# FE-Study on the Effect of Gradient Concrete on Early Constraint and Crack Risk
## 2 Finite Element Model and Material Properties Used in Analysis (Materials and Methods)
### Material Properties
#### 2.2.3 Requirements for Durable Strong Surface Layer Concrete  
The surface of the concrete is exposed to freeze and thaw attack as well as carbonation. For high water saturation without de-icing agent exposure class XF3 applies according to EN 206 [2]. For cyclic wet and dry concrete XC4 applies. EN 206 [2] distinguishes between designed concrete and prescribed concrete. Although the concept of gradient concrete is better suited to the background of designed concrete, the material properties for the durable surface layer concrete are chosen in accordance to the recommendation for limiting values of concrete in EN 206, Annex F as shown in Table 1.  
To fulfill the requirements for the exposure classes the minimum cement content is 320 kg/m\({}^{3}\) and the maximum w/c-ratio is 0.5. The surface layer requires a high durability. In a performance-based design concept, the aim is to predict the carbonation depth in concrete depending on the concrete age. One concept that is based on experimental observations and theoretically derived by diffusion theory can be found in the literature [4; 5; 6; 39]. In this theory, carbon dioxide is diffused into concrete and reacts with non-carbonated calcium minerals in the concrete at the \"moving carbonation boundary\". That boundary is at a certain distance from the concrete surface. If the carbon dioxide content between the concrete surface and the moving boundary is assumed to be linear the flux can be described using a diffusion law [39], as shown in Equation (6):  
\[J=D\cdot\Delta c/x_{\alpha} \tag{6}\]  
\(J\dots\) flux of CO\({}_{2}\) into the concrete, kg m\({}^{-2}\) s\({}^{-1}\)  
\(D\dots\) diffusion coefficient (CO\({}_{2}\)), m\({}^{2}\) s\({}^{-1}\)  
\(\Delta c\dots\) difference in CO\({}_{2}\) content between the concrete surface and the moving boundary, kg(CO\({}_{2}\)) m\({}^{-3}\)  
\(x_{\alpha}\dots\) distance of the moving carbonation boundary from the concrete surface, m  
The carbon dioxide flux into concrete is balanced with the rate of mass growth of the bound CO\({}_{2}\), as shown in Equation (7):  
\[J=a\cdot dx_{\alpha}/dt \tag{7}\]  
\(J\dots\) flux of CO\({}_{2}\) into the concrete, kg m\({}^{-2}\) s\({}^{-1}\)  
\(a\)\(\dots\) CO\({}_{2}\) binding capacity of the concrete, g(CO\({}_{2}\)) m\({}^{-3}\)  
\(t\dots\) time, s \begin{table}
\begin{tabular}{c c c} \hline
**Exposure Class Requirement** & **XC4** & **XF3** \\ \hline Maximum w/c & 0.50 & 0.50 \\ Minimum strength class & C30/37 & C30/37 \\ Minimum cement content (kg/m\({}^{3}\)) & 300 & 320 \\ Minimum air content (\%) & - & 4.0 \\ Other requirements & - & -1 \\ \hline \end{tabular} \({}^{1}\) Aggregate in accordance with EN 12620 [38] with sufficient freeze-thaw resistance.  
\end{table} 
Table 1: Recommended limiting values for composition and properties of concrete to meet the desired material properties for exposure classes. Carbonation depth \(x_{\alpha}\) is assumed zero at time zero. Combining Equations (6) and (7) and integrating over time, the following solution is obtained, see Equation (8):  
\[x_{\alpha}=(2D\cdot\Delta c\cdot t/a)^{0.5} \tag{8}\]  
For constant conditions and constant material properties the depth of carbonation is therefore proportional to the square root of time. The units are changed to millimeters and years to be in a practical range of values, as can be seen in Equation (9):  
\[x_{\alpha}=k_{NAC}\cdot t^{0.5} \tag{9}\]  
\(t\dots\) concrete age, a  
\(k_{NAC}\dots\) carbonation rate (\(2D\ \Delta c/a\))\({}^{0.5}\), mm a\({}^{-0.5}\)  
To account for environmental conditions (humidity, CO\({}_{2}\) concentration, curing, wetting effects) additional factors are applied in [4; 5; 6]. According to [4; 5; 6] the performance-based resistance to carbonation is highly dependent on the cement type. Cement CEM I (high hydration energy) has good carbonation resistance while CEM III (low hydration energy) has a poor one.