# Non-Classical Model of Dynamic Behavior of Concrete
## 2 Model of Concrete
### Cracking and Crushing Mechanism  
The applied deformation model of concrete enables the simplified modeling of the failure mechanism. This mechanism results from the applied softening rule, which assumes the gradual loss of material load-capacity until attaining the residual stress state during the tension or compression processes. In Figure 4, basic cases of loading and unloading paths in the deformation processes of the assumed model of concrete are schematically shown. Three different states of the failure mechanism can be defined at the moment of attainment of the minimal (residual) value of the evolution parameter:  
\[ \begin{array}{l}K_{mt}=K_{m}\qquad\left\{\begin{array}{ccc} if&\sigma_{0}<0&\textit{cracking state}\\ if&0<\sigma_{0}<\sigma_{0}^{*}&\textit{semi-cracking state}\\ \end{array}\right.\\ K_{mc}=K_{mt}=K_{m}\quad if&\sigma_{0}\geq\sigma_{0}^{*}&\textit{crushing state}\end{array} \right\} \tag{38}\]  
The cracking state can be attained in the monotonic tension process (Figure 4a) or in the cyclic, convertible unloading from the compression process that reverses the loading in the tension process (Figure 4b,c). The cracking state does not reduce the compression strength, and the reloading process in compression is possible after closing the generalized, volumetric crack. The crack opening and closing mechanisms are determined by the following conditions:  
\[ \begin{array}{l}\varepsilon_{0}<\varepsilon_{0}^{*}\ -\textit{crack opening}\\ \varepsilon_{0}\geq\varepsilon_{0}^{*}\ -\textit{crack closing}\end{array}  \tag{39}\]  
where \(\varepsilon_{0}^{*}\) is the last converged limit volumetric strain corresponding to the instant of attainment of the residual stress state in the tension process. The semi-failure state, being analogical to the cracking state, is characterized by its capability to the re-compressive process if the current volumetric deformation is greater than the limit volumetric strain \(\varepsilon_{0}\geq\varepsilon_{0}^{*}\), attained at the instant of the semi-failure state of concrete in the previous cycle of deformation (Figure 4e,f). Figure 4: Schemes of approximate modeling of cracking and crushing for concrete (description in text): (**a**) crack opening in tension path 0-1-2-3-4 / re-compression after crack closing path 5-6-7-8 / unloading path 8-9 / secondary crack opening path 9-10 / secondary crack closing and re-compression path 11-12-13; (**b**) compression path 0-1-2 / unloading and tension path 2-3-4-5 / crack opening path 5-6 / secondary loading and crack closing path 6-7 / re-compression path 7-8-9-10; (**c**) advanced compression to material softening range path 0-1-2-3 / unloading with current deformation modulus and tension path 3-4-5 / crack opening path 5-6 / secondary loading and crack closing path 6-7 / re-compression path 7-8-9; (**d**) attaining of the crushing state in the material softening process with local unloading and re-loading paths; (**e**) advanced shear stress state: loading path 0-1-2-3 / unloading path 3-4-5 / loss of load capacity with associated tension path 5-6 / re-loading path 6-7 with associated compression or tension; (**f**) attaining of the semi-failure state in the shear stress state path 0-1-2-3 / loss of load capacity 3-4 / re-loading path 4-5 with associated compression or tension. The crushing state determines the total loss of stress carrying capacity of the concrete (Figure 4d). The assumed interpretation of the cracking and crushing mechanism can be described by means of the indicator defined as the unitary, active cross-section of concrete:  
\[\delta=\left\{ \begin{array}{ccc}\delta_{c}&if&\sigma_{0}\geq 0\\ \delta_{c}\delta_{t}&if&\sigma_{0}<0\end{array} \right. \tag{40}\]  
where \(\delta_{c}=1\) and \(\delta_{t}=1\) are the initial values of the partial indicators that change their values only once, just at the moment when the residual stress states are attained  
\[ \begin{array}{ccc}\delta_{c}=0&if&K_{mc}=K_{m}\\ \delta_{t}=0&if&K_{mt}=K_{m}\end{array}