## 4 The Compressive Constitutive Relationship of Steel Slag Micronized UHPC
### Simulation Model Establishment and Verification  
To validate the rationality of the proposed constitutive relation models in this study, comprehensive nonlinear analysis of prism specimens was conducted using finite element software. Uniaxial compression tests were simulated of fibre-reinforced prisms using finite element software ABAQUS, with an analysis model of 100 mm \(\times\) 100 mm \(\times\) 300 mm prism (consistent with the size of the test specimen); the concrete unit was a three-dimensional Hexahedron reduction unit, which was divided by a global grid with a size of 10. In order to prevent the local concrete from being destroyed ahead of time due to stress concentration and to more realistically simulate the actual situation of uniaxial compression, the upper and lower ends of the prism were, respectively, provided with a size of 150 mm \(\times\) 150 mm \(\times\) 10 mm rigid pad simulation loading device with elastic modulus set to 2.08 \(\times\) 10\({}^{5}\) MPa. The upper and lower ends of the specimen were in contact with the rigid pad through rough friction, and the vertical loading point was coupled with the plane of the rigid pad end. The loading method adopted displacement control loading. The concrete expansion angle was taken as 30 degrees, the eccentricity was 0.1, the ratio of biaxial compressive strength to uniaxial compressive strength was 1.16, the influence parameter of concrete yield form was 0.667, and the viscosity coefficient was 0.005. In the calculation, the constitutive equation of concrete under compression adopted the above steel fibre-reinforced steel slag powder UHPC uniaxial compression fitting stress-strain curve relationship. The tension stress-strain relationship adopted the Constitutive equation suggested in Appendix C of GB50010-2010 Code for Design of Concrete Structures. The compression inelastic strain \(\varepsilon_{c}^{in}\) and tension cracking strain \(\varepsilon_{l}^{in}\) can be calculated according to the following Equation (18):  
\[\varepsilon_{c}^{in}=\varepsilon_{c}-\frac{\sigma_{c}}{E_{0}}\hskip 28.452756 pt \varepsilon_{c}^{in}=\varepsilon_{t}-\frac{\sigma_{t}}{E_{0}}\]  
where \(\varepsilon_{c}^{in}\) and \(\varepsilon_{t}^{in}\) represent the inelastic strain in compression and the tensile cracking strain, respectively; \(\sigma_{c}\) and \(\varepsilon_{c}\) are obtained from the uniaxial compressive stress-strain constitutive equation of steel fibre-reinforced steel slag micro powder UHPC mentioned above; \(\sigma_{t}\) and \(\varepsilon_{t}\) are obtained from [25]. The damage factor \(d\) refers to the CDP model damage factor calculation formula proposed by [PERSON] [31]:  
\[d=1-\sqrt{\frac{\sigma}{E_{0}\varepsilon}} \tag{19}\]  
where \(\sigma\) and \(\varepsilon\) obtained from the constitutive equation proposed in this article, and \(E_{0}\) is the initial elastic modulus of concrete  
In Figure 8(a), it can be noted that the simulated prism specimens exhibit the most significant damage in the middle of the side surface, with a radial distribution of damage intensity. Under the applied load, the central region experiences compressive failure first, and some specimens display an \"X\" pattern of failure, which aligns with the actual experimental observations. In Figure 8(b), it can be observed that after loading, the maximum stress and slight expansion deformation occur in the middle of the concrete on both sides of the specimen, while the stress and deformation at the ends of the concrete are relatively small. The stress distribution on the side surface of the specimen demonstrates a radial pattern, which accurately reflects the actual damage scenario. The partial uniaxial compressive stress-strain simulation curves and experimental curves of UHPC specimens mixed with steel fibre and steel slag powder are shown in Figure 10. It can be seen that the numerical simulation curves and the overall trend of the experimental curves are in good agreement with different fibre contents.