissipate \(G_{IC}\). This means that, in the proposed model, there is no relation between the nonlinear strain (softening branch of the stress-strain curve in tension) and real crack propagation.