# The Application of Numerical Ductile Fracture Simulation in the LBB Evaluation of Nuclear Pipes
## 1 Introduction  
Since the 1980s, with the development of elastic-plastic fracture mechanics and experience in nuclear power plant operations, researchers have reached the conclusion that nuclear pipes show a large amount of leakage before breakage. It has also been proven that the leakage can be detected by leakage monitoring systems long before the pipe breaks down, and there will be enough time to fix or replace the cracked pipe; thus, major accidents can be prevented [1, 2, 3]. The design of nuclear pipes based on this conclusion is called the LBB design concept, in which it is not necessary to consider the dynamic effect due to pipe breaks, and the construction cost can be significantly reduced by eliminating unnecessary pipe with restraints and jet impingement devices. The application of the LBB design concept needs to determine the critical load level at which a cracked nuclear pipe suddenly breaks down, which requires the J-R curves of cracked pipes. The J-integral, as a parameter characterizing the energy release rate and a core criterion in elastic-plastic fracture mechanics, is widely used in crack propagation analysis of engineering structures, the fracture toughness evaluation of materials, and structural integrity assessments under complex stress conditions. Unlike the stress intensity factor (K), which is limited to linear elastic fracture mechanics (LEFM), the J-integral extends its applicability to nonlinear material behavior, making it an indispensable tool for evaluating fractures in ductile materials. By quantifying the energy per unit area associated with crackpropagation, the J-integral serves as a unified parameter for fracture toughness assessment, fatigue analysis, and failure prediction. Its path independence ensures robustness in numerical simulations and experimental measurements. The material's ability to resist crack propagation under elastic-plastic conditions can also be expressed using the J-integral, namely the J-R curve. In practical application, the specimens listed in ASTM E 1820 [4](e.g., CT specimens) are usually used to evaluate the J-R curves of nuclear pipes. Due to the high crack-tip constraint effect in these specimens with a deep crack length dominated by bending load, an LBB evaluation based on these specimens usually causes a lot of conservatism [5]. However, considering the high cost and implementation difficulties of full-scale nuclear pipe testing, two-parameter fracture constraint theories (e.g., J-Q) [6; 7; 8] have been developed to transform the J-R curves obtained from standard specimens to those of full-scale pipes. These theories can characterize the crack-tip fields and quantify constraint levels for various geometries and loading configurations in elastic-plastic material. However, lots of research shows that these theories lose their validity at large-scale yielding [9; 10]. Instead of standard specimens, several nonstandard specimens with a more similar crack-tip constraint to that of full-scale pipes have been developed and can more effectively predict the fracture behaviors of nuclear pipes [11; 12]. Nonetheless, lots of test work is still required to obtain the J-R curves, including the development of the specific test methodology and fixtures for nonstandard specimens. A numerical ductile fracture simulation provides an efficient approach for predicting the J-R curves of nuclear pipes. Many damage models have been developed for FE ductile fracture simulation [13; 14]. The widely used damage models can be primarily categorized as continuum damage mechanics (CDM) models or micromechanical damage models. CDM, also referred to as macroscopic damage mechanics, is a phenomenological approach based on continuum mechanics and thermodynamics. CDM employs internal variables to characterize the irreversible evolution of microstructural defects (e.g., microcracks, voids) within materials and also considers the damage process as a dissipative process coupled with plastic deformation, creep, and other mechanisms. [PERSON] et al.