# Simulation of Crack Propagation in Reinforced Concrete Elements
## 3 Crack Propagation Model (CPM)
### Element Size and Autocorrelation  
The concrete bar of the CPM consists of a finite elements series, with each element being assigned a random tensile strength \(f_{\rm ct,i}\) according to the applied distribution. As shown in Section 3.2 (Figure 6), each tensile strength value is assigned according to a mechanically reasoned representative element size \(\Delta x\), which is often defined based on the maximum aggregate size \(d_{\rm g}\). Here, \(\Delta x\) is assumed to be in the range of \(2.d_{\rm g}\) to \(3.d_{\rm g}\)[3, 22, 23]. A maximum aggregate size of 16 mm results in a representative element size \(\Delta x\) of about 32-48 mm. The element size affects directly the prediction accuracy of the model and the accuracy of the crack spacing values, as a crack can only emerge in the mid-length area of each element. Therefore, to enable the simulation of fine cracking patterns, the element length \(l_{\rm e}\) should also be very small. To consider these aspects and simultaneously achieve a nearly mesh-independent model, a smaller element length \(l_{\rm e}\) is selected and the concrete tensile strengths of the adjacent elements within a correlation length \(L_{\rm x}\) are correlated to avoid mechanically inconsistent oscillation of the tensile strengths in the smaller elements. This approach is known as spatial autocorrelation [24]. The expression of the linear dependence or correlation between two points \(x_{1}\) and \(x_{2}\) is described by the correlation function \(\rho(x_{1},x_{2})\). A common form is the quadratic exponential (Gaussian) correlation function [4, 25, 26]. \[\rho(x_{1},x_{2})\ =\ e^{-(\frac{|x_{1}-x_{2}|}{L_{\rm x}})^{2}} \tag{7}\]  
Figure 8 illustrates the correlation function according to Equation (7). For \(\rho(x_{1},x_{2})=0\), there is no linear correlation between \(x_{1}\) and \(x_{2}\), whereas \(\rho(x_{1},x_{2})=1\) shows a full dependency of both points. Figure 7: Fracture tensile strength distribution and the corresponding specimen tensile strength distribution. By multiplying the correlation function with the standard deviations \(\sigma(x_{1})\) and \(\sigma(x_{2})\) (here \(\sigma(x_{1})=\sigma(x_{2})=\sigma\)), the value of covariance is obtained and the positive symmetric covariance matrix \(C\) can be composed:  
\[\mathbf{C}\ =\ \left[ \begin{array}{cccc}\sigma^{2}&\rho_{12}\sigma^{2}&\cdots& \rho_{1\text{m}}\sigma^{2}\\ \rho_{21}\sigma^{2}&\sigma^{2}&\cdots&\rho_{2\text{m}}\sigma^{2}\\ \vdots&\vdots&\ddots&\vdots\\ \rho_{\text{m}1}\sigma^{2}&\rho_{\text{m}2}\sigma^{2}&\cdots&\sigma^{2}\\ \end{array} \right] \tag{8}\]  
The generation of normally distributed correlated random numbers can be performed by means of the Cholesky decomposition. When using the Cholesky decomposition, normally distributed uncorrelated random values (\(a_{1},a_{2},\dots\ a_{\text{m}}\)) with a mean value of zero and a standard deviation of one must first be generated. These represent the size of the scattering area. The Cholesky decomposition of the correlation matrix leads to:  
\[\mathbf{C}\ =\ \mathbf{L}\cdot\mathbf{L}^{\text{T}} \tag{9}\]  
Subsequently, the normally distributed random values (\(a_{1}\), \(a_{2},\dots\ a_{\text{m}}\)) are multiplied with the triangular matrix \(\mathbf{L}\):  
\[\left[ \begin{array}{c}a_{1}^{*}\\ a_{2}^{*}\\ \vdots\\ a_{\text{m}}^{*}\end{array} \right]\ =\ \left[ \begin{array}{cccc}L_{11}&0&\cdots&0\\ L_{21}&L_{22}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ L_{\text{m}1}&L_{\text{m}2}&\cdots&L_{\text{mn}}\\ \end{array} \right]\cdot\left[ \begin{array}{c}a_{1}\\ a_{2}\\ \vdots\\ a_{\text{m}}\end{array}