Crack Growth Analytical Model Considering the Crack Growth Resistance Parameter Due to the Unloading Process

Abstract

Crack growth analysis is essential for probabilistic damage tolerance assessment of aeroengine life-limited parts. Traditional crack growth models directly establish the stress ratio–crack growth rate or crack opening stress relationship and focus less on changes in the crack tip stress field and its influence, so the resolution and accuracy of maneuvering flight load spectral analysis are limited. To improve the accuracy and convenience of analysis, a parameter considering the effect of unloading amount on crack propagation resistance is proposed, and the corresponding analytical model is established. The corresponding process for acquiring the model parameters through the constant amplitude test data of a Ti-6AL-4V compact tension specimen is presented. Six kinds of flight load spectra with inserted load pairs with different stress ratios and repetition times are tested to verify the accuracy of the proposed model. All the deviations between the proposed model and test life results are less than 10%, which demonstrates the superiority of the proposed model over the crack closure and Walker-based models in addressing relevant loading spectra. The proposed analytical model provides new insights for the safety of aeroengine life-limited parts.

1. Introduction

Probabilistic damage tolerance assessment is applied in the design of aeroengine life-limited parts to prevent catastrophic accidents caused by anomalies in processes such as machining and manufacturing [1,2]. In probabilistic damage tolerance assessment, crack growth analyses of large samples are carried out at different locations of the life-limited parts in the operating environment [3], in which analytical methods are usually used to increase the analytical speed. As typical life-limited parts, compressor disks often experience a variable amplitude load spectrum at high stress levels in the operating environment [4]. Therefore, to achieve damage tolerance assessment, it is necessary to carry out accurate and efficient crack growth analysis of such a load spectrum.

For crack growth analytical analysis, Paris et al. proposed the classical Paris formula based on the relationship between the range of the stress intensity factor $\Delta K$ and the crack growth rate $\mathrm{d}a/\mathrm{d}N$ [5]. Afterward, for the situation in which the crack growth rate $\mathrm{d}a/\mathrm{d}N$ varies with the stress ratio $R$ (the ratio of the minimum value ${\sigma }_{\min }$ to the maximum value ${\sigma }_{\max }$), Walker presented Walker’s formula [6], $\mathrm{d}a/\mathrm{d}N=C{\left[{\left(1-R\right)}^m\Delta K\right]}^n$, and then established a link between $R$ and $\mathrm{d}a/\mathrm{d}N$, as shown by the blue line in Figure 1. During the same period, Elber discovered and analyzed the crack closure phenomenon [7]. By defining the crack opening stress ${\sigma }_{\mathrm{op}}$ as the “stress level at which the crack is just fully open” [8] and modifying $\Delta K$ to the range of the effective stress intensity factor $\Delta K_{\mathrm{eff}}$, the Elber formula can be obtained: $\mathrm{d}a/\mathrm{d}N=C{\left(\Delta K-\Delta K_{\mathrm{op}}\right)}^n$. Based on Elber’s work, Newman et al. [9] and Yamada et al. [10] further enriched and improved the theory of plastic-induced crack closure through simulation and experiments. Through the above efforts, crack growth analysis was further focused on the relationship between $R$ and ${\sigma }_{\mathrm{op}}$, as shown by the purple line in Figure 1.

Although some progress has been made in crack growth analysis through the above studies, these models with $R$ as the basic analysis unit still have some limitations, which limit their accuracy under the variable amplitude loads of an engine. The reason for this limitation is that the application of these analytical models under load spectra is a direct extension of the results for constant amplitude tests without considering the stress change at the crack tip in a variable amplitude load environment. This extrapolation may not always be effective. In the Walker-based model process, the load spectrum needs to be processed by the rain-flow counting method, which leads to loss of crack tip load information and a decrease in the analysis resolution. For the crack closure model, Chris Wallbrink et al. showed that the correlation between the crack opening stress in a complex variable amplitude load environment and the crack opening stress under a constant amplitude load is insignificant [11,12]. When a crack is subjected to many varying amplitude loads, in the case of crack opening under certain load spectra, the deviation of the prediction results of the crack closure model increases [13]. For rotating machinery such as aeroengines, many loads fluctuating between 30% and 100% of the maximum load [4,14] are applied during operation, resulting in cracks that are mostly in the open stage, which leads to a decrease in the accuracy of the traditional crack closure model. Therefore, traditional analytical models need to be improved to enhance the resolution and accuracy of load spectral analysis.

Some studies related to the crack tip stress field under unequal amplitude loads have been performed. For example, Enrico Salvati et al. showed that when a crack opens but the residual compressive stress caused by the previous unloading process is large, subsequent crack growth is still hindered [15]. Zhang et al. established a virtual crack annealing model based on the assumption that the residual compressive stress field at the crack tip is generated before crack closing and discussed the crack opening stress based on this model [16]. The above studies suggest that the influence of the preceding load history (especially the unloading process) on subsequent cracking may be worth considering in crack growth analysis.

An analytical model considering the effect of the unloading process on crack growth is proposed in this study to improve the convenience and accuracy of crack growth analysis. In the framework of crack growth analysis, the connection established in this study is shown by the red line in Figure 1. The sections of this paper are as follows: In Section 2, the relationship between the crack tip stress field and crack closure is shown in a simulation, and an analytical model considering the effect of the unloading capacity is proposed. In Section 3, constant amplitude load tests under different $R$ values are carried out on Ti-6AL-4V compact tension (CT) specimens, and the method of obtaining the parameters of the new analytical model is demonstrated. In Section 4, combined flight load spectra are designed to evaluate the effects of the new analytical model, the crack closure model, and the Walker-based model. The conclusions are summarized in Section 5.

2. The Model Considering the Relationship between the Unloading Capacity and Crack Growth Resistance

This section is divided into three parts. In Section 2.1, the relationship between crack closure and crack tip stress field is illustrated by finite element simulation. In Section 2.2, the stress distribution along the crack growth path is extracted through the finite element model to describe the superimposed reverse stress field caused by entire unloading. In Section 2.3, a hindering effect is employed to characterize the effect of the superimposed reverse stress field, which leads to the model proposed in this paper.

2.1. Relationship between the Crack Tip Stress Field and Crack Closure

The crack closure phenomenon based on the strip yield model is described as follows: when the cyclic load is unloaded to the minimum value, the deformation in the linear elastic region needs to be recovered, thus squeezing the plastic region and residual plastic deformation region near the crack, resulting in closure of the upper and lower surfaces of the crack tip [17]. That is, the crack closure phenomenon is the result of the stress field redistribution at the crack tip.

A simulation was performed to further simulate the stress and displacement information at the crack tip. The shape of the CT specimen is shown in Figure 2a, where $W=25$ mm and $B=5$ mm were selected to ensure $W/20\le B\le W/4$. A 3 mm crack was prefabricated both in the sample and in the simulation. The simulation model is shown in Figure 2b. The X-axis is the direction of crack growth, and the Y-axis is the direction of load application. Considering the symmetry of the specimen geometry, only half of the specimen was simulated using adequate boundary conditions. The notch for measuring crack opening displacement away from the crack was also omitted for simplicity. A triangular wave load with a maximum value of 4 kN and a stress ratio of 0.1 was applied at the pin. A rigid surface was placed at the symmetry axis to avoid overlapping of the crack surfaces during unloading. The mesh in the crack growth region was refined (the minimum mesh size was $20 \mathsf{\mu }\mathrm{m}$) to obtain a high-precision stress gradient. The plane stress state was used for CT specimen analysis with B/W = 1/2 and 1/4 [18,19]. In this study, the B/W = 1/5, which is closer to the plane stress condition. The full integration is considered to be the preferred method for analyzing crack growth [18] and has been applied in similar simulations [19]. In conclusion, 6651 fully integrated 4-node plane stress element (CPS4) was used for analysis.

The Ti-6AL-4V alloy, which is commonly used and analyzed in aerospace engineering [20,21,22], was used for the analysis. The Ti-6AL-4V alloy used in the study was manufactured through vacuum arc remelting (VAR) three times. The chemical composition of Ti-6AL-4V is shown in

Table 1. Chemical composition of Ti-6AL-4V [23].

Element	Composition (%) (mass/mass)
Nitrogen	0.05
Carbon	0.08
Hydrogen	0.015
Iron	0.30
Oxygen	0.20
Aluminum	5.5–6.75
Vanadium	3.5–4.5
Yttrium	0.005
Titanium	Balance

[23]. The material properties of Ti-6AL-4V were tested by a national special metal structural materials quality inspection and testing center, and the material parameters were obtained, as shown in

Table 2. Material properties of Ti-6AL-4V.

Material Parameter	Value
Elastic modulus $E$	112 GPa
Proof strength plastic extension $R_{\mathrm{p}0.2}$	836 MPa
Tensile strength $R_{\mathrm{m}}$	905 MPa
Poisson’s ratio $\gamma$	0.31
Fracture toughness $K_{\mathrm{c}}$	106.52 $\mathrm{MPa}\sqrt{\mathrm{m}}$

. Kumar P. et al. modeled Ti-6AL-4V as elastic–perfectly plastic because the material does not exhibit significant levels of work hardening [24]. With reference to the application of elastic–perfectly plastic [25] and isotropy [26] in Ti-6AL-4V crack simulation, the material was assumed to be elastic–perfectly plastic and isotropic. Node release technology was used for simulation in ABAQUS. The time versus crack length criterion was used as the crack growth criterion. According to [27], the crack was set to propagate every two cycles. To simulate the possible contact of the crack surface, the rigid surface was set as the master contact surface, and the crack surface was set as the slave contact surface. A constraint was imposed in the Y direction for the part of the slave contact surface far from the analysis region. The friction coefficients of the two contact surfaces were set to 0.

The results of crack propagation to 6.5 mm are shown in Figure 3. An “unloading-reloading” segment of the triangular wave load is shown in Figure 3. Four different unloading moments were selected for analysis. The unloading degrees at moments “A”, “B”, “C”, and “D” are 0%, 50%, 80%, and 100%, respectively. The component of the stress in the Y direction, rather than the von Mises stress, was adopted to better reflect the influencing factor of crack opening and closure. The stress contours and displacements at the crack tip at moments “A”, “B”, “C”, and “D” are shown in Figure 4. The displacement at the crack tip is schematically magnified to facilitate observation of opening and closing at the crack tip.

At moment “A”, the load reaches its maximum, and the tensile stress at the crack tip is the largest. During the unloading process, the positive stress field decreases and the reverse stress field increases. At moment “B”, the crack is not closed but a reverse compressive stress is generated at the crack tip. At moment “C”, the crack slightly closes, and the reverse stress field at the crack tip further increases. At moment “D”, the crack is mostly closed, and the reverse stress field reaches its maximum. Figure 4 shows that the reverse stress is generated before the crack closes, and the two do not synchronously change. The change in the stress field is more worthy of attention than whether the crack is closed.

2.2. Relationship between the Reverse Loading and the Reverse Stress Fields

The distributions of the Y-axis component of the stress along the crack growth path at different moments were extracted when the crack length was 6.5 mm, as shown in Figure 5a. As the unloading amount increases, the forward stress field at the crack tip decreases, and the reverse stress field increases. By subtracting the stress distributions at moments “B”, “C”, and “D” from the stress distribution at moment “A” (the maximum load moment), diagrams of the stress variation relative to the maximum stress moment distribution were obtained, as shown in Figure 5b. These stress variation diagrams can be regarded as the reverse stress fields superimposed on the original stress field due to reverse loading. As the reverse loading increases, the reverse stress fields also increase, regardless of whether the crack is closed. Notably, when the geometric properties and material characteristics change, the law that the reverse stress fields increase with the increase of the reverse loading is still valid.

2.3. Analytical Model Considering the Effect of the Unloading Capacity

Figure 5b shows the reverse stress field of crack superposition due to the unloading process. However, finite element simulation of crack growth for any course would result in a lot of time cost, and the obstacle effect of the reverse stress field on subsequent crack growth is difficult to directly quantify through a finite element model. Therefore, a virtual crack growth resistance parameter is proposed to represent the hindering effect of the superimposed reverse stress field caused by the entire unloading process on subsequent crack growth. The corresponding unloading-induced cracking resistance (UCR) model is established to facilitate engineering calculations. The detailed introduction of the model is as follows:

The schematic of the model under constant amplitude loads with high and low stress ratios is shown in Figure 6a, where the unloading process is indicated by the purple lines. The value of the entire unloading (reverse loading) capacity is defined as $\Delta {\sigma }_{\mathrm{un}}$, which represents the difference between the stress value before and after unloading. It is assumed that the reverse stress field is superimposed on the crack tip due to the entire unloading process, which in turn counteracts the effect of partial reloading, thus causing crack growth resistance to crack growth. This assumed crack growth resistance effect is indicated by the crack growth resistance parameter $\Delta {\sigma }_{\mathrm{r}}$. The load that is counteracted is shown in the red line in Figure 6a during the loading process. The crack growth threshold considering the effect of $\Delta {\sigma }_{\mathrm{r}}$ is set as $\Delta {\sigma }_{\mathrm{th},\mathrm{r}}$, which is represented by red dots, and the load segment exceeding $\Delta {\sigma }_{\mathrm{th},\mathrm{r}}$ is regarded as the effective load $\Delta {\sigma }_{\mathrm{eff},\mathrm{r}}$ in the UCR model, which is indicated by blue lines. The parts that do not need to be emphasized are shown in black. Figure 6b shows the UCR model under simple variable loading. The different colored lines and red dots have the same meaning as those in Figure 6a. Under the same conditions, these stress-related parameters can be expressed in the form of the stress intensity factor $K$, such as $\Delta K_{\mathrm{s}}$, $\Delta K_{\mathrm{r}}$, $\Delta K_{\mathrm{th},\mathrm{r}}$, and $\Delta K_{\mathrm{eff},\mathrm{r}}$.

The crack closure model is compared with the UCR model in Figure 6. In Figure 6a, $\Delta {\sigma }_{\mathrm{th},\mathrm{r}}$ increases with increasing stress ratio $R$. In the crack closure model, the crack opening stress $\Delta {\sigma }_{\mathrm{op}}$, which represents the crack growth threshold, also increases with increasing $R$. That is, the UCR model and the crack closure model are consistent in terms of the variation in the threshold parameters with $R$. In the load spectrum of Figure 6b, the dividing line of whether the crack is open or not is indicated by an orange dashed line. All loading segments above the orange dashed line are counted as damage in the crack closure model. In contrast, the UCR model accounts for the obstruction caused by unloading after crack opening, i.e., the red lines above the orange dashed line. This corresponds to the difference between the UCR model and the traditional crack closure model.

According to the relationships among the various parameters shown in Figure 6, a new analytical model in the form of the stress intensity factor is obtained, as shown in Equation (1):

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}a}{\mathrm{d}N}}=C{\left(\Delta K_{\mathrm{eff},\mathrm{r}}\right)}^n\\ \Delta K_{\mathrm{eff},\mathrm{r}}=\Delta K_{\mathrm{load}}-\Delta K_{\mathrm{r}}\\ {\displaystyle \frac{\Delta K_r}{\Delta K_{\mathrm{un}}}}=f\left(u\right)\end{array}\right.$$

(1)

where $a$ is the crack length; $N$ is the cycle, which is a single loading and unloading process in this model; $C$ and $n$ are undetermined parameters of the UCR model; $\Delta K_{\mathrm{load}}$ indicates the stress intensity factor corresponding to the loading process; $u$ is the dimensionless value of the unloading capacity, with $u=\Delta K_{\mathrm{un}}/K_{\max }=\Delta {\sigma }_{\mathrm{un}}/{\sigma }_{\max }$; and f(·) is a function.

For the selection of the form of function $f\left(·\right)$, Irwin’s reverse plastic zone formula [28] can be referred to as follows: $r=1/\alpha \pi \times {\left(\Delta K/2{\sigma }_{\mathrm{ys}}\right)}^2=g\left(\Delta K^2\right)$, where $r$ is the size of the reverse plastic zone. The parameter $r$ is usually used as an approximate representation of the resistance in overload models [29,30]. $r$ is quadratically related to $\Delta K$, that is, quadratically related to $\Delta \sigma$. Analogously, $f\left(u\right)$ can be set in polynomial form with the highest degree of 2: $f\left(u\right)=a*u^2+b*u+c$, where $a, b, c$ are undetermined parameters. When unloading does not occur, there is no crack growth resistance, i.e., $f\left(u\right)=0$ when $u=0$. Thus, $c=0$. Then, Equation (1) can be written as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{\mathrm{d}a}{\mathrm{d}N}}=C{\left(\Delta K_{\mathrm{eff},\mathrm{r}}\right)}^n\\ \Delta K_{\mathrm{eff},\mathrm{r}}=\Delta K_{\mathrm{load}}-\Delta K_{\mathrm{r}}\\ {\displaystyle \frac{\Delta K_r}{\Delta K_{\mathrm{un}}}}=a*u^2+b*u\end{array}\right.$$

(2)

3. Experiment and Model Parameter Acquisition

3.1. Specimen, Machine, and Test Process

Constant amplitude load crack growth tests of CT specimens under different stress ratios were carried out to obtain the parameters of the UCR model. The shape, size, and material of the specimen were the same as those described in Section 2. The test equipment is shown in Figure 7. A 100 $\mathrm{kN}$ servo-hydraulic testing machine was selected for the test. The stress ratios in the test were 0.1, 0.4, and 0.7. The load waveform was a triangular wave with a test frequency of 5~20 Hz. The test was carried out at 200 °C in a high temperature test chamber. Sharp cracks were prefabricated using the load reduction method recommended by ASTM Standard E647 [31]. A Type 3541 model crack opening displacement (COD) gauge by the Epsilon Technology Corporation was used to measure the constant amplitude load test data. The $a-N$ and $\mathrm{d}a/\mathrm{d}N-\Delta K$ curves under different stress ratios were obtained according to the procedure of ASTM Standard E647 Annex A1 [31]. The digital image correlation (DIC) system of the Xintuo 3D full-field measurement and analysis system (version 9.5) was used to detect the crack length in the combined load tests presented in Section 4. The DIC system provides the observation platform, and the crack measurement is optical measurement by camera in the DIC system. The algorithm in the DIC system provides crack length proportional conversion, synchronous shooting, horizontal calibration, and other functions. The crack length calibration of the COD system and DIC system was carried out in a constant amplitude test. The crack length was obtained by converting the COD measurement results according to ASTM E647. Comparing the crack length obtained by the DIC system with that obtained by the COD system, it is found that they are in good consistency, which shows the accuracy of the DIC system measurement. Two tests were conducted for each condition.

3.2. Acquisition of Model Parameters

The crack growth analysis region was selected as follows: Forth et al. [32], Newman et al. [33], and Ruschau et al. [34] showed that the load reduction method in ASTM Standard E647 causes a load history effect (elevated threshold) on the near-threshold crack growth rate behavior. In this research, cracks were prefabricated by the load reduction method, so the near-threshold region was partially affected. In addition, according to the Federal Aviation Administration Advisory Circular 33.14-1, hard alpha anomalies of aeroengines can be produced during the manufacturing process [1]. Regarding this hazardous situation, engineers often ignore crack initiation and directly adopt the linear growth stage in the analysis of crack growth to obtain conservative results. Therefore, for accurate analysis and to meet the requirements of the research background, the near-threshold stage of crack growth was truncated. In addition to ignoring the near-threshold stage, Reference [35] showed that the unstable crack growth region is relatively short and that ignoring this stage will not produce obvious errors, which also indicates the feasibility of omitting the near-fracture-toughness stage. In summary, this research focuses on the analysis of the linear stage of crack growth. The linear stage characteristics of crack growth for $R=0.1,0.4,0.7$ obtained through experiments are represented in the form of hollow points in Figure 8.

The hollow points obtained under different stress ratios were fitted, and the corresponding curves of the Paris equations were obtained, as shown by the lines in Figure 8. The process of obtaining the undetermined parameters of Equation (2) and the effective stress intensity factor curve was as follows:

(1)

One crack growth rate ${\left(\mathrm{d}a/\mathrm{d}N\right)}_1$ was selected such that $y={\left(\mathrm{d}a/\mathrm{d}N\right)}_1$ passed through the curves of the Paris equations under different $R$. The intersection points are indicated as black dots. The X-axis values of the intersection points corresponding to $R=0.1,0.4,0.7$ are $\Delta K_1$, $\Delta K_2$, and $\Delta K_3$, respectively.

(2)

Since the crack growth under different stress ratios obeys Equation (2), $\Delta K_{\mathrm{eff},\mathrm{r}}$ is equal when ${\displaystyle \frac{\mathrm{d}a}{\mathrm{d}N}}$ is constant. Therefore, all three black intersection points correspond to the same $\Delta K_{\mathrm{eff},\mathrm{r}}$. This unknown $\Delta K_{\mathrm{eff},\mathrm{r}}$ is schematically represented by a red dot in Figure 8. Since $\Delta K_{\mathrm{load}}=\Delta K_{\mathrm{un}}=\Delta K$ under constant amplitude loading, the following can be obtained from Equation (2):

$$\Delta K_{\mathrm{eff},\mathrm{r}}=\Delta K_1-\Delta K_{\mathrm{r}1}=\Delta K_2-\Delta K_{\mathrm{r}2}=\Delta K_3-\Delta K_{\mathrm{r}3}$$

(3)

where $\Delta K_{\mathrm{r}1},\Delta K_{\mathrm{r}2},\Delta K_{\mathrm{r}3}$ are the $\Delta K_{\mathrm{r}}$ at $R=0.1,0.4,0.7$, respectively.

Equation (4) can be obtained by expanding Equation (3) according to Equation (2):

$$\begin{array}{cc}\Delta K_{\mathrm{eff},\mathrm{r}}& =\Delta K_1 \left(1-a*u_1^2-b*u_1\right)\\ & =\Delta K_2\left(1-a*u_2^2-b*u_2\right)\\ & =\Delta K_3\left(1-a*u_3^2-b*u_3\right)\end{array}$$

(4)

where $u_1,u_2,u_3$ are the dimensionless values of the unloading capacity at $R=0.1,0.4,0.7$, respectively.

Equation (5) can be obtained by simplifying Equation (4):

$$\left\{\begin{array}{c}\left(\Delta K_1{u}_1^2-\Delta K_2{u}_2^2\right)a+\left(\Delta K_1{u}_1-\Delta K_2{u}_2\right)b=\Delta K_1-\Delta K_2\\ \left(\Delta K_3{u}_3^2-\Delta K_2{u}_2^2\right)a+\left(\Delta K_3{u}_3-\Delta K_2{u}_2\right)b=\Delta K_3-\Delta K_2\end{array}\right.$$

(5)

By solving Equation (5), the values of parameters a and $b$ can be obtained.

(3)

After the values of parameters a and b were obtained, the hollow points under different stress ratios were processed via $\Delta K_{\mathrm{eff},\mathrm{r}}=\Delta K_{\mathrm{load}}-\Delta K_{\mathrm{r}}$ in Equation (2). The corresponding results are shown in the form of solid points in Figure 9. By fitting all the solid points, the crack growth parameters $C$ and $n$ in Equation (2) can be obtained. The corresponding $\Delta K_{\mathrm{eff},\mathrm{r}}$ line is shown as the red line in Figure 9.

The validity of the obtained fitting line was tested based on the determination coefficient $R^2$, and the process for acquiring $R^2$ [36] is shown in Equation (6):

$$\left\{\begin{array}{l}{R}^2=1-{\displaystyle \frac{SSR}{SST}}\\ SSR=\sum {\left(y_i-y_{i, \mathrm{egression}}\right)}^2\\ SST=\sum {\left(y_i-\overline{y}\right)}^2\end{array}\right.$$

(6)

where SST is the total sum of squares; SSR is the regression sum of squares; $y_i$ represents the original parameter point; $\overline{y}$ is the mean of the original parameter points; and $y_{i, \mathrm{regression}}$ is each parameter point after linear regression.

Notably, in step (1), since the slopes of the Paris lines under different stress ratios are slightly different, taking different 0 lines gives slightly different parameters C and n at the end. To reduce the interference of unstable crack growth and be more consistent with the linear stage of crack growth, limiting the selection range of the $y=\mathrm{d}a/\mathrm{d}N$ line such that all the values of $K_{\max }$ corresponding to the intersection points are less than 1/2 $K_{\mathrm{c}}$ is recommended. Within this range, the deviations of C and n caused by different $y=\mathrm{d}a/\mathrm{d}N$ lines are less than 2% and 1‰, respectively. The $R^2$ values obtained through Equation (6) are all greater than 0.98. This result shows that the method has good robustness.

The $R^2$ results, which are all close to 1, indicate that processing of the experimental data by the model described in Equation (2) can reasonably collapse different test data into an almost unique relation. The crack growth model obtained via the above process is shown in Equation (7):

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}a}{\mathrm{d}N}}=9.41\times {10}^{-11}{\left(\Delta K_{\mathrm{eff},\mathrm{r}}\right)}^{2.69}\\ \Delta K_{\mathrm{eff},\mathrm{r}}=\Delta K_{\mathrm{load}}-\Delta K_{\mathrm{r}}\\ {\displaystyle \frac{\Delta K_r}{\Delta K_{\mathrm{un}}}}=0.207u^2+0.011u\end{array}\right.$$

(7)

The above UCR model parameters are acquired based mainly on test data of the material obtained under different constant amplitude stress ratios, and crack opening stress or other test data are not required. The UCR model can be built by utilizing data from existing handbooks such as those in Reference [37] and is therefore very beneficial for engineering applications.

4. Results of Different Models under Combined Loads

In this section, several combined load spectra were established based on the load history of an aeroengine during maneuvering flight [4], and the combined load spectra were tested to verify the accuracy of the obtained UCR model in predicting the load spectrum and to compare the prediction effects of different crack growth models.

The load form is shown in Figure 5b. The method for constructing the load block is schematically shown in Figure 10. The load block was constructed by inserting several repeated constant amplitude loads with higher $R$ into a load with $R=0.1$. The maximum load was 4 $\mathrm{kN}$. The stress ratios of the inserted load pairs $R_{\mathrm{insert}}$ were 0.3, 0.6, and 0.8. The number of repetitions of the inserted loads $N_{\mathrm{insert}}$ were 11 and 22. Six kinds of load spectra were obtained by combining different $R_{\mathrm{insert}}$ and $N_{\mathrm{insert}}$. The corresponding crack closure model and crack driving parameter $K^{*}$ (Walker-based) model were obtained from References [38,39] and Reference [40], respectively, as the control groups. The crack growth data of the control groups were also collapsed according to the processing instructions in the above literature, as shown in Figure 11. The UCR model, the crack closure model, and the Walker-based model correspond to the three technical paths in Figure 1. Considering there is no overload environment and the effect of pure underloads on crack growth rate is negligible for Ti-6AL-4V [30], the effects of overload and underload are not considered in this paper.

Except for the load spectrum, the specimens, machines, test processes, and processing standards are the same as those in Section 3. The initial length of the crack analysis was 6.5 mm. According to the fracture toughness $K_{\mathrm{c}}$ in Table 2, the critical crack length as the calculation end point is 17.4 mm. Two tests were conducted for each condition. The different combined loads and corresponding crack growth results are shown in Figure 12. The left figures show the specific load blocks. In the left figures, the boundary between crack opening and closing is marked as orange dashed lines, and the $\Delta K_{\mathrm{th},\mathrm{r}}$ at each load is marked as a red dot. The right figures show the crack growth results obtained by the different analytical models.

As shown in Figure 12a–d, when $R_{\mathrm{insert}}=0.3$, the crack closure model greatly deviates from the test results. However, the UCR model and the Walker-based model have better prediction results. According to the description of Figure 6b in Section 2.3, compared with the UCR model, the crack closure model fails to consider the effect of unloading when a crack is opening, thus resulting in a larger deviation. The difference in the prediction results proves the superiority of the UCR model over the crack closure model when the crack is close to closure but not closed due to unloading.

Figure 12e–h shows that when $R_{\mathrm{insert}}=0.6$, the prediction results of the three models are similar. The prediction results of the crack closure model are improved compared with those for the $R=0.3$ group. When the $R$ of the inserted load pairs is relatively high, the unloading capacity in the intermediate process is not large, resulting in a smaller $\Delta K_{\mathrm{r}}$. This change makes the crack closure model close to the UCR model.

As shown in Figure 12i–l, when $R_{\mathrm{insert}}=0.8$, the results of the UCR model and the crack closure model are similar and good, while the prediction results of the Walker-based model are not ideal. When $R_{\mathrm{insert}}>0.7$, $\Delta K_{\mathrm{r}}$ is small because of the small $\Delta K_{\mathrm{un}}$, so the effective load ($\Delta K_{\mathrm{eff}}$ in the crack closure model, $\Delta K_{\mathrm{eff},\mathrm{r}}$ in the UCR model) is approximately equal to the loading capacity ($\Delta K_{\mathrm{load}}$), resulting in the similarity between the UCR model and the crack closure model. However, the Walker-based model directly derives the damage from $R$. The converted effective load at high stress levels is not equal to the loading capacity, which may be the source of deviation.

The predictions of all the models are summarized in Figure 13, where the X-axis is the real crack growth life (the average life in two tests) and the Y-axis is the predicted crack growth life. All deviations between the proposed model and test life results are less than 10%. Generally, the prediction accuracy of the UCR model is greater than that of the crack closure model and the Walker-based model, indicating the effectiveness of the UCR model.

In this validation test, relatively simple combined loads were utilized to illustrate the differences in the effects of different models. The more complex random load spectra and overload spectra involve more complex crack tip stress field changes, requiring an extension of the UCR model. These complex environments need to be further investigated.

5. Conclusions

In the probabilistic damage tolerance assessment of aeroengine life-limited parts, crack growth under the maneuvering flight loading spectrum needs to be accurately analyzed. The traditional crack growth models focus less on the change of crack tip stress field and its influence, so the resolution and accuracy in maneuvering flight load analysis are limited. An analytical model, i.e., the UCR model, is proposed based on the effect of the stress field generated by the unloading process rather than whether the crack is closed. The accuracy of the new model and the superiority of the model over the traditional crack closure model and Walker-based model are verified by combined load spectrum tests. The main conclusions are as follows:

(1)

A reverse stress field is superimposed on the crack tip during unloading. The intensity of the reverse stress field increases with increasing unloading quantity. The crack closure phenomenon follows the change in the stress field at the crack tip.

(2)

A new parameter was proposed to characterize the effect of the reverse stress field caused by the entire unloading process on subsequent crack growth. The corresponding analytical model was established, and the intrinsic relationship between the UCR model and the traditional crack closure model was determined. The method of obtaining UCR model parameters using constant amplitude load test data under different stress ratios was shown. This acquisition process is relatively simple and can be carried out with existing test data in manuals.

(3)

Six combined load spectra were established based on the load history of an aeroengine during maneuvering flight. The UCR, the crack closure, and the Walker-based models were evaluated through spectrum tests. The verification results show that because the crack closure model does not consider the influence of the unloading process after crack opening, the simulation deviation is large when the crack is close to closure but not closed due to unloading. The deviation of the Walker-based model increases when the $R$ of the inserted loads is relatively high. According to the combined load spectrum test in this study, the results of the UCR model are satisfactory.

In conclusion, this work provides a reference for exploring the damage tolerance method and supports the optimal design of life-limited parts.