Quantitative study of high-frequency fatigue hole-edge stresses in nickel-based single-crystal superalloy

In this paper, the influence of air film holes on the stress distribution of nickel-based single-crystal superalloy blades is investigated, a simulated test piece is designed for the analysis and test of the stress distribution of air film holes, and a method of quantitative inverse inference of the stress at the edge of the air film holes is proposed. Finite element simulation of high circumferential vibration fatigue of the simulated test piece was carried out by ANSYS to obtain the gradient of stress distribution at the hole edge. The EBSD technique was used to measure the dimensions of the plastic zone in the region near the fracture, and the quantitative fatigue stresses were extrapolated using a crack tip plastic zone model with Tresca yield criterion. The compressive stress exerted by the femtosecond laser perforation method on the air film aperture wall was detected by the nanoindentation technique, and the residual stress was analysed using the S. Suresh computational model. The results of this study show that based solely on finite element simulation, it is not possible to completely simulate the effect of residual stresses caused by machining on fatigue fracture, which in turn leads to a lack of conformity between the results of finite element simulation and the results of the inverse plastic zone size extrapolation; however, the stress magnitude at the edge of the hole can be well computed by combining the stress concentration coefficients from finite element simulation with the stresses from the inverse plastic zone size extrapolation after taking into account the residual stresses from machining. The quantitative analysis results of this method have an error within 1.3 times the dispersion band compared to the test data.


Introduction
Nickel-based single-crystal high-temperature alloys are widely used in the aerospace industry due to their excellent high-temperature mechanical properties [1][2] .Nickel-based single-crystal high-temperature alloys have excellent thermal resistance properties, high strength at high temperatures, good corrosion resistance, excellent fatigue properties, and good organisational stability, which can meet the requirements of turbine blades in harsh conditions, and are widely used in the hot end of the modern turbofan engine, which is the main material for the manufacture of advanced aero-engine and gas turbine blades [3][4][5][6] .DD6 single-crystal high-temperature alloy is the second generation of nickel-based single-crystal high-temperature alloy independently developed by China [7][8] , which is usually used as the material for turbine rotor blades.Fatigue failure in the turbine blade is often related to vibration, but different from other fatigue, vibration fatigue test refers to the alloy and components in the vibration, impact, noise and so on dynamic alternating loads, so that the structure resonates with the fatigue damage phenomenon [9] .DD6 single-crystal high-temperature alloy as a new type of high-temperature alloys, its microstructure and chemical composition and the ordinary high-temperature alloys there are different, so its fracture characteristics and damage mechanism [9] .So its fracture characteristics and damage mechanism is also different from the unusual high-temperature alloys, so the vibration fatigue properties and fracture mechanism of DD6 nickel-based single-crystal high-temperature alloys have important significance in guiding the engineering application of DD6 single-crystal high-temperature alloys [10] .Under the continuous development of modern computer technology, the use of finite element simulation can solve a wide range of complex mechanical problems [11] .In this paper, we mainly apply the combination of finite element simulation and the size of the plastic zone to quantitatively back-propagate the fatigue stresses of high-frequency vibration of DD6 single-crystal superalloy.

Experimental
In this study, nickel-based single-crystal superalloy were prepared in a super directional solidification furnace through the helical crystal selection method with various compositions (as shown in Table 1) and crystallographic orientations set as [001].The as-cast single-crystal alloys were treated with a solution at 1300°C for six hours and underwent a two-stage aging treatment at 1120°C for four hours and 870°C for 32 hours.This was done to achieve the elimination of eutectic and reduce a certain amount of segregation and precipitation in the γ'-strengthened phases through aging.The simulated parts were designed according to the stresses on the edge of the air film aperture of a certain type of blade.The DD6[001] orientation high frequency vibration fatigue test was carried out, and the high frequency fatigue simulator fracture was obtained.The total length of the test piece is 120 mm, the angle of the air film hole is 40°oblique hole (angle with X-axis), and the diameter of the hole is 0.4 mm, as shown in figure 1a.Due to the quantitative analysis of the blade vibration stress, the loading of the simulated piece is also tested by vibration fatigue, and the 1st-order vibration mode of the test is as shown in figure 1b.The test temperature is 850 ℃, and the maximal stress in the working area is 320 MPa, and the stress ratio is R=-1.

Figure 1. (a)
Dimensioned view of simulated parts with (b) first order vibration pattern.The fracture was observed using an optical microscope and the fracture was cleaned by sonication in anhydrous ethanol for 15 minutes.A Zeiss scanning electron microscope was used to observe the fracture morphology.The specimens subjected to EBSD characterization were treated with a focused ion beam system (FIB) and vibratory polishing, respectively.Vibratory polishing specimens were first cut perpendicular to the fracture along the direction of main crack propagation by wire cutting, and then vibratory polished with silica suspension for 6 h after mechanical polishing treatment.Electron backscatter diffraction (EBSD) characterization was performed using a field emission scanning a b electron microscope GeminiSEM300 with Aztec data collection software.Finite element simulations were performed using ANSYS.Since the tests simulated high week vibration fatigue, the material was given to be built using orthogonal anisotropic model.shows the direction of crack extension at the pore edge of the air film after testing of the simulated part.Figure 2a demonstrates the stress distribution at the air film pore edge after modulus simulations have been carried out using ANSYS.As the simulation is for high week vibration fatigue, the material of DD6 nickel-based single crystal high temperature alloy is given after adopting the model of linear elastic orthogonal anisotropy.It can be found that after applying the vibration of R equals -1, the stress at the pore edge shows obvious stress concentration, and the law of stress gradient distribution is the same as that of the simulated part in figure 2b at the cracking location of the air film pore edge.The stress values of the finite element simulation at the position of the strain gauge patch during the test are taken as σ0, the stress values of the finite element simulation at each point along the direction of crack extension during the test are taken as σx (x=1,2,3), and the stress concentration coefficients at the corresponding points are taken as kt=σx / σ0, and the stress concentration coefficients change with the change of the length of the crack originating from the edge of the pore as shown in Figure 3.It can be found that after approaching about 0.8 mm, the stress concentration situation affected by the air-membrane pore basically disappeared.

Fracture morphology analysis
In order to determine the location of the source area of the fracture so as to analyze the process of crack extension and perform effective quantitative analysis, the port was observed by optical microscopy and scanning electron microscopy.As shown in Figure 4, it can be seen that the fracture originated from the side of the intersection angle between the air membrane pore and the outer wall, but the crack did not extend along the wall of the air membrane pore, which has the most serious stress concentration situation as simulated by the finite element in Figure 2, and it did not fracture along the plane parallel to the central axis of the air membrane pore as in the static tensile process.Through further microscopic observation, the fracture can be classified into source zone, extension zone and transient fracture zone according to the fracture morphology.The source area is the region wrapped within the first fatigue arc at the edge of the air film hole, in which no obvious fatigue strips are observed, and the whole is smooth, and the region is mainly affected by the fatigue crack sprouting, and after accumulating a large amount of energy, it reaches the crack rupture toughness KC and undergoes a rapid deconvolutional rupture, which leaves the plane behind; the expansion area is the area from the first fatigue arc at the boundary of the source area outward to the last fatigue arc, which contains a large number of parallel fatigue arcs, which are parallel to the last fatigue arc, and which contains a large number of parallel fatigue arcs.region contains a large number of parallel fatigue arcs and strips, fatigue strips are due to cyclic changes in stress during the fatigue process and leave the microscopic morphology; instantaneous fracture zone is the last fatigue arcs after all the region, the region consists of smooth fracture ramps and planes with obvious tear ridges, and in the width direction of the two sides of the fracture is mainly the expansion of ramps, which is due to the size difference between the width and the thickness, which This is due to the size difference between width and thickness, which leads to the cracks on both sides are more likely to expand along the plane of the maximum shear stress component, while in the middle of the fracture, because the cracks have already expanded for a certain distance in the fatigue sprouting and expansion stages, forming a gap of a certain width, leading to the crack gap becoming the dominant of the subsequent crack expansion in the thickness direction, which results in the instantaneous fracture planes leaving the tear ridges as the main morphology on the fracture.In the existing studies by other scholars, the static oxidation temperature of DD6 nickel-based single-crystal superalloy is above 1000°C, while in this study, obvious oxidation occurs in the DD6 nickel-based single-crystal superalloy at 850°C.Comparing the transient fracture zone and fatigue morphology region, it is not difficult to find out that the oxidation situation is related to the fracture zone and fatigue morphology zone.It is not difficult to find that the oxidation condition is caused by a combination of fracture exposure time, test temperature and stress.This point will also be corrected in the process of quantitative back extrapolation of fatigue fracture stresses.

Quantitative back extrapolation of fatigue stress
Based on the relationship between the size of the plastic zone and the stress intensity factor in Irwin's plastic zone model, and also based on the Tresca shear stress criterion, yielding occurs when the maximum shear stress is equal to the shear yield stress of the material under complex loading conditions, that is, the left side of 0.5(σi -σj) = τys (i,j=1,2,3) (takes the maximum value.In unidirectional stretching, 0.5(σi-0) = τys, then there is τys = 0.5σys.Thus, the Tresca criterion can be written as the left-hand side of 0.5(σi -σj) = σys (i,j=1,2,3), taking its larger value.Substituting each stress components of σ1=K Ⅰ /(2πr) 0.5 cos 0.5ɵ(1+sin 0.5ɵ) σ2=K Ⅰ /(2πr) 0.5 cos 0.5ɵ(1-sin 0.5ɵ) σ3= 0 (plane stress state) σ3= 2υK Ⅰ /(2πr) 0.5 cos 0.5ɵ (plane strain state) into 0.5(σi -σj) = σys (i,j=1,2,3), the equation for the boundary of the plastic zone is obtained, i.e., the relationship between the size of the plastic zone rp and the stress intensity factor K rp(ɵ) = K Ⅰ 2 /(2πσys 2 ) ［cos 0.5ɵ(1+sin 0.5ɵ)］ 2 (plane stress state) The above equation is obtained in the plane strain state by considering the cyclic loading coefficient [12] and the plastic constraint coefficient p.c.f after substitution into K = Yσ(πa) 0.5 to obtain the equation as follows: The increment value of σ = 4.25σys/Kt (τp/Y 2 σ) 0.5 The EBSD technique was used to characterize the fracture, and since the fracture surface as well as the oxide layer appeared at 850°C, a parameter of 0.5886 was taken as the correction factor (i.e., the molar volume ratio of nickel/nickel oxide) for the correction of the oxide layer thickness, and points were taken for the characterization along the schematic in Fig. 6a, and the results of the calculations are shown in Figs.6b and c.It can be found that the error in the quantitative back extrapolation of fatigue stresses by the size of the plastic zone of the fracture in conjunction with the finite-element simulation of the stress concentration factor, the error of quantitative inverse fatigue stress is within 1.2 times, and the error is within the tolerance range of quantitative analysis.

Conclusions
(1) By performing finite element simulations in ANSYS, a more accurate stress concentration factor can be obtained by using orthogonal anisotropy for the assignment of materials to nickel-based single crystal high temperature alloys due to the high week fatigue of test conditions.
(2) Since finite element simulation cannot show the residual stress caused by the process, the nanoindentation technique is used for the analysis of the residual stress at the edge of the hole, and the gradient distribution law of the residual stress can be obtained very accurately.
(3) After correcting for the effect of residual stresses brought about by the processing of the air film pore edges, the stress concentration coefficient Kt obtained from the plastic zone size rp and finite element simulation can be used to quantitatively invert the stress range of the rupture, with a computational error of less than 1.3 times.

3 .Figure 2 .
Figure 2. (a) shows the stress distribution at the hole edge obtained after finite element simulation in ANSYS.(b) shows the direction of crack extension at the pore edge of the air film after testing of the simulated part.Figure2ademonstrates the stress distribution at the air film pore edge after modulus simulations have been carried out using ANSYS.As the simulation is for high week vibration fatigue, the material of DD6 nickel-based single crystal high temperature alloy is given after adopting the model of linear elastic orthogonal anisotropy.It can be found that after applying the vibration of R equals -1, the stress at the pore edge shows obvious stress concentration, and the law of stress gradient distribution is the same as that of the simulated part in figure2bat the cracking location of the air film pore edge.The stress values of the finite element simulation at the position of the strain gauge patch during the test are taken as σ0, the stress values of the finite element simulation at each point along the direction of crack extension during the test are taken as σx (x=1,2,3), and the stress concentration coefficients at the corresponding points are taken as kt=σx / σ0, and the stress concentration coefficients change with the change of the length of the crack originating from the edge of the pore as shown in Figure3.It can be found that after approaching about 0.8 mm, the stress concentration situation affected by the air-membrane pore basically disappeared.

Figure 3 .
Figure 3. Distribution of stress concentration factor at the edge of air film holes of nickel-based single-crystal superalloy simulated parts

Figure 4 .
Figure 4. Fracture morphology.(a) Macroscopic optical microscope image.(b) Macroscopic scanning electron microscope image.(c) Extended fatigue arc microscopic morphology.(d) Microscopic morphology of the source region.(e) Boundary morphology of fatigue extended zone.(f) Rapid expansion zone topography.Meanwhile, obvious oxidation behavior can be observed in the fatigue source and extension zones of the fracture, while almost no oxidation occurs in the transient fracture zone, as shown in figure 5.In the existing studies by other scholars, the static oxidation temperature of DD6 nickel-based single-crystal superalloy is above 1000°C, while in this study, obvious oxidation occurs in the DD6 nickel-based single-crystal superalloy at 850°C.Comparing the transient fracture zone and fatigue morphology region, it is not difficult to find out that the oxidation situation is related to the fracture zone and fatigue morphology zone.It is not difficult to find that the oxidation condition is caused by a combination of fracture exposure time, test temperature and stress.This point will also be corrected in the process of quantitative back extrapolation of fatigue fracture stresses.

Figure 5 .
Figure 5. Results of EDS characterization of the fracture.(a) EDS line detection range.(b) EDS line detection results.(c) EDS surface detection area.(d) Distribution of oxygen elements detected by EDS surface.

Figure 6 .
Figure 6.(a) Schematic representation of the EBSD characterization location; (b) Plot of the quantitative back extrapolation results of fatigue stresses compared to the experimental monitoring values; (c) Error plot.