## 3 Models
### Models for Concrete
#### 3.2.1 Early-Age Thermal Model  
Hydration is a thermo-activated phenomenon. The structure thus heats up under the effect of internal heat sources, especially due to its massive nature. Exhaustion of the reaction and exchanges with the outside cause the structure to cool until reaching equilibrium. At the same time, the material is subject to various volumetric variations, and the mechanical performance of the material improves. At this stage, two major assumptions were adopted:  
* the thermal problem was treated independently from the mechanical problem;
* temperature variations were assumed to be identical in each section of the beam (independence with respect to the longitudinal axis \(x\)).  
_Heat evolution_  
The three following primary equations in this problem [6] must now be solved:  
* The energy balance equation:  
\[C\overset{\bullet}{T}=\
abla(k\
abla T)+L\overset{\bullet}{\xi} \tag{1}\]  
\(\overset{\bullet}{x}=dx/dt\), in which \(T\) is the temperature (in \({}^{\circ}\)K), \(k\) the thermal conductivity (W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\)), \(L\) the latent hydration heat (J\(\cdot\)m\({}^{-3}\)), and \(C\) the volumetric thermal capacity (J\(\cdot\)m\({}^{3}\cdot\)K\({}^{-1}\)).  
* \(\xi\) is the degree of hydration, as given by the Arrhenius equation:  
\[\overset{\bullet}{\xi}=A(\xi)\exp\left(-\frac{E_{a}}{RT}\right) \tag{2}\]  
\(E_{a}\) is the activation energy (J\(\cdot\)mol\({}^{-1}\)), \(R\) the ideal gas constant (\(R=8.3145\) J\(\cdot\)K\({}^{-1}\cdot\)mol\({}^{-1}\)), and \(A(\xi)\) the chemical affinity (s\({}^{-1}\)).  
\(\xi\) evolves between \(\xi_{0}\), the percolation threshold (close to 0), and \(\xi_{\infty}\), the degree of hydration upon completion of the hydration reaction. To describe the corresponding state of the medium, the maturity \(M\) is introduced [13]:  
\[\text{For }\xi>\xi_{0}\ M=\left\langle\frac{\xi-\xi_{0}}{\xi_{\infty}-\xi_{0}} \right\rangle_{+} \tag{3}\]  
\(<>_{+}\) is the positive part operator. \(M\) evolves between 0 (beginning of the process) and 1 (end of hydration).  
In introducing the maturity concept into the expression derived by [PERSON] et al. (2014), the chemical affinity, \(A(\xi)\), is given by:  
\[A(\xi)=\frac{A_{i}+(A_{p}-A_{i})\sin\left[\frac{\pi}{2}\left(1-\left\langle\frac {M_{p}-M}{M_{p}}\right\rangle_{+}\right\rangle\right]}{\left[1+\xi\left\langle \frac{M-M_{p}}{1-M_{p}}\right\rangle_{+}^{4}\right]}-\left(\frac{A_{p}}{1+ \xi}\right)\left\langle\frac{M-M_{p}}{1-M_{p}}\right\rangle_{+} \tag{4}\]  
\(A_{i}\) is the initial affinity, \(\zeta\) governs the deceleration phase of the hydration process, and \(A_{p}\) is the maximum value of the affinity function attained for \(M=M_{p}\).  
* The external exchange is solved as follows: \[\varphi=h(T_{s}-T_{ext})n\] (5)  
\(\varphi\) is the heat flux (W\(\cdot\)m\({}^{-2}\)), \(n\) the vector normal to the surface, \(T_{s}\) the surface temperature (K), and \(T_{ext}\) the external temperature (K). \(h\) is the exchange coefficient whose value changes when the mold is removed from 2, calculated by taking into account the insulation produce by the formwork, to 10 W\(\cdot\)m\({}^{-2}\cdot\)K\({}^{-1}\)([PERSON], 2015).  
Figure 2a shows the results obtained after identifying the parameters (see Appendix A, Table A1 for their numerical values) involved in the temperature evolution both at the beam center and near the surface. A good level of agreement is obtained with the measurements performed in situ.