**1. Introduction**

Ground granulated blast furnace slag (GGBFS) is widely used in various engineering applications to replace the ordinary Portland cement (OPC) [1–3]. However, GGBFS is sensitive to curing conditions and exhibit slow strength development [4–8]. In order to predict the strength behavior in the concrete with GGBFS, much research have been performed based on cement hydration phenomena [9]. Moreover,

it is necessary to estimate the mechanical properties of GGBFS concrete, such as compressive strength, splitting tensile strength, elastic modulus, creep, shrinkage, etc. Among the mechanical properties used in the design, compressive strength is the most important. There are different models for strength development, but among them, the maturity model is the best model. This model is based on the age concept. It has been widely used to evaluate the compressive strength of the concrete on the assumption of linear relationship with temperature [10–12] or a nonlinear relationship with the chemical reaction rate of the cement [13]. By considering curing temperature range and the accuracy of the prediction result, the equivalent age model is widely used for the interpretation of strength development, which incorporates the chemical reaction rate of the cement [13]. In the chemical reaction rate model, it is considered that apparent activation energy (*Ea*) is a key parameter with the curing temperature on the hydration reaction [14–20]. The *Ea* is indirectly proportional to compressive strength, as suggested in ASTM C 1074-11 [21]. The application of *Ea* in the equivalent age model for normal concrete and the related strength prediction model is reasonably agreed with the test results. However, the concrete with GGBFS exhibits several differences compared to normal concrete in regards to the strength development attributed to a reduction of compressive strength in early age, caused by retardation of the setting. When the GGBFS comes into contact with concrete pore solution, the impermeable acid film surrounds the particles on the surface is destroyed; therefore, the chemical reaction in concrete would start. The delay in initial compressive strength of concrete with GGBFS is owing to insufficient alkalinity of cement paste [4–6,22,23]. In addition, Escalante et al. [4,5] studied the hydration of Portland cement with GGBFS under curing conditions. The hydration reaction was measured for six months at 10 ◦C, 30 ◦C and 50 ◦C curing temperature of cement pastes with 30%, 50% GGBFS replacement in 0.5 and 0.35 W/C ratio. The highest hydration reaction rate was found with 30% GGBFS at 50 ◦C in 0.5 W/C, while the lowest was shown by 50% GGBFS replacement at 10 ◦C in 0.35 W/C. Thus, a prediction of concrete compressive strength development with GGBFS by using a full maturity model by evaluating *Ea* is required. The present study is aimed to predict the compressive strength development of the concrete with GGBFS by calculating the *Ea*.

#### **2. Prediction of the Compressive Strength Based on Maturity Theory**

#### *2.1. Maturity*

Maturity is a function that quantitatively expresses the effect of curing temperature and time on strength development of the concrete. Therefore, the maturity theory by considering curing temperature and time function can be defined as:

$$M = \int\_0^t H(T)dt,\tag{1}$$

where *M* and *h*(*T*) is maturity and maturity function, respectively. *T* denotes curing temperature over the age of *t*.

The maturity function is expressed in linear Equation (2) considering temperature and the age [10], whereas, the equivalent age can be calculated by Arrhenius chemical reaction rate (Equation (3)) [12,13,24].

$$\sum\_{0}^{t} M\_{5} = \sum \left( T - T\_{0} \right) \Delta t\_{\prime} \tag{2}$$

where *Ms* is maturity at age *t. T* is the average temperature (◦C) of the concrete during the time interval, and *T*<sup>0</sup> is reference temperature (−10 ◦C) [11].

$$
\lambda k\_T = A \cdot \exp^{\frac{k}{R \cdot T}},
\tag{3}
$$

where *kT* is the reaction rate constant, *A* is proportionality constant, *Ea* is apparent activation energy (kJ/mol), *R* is gas constant (8.314 J/mol·K), and *T* is the absolute temperature (Kelvin). Thus, the equivalent age (*te*) can be calculated by the following equation:

$$t\_{\varepsilon} = \frac{\int\_{0}^{t} H(T)dt}{H(T\_{r})},\tag{4}$$

where *H*(*T*) is maturity function at different curing temperature (*t)*, whereas, *H*(*Tr*) represents the maturity function at fixed curing temperature, i.e., 20 ◦C (293 K). The equivalent age means curing time at standard temperature (20 ◦C). By considering *Ea*, *te* can be derived as:

$$t\varepsilon = \int\_0^t \exp^{\frac{E\_0}{\hbar} \cdot \left(\frac{1}{T\_r} - \frac{1}{T}\right)} dt,\tag{5}$$

where *T* is the average curing temperature of the concrete during a time interval and *Tr* is the absolute or fixed temperature, i.e., 20 ◦C (293 K).

The strength development analysis using maturity was carried out by considering curing temperature and duration. Other researchers have suggested a hyperbolic regression model (Equation (6)) which gives more accurate predictions compared to an exponential function [24–27]. In addition, the following model was implemented by introducing a third variable to explain the dormant period during the hydration process of cement.

$$S = \frac{S\_\text{\tiny\mu k} (t\_\text{\textit{t}} - t\_\text{0})}{1 + kr(t\_\text{\textit{t}} - t\_\text{0})}.\tag{6}$$

*S* is predicted compressive strength, whereas, *Su* represents the obtained experimental compressive strength at 28 days of curing. *kT* is the reaction rate constant at curing temperature (*T*). *te* and *t*<sup>0</sup> represent the equivalent age and the age when compressive development starts (final setting time), respectively.
