Abstract
Purpose
The purpose of this paper is to apply the concept of equivalent initial flaw size (EIFS) to the anisotropic nickel-based single crystal (SX) material, and to predict the fatigue life on this basis. The crack propagation law of SX material at different temperatures and the weak correlation of EIFS values verification under different loading conditions are also investigated.
Design/methodology/approach
A three-parameter time to crack initial (TTCI) method with multiple reference crack lengths under different loading conditions is established, which include the TTCI backstepping method and EIFS fitting method. Subsequently, the optimized EIFS distribution is obtained based on the random crack propagation rate and maximum likelihood estimation of median fatigue life. Then, an effective driving force based on anisotropic and mixed crack propagation mode is proposed to describe the crack propagation rate in the small crack stage. Finally, the fatigue life of three different temperature ESE(T) standard specimens is predicted based on the EIFS values under different survival rates.
Findings
The optimized EIFS distribution based on EIFS fitting - maximum likelihood estimation (MLE) method has the highest accuracy in predicting the total fatigue life, with the range of EIFS values being about [0.0028, 0.0875] (mm), and the mean value of EIFS being 0.0506 mm. The error between the predicted fatigue life based on the crack propagation rate and EIFS distribution for survival rates ranges from 5% to 95% and the experimental life is within two times dispersion band.
Originality/value
This paper systematically proposes a new anisotropic material EIFS prediction method, establishing a framework for predicting the fatigue life of SX material at different temperatures using fracture mechanics to avoid inaccurate anisotropic constitutive models and fatigue damage accumulation theory.
Keywords
Citation
Wen, Z., Li, F. and Li, M. (2023), "Evaluation method of equivalent initial flaw size and fatigue life prediction of nickel-based single crystal superalloy", Multidiscipline Modeling in Materials and Structures, Vol. 19 No. 6, pp. 1311-1338. https://doi.org/10.1108/MMMS-08-2023-0256
Publisher
:Emerald Publishing Limited
Copyright © 2023, Emerald Publishing Limited
crack length, mm
initial crack length, mm
average initial crack length, mm
detectable crack length, mm
critical crack length, mm
reference crack length, mm
the value of 5% probability of transcending number and 95% confidencelevel, mm
Paris formula fitting coefficient
- Cov
noise value
the data set
error factors
probability density function of the initial crack length
the formula for calculating fatigue life
geometric correction factor for stress intensity factor
EIFS probability density function
update EIFS probability density function
undetermined coefficient in crack length expression
maximum stress intensity factor,
three stress intensity factors,
fracture toughness of type I crack,
effective stress intensity factor,
the mean value of EIFS probability density function
number of cycles (or fatigue life)
number of cycles under the upper and lower limits of the integral are the
the median number of cycles
fitting coefficient between the crack length growth rate
fitting coefficient for type ith stress level (or condition)
the probability of detected crack length
equivalent variable of EIFS value
equivalent variable of EIFS value with error
random crack propagation parameter
upper limit for EIFSD
anisotropy correction factor
maximum stress,
variance and mean of variable V
TTCI or EIFS distribution function parameters
total number of test samples
measurement error caused by temperature
Nomenclature
1. Introduction
The nickel-based single crystal superalloys (SX) are remarkable for their resistance to mechanical and chemical degradation at temperatures up to 1000 °C and beyond, which find application in a range of jet engines and land-based turbines. As a crucial component of engines, the research on material properties, the study of material properties and strength life of single crystal turbine blades is very important. In terms of simple fatigue loads, extensive research has been conducted on material failure mechanisms, deformation mechanisms and life models construction etc. (Nazé et al., 2022; Le Graverend et al., 2014; Wang et al., 2023), but the analysis results are often far from the test data and lack repeatability. The reason is that the fatigue life is controlled by three coupling factors of material, environment and manufacture, and it is very complicated to describe it accurately.
At present, a lot of research has been conducted on the mechanism of the influence of microstructure on strength of SX materials. During the in situ tension process, microcracks often generate between the carbide and the matrix, and the carbide itself also cracks to accelerate the propagation of the main crack (Ma et al., 2010). In the process of high temperature creep, microcracks usually occur at the interface between the TCP phase and the γ matrix phase or in the stress concentrated interaction region (Liu et al., 2017; Tan et al., 2013; Tian et al., 2014). Compared with tensile and creep conditions, fatigue crack initiation and propagation are more complicated. The discontinuous or non-uniform microstructure (such as porosity, carbide or eutectic) on the surface or subsurface of single crystal materials can easily induce irreversible cyclic dislocation slip during cyclic loading, resulting in local plastic strain accumulation and crack initiation (Cervellon et al., 2020). The specific cracking location mainly depends on the applied stress, temperature, frequency, as well as the size, shape and position of the defect (Ormastroni et al., 2020; Cervellon et al., 2017, 2018; Lamm and Singer, 2007; Jiang et al., 2018). In general, crack initiation occurs in the largest pores in the alloy (Ormastroni et al., 2020; Cervellon et al., 2018; Lamm and Singer, 2007; Jiang et al., 2018). In addition, the decrease of the distance between the pore and the surface or the increase of pore irregularity will also promote the initiation of cracks (Lamm and Singer, 2007; Jiang et al., 2018). When the pores in the alloy are eliminated by hot isostatic pressing, fatigue cracks may also be generated at the carbide (Cervellon et al., 2018; Yi et al., 2007), which was observed by Lu et al. (2013) and (Cervellon et al., 2017) through in situ tensile at high temperature and low-cycle fatigue response. For longer duration tests (with low stress or frequency), the oxide layer may be the main crack initiation site (Cervellon et al., 2018; Rémy et al., 2013). Similarly, the initial defects can significantly affect the very-high cycle fatigue (VHCF) behavior of SX materials (Cervellon et al., 2018, 2020). In general, there are defects in microstructure of monocrystalline materials and crystal anisotropy of the materials, resulting in more errors in the damage accumulation description model. It cannot be ignored that cracks or crack-type defects may occur during the processing of mechanical structures, and these damages will gradually form cracks during the test.
The definition of crack initiation in structural components is currently uncertain. Some scholars suggested that the crack initiation length should be 0.1 mm, but in most cases, when microcracks reach this scale, it will steadily propagate along the material cross-section (Claude et al., 2013). The boundary between crack initiation and propagation is not clear, and there is no precise description method for the life of the initiation stage. Therefore, fracture mechanics often encounters bottlenecks in predicting the full fatigue life. The concept of equivalent initial flaw size (EIFS) is introduced to quantify the initial defects caused by materials, structures and manufacturing processes. As a starting point for life calculation, the whole fatigue life of a structure is only expressed as the crack growth stage (Rice and Broek, 1978). The EIFS theory was first proposed by Yang and Manning (1980), Rudd and Gray (1976), and then Wang (1982) applied it to the structural durability analysis, considering that the initial damage of the structure cannot be directly measured and assuming it as an imaginary crack (
The crack propagation behavior of SX materials is significantly different from that of common metals. Previous studies (Sakaguchi et al., 2019; Reed, 2008) have reported that crack initiation and propagation are closely related to crystal orientation, load and temperature, and depend on the strength of the strengthening phase (γ' phase) and matrix phase (γ phase). Traditional linear elastic fracture mechanics (LEFM) cannot simply determine the driving force of crack growth based on the stress intensity determined by ΔK, although the crack growth life can be combined with Paris' law to establish the crack growth rate based on ΔK (Zhang et al., 2019; Musinski and McDowell, 2012; Paris, 1963). These models are often described in terms of damage parameters such as maximum plastic strain and maximum decomposing shear stress (Zhang et al., 2019; Hong et al., 2011). In addition, SX materials exhibit different fracture modes at different temperatures and crystal direction, usually type I fracture perpendicular to the loading axis and mixed crack propagation along the crystal surface (Li et al., 2023), which further increases the difficulty of description.
To sum up, the application of EIFS mainly focuses on isotropic materials, and the inferred EIFS values are closely related to the structure and experimental environment. Therefore, this article applies the principle of probabilistic fracture mechanics to derive the EIFS theory and establishes a new universal EIFS distribution (EIFSD) that can describe the initial fatigue quality of the structure to reduce the dependence of the experimental environment. The rationality of EIFS theory is verified through SX material fatigue crack propagation experiments, and the fatigue life is predicted using fracture mechanics to avoid inaccurate anisotropic constitutive models and fatigue damage accumulation theory. This method can quantitatively evaluate the initial damage state of the material, provide a reference for guiding the manufacture of single crystal blades and provide a new idea for further analyzing the fatigue life of stress concentrated parts of blades.
2. Theoretical formulation
2.1 Initial flaw size
After the actual structure is manufactured, there are often defects, inclusions, micropores, notches, steps and other materials or geometric discontinuities. Suppose that the probability of a crack with equivalent crack size a being detected in a single inspection is denoted as
Therefore, the probability that a defect of size x exists and is detected is
If
In the above equation,
Furthermore, crack length values that meet different confidence levels and survival rates can be determined based on
2.2 EIFS probability distribution
Considering that the artificial determination of crack length during non-destructive testing cannot comprehensively consider the original state of materials and structures, the EIFSD is used instead. This method equivalently represents the entire life of the structure as crack propagation life, as shown in Figure 1. The damage before the small crack is regarded as EIFS, and the fatigue life prediction is carried out in combination with the crack growth rate at the small crack stage. The function expression is as follows:
It is often very difficult to obtain the EIFSD by obtaining
The solution of
Fit all fracture observation points under the same stress to obtain
Perform indefinite integration on Eq. (8), and when
Assuming that at time
Set the upper limit for EIFSD to xu, and according to the above equation, the lower bound of the TTCI distribution (TTCID) can be obtained as:
Specifying
Assuming that TTCID follows the Weibull distribution and EIFS is treated as a random variable x, from a statistical perspective, the EIFSD and TTCID are compatible. The probability density and cumulative probability distribution functions of EIFS can be obtained as shown in Eq. (14), where αi,βi and εi are the shape parameter, scale parameter and lower bound of the TTCID function under the i-th stress, respectively.
TTCI backstepping method
To determine αi,βi and εi and crack propagation parameters (
Among them, r is the
In order to determine the parameter α, considering the dimension of the cumulative TTCID as 1, and the random variable W is constructed:
And Eq. (14) can be rewritten as:
In this case,
Taking the quadratic logarithm on both sides of Eq. (18) can be simplified to the least squares form (
EIFS fitting method
Under the specified (or optimal) reference crack length
Arrange
Changing
2.3 Random crack propagation probability and EIFSD updating
2.3.1 Description of random crack growth rate
From the above description, it can be concluded that the power exponential expression of crack propagation rate concentrates on the dispersion of the fitted data on the Q value after a fixed index b. Therefore, Eq. (8) can be further rewritten as:
Undimensionalise a(t) and take the logarithm:
There are many parameters to be considered in the three-parameter Weibull distribution. In order to achieve roughly the same accuracy and reduce parameter estimation, the EIFS value changing over time is further transformed into a lognormal distribution to facilitate the derivation of the safety crack length under different failure rates. Moreover, the lognormal distribution can predict the lower average failure rate faster than the two-parameter Weibull distribution (Tuegel et al., 2018). Assuming lnV follows a normal distribution (V follows a normal distribution), its mean μ and variance
Thus,
If
Since
When
The probability density function of crack length a can be expressed as:
Similarly, when b = 1, Eqs (31) and (33) can be written as:
From the above equations, it can be seen that when the initial crack length
2.3.2 EIFS maximum likelihood updating
In order to further improve the accuracy of EIFS and apply its inferred fatigue life in engineering, the update process of the original EIFS distribution form is shown in Figure 3. According to the joint optimization criteria of TTCID and EIFSD, fitting can obtain parameters (α,Q,β,ε,
Let the EIFS variable be
Among the research methods of EIFS obtained by many scholars, the MLE method is often used to obtain the distribution parameters of the Weibull distribution. Thus, the likelihood function
In Eqs (37)-(38),
In the process of experiment, real-time monitoring of crack length is obtained, and then the lognormal distribution of parameter X is obtained according to the above function formula. The estimation of its lifetime N was as follows:
Assuming the mean and variance of variables EIFS for
The measurement error caused by temperature change is denoted as ω, the measured experimental data (including the real results of various error effects) are introduced with the noise value of 0.1, and the sample value of the updated variable V is obtained by selecting the optimal TTCID parameters (α, β, ε) and the median life Nm corresponding to the reference crack length a, combined with the probability reliability formula. And then
Among them,
2.3.3 Critical crack length distribution
According to linear elastic fracture mechanics, the maximum stress intensity factor
Considering the geometric correction parameters of a finite plate, the crack length can be expressed as:
From Eqs (44)-(45), it can be seen that when fracture toughness
From a statistical perspective, the unbiased estimation of the
The above derivation process is sorted into a flowchart, as shown in Figure 4.
3. Materials and experiment
Since turbine blades often work at different temperatures, the SX material was used in this test, and its elastic properties at different temperatures are shown in Appendix. ASTM E647-15 (ASTM, 2015) mainly recommends three types of specimens, namely, compact tension [C(T)] specimens, middle tension [M(T)] specimens and eccentrically-loaded single edge tension [ESE(T)] specimens for crack growth studies of materials. ESE(T) samples were used in this study, which has the main advantage of providing additional workspace while reducing the amount of processed material. The design reduces the T-stress (the stress parallel to the crack surface) that affects the crack propagation angle, making the crack fracture path more self-similar than other specimens. The size of the specimen is 92.5 × 25 × 2 mm, and the geometric length (tensile) direction is cut along the single crystal blank [001] orientation. A small notch was created on the ESE(T) specimens to induce crack initiation, and the notch was made by EDM. The length and width of the notch were 7.5 and 1.5 mm respectively, and the sharp angle is 30°. Prior to the experiment, the machining caused obvious residual stress and rough surface on the surface of the specimens, which had a large impact on the crack propagation behavior. Therefore, this experimental piece was ground with low stress and matched with manual polishing. The tensile strength, yield strength, elongation of the material, as well as the geometry and physical objects of the ESE(T) specimen are shown in Figure 5.
It can be seen from Figure 5(a) that the tensile properties of the material vary greatly at different temperatures (Committee for Compilation of Materials, 2010). In order to explore the applicability of the EIFS theory of SX materials at different temperatures, and minimize the influence caused by the difference in mechanical properties of materials, while taking into account the accuracy of crack observation and test efficiency, crack propagation tests were carried out at three low-medium temperatures. The maximum test stress
4. Results and discussion
4.1 Crack propagation path and effective driving force
Subcritical crack propagation behavior is studied by using fracture mechanics under different isothermal cyclic loading conditions. Unlike traditional cast polycrystalline materials, the fatigue crack propagation of nickel-based monocrystalline superalloys is along the strong slip band. The crack plane is crystal plane and inclined to the stress axis. Therefore, its cracking patterns are mixed, consisting of modes I, II and III. Typical crystallographic mixed mode cracks at these three different temperatures are shown in Figures 6 and 7 (only several typical fracture modes are shown in the figures). The results indicate that the SX material exhibits fatigue failure at room temperature on the {111} crystal plane, which is inclined to the width and thickness of ESE(T) specimen by about 45°. As the temperature increases, type I cracks perpendicular to the loading axis appear, and the proportion of 650 °C exceeds 450 °C.
Due to the complex geometric shape of cracks, the stress intensity factor solution for type I cracks based on isotropic materials in the ASTM standards does not meet the limited crack inclination angle and material property range. It is clearly necessary to conduct stress intensity factor analysis and calculation for anisotropic SX materials, and provide the stress intensity factor solution for anisotropic ESE (T) specimens with inclined cracks. The expression for the type I stress intensity factor of isotropic ESE(T) specimen provided according to ASTM E647-15 standard (ASTM, 2015):
In Eq. (48), L is the applied load (unit: N);
By analyzing the fracture morphology, cracks often exhibit two mutually perpendicular slip systems (such as SS1 and SS4 in Figures 7(a)–(b)). When considering three-dimensional octahedral crystal oblique cracks and type I cracks comprehensively (Figure 7(c)), based on the analytical SX material stress field expression (Chan et al., 1987) combined with Eq. (52), the KI, KII, KIII and effective stress intensity factor
Among them,
The
The expression of the equivalent stress intensity factor shown in Table 2 can be obtained by combining the flexibility matrix parameters with Eqs (53)-(56).
The discussion of crystal orientation and declination in combination Eq. (53) and Appendix can be concluded that the crack inclination angle has little effect on polycrystalline materials within 30°, but the critical point is not given. Furthermore, the two-dimensional moment balance equation for crack tip load is obtained as follows:
As the crack length a and inclination angle(β > 30°) becomes larger,
4.2 Solution of EIFSD
In the solution results,
The TTCI backstepping method and EIFS fitting method take the minimum values after 393 iterations and 465 iterations, respectively, and the probability density and distribution function of EIFS can be obtained according to the solving parameters, as shown in Figure 11. It can be concluded from the figure that the average rank distribution can functionalize EIFS and exhibit a probability distribution. Compared to the TTCI backstepping method, the EIFS fitting method yields a data point fitting function that is relatively close to the average rank distribution value, which indirectly indicates that the SEE value in Eqs (20) and (23) is more inclined for the EIFS fitting method. According to the distribution functions, the "double 95" equivalent initial crack size
Substitute the parameter values into the above equation, and the calculated values for the TTCI backstepping method and EIFS fitting method are 0.1389 mm and 0.1279 mm, respectively. Similarly, the EIFS values are mainly concentrated within the average rank probability range of 20–80%, with distribution intervals of [0.0072, 0.0399] and [0.0114, 0.0803] (mm), respectively.
By dispersing the EIFSD, the median life (
It can be concluded that the average EIFS value is 0.0842 mm in Eq.(60), and the data are mainly concentrated in the average rank probability of 20%–80%, and the interval of distribution in the data set can be solved as [0.0212, 0.0736] (mm). It should be noted that, when EIFS≈0.25 mm, the distribution function value is close to 1 in the TTCI backstepping method. In other words, the probability expression of the TTCI backstepping method cannot cover the range of EIFS values predicted by the MLE method. In the MLE method, the calculation interval [0.0316, 0.1128] (mm) contains most of the results of TTCI backstepping calculation. Therefore, an improved TTCI backstepping method is proposed in this paper, which is mainly used to determine the initial EIFS distribution, and the EIFS distribution in the EIFS fitting - MLE method is more suitable, with wider range and higher precision. Similarly, the EIFS values calculated by EIFS fitting method can be fitted and updated with the EIFS distribution of specific specimens. The updated probability density expression is:
The schematic diagram of its probability density and cumulative distribution function is shown in Figure 12, and
4.3 Fatigue life prediction
Determine the probability distribution of the critical crack length ac using the KIC that satisfies the probability distribution (Eq. (47)), and only the average value of the same temperature is considered to simplify the analysis process. According to the test results, the
It can be clearly seen from the figure that the error between the predicted life of the EIFS fitting method and the test life is very small. The EIFS fitting - MLE method prediction results are better, and the corresponding lives between different temperatures are very close, with the vast majority of results within the double error band range. This is mainly because, on the one hand, the function expression of crack propagation with high reliability of standard parts is adopted, and repeated experiments are combined to avoid accidental errors and large backstepping errors, resulting in better prediction performance than ordinary test pieces. On the other hand, considering that the EIFS values at different temperatures are the same and meet different guarantee rates, the use of universal EIFS is repeatedly optimized, which can better match the experimental data, and the predicted life of EIFS can be evenly distributed on both sides of the test life.
Undoubtedly, the two methods based on TTCI theory can effectively avoid the problem of EIFS values changing with temperature. It fits the EIFS values obtained under various conditions into a specific Weibull distribution or lognormal distribution. Noted that the fatigue life prediction data not only relies on the average EIFS value, but also satisfies the EIFS survival rate of 5–95% in the data. The TTCI backstepping method error bar is longer than that of the EIFS fitting method. In the prediction results at room temperature, the vast majority results for TTCI backstepping method are within double error band range, but it exceeds double error band at 450 and 650 °C. Although the MLE method has a relatively large deviation compared to other methods in the absence of a prior distribution, most of it is also within a 3-fold error band, but it is clearly temperature related, especially the error between room temperature and high temperature, which is better than TTCI backstepping method at room temperature. Therefore, the advantages of MLE method and EIFS fitting method can better reflect the overall distribution of EIFS, making it more widely applicable.
Fortunately, the range of predicted total fatigue EIFS values based on MLE method and EIFS fitting optimization is approximately [0.0028, 0.0875] (mm), which has little impact on life prediction. Further, although the conventional TTCI backstepping method has a larger error than the optimized EIFS fitting-MLE method, it is better than its simplicity and can still be used in engineering, but the life prediction is generally more aggressive.
5. Conclusion
Based on the background of EIFS quantitative characterization of initial defects, the fatigue life of ESE(T) specimens made of anisotropic SX material at three different temperatures is predicted in this paper. The conclusions are as follows:
TTCI method considering different reference crack lengths and test environments can effectively solve the chronic disease of temperature-related EIFS value. To simplify the analysis process and ensure that b = 1, randomness is imposed on the coefficient Q. According to TTCI backstepping method, the optimal Q, reference crack ar and approximate EIFS values are obtained, and then the random crack growth coefficient is derived, which is more reasonable than the deterministic crack propagation rate to describe crack behavior.
The crack growth path of SX material at different temperatures presents a mixed form of straight crack and crystal plane crack. Based on the anisotropic correction and the effective driving force
The optimized EFSD based on MLE method and EIFS fitting method has the highest accuracy in predicting the total fatigue life, with the range of EIFS values being about [0.0028, 0.0875] (mm), and the mean value of EIFS being 0.0506 mm. The fatigue life calculated by combining the EIFSD with the crack propagation rate under 5–95% survival rates is within a twice dispersion band of the experimental error. The TTCI backstepping method is not excluded in practical engineering application, because it can effectively avoid the problem of large life prediction deviation, and it is more convenient to operate.
The total fatigue life is derived from the effective EIFS values, which are derived from fatigue crack growth data at different temperatures. From this perspective, the results of this paper also provide a reference value for the theory in the quantitative evaluation of the initial fatigue quality of aero engine materials. They are more prone to fatigue cracks when used in high-temperature environments, and the combination of EIFS evaluation at room temperature and fracture mechanics can effectively evaluate the high-temperature fatigue life.
Figures
Figure 1
Fatigue life predicted by EIFS method
Figure 2
Schematic representation of TTCI theory in order to obtain the EIFSD
Figure 3
EIFS distribution renewal scheme and life prediction
Figure 4
The flowchart for the equations
Figure 5
(a) Mechanical properties of materials and (b)–(c) Specimen drawings and objects (unit: mm)
Figure 6
Crack propagation forms at different temperatures
Figure 7
Fracture schematic diagram of ESE(T) specimen with crack inclination angle: (a) different slip planes and slip directions; (b) two typical slip directions for crack propagation; (c) mixed crack propagation mode; (d) different orientations of material; (e) The inclination angle of the crack along the thickness direction; (f) load and moment balance diagram at crack tip
Figure 8
Type I stress intensity factor characterizes the inclined cracks
Figure 9
Crack propagation fitting lines at different temperatures: (a)–(b) R.T.; (c)–(d) 450 °C; (e)–(f) 650 °C
Figure 10
EIFSD parameter values: (a) TTCI backstepping method and (b) EIFS fitting method
Figure 11
EIFS probability distribution function curve
Figure 12
EIFS probability density and cumulative distribution
Figure 13
Fatigue life prediction results
Figure A1
Influence of different crystal orientation angles on the propagation behavior of type I crack
Figure A2
Effect of different crack inclination on crack propagation behavior of type I and type II
Test matrix for ESE (T) specimens at different temperatures
| Temperature | Maximum stress | Stress ratio | Frequency | Design number | Valid number |
|---|---|---|---|---|---|
| 25 °C | 50 MPa | 0.1 | 5 Hz | 7 | 6 |
| 450 °C | 7 | 6 | |||
| 650 °C | 7 | 6 |
Source(s): Authors own work
Equivalent stress intensity factors for ESE(T) specimens at different temperatures
| Temperature | Function expression of | |||
|---|---|---|---|---|
| 25 °C | conjugate complex | 4.10 | 1.94 | |
| 450 °C | 4.10 | 2.02 | ||
| 650 °C | 4.10 | 2.07 |
Source(s): Authors own work
Elastic constants
| Temperature/°C | Elasticity modulus/GPa | Poisson's ratio | Shear modulus/GPa |
|---|---|---|---|
| 25 | 131.5 | 0.344 | 137.0 |
| 700 | 107.0 | 0.374 | 100.2 |
| 760 | 105.5 | 0.377 | 105.0 |
| 850 | 98.0 | 0.383 | 60.6 |
| 980 | 80.5 | 0.390 | 80.4 |
| 1070 | 69.5 | 0.399 | 74.2 |
| 1100 | 67.5 | 0.413 | 63.8 |
Source(s): Authors own work
Table A1 shows the elastic constants of SX material at different temperatures.
Verify the influence of crack propagation formula under the coupling of crack inclination angle and crystal orientation (Pieracci, 1995), and select two types of crack inclination angles (β = 0 and 30°) and two crystal orientations (α = 0 and 45°) were combined in pairs, and the crack length l was projected in the perpendicular tensile direction of the crack geometry. Within the range of the crack length occupying the total width of the specimen [0.2, 0.8], the isotropic empirical formula for CT specimens in ASTM E466 was used to obtain the conclusion that the quintic function of the empirical formula can be fitted to obtain that different crystal orientation angles have no effect on the propagation behavior of type I cracks (Figure A1). Subsequently, the interference of crystal orientation deflection angle α on type I and II cracks is excluded, and the functional relationship between
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Acknowledgements
The research was supported by the National Natural Science Foundation of China (52105147), National Science and Technology Major Project (J2019-IV-0011–0079), and Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (No. CX2021068).