## 4 The Compressive Constitutive Relationship of Steel Slag Micronized UHPC
### Determination of Constitutive Relationship  
To establish the constitutive relationship equation for the compressive behaviour of the concrete, different scholars have proposed various constitutive models. In this study, after a comprehensive analysis of different constitutive models, As shown in Table 6, we selected the constitutive equation proposed by the following researchers for comparison in order to find the most suitable formula for expressing the stress-strain relationship of steel fibre-doped steel slag micronized UHPC. This experiment uses the dimensionless coordinate \(x=\varepsilon\). The typical UHPC axial compression stress-strain curve is shown in Figure 7.  
Figure 6: Comparison between standard and experimental curves. Figure 7: Typical uniaxial compression stress–strain curve. Using the three aforementioned models, along with the model proposed in this study, non-dimensionalized nonlinear fitting of the experimentally obtained stress-strain curves was performed, as shown in Figure 8.  
1. The ascending segment of the Guo Zhenhai model exhibited a good fit with the experimental curve, with \(1<a<1.5\), while the descending segment displayed a large fitting deviation. 2. The [PERSON] model showed only a few instances of good fit in the ascending segment, with large deviations in other cases, and the stress in the descending segment exceeded that of the experimental curve. 3. The Ju Yanzhong model underestimated the stress in the ascending segment and exhibited a larger fitting deviation compared to the experimental curve, but it demonstrated a better fit and smaller deviations compared to the previous two models. Although the ascending segment of the Guo Zhenhai model showed a good fit, the value of parameter a1 did not exhibit a clear relationship with the steel fibre and coarse aggregate content, making it difficult to predict. Therefore, to construct a reasonable constitutive relationship model for the ascending segment of steel slag micronized UHPC under compression, a fourth-degree polynomial equation was fitted. The proposed constitutive model for the ascending segment of steel slag micronized UHPC is given by Equation (10):  
\[y=a_{0}+a_{1}x+a_{3}x^{3}+a_{4}x^{4} \tag{10}\]  
As shown in Figure 7, the boundary conditions for the equation are as follows:  
\[x=0,\;y=0;\;x=1,\;y=1,\frac{\mathrm{d}y}{\mathrm{d}x}=0 \tag{11}\]  
The parameters \(a_{0}\), \(a_{1}\), \(a_{3}\), and \(a_{4}\) are undetermined in Equation (10). Substituting Equation (11) into Equation (10),  
\[a_{0}=0\;\;\;\;\;\;a_{1}+3a_{3}+4a_{4}=0\;\;\;\;\;\;\;a_{1}+a_{3}+a_{4}=1 \tag{12}\] \begin{table}
\begin{tabular}{c c c} \hline
**Presenter** & **The Relationship Equation of This Structure** & **Parameter Meaning** \\ \hline [PERSON] [29] & \(y=\left\{\begin{array}{ll}\frac{Ax-x^{2}}{1+(A-2)x^{2}}&0\leq x<1\\ \frac{kx}{1+(k-2)x+x^{2}}&x>1\end{array}\right\}\) & (7) & \(V_{f}\): Steel fibre volume fraction \\ [PERSON] [29] & \(A=\frac{(2.99\sqrt{f_{c}}+14.27)(1.46+25.4V_{f}+0.00779f_{c})}{f_{c}}\) & \(k:\) descending segment parameters \\ \[PERSON] [30] & \(y=\left\{\begin{array}{ll}cx+(4-3c)x^{3}+(2c-3)x^{4},&0\leq x\leq 1\\ \frac{tx}{1+(t-2)x+x^{2}},&x>1\end{array}\right\}\) & (8) & \(V_{S}\): Steel fibre volume fraction \\ & \(c=1566\,V_{S}^{3}-7.778V_{S}^{2}+2.14V_{S}+1.318\) & \(t\): descending segment parameters \\ & \(t=-3444\,V_{S}^{3}+317.2V_{S}^{2}-4.106V_{S}+0.1128\) & & \\ [PERSON] & \(y=\left\{\begin{array}{ll}ax+(3-2a)x^{2}+(a-2)^{3},&0\leq x<1\\ \frac{x}{b(x-1)^{2}}&,&1\leq x\end{array}\right\}\) & (9) & \(a:\) Ascending segment parameters \\ & \(a=2.4-0.0125f_{c}.\;\;\;\;b=0.157f_{c}^{0.795}-0.905\) & & \\ \hline \end{tabular}
\end{table} 
Table 6: Table of constitutive equation relation.