# Fatigue Crack Growth in Managing Steel Obtained by Selective Laser Melting
## 3 FCG Analysis Based on Plastic CTOD Range
### Numerical Results  
Figure 6 presents a typical numerical result of CTOD as a function of the remote stress, obtained for an initial crack length \(\mathrm{a}_{0}=24\) mm, followed by 20 crack propagations (\(\Delta\mathrm{a}=160\)\(\mathrm{\SIUnitSymbolMicro m}\)); the CTOD was assessed at the first node located behind the crack tip, at a distance of 8 \(\mathrm{\SIUnitSymbolMicro m}\) from the tip, as is schematically indicated, which presents the most sensitivity to crack tip phenomena. The crack is closed between A and B, for relatively low loads, i.e., the CTOD equals zero. The load increase opens the crack at point B. After point B, the crack opening evolves linearly with load increase up to point C, which is the boundary of the linear elastic regime. The range of loads between the opening load and the onset of plastic deformation was used to predict the fatigue threshold, as will be described later. A progressive increase of plastic deformation is found between points C and D, achieving its maximum value for the maximum load. The load decrease generates reversed elastic deformation with the same rate observed during loading. The subsequent load decrease produces reversed plastic deformation. Figure 6 also plots the variation of plastic CTOD. Plastic deformation initiates at point C, achieving its maximum value at maximum load. The plastic CTOD range, \(\mathrm{\delta}_{\mathrm{p}}\), which is correlated with fatigue crack propagation rate, is shown in Figure 6. It should be noted that crack closure is implicit in the value of \(\mathrm{\delta}_{\mathrm{p}}\). Increasing the crack closure phenomenon reduces the effective range of stress, also reducing the total CTOD and the plastic CTOD. In the absence of crack closure, the entire load cycle is felt by the crack tip. The plastic CTOD range also excludes the elastic deformation, which is not supposed to affect FCG. Figure 7 presents the evolution of plastic CTOD range, \(\mathrm{\delta}_{\mathrm{p}}\), with load range quantified by \(\mathrm{\Delta}\mathrm{K}_{\mathrm{eff}}\) (\(=\mathrm{K}_{\mathrm{max}}-\mathrm{K}_{\mathrm{open}}\)), i.e., the effective range of stress intensity factor. The remote loads applied were those used in the experimental FCG tests of the SLM material. The numerical tests were repeated for different materials keeping constant the specimen geometry, the crack lengths and loading. The increase in the load level increases \(\mathrm{\delta}_{\mathrm{p}}\), as might be expected. However, for the SLM material, the plastic deformation level is relatively low, being about one order of magnitude lower than the values obtained for the other materials. In fact, the values of plastic CTOD range, \(\mathrm{\delta}_{\mathrm{p}}\), are lower than 0.1 \(\mathrm{\SIUnitSymbolMicro m}\). As can be seen in Table 3, this material has a relatively high yield stress, which explains the low level of plastic deformation. This smallness makes the results very sensitive to errors of material modeling or geometry. Anyway, there is a well-defined variation of \(\mathrm{\delta}_{\mathrm{p}}\) with load range. Figure 6: Evolution of Crack Tip Opening Displacement (CTOD) with load (\(\mathrm{a}_{0}=24\) mm; plane stress). The experimental results showed a relatively low influence of specimen thickness and stress on FCGR, which indicates that the crack closure phenomenon may have a low relevance. Figure 8 presents the evolution of crack closure with the crack length. The crack closure level was quantified by two parameters:  
\[\mathrm{U_{clos}}\;=\;\frac{\mathrm{F_{open}-F_{min}}}{\mathrm{F_{max}-F_{min}}}\times 100 \tag{5}\]  
where \(\mathrm{F_{open}}\) is the crack opening load, and  
\[\mathrm{R_{eff}}\;=\;\frac{\mathrm{F_{open}}}{\mathrm{F_{max}}} \tag{6}\]  
The first parameter quantifies the percentage of load range during which the crack is closed, while the second one is the fraction of crack opening load divided by the maximum load. The crack closure level increases with crack length up to 21% (or 25% for \(\mathrm{F_{open}/F_{max}}\)).