## 4 Dynamic Damage Constitutive Model  
### Visco-Elastic Damage Constitutive Model (V-E Model)  
As a heterogeneous, anisotropic multiphase composite material, there were many initial defects such as micro-cracks and micro-holes in the interior of concrete, and the strength distribution was not uniform. In the process of dynamic loading, micro-cracks and micro-holes were formed, developed and aggregated to form macroscopic cracks, which eventually led to breakage. [PERSON] et al. [31] proposed that the concrete damage was caused by the development and accumulation of cracks, which was a continuous process. The distribution of cracks conformed to the Weibull distribution. Meanwhile, under the dynamic load, pore water in concrete had viscosity effect (Stefan effect) [32].  
For the convenience of analysis, it was assumed that concrete was divided into microelements with different defects. The size of microelements should be very small and meet the dual nature of size [33]. Their dimensions were large enough to contain enough microcracks, microholes and other micro information on the microscopic level. Meanwhile, their size was small enough to be considered as a particle of continuous damage mechanics on the macroscopic level.  
As the concrete unit had both viscosity effect and statistical damage characteristics, the BFRC specimens can be set up as a union of viscous component and damaged component, as shown in Figure 9.  
Where, \(\sigma\) is the axial stress applied to the microelement. From the parallel relation between the damaged component \(D\) and the viscous component \(\eta\), \(\sigma\) is the sum of the two parts' stresses, therefore;  
\[\sigma=E\varepsilon+\eta\dot{\varepsilon} \tag{2}\]  
According to the strain equivalence principle of [PERSON], the strain produced by the damaged material under the action of effective stress is equivalent to that produced by the same material under the non-destructive injury. Moreover, the damage only occurs on the spring element, and not on the viscous body. Accordingly, the following concrete damage constitutive model was established:  
\[\sigma=(1-D)E\varepsilon+\eta\dot{\varepsilon} \tag{3}\]  
The damage of concrete was continuous under the action of loads. Assume the strength of each microelement conforms to probability distribution \(\varphi(\varepsilon)\), then:  
\[\frac{\mathrm{d}D}{\mathrm{d}\varepsilon}=\varphi(\varepsilon) \tag{4}\]  
\(\varphi(\varepsilon)\) follows the three-parameter Weibull distribution:  
\[\varphi(\varepsilon)=\frac{m}{\alpha}(\varepsilon-\gamma)^{m-1}\exp[-\frac{ \left(\varepsilon-\gamma\right)^{m}}{\alpha}] \tag{5}\]  
Figure 9: Visco-elastic damage constitutive model.  
Table 4, and Figures 10 and 11 show that the model calculated curves of specimens fitted well with the test curves, showing that the constitutive model established can basically accurately describe the dynamic mechanical response law of BFRC.  
Figure 11: The comparison between test curves and fitting curves at different temperatures (fiber contents: 0.2%, impact velocity: 8.8 m/s). (**a**) Temperature: 20 \({}^{\circ}\)C; (**b**) temperature: 200 \({}^{\circ}\)C; (**c**) temperature: 400 \({}^{\circ}\)C; (**d**) temperature: 600 \({}^{\circ}\)C; (**e**) and temperature: 800 \({}^{\circ}\)C.  
Figure 10: The comparison between test curves and fitting curves with different fiber contents (temperature: 20 \({}^{\circ}\)C, impact velocity: 8.8 m/s). (**a**) Fiber content: 0.0%; (**b**) fiber content: 0.1%; (**c**) fiber content: 0.2%; (**d**) fiber content: 0.3%; (**e**) and fiber content: 0.4%.