## 2 Establishment of Constitutive Model  
The Zhu-Wang-Tang (ZWT) intrinsic model was proposed by [PERSON], [PERSON], and [PERSON], the famous Chinese mechanics scholars, which can simulate the nonlinear viscoelastic intrinsic relationship of engineering plastics in the range of quasi-static load to impact load well. The main application objects of the original ZWT principal structure model are engineering plastics, including epoxy resin, plexiglass PMMA, polycarbonate PC, nylon, ABS, and PBT, etc. Concrete and rock is a typical non-homogeneous material, and there are a large number of defects inside the material, such as particle boundaries, pores, fissures and soft media, etc. The influence of damage on dynamic strength needs to be considered under high-strain-rate conditions. For the original ZWT instantonal model, it contains eight parameters, and there will be more parameters after considering the damage, and some equations of the original ZWT instantonal models do not work under the high-strain-rate conditions. Therefore, it is necessary to improve and simplify the ZWT instantonal model so that it can be applied to non-homogeneous materials. The development of a damage type viscoelastic instantonal model for soft and hard materials under high-strain-rate conditions is of practical significance for the study of the mechanical response of non-homogeneous materials under high-strain-rate conditions. Instantonal models are models that describe the short-term mechanical changes of materials and are mainly used in the field of dynamics. Considering the strain hardening and strain rate hardening of the material in dynamic experiments, the existing Zhu-Wang-Tang constitutive model can be well expressed. The ZWT constitutive model is the equation expression used to express the impact resistance of materials, and the formula is as follows:  
\[ \begin{split}\sigma=\sigma_{e}+\sigma_{m1}+\sigma_{m2}& =E_{0}\varepsilon+\alpha\varepsilon^{2}+\beta\varepsilon^{3}+E_{1} \int_{0}^{t}\frac{\bullet}{\varepsilon}(\tau)\exp\Bigl{(}-\frac{t-\tau}{ \theta_{1}}\Bigr{)}d\tau+\\ E_{2}\int_{0}^{t}\frac{\bullet}{\varepsilon}(\tau)\exp\Bigl{(}- \frac{t-\tau}{\theta_{2}}\Bigr{)}d\tau\end{split}  \tag{1}\]  
\(\alpha\), \(\beta\), \(E_{0}\) represent the elastic constant of nonlinear spring; two integral terms represent the [PERSON] body with different relaxation degrees, where \(\theta_{1}\) represents relaxation time of the [PERSON] viscoelastic response at low strain rate, \(\theta_{2}\) represents the relaxation time of the [PERSON] viscoelastic response at high strain rate, \(E_{1}\), \(E_{2}\) represent the elastic constants of the [PERSON] body. The constitutive model can be expressed by the mechanical model composed of a spring sticky pot, as shown in Figure 1.  
In view of the actual mechanical properties under dynamic loading summarized in the experiment and the physical significance of the \"Zhu-Wang-Tang\" constitutive model, the following simplifications or improvements are made as follows:  
(a) Simplification I. For the first term of the equation of \(\sigma_{e}=E_{0}\varepsilon+\alpha\varepsilon^{2}+\beta\varepsilon^{3}\), the meaning is strain rate-independent equilibrium stress, which describes the nonlinear elasticity of the material. Often, the elastic part in dynamic experiments can be considered as linear. The trinomial of the first term can be simplified into monomial, that is, \(\sigma_{e}=E_{0}\varepsilon\); the nonlinear spring in the corresponding ZWT constitutive model becomes a linear spring. With this method, the constitutive model is simplified, and the simplified formula of the ZWT constitutive model is:  
\[ \begin{split}\sigma=\sigma_{e}+\sigma_{m1}+\sigma_{m2}& =E_{0}\varepsilon+E_{1}\int_{0}^{t}\frac{\bullet}{\varepsilon}( \tau)\exp\Bigl{(}-\frac{t-\tau}{\theta_{1}}\Bigr{)}d\tau+\\ E_{2}\int_{0}^{t}\frac{\bullet}{\varepsilon}(\tau)\exp\Bigl{(}- \frac{t-\tau}{\theta_{2}}\Bigr{)}d\tau\end{split}  \tag{2}\]  
(b) Simplification II.