## 2 RPC-FST Model Based on ALE Method
### Establishing the Model  
The Jones-Wilkins-Lee equation of state (JWL EOS) defines the pressure as a function of the relative volume \(V\) and internal energy \(E_{0}\). TNT has been established as material by using the JWL EOS method. It can be expressed as:  
\[P=A(1-\frac{\omega}{R_{1}V})e^{-R_{1}V}+B(1-\frac{\omega}{R_{2}V})e^{-R_{2}V}+ \frac{\omega E_{0}}{V} \tag{1}\]  
where \(A\), \(B\), \(R_{1}\), \(R_{2}\), and \(\omega\) are the parameters related to the TNT type. The air domain is established with the equation of state, which represents the relationship between the element pressure and the internal energy given in Equation (2). \[P=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+(C_{4}+C_{5}\mu+C_{6}\mu^{2})E \tag{2}\]  
where \(P\) is the pressure, \(C_{i}\) (\(i=1,2,\ldots,6\)) values are constant, and \(E\) is the internal energy per unit volume. The plastic kinematic model is an elastic-plastic model with kinematic and isotropic hardening. Therefore, the plastic kinematic model is selected for steel tube materials under impact load. The strain rate effect on steel material can be incorporated by the Cowper and Symonds (CS) model (Equation (3)), which scales the yield stress with the factor. \[DIF=1+(\frac{\dot{e}}{C})^{1/p} \tag{3}\]  
where \(\dot{e}\) is the strain rate. \(C\) and \(p\) denote strain rate parameters for the CS strain rate model. In this study, \(C\) and \(p\) are set as 6844 s\({}^{-1}\) and 3.91, respectively. This combination accurately predicts the dynamic response of concrete-filled steel tubular columns, as verified by tensile experiments [28]. The constants mentioned above will be explained in detail later in Section 3. RPC has extremely high compressive strength and fracture energy [29; 30; 31]. There is a lack of accurate model parameters to describe the characteristics of RPC materials in RPC-FST columns. The K&C model is considered a promising dynamic constitutive model [15], and thus can be used to reveal the material properties under impact and explosion loads. The K&C model is modified according to the characteristics of RPC without coarse aggregate. The K&C model is specifically used to calculate the concrete structural response under blast and impact loadings. It is comprised of two parts--volume response and deviatoric response. In volume response, the volume change of materials under different pressures is observed by applying stress. The equation of state *EOS_TABULATED_COMPRESSION correlates the pressure \(p\) and the volumetric strain \(\varepsilon_{V}\). In deviatoric response, three failure surface strength models--yield failure surface \(\Delta\sigma_{y}\), maximum failure surface \(\Delta\sigma_{m}\), and residual failure surface \(\Delta\sigma_{r}\)--have to be characterized via \(a_{0y}\), \(a_{1y}\), \(a_{2y}\), \(a_{0}\), \(a_{1}\), \(a_{2}\), \(a_{1f}\), and \(a_{2f}\). The deviatoric stresses remain elastic during loading or reloading until the stressachieves the yield failure surface. Then, the deviatoric stress further increases until the maximum failure surface is reached. Beyond this stage, the response can be perfectly plastic or softened to the residual failure surface. The damage function captures the hardening and softening behavior of three failure surfaces. According to the characteristics of 5% porosity of RPC material in RPC-FST, the modified K&C model process is shown in Figure 2.  
The dynamic increase factor (_DIF_) relationship proposed by [PERSON] [32] is adopted, which considers the strain rate effect of RPC. The _DIF_ of the RPC compressive strength can be expressed as in Equation (4). \[DIF=a\left(\frac{\dot{\epsilon}}{\dot{\epsilon}_{s}}\right)^{b}[\dot{\epsilon} ]\leq\dot{\epsilon}\leq 317s^{-1} \tag{4}\]  
where (\(\dot{\epsilon}\)) is the critical strain rate of _DIF_. If the strain rate is lower than (\(\dot{\epsilon}\)), the strain rate effect on the compressive strength can be neglected.