# Non-Classical Model of Dynamic Behavior of Concrete
## 2 Model of Concrete
### Dynamic Strength Criterion  
The basis of the model is the method of determination of the initial dynamic yield surface. For this purpose, the integral criterion of dynamic strength for concrete in complex stress states, is applied in the form proposed by [PERSON] [40]:  
\[\int_{0}^{t_{d}}[\psi(t)]^{\alpha_{c}}dt=t_{c0} \tag{8}\]  
where  
\[\psi(t)=\frac{\sigma_{ij}(t)}{\sigma_{ij}^{0}}=\frac{\sigma_{int}(t)}{\sigma_ {int}^{0}} \tag{9}\]  
is the proportionality function of any unrestricted elastic stressing process \(\sigma_{ij}^{t}=\sigma_{ij}(t)\) to stress state \(\sigma_{ij}^{0}\) satisfying the equation of static limit surface \(F\!\left(\sigma_{ij}^{0},K=1\right)=0\), expressing also the proportionality  
Figure 1: Limit surface for \(K=1\), \(\varphi_{t}=0.1\), \(\varphi_{cc}=1.1\): (**a**) section through the octahedral plane; (**b**) section through the Rendulic plane; compression curve for \(\varphi=0\), tension curve for \(\varphi=\frac{\pi}{3}\); (**c**) plane stress state; (**d**) reduced plane stress state. of the stress intensity (invariant of stress deviator) \(\sigma_{int}(t)=\sigma_{int}\!\left(\sigma_{ij}^{t}\right)=\sqrt{3J_{2}}\) corresponding with the dynamic stress state function \(\sigma_{ij}^{t}\) to stress intensity \(\sigma_{int}^{0}\) corresponding with stress state \(\sigma_{ij^{\prime}}^{0}\) satisfying the equation of static limit surface. Using the assumptions from Equation (9), the proportionality function \(\psi(t)\) is determined based on the static yield surface according to Equation (1) for \(K=1\):  
\[\psi(t)=\frac{2\!\left[\frac{\tau_{0}}{\rho\left(\psi\right)}\right]^{2}}{2a \frac{\tau_{0}}{\rho\left(\psi\right)}-b\sigma_{0}}\left\{\sqrt{1-\frac{4\! \left(a^{2}-c\right)\!\left[\frac{\tau_{0}}{\rho\left(\psi\right)}\right]^{2} }{\left[2a\frac{\tau_{0}}{\rho\left(\psi\right)}-b\sigma_{0}\right]^{2}}-1} \right\}^{-1} \tag{10}\]  
Material constants \(a_{c}\) and \(t_{c0}\) were determined on the base of approximation of dynamic experimental results for uniaxial compression of concrete, and according to [PERSON] [40], have the following values:  
\[\alpha_{c}=17.75\,\ t_{c0}=0.180\ s \tag{11}\]  
Symbol \(\psi(t)\) denotes:  
\[\psi(t)=\left\{ \begin{array}{ccc}\psi(t)&for&t\leq t_{*}&or&\psi(t)\geq 1 \\ 0&for&t>t_{*}&if&\psi(t)<1\end{array} \right. \tag{12}\]  
and \(t_{*}\) is the time of attaining the static yield surface during the first stressing cycle, as shown in Figure 2a. Time instant \(t_{d}\) of attaining the dynamic limit surface \(F\!\left(\sigma_{ij}^{t_{d}},K=\psi_{d}\right)=0\) is determined by the stress state \(\sigma_{ij}^{t_{d}}=\sigma_{ij}(t_{d})\) satisfying the dynamic strength criterion for concrete according to Equation (8):  
\[\psi_{d}=\psi(t_{d})\geq 1 \tag{13}\]  
In Figure 2a,b, the interpretation of the dynamic strength criterion integration method is shown, both as the scheme of attaining the initial dynamic yield surface (Figure 2b), and during the variable stressing process (Figure 2a). Determination of the dynamic strength coefficient (13) allows its use for the assumption of identical dynamic strength hardening for basic stress states:  
\[\psi_{d}=\psi_{c}^{d}=\psi_{t}^{d}=\psi_{cc}^{d}\,\ \psi_{c}^{d}=\frac{f_{c}^{d} }{f_{c}}\,\ \psi_{t}^{d}=\frac{f_{t}^{d}}{f_{t}}\,\ \psi_{\alpha}^{d}=\frac{f_{cc}^{d} }{f_{cc}} \tag{14}\]  
where \(f_{c}^{d}\), \(f_{t}^{d}\), \(f_{cc}^{d}\) are the dynamic strengths of concrete for uniaxial compression, uniaxial tension, and biaxial compression, and further extension of this assumption into any stress state \(\psi_{d}=\frac{f_{ij}^{d}}{f_{ij}}\). Thus, the dynamic strength coefficient \(\psi_{d}\) can be treated as the scaling parameter defining basic dynamic strengths of concrete:  
\[\varphi_{c}^{d}=\psi_{d}\varphi_{c}\equiv\psi_{d}\,\ \varphi_{t}^{d}=\psi_{d} \varphi_{t}\,\ \varphi_{cc}^{d}=\psi_{d}\varphi_{cc} \tag{15}\]  
The proposed criterion of dynamic strength is used as the basic component of non-classical constitutive model of concrete.