# Modeling of Fatigue-Strength Development
### The Maturity Method
#### 1.2.2 Effects of Temperature on Concrete Strength Development  
The apparent activation energy, denoted as E in Equation (2), defines the dependence of the rate of concrete strength development on the cure temperature. Noting the challenge associated with accurate determination of the activation energy for the equivalent age function, [PERSON] and [PERSON] [46], proposed its replacement with the rate constant, k, which is given by the initial slope of the strength versus cure duration plots for concrete samples cured at isothermal temperature conditions. The simplified exponential relationship, Equation (3), describes the variation of the rate constant with cure temperature.  
\[\mathrm{k}=\mathrm{A_{o}e^{BT}} \tag{3}\]  
where,  
k = Rate constant, the initial slope of strength versus duration curve (1/days)  
A\({}_{\mathrm{o}}\) = The value of the rate constant at 0 \({}^{\circ}\)C (1/days)  
B = The temperature sensitivity factor, (1/\({}^{\circ}\)C)  
T = Concrete temperature (\({}^{\circ}\)C)  
In Equation (3), the temperature sensitivity factor, B, indicates the effects of the cure temperature on the rate constant and, by extension, the effect of temperature on the rate of strength gain at variable cure temperature conditions. The temperature sensitivity factor for a particular batch of concrete is determined by fitting the simplified exponential function, Equation (3), to the plot of the rate constant versus the cure temperatures. Replacement of the activation energy with the rate constant in the Arrhenius maturity function, Equation (2), yielded the simplified exponential maturity function presented as Equation (4).  
\[\mathrm{t_{e}}=\sum_{0}^{t}\propto\times\Delta\mathrm{t} \tag{4}\]  
where,  
\(\mathrm{t_{e}}\) = The equivalent age at the reference temperature (hours or days)  
\(\propto\) = Age conversion factor, given as \(\mathrm{e^{B(T-T_{r})}}\)  
B = The temperature sensitivity factor, (1/\({}^{\circ}\)C)  
T = Average concrete temperature during time interval, \(\Delta\mathrm{t}\) (\({}^{\circ}\)C)  
T\({}_{\mathrm{r}}\) = The reference temperature, (20 \({}^{\circ}\)C in Europe and 23 \({}^{\circ}\)C in North America)  
\(\Delta\mathrm{t}\) = Time interval (hours or days)  
The age conversion factor in Equation (4) defines the temperature sensitivity of the maturity function [46]. In a study of stiffness development in a foamed bitumen asphalt, [PERSON] et al. [47], observed that at the same equivalent age, foamed bitumen asphalt samples cured variously at 5 \({}^{\circ}\)C, 20 \({}^{\circ}\)C, and 40 \({}^{\circ}\)C had different stiffness values. This observation defies the traditional definition of maturity, which anticipates concrete with the same equivalent age to have the same stiffness values regardless of the cure histories. To address this shortcoming in the equivalent age method, [PERSON] et al. [47] proposedthe modification of Equation (4) by the introduction of cure temperature into it. This modification was guided by an observation that the ultimate stiffness of the foamed bitumen asphalt had an almost linear relationship with the cure temperature. The modification yielded a new maturity index given by Equation (5).  
\[\text{M}=\sum_{0}^{\text{t}}\propto\times\Delta\text{t}\times\text{T} \tag{5}\]  
where,  
M = Maturity index at age t (\({}^{\circ}\)C-hours or \({}^{\circ}\)C-days)  
\(\propto\) = Age conversion factor, given as \(\text{e}^{\text{B(T-T_{\text{r}})}}\)  
B = The temperature sensitivity factor, (1/\({}^{\circ}\)C)  
T = Average concrete temperature during time interval, \(\Delta\)t (\({}^{\circ}\)C)  
T\({}_{\text{r}}\) = The reference temperature, (20 \({}^{\circ}\)C in Europe and 23 \({}^{\circ}\)C in North America)  
\(\Delta\)t = Time interval (hours or days)