**Analytical and Numerical Crack Growth Analysis of 1:3 Scaled Railway Axle Specimens**

**David Simunek 1,\*, Martin Leitner 1, Jürgen Maierhofer 2, Hans-Peter Gänser <sup>2</sup> and Reinhard Pippan <sup>3</sup>**


Received: 28 December 2018; Accepted: 29 January 2019; Published: 3 February 2019

**Abstract:** This paper deals with experimental fatigue crack propagation in rotating bending loaded round bar specimens as well as an analytical and numerical analysis of the residual lifetime. Constant amplitude (CA) load tests are performed with the surface crack length being evaluated using an optical measurement system. Fracture surfaces are microscopically analyzed to determine crack growth in depth as well as the crack shape. In spite of identical testing conditions, the experimental results show some scatter in residual lifetime, which is mainly caused by different residual stress states. Although X-ray residual stress measurements reveal only minor values, a superposition of the residual stress state with the load-induced stress leads to a significant impact on the residual lifetime calculations, which explains the experimental scatter. Numerical analyses are conducted to consider the residual stress state and their effect on crack propagation by different options. Considering the residual stress distribution in depth within the residual lifetime assessment, the deviation to the most conservative experiment is reduced from +48% to +2%. In conclusion, the results in this paper highlight that it is of utmost importance to consider local residual stress conditions in the course of a crack propagation analysis in order to properly assess the residual lifetime.

**Keywords:** fatigue crack growth; railway axle; semi-elliptical crack; residual stresses

#### **1. Introduction**

Fatigue crack propagation is generally influenced by a multitude of different effects. With common crack propagation material parameters, determined using laboratory specimens, crack growth in full-scale components can be additionally affected by the manufacturing process and the operational loads. Load sequences, stress concentration due to notches and press fits as well as manufacturing-induced residual stress states majorly affect the crack propagation during service. Thus, the residual lifetime estimation of railway axles is still a demanding task and may also lead to a non-conservative assessment in the case of inadequate information. Gänser et al. [1] describe the issue of transferability from small-scale laboratory specimens to full-scale components. Many papers deal with fatigue crack growth behavior and assessment methods in railway axles. Numerically based stress intensity factor solutions for fatigue cracks in rotating bending loaded railway axles are given by Beretta et al. [2], Madia et al. [3,4] and Luke et al. [5,6], where various sections of a railway axle (T- and V-notches and the axle body) are analyzed as well as the influence of press fits on stress intensity factor (SIF) solutions. In [4] a comprehensive collection of different stress intensity factor solutions from several authors is presented. An overview on safe life and damage tolerance methods for railway axles,

failure scenarios and causes is provided by Zerbst et al. in [7,8]. Special focus on fatigue and crack growth in railway axles under corrosion is presented in [9,10].

Contributing to the ongoing research on this topic, this paper deals with the fatigue crack propagation behavior in round bar specimens on a scale of 1:3 extracted from railway axle blanks. The investigations are performed within the framework of the international project 'Probabilistic fracture mechanics concept for the assessment of railway wheelsets' (Eisenbahnfahrwerke 3, EBFW3) aiming of the transferability of crack propagation parameters determined on standard laboratory specimens to full-scale test axles [11,12]. In the framework of this project, extensive fatigue crack propagation experiments in 1:1 axles, 1:3 axle specimens and single edge notch bending (SENB) specimens have been performed. This paper focuses on the comparison of 1:3 and SENB results. Special attention is denoted on the effect of residual stresses because they affect the crack propagation rate as well as the crack shape. Besides the stress intensity factor range Δ*K*, the crack propagation rate is mainly influenced by the stress intensity factor ratio *R*, see Equation (1).

$$R = \frac{K\_{min}}{K\_{max}} \tag{1}$$

As shown, the ratio *R* depends on the minimum and maximum stress intensity factor *Kmin* and *Kmax* respectively. In practice, local conditions may influence the local stress and strain fields. One issue is the proper determination of the local stresses and stress intensity factors. In the case of real components such as railway axles, one challenge is the fact that residual stresses and press fit induced stresses, which lead to a residual stress intensity factor *Kres*. This factor superimposes with the external load, thereby influencing the local minimum and maximum effective stress intensity factors *Kmin,eff* and *Kmax,eff* leading to a change of the load-dependent ratio *R* to an local effective ratio *Reff* at the crack tip, see Equation (2). Although Δ*K* is well known, the superposition leads to a local change of *Kmin* and *Kmax*, which may significantly affect the residual lifetime due to acceleration or delay of the crack growth rate.

$$R\_{eff} = \frac{K\_{min,eff}}{K\_{max,eff}} = \frac{K\_{min} + K\_{res}}{K\_{max} + K\_{res}} \tag{2}$$

Other crucial issues are the determination of the material properties, load sequence effects as well as the crack propagation models, used for the estimation of the residual lifetime. A review of several crack propagation models under constant and variable amplitude loading is given by Beden et al. in [13]. The original Paris/Erdogan model [14] is a commonly applied method, which is comparatively easy to handle for a simple crack growth estimation; however, one disadvantage is that it does not cover the effect of the load ratio. Hence, a high deviation of the estimated residual lifetime may result. Walker [15] modified the Paris/Erdogan equation to account for the influence by the load ratio. Crack propagation models from Erdogan/Ratwani [16] or the NASGRO equation according to Forman/Mettu [17] also consider the load ratio at residual lifetime estimations and may lead to more accurate assessments. Additionally, the NASGRO equation, see Equation (3), considers crack closure mechanisms similar to Newman [18], which is included in the factor *Flc* additionally depending on the effective ratio *Reff*, see Appendix B.

$$\frac{da}{dN} = \mathbb{C} \cdot F\_{\text{lc}} \cdot \Delta K^m \cdot \frac{\left(1 - \frac{\Delta K\_{\text{th}}}{\Delta K}\right)^p}{\left(1 - \frac{K\_{\text{max}}}{K\_c}\right)^q} \tag{3}$$

Maierhofer et al. [19] modified the NASGRO equation for physically short cracks according to Equation (4) considering that short cracks can grow even though the stress intensity factor range is below the long crack threshold value Δ*Kth,LC*, see Equation B1 for Δ*Kth*(*Reff*, Δ*a*) in Appendix B. Note that in Equation Error! the transition region III of the fatigue crack curve is not considered due to neglecting the parameter *q* (*q* = 0). In the remainder of this article, the short crack model (SCM)

according to Maierhofer is used for analytical residual lifetime estimation. Detailed information about crack closure mechanisms and short crack behavior is provided in [19–22].

$$\frac{da}{dN} = \mathbb{C} \cdot F\left(R\_{eff}, \Delta a\right) \cdot \Delta K^{m-p} \cdot \left(\Delta K - \Delta K\_{th} \left(R\_{eff}, \Delta a\right)\right)^{p} \tag{4}$$

The differences between the NASGRO equation and its modification according to Maierhofer are in the determination of the crack velocity factor *F* and *Flc* respectively as well as the fatigue crack propagation threshold Δ*Kth*. For detailed information see [19].

In general, round bars under pure rotating bending exhibit a semi elliptical crack front [3,23,24]. In Figure 1, representative fractographies of semi elliptical cracks in 1:1 railway axle specimens are illustrated.

**Figure 1.** Two typical fractographies of semi elliptical cracks in rotating bending loaded 1:1 railway axle specimens: (**a**) Moderate visibility of beach marks; (**b**) Improved visibility of beach marks.

Furthermore, in some cases deviations from typical reported fracture surfaces are noticed. Figure 2 exhibits two extreme examples of such discrepancies due to local residual stress fields. While in Figure 2a, two-thirds of the crack growth period exhibit semi elliptical crack extension (as can be seen from beach marks), the final fracture surface shows minor deviation of the semi elliptical shape. It seems that crack growth in depth is retarded compared to surface. In Figure 2b, one sided near surface crack growth over a wide range of the fracture surface is shown. The effective stress intensity factor range is not exceeding the threshold on one side of the initial notch, hence unsymmetrical and non-semi-elliptical crack propagation can be observed.

**Figure 2.** Examples for the deviation from semi elliptical crack front in rotating bending loaded 1:1 railway axle specimens: (**a**) Symmetrical crack growth; (**b**) Unsymmetrical crack propagation.

While the depicted examples of Figure 2 are curiosities, the influence of local residual stress fields on crack propagation is clearly visible. Note that the illustrated fracture surfaces in Figure 1 do not give information about the quantity of the residual stress field, but an indication of the homogeneity. In this paper, only semi-elliptical crack propagation, as illustrated in Figure 1, is considered, being the focus of this article. In the analytical approach, semi-elliptical crack growth is assumed and hence divergences of a semi-elliptical shape are not provided. Numerical methods in this case are more flexible and offer an opportunity for non-semi-elliptical crack growth estimation. In summary, this paper scientifically contributes to the following research topics:


#### **2. Materials and Methods**

The material used for all experimental investigations is the railway axle steel EA1N, which is a normalized 0.35% carbon steel with a minimum yield stress *ReH* ≥ 320 MPa, see [25]. Figure 3 shows the investigated specimen geometries. Single edge notch bending specimen (SENB) tests are performed to determine crack propagation parameters at different stress ratios. These parameters are input values for the analytical and numerical fatigue crack propagation assessment in order to estimate the crack growth behavior in round bar specimens with a semi-elliptical crack front. These specimens are extracted from railway axle blanks with a scale of one-third; hence, denoted as 1:3-scale round bar specimens. The model predictions are finally compared to fatigue crack growth experiments under rotating bending. Details about manufacturing of the specimens are depicted in Appendix A.

**Figure 3.** (**a**) Illustration of the investigated single edge notch bending (SENB) specimen with straight initial notch and (**b**) 1:3-scaled axle specimen with semi-elliptical initial notch.

#### *2.1. Single Edge Notch Bending (SENB) Specimens*

As introduced, experimental fatigue crack growth tests with SENB specimens have been performed to determine fatigue crack growth parameters. The specimens exhibit a thickness *t* = 6 mm, the width *W* = 50 mm, and a length *L* = 250 mm (see Figure 3a). All specimens are tested under four-point bending at different load ratios under laboratory conditions at room temperature. Crack growth was measured by using the direct current potential drop (DCPD) technique [26–28]. An initial notch with a depth of *a*<sup>0</sup> = 10 mm was spark eroded and sharpened by polishing the notch root with a razor blade and a diamond paste. All specimens were fatigue pre-cracked under compression at a load ratio of *R* = 20 [29,30]. The stress intensity factor solution for the SENB specimen was used based on ISO 12108, see [31], and determined according to Equations (5) and (6).

$$K\_I \left(\frac{a}{\mathcal{W}}\right) = \frac{F\_B}{t \cdot \mathcal{W}^{1/2}} \cdot \lg\left(\frac{a}{\mathcal{W}}\right) \tag{5}$$

$$\lg\left(\frac{a}{W}\right) = 3 \cdot (2 \cdot \tan\theta)^{\frac{1}{2}} \cdot \left[\frac{0.923 + 0.199 \cdot (1 - \sin\theta)^4}{\cos\theta}\right] \\
\text{with} \\
\theta = \frac{\pi \cdot a}{2 \cdot W} \tag{6}$$

Figure 4a illustrates the results of the crack growth experiments at a load ratio *R* = −1, in Figure 4b results and fitted data using Equation (4) at load ratios from *R* = −1 to *R* = 0.7 are depicted. As shown in Figure 4a, the short crack effect is clearly visible at the beginning of the experiment. Further details are provided in [19].

**Figure 4.** Experimental crack growth diagrams of SENB samples; (**a**) test data of eight specimens at a stress ratio *R* = −1; (**b**) Comparison of results and fit for different stress ratios from *R* = −1 to *R* = 0.7.

The material crack propagation parameters of the fitted data according to Figure 4b are shown in Table 1. The Coefficients for the crack opening function f according to Newman [18] are determined with *σmax*/*σ<sup>F</sup>* = 0.3 and the constraint factor *α* = 2.5. Detailed description about the parameters and influence on the crack growth curve are given in Appendix B.

**Table 1.** Crack growth material parameters of fitted SENB experiments.


#### *2.2. Round Bar Specimens (1:3 Scaled Railway Axle Specimens)*

Round bar specimens with a testing diameter *d* = 55 mm (scale of 1:3 to a real railway axle) have been tested in a rotating bending test rig. The geometry of the samples is depicted in Figure 3b and detailed information on the manufacturing procedure is given in [32]. The nomenclature of the semi-elliptical crack front is illustrated in Figure 5. The crack length on the surface of the specimen is observed by an in-situ optical measurement system. Crack propagation tests are performed starting with surface crack lengths between 2*s* = 4 to 5 mm up to a final value of about 2*s* = 18 mm, which corresponds to the limit of the optical measurement system. Detailed information about the optical crack length acquisition system and calibration is given in [33]. A geometrical recalculation of the projected surface crack length (shortest distance between *S*<sup>1</sup> and *S*2) to the real surface arc crack length 2*s* is performed in order to properly evaluate the crack growth behavior.

After testing, all samples are cooled down in liquid nitrogen atmosphere and fractured. Optical microscopical analyses of the fracture surfaces are conducted to evaluate the crack shape evolution of the semi-elliptical crack front. To this purpose, a self-written software code is established to measure the semi-elliptical crack front evolution at the fracture surfaces. Starting from the initial notch, beach marks and the final fracture surface are evaluated by non-linear least-squares curve fitting so that the axes of the semi-elliptical crack front can be determined. The data points of the investigated specimens provide information of crack depth evolution depending on the surface crack length (*a* = *f*(2*s*) and *a* = *f*(2*c*) respectively). In Figure 6, the evolution of the crack shape *a* = *f*(2*c*) for 10 tested round bar specimens is depicted. The variation of the shape between the different specimens is small, hence a single fitting function has been determined.

**Figure 5.** Nomenclature of the semi-elliptical crack.

**Figure 6.** Illustration of crack depth as a function of surface crack length 2*c*.

Based on the fitted crack shape evolution, three dimensional (3D) finite element (FE) models of the specimen with different crack shapes in Abaqus are built up to investigate the stress intensity factor KI along the crack front under rotating bending. Note that a fatigue crack in a shaft under rotating bending is a Mode I case and the crack grows perpendicular to the maximum principal stress. The other opening modes are negligible. In literature, many papers deal with numerical investigations of semi-elliptical surface cracks in round bars and their crack shape evolution, see [34–41]. In this study, the surface crack length was extended with an increment of Δ*c* = 0.25 mm and the associated crack depth of the semi-elliptical crack front was adapted by the function of the fitted data points from fractography, see Figure 6. For each increment (crack length), a new numerical model is built up. In the FE model, the four-point rotating bending load was realized by an alternating bending moment. There are different numerical methods to evaluate the stress intensity factor (SIF) [42]. Courtin et al. [43] compared different numerical techniques for estimating the SIF and show the advantages of the J-Integral approach [44] using Abaqus (Version 6.14, Dassault Systemes Simulia Corp., Providence, RI, USA) [45]. The J-Integral can also be adopted for 3D crack problems. In this case, rings of elements surrounding the crack line need to be defined. Abaqus automatically identifies the contours around the crack line, whereas the first contour includes all nodes on the crack line. The first few contours are not recommended for SIF evaluation and may lead to inaccurate results [45]. Hence, a tube-shaped partition along the crack line was generated and the rings of elements have been meshed so that nine contours for evaluation were available, see Figure 7. The model is built up with hexahedral elements exhibiting reduced integration scheme and an element length of approximately 0.15 mm along the crack front. In radial direction the elements exhibit a size of 0.05 mm to ensure an accurate computation result.

**Figure 7.** Mesh around crack front for contour integral evaluation; (**a**) Tubular mesh around crack line; (**b**) Detail view of contours at surface

The J-Integral leads to inaccurate results for contour integral evaluation on free surfaces (at the end of the crack front) due to the boundary layer effect [45,46]. This means that the 1/√*r*-singularity of the stress field on the surface domain is not fulfilled. To this purpose, the SIF-values from contour integrals below the surface are fitted and extrapolated to the surface in order to properly evaluate the stress intensity factor Δ*KS* at the surface points *S*<sup>1</sup> and *S*2, which majorly influence the crack growth behavior. In the case of linear elastic fracture mechanics, the J-Integral is equivalent to the strain energy release rate G and the stress intensity factor *KI* can be determined according to Equation (7),

$$K\_I = \sqrt{I \cdot E'} \tag{7}$$

whereas *E*' is the Young's modulus and related to *E*' = *E* at plane stress or *E*' = *E*/(1−*ν*2) at plane strain condition. In Figure 8, the results of the stress intensity factor range Δ*K* for the surface points *S*1, *S*<sup>2</sup> and the crack depth point *A* as a function of the surface crack length 2*s* are depicted. Based on the measured surface crack length, the SIF-range for the surface Δ*KS* and depth Δ*KA* is evaluated and used to generate *ds/dN*-Δ*K* diagrams for comparing the crack growth behavior of the different specimens.

**Figure 8.** Stress intensity factor of the surface points (*S*<sup>1</sup> and *S*2) and the depth (point *A*) as a function of the surface crack length *2s*.

Although all experiments show similar slopes and minor shift of the different curves in the Paris region (see Figure 9), a maximum deviation of 1.65 at residual lifetime between Experiment #3 and #4 was observed (evaluated at a surface crack length *2s* = 15 mm).

**Figure 9.** Results of rotating bending experiments: (**a**) Surface crack length in dependence of load-cycles and (**b**) Crack growth rate vs. Δ*KS*.

The comparison of fitted data points from SENB specimens and round bar specimens is depicted in Figure 10 and shows that the crack growth curves of the round bar samples are in the range of SENB specimens between load ratios *R* = −1 and *R* = −0.5. Although all round bar specimens are tested at *R* = −1, a shift of the crack growth curves can be observed.

**Figure 10.** Comparison *da/dN* vs. Δ*KS* of round bar specimens to SENB.

Figure 10 shows that experiments #1–#3 of the round bar samples tend to a load ratio higher than the nominal one; thus, leading to higher crack growth rates compared to SENB specimens at *R* = −1 and to the round bar sample of experiment #4. The shift of the load ratio can be influenced by internal or external superimposed mean stresses. Internal residual stresses act like mean stresses and thus may lead to a shift in load ratio. The samples are extracted from railway axle blanks with a diameter of 190 mm, for further details see also [32]. Whereas 1:1 railway axles usually exhibit compressive axial residual stresses 10–20 mm below the surface and minor tensile stresses below (see [1,47,48]), the residual stress distribution is significantly influenced due to cutting and machining small scale specimens out of these blanks. Hence, X-ray diffraction (XRD) residual stress measurements are performed on round bar specimens up to a maximum depth of 2.5 mm at the position of the notch. The measurements reveal high compressive residual stresses up to −250 MPa on the surface in axial direction due to machining, but almost immediately reducing to zero at a depth of approximately 100 μm. At depths below 100 μm, comparably minor axial tensile residual stresses are measured. Schindler [48] neglected the compressive residual stress peak directly on the surface in calculations due to the fact that right below the effect is not present and problems in calculating the crack growth

rate occur. The averaged XRD residual stress values *σres* of three different measurements are depicted in Table 2. The measurements are performed from 0.1 mm to a maximum depth of 2.5 mm at three different specimens. Each measurement point exhibits some scatter, hence the minimum (best case) and maximum values (worst case) are specified additionally.



These measurement results are considered for the analytical and numerical assessment within the subsequent chapters. Additionally, performed XRD measurements of the SENB specimens showed negligible residual stresses, see [29]. Hence, the defined load ratio of the SENB results can be taken as reference without further modification of the local residual stress state.

#### **3. Results**

As shown in the previous chapters, parameters for different crack growth models are generated based on experimental fatigue crack growth tests with single edge notched bending (SENB) specimens. Analytical and numerical tools for residual lifetime estimation are used to compare calculations to experimental investigations of the round bar specimens.

#### *3.1. Analytical Residual Lifetime Estimation*

The analytical assessments are performed with INtegrity Assessment for Railway Axles (INARA) (Version 19-3-2018\_13-47, Materials Center Leoben Forschungs GmbH, Leoben, Austria), a software tool to analyze semi-elliptical crack propagation in railway axles within the scope of the research project "Eisenbahnfahrwerke 3". The crack propagation model, used for all analytical calculations, is the short crack model according to Maierhofer et al. [19] and the parameters are determined from SENB specimen results, see Section 2.1. The stress intensity factor solutions for the semi-elliptical crack front in the solid round bar is based on numerical calculations according to Varfolomeev [49] and also reported in [4,50]. The finite element software Abaqus [45] was used for stress intensity factor determination along the crack front for different crack aspect ratios, crack depths and position of the shaft. Based on these results, polynomial influence functions were generated and implemented in the software tool. According to Equation (8), the stress intensity factors are determined [4,49,50]:

$$K\_{l}\left(\frac{a}{\varepsilon},\frac{a}{\overline{\mathbb{K}}\_{\boldsymbol{a}}},\boldsymbol{\phi}\right) = \sqrt{\pi \cdot a} \sum\_{m=0}^{4} \sum\_{n=0}^{4} \left[D\_{mn}^{(1)} \cdot f\_{mn}^{(1)}\left(\frac{a}{\varepsilon},\frac{a}{\overline{\mathbb{K}}\_{\boldsymbol{a}}},\boldsymbol{\phi}\right) + D\_{mn}^{(2)} \cdot f\_{mn}^{(2)}\left(\frac{a}{\varepsilon},\frac{a}{\overline{\mathbb{K}}\_{\boldsymbol{a}}},\boldsymbol{\phi}\right)\right] \cdot \left(\frac{a}{\overline{\mathbb{K}}\_{\boldsymbol{a}}}\right)^{m+n} \cdot \left(\frac{a}{\varepsilon}\right)^{-n} \tag{8}$$

First, crack growth calculations without any consideration of residual stresses have been performed. The results highlight that the estimation is satisfying for experiment #4, but non-conservative for the results of experiments #1–#3. Consequently, the averaged results of residual stress measurement #1, see Table 2, are considered as a mean stress state in the analysis. The consideration of such mean stresses leads to a variation of the local load ratio and thus changes the crack growth rate. The results without and with consideration of residual stresses are depicted in Figure 11 in comparison to the experiments.

**Figure 11.** Comparison of analytical assessments and rotating bending experiments #1–#4.

Although the residual stresses considered are quite small, the influence on the residual lifetime is shown to be significant. Here, a mean stress of *σ<sup>m</sup>* = 6.1 MPa for the best case situation shows a reduction of 16% estimated residual lifetime at a final surface crack length of 2*s* = 18 mm. For comparison, in the worst case with *σ<sup>m</sup>* = 20.5 MPa mean stress, the residual lifetime is even 41% lower. Although the assumption of average constant residual stresses as a mean stress state is an approximation, the results of the evaluated residual lifetime in Figure 11 show sound accordance with the experiments. The analytically estimated crack shape evolution of the semi-elliptical crack front compared to the fitted data points of the fracture surface analysis is depicted in Figure 12. The comparison exhibits some deviation of the crack shape evolution, whereas the experiments show a pronounced parabolic shape evolution after crack initiation, the calculations exhibit a less pronounced development. However, a maximum deviation of only 6% between the fracture surface analysis and the assessment without any residual stress consideration is observed at an *a*/*RS*-ratio of about 0.37.

**Figure 12.** Crack shape evolution compared to fracture surface analysis.

Note that residual stresses are considered as constant mean stresses over the cross section for simplification, which does not describe the real residual stress distribution. Notwithstanding, satisfying results are achieved in case of the investigated round bar specimens.

As mentioned, residual stresses are measured up to a maximum depth of 2.5 mm by X-ray diffraction (XRD). For depths below, no further information is available due to measurement limitations. Based on the measured data points, residual stress distributions up to a depth of 12 mm are extrapolated to achieve an improved crack shape evolution in the course of the crack growth calculations.

#### *3.2. Improved Analytical Assessment Based on Residual Stress Distribution*

The influence of minor residual stresses considered as averaged overlapping constant mean stresses is shown in the preceding section. In this section, residual stress depth profiles are generated based on XRD measurements and their influence on the residual lifetime and crack shape evolution is analyzed. To this purpose, a multitude of different radial symmetrical stress distributions are investigated. Figure 13 illustrates two residual stress distributions up to a depth of 8 mm, which are estimated based on the measured XRD data points up to a depth of 2.5 mm. Additionally, the mean values of the data points from XRD measurements are depicted.

**Figure 13.** Illustration of measured and fitted residual stress distributions for assessments.

These two residual stress distributions are subsequently considered within the analytical approaches. The analysis reveals an improved crack shape evolution compared to the preceding calculations assuming a constant mean stress. Figure 14a depicts the two distributions compared to the fracture surface analysis and the calculations with constant mean stresses. The comparison highlights only a minor overestimation of the a/c-ratio using the two residual stress distributions compared to the experiments, which leads to conservative results for a crack growth assessment. A maximum deviation of 2.5% was observed at an *a*/*RS*-ratio of 0.19 for the evaluation including residual stress distribution #1.

**Figure 14.** (**a**) Influence of residual stress distributions on crack shape evolution; (**b**) and residual lifetime.

In summary, the results of the residual lifetime assessments show sound accordance with the experimental investigations if the estimated residual stress depth profiles are included. Furthermore, it is highlighted that even comparably minor residual stresses in depth may significantly affect the crack growth rate and the residual lifetime estimation. Hence, it is of utmost importance to incorporate the exact stress conditions, such as local residual stress states, in the crack propagation analysis to ensure a proper fatigue assessment and to avoid non-conservative results.

#### *3.3. Numerical Residual Lifetime Estimation*

The numerical analyses are conducted with Franc3D (FRacture ANalysis Code 3D Version 7.1.0.2, Fracture Analysis Consultants, Inc., Ithaca, NY, USA), which is a 3D finite element fracture analysis software to simulate crack growth [51]. It is used in combination with a general Finite Element program, such as Abaqus in this case. The crack free model is built up and meshed in Abaqus including boundary and loading conditions. The input file is imported to Franc3D and a sub-model technique is used for the round bar specimen. A semi-elliptical crack is inserted and re-meshed automatically by Franc3D considering the singularity at the crack tip by 3D quarter point singular elements, for detailed information see [52]. Based on this model, crack growth is simulated with the stress and strain analyses conducted by Abaqus and crack extension and re-meshing being done by Franc3D. Different consideration of crack face traction and surface residual stresses and their influence on the residual lifetime are analyzed. In Franc3D, different crack growth models are deposited. In the case of the investigated round bar specimens, all analyses are based on the NASGRO model [17]. Similar to the previously described analytical calculations with INARA, the influence of the residual stress state on the residual lifetime is observed. A comparison of the numerical results based on the NASGRO model with and without consideration of the residual stress condition is provided in Figure 15. The results of the assessment for the surface crack length excluding any residual stress influence are similar to the analytical evaluation with INARA.

**Figure 15.** Results of numerical assessments compared to experiments.

Franc3D provides different possibilities for considering residual stresses in crack growth simulations. In the case of the round bar specimen, three options are used for crack propagation analyses. First the mean value (*σres* = 13.3 MPa) from XRD measurements is considered as a constant crack face pressure (CCFP), which allows to apply a uniform pressure or tensile load on the crack face. The results show sound accordance with the experimental investigations. Another option is to respect only surface residual stresses. To this purpose, only XRD measurement points up to a depth of 2.5 mm are taken into account (NASGRO SRS). The residual lifetime at a surface crack length 2*s* = 18 mm was slightly non-conservative for experiment #2 and #3; however, it is shown to work well in the case of experiment #1. Finally, the residual stress distribution #1 is considered within the numerical analysis as a 1D radial symmetrical stress distribution (NASGRO *σres* distribution #1). This computation leads to almost the same results as for the assessments with constant crack face pressure (NASGRO SRS). Similar to the analytical assessments, minor residual stresses, considered in calculations, significantly reduce the residual lifetime.

Finally, the crack shape evolution in the course of the different numerical analyses has also been investigated. The results are depicted in Figure 16a and significant different evolutions of the *a*/*c*-ratio for the different simulations are observed. The decrement of *a*/*c*-ratio is generally higher in the first crack growth steps, compared to the analytical assessments by INARA. An improved shape evolution is observed for constant crack face pressure (CCFP) as well as in the case of considering the residual stress distribution. On the contrary, the assessment including only surface residual stresses exhibits higher deviation. This seems logical due to the fact that in this case the crack propagation is only influenced at the surface, where tensile residual stresses accelerate the crack growth, whereas in depth the load ratio is basically not affected.

**Figure 16.** (**a**) Crack shape evolution of numerical computations with Franc3D; (**b**) Comparison of SCM model (analytically by INARA) and NASGRO model (numerically by Franc3D) to the experimental fit.

The deviations compared to the experimental investigations are slightly higher than the crack shape evolution according to INARA. Especially in the first few crack growth steps between *a*/*RS* = 0.06–0.18 a steep decrease can be observed. Anyway, except the assessment with surface residual stresses (SRS), a maximum deviation of 9% was noticed at *a*/*RS* ≈ 0.17 for the analysis without any residual stresses, see Figure 16a. As depicted in Figure 16b, the consideration of residual stress distribution #1 leads to a maximum deviation of 3.1%.

Figure 16b illustrates the crack shape evolutions of the analytical and numerical assessments with and without the consideration of the residual stress distribution compared to the experimental investigations. For both assessment methods, an improved estimation of the crack shape evolution is achieved.

#### **4. Discussion**

Although minor residual stresses in depth at round bar specimens are measured, the influence can be significant. Table 3 shows the results of the experimental investigated residual lifetime compared to calculated residual lifetime estimations evaluated at a surface crack length of 2*s* = 15 mm. For that purpose, the lifetime of the most conservative experiment and the mean lifetime value of all experiments are depicted.

**Residual Stress Condition SCM (INARA) NASGRO (Franc3D) Experiments (Mean of All Tests) Experiments (Most Conservative Test)** *<sup>W</sup>*/*<sup>o</sup> <sup>σ</sup>res* 2.26 <sup>×</sup> 106 2.28 <sup>×</sup> 106 1.88 <sup>×</sup> 106 1.53 <sup>×</sup> <sup>10</sup><sup>6</sup> *<sup>σ</sup>res* distribution #1 1.56 <sup>×</sup> <sup>10</sup><sup>6</sup> 1.56 <sup>×</sup> 106

**Table 3.** Comparison of residual lifetime for calculations and experiments at 2*s* = 15 mm.

A comparison of the experimental mean value shows that both calculation methods (SCM and NASGRO) with residual stress distribution #1 lead to a conservative assessment and exhibit a deviation of −17%. On the contrary, estimation without considering residual stresses results in a non-conservative assessment with a deviation of +20% compared to the mean value of the experiments.

The residual lifetime estimation considering residual stress distribution #1 is in sound accordance with the most conservative experiment with a minor difference of only +2%. Again, neglecting residual stresses within the lifetime assessment leads to a significant overestimation of +48% in residual lifetime, which proves the importance of considering local residual stress states in the crack propagation analysis.

In general, X-ray diffraction measurements are limited in depth. Other methods, such as the cut-compliance method, exhibit the potential for residual stress distribution over the total cross section. To that purpose, cut-compliance measurements are planned to determine the residual stress distributions at real railway axles. Furthermore, the applicability of the numerical crack growth approach presented in this paper will be validated for real components exhibiting varying residual stress conditions, which will lead to non-semi-elliptical crack fronts as shown Figure 2.

#### **5. Conclusions**

Analytical and numerical assessment of rotating round bar specimens was conducted and the influence of residual stresses on crack propagation was analyzed. Based on X-ray diffraction measurements, residual stresses are determined and included in calculations. The consideration showed an improved assessment of residual lifetime for the investigated specimens. Based on fracture surface analysis, the crack shape evolution of the semi-elliptical crack front was observed and compared to calculations. In addition, the consideration of residual stress distributions showed an enhanced crack shape evolution. Based on the investigated assessment and experimental analyses, the following conclusions can be drawn:


**Author Contributions:** Conceptualization, D.S. and M.L.; investigation, D.S.; J.M.; methodology, D.S.; software, D.S., M.L., J.M. and H.-P.G.; validation, D.S. and M.L.; data curation, D.S. and M.L.; writing—original draft preparation, D.S.; writing—review and editing, M.L., J.M., H.-P.G. and R.P.

**Funding:** Scientific support was given within the framework of the COMET K2-Programme, whereby the Austrian Federal Government represented by Österreichische Forschungsförderungsgesellschaft mbH and the Styrian and the Tyrolean Provincial Government, represented by Steirische Wirtschaftsförderungsgesellschaft mbH and Standortagentur Tirol, is gratefully acknowledged.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A**

Both SENB and 1:3 scaled specimens are extracted out of railway axle blanks. The distance of the initial notch for both types of specimen to the primary surface of the blank is equal (20 mm) to guarantee similar microstructure, as schematically illustrated for 1:3 scaled specimens in Figure A1. Although the microstructure should be comparable for both SENB and 1:3 scaled specimens, the residual stress state is influenced by the truncated size of the final sample.

**Figure A1.** Extraction of 1:3 scaled specimens out of a railway axle blank.

#### **Appendix B**

The short crack behavior can be considered according Maierhofer s modification [19] of the NASGRO-model. Based on Equation (4) in Section 1 and the crack growth material parameters of Section 2.1 (Table 1), the crack growth rate can be determined by accounting for the built-up of the crack growth threshold value in dependence of the crack extension Δ*a*. The threshold value Δ*Kth* can be determined according Equation (A1), which describes the transition from the effective threshold Δ*Kth,eff* to the long crack growth threshold Δ*Kth,lc* based on the fictitious length scales *l*<sup>1</sup> and *l*2. The constraint factors are denoted as *ν*<sup>1</sup> and *ν*2.

$$
\Delta K\_{th} = \Delta K\_{th,eff} + \left(\Delta K\_{th,lc} - \Delta K\_{th,eff}\right) \cdot \left[1 - \left(\nu\_1 \cdot \exp\left(-\frac{\Delta a}{l\_1}\right) + \nu\_2 \cdot \exp\left(-\frac{\Delta a}{l\_2}\right)\right)\right] \tag{A1}
$$

The long crack growth threshold can be determined with Equation (A2). Newman s crack opening function *f* [18] and the Newman coefficient *A*<sup>0</sup> are used. Δ*Kth,0* is the long crack growth threshold at a load ratio *R* = 0. The curve control coefficient *Cth* depends on the load ratio. In the case that the investigated material *Cth* for positive *R*-ratios is specified in Table 1, for negative *R*-values (*R* < 0) *Cth* = 0.

$$
\Delta K\_{th,lc} = \frac{\Delta K\_{th,0}}{\left[\frac{1-f}{(1-A\_0)\cdot(1-R)}\right]^{(1+C\_{th}\cdot R)}}\tag{A2}
$$

The crack velocity factor *F* is determined according to Equations (A3) and (A4),

$$F = 1 - (1 - F\_{l\varepsilon}) \cdot \left[ 1 - \left( \nu\_1 \cdot \exp\left(-\frac{\Delta a}{l\_1}\right) + \nu\_2 \cdot \exp\left(-\frac{\Delta a}{l\_2}\right) \right) \right] \tag{A3}$$

whereas *Flc* describes the behavior for long cracks.

$$F\_{lc} = \left(\frac{1-f}{1-R}\right)^m \tag{A4}$$

#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## **A Fatigue Life Prediction Model Based on Modified Resolved Shear Stress for Nickel-Based Single Crystal Superalloys**

#### **Jialiang Wang 1,2, Dasheng Wei 1,2, Yanrong Wang 1,2,\* and Xianghua Jiang 1,2**


Received: 4 January 2019; Accepted: 29 January 2019; Published: 2 February 2019

**Abstract:** In this paper, the viewpoint that maximum resolved shear stress corresponding to the two slip systems in a nickel-based single crystal high-temperature fatigue experiment works together was put forward. A nickel-based single crystal fatigue life prediction model based on modified resolved shear stress amplitude was proposed. For the four groups of fatigue data, eight classical fatigue life prediction models were compared with the model proposed in this paper. Strain parameter is poor in fatigue life prediction as a damage parameter. The life prediction results of the fatigue life prediction model with stress amplitude as the damage parameter, the fatigue life prediction model with maximum resolved shear stress in 30 slip directions as the damage parameter, and the McDiarmid (McD) model, are better. The model proposed in this paper has higher life prediction accuracy.

**Keywords:** fatigue; nickel-based single crystal superalloy; life modeling; resolved shear stress

#### **1. Introduction**

Nickel-based single crystal superalloy materials are mainly used in engine turbine blades, and their working environment is very harsh, which is why they are one of the components with the most structural failures. Many authors have conducted a number of experimental and theoretical studies on turbine blades. Some authors have focused on experiments and microscopic observations to investigate the fatigue behavior of nickel-based single crystals with different orientations, and some nickel-based single crystal fatigue life prediction models have been proposed. However, these have mostly been improved models of isotropic material life prediction and are often phenomenological life prediction models. Based on experiments, other scholars have conducted in-depth studies on the deformation behavior of nickel-based single crystals and have established some nickel-based single crystal constitutive models to simulate deformation, finally realizing prediction of fatigue life.

A large amount of literature has described the effect of resolved shear stress on fatigue. Ref. [1] mentions that plastic deformation mainly concentrates on the octahedral slip plane at low and medium temperatures, but both octahedral slip systems and cubic slip systems are activated at high temperatures (>600 ◦C). According to [2], for directional solidification superalloys, it is found that the crack mainly lies in the primary octahedral slip systems when the temperature is between 500 ◦C and 600 ◦C. The fracture surface presents crystallographic features. When the temperature is higher than 700 ◦C, the I type crack is dominant. With temperatures between 600 ◦C and 700 ◦C, the fracture surface has the two aforementioned fracture characteristics. Ref. [3] mentions that at lower temperatures, octahedral slip systems will cause crystallographic fracture. At higher temperatures the 'wavy' slip will cause the fracture to be perpendicular to the loading direction and crack initiation sites will be mostly located on micro-holes near the subsurface. Most crack initiations comprise multiple sources. Ref. [4] considers that at room temperature and at 300 ◦C the cracks exhibit a non-crystallographic expansion mode and the crack initiation sites are mostly located on the persistent slip band. At 600 ◦C, cracks tend to expand along the crystallographic slip plane, and cracks propagating along slip lines on surfaces of specimens have been observed. Sliding surfaces and surface slip lines correspond to the primary octahedral slip system, so resolved shear stress plays an important role in crack initiation and propagation.

A large amount of literature describes the activation of nickel-based single crystal slip systems. For the SC16 nickel-based single crystal superalloy, [5] considers that the primary octahedral slip system is activated in the [001] loading direction at a temperature of 950 ◦C. Plastic deformation was found on the corresponding crystallographic plane. Ref. [6] mentions that inhomogeneous planar dislocations distributed in strip form have been observed from room temperature to 800 ◦C, and that the dislocation structure became gradually homogeneous as the temperature was further increased. For the nickel-based single crystal superalloys, it is believed that with increases in experimental temperature, the cubic slip system is gradually activated when loaded in the [001] direction [7]. As the temperature increases further, the quantity of the cubic slip also increases further, and the quantity of the octahedral slip gradually decreases. In Ref. [8], for nickel-based single crystal superalloys, it has been noted that inelastic deformation corresponding to the [001] loading direction is dominated by octahedron slip systems and that inelastic deformation corresponding to the [111] loading direction is dominated by cubic slip systems. Secondary octahedral systems have been observed only after a long period of creep deformation. Creep fatigue interaction experiments have been conducted for the SRR99 nickel-based single crystal superalloy at 950 ◦C, and the phenomenological model of the nickel-based single crystal superalloy has been proposed based on the isotropous constitutive model. It can be seen that with an increase in temperature the primary octahedral slip system and the cubic slip system are both activated, and under the influence of creep, the secondary octahedral slip system is also activated. Therefore, at higher temperatures, there is a certain influence of the three slip systems on the fatigue life of nickel-based single crystals.

Some literature has determined the activated slip system by observing slip lines on the surface of specimens. In Ref. [9], monotonic tensile tests were carried out for PWA1480 notch specimens at room temperature. The primary octahedral slip system was considered active. The maximum resolved shear stress on the specimen surface near the notch was calculated. Slip lines based on maximum resolved shear stress were consistent with experimental observations of the surface. In Ref. [10], a cylindrical indentation experiment using PWA1480 at room temperature was carried out. It was also noted that the primary octahedral slip system was activated. The numerical calculation results of the surface slip lines of the specimens were consistent with experimental observations. For the PWA1480 notch specimens [11], a monotonic tensile experiment was carried out at room temperature, obtaining the same conclusions as those drawn in [9,10]. In Ref. [12], a monotonic tensile experiment of copper single crystal notch specimens at room temperature was conducted. It was noted that the primary octahedral slip system was activated, and it was postulated that single slips, double slips, and multiple slips might be generated with different loading directions, theoretically. However, there are manufacturing deviations and loading direction deviations due to the installation process of the specimens causing there to be only a single slip with a [001] loading direction. Thus, in the beginning, the largest resolved shear stress in primary octahedral slip systems plays a significant role. It can be seen that only the primary octahedral slip system is activated at room temperature and the corresponding maximum resolved shear stress plays a major role.

The theoretical calculation methods regarding resolved shear stress and shear strain of nickel-based single crystals are listed in the theoretical formulae section. Eight kinds of nickel-based single crystal fatigue life prediction models are listed in the next section, and the viewpoint that maximum resolved shear stress corresponding to the two slip systems in a nickel-based single crystal high temperature fatigue experiment work together is proposed. A nickel-based single crystal fatigue life prediction model based on modified resolved shear stress amplitude is proposed. In the third section, for the four groups of fatigue data, eight classical fatigue life prediction models are compared with the model proposed in this paper. The advantages and disadvantages of the current several nickel-based single crystal fatigue life prediction models and the model proposed in this paper are separately described and discussed.

#### **2. Methods**

#### *2.1. Elastic Stress and Strain Calculation of Nickel-Based Single Crystal*

The coordinate system consisting of the three principal axes is the material coordinate system *oxyz*. Correspondingly the calculation coordinate system is defined by *ox'y'z'*. The property of each material axis is described by three elastic parameters, respectively; these are the elastic modulus *E*, Poisson's ratio υ, and shear modulus *G*, where *G* = *<sup>E</sup>* <sup>2</sup>(1+*ν*). According to the theory of elastic mechanics, in the material coordinate system, the stress-strain relation is ε = *C*σ or σ = *D*ε, in which *C* is the flexibility matrix and *D* is the elasticity matrix. ε and σ represent the strain and stress vectors, respectively:

$$
\boldsymbol{\varepsilon} = \begin{bmatrix} \boldsymbol{\varepsilon}\_{\text{x}} \\ \boldsymbol{\varepsilon}\_{\text{y}} \\ \boldsymbol{\varepsilon}\_{\text{z}} \\ \boldsymbol{\gamma}\_{\text{xy}} \\ \boldsymbol{\gamma}\_{\text{yz}} \\ \boldsymbol{\gamma}\_{\text{xz}} \end{bmatrix}, \boldsymbol{\sigma} = \begin{bmatrix} \boldsymbol{\sigma}\_{\text{x}} \\ \boldsymbol{\sigma}\_{\text{y}} \\ \boldsymbol{\sigma}\_{\text{z}} \\ \boldsymbol{\sigma}\_{\text{xy}} \\ \boldsymbol{\sigma}\_{\text{yz}} \\ \boldsymbol{\sigma}\_{\text{z}} \end{bmatrix}, \tag{1}
$$

Nickel-based single crystal material is one type of the commonly used materials utilized for turbine blades and is an orthotropic cubic symmetric material. In the three major axis directions of the material coordinate system, the elastic parameters of the material are equal, respectively. The elastic parameters in the three directions are: *Ex* = *Ey* = *Ez* = *E*, *υxy* = *υyz* = *υzx* = *υ*, and *Gxy* = *Gyz* = *Gzx* = *G*. *li*, *mi*, and *ni* are the cosines of the angles between the material coordinate system and the calculation coordinate system, respectively. The specific correspondence relationship is shown in Table 1.


**Table 1.** Direction cosines between coordinate axes in the different coordinate systems.

According to geometry relationships, the transformation matrix *A* and *B* of the material coordinate system and the calculation coordinate system can be expressed as

$$A = \begin{bmatrix} l\_1^2 & m\_1^2 & n\_1^2 & 2l\_1m\_1 & 2m\_1n\_1 & 2l\_1n\_1 \\ l\_2^2 & m\_2^2 & n\_2^2 & 2l\_2m\_2 & 2m\_2n\_2 & 2l\_2n\_2 \\ l\_3^2 & m\_3^2 & n\_3^2 & 2l\_3m\_3 & 2m\_3n\_3 & 2l\_3n\_3 \\ l\_1l\_2 & m\_1m\_2 & n\_1n\_2 & l\_1m\_2 + l\_2m\_1 & m\_1n\_2 + m\_2n\_1 & l\_1n\_2 + l\_2n\_1 \\ l\_2l\_3 & m\_2m\_3 & n\_2n\_3 & l\_2m\_3 + l\_3m\_2 & m\_2n\_3 + m\_3n\_2 & l\_2n\_3 + l\_3n\_2 \\ l\_1l\_3 & m\_1m\_3 & n\_1n\_3 & l\_1m\_3 + l\_3m\_1 & m\_1n\_3 + m\_3n\_1 & l\_1n\_3 + l\_3n\_1 \end{bmatrix} \tag{2}$$

Furthermore, *B* = *A*−1<sup>T</sup> . By using the transformation matrix, stress and strain in the calculation coordinate system can be obtained [13]:

$$
\sigma' = A\sigma,\tag{3}
$$

$$
\varepsilon' = \mathbf{B}\varepsilon,\tag{4}
$$

From <sup>σ</sup> = *<sup>D</sup>*<sup>ε</sup> and <sup>σ</sup> = *<sup>D</sup>xyz*ε , *<sup>D</sup>xyz* = *ADA*<sup>T</sup> can be obtained; similarly, *<sup>C</sup>xyz* = *BCB*<sup>T</sup> can be obtained. In actual engineering analysis, the material coordinate system and the calculation coordinate system are often not uniform and between them there is a certain angle. The coordinate transformation matrix *A* can be used to convert the elasticity matrix *D* in the material coordinate system to the elasticity matrix *<sup>D</sup>xyz* in the calculation coordinate system. *<sup>D</sup>xyz* contains 21 different elements:

$$D\_{x'y'z'} = A\mathbf{D}\mathbf{A}^T = \begin{bmatrix} \mathbf{C}\_{11} & \mathbf{C}\_{12} & \mathbf{C}\_{13} & \mathbf{C}\_{14} & \mathbf{C}\_{15} & \mathbf{C}\_{16} \\ & \mathbf{C}\_{22} & \mathbf{C}\_{23} & \mathbf{C}\_{24} & \mathbf{C}\_{25} & \mathbf{C}\_{26} \\ & & \mathbf{C}\_{33} & \mathbf{C}\_{34} & \mathbf{C}\_{35} & \mathbf{C}\_{36} \\ & & & \mathbf{C}\_{44} & \mathbf{C}\_{45} & \mathbf{C}\_{46} \\ & & & & \mathbf{C}\_{55} & \mathbf{C}\_{56} \\ & & & & & \mathbf{C}\_{66} \end{bmatrix} \tag{5}$$

When the elastic moduli *E*[001], *E*[011], and *E*[111] are known, the three independent material parameters *E*, υ, and *G* can be calculated using the coordinate transformation.

#### *2.2. Nickel-Based Single Crystal Resolved Shear Stress and Resolved Shear Strain*

In the material coordinate system, the relationship between resolved shear stress and the stress tensor is τ(*a*) = σ : *P*(*a*) , in which *P*(*a*) = <sup>1</sup> <sup>2</sup> (*m*(*a*)*n*(*a*)<sup>T</sup> <sup>+</sup> *<sup>n</sup>*(*a*)*m*(*a*)T), *<sup>m</sup>*(*a*) is the sliding direction of the α slip system, and *n*(*α*) is the normal direction of the plane of the α slip system. The solution to the resolved shear stresses of the 12 primary octahedral slip systems is showed in Equation (6). The solution to the resolved shear stresses of the 12 secondary octahedral slip systems is shown in Equation (7). The solution to the resolved shear stresses of the six cubic slip systems is shown in Equation (8). The corresponding resolved shear strains of the different slip systems were calculated by a similar formula to that used for the resolved shear stresses.

$$
\begin{Bmatrix}\tau\_1\\\tau\_2\\\tau\_3\\\tau\_4\\\tau\_5\\\tau\_6\\\tau\_7\\\tau\_8\\\tau\_9\\\tau\_{10}\\\tau\_{11}\\\tau\_{12}\end{Bmatrix} = \frac{1}{\sqrt{6}} \begin{Bmatrix}1&0&-1&1&-1&0\\0&-1&1&-1&0&1\\1&-1&0&0&-1&1\\-1&0&1&1&-1&0\\-1&1&0&0&-1&-1\\0&1&-1&-1&0&-1\\1&-1&0&0&-1&-1\\0&1&-1&-1&0&1\\1&0&-1&-1&-1&0\\0&-1&1&-1&0&-1\\-1&0&1&-1&-1&0\\\\-1&1&0&0&-1&1\end{Bmatrix} \begin{Bmatrix}\sigma\_{xx}\\\\\sigma\_{yy}\\\\\sigma\_{zz}\\\\\sigma\_{xy}\\\\\sigma\_{yz}\\\\\sigma\_{zx}\\\end{Bmatrix} \tag{6}
$$

*Metals* **2019**, *9*, 180

⎧

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

τ<sup>13</sup> τ<sup>14</sup> τ<sup>15</sup> τ<sup>16</sup> τ<sup>17</sup> τ<sup>18</sup> τ<sup>19</sup> τ<sup>20</sup> τ<sup>21</sup> τ<sup>22</sup> τ<sup>23</sup> τ<sup>24</sup> ⎫ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ <sup>=</sup> <sup>1</sup> 3 √2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ −1 2 −11 1 −2 2 −1 −1 1 −2 1 −1 −1 2 −21 1 −1 2 −1 −1 −1 −2 −1 −12 2 −1 1 2 −1 −1 −12 1 −1 −12 2 1 −1 2 −1 −1 −1 −2 −1 −1 2 −1 −11 2 2 −1 −11 2 −1 −1 2 −1 1 −1 2 −1 −1 2 −2 −1 −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ σ*xx* σ*yy* σ*zz* σ*xy* σ*yz* σ*zx* ⎫ ⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎭ (7) ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ τ<sup>25</sup> τ<sup>26</sup> τ<sup>27</sup> τ<sup>28</sup> τ<sup>29</sup> τ<sup>30</sup> ⎫ ⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎭ <sup>=</sup> <sup>1</sup> √2 ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ 0001 0 1 0001 0 −1 0001 1 0 0001 −1 0 0000 1 1 0000 1 −1 ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎧ ⎪⎪⎪⎪⎪⎪⎪⎨ ⎪⎪⎪⎪⎪⎪⎪⎩ σ*xx* σ*yy* σ*zz* σ*xy* σ*yz* σ*zx* ⎫ ⎪⎪⎪⎪⎪⎪⎪⎬ ⎪⎪⎪⎪⎪⎪⎪⎭ (8)

The formula σ*ij* = *liiljj*σ*ij* can be used to convert the stress tensor in the *oxyz* coordinate system to the *ox'y'z'* coordinate system. The *ox'* axis in the new coordinate system *ox'y'z'* corresponds to the normal direction of the slip plane and hence the normal stress of the slip plane can be calculated as σ*xx* = *lxilxj*σ*ij*, in which *lx<sup>x</sup>* = → *ox* · → *ox* " " " → *ox* " " " "× " " " → *ox* " " " , *lx<sup>y</sup>* = → *ox* · → *oy* " " " → *ox* " " " "× " " " → *oy* " " " , and *lx<sup>z</sup>* = → *ox* · → *oz* " " " " → *ox* " " " "× " " " → *oz* " " " . The direction cosines

" " between the old and new coordinate systems are shown in Table 2.

**Table 2.** Direction cosines between the two coordinate systems.


#### *2.3. Nickel-Based Single Crystal Fatigue Life Prediction Model*

This article lists eight classic nickel-based single crystal fatigue life prediction models (see Table 3); parameters in these models may be calculated using Equations (6)–(8). The damage parameter of model (1) is the maximum resolved shear stress amplitude, which may be obtained by taking the maximum value of the 30 resolved shear stress amplitudes. The damage parameter of model (2) is the maximum value of the 12 resolved shear stress amplitudes corresponding to the primary octahedral slip system. The model (3) damage parameter is the maximum resolved shear strain amplitude, which may be obtained by taking the maximum value of the 30 resolved shear strain amplitudes. The damage parameter of model (4) is the maximum value of the 12 resolved shear strain amplitudes corresponding to the primary octahedral slip system. The model (5) damage parameter is the maximum shear stress range. Model (6) obtains the resolved shear strain range, the maximum resolved shear stress, the normal strain amplitude, and the maximum normal stress respectively corresponding to the 30 slip directions. The left side of model (6) is first obtained, the maximum value of which is taken as the damage parameter. For model (7), the resolved shear stress amplitudes and corresponding maximum normal stresses are obtained respectively considering the 30 slip directions. The left side of model (7) is first obtained, the maximum value of which is taken as the damage parameter. For model (8), the maximum resolved shear stress amplitude and the maximum normal stress in the 30 slip directions are calculated, the combination of which is used as the damage parameter. The difference between

model (7) and model (8) is that for the former the resolved shear stress amplitude and the maximum normal stress correspond, while for the latter resolved shear stress amplitude does not necessarily correspond to the maximum normal stress.

**Table 3.** Nickel-based single crystal fatigue life prediction model [14,15].


#### *2.4. Modified Life Prediction Model*

A single crystal fatigue life prediction model based on a modified resolved shear stress amplitude is proposed. *M*<sup>1</sup> and *M*<sup>2</sup> are the maximum and median values of the Schmid factor corresponding to the maximum resolved shear stress in the primary octahedral slip system, the secondary octahedral slip system, and the cubic slip system, respectively, which are obtained from the following two equations:

$$M\_1 = \max \{ \Delta \tau\_{\text{oct\\_prim}} \Delta \tau\_{\text{oct\\_soc}} \Delta \tau\_{\text{cub}} \} / \frac{\Delta \sigma}{2} \tag{9}$$

$$M\_2 = \text{median}\{\Delta\tau\_{\text{oct\\_prim}}\Delta\tau\_{\text{oct\\_sec}}\Delta\tau\_{\text{cub}}\}/\frac{\Delta\sigma}{2} \tag{10}$$

The nickel-based single crystal fatigue life prediction model can be obtained as follows:

$$\frac{\Delta \sigma}{2} \times \frac{\left(M\_1 + M\_2\right)}{2} = a \left(N\_{\rm f}\right)^b \tag{11}$$

According to fatigue experiment data about different loading directions of nickel-based single crystal materials, compared to other classic nickel-based single crystal fatigue life models, the fatigue life prediction accuracy of the proposed model is higher. Compare the Schmid factor with the modified factor (*M*<sup>1</sup> + *M*2)/2, as shown in Figure 1. Figure 1a corresponds to the maximum Schmid factor only considering the primary octahedral slip system. Figure 1b corresponds to the maximum Schmid factor only considering the secondary octahedral slip system. Figure 1c corresponds to the maximum Schmid factor considering the cubic slip system. Figure 1d corresponds to the maximum Schmid factor considering all three slip systems. In Figure 1e, compared to Figure 1d, only the primary octahedral slip system and the cubic slip system are activated. Figure 1f corresponds to the model proposed in this paper, namely, the modified factor (*M*<sup>1</sup> + *M*2)/2. The plane formed by the *x*-axis and *y*-axis corresponds to the standard projection plane. Because of the spatial symmetry of the nickel-based single crystal material, all the loading directions can be represented by one point in the black line zone.

**Figure 1.** Schmid factors and modified factor: (**a**) primary octahedral slip system; (**b**) secondary octahedral slip system; (**c**) cubic slip system; (**d**) all three slip systems; (**e**) primary octahedral slip system and cubic system; and (**f**) modified factor.

When considering only the primary octahedral slip system, the Schmid factor corresponding to the [111] direction is less than that for the [001] and [011] directions. The experiment data shows that the stress level in the [111] direction is greater than that in the [001] and [011] directions in the uniaxial loading with the same fatigue life. However, when the fatigue life prediction method only considering the resolved shear stress of the primary octahedral slip system is adopted, the calculated resolved shear stress in the [111] direction tends to be smaller. According to [14,16], three slip systems for low cyclic fatigue at 648 ◦C can be activated for PWA1480 material and it has been concluded that the maximum resolved shear stress amplitude model has good prediction effect. For the primary octahedral slip system and the cubic slip system, the maximum Schmid factor for the [111] loading direction is greater than the maximum Schmid factor for the [001] and [011] loading directions. For the three slip systems, the maximum Schmid factor for the [111] loading direction is equal to the maximum Schmid factor for the [001] and [011] loading directions. However, for the monotonic tensile tests, the stress loading in the [111] direction is often greater than that in the [001] and [011] directions with the same fatigue life. Therefore, the nickel-based single crystal fatigue life prediction model which considers the three slip

systems or the primary octahedral slip system and the cubic slip system is often inconsistent with its experimental results; this will be discussed further.

#### **3. Results**

For the above nine nickel-based single crystal fatigue life prediction models, the high cyclic fatigue data for DD6 material at 700 ◦C and 800 ◦C, the low cycle fatigue data for PWA1480 material at 648 ◦C, and the high cycle fatigue data for PWA1484 material at 593 ◦C were used to predict uniaxial fatigue life. The plasticity effect was not considered. The basic material properties of the three materials are shown in Table 4.


**Table 4.** Material properties.

On the one hand, the fatigue life prediction model was selected by comparing the adjusted coefficient of determination (Adj. R\_Square), where the coefficient of determination is *<sup>R</sup>*<sup>2</sup> = <sup>1</sup> − ∑*n <sup>i</sup>*=<sup>1</sup> (*yi*−- *y i* ) 2 ∑*n <sup>i</sup>*=<sup>1</sup> (*yi*−*yi* ) <sup>2</sup> and the adjusted coefficient of determination is Adj.R \_Square <sup>=</sup> <sup>1</sup> <sup>−</sup> <sup>∑</sup>*<sup>n</sup> <sup>i</sup>*=<sup>1</sup> (*yi*−- *y i* ) 2 /(*n*−*k*−1) ∑*n <sup>i</sup>*=<sup>1</sup> (*yi*−*yi* ) 2 (*n*−1) . *ni* is the experimental sample freedom and *k* is the number of explanatory variables in the model (excluding the constants in the model). On the other hand, the fatigue life prediction model was selected by comparing the size of the fatigue life dispersion zone. In view of the fact that the calculation results are based on pure elastic static analysis, the prediction results of the single crystal life prediction models, including the elastic strain, tend to be poor, as shown by Model (3) and Model (4) in Table 3. Fatigue life prediction results of Model (3) and Model (4) are not listed.

#### *3.1. High Cycle Fatigue Life Prediction for DD6 Material*

#### (1) DD6 material at 700 ◦C

High cyclic fatigue data for DD6 material at 700 ◦C was selected for analysis [17]. The experiments were carried out according to the loading directions of [001], [011], and [111], with a stress ratio of −1. The results are shown in Figures 2 and 3.

For the high cyclic fatigue experiment using DD6 material at 700 ◦C with a stress ratio of −1, the fatigue life model with stress amplitude as the damage parameter, the fatigue life model with the maximum resolved shear stress amplitude considering 30 slip systems as the damage parameter, the Fin model, the McD model, and the model proposed in this paper produced relatively higher Adj. R\_Square values and better life prediction results. The results of the fatigue life prediction model which only uses the resolved shear strain as the damage parameter were poor. The fatigue life prediction results obtained using the model which only considers the maximum resolved shear stress of the primary octahedral slip system as the damage parameter, and the Chu-Conle-Bonnen (CCB) model, were worse. The fatigue life prediction results of most data points using modified maximum resolved shear stress amplitude as the damage parameter proposed in this paper were within three times of the dispersion band. The fatigue life prediction model proposed in this paper can be used to predict fatigue life with high accuracy for high cyclic fatigue experimental data for DD6 material at 700 ◦C with a stress ratio of −1.

**Figure 2.** Fatigue life predictions for DD6 material at 700 ◦C: (**a**) stress amplitude; (**b**) maximum resolved shear stress amplitude; (**c**) maximum resolved shear stress amplitude (primary octahedral slip system); (**d**) McD model; (**e**) Fin model; (**f**) CCB model.

**Figure 3.** Fatigue life prediction for DD6 material at 700 ◦C using the proposed model.

(2) DD6 material at 800 ◦C

High cyclic fatigue data for DD6 material at 800 ◦C was selected for analysis [17]. Experiments were carried out according to the loading directions of [001], [011], and [111], and the stress ratio was −1. The fatigue life prediction results are shown in Figures 4 and 5.

For the high cyclic fatigue experiment for DD6 material at 800 ◦C with a stress ratio of −1, the fatigue life model with stress amplitude as the damage parameter, the fatigue life model with maximum resolved shear stress amplitude considering the 30 slip systems as the damage parameter, the Fin model, the McD model, and the model proposed model in this paper gave relatively higher Adj. R\_Square values and better life prediction results. The comparison results are the same as for DD6 material at 700 ◦C. The fatigue life prediction results of all the data points using modified maximum resolved shear stress amplitude as the damage parameter proposed in this paper were within three times of the dispersion band. The fatigue life prediction model proposed in this paper can be used to predict fatigue life with high accuracy for high cyclic fatigue experimental data for DD6 material at 800 ◦C with a stress ratio of −1.

**Figure 4.** *Cont.*

**Figure 4.** Fatigue life predictions for DD6 material at 800 ◦C: (**a**) stress amplitude; (**b**) maximum resolved shear stress amplitude; (**c**) maximum resolved shear stress amplitude (primary octahedral slip system); (**d**) McD model; (**e**) Fin model; (**f**) CCB model.

**Figure 5.** Fatigue life prediction for DD6 material at 800 ◦C by the proposed model.

#### *3.2. Low Cycle Fatigue Life Prediction for PWA1480 Material*

Analysis was performed for PWA1480 material with 648 ◦C low cycle fatigue data [14]. In order to facilitate comparison with the literature, experiments were conducted using the [001], [011], [111], and [213] loading directions, respectively, with different strain ratios controlled. The stress tensor

and resolved shear stress were obtained by using the elasticity matrix [14]. The fatigue life prediction results are shown in Figures 6 and 7.

**Figure 6.** Fatigue life predictions for PWA1480 material at 648 ◦C: (**a**) stress amplitude; (**b**) maximum resolved shear stress amplitude; (**c**) maximum resolved shear stress amplitude (primary octahedral slip system); (**d**) McD model; (**e**) Fin model; (**f**) CCB model.

**Figure 7.** Fatigue life prediction for PWA1480 material at 648 ◦C using the proposed model.

For the low cyclic fatigue experiment using PWA1480 material at 648 ◦C with different strain ratios, the fatigue life model with stress amplitude as the damage parameter, the fatigue life model with maximum resolved shear stress amplitude considering the 30 slip systems as the damage parameter, the CCB model, the McD model, and the model proposed in this paper produced relatively higher Adj. R\_Square values and better life prediction results. The fatigue life prediction results of most data points using modified maximum resolved shear stress amplitude as the damage parameter proposed in this paper were within three times of the dispersion band. The fatigue life prediction results of all the data points were within ten times of the dispersion band. The fatigue life prediction model proposed in this paper can be used to predict fatigue life with high accuracy for low cyclic fatigue experimental data for PWA1480 material at 648 ◦C with different strain ratios.

#### *3.3. Fatigue Life Prediction of PWA1484 Material*

Analysis was performed on the PWA1484 material at 593 ◦C with a stress ratio of 0.1 fatigue data [15]. Experiments were carried out according to the loading directions of [001], [011], and [111], respectively. The experiment loading frequency was 30 Hz. The fatigue life prediction results are shown in Figures 8 and 9.

**Figure 8.** *Cont.*

**Figure 8.** Fatigue life predictions for PWA1484 material at 593 ◦C: (**a**) stress amplitude; (**b**) maximum resolved shear stress amplitude; (**c**) maximum resolved shear stress amplitude (primary octahedral slip system); (**d**) McD model; (**e**) Fin model; (**f**) CCB model.

**Figure 9.** Fatigue life prediction for PWA1484 material at 593 ◦C using the proposed model.

For the fatigue experiment data for PWA1484 material at 593 ◦C with a stress ratio of 0.1, due to the large data dispersion, the fatigue life prediction results using the eight models mentioned above and the model proposed in this paper were worse. The life prediction model could not ensure that the prediction effect was located within ten times of dispersion. However, compared with the life prediction results of the other eight kinds of damage parameters, the Adj. R\_Square value of the model proposed in this paper was higher and was only inferior to that value for the model with maximum resolved shear stress of the primary octahedral slip system. This was mainly due to the lower experimental temperature which resulted in the primary octahedral slip system playing a dominant role and the role of the secondary octahedral slip system and the cubic slip system not being obvious.

#### **4. Discussion**

Based on the prediction results of the four groups of nickel-based single crystal fatigue life data, all the fatigue life prediction models are summarized in Table 5.


**Table 5.** Nickel-based single crystal fatigue life prediction models.

Ref. [18] focuses on a fatigue experiment using thin plates with cooling holes at 900 ◦C for DD6 material, establishing the nickel-based single crystal plastic constitutive model and combining it with the critical distance method to predict its fatigue life. The fatigue life prediction method used in [18] which is based on a nickel-based single crystal fatigue constitutive model is complicated. It takes a long time to calculate the stress and strain of single crystal hollow blades with complex cooling structures. In addition, the program in [18] is unstable and is not easy to apply in engineering. Ref. [19] uses a modified Mücke's anisotropic model to predict fatigue life. Using a solution of a nonlinear equation to determine the model parameters in [19] is a tedious process. The multiaxial fatigue life prediction of a CMSX-2 nickel-based single crystal material at 900 ◦C with stress controlled for is performed in [20]. The result of the fatigue life prediction model with stress as the damage parameter is generally better than the fatigue life prediction model with strain as the damage parameter, which is consistent with the conclusion in this paper. Ref. [14,15,20] have adopted the same idea as this paper, selecting appropriate damage parameters and establishing nickel-based single crystal fatigue life prediction methods.

However, the model proposed in this paper does not consider the effect of plastic, which has no effect on predictions of high cycle fatigue life. The fatigue life prediction results are better than other models. For predictions of low cycle fatigue life, the material has often yielded, and the experiments are often strain controlled. The stress tensor and resolved shear stress are obtained by using the elasticity matrix; thus, the stress levels of some data points in Figure 6a will appear larger. In the absence of a stress gradient, the model proposed in this paper still has a good prediction accuracy of low cycle fatigue life. Although the corresponding damage parameters are fictitious, they still have a good one-to-one correspondence with life. Due to the deficiency of data, the effects of average stress and stress gradient are not considered in this paper. Ref. [21–23] have established life prediction models for the effects of stress gradient for uniaxial and multiaxial fatigue, respectively. As a next step, the right side of the model proposed in this paper can be modified to introduce the influence of average stress [24], and the stress gradient factor can be introduced into the model based on the nickel-based single crystal notch experiment [21].

In Ref. [25], monotonic tensile tests of a MD2 nickel-based single crystal superalloy at room temperature were conducted and it was observed that the cracks often initiated near the inclusions or holes. The initiation position was often located on the surface or near the surface of the specimen. In Ref. [5], monotonic tensile tests for SC16 at 950 ◦C were conducted, and it was concluded that micro-cracks on the surface and the casting holes of the specimens were often potential crack initiation sites. In Ref. [26], notch ultra-high cycle fatigue experiments for CMSX-4 and CM186LC at 850 ◦C were conducted, and crack initiation was often related to the interaction of the persistent slip band (PSB) with casting holes or carbides. At the same time, the life of crack initiation accounts for most of the total life. The actual life of the specimen often depends on the state of the material at the crack initiation position. Casting holes and inclusions often produce large stress concentrations, especially near the surface. Therefore, it is appropriate to predict the high cycle fatigue life of the material with crack initiation near the casting hole or the inclusion with stress as the damage parameter. To facilitate applications within engineering, it is still applicable to extend this method to low cycle fatigue life prediction.

#### **5. Conclusions**


**Author Contributions:** Methodology, J.W., D.W. and Y.W.; Formal analysis, X.J.; Data curation, X.J.; Writing—original draft preparation, J.W.; Writing—review and editing, D.W. and Y.W.

**Funding:** This research was funded by the National Natural Science Foundation of China (grant No. 51475024).

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **Nomenclature**


#### **References**


© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Precipitate Evolution and Fatigue Crack Growth in Creep and Artificially Aged Aluminum Alloy**

#### **Chi Liu 1, Yilun Liu 1, Liyong Ma 2,\*, Songbai Li 1, Xianqiong Zhao <sup>1</sup> and Qing Wang <sup>1</sup>**


Received: 8 November 2018; Accepted: 4 December 2018; Published: 7 December 2018

**Abstract:** The fatigue performance of high-strength Al-Cu-Mg alloys is generally influenced by the process of creep age formation when applied to acquire higher strength. The results show that creep aging accelerates the precipitation process, leading to a more uniform precipitation of strengthening phases in grains, as well as narrowed precipitation-free zones (PFZ). Compared with the artificially aged alloy, the yield strength and hardness of the creep aged alloy increased, but the fatigue resistance decreased. In the low stress intensity factor region (Δ*<sup>K</sup>* ≤ 7 MPa·m1/2), the fatigue crack propagation (FCP) rate was mainly affected by the characteristics of precipitates, and the fatigue resistance noticeably decreased with the increased creep time. In a 4 h creep aged alloy, the microstructure was dominated by Cu-Mg clusters and Guinier-Preston (GP) zones, while S" phases began to precipitate in the matrix, showing better fatigue resistance. After aging for 24 h, the needle-shaped S' phases were largely precipitated and coarsened, which changed the mode of dislocation slip, reduced the reversibility of slip, and accelerated the accumulation of fatigue damage. In stable and rapid crack propagation regions, the influence of precipitates on the FCP rate was negligible.

**Keywords:** precipitates; fatigue crack growth; creep aging; artificial aging

#### **1. Introduction**

A new aerospace high-strength Al-Cu-Mg alloy called AA2524, developed after the 2024 and 2124 aluminum alloys, exhibits excellent fatigue resistance [1–3]. Combined with creep age forming (CAF) technology, currently, AA2524 primarily finds its application in the manufacturing of wing coverings, fuselage panels, and other components [4,5], e.g., the upper wing skin of civil aircrafts such as the Airbus A330/340/380 [6], the wing panels of military aircrafts such as the B-1B bomber, C-17, and F-35 [7], as well as tank panels and melon petals in the American Saturn-5, Hercules-4, and Ariane-5 launch vehicles of the European Space Agency [4,8]. Creep age formation is a process that takes advantage of the creep property of metals to synchronize the forming and aging treatments. The formed products achieve excellent structural integrity and low residual stress [9–12]. Many scholars have conducted substantial research on the mechanical properties and springback prediction of creep aged materials. Zhan et al. [10] established a creep constitutive model and analyzed the relationship between the change of stress and precipitates and the aging strengthening in the creep aging process. Jeshvaghani et al. [13] asserted that for a 7075 aluminum alloy that has been creep aged at high and low temperatures in sequence, the springback rate decreased, and the exfoliation corrosion resistance was improved. Xu et al. [14] researched the creep aging behavior of AA2524 with the presence of pre-strain. The results showed that the increase of pre-strain can reduce the average size of S phases in the creep aged alloy and increase its density and uniformity, leading to a shortened time for peak aging and improved strength.

The current research on creep aged alloys focuses mainly on conventional mechanical properties and less on the fatigue properties. Influenced by frequent takeoff, landing, and airflow, creep aging-formed fuselage skin, wings, and other components are the most vulnerable to fatigue failure [15]. The fatigue crack propagation (FCP) resistance of aerospace structural components is an extremely important indicator. Yin [16] and Shou [17] et al. discussed the influence of grain size on the crack growth rate of 2524 aluminum alloy. Yin suggested that within the range of low stress intensity factor Δ*K*, the crack closure effect of the coarse grain samples was greater than that of the fine grain samples, and the grain refinement degraded the fatigue resistance. Shou demonstrated that when the grain size was between 50 and 100 μm, the crack growth rate was relatively low, and the crack growth path became more zigzagged. Srivatsan et al. [18] studied the effect of test temperature on the high-cycle fatigue and fracture properties of AA2524, indicating that the fatigue life decreased with the increasing test temperature. Baptista et al. [19] introduced an enhanced two-parameter exponential equation model to describe the subcritical FCP behavior of 2524-T3 aluminum alloy, which performed better than other test models. However, the aforementioned studies focused on the materials without thoroughly examining the impact of the creep age forming process.

Liu studied the fatigue behavior of creep aged AA2524 at 180 ◦C and suggested that the crack growth resistance of the alloy was reduced after treatment [20]. On the contrary, the research of Wenke Li [21] showed that the fatigue life of AA2524 was improved after creep aging. However, his research only considered the fatigue performance of the alloy under high stress and single stress, thus the results are not considered fully representative. Therefore, it is necessary to further study the effect of microstructure evolution on the fatigue performance of the creep formed alloy.

This current study achieved initial results. In this paper, the influence of creep aging and artificial aging on the microstructure, conventional mechanical properties, and FCP resistance of AA2524 is discussed. The goals of the present work are to characterize and correlate the evolution of precipitates and FCP resistance with creep age forming, and to provide a theoretical basis for AA2524 creep age forming technology.

#### **2. Material and Experiments**

The experimental material was a 5-mm-thick plate of AA2524-T3 alloy (Southwest Aluminum Group Co., Ltd., Chongqing, China), with chemical compositions (in wt.%) as follows: 4.26% Cu, 1.36% Mg, 0.57% Mn, 0.024% Zn, 0.01% Ti, 0.002% Cr, 0.089% Si, and a balance of Al. The CAF tests were completed in a vacuum autoclave. This process is shown in Figure 1. The creep temperature was set at 160 ◦C at a heating rate of 1.5 ◦C/min, and the creep aging times were 4 h, 9 h, and 24 h. The artificial aging test was performed using the same aging times and temperature, but without external stress applied, which is called stress-free aging (SFA).

**Figure 1.** The process of creep age forming.

TEM observations were conducted on a Tecnai G220 (200 kV) transmission electron microscope (United States FEI limited liability company, Hillsboro, OR, USA). TEM samples were mechanically thinned to approximately 60~80 μm, then punched into 3-mm diameter discs and polished in an MTP-1 twin-jet electro-polisher in a 30% HNO3 and 70% CH3OH mixed solution at −30 ◦C~−25 ◦C with a voltage of 15 V.

Tensile tests and FCP tests were performed on a MTS810-50 KN (MTS Systems Corporation, Eden Prairie, MN, USA) electro-hydraulic servo fatigue machine. Tensile specimens were cut in the rolling direction of the plate and tested in accordance with ASTM-E8M-2004 at room temperature, with a strain rate of 2 mm/min, resulting in an average of three samples. The FCP rate tests were conducted in ambient air at a room temperature of 18 ◦C–25 ◦C and a relative humidity of 40–60% in accordance with ASTM-E647. The FCP specimens were of the compact tension (CT) geometry, as shown in Figure 2. FCP tests were characterized for constant amplitude loading at a frequency of 10 Hz and a stress ratio of 0.5. Crack length was measured by using a crack opening displacement (COD) gauge, as shown in Figure 3. The hardness tests were completed on a HVS-1000Z Vickers micro digital hardness machine (Huayin Testing Instrument Co., Ltd., Yantai, China) with a holding pressure of 3 KN for 15 s. The average of five test points per sample was taken as the hardness value.

**Figure 2.** Compact tensile specimen (mm).

**Figure 3.** Fatigue crack propagation test with a crack opening displacement (COD) gauge.

#### **3. Results and Discussion**

#### *3.1. Microstructure*

The microstructures of AA2524 after creep aging and artificial aging are shown in Figure 4. The alloy contained rod-shaped Mn-rich phases (Figure 4b,d) and Al20Cu2Mn3 (T phases) [22–24] with a size range of 0.2 μm~0.5 μm. These phases were formed during a homogenizing treatment and hot rolling, and were not re-dissolved in the subsequent heat treatment.

**Figure 4.** TEM bright field images and selected area electron diffraction (SAED) patterns along a <100>Al zone axis for AA2524 under different aging conditions: (**a**) creep aged (CAF)-4h; (**b**) stress-free aged (SFA)-4h; (**c**) CAF-9h; (**d**) SFA-9h; (**e**) CAF-24h; (**f**) SFA-24h.

Precipitates were not observed in the TEM bright field images of the alloy that was creep aged for 4 h (CAF-4h) (Figure 4a), but it was discovered that tiny S" phases began to appear in the selected area electron diffraction (SAED) pattern in a <100>Al direction (Figure 5a). The S" phase is a precursor of the S phase, and generally occurs in the early aging stage of the Al-Cu-Mg alloy. It is difficult to observe the S" phases in the TEM bright field image [25] because they are only several nanometers in size and there is an ambiguous interface with the matrix. As these S" phases appeared, a large number of dislocation loops and helical dislocation lines were observed in the grains (CAF-4h) (Figure 4a). However, no obvious precipitate was observed in the alloy that was stress-free aged for 4 h (SFA-4h) (Figure 4b), and there were no apparent diffraction spots or streaks in the SAED pattern [26]. Therefore, it can be determined that the microstructures in the SFA-4h alloy were mainly Cu-Mg clusters and Guinier-Preston (GP) regions.

**Figure 5.** SAED patterns along a <100>Al zone axis and corresponding schematic diagrams for AA2524: (**a**,**b**) CAF-4h; (**c**,**d**) CAF-24h.

Large quantities of needle-shaped, fine transition phases—the densely distributed S' phases—were observed (Figure 4c) in the alloy that was creep aged for 9 h (CAF-9h), with sizes ranging from 150~250 nm. The precipitates in the alloy that was stress-free aged for 9 h (SFA-9h) were still not evident in the bright field images (Figure 4d). After creep aging for 24 h (CAF-24h), the precipitates continued to grow and thicken, reaching sizes of about 200~400 nm (Figure 4e), and the clearance between precipitates widened significantly. From the SAED pattern in Figure 5c, it is inferred that the main strengthening phase in grains occurs in the S' phase. In the alloy that was stress-free aged for 24 h (SFA-24h), precipitates with sizes of about 200 nm~300 nm occurred (Figure 4f), which were smaller than those found in the CAF-24h alloy (Figure 4e). The precipitation behavior shows that the presence of stress during creep aging promoted the precipitation. Some scholars contend that a great deal of nucleation precipitates from the high-density dislocations that are caused by creep stress. Then, these dislocations act as fast diffusion channels to aggregate the solute atoms toward heterogeneous nucleation, thus promoting the growth of S phases [27].

Figure 6 shows the grain boundary feature of AA2524 samples that were creep aged and artificially aged for 9 h and 24 h. There was no apparent precipitation-free zone (PFZ) at the grain boundary (Figure 6a,c) in either the CAF-9h or SFA-9h alloys, and the precipitates at the grain boundary showed discontinuous distribution. After aging for 24 h, a distinct PFZ (Figure 6d) with a width of about

184 nm appeared at the grain boundary in the SFA-24h sample, while the PFZ (Figure 6b) in the CAF-24h sample was 140 nm wide—narrower than that of the stress-free aged sample. From this, it can be explained that creep stress generates a large number of dislocations in grains. These dislocations promote the surrounding preferential precipitation, resulting in more uniform precipitation kinetics of the strengthening phases in grains and grain boundaries, thus narrowing the PFZ.

**Figure 6.** TEM images of AA2524 at the grain boundary under different treatment: (**a**) CAF-9h; (**b**) CAF-24h; (**c**) SFA-9h; (**d**) SFA-24h.

#### *3.2. Conventional Mechanical Properties*

Figures 7 and 8 show the conventional mechanical properties for AA2524 under creep aging and stress-free aging, respectively. As the aging time increased, the hardness of AA2524 first increased and then decreased (Figure 7), clearly reflecting characteristics that occur at three aging stages: under aging (4 h of aging), peak aging (9 h of aging), and over aging (24 h of aging). For the stress-free aged alloy, the hardness decreased within the aging time of 0 h~4 h (Figure 7) and increased from 9 h to 24 h. However, the hardness changed slightly during the 4 h~9 h range of time, indicating an apparent plateau region. The study done by Ringer et al. [28] on the low Cu/Mg ratio Al-Cu-Mg alloy confirmed that the hardening curves of the Al-Cu-Mg alloy, at an aging temperature ranging from 100 ◦C to 240 ◦C and within an α-S phase region, displayed an obvious plateau region. Experimental results in the present paper are consistent with this conclusion. In contrast, there was no hardness plateau region under creep aging. After aging for 9 h, the hardness of the CAF-9h alloy rapidly reached a peak value of 159 HV, 16.9% higher than that of the SFA-9h alloy (136 HV). This indicates that creep aging significantly accelerates the hardening rate and improves the hardness of the alloy.

**Figure 7.** Hardness of AA2524 under creep age forming and artificial aging.

**Figure 8.** Yield strength and elongation of AA2524 under creep age forming and artificial aging.

After 4 h of aging, the yield strength (σ0.2) of AA2524 had significantly degraded compared to AA2524-T3, the 0 h artificially aged alloy (Figure 8). With the increased aging time, the yield strength of the alloy was on the rise. With the same aging time, the yield strength of the creep aged alloys was higher than that of the artificially aged alloys. The yield strengths of the CAF-9h and CAF-24h alloys were 13.7% and 7% higher than those of the SFA-9h and SFA-24h alloys, respectively. All of the alloys were of superior plasticity, with an elongation (δ) above 15%. The variation trend of elongation was opposite that of the change in yield strength. Elongation of the 4 h aged alloys reached its highest point at 27%, followed by a downward trend. Compared with artificial aging, creep aging degraded the elongation of the alloy. After aging for 9 h, the elongation of the CAF-9h alloy was 18.2% lower than that of the artificially aged 9 h (AA-9h) alloy.

The changes in the mechanical properties of the alloy are closely related to its microstructure characteristics [29]. According to the aforementioned TEM observations, the main strengthening phases of AA2524 are the needle-shaped S (S ) phases, which can improve the strength and hardness but reduce the plasticity. The sizes of precipitates in the CAF-9h alloy (Figure 4c) were larger than those in the SFA-9h alloy (Figure 4d), and the effect of precipitation strengthening was more pronounced. Therefore, creep aged alloys boast higher tensile strength and hardness than stress-free aged alloys, but lower elongation.

#### *3.3. Fatigue Crack Growth Behavior*

Figure 9 shows the FCP rate of AA2524 under different creep aging conditions. The fatigue crack growth behavior can be divided into three stages. In the low stress region of <sup>Δ</sup>*<sup>K</sup>* ≤ 7 MPa·m1/2, the FCP rates of different creep aged specimens varied significantly, among which the FCP rate of the CAF-4h specimen was the lowest. With the increased aging time, the FCP rate was accelerated, and the FCP rate of the CAF-24h specimen was the highest. This difference was mainly related to the characteristics of precipitates and dislocation slips in the alloy matrix. However, with the increase of the stress intensity factor range Δ*K*, the FCP rates in the Paris region and the rapid fracture region tended to be consistent under different aging treatments, indicating that the effect of precipitate features in the alloy matrix on crack propagation resistance was no longer pronounced.

**Figure 9.** Fatigue crack growth rates of AA2524 under different creep age forming conditions.

The CAF-4h alloy mainly contained fine S" phases (Figure 4a), Cu-Mg clusters, and GP zones. These coherent clusters promoted the planar slip of dislocations under cyclic loading, which greatly increased the reversibility of dislocation slip, reduced the accumulation of fatigue damage, and improved the fatigue resistance of the alloy. The precipitates in both the CAF-9h and CAF-24h alloys were mainly needle-shaped S phases (Figure 4c,e), with larger sized S phases in the CAF-24 h alloy. These needle-shaped coarse phases changed the dislocation slip mode from single slip to cross slip, which degraded the reversibility of dislocation slip and accelerated the FCP rate. A PFZ appeared in the grain boundary of the CAF-24h alloy (Figure 6b), leading to a decrease in the grain boundary strength. These softer PFZs caused stress concentration around the grain boundary and accelerated the FCP rate [30]. To summarize, PFZs and larger needle-shaped precipitates lead to the highest FCP rate of the CAF-24h alloy.

In the stable crack growth stage—namely when <sup>Δ</sup>*<sup>K</sup>* = 7~16 MPa·m1/2—the Paris region precipitates, with a higher Δ*K* level, showed a weakened effect on the FCP rate. Figure 9 displays that the crack growth rate of the CAF-9h sample was slightly higher, but the difference in FCP rates under these three conditions was much smaller than that of the low <sup>Δ</sup>*<sup>K</sup>* level. When <sup>Δ</sup>*<sup>K</sup>* ≥ 16 MPa·m1/2, <sup>d</sup>*a*/d*<sup>N</sup>* ≥ <sup>10</sup>−<sup>3</sup> mm/cycle, and the d*a*/d*N*-Δ*<sup>K</sup>* curves showed an apparent knee, this indicated that the crack propagation had entered a rapid growth stage.

Figure 10 shows a comparison between the FCP rates of creep aged and stress-free aged AA2524 for aging times of 4 h, 9 h, and 24 h. In Figure 10a, the FCP rates of 4 h aged alloys under two different aging conditions were very close because both alloys contained Cu-Mg clusters and GP zones (Figure 4a,b). Although fine S" phases could be found in the SAED pattern of the CAF-4h alloy, these phases had little influence on the FCP rate of the alloy due to the S" phases remaining coherent with the matrix. As seen in Figure 10b,c, after aging for 9 h and 24 h, the FCP rates of the creep aged alloys were significantly higher in the low stress intensity factor region than those of the stress-free aged alloys.

**Figure 10.** Comparison of the fatigue crack propagation (FCP) rates of creep aged and stress-free aged AA2524 for various aging times: (**a**) 4 h; (**b**) 9 h; (**c**) 24 h.

After 9 h of aging, needle-shaped S phases that were semi-coherent with the matrix (Figure 4c,d) mainly precipitated. However, the presence of stress in the creep aging process promoted the precipitation, so the sizes of the S phases in the CAF-9h alloy were larger. Under the cyclic loading, dislocations could not cut through part of the coarse S phases and instead bypassed them, thus reducing the reversibility of cyclic slip, accumulating a large plastic deformation at the crack tip, and promoting crack propagation. In the region of <sup>Δ</sup>*<sup>K</sup>* ≤ 7.5 MPa·m1/2, the fatigue crack propagation resistance of the CAF-9h alloy was lower than that of the SFA-9h alloy.

After 24 h of aging, the sizes of precipitates showed a similar rule (Figure 4e–f). In the low stress region, the FCP rate of the SFA-24h alloy was lower than that of the CAF-24h alloy. However, the PFZ in the grain boundaries of the SFA-24h alloy (Figure 6d) was wider than it was (Figure 6b) in the CAF-24h alloy, which accelerated the FCP rate to some extent. Under the influence of both precipitates and PFZs, the effect of microstructures on the FCP resistance disappeared in advance. Therefore, the d*a*/d*N*-Δ*K* curves of the two alloys (SFA-24h and CAF-24h) were basically consistent when <sup>Δ</sup>*<sup>K</sup>* ≥ 7 MPa·m1/2.

In the lower stress region, we chose to analyze the corresponding fatigue fracture at <sup>Δ</sup>*<sup>K</sup>* = 6 MPa·m1/2, in which the crack was in the stable growth stage. Under the SEM, a large area of regular and parallel fatigue striations could be observed, as shown in Figure 11. When encountering the particles in the matrix, the striations would bypass the particles and continue to expand (Figure 11b,c), as the crack propagation direction conveys with the dotted arrow in Figure 11. The width of fatigue striations marked in Figure 11 shows the general rule that the width of fatigue striations increases with the increasing creep time. The fatigue striation width of the creep aged specimen was larger than that of the stress-free aged one when tested with the same aging time. The fatigue striation was caused by the repeated sharpening of the crack tip during the cyclic loading. The fatigue striation width corresponds to the length of a certain cyclic fatigue crack propagation, which can represent the fatigue crack growth rate to some extent. That is, the larger the fatigue striation width, the higher the fatigue crack growth rate. This conclusion is also consistent with the previous d*a*/d*N*-Δ*K* curve analysis.

**Figure 11.** Fatigue striation morphology at fractures of different specimens when <sup>Δ</sup>*<sup>K</sup>* = 6 MPa·m1/2: (**a**) CAF-4h; (**b**) CAF-9h; (**c**) CAF-24h; (**d**) SFA-4h; (**e**) SFA-9h; (**f**) SFA-24h.

It can be concluded that the FCP rate of AA2524 in the low stress region is mainly affected by the precipitate features. With the increase of Δ*K*, precipitates gradually show a weakened effect on the FCP rate, while PFZs in the grain boundaries accelerate the FCP of the alloy to some extent.

#### **4. Conclusions**


**Author Contributions:** C.L. and Y.L. conceived and designed the experiment; C.L., Q.W. performed the experiments; C.L., L.M., and X.Z. analyzed the data; S.L. and X.Z. contributed reagents, materials, and analysis tools; C.L. wrote the paper; M.L. was responsible for the revision.

**Funding:** This work was supported by the National Natural Science Foundation of China (No. 51375500), the Fundamental Research Funds for the Central Universities of Central South University (No. 2015zzts038), Hebei Provincial Science and Technology Plan Self-Financing Project (No. 17211828), Hebei Province Higher Education Science and Technology Research Youth Fund Project (QN2018013), and the Hunan Science and Technology Plan Project (No. 2016GK2005).

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Fatigue Damage Accumulation Modeling of Metals Alloys under High Amplitude Loading at Elevated Temperatures**

#### **Jarosław Szusta \* and Andrzej Seweryn**

Faculty of Mechanical Engineering, Bialystok University of Technology, 45C Wiejska Str., 15-351 Bialystok, Poland; a.seweryn@pb.edu.pl

**\*** Correspondence: j.szusta@pb.edu.pl; Tel.: +48-085-746-9300; Fax: +48-085-746-9210

Received: 16 November 2018; Accepted: 3 December 2018; Published: 6 December 2018

**Abstract:** This article presents an approach related to the modeling of the fatigue life of constructional metal alloys working under elevated temperature conditions and in the high-amplitude load range. The article reviews the fatigue damage accumulation criteria that makes it possible to determine the number of loading cycles until damage occurs. Results of experimental tests conducted on various technical metal alloys made it possible to develop a fatigue damage accumulation model for the LCF (Low Cycle Fatigue) range. In modeling, the material's damage state variable was defined, and the damage accumulation law was formulated incrementally so as to enable the analysis of the influence of loading history on the material's fatigue life. In the proposed model, the increment of the damage state variable was made dependent on the increment of plastic strain, on the tensile stress value in the sample, and also on the actual value of the damage state variable. The model was verified on the basis of data obtained from experiments in the field of uniaxial and multiaxial loads. Samples made of EN AW 2024T3 aluminum alloy were used for this purpose.

**Keywords:** elevated temperature; low cycle fatigue; damage accumulation; uniaxial and multiaxial loading

#### **1. Introduction**

During operation, machine parts are frequently subjected to fatigue loading at elevated temperatures. Such operating conditions have an adverse effect on the materials' behavior. An elevated temperature accelerates wear processes and contributes to increasing the rate of fatigue crack propagation [1,2]. Aluminum alloys, e.g., EN AW 2024T3, are among the materials used to make constructions capable of working at an elevated temperature. It is widely used in the motorization and aviation industries, which place strong emphasis on ensuring high strength and light weight at the same time [3]. These features make it suitable for the manufacture of parts working in close proximity to combustion and jet engines where elevated temperature zones are present. In the work of Szusta and Seweryn [4,5], the influence of elevated temperature on the fatigue life of this material was tested. The authors demonstrated that strength parameters decreased as the test temperature increased. Based on the tests conducted, the authors determined the parameters of the Manson-Coffin fatigue life curve as a function of temperature:

$$\frac{\Delta \varepsilon\_{\rm eq}}{2} = \frac{\sigma\_{\rm f}'(T)}{E(T)} (2N\_{\rm f})^{b(T)} + \varepsilon\_{\rm f}'(T) (2N\_{\rm f})^{c(T)},\tag{1}$$

where *N*<sup>f</sup> is the number of cycles until failure; *E*(*T*) is Young's modulus determined at temperature *T*; *σ*f (*T*) and *ε*<sup>f</sup> (*T*) are, respectively, coefficients of the elastic fatigue life curve and plastic fatigue life curve for the analyzed temperature *T*; and *b*(*T*) and *c*(*T*) are, respectively, exponents of the elastic

and plastic fatigue life curve for temperature *T*. Furthermore, based on observations of the material's cracking mechanisms at an elevated temperature, the authors proposed a semi-empirical model for estimating the material's fatigue life. In this model, it was accepted that the material's fatigue life drops as the test temperature (isothermal) increases. The lower the material's melting temperature, the more substantial this drop is. Moreover, it was assumed that there exists a limit temperature *T*<sup>m</sup> at which the given material no longer has any immediate strength. Therefore, knowing the value of this temperature and the function defining the reduction in fatigue life as temperature increases, calculations of the number of cycles until crack initiation can be performed with engineering accuracy for a given temperature level and strain amplitude. For uniaxial tensile-compressive loads, the function of the temperature's influence on the material's fatigue life has been defined as follows:

$$N\_{\rm f}(T) = AT + B\_{\prime} \tag{2}$$

where *A* and *B* are material constants determined in periodical tensile-compression tests at temperature *T*.

In this model, the material constant *B* is a function of the material's recrystallization temperature *T*<sup>m</sup> (*B* = −*AT*m). The value of parameter *A* is calculated on the basis of the experimentally determined number of cycles to failure (obtained for a given load course at room temperature) related to the value of temperature difference: the material recrystallization and room temperature. Parameter *A* is determined from the following equation:

$$A\left(\frac{\Delta\varepsilon\_{\rm eq}}{2}\right) = -\frac{N\_{\rm f}\left(RT, \frac{\Delta\varepsilon\_{\rm eq}}{2}\right)}{\left(T\_{\rm m} - RT\right)},\tag{3}$$

where *N*<sup>f</sup> - *RT*, Δ*ε*eq 2 is determined from Manson–Coffin–Basquiun equation at room temperature *RT* for the set total strain amplitude <sup>Δ</sup>*ε*eq <sup>2</sup> .

Modified cast irons and cast steels, which are used in the manufacture of brake disks, are another group of materials that can be exploited at elevated temperatures. In the works of Samec et al. [6], Peve et al. [7], and Li et al. [8], the authors analyzed the fatigue life of the following materials: EN-GJS-500-7, EN-GJL-250, and Cr-Mo-V. Material cracking mechanisms were studied within the LCF range at temperature up to 700 ◦C, and the influence of temperature on the material's physical properties was defined.

Alloy steels are another group of materials used to make constructions which are exposed to the action of elevated temperatures. Here, some examples include 8Cr-2WvTa, (RAFM) JLF-1, or 617M steels, which are used to build thermoreactors and machine parts exposed to the action of high temperatures. In the works of Ishii et al. [9], Mariappan et al. [10], and Shankar et al. [11], the characteristics of these steels' cyclic properties were determined as a function of temperature. Degradation mechanisms of materials subjected to the action of fatigue loads and temperatures were analyzed. In the article by Ishii et al. [9], an approach to modeling fatigue life of the material was also presented, accounting for the influence of creep on the material's fatigue life:

$$
\varepsilon\_{\rm eq}(N\_{\rm f}, T) = \varepsilon\_{\rm e}(N\_{\rm f}, T) + \varepsilon\_{\rm pl}(N\_{\rm f}, T) + \varepsilon\_{\rm cr}(N\_{\rm f}, T), \tag{4}
$$

where *ε*eq is the equivalent strain, *ε*<sup>e</sup> is the elastic strain, *ε*pl is the plastic strain, and *ε*cr is the creep-induced strain.

Works by Nagode and Zingsheim [12] and Nagode and Hack [13] present the results of experimental tests performed on steels for work at elevated temperatures: 10 CrMo 9 10 and X22CrMoV121. Fatigue damage accumulation models of the material under non-isothermal conditions at elevated temperatures were developed on the basis of these results. These models were developed for estimation of the fatigue life of materials which undergo changes of the load amplitude and temperature transient over the course of its exploitation. These models do not account for the influence of creep or hardening of the material on its fatigue life. The authors presented a stress and strain approach in which it is assumed that the instant working temperature will be accounted for. In models, stabilized hysteresis loops were described by means of the Ramberg–Osgood equation. The damage parameter *P*SWT was identified similarly as in the SWT (Smith–Watson–Topper) model:

$$\begin{split} P\_{\text{SWT}} &= \sqrt{\left(\sigma\_{\text{f}}^{\prime\prime}(T\_{\text{e}})(2\text{N}\_{\text{f}})^{2b(T\_{\text{e}})} + \sigma\_{\text{f}}^{\prime}(T\_{\text{e}})E(T\_{\text{e}})\right) \varepsilon\_{\text{f}}^{\prime}(T\_{\text{e}})(2N\_{\text{f}})^{b(T\_{\text{e}}) + c(T\_{\text{e}})}}{\left(\frac{\sigma\_{\text{f}}^{\prime}(T\_{\text{e}})}{K^{\prime}(T\_{\text{e}})}\right)^{\frac{1}{\mu^{\prime}(T\_{\text{e}})}}}, \\ c\_{\text{f}}^{\prime}(T\_{\text{e}}) &= \frac{\left(\frac{\sigma\_{\text{f}}^{\prime}(T\_{\text{e}})}{K^{\prime}(T\_{\text{e}})}\right)^{\frac{1}{\mu^{\prime}(T\_{\text{e}})}}}{\frac{b^{\prime}(T\_{\text{e}})}{\mu^{\prime}(T\_{\text{e}})}}, \end{split} \tag{5}$$

where *T*<sup>e</sup> is the actual temperature; *σ* f(*T*e) and *b*(*T*e) are, respectively, the coefficient and exponent of the fatigue life curve at the tested temperature; *K* (*T*e) and *n* (*T*e) are, respectively, the coefficient and exponent of the cyclic hardening curve at the tested temperature; and *ε* f(*T*e) and *c*(*T*e) are, respectively, the coefficient and exponent of the plastic fatigue life curve.

Elements of gas turbines, e.g., rotor blades working under extremely difficult conditions, are also made from Inconel 718 alloy. Test results for this material was given in the work of Schlesinger et al. [14]. Here, a dispersion-hardening nickel-chromium alloy was analyzed with respect to fatigue loading at temperatures up to 650 ◦C. Similarly, in a different study [15], the same authors studied the Inconel 718 alloy. They proposed a model for estimation of the fatigue life of material working at elevated temperatures. The Manson–Coffin equation was used during modeling, in which parameters of the fatigue life curve were made functionally dependent on the structure and grain size in the given material *G*, test temperature *T,* and strength parameter *S*:

$$\frac{\Delta \text{eq}(\text{G}, T, \text{S})}{2} = \frac{\sigma\_{\text{f}}'(\text{G}, T, \text{S})}{E(T)} (2\text{N}\_{\text{f}})^{b(\text{G}, T, \text{S})} + \epsilon\_{\text{f}}'(\text{G}, T)(2\text{N}\_{\text{f}})^{c(\text{G}, T)} \tag{6}$$

Materials adapted for manufacturing tools like casting molds and forging dies play an important role among materials intended for work at elevated temperature. In the study by Tunthawiroon et al. [16], the single-phase Co-29Cr-6Mo steel alloy, intended for aluminum casting molds, was tested. The study included analysis of the alloy's cracking mechanisms at temperatures up to 700 ◦C. A simple fatigue damage accumulation model at elevated temperatures was also proposed based on the Arrhenius equation:

$$D = N\_{\rm f}^{-1} = A \exp(Q/RT) \tag{7}$$

where *N*<sup>f</sup> is the number of cycles until crack initiation, *T* is the test temperature, *R* and *A* are material constants, and *Q* is the activation energy.

Similarly, the works of Gopinath et al. [17] and He et al. [18] present the results of experimental tests conducted on 720Li and HAYNES HR-120 superalloys within the low-cycle fatigue range at temperatures up to 980 ◦C. The material was tested under uniaxial tensile-compressive loads. The mechanisms accompanying the cracking of these materials were defined.

ACI HH50 austenitic stainless steels can also work at elevated temperatures. The results given in the work by Kim and Jang [19] can serve as evidence of this. Due to its high resistance to elevated temperatures, high fatigue strength, and resistance to pitting and corrosion, the material in question finds applications in parts of combustion engines and power unit structures of nuclear power plants. The paper investigates the material's degradation mechanisms and presents a model estimating fatigue

life in the LCF range. The model assumes that fatigue crack initiation will occur when damage reaches a limit value given by the following function:

$$\begin{cases} \frac{N\_{\text{f}}}{N\_{\text{0}}} \Big) \Big( \frac{\Delta \mathbf{r}\_{\text{P}}}{\Delta \mathbf{r}\_{\text{p}0}} \Big) \sinh \Big( \frac{\Delta \mathbf{r}\_{\text{P}}}{\Delta \mathbf{r}\_{\text{p}0}} \Big) = 1, & \text{where} \\\ N\_{\text{0}} = \frac{\text{avC}}{\varepsilon\_{0} \exp \left( \frac{Q}{RT} \right)}, & \\\ \Delta \varepsilon\_{\text{p}0} = \frac{\varepsilon\_{0} \exp \left( \frac{-Q}{RT} \right)}{\nu}, & \end{cases} \tag{8}$$

where *α* is the material constant associated with dislocation movement over the course of the material loading process, *<sup>ν</sup>* is the frequency of load change during the test, . *ε*<sup>0</sup> and C are material constants independent of test temperature, *Q* is activation energy, *R* is the gas constant, and *T* is the test temperature.

The non-linear creep damage accumulation model under uniaxial loads at elevated temperatures (700 ◦C) that was also developed on the basis of tests performed on stainless steel X-8-CrNiMoNb-16-16 was presented by Pavlou 2001 [20]. The model was developed and verified by other researchers. In the work of Grell et al. [21], attempts have been made to estimate durability under the conditions of uniaxial isothermal loads (constant and incremental) at elevated temperatures using the dependencies proposed by Pavlou 2001 [20]. The work demonstrates greater accuracy in predicting durability with the use of non-linear models of the accumulation of defects in relation to linear models.

Titanium alloys are the next group of materials used to manufacture components working at elevated temperatures. One study [22] analyzes the influence of different methods of the surface hardening of the Ti-6Al-4V titanium alloy on the fatigue life of this material under uniaxial tensile-compressive loading conditions at temperatures up to 555 ◦C. The influence of temperature on the evolution of the cyclic properties of the TNB-V2 titanium alloy is also presented in the work [23]. Specimens of the material were subjected to constant-amplitude strains within the low-cycle loading range at temperatures within the range of 550–850 ◦C. Due to its good mechanical properties at elevated temperatures, this alloy finds applications in motorization and aviation and in engine parts. The authors tested the material within the entire safe temperature range, and determined that, in the case of this alloy, the Manson–Coffin fatigue life model does not allow for the prediction of the material's fatigue life at elevated temperatures due to the length of the phase between fatigue crack initiation and its propagation.

The fatigue damage accumulation model proposed in this paper was designed to predict the fatigue life of the material operated at elevated temperatures. The concept of elevated temperature can be defined here, according to the works by Chen et al. [24], as the temperature corresponding to two-thirds of the melting point of the bulk material *T*c. In most cases, as temperature increases, a material's strength and fatigue properties are reduced in a predictable manner until a certain threshold temperature value is reached [25]. Under these conditions, the operation of technical machinery may be safe when the material's response to loading conditions and the working environment are known. However, if the conventional upper threshold of elevated temperature is crossed even slightly, the material's properties cannot be predicted directly. The material's strength suddenly drops (in a non-linear manner), and use of machinery under such conditions may pose a safety risk.

Uncontrolled loss of durability may lead to costly failures, standstills on process lines, and sometimes even to the death of those nearby when a structural component cracks. This is why information about how long and under what loading and temperature conditions a structural component can work safely is important. The fatigue damage accumulation models of materials working under low-cycle loading conditions at elevated temperatures presented in the literature are modified functions formulated to estimate the material's fatigue life at room temperature. The equations mentioned earlier can serve as an example: Manson–Coffin–Basquin, Ramberg–Osgood, and Smith–Watson–Topper. In these models, material parameters are determined independently for each of the analyzed temperatures. The process of calculating these parameters is not complicated in

itself; however, preparing the data for determining them is very expensive and time-consuming and requires specialized testing apparatus. These models are dedicated for specific loading cases and metal alloys. Due to the low number of functioning fatigue criteria in this scope, there is a justified need to create new models that will provide a more accurate description of the material cracking process, thus enabling calculation of fatigue life with greater accuracy. That is also why it this paper set out to create a new fatigue damage accumulation model that will make it possible to determine the number of cycles until failure of a material working under conditions of fatigue loads of constant amplitude at elevated temperatures with engineering accuracy.

#### **2. Materials and Methods**

#### *2.1. Fatigue Life of EN AW-2024T3 Aluminum Alloy at Elevated Temperatures*

To develop a fatigue damage accumulation model, it is indispensable to perform experimental tests. They constitute the physical basis of formulated numerical dependencies. The fatigue damage accumulation model proposed in this paper was created based on an experiment in which EN AW-2024T3 aluminum alloy samples were subjected to the action of high-amplitude fatigue loading in the form of uniaxial tension-compression and complex loading (tension-compression with simultaneous torsion) at elevated temperatures. More information about the experiment can be found in other works [4,5].

Experimental tests of fatigue life were carried out in accordance with the guidelines included in ISO 6892-1, ISO 12106: 2017, ASTM E8-04, ASTM E 606, GB/T 15248-2008, GB/T4338-2006, and (J IS) Z2201. Cylindrical and tubular test specimens (Figure 1) were made using machining. The working surfaces of the specimens were polished until satisfactory smoothness was obtained.

**Figure 1.** Diagram of test specimens for determination of the fatigue life of the material at elevated temperatures: (**a**) cylindrical specimen used in uniaxial loading tests (unit: mm) and (**b**) tubular specimen used in multiaxial loading tests (unit: mm).

The scope of performed tests covered, first of all, monotonic tests determining the influence of elevated temperatures on the mechanical properties of the EN AW-2024 T3 aerospace aluminum alloy. Table 1 presents the obtained results.


**Table 1.** Material parameters of the EN AW 2024 T3 alloy obtained for different temperature values [4].

Next, fatigue tests were carried out. Samples were subjected to uniaxial loading, periodically varying (tension-compression), until their failure at different set temperatures. Tests were performed at temperatures of 20, 100, 200, and 300 ◦C and at a constant value of the strain change range *ε*, frequency *f* = 1 Hz, and *R*<sup>ε</sup> = −1. The following values of control variable *ε* were applied: 0.015, 0.01, 0.0095, 0.008, and 0.006. The tests were performed on three samples for every value of the control variable's range and every test temperature.

Based on the performed tests, parameters describing the process of cyclic deformation of the material (as expressed by the Ramberg–Osgood equation) at elevated temperatures were determined by the following equation:

$$
\varepsilon\_{\mathbf{a}} = \varepsilon\_{\mathbf{a}}^{\mathbf{c}}(T) + \varepsilon\_{\mathbf{a}}^{\mathbf{P}}(T) = \frac{\sigma\_{\mathbf{a}}}{E(T)} + \left(\frac{\sigma\_{\mathbf{a}}}{K(T)}\right)^{1/n(T)} \tag{9}
$$

where *σ*<sup>a</sup> is the amplitude of normal stress induced by the action of periodic axial force, *K*(*T*) and *n*(*T*) are the coefficient and exponent of Ramberg-Osgood cyclic strain curve, which are dependent on temperature.

Table 2 presents the calculated parameters of the cyclic strain curve.

**Table 2.** Ramberg–Osgood equation parameters for the case of cyclic tension of the EN AW 2024T3 alloy at different temperature values [4].


In the next stage, parameters of the Manson–Coffin–Basquin strain-based fatigue life curve were determined. Its parameters, calculated for various temperatures, are given in Table 3.

**Table 3.** Coefficients and factors of the fatigue curve for the EN AW-2024T3 alloy at different temperature values [4].


Next, samples were subjected to the action of cyclic loading with torque moment and tensile or compressive force (during loading of samples, a constant quotient of maximum longitudinal and shear strain was assumed over the course of a loading cycle: *<sup>ε</sup>*a/*γ*<sup>a</sup> <sup>=</sup> <sup>√</sup>3). In all cases, loading histories formed oriented, multi-segment loops. Their configuration and parameters are given in Figure 2. Just as in the case of uniaxial tests, the loading process was controlled by means of increments of strain tensor components.

**Figure 2.** Evolutions of multiaxial loadings and the values of the control variable corresponding to them under the assumption of a constant quotient of maximum strains *ε*max = 0.003 and *γ*max = 0.00173 [5].

Tubular specimens (Figure 1b) were used in the experiment, and they were subjected to proportional and non-proportional tension-torsion cyclic loading with cycle asymmetry factor *R* =0(*ε*min/*ε*max; *γ*min/*γ*max). Fatigue tests were conducted for different combinations of axial and shear strains as well as for four temperature values: 20, 100, 200, and 300 ◦C. The specimen's kinematic input was controlled by means of the MTS 632.68F-08 (Eden Prairie, MN, USA) biaxial extensometer, and increments of strain components (linear and shear), averaged over a 25 mm-long segment of the measuring base, were used for this purpose. Three tests were conducted for every load with frequency of load changes *f* = 1 Hz.

The crack initiation moment was determined on the basis of the analysis of the maximum force value in the loading cycle. A 10% reduction of the force value relative to its maximum value at a given strain level was accepted as the criterion for crack initiation.

A compilation of the results of experimental tests (number of loading cycles until crack initiation) is given in Table 4.

The registered hysteresis loops for selected multiaxial loading configurations (individual loading components) have been presented in Figure 3.


**Table 4.** Results of experimental fatigue life tests for multiaxial loading configurations (tension/compression with torsion).

**Figure 3.** Hysteresis loops obtained at half of the fatigue life of the loading progression: (**a**) axial component for RS0; (**b**) non-dilatational component for various elevated temperature values for RS0; (**c**) axial component for RS45;(**d**) non-dilatational component for various elevated temperature values for RS45.

#### *2.2. Modeling of Damage Accumulation and Fracture of Material at Elevated Temperatures*

Preliminary tests based on the cyclic loading of a flat sample and observation of the amount of energy dissipating from the material due to the action of plastic strains were performed in order to investigate the fatigue damage accumulation process. The energy released as a result of fatigue loading was in the form of heat and was observed through a thermal imaging camera. Tests were performed at room temperature on the stand shown in Figure 4.

**Figure 4.** Test stand: 1—Fatigue testing machine, 2—thermal imaging camera IR, 3—*Aramis* video system DIC (Digital Image Correlation), 4—test sample.

During the tests, changes in the deformation fields (Aramis 4M-DIC system) and temperature fields (Cedip Titanium-IRT system) were recorded simultaneously. On their basis, areas of plastic deformation were identified and the distribution of plastic strain and energy converted into heat was determined. As can be seen in Figure 5, the temperature distribution corresponds to the strain distribution. In connection with the above, it can be assumed that the change in the temperature of the object caused by the action of external load is an indicator of damage generated during the loading process.

An example of temperature and strain fields is presented in Figure 5. Those are the fields corresponding to the localization of the plastic strain. This localization was manifested by the non-uniform temperature field and by the heterogeneity of the strain field on the specimen surface. The evolution of the temperature field due to the plastic deformation was used to determine the evolution of the field of heat dissipated by the specimen. The distribution of the heat and the plastic work on the surface of the deformed specimen was used to determine the failure mechanism.

The amount of heat emitted over the course of the tests can be related to the number of faults (damage) appearing in the material as a result of the action of external load. As the number of damages in the material increases, the object's temperature increases. Figure 6 presents the function of the maximum sample temperature at the crack initiation site as a function of test time (number of loading cycles).

**Figure 5.** Fields of strain and temperature at the selected time during cyclic loading (five cycles until failure).

**Figure 6.** Evolution of temperature over the course of the process of cyclic deformation of a flat sample ENAW 2024T3.

As can be seen in the initial phase of the loading process, damage accumulated according to the linear hypothesis of damage summation. At a certain time in the loading process, there was a sharp spike in the amount of damage, observed as an increase in released thermal energy. Damage then increased exponentially. In the tensile half-cycle of loading, the object's temperature increased as a result of growing damage. Meanwhile, the compressive half-cycle slowed down the damage accumulation process, resulting in a drop of the sample's temperature.

Based on observations of fatigue cracking planes obtained over the course of experimental tests, it was noted that slip lines form first on the samples' polished surfaces under the influence of cyclic loadings at elevated temperature, even with values less than that of fatigue strength, and they transform into fatigue slip bands (numerous parallel faults) as the number of loading cycles grew (Figure 7).

**Figure 7.** Slip bands (parallel faults) observed on the inner surface of a damaged tube sample (RS0—100 ◦C).

Similarly, as in the case of using room temperature [26], a fatigue damage accumulation criterion can be proposed based on the assumption that the material's damage state mainly depends on variable slips on physical planes and the normal stresses on these planes.

Considering the above, this paper proposes a damage accumulation model intended for the analysis of the fatigue life of constructional elements working under uniaxial and multiaxial loading conditions at elevated temperatures. This model assumes that permanent slip bands are the site of crack nucleation and defect formation [26], where crack initiation is the result of the accumulation of this damage. In this case, damage is induced by the action of the stress vector's normal component. Here, it is accepted that tensile stress in the loading cycle is responsible for generating new faults and accelerating the development of existing faults (temperature increase in loading cycle; Figure 6), while compressive stress retards the damage accumulation process (temperature drop in loading cycle; Figure 6). This fact was accounted for in the definition of the damage state variable d*ω*n.

The presented law of damage accumulation induced by plastic strains under elevated temperature conditions was linked to the stress-based damage accumulation function Ψp, the value of which is dependent on the value of normal stresses *σ*n, damage state variable *ω*<sup>n</sup> on the physical plane, and temperature *T*. The increment of the damage state variable on the physical plane d*ω*n, as caused by the development of plastic strains at elevated temperature *T*, was made dependent on the increment of plastic shear strains d*γ*<sup>n</sup> <sup>p</sup> on the same plane [27], namely:

$$\begin{cases} \operatorname{ d}\omega\_{\mathrm{n}}(\boldsymbol{\sigma},\operatorname{d\varepsilon},T) = A\_{\mathrm{P}}(T)\,\,\Psi\_{\mathrm{P}}(\sigma\_{\mathrm{n}},\omega\_{\mathrm{n}},T)\left|\operatorname{d}\gamma\_{\mathrm{n}}^{\mathrm{P}}\right| & \text{for } \sigma > 0 \text{ i } \mathrm{d}\varepsilon > 0\\\ \operatorname{ d}\omega\_{\mathrm{n}}(\boldsymbol{\sigma},\operatorname{d\varepsilon},T) = 0 & \text{for } \sigma \le 0 \text{ i } \mathrm{d}\varepsilon \le 0 \end{cases} \tag{10}$$

where *A*p(*T*) is the material variable describing the evolution of the material's plastic properties depending on the actual stress state and temperature. In numerical simulations, the following values of parameter *A*<sup>p</sup> were accepted: *A*p(20 ◦C) = 0.55; *A*p(100 ◦C) = 1.4; *A*p(200 ◦C) = 1.8; and *A*p(300 ◦C) = 1.95.

The damage accumulation mechanism described by this law is presented in Figure 8.

**Figure 8.** Generation of defects on the physical plane by slip band [22].

Similar to the dependencies presented in the work [26], the damage accumulation function *Ψ*<sup>p</sup> can be proposed in the following form:

$$\Psi\_{\mathbb{P}}(\sigma\_{n\prime}\omega\_{n\prime}T) = \left(1 - \frac{1}{3}R\_{\sigma}(\sigma\_{n\prime}\omega\_{n\prime}T)\right)^{1/c(T)}\tag{11}$$

where *R*σ is the stress function of cracking given by the condition of maximum normal stresses, which is dependent on the test temperature:

$$R\_{\sigma}(\sigma\_{n}, \omega\_{n}, T) = \frac{\sigma\_{n}}{\sigma\_{\mathfrak{c}}(\omega\_{n}, T)} \tag{12}$$

*c*(*T*) is the exponent present in the Manson–Coffin–Basquin equation, which is also dependent on temperature (values of parameter *c*(*T*) used in numerical simulations are given in Table 3).

The crack initiation criterion can be written in the form of the condition of maximum normal stresses (linked to the physical plane), namely:

$$R\_{\mathbf{f}\sigma} = \max\_{\left(\mathbf{n}\right)} R\_{\sigma} \left(\sigma\_{\mathbf{n}}, \omega\_{\mathbf{n}\_{\sigma}}, T\right) = \max\_{\left(\mathbf{n}\right)} \left(\frac{\sigma\_{\mathbf{n}}}{\sigma\_{\mathbf{c}}(\omega\_{\mathbf{n}\_{\sigma}}, T)}\right) = 1 \tag{13}$$

where *R*f<sup>σ</sup> is the stress cracking coefficient dependent on the state of stress and damage in the material as well as on the test temperature *T*, and *σ*<sup>c</sup> is the actual value of normal failure stresses for the material at the given temperature, which is dependent on the damage state variable and temperature, namely:

$$
\sigma\_{\mathfrak{c}}(\omega\_n, T) = \sigma\_{\mathfrak{c}0}(T)(1 - \omega\_n), \tag{14}
$$

where *σ*c0(*T*) is the critical stress for undamaged material, which is dependent on temperature. The value of parameter *σ*c0(*T*) can be equated with the value of the coefficient of the Manson–Coffin fatigue life curve *σ*<sup>f</sup> (*T*). Critical stress values according to Table 3 were used to calculate the accumulation of fatigue damage by means of the proposed model.

An additional cracking condition should also be introduced, in which it is accepted that crack initiation will occur when the damage state variable induced by plastic strains on any physical plane reaches critical value, that is:

$$\max\_{\mathbf{(n)}} \omega\_{\mathbb{N}} = 1.\tag{15}$$

At that moment, the material has no strength (*σ*<sup>c</sup> = 0).

In the presented model, stress and strain values were determined independently for each test temperature by using the generalized Hooke's law, the Huber-von Mises yield criterion, the gradient flow law, and Mroz–Garud multiple-surface material hardening model associated with the yield criterion. They are an integral part of the numerical model of damage accumulation, making it possible to determine stress and strain state in any loading state [28].

When estimating the material's fatigue life, the principle of additivity of components of elastic and plastic strain increment tensors was adopted. Increments of elastic strain were determined by means of the generalized Hooke's law, and increments of plastic strain were calculated by means of the gradient flow law, associated with the yield surface, namely (*T* = const):

$$\mathbf{d}\varepsilon\_{\mathrm{ij}} = \frac{1+\nu(T)}{E(T)}\mathbf{d}\sigma\_{\mathrm{ij}} - \frac{\nu(T)}{E(T)}\mathbf{d}\sigma\_{\mathrm{kk}}\delta\_{\mathrm{ij}} + \mathrm{d}\lambda(T)\frac{\partial f(T)}{\partial \sigma\_{\mathrm{ij}}},\tag{16}$$

where both the proportionality coefficient d*λ* and the elasticity constants *E* and *ν* are dependent on temperature. In numerical simulations, the values of parameters determined in monotonic tensile tests were used according to Table 1. The material's hardening curve was plotted according to the parameters given in Table 2.

Yield surface *f* = 0 was determined using the Huber-von Mises criterion:

$$f = \frac{3}{2}(s\_{ij} - a\_{ij})\left(s\_{ij} - a\_{ij}\right) - R(T)^2 = 0,\tag{17}$$

where *R*(*T*) defines the temperature-dependent size of the yield surface.

#### **3. Results**

Simulations of the fatigue life of the aerospace aluminum alloy within the range of uniaxial loads (tension-compression) at elevated temperatures were conducted for the modified Manson-Coffin-Basquin model (Equation (1)), semi-empirical model S-S\_1 (Equation (2)), and the proposed approach S-S\_2 (Equation (10)). Damage accumulation simulations were carried out for four different temperatures and five strain levels. The results of the simulations are provided in Table 5.

**Table 5.** Experimental data and results of numerical simulations of fatigue damage accumulation for uniaxial loads at elevated temperatures.


Figure 9a presents the evolution of the damage state variable over the course of the fatigue loading process, plotted based on Equation (10) for the analyzed levels of temperature and strain *ε*<sup>a</sup> = 0.015. As can be seen in its evolution, the fatigue damage accumulation process progresses in a near-linear manner in the first loading stage until it reaches a certain limit state, and after this limit state is crossed, there is a sudden spike in damage. This was confirmed by thermograms (Figure 9b), which show a sudden temperature increase in the final phase of periodic loading. The characters of the fatigue damage accumulation process and of the evolution of the sample's temperature were similar over the course of cyclic loading. It can be surmised that the object's temperature may be an indicator of the number of damage accumulated over the course of the material loading process.

**Figure 9.** (**a**) Evolution of the damage state variable over the course of the action of fatigue load for different temperatures and strain levels *ε*a = 0.015. (**b**) Evolution of sample temperature over the course of cyclic loading at RT–Room Temperature.

Temperature influences the rate at which the material subjected to the action of cyclic loads reaches the state of "damage saturation". Samples loaded at 300 ◦C reached the limit state after working 0.26 of the number of cycles until failure, while at room temperature, this state was reached after 0.38 of the number of cycles until failure. This state can be explained by the fact that movement of dislocations is facilitated as temperature increases, since they do not encounter obstacles that could slow them down on their path due to the elevated mobility of atoms. After crossing the "limit" state, further damage accumulation progresses exponentially. Microcracks appear in the material, leading to the material's decohesion as they are joined.

Figure 10 presents a comparison of numerical calculations of fatigue life (for the modified Manson-Coffin-Basquin model (Equation (1)), semi-empirical model S-S\_1 (Equation (2)), and the proposed approach S-S\_2 (Equation (10)) with experimental data for uniaxial tension-compression of ENAW2024T3 aluminum alloy samples at elevated temperature: (a) 20 ◦C; (b) 100 ◦C; (c) 200 ◦C, (d) 300 ◦C.

**Figure 10.** Comparison of numerical simulations of fatigue life with experimental data for uniaxial tension-compression of ENAW2024T3 aluminum alloy samples at elevated temperature: (**a**) 20 ◦C; (**b**) 100 ◦C; (**c**) 200 ◦C, (**d**) 300 ◦C.

Next, the proposed fatigue damage accumulation model was also verified for the case of multiaxial loads. The results of original experimental tests performed on the EN AW-2024T3 aerospace aluminum alloy were used for this purpose [5]. Figure 11 presents normalized evolutions of the damage state variable, plotted for load paths RS0 and RS90 at the analyzed temperature. A comparison between the results of experimental tests and of numerical simulations using the proposed SS2 approach (Equation (10)) and the SS1 semi-empirical model (Equation (2)) described in Reference [5] is given in Figure 12 and in Table 6.

**Figure 11.** Evolution of damage state variable normalized with respect to cycles until crack initiation at the analyzed temperature for load path: (**a**) RSO; (**b**) RS90.

**Figure 12.** Comparison of the results of numerical simulations (strain-based approach) with experimental data for the EN AW-2007 aluminum alloy for complex loading histories.


**Table 6.** Comparison of the results of numerical simulations using the SS1 model (Szusta Seweryn 2015) and SS2 model (Equation (10)) and experimental tests of the number of cycles *N*<sup>f</sup> for different loading paths (EN AW-2024 T3).

The conducted simulations indicate good consistency between results obtained using the proposed strain-based model and the authors' original experimental results, within the scope of both uniaxial and complex loading.

#### **4. Conclusions**

This paper presents an approach that makes it possible to estimate the fatigue life of materials working under conditions of cyclic loading at elevated temperatures. An undoubted advantage of the model is its capability of accounting for the loading history in the fatigue damage accumulation process. This is done when determining stress or strain tensor components depending on the method of effecting force (force/displacement). The tests verifying the developed model show that, in the initial loading phase, as a result of the action of cyclic loads, damage accumulates gradually with every cycle of high-amplitude loading until the material reaches a certain limit state. After this state is exceeded, the "damage density" is high enough that the material is no longer able to safely carry the load applied to it. The process related to the appearance of microcracks in the material then occurs, and this leads to a sharp rise in damage during further loading. After that, faults increase exponentially until macrocracks form, leading to decohesion of the material. Elevated temperatures foster damage accumulation by facilitating the yielding of the material; dislocations can move more easily within the material. Moreover, the material's temperature changes the character of cracking and the time until failure.

**Author Contributions:** J.S. was the author of the concept of the article, made calculations, and analyzed the results. A.S. was a scientific consultant.

**Funding:** The investigation described in this paper is part of the research project no. G/WM/5/2017 Sponsored by the Polish National Science Centre and realized in Bialystok University of Technology.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


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