## 2 The Modified Dynamic Constitutive Model of Polypropylene Concrete
### The Modified Z-W-T Nonlinear Viscoelastic Model  
The Z-W-T nonlinear viscoelastic model can be improved by considering how the dynamic mechanical properties of polypropylene concrete are affected by temperature and impact load under real-time high temperatures:  
1. The initial stage of the stress-strain curve of concrete under impact loading was nearly linear elastic [29], meaning that part I of Equation (1) can be approximately converted into a linear polynomial, as shown in Equation (2). \[E_{0}\varepsilon+\alpha\varepsilon^{2}+\beta\varepsilon^{3}\to E_{0}\varepsilon\] (2)
2. Part II of Equation (1) consists of two Maxwell element relaxation functions and have large differences in the relaxation time (\(\theta_{1}\) and \(\theta_{2}\)), where the Maxwell element with relaxation time \(\theta_{1}\) describes the mechanical behavior of the material at a low strain rate, and the Maxwell element with relaxation time \(\theta_{2}\) describes the viscoelastic behavior of  
Figure 1: The expression of Z-W-T nonlinear viscoelastic model. The expression of Z-W-T nonlinear viscoelastic model. (**a**) The main expression of the Z-W-T non-linear viscoelastic model. (**b**) The approximately converting of part I in Equation (1). (**c**) Re-place the low-frequency Maxwell element in the Z-W-T non-linear viscoelastic mode. the material at a high strain rate (the order of magnitude for \(\theta_{1}\) and \(\theta_{2}\) are 10-10\({}^{2}\) s and 10\({}^{-4}\)-10\({}^{-6}\) s, respectively). Other studies have shown that the mechanical properties of concrete material were obviously affected by the strain rate especially and they were sensitive at high strain rates [26].The strain rate of polypropylene concrete under the impact load 10\({}^{2}\) s\({}^{-1}\)[30] resulted in a short observation time; in this case, the low-frequency Maxwell element could not be relaxed, showing linear springs characteristics, whereas the Maxwell element with a relaxation time of \(\theta_{2}\) described the viscoelastic mechanical behavior of the material under high strain rate conditions. Therefore, a simple spring can be used to replace the low-frequency Maxwell element in the Z-W-T non-linear viscoelastic model in this situation (Equation (3)), and under a high strain rate, the Z-W-T nonlinear viscoelastic model can be expressed by Figure 1b. Equation (4) shows the equivalent treatment of two parallel elastomers and the adjusted Z-W-T nonlinear viscoelastic model (Equation (5)) is represented in Figure 1c. Although the final expression form of the modified Z-W-T nonlinear viscoelasticity model was similar to that of the Kelvin-Voigt model [31], the derivation processes of the two models were not the same. \[E_{1}\int_{0}^{t}\dot{\epsilon}(t)\exp\biggl{(}-\frac{t-\tau}{\theta_{1}} \biggr{)}d\tau\to E_{1}\epsilon\] (3) \[E_{0}\epsilon+E_{1}\epsilon\to E_{a}\epsilon\] (4) \[\sigma=E_{a}\epsilon+E_{2}\int_{0}^{t}\dot{\epsilon}(t)\exp\biggl{(}-\frac{t- \tau}{\theta_{2}}\biggr{)}d\tau\] (5)
3. As a heterogeneous material, polypropylene concrete contains a large number of random polypropylene fibers and pores [32]. Therefore, damage factors should be considered when studying dynamic damage constitutive models of polypropylene concrete under impact load and thermal conditions. From the perspective of continuous damage mechanics, polypropylene concrete is assumed to be a continuous medium [33]. The composite damage amount D was introduced to measure the degree of damage experienced by the polypropylene concrete. In this case, Equation (6) describes a relationship according to the principle of strain equivalence [34]. Additionally, the modified Z-W-T non-linear viscoelastic model with the damage variable D (Equation (7)) can be obtained by substituting Equation (5) into Equation (6). \[\sigma_{a}=(1-D)\sigma_{r}\] (6) where \(\sigma_{a}\) represents effective stress, \(\sigma_{r}\) represents the nominal stress, and \(D\) represents the damage variable.