# Simulation of Crack Propagation in Reinforced Concrete Elements
## 3 Crack Propagation Model (CPM)
### Distribution of Tensile Strength  
The experimental determination of tensile strength has been the subject of recent research (e.g., [8, 9]), and several indirect and direct testing methods exist. Uniaxial concrete tensile strength is determined most effectively using a direct (uniaxial) tensile test (Figure 4). To determine the scattering of the concrete tensile strength of a single batch, the statistical distribution of several direct tensile tests is required. This scattering is caused by a wide variety of reasons, such as flaws due to insufficient compaction, the presence of inhomogeneity and anisotropy caused by aggregates, as well as internal stress states caused by shrinkage and hydration of cement. To describe the distribution of tensile strength, a probability function is required. In [10], a logarithmic normal distribution is assumed. In contrast, [11] and some codes in [12, 13] assume a normal distribution, while a summary is provided in [14]. Both distributions are shown in Figure 5 (probability density function) for an expected tensile strength value of 2.0 N/mm\({}^{2}\) and coefficients of variation of \(v_{\text{k,1}}=10\%\) and \(v_{\text{k,2}}=20\%\). Figure 3: Cracked element in the CPM. It can be seen that both distributions are very similar for a coefficient of variation \(v_{\rm k,1}=10\%\), with an overlap of 96.2%. For \(v_{\rm k,2}=20\%\), the overlap is 92.4%, but the visible deviation in Figure 5 is significantly larger than for \(v_{\rm k,1}=10\%\). Consequently, the logarithmic normal distribution for \(v_{\rm k}<20\%\) can be approximately represented by a normal distribution. For ranges of variation for different expected values, a distinction must be made between a constant standard deviation and a constant coefficient of variation. For the concrete tensile strength--in contrast to the concrete compressive strength (standard deviation constant [15; 16])--the coefficient of variation is considered a constant quantity. With regard to the scattering of the tensile strength at failure, evaluations in [11; 17; 18] show a coefficient of variation \(v_{\rm k}<10\%\). In Model Code 2010 [12] and Eurocode 2 [13], the coefficient of variation of the tensile strength of the component is \(v_{\rm k}=18\%\) (see [4]). Similar results are demonstrated in [19], by considering the influences of different concrete mixtures [17]. Thus, for one concrete batch, a coefficient of variation \(v_{\rm k}<10\%\) seems to be realistic. For several concrete batches in a component or in the absence of tensile tests, \(v_{\rm k}=18\%\) may be reasonable. For the probability function of the tensile strength, however, a normal distribution can be assumed approximately in both cases. Figure 4: Direct tensile tests (cylinder \(\mathcal{O}150\times 300\) mm, \(\text{C}30/37\), \(d_{\rm g}=16\) mm). Figure 5: Comparison between the normal distribution and log-normal distribution of tensile strength, with coefficients of variation of \(v_{\rm k}=10\%\) and \(v_{\rm k}=20\%\). The aforementioned distribution function applies to the tensile strength values obtained from several uniaxial tensile tests and cannot be directly implemented to define a tensile strength distribution in the CPM to simulate a tensile bar. This is due to the fact that in uniaxial tensile tests (of unnotched specimens), the tensile strength applies to the weakest point of the specimen. In the following, this value will be designated as the fracture tensile strength\(f_{\text{ct,f}}\). Previous experimental investigations (e.g., [17; 20]) show a difference between the distribution of the tensile strength within a specimen and the distribution of the fracture tensile strength\(f_{\text{ct,f}}\). Regarding this point, [PERSON] [3] defined a term for the specimen tensile strength\(f_{\text{ct}}\), which is explained below. Figure 6a shows the fracture process in a uniaxial tensile specimen. The existing micro-cracks accumulate at the weakest point of the specimen shortly before the maximum load is reached [21].