## 1 Introduction  
Gas-wave machines provide refrigeration through the movement of gas flow waves. These are widely used in industrial fields, such as natural gas dehydration, aviation engines, and food freezing and preservation [1; 2; 3]. The oscillating tube is the core component of the gas-wave machine. When an oscillating tube is manufactured using the hub and blade brazing method, the brazed joint is subjected to complex and alternating loads during operation, making it prone to fatigue failure [4]. In this paper, brazed joints with various angles were designed for the brazed parts of an oscillating tube. Based on the cohesive zone model (CZM) [5; 6; 7; 8], fatigue strength research on the brazed joints of gas-wave machine oscillating tubes has been conducted.  
[PERSON] [9] and [PERSON] [10] first proposed the application of a CZM to study the fracture behavior of brittle materials. [PERSON] [11] proposed a bilinear CZM suitable for crack propagation of brittle and metal alloy materials. [PERSON] [12] used the CZM to simulate fatigue crack propagation in low-carbon steel brazed joints and conducted strain-controlled fatigue tests on brazed samples. The results showed that the larger the strain amplitude, the faster the crack propagated along the brazed layer. [PERSON] [13] employed an irreversible CZM to simulate the fatigue crack growth in ETS (Compact-Tension-Shear) samples under different loading modes. The findings revealed that the crack growth rate during overload was approximately \(7\times 10^{-3}\) mm/cycle, significantly higher than that under baseline loading conditions. [PERSON] [14] developed a two-dimensional CZM subroutine to simulate the fatigue crack propagation of composite materials under normal,tangential, and mixed loading. This revealed the mechanisms of crack-tip opening, sliding, and mixed-mode propagation. [PERSON] [15] employed a bilinear cyclic CZM to investigate the high-cycle fatigue of 7050-T7451 aluminum alloy. By comparing the simulated and experimental S-N curve, it was found that the maximum error in the fatigue life at the 100,000 th cycle was 20%, indicating that the CZM was suitable for simulating the fretting fatigue of aluminum alloys and predicting their life cycle.  
[PERSON] [16] investigated the influence of environmental fatigue on the mechanical properties of composite materials through nanoindentation tests, incorporating a finite element model to facilitate the analysis.  
[PERSON] [4] studied the fatigue crack growth process in the interface layer of 304 stainless steel/T2 copper T-type brazed joints using a bilinear CZM. The results indicate that as the cyclic displacement increased, the crack initiation decreased, and the crack growth rate increased. In addition, the interface damage, crack length, and fatigue load increased with the thickness of the base material.  
At present, there is a significant amount of research on the interfacial strength of composites and metal alloys, but relatively little research has been conducted on the interfacial strength of butt-brazed joints at different angles. In this article, the oscillating tube of a gas-wave machine was manufactured using the hub and blade brazing methods. For the brazed parts of the oscillating tube, brazed joints with different butt angles were designed and prepared for tensile fatigue tests. Based on the bilinear CZM, a cyclic CZM VUMAT subroutine was developed. Using the ABAQUS finite element software (abaqus 6.14.4), the fatigue crack initiation and propagation of the butt-brazed joints were simulated. On this basis, the fatigue life of the DN285 \(\times\) 450 mm oscillating tube finite element model was verified, providing theoretical guidance for the design and manufacture of the brazed structure of the gas-wave machine oscillating tube.