Interface strength investigations of 304 stainless steel/T2 red copper T-type brazed joint based on cohesive zone model

Gas wave refrigerators are a type of refrigeration equipment with broad application prospects. The oscillating tube, its core component, was manufactured by brazing to improve production efficiency. To study the brazing interface strength, a small 304 stainless steel/T2 red copper T-type brazed joint specimen was designed. The tensile process of the brazed joint was simulated using ABAQUS software based on the bilinear cohesive zone model (CZM), and the results were compared with the experimental tensile results. Damage critical stress and critical fracture energy of σmax = 40MPa and φc = 35 mJ mm−2, respectively, were obtained. A VUMAT subroutine was used to obtain the fatigue life of the specimen, and the errors with the experiment were 1.8% and 5.6%, respectively. It was proved that the CZM can simulate the tensile and fatigue loading processes of T-type brazed joints. The microstructure of the fatigue fracture of the specimen was analyzed using scanning electron microscopy. In addition, the effects of cyclic displacement and base metal thickness on the interface fatigue damage process were simulated and analyzed. The results showed that with an increase in cyclic displacement, the crack initiation life decreased and the crack growth rate increased; the interface damage, the crack length, and the fatigue load increased with the increase in base metal thickness.


Introduction
Gas wave refrigerators energy exchange is through shock wave. which have the advantages of high refrigeration capacity, small size, ability to run with liquid, and broad application prospects [1]. The oscillating tube is the core part of the gas wave refrigerator; in the past, it was mainly manufactured by integral machining, which was difficult to process. In this study, an oscillating tube was manufactured by brazing the hub and blade to improve the production efficiency.
During the operation of a gas wave refrigerator, fatigue cracks easily occur at the brazed joint between the oscillating tube and hub owing to prolonged gas shock. With the continuous effect of the load, the cracks expand and cause equipment damage. At present, research on gas wave refrigerators mainly focuses on the gas flow principle, structural optimization, refrigeration efficiency, etc [2][3][4][5][6]. However, research on the fatigue crack growth process of the brazing part of a gas-wave refrigerator is scarce. Therefore, it is of great significance to study the strength of an oscillating tube brazed joint for the safe operation of a gas wave refrigerator. Owing to the large volume of the gas wave refrigerator, it is difficult to conduct an overall analysis of the interface strength of the oscillating tube brazed joints through tests. In this study, a small specimen of the gas wave refrigerator brazed parts is designed, a cohesive zone model (CZM) is proposed, and the fatigue crack growth process of the interface layer of 304 stainless steel/T2 red copper T-type brazed joints is studied.
CZM was first proposed by Dugdale [7] and Barenblatt [8] as a tool for studying crack growth in the interface layer of materials. At present, according to different crack propagation modes and the relationship between the external force and displacement, there are five CZMs: cubic polynomial CZM, exponential CZM, trapezoidal CZM, constant CZM, and bilinear CZM. Bilinear CZM is widely used because of its simplicity and efficiency [9,10]. Eliáš [11] proposed a cyclic CZM to simulate the fatigue compression crack growth process of polycrystalline aluminum. The fracture dynamics equation was established, and the simulation results agreed well with the theoretical results. Rocha [12] studied the influence of different interface materials on the tensile strength of a lap joint through experiments and, based on the CZM, a numerical simulation was carried out. It was found that the error between the square stress damage equation and the experimental result was the smallest. Choi [13] proposed a fatigue damage model based on CZM, analyzed the influence of cohesive parameters on fatigue damage, simulated a three-point bending fatigue experiment, and compared it with experimental data. The fatigue number and cyclic load were in good agreement with the experimental results. Pirundi [14], based on the CZM and two-dimensional fatigue crack growth, compiled a three-dimensional fatigue subprogram to predict the fatigue crack growth of aluminum-bonded joints, which is close to the theoretical value. Ghovanlou [15] studied the fatigue crack growth of low-alloy steel copper base brazed joints, developed an ABAQUS Python program, coupled the cyclic damage evolution law to the bilinear CZM, and predicted the crack growth life of the joints. Nijin [16] used cyclic CZM to obtain the fatigue life and crack growth process of an aluminum alloy CTS specimen by adjusting the parameters of the fatigue equation, which was in good agreement with the experimental results. Wei [17] experimentally obtained the fatigue life of a 30CrNi2MoV material, verified the simulation accuracy based on cyclic CZM, and predicted the fatigue crack growth of the fracturing pump. Zhou [18] used an SS304/BNi-2 brazed joint to carry out tensile tests to obtain the critical fracture energy, determined the critical stress of the brazed joint by comparing simulation and test results, and predicted the peeling process of a brazed joint using CZM.
At present, research on CZM mainly focuses on the interface strength of composite materials and metal alloys, whereas research on the interface strength of T-type brazed joints is scarce. Tensile and fatigue tests were carried out on 304/T2 red copper T-type brazed joints in this study. The tensile process of brazed joints was simulated using CZM, and the VUMAT subroutine was compiled to study the fatigue crack growth process of brazed joints. The research results have a certain guiding significance in the manufacturing of brazed structures for gas wave refrigerators.

Specimen preparation
Referring to the ASTM D1876-08 T-peel test of the adhesive, a small T-type 304/T2 red copper brazed joint was designed using two L-type plates made of 304 stainless steel with a thickness t of 4 mm. The brazing filler metal was T2 red copper with a thickness h of 0.08 mm. Before brazing, the surface of the L-type plate was treated. Sandpaper was used to polish the connecting surface of the plate to improve the flatness, remove dirt, burrs, and oxide film on the surface, and ensure that the brazing surface has good wettability. Next, the polished L-type plate was cleaned with acetone solution to further remove residual impurities on the surface, and subsequently sealed for brazing. Tables 1 and 2 list the chemical composition of 304 stainless steel and T2 copper. Figures 1  and 2 show the structural dimensions of the T-type brazed joint specimen and T2 copper strip, respectively. Table 3 lists the specimen material parameters. Figure 3 shows a PX-10 electro-mechanical tensile fatigue testing machine. To investigate the static strength performance of the brazed joint, the T-type brazed joint test was conducted, with the displacement rate of 0.5 mm min −1 , and the load was stopped when it reached 6 mm.

Fatigue test
To obtain the fatigue life of T-type brazed joints under different cyclic displacements, the two cyclic loading displacements were taken as 0.6 mm and 0.65 mm. The cycle ratio R = 0, frequency f = 10 Hz, and the triangular waveform were selected. Figure 4 shows the load displacement curve of the tensile test of the specimen. It can be seen from the figure that the load first increases and then decreases with the increase in displacement, and the maximum loads are 1340.5, 1396, and 1317 N, respectively. When the load reached the maximum value, owing to the damage of the interface layer, the crack expanded after initiation, and the load decreased and became stable. The stable loads     Table 4 lists the fatigue cycle life values and cyclic loads of the specimens. Figure 6 shows the fracture morphology of the fatigue crack growth in the T-type brazed joint. As shown in the figure, the fatigue process is divided into three stages: crack initiation, stable crack growth, and transient crack breaking. In the crack initiation stage, owing to plastic fracture during fatigue loading, there were small holes and a small number of fatigue stripes at the fracture surface. In the stable crack growth stage, numerous compact and uniform fatigue stripes appeared. In the transient crack fracture stage, the number of fatigue stripes decreased, the number of dimples increased significantly, and the plastic fracture accelerated. At this time, the brazed joint underwent a transient fracture under cyclic loading, and the interface layer was completely destroyed.

. Selection of CZM
This study simulated the static tensile and fatigue loading processes of crack growth using a bilinear CZM. Figure 7 shows the bilinear CZM. Figure 7(a) shows the static loading process, and figure 7(b) shows the cyclic loading process. The curve is divided into two stages: a linear elastic stage and a damage stage. When the loading displacement is at 0 d d < < , 0 the material will not be damaged; at this time, the slope is the initial stiffness K, and the stress corresponding to the damage critical stress is s .

Static damage model
Considering that materials are subjected to loads in multiple directions, the square stress criterion was selected to determine the initial time of material damage, as follows [20,21]: where s , n t , s t t represent the stress in the normal, first tangential, and second tangential directions, respectively; s t t , , n s t max max max represent the damage critical stresses in the normal, first tangential, and second tangential directions, respectively; and the < > Macaulay symbol is represented as follows: In the damage evolution stage, the Benzeggagh-Kenane (B-K) criterion based on energy was used to characterize the crack growth, as follows [22]:

Determination of cohesive parameters
According to the parameter relationship of the bilinear CZM, three parameters must be determined: the initial stiffness K of the cohesive zone, the damage critical stress s , max and the critical fracture energy j . c The initial stiffness K is expressed as follows [23]: where E is the elastic modulus of the cohesive element and t is the thickness of the cohesive element. The damage critical stress s max based on the yield strength of the material R eL or tensile strength s b takes an appropriate value [24]. For elastoplastic materials, the total energy is obtained from the peel test [25]. For the T-peel specimens, the expression is as follows: where P is the stable load loaded on the specimen obtained through the experiments, and b is the width of the stripping arm. The total energy j consists of the critical fracture energy j c and the plastic dissipation energy j . 0 Because it is difficult to obtain the plastic dissipation energy j 0 from experiments, the value can be determined by combining the experiments and finite element software for the case of j c <j.

Implementation of cyclic CZM in VUMAT
A VUMAT subroutine was used to simulate the fatigue loading of the brazed joint. The total damage of the interface layer considers the static damage during the first cyclic loading and the fatigue damage during the subsequent cyclic loading, as follows: where D is the total damage, D s is the static damage increment during the first cyclic loading, and  D c is the fatigue damage corresponding to each strain increment. When the critical displacement of damage reaches, D = 1, the cohesive element was removed, and the crack was initiated and propagated.
When the loading displacement exceeds the critical displacement d , 0 the static damage increment was calculated as follows [26]: The calculation equation of fatigue damage evolution equation is as follows [27]: where  Du is the strain increment; Du is the current strain value; d 0 is the critical displacement; s is the current interface strength; a b C , , f are the fatigue damage parameters. H(x) represents the Heaviside function. It is defined as follows: The interface layer between the total damage D of the interface layer and the displacement is as follows: The VUMAT subroutine based on ABAQUS software was compiled and the following cohesive material parameters were inputted: Young's modulus, Poisson's ratio, damage critical stress, critical fracture energy, and initial stiffness. Next, the failure deletion of cohesive elements under external forces was calculated by external subprograms. The damage equation and failure criterion of the cohesive element were constructed to realize crack growth in the interface layer. The calculation steps are shown in figure 8. Figure 9 shows the finite element model and local grid diagram of the T-type brazed joint. The base metal uses the plane strain element CPE4R, and material parameters were determined according to table 3. The plastic behavior of 304 stainless steel was obtained according to the literature [18]; the interface layer uses the twodimensional cohesive element COH2D4, the parameters of which are defined in the next section. The base metal and the interface layer use Tie binding, the lower end of the model uses fixed constraints, and the upper and lower ends of the specimen apply horizontal constraints, vertical static and cyclic displacement loads were applied on the upper end of the model. The fatigue life and crack growth under cyclic displacement were calculated by the ABAQUS associated VUMAT subroutine. According to the proportional relationship between the cohesive element and the base metal element, the mesh size of 4:1 was selected. After the verification of mesh irrelevance, the base metal mesh size of 0.4 mm and the weld mesh of 0.1 mm determined.

Results analysis
According to equation (5), total energy j=184.1 mJ mm −2 . Under the condition of j c <j, the damage critical stress s max = 40 MPa and critical fracture energy j c = 35 mJ mm −2 were determined. Figure 10 shows a comparison between the tensile test and the CZM simulation results. The maximum simulated load was 1333.72 N, and the stable load was 944.66 N. Table 5 shows the error between the tensile test and simulation values. The maximum load error was 1.3% and the stable load error was 2.6%. This indicates that CZM can simulate the tensile process of brazed joints effectively.
In equation (8), the fatigue parameters were selected according to the values recommended in the literature [26]. Figures 11(a) and (b) show the comparison between the load life curve fatigue test and the simulated values under cyclic displacement U = 0.6 mm and U = 0.65 mm. It was found that the simulated cyclic load decreased uniformly with an increase in the number of cycles, indicating that the change trend of the simulated load curve under cyclic displacement was close to the test result. This was because more cohesive elements were deleted during the cycle, and the effective area of the interface layer decreased. Figure 12 shows a comparison between  the total number of cycles test and the simulated values, which is in good agreement with the test value. Table 6 presents the simulation errors of the fatigue cycle life test. The simulated life was 255000 cycles and 205000 cycles respectively with errors of 1.8% and 5.6%, respectively. This proves that the fatigue loading VUMAT subprogram can simulate the fatigue crack growth process of the interface layer of the brazed joint.

Fatigue crack growth analysis
The damage failure process of the interface layer cohesive element can be obtained using the VUMAT subroutine. Figure 13 shows the failure of the cohesive element at different stages with a cyclic displacement U = 0.6 mm. Figure 13(a) shows the crack initiation stage. The total damage of the first cohesive element D = 1 after 17500 cycles, failure deletion and crack initiation occurred. Figure 13(b) shows the crack growth over 60000 cycles. With the continuous accumulation of fatigue, the cohesive element was deleted and the crack length   increased. According to table 6, when the number of cycles reaches 255000, the cohesive element completely fails, and the brazed joint eventually experiences fatigue fracture failure.
The first cohesive element was extracted according to figures 14 and 15 shows a comparison of the fatigue crack initiation life of the first cohesive element under different cyclic displacements. It was found that with the increase in cyclic displacement, the total damage D of the cohesive element increased, resulting in the deletion of cohesive element failure and crack initiation of the interface layer in a short number of cycles. The crack initiation life of U = 0.65 mm is 17000 cycles. The damage accumulation of the cohesive element was accelerated owing to the increase in cyclic displacement, and the crack initiation life decreased by 2.9% compared with U = 0.6 mm. Figure 16 shows a comparison of the fatigue crack growth rates under different cyclic   displacements. It was found that with an increase in cyclic displacement, the damage to cohesive elements and the fatigue crack growth rate increased. The VUMAT subroutine combined with the CZM can be used to obtain the damage distribution of the interface layer of the brazed joint in different cycles. Figure 17 shows the orientation of interface damage distribution and figures 18(a) and (b) show the interface damage curves of different cycles at U = 0.6 mm and   U = 0.65 mm, respectively. As shown in the figure, the total damage D of the cohesive element increases with an increase in the number of cycles at both cycle displacements, the interface damage decreases gradually as the distance from the first cohesive element increases, and the interface damage increases for the same number of cycles when the cyclic displacement increases. The damage distribution conformed to the rule of maximum stress at the crack tip.

Effect of base metal thickness on fatigue life
The cycle displacement U = 0.6 mm, cycle ratio R = 0, and frequency f = 10 Hz were selected to analyze the influence of the base metal thickness of the brazed joint (t = 1, 2, 3, and 4 mm) on the fatigue life and crack  growth. Figure 19 shows the effect of the base metal thickness on the fatigue crack length. With an increase in the base metal thickness, the fatigue crack length also increased for the same number of cycles. When the number of cycles reaches 255000, the fatigue crack length increases from 0.9 to 10 mm, and the corresponding fatigue life is shortened. It is believed that because the strength and stiffness of the base metal of the brazed joint increased and the plastic deformation reduced after analysis, the brazed weld become the weak point of the brazed joint; therefore, the brazed weld is prone to tear failure under the same cycles. Figure 20 shows the effect of the base metal thickness on the fatigue crack growth rate. The fatigue crack growth rate increases with an increase in the base metal thickness. Figure 21 shows the distribution of the fatigue load for different base metal thicknesses. It was found that the maximum fatigue load increased with an increase in base metal thickness. However, owing to the increase in thickness, the fatigue crack length and load drop speed increased. Figure 22 shows the effect of different base metal thicknesses on the fatigue load drop rate. It was found that the rate of fatigue load reduction increased with an increase in base metal thickness. This is because, as the thickness increases, the stiffness increases, the plastic deformation decreases, and the crack growth length increases under the same number of cycles, leading to a decrease in the interface layer area and accelerated load descent.

Conclusions
In this study, a small 304 stainless steel/T2 red copper T-type brazed specimen was designed to study the fatigue crack growth process of the brazing interface between the hub and blade of the oscillating tube of a gas wave refrigerator The main conclusions are as follows.
(1)The load displacement curves of the specimen and the corresponding fatigue life under cyclic displacement were obtained experimentally. The fracture morphology of the fatigue specimens at different stages was analyzed using SEM images.
(2)From the simulation of the tensile process of brazed joints using CZM, the maximum load and stable load were 1333.72 and 944.66 N, respectively. Compared to the experimental results, the errors were 1.3% and 2.6%, respectively. The cohesive element parameters s max =40 MPa and j c = 35 mJ mm −2 were determined. It was demonstrated that the CZM can simulate the tensile process of T-type brazed joints effectively. Utilizing the VUMAT fatigue subroutine, the fatigue life under cyclic displacement was simulated, and the errors with the average value of the fatigue test were 1.8% and 5.6%, respectively. This proves the accuracy of the proposed subroutine.  (3)Under different cyclic displacements, it was found that the crack initiation life decreased, and the crack growth speed increased with an increase in cyclic displacement. Simultaneously, the interfacial damage of the brazing zone was analyzed. It was found that the crack-growth length and interface-layer damage increased with an increase in the number of cycles.
(4)For different base metal thicknesses, it was found that the crack growth length and crack growth rate increased with the increase in thickness, the stiffness increased, and the fatigue load also increased as a whole. However, owing to the increase in crack length under the same number of cycles, the load drop rate increased.