# The Application of Numerical Ductile Fracture Simulation in the LBB Evaluation of Nuclear Pipes
## 4 LBB Analysis and Discussion  
Based on the LBB design philosophy, it is essential to ensure that pipe leaks can be detected by the monitoring system before the pipe completely breaks down. This requires the pipe material to possess sufficient toughness. Austenitic stainless steels such as TP304L exhibit excellent toughness, allowing them to maintain crack resistance even after crack initiation, thereby demonstrating stable crack propagation behavior. This characteristic provides a critical time window for leak detection and pipe repair, effectively preventing catastrophic failures. Under increasing load, the crack undergoes stable propagation until reaching a critical state. When the J-integral at the crack tip reaches the material's fracture toughness, the crack transitions into unstable propagation. This critical point marks the onset of unstable crack growth, beyond which rapid structural failure occurs. Therefore, accurately predicting the critical load level for unstable crack propagation is of significant engineering importance.  
The elastic-plastic fracture mechanics approach based on crack driving force is commonly employed to locate the unstable crack growth point [33; 34]. This method analyzes the mechanical behavior of cracked structures under external loading, where the applied J-integral characterizes the crack driving force, while the material's resistance to crack growth is described by the J-R curve (\(J_{R}=J_{R}(a)\)). During stable crack growth, the rate of increase in the crack driving force remains lower than that of the material's resistance. However, once the driving force's growth rate exceeds that of the resistance, unstable crack propagation occurs. This critical condition can be expressed as  
\[\frac{\partial J(M,a)}{\partial a}=\frac{\partial J_{R}(M,a)}{\partial a} \tag{8}\]  
and can be graphically represented using a crack driving force diagram (CDFD). Figure 13 shows the CDFD for a pipe with an initial crack length of 80 mm. The driving force \(J(M,a)\) is a function of applied load \(M\) and crack length \(a\), and \(J_{R}\) is the J-R curve of the cracked structure obtained from the fracture resistance test. Unstable crack growth occurs when the driving force \(J(M,a)\) is tangent to the J-R curve.  
Figure 12: Comparison of test and simulation results of full-scale pipes: (**a**) load–CMOD curves; (**b**) J-R curves.  
In this study, the driving force \(J(M,a)\) for a cracked nuclear pipe is obtained according to the GE/EPRI equation [35; 36]. Figure 14 shows the CDFD analysis based on the J-R curves from the fracture resistance test of a 0.5T-CT specimen, fracture resistance test of full-scale pipe (\(2\theta=120^{\circ}\)), and numerical simulation of full-scale pipe (\(2\theta=120^{\circ}\)). \(J_{inst}\) represents the point where the crack driving force curve (\(J(M,a)\)) intersects with the J-R curve. The determined values of \(J_{inst}\) and \(M_{max}\) are shown in Figure 15 It can be seen for the pipe with a large initial crack length (\(2\theta=180^{\circ}\)), the critical load levels predicted based on the J-R curve of the CT specimen are similar to those predicted based on the J-R curve of full-scale pipe, while for the pipe with a short initial crack length (\(2\theta=60^{\circ}\)), the use of the CT specimen to predict critical load levels will cause lots of conservatism. Figure 15 shows that the critical load levels predicted based on the numerical simulations agree quite well with those predicted based on the fracture resistance tests, which demonstrates the validity of numerical ductile fracture simulation in the LBB evaluation of nuclear pipes.  
Figure 14: CDFD analysis: **(a)** based on the J-R curves from the fracture resistance test of a 0.5T-CT specimen; **(b)** based on the J-R curves from the fracture resistance test of a full-scale pipe (\(2\theta=120^{\circ}\)); **(c)** based on the J-R curves from the numerical simulation of a full-scale pipe (\(2\theta=120^{\circ}\)).  
Figure 13: Crack driving force diagram (CDFD).