## 4 Damage Constitutive Model
### Model Establishment  
To investigate the effect of the stainless steel slag dosage on the damage of the binder by utilizing Equation (3) in conjunction with the experimental stress-strain curves, the corresponding parameters were obtained through nonlinear regression using Matlab2022b. A detailed comparison of these calculated parameters with values derived from Equations (6) and (7) is presented in Table 3a. It is noteworthy that the small elastic modulus of the mixture resulted in variation coefficients exceeding one for parameters \(a\) and \(b\), indicating a discrepancy between the calculated and fitted values. This incongrurity suggests that the calculated parameters inadequately represent the experimental curves. Consequently, we decided to adopt the fitted parameters to accurately fit the experimental curves. Parameters \(E\), \(a\), \(b\), and \(t\) were substituted into Equation (2) and compared with the experimental outcomes. The finalized constitutive model, which includes parameter \(t\), is outlined in Table 3b. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & & & & \\ \hline
**Specimen Identification** & \(\mathbf{a}\)**Fitted** & \(\mathbf{a}\)**Calculated** & \(\mathbf{\mathrm{Ratio}}\) & \(\mathbf{b}\)**Fitted** & \(\mathbf{b}\)**Calculated** & \(\mathbf{\mathrm{Ratio}}\) \\
**Number** & **Value** & **Value** & **Value** & **Value** & **Value** & **Ratio** \\ \hline A-0 & 0.0072 & 0.0679 & 0.10 & 1.838 & 2.695 & 0.68 \\ A-3 & 0.0088 & 0.0554 & 0.16 & 1.601 & \(-\)2.367 & \(-\)0.68 \\ A-6 & 0.0108 & 17.2923 & 0.0006 & 1.389 & \(-\)40.601 & \(-\)0.03 \\ A-9 & 0.0105 & 0.0311 & 0.34 & 1.209 & 1.705 & 0.71 \\ A-12 & 0.0082 & 0.0058 & 1.43 & 1.111 & 0.717 & 1.55 \\ \hline Mean value & & & 0.41 & & & 0.45 \\ Coefficient of variation & & 1.28 & & & 1.69 \\ \hline \multicolumn{6}{c}{**(b)**} \\ \hline
**Specimen Identification** & \multicolumn{3}{c}{**Damage Constitutive Model**} & \(\mathbf{\mathrm{t}}\) & \(\mathbf{\mathrm{r^{2}}}\) & **RMSE 1**  \\
**Number** & \multicolumn{3}{c}{**Damage Constitutive Model**} & \(\mathbf{\mathrm{t}}\) & \(\mathbf{\mathrm{r^{2}}}\) & **RMSE 1**  \\ \hline A-0 & \(\sigma=\left\{\begin{matrix}861\varepsilon&\varepsilon<0.0066\\ 861\varepsilon\exp\left[-\left(\frac{\varepsilon-0.0066}{0.0072}\right)^{1.8 38}\right]&\varepsilon\geq 0.0066\\ \end{matrix}\right.\) & 0.0066 & 0.967 & 0.396 \\ \hline A-3 & \(\sigma=\left\{\begin{matrix}1232\varepsilon&\varepsilon<0.0070\\ 1232\varepsilon\exp\left[-\left(\frac{\varepsilon-0.0070}{0.0088}\right)^{1.6 01}\right]&\varepsilon\geq 0.0070\\ \end{matrix}\right.\) & 0.0070 & 0.869 & 1.55 \\ \hline A-6 & \(\sigma=\left\{\begin{matrix}1180\varepsilon&\varepsilon<0.0072\\ 1180\varepsilon\exp\left[-\left(\frac{\varepsilon-0.0072}{0.0106}\right)^{1. 389}\right]&\varepsilon<0.0072\\ \end{matrix}\right.\) & 0.0072 & 0.922 & 1.01 \\ \hline A-9 & \(\sigma=\left\{\begin{matrix}1130\varepsilon&\varepsilon<0.0074\\ 1130\varepsilon\exp\left[-\left(\frac{\varepsilon-0.0074}{0.0105}\right)^{1. 209}\right]&\varepsilon\geq 0.0074\\ \end{matrix}\right.\) & 0.0074 & 0.982 & 0.430 \\ \hline A-12 & \(\sigma=\left\{\begin{matrix}844\varepsilon&\varepsilon<0.0078\\ 844\varepsilon\exp\left[-\left(\frac{\varepsilon-0.0078}{0.0082}\right)^{1.111 }\right]&\varepsilon\geq 0.0078\\ \end{matrix}\right.\) & 0.0078 & 0.957 & 0.392 \\ \hline \multicolumn{6}{c}{**(c)**} \\ \hline
**\(\mathbf{p}\)-Value of \(\mathbf{a}\)** & \multicolumn{3}{c}{**\(\mathbf{p}\)-Value of \(\mathbf{b}\)**} \\ \hline A-0 & \(7.1434\times 10^{-153}\) & & \(5.8342\times 10^{-88}\) \\ A-3 & \(2.141\times 10^{-162}\) & & \(1.0824\times 10^{-82}\) \\ A-6 & \(5.6294\times 10^{-202}\) & & \(6.1464\times 10^{-130}\) \\ A-9 & \(5.0837\times 10^{-282}\) & & \(1.6415\times 10^{-220}\) \\ A-12 & \(6.8385\times 10^{-202}\) & & \(6.384\times 10^{-153}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: (a) Fitting and calculating values. (b) Damage constitutive model and correlation coefficients. (c) The \(p\)-value of the parameter.