# Simulation of Crack Propagation in Reinforced Concrete Elements
## 3 Crack Propagation Model (CPM)
### Bond-Slip Relationship  
The bond-slip behaviour between the concrete bar and reinforcement bar is simulated in the CPM using springs with a stiffness equal to:  
\[\text{c}\ =\ (\tau_{\text{b}}(s)/s)\cdot u_{\text{s}}\cdot l_{\text{e}} \tag{13}\]  
where \(\tau_{\text{b}}(s)\) is the local bond stress; \(s\) is the local slip (relative displacement between concrete and rebar); \(u_{\text{s}}\) is the perimeter of the reinforcement bar; and \(l_{\text{e}}\) is the element length.  
A bond-slip relationship according to Equation (14) is chosen in the first instance, with which a large variety of bond laws can be integrated:  
\[\tau_{\text{b}}(s)\ =\ \tau_{\text{b,max}}\cdot\left(\frac{s}{s_{1}}\right)^{\alpha} \tag{14}\]  
This correlation corresponds to the ascending branch of the bond law according to Model Code 2010 [12]. For calculations in the serviceability limit state, using the ascending branch of the bond-slip relationship is sufficient according to [28]. This considers the expected maximum crack width of 0.4 mm, which corresponds to a slip of 0.2 mm at both sides of the cracks, which is clearly within the ascending branch of the bond-slip relationship.  
Figure 9: Element length \(l_{\text{e}}\) vs. correlation length \(L_{\text{x}}\) for (**a**) uncorrelated and (**b**) correlated distributions along the model axis.  
For the general form of the bond-slip relationship according to Equation (14), some reference values based on the technical literature are listed in Table 1 and are illustrated in Figure 10a for concrete strength class C30/37.  
Even if the real bond behaviour differs significantly from the bond-slip relationship in Equation (14), user-defined bond-slip relationships can be defined in the CPM using slip values and the corresponding bond stress values. Between the points, a linear interpolation is used to determine the missing values. The only prerequisite here is the starting point at the coordinate origin (\(\tau_{\text{b}}(s=0)=0\)). An exemplarily user-defined bond-slip relationship is shown in Figure 10b for a splitting failure.  
A reduction of the local bond stress in the vicinity of a separation crack, e.g., due to internal cracking, can be considered by using a linear reduction factor according to [12]:  
\[\tau_{\text{b,red}}(s,\ x)\ =\ \lambda(x)\cdot\tau_{\text{b}}(s) \tag{15}\]  
where: \(\lambda(x)\) is the factor for a reduction of the local bond stress at distance \(x\) from the separation crack; \(x\) is the distance from the crack; and \(\tau_{\text{b}}(s)\) is the local bond stress (e.g., according to Equation (14)).  
In Model Code 2010 [12], \(\lambda(x)\) is defined by:  
\[\lambda(x)\ =\ 0.5\cdot x/\phi\leq 1.0 \tag{16}\]  
As an alternative to the linear reduction factor according to Equations (15) and (16), it is also possible to take into account the zone of the disturbed bond using a regionally separated bond stress-slip relationship according to [33; 34; 35]. It must be noted that the numerical calculation effort and the computation time increase significantly with the used model.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Bond Conditions** & \(\mathbf{\tau_{\text{b,max}}}\) & \(\mathbf{s_{1}}\) & \(\mathbf{\alpha}\) \\ \hline MC2010 [12] & Good & \(2.5\cdot\sqrt{f_{\text{cm}}}\) & 1.0 & 0.40 \\ MC2010 [12] & All other cases & \(1.25\cdot\sqrt{f_{\text{cm}}}\) & 1.8 & 0.40 \\ [PERSON] [29] & Mean & \(0.22\cdot f_{\text{cm}}^{0.95}\) & 1.0 & 0.30 \\ König and Tue [30] & & \(0.31\cdot f_{\text{cm}}\) & 1.0 & 0.30 \\ [PERSON] [31] & Good & \(0.95\cdot f_{\text{cm}}^{2/3}\) & 1.0 & 0.12 \\ [PERSON] [32] & Good & \(0.45\cdot f_{\text{cm}}\) & 1.0 & 0.40 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters for the ascending branch of the bond–slip relationship for normal-strength concrete.  
Figure 10: Bond laws: (**a**) according to Equation (14) and Table 1; (**b**) user-defined.