# Microscopic Multiple Fatigue Crack Simulation and Macroscopic Damage Evolution of Concrete Beam
## 2 Microscopic Multiple Crack Simulation and Fatigue Crack Growth Law
### Fatigue Crack Growth Law  
The fatigue crack propagation law is formulated by [PERSON] and [PERSON] based on experiments [33], which points out the relationship between stress intensity factor and crack growth rate, namely  
\[\frac{da}{dN}=C(\Delta K)^{m}. \tag{1}\]  
Here \(da/dN\) refers to the crack growth rate. \(a\) is the current crack length, and \(\Delta K\) is the incremental stress intensity factor. \(C\) and \(m\) are the empirical fatigue constants obtained from material tests. In this paper, a modified Paris law is introduced to govern the growth behavior of microcracks. The whole crack growth process is divided into three stages as displayed in Figure 4.  
1. In the low rate zone (Stage I) \(\Delta K<\Delta K_{\text{th,1}}\), the crack is set not to propagate because the rate is very low. The fatigue threshold is chosen as \[\Delta K_{\text{th,1},J}=0.2K_{\text{C},J}.\] (2)
2. In the medium rate zone (Stage II) \(\Delta K>\Delta K_{\text{th,1}}\) and \(\Delta K<\Delta K_{\text{th,2}}\), the crack growth follows the Paris law [33]. \[\frac{da_{I}}{dN}=C(\Delta K_{I})^{m}.\] (3) According to the experiments, the value of \(m\) for concrete is generally between 1.0 and 2.5. In order to make the crack growth rate lies between \(10^{-5}\) mm/c and \(10^{-2}\) mm/c [12; 34], the empirical parameters C and m are adopted as \(C=10^{-8}\), \(m=1.5\).
3. In the high rate zone (Stage III) \(\Delta K>\Delta K_{\text{th,2}}\), the crack will quickly propagate in an unstable fashion, finally lead to rupture. For the convenience of numerical simulation, here in this zone, the growth rate is practically set to be \(1.2\times 10^{-3}\) mm/c. Also, the 2 nd fatigue threshold \(\Delta K_{\text{th,2}}\) is set to be \[\Delta K_{\text{th,2},J}=0.8K_{\text{C},J}.\] (4)  
Microcracks subjected to repetitive load, start to propagate, though advancement of some cracks can be ignored. Subsequently, these cracks may coalesce into several macrocracks and finally reach to the stage of final rupture. In the actual computing process, one 'computational step' is not equal to one 'load cycle step'. Since the crack shows a'significant' growth only after the load has been repeated  
Figure 4: Fatigue crack growth rate.  
for a certain number of cycles, \(\Delta a\)-controlled rather than \(N\)-controlled strategy is used here. In each step, marked by superscript \(i\), the growth length of the crack that has the highest rate, assuming to be the \(\beta^{i}\)-th crack, is controlled to be \(\Delta a^{i}\). The number of load cycles \(\Delta N^{i}\) is determined by the following formula  
\[\Delta N^{i}=\frac{\Delta a^{i}}{C(\Delta K_{\beta})^{m}}. \tag{5}\]  
Then, incremental lengths for all cracks in this step could be calculated as  
\[\Delta a_{I}^{(i)}=C(\Delta K_{I}^{(i)})^{m}\Delta N^{(i)}. \tag{6}\]  
Summing up to get the total fatigue life  
\[N=\sum\Delta N^{i} \tag{7}\]  
Displayed in Figure 5 is the flowchart of numerical algorithm for multiple fatigue crack simulation. It shall be emphasized that both macroscopic and microscopic models are created such that a multi-scale model is further created via interface linking. Fatigue crack growth law is subsequently adopted to study the multiple microcrack growth, coalescence and macro crack formation. After the final rupture is reached, result extraction and data analysis is made possible.