## 2 Model and Methodology  
### Concrete Compressive Strength Development over Time Model  
A lot of researchers have introduced a relationship between the concrete compressive strength and curing time with different logarithmic equations [23; 33; 34; 35]. [PERSON], [PERSON] [23] adopted the following equation to predict the compressive strength of high-performance concrete, and then the results were compared to the results by ANN:  
\[f^{\prime}_{c}(t)=AY^{B}.[C\ln(t)+D]. \tag{1}\]  
where \(f^{\prime}_{c}(t)\) is the compressive strength of high-performance concrete at \(t\) days, \(t\) is age at test, Y is water-to-binder ratio and A, B, C, and D are regression coefficients. However, in practical concrete engineering, the strength evaluation of concrete is generally based on the strength of curing after 28 days. Many concrete properties have a great relationship with the strength after 28 days [36]. Besides, there are many factors affecting the 28-day compressive strength of concrete. Therefore, the model in this study is built based upon the 28-day compressive strength of concrete and the modified logarithmic relationship established in the literatures. Moreover, in order to reflect the characteristics of different kinds of concrete, the parameter \(\lambda\) is introduced to the model. The concrete compressive strength development over time model is written as:  
\[S_{t}=S_{28}(1+\lambda\ln\frac{t}{28}). \tag{2}\]  
where \(S_{28}\) is the concrete compressive strength after \(t\) days. Also, \(\lambda\) expresses the factor of the concrete compressive strength in a later period. As the development of concrete progressed, more and more kinds of gelled materials were used in concrete. In summary, there are four main kinds of gelled materials: cement, slag, fly ash and limestone flour. Also, the gelled materials greatly influence the compressive strength of concrete [36]. In general, the use of slag and fly ash is unfavorable to the early age compressive strength development of concrete compared to cement while it is beneficial to the long-term compressive strength development of concrete compared to cement. Thus it is reasonable that the 28-day compressive strength (\(S_{28}\)) is dictated by the contribution of cement, slag, fly ash and limestone flour. \[S_{28}=S_{28C}+S_{28 SL}+S_{28 FA}+S_{28 LF}. \tag{3}\]  
where \(S_{28C}\) is the 28-day concrete compressive strength contribution of cement, \(S_{28 SL}\) is the 28-day concrete compressive strength contribution of slag, \(S_{28 FA}\) is 28-day concrete compressive strength contribution of fly ash and \(S_{28 LF}\) is 28-day concrete compressive strength contribution of limestone flour. A positive linear correlation between the content and the 28-day compressive strength contribution of each gelled material means that the impact index of gelled material \(x\) (\(i_{x}\)) is equal to 1, \(x\) can be cement, slag, fly ash or limestone flour. \[S_{28x}=S_{x}q_{x}. \tag{4}\]  
where \(S_{28x}\) is the 28-day concrete compressive strength contribution of gelled material \(x\), \(S_{x}\) is the 28-day compressive strength of gelled material \(x\), \(q_{x}\) is the percentage content by weight of gelled material \(x\). Substitute Equation (4) into Equation (3) and we get Equation (5). The 28-day compressive strength of concrete (\(S_{28}\)) is derived as follows:  
\[S_{28}=S_{C}q_{C}+S_{SL}q_{SL}+S_{FA}q_{FA}+S_{LF}q_{LF}. \tag{5}\]  
where \(S_{C}\) is the 28-day compressive strength of cement, \(S_{SL}\) is the 28-day compressive strength of slag, \(S_{FA}\) is the 28-day compressive strength of fly ash and \(S_{LF}\) is 28-day compressive strength of limestone flour. \(q_{C}\), \(q_{SL}\), \(q_{FA}\), and \(q_{LF}\) are the percentage of each gelled material content which can be expressed as follows:  
\[q_{x}=\frac{m_{x}}{m_{C}+m_{SL}+m_{FA}+m_{LF}}\times 100\%. \tag{6}\]  
where \(m_{x}\) is the quantity of gelled material.