**Computing the Growth of Small Cracks in the Assist Round Robin Helicopter Challenge**

#### **Rhys Jones 1,2,\* , Daren Peng <sup>1</sup> , R.K. Singh Raman 1,3 and Pu Huang <sup>1</sup>**


Received: 17 June 2020; Accepted: 8 July 2020; Published: 14 July 2020

**Abstract:** Sustainment issues associated with military helicopters have drawn attention to the growth of small cracks under a helicopter flight load spectrum. One particular issue is how to simplify (reduce) a measured spectrum to reduce the time and complexity of full-scale helicopter fatigue tests. Given the costs and the time scales associated with performing tests, a means of computationally assessing the effect of a reduced spectrum is desirable. Unfortunately, whilst there have been a number of studies into how to perform a damage tolerant assessment of helicopter structural parts there is currently no equivalent study into how to perform the durability analysis needed to determine the economic life of a helicopter component. To this end, the present paper describes a computational study into small crack growth in AA7075-T7351 under several (reduced) helicopter flight load spectra. This study reveals that the Hartman-Schijve (HS) variant of the NASGRO crack growth equation can reasonably accurately compute the growth of small naturally occurring cracks in AA7075-T7351 under several simplified variants of a measured Black Hawk flight load spectra.

**Keywords:** small cracks; helicopter flight load spectra; FALSTAFF flight load spectra; fatigue crack growth

### **1. Introduction**

It is now known that "ab initio" design and aircraft sustainment [1,2] are best tackled using different computational tools. United States Air Force (USAF) airworthiness standard MIL-STD-1530D [3] states that analysis is the key to both damage tolerant design and to assessing the economic life of military aircraft. MIL-STD-1530D also states that the primary role of testing is "to validate or correct analysis methods and results and to demonstrate that requirements are achieved". The USAF-McDonnell Douglas study into the economic life of USAF F-15 aircraft [1] was arguably the first to reveal that sustainment analyses need to use the short crack da/dN versus ∆*K* curve, and not the da/dN versus ∆*K* curve determined as per the US American Society for Testing and Materials (ASTM) fatigue test standard ASTM E647-13a [4]. (The term durability is defined in MIL-STD-1530D [3] as: "Durability is the attribute of an aircraft structure that permits it to resist cracking, corrosion, thermal degradation, delamination, wear, and the effects of foreign object damage for a prescribed period of time". MIL-STD-1530D [3] defines the term economic service life: The economic service life is the period during which it is more cost-effective to maintain, repair, and modify an aircraft component or aircraft than to replace it.) This conclusion is now echoed in ASTM E647-13a, Appendix X3. Whereas the ability of various crack growth equations to capture the growth of long cracks under a representative helicopter flight load spectrum has been studied [5–7] as part of the "Helicopter Damage Tolerance Round-Robin" challenge [8], there are few studies into the ability of crack growth equations to model small crack growth of small under helicopter flight load spectra. This shortcoming is particularly important since the durability/economic "initial flaw assumptions" contained in the US Joint Services Structures Guidelines JSSG2006 [9], which in MIL-STD-1530D are termed as equivalent initial damage sizes (EIDS), are typically 0.05 inches (0.127 mm). Indeed, this size of EIDS is also referenced in Structures Bulletin EZ-19-01, which presents the USAF approach to the Durability and Damage Tolerance Certification for Additive Manufacturing of Aircraft Structural Metallic Parts [10]. The importance of a validated predictive capability is highlighted in MIL-STD-1530D Sections 5.2.5 and 5.2.6, which state that a damage tolerance and a durability analysis must be performed for all aircraft, and in Section 5.3.4 which states that the purpose of full scale fatigue testing is to "validate or correct the analysis". MIL-STD-1530D also states that a factor of 2 is to be used on these analyses. The Australian Defence Science and Technology (DST) Group's small crack Helicopter Round Robin Challenge [11,12] is, to the best of the author's knowledge, the first attempt to address this shortcoming, i.e., the lack of a validated analysis for small crack growth under a helicopter flight load spectrum.

It has previously been shown [7] that the Hartman-Schijve (HS) crack growth equation [2], which is an extension of a concept first proposed in [13], accurately predicted crack growth in the "Helicopter Damage Tolerance Round-Robin" challenge [8]. It has also been shown [2,7,14–18] that this formulation can also predict small crack growth under maritime aircraft, combat aircraft, and civil aircraft flight load spectra, and that the small crack equation needed for a durability/economic life analysis can often be determined from the associated long crack equation by setting the fatigue threshold term to a small value. The HS equation has also been shown to hold for crack growth in adhesively-bonded joints, bonded wood structures, and both bridge and rail steels [19–25], as well as for delamination crack growth in composite structures [26–32]. Consequently, the focus of the present paper is to examine if the HS crack growth equation can also capture crack growth in the DST Advancing Structural Simulation to drive Innovative Sustainment Technologies (ASSIST) small crack Helicopter Round Robin Challenge.

The general form of the HS equation used in this paper is as given in [2], viz:

$$\text{da/dN} = D \text{ (}\Delta\text{\kappa)}^{\text{\textquotedblleft}} \text{ }\tag{1}$$

where *a* is the crack length/depth, *N* is the number of cycles, *D* is a material constant and *n* is another material constant that is often approximately 2. The crack driving force ∆κ used in Equation (1) was first suggested by Schwalbe [33], viz:

$$
\Delta\kappa = (\Delta K - \Delta K\_{\text{thr}})/(1 - K\_{\text{max}}/A)^{1/2} \tag{2}
$$

here *K* is the stress intensity factor, *K*max and *K*min are the maximum and minimum values of the stress intensity factor seen in the cycle, ∆*K* = (*K*max − *K*min) is the range of the stress intensity factor that is seen in a cycle, ∆*K*thr is the "effective fatigue threshold", and *A* is the cyclic fracture toughness. As per [2,14,16,18], the values of the terms ∆*K*thr and *A* are chosen to fit the measured data. As further explained in [34], the term ∆*K*thr is related to the ASTM E647-13a definition of the fatigue threshold <sup>∆</sup>*K*th, namely the value of <sup>∆</sup>*<sup>K</sup>* at a value of da/dN of 10−<sup>10</sup> <sup>m</sup>/cycle, by:

$$
\Delta K\_{\rm th} = \Delta K\_{\rm thr} + (10^{-10}/D)^{1/n} \tag{3}
$$

as a general rule, crack growth predictions made using Equations (1) and (2) are quite sensitive to the value used for ∆*K*thr, and relatively insensitive to the value of *A*.

The HS equation has also been shown [34–38] to capture the growth of both small and long cracks in additively manufactured materials (AM), and has an ability to account for the effect of residual stresses in both conventionally and additively manufactured materials [39]. It has also been shown to be able to capture the effect of surface roughness on the fatigue life of a component [39]. This finding

is particularly important given the statement by the Under Secretary of Defense, Acquisition and Sustainment [40] that "AM parts can be used in both critical and non-critical applications", and the statement in the USAF Structures Bulletin EZ-19-01 [10] that for AM parts that the most difficult challenge is to establish an "accurate prediction of structural performance" specific to durability and damage tolerance (DADT). As such it is envisaged that if it can be shown that the HS equation can be shown to reasonably accurately compute the growth of small cracks subjected to helicopter flight load spectra, then it may be useful for assessing if an AM (helicopter) replacement part, or an AM repair to a helicopter part, meets the durability requirement inherent in the Structures Bulletin EZ-19-01. part, or an AM repair to a helicopter part, meets the durability requirement inherent in the Structures Bulletin EZ-19-01. **2. Materials and Methods**  The majority of references quoted in this paper are taken from peer reviewed journals. The refereed conferences, proceedings, and texts referenced are either publicly available, or available from Google searches. Thirty-nine of the journal papers referenced are in journals that are either listed in SCOPUS or in the World of Science (WOS). The book chapters referenced are listed in SCOPUS. In

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to durability and damage tolerance (DADT). As such it is envisaged that if it can be shown that the

#### **2. Materials and Methods** the case of conference papers, one is in the Proceedings of 13th International Conference on Fracture

The majority of references quoted in this paper are taken from peer reviewed journals. The refereed conferences, proceedings, and texts referenced are either publicly available, or available from Google searches. Thirty-nine of the journal papers referenced are in journals that are either listed in SCOPUS or in the World of Science (WOS). The book chapters referenced are listed in SCOPUS. In the case of conference papers, one is in the Proceedings of 13th International Conference on Fracture (ICM13), two are contained in the Proceedings of the 1st Virtual European Conference on Fracture, two are available on Research Gate; seven references are available on various US Department of Defense DTIC websites, one is available on the American Helicopter Society website, and another is on the US Pentagon website. Keywords that were used in these searchers were durability, damage tolerance, Hartman-Schijve, small cracks, additive manufacturing, crack growth in operational aircraft, and aircraft certification. (ICM13), two are contained in the Proceedings of the 1st Virtual European Conference on Fracture, two are available on Research Gate; seven references are available on various US Department of Defense DTIC websites, one is available on the American Helicopter Society website, and another is on the US Pentagon website. Keywords that were used in these searchers were durability, damage tolerance, Hartman-Schijve, small cracks, additive manufacturing, crack growth in operational aircraft, and aircraft certification. The paper begins by using the HS equation [2] to compute crack growth in an AA7075-T7352 specimen under a FALLSTAFF (which is an industry standard combat aircraft spectrum) flight load spectrum. It is then used evaluate crack growth under several variants of a US Army Blackhawk spectrum.

The paper begins by using the HS equation [2] to compute crack growth in an AA7075-T7352 specimen under a FALLSTAFF (which is an industry standard combat aircraft spectrum) flight load spectrum. It is then used evaluate crack growth under several variants of a US Army Blackhawk spectrum. **3. Crack Growth in 7075-T7351** 

#### **3. Crack Growth in 7075-T7351** *3.1. Crack Growth under a FALSTAFF Flight Load Spectrum*

#### *3.1. Crack Growth under a FALSTAFF Flight Load Spectrum* Before we can compute crack growth in the Helicopter Challenge, we need to establish the

Before we can compute crack growth in the Helicopter Challenge, we need to establish the constants in the HS equation. To do this we examined the crack length histories given in [41] for the growth of through-the-thickness cracks in a 6.35 mm thick middle tension (MT) panel, with a rectangular cross section, tested under a FALSTAFF flight load spectrum. A plan view of the specimen geometry is shown in Figure 1. The specimens were pre-cracked to a length of approximately 2 mm before the main fatigue test. The specimens were then tested under FALSTAF, an industry-standard combat aircraft load spectrum. The test loads were applied at a frequency of 10 Hz, see [41]. The maximum load in the spectrum was 60 kN. This corresponds to a remote stress of 157.48 MPa in the working section. One block of FALSTAFF load spectrum consisted of 9006 cycles. This equates to 100 equivalent flight hours. The various crack growth histories for the 25 tests performed in [41] are shown in Figure 2. constants in the HS equation. To do this we examined the crack length histories given in [41] for the growth of through-the-thickness cracks in a 6.35 mm thick middle tension (MT) panel, with a rectangular cross section, tested under a FALSTAFF flight load spectrum. A plan view of the specimen geometry is shown in Figure 1. The specimens were pre-cracked to a length of approximately 2 mm before the main fatigue test. The specimens were then tested under FALSTAF, an industry-standard combat aircraft load spectrum. The test loads were applied at a frequency of 10 Hz, see [41]. The maximum load in the spectrum was 60 kN. This corresponds to a remote stress of 157.48 MPa in the working section. One block of FALSTAFF load spectrum consisted of 9006 cycles. This equates to 100 equivalent flight hours. The various crack growth histories for the 25 tests performed in [41] are shown in Figure 2.

**Figure 1.** Schematic diagram of the MT specimen used. **Figure 1.** Schematic diagram of the MT specimen used.

The value of *A* was taken from that given in [14] for tests on small cracks in AA7075-T7351, viz: *A* =

The similarity between the da/dN versus ΔK crack growth curves associated with AA7075-T6

performed on cracks in an MT panel.

111 MPa √m. A similar value is given in [43]. The resultant measured and computed crack growth histories are shown in Figure 2, and the values of *A* and Δ*Kthr* used in the analysis are given in Table 1. Figure 2 reveals excellent agreement between the measured and computed crack growth histories. Figure 2 also reveals that, as reported in [2,15,16,18,38,44], the scatter in the crack growth histories can be captured by allowing for variability in the term Δ*Kthr*. Figure 3 presents the crack growth

**Figure 2.** Plot showing the cycle-by-cycle analysis of MT specimens under FALSTAFF load spectrum. **Figure 2.** Plot showing the cycle-by-cycle analysis of MT specimens under FALSTAFF load spectrum.

**Table 1.** Values of *A* and ∆*K*thr used in Figure 2 when computing the crack growth curves for the various tests. **Test** *A* **(MPa √m) ∆***K***thr (MPa √m)** G 111 1.3 K 111 1.79 L 111 1.1 N 111 1.1 U 111 1.48 V 111 1.6 X 111 1.45 Y 111 1.6 G 111 1.3 The similarity between the da/dN versus ∆K crack growth curves associated with AA7075-T6 and AA7075-T7351 meant that the values of the constants D and n in Equation (1) for AA7075-T7351 could be taken to be as given in [14] for AA7075-T6, namely: *D* = 1.86 × 10−<sup>9</sup> (MPa−<sup>2</sup> cycle−<sup>1</sup> ), and *n* = 2. The value of *A* was taken from that given in [14] for tests on small cracks in AA7075-T7351, viz: *<sup>A</sup>* <sup>=</sup> 111 MPa <sup>√</sup> m. A similar value is given in [42]. The resultant measured and computed crack growth histories are shown in Figure 2, and the values of *A* and ∆*K*thr used in the analysis are given in Table 1. Figure 2 reveals excellent agreement between the measured and computed crack growth histories. Figure 2 also reveals that, as reported in [2,15,16,18,38,43], the scatter in the crack growth histories can be captured by allowing for variability in the term ∆*K*thr. Figure 3 presents the crack growth history plotted using log-linear axes. Figure 3 reveals that crack growth in these 25 tests could be approximated as being exponential. As explained in [2] this is due to the test program being performed on cracks in an MT panel.


**Table 1.** Values of *A* and ∆*K*thr used in Figure 2 when computing the crack growth curves for the various tests.

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**Figure 3.** The crack growth history plotted using log-linear axes. **Figure 3.** The crack growth history plotted using log-linear axes.

#### *3.2. Short Cracks in 7075-T7351 3.2. Short Cracks in 7075-T7351 3.2. Short Cracks in 7075-T7351*

Having determined the crack growth equation for AA7075-T7351 a comparison between the *R* = 0.8 AA7075-T73 short crack da/dN versus ΔK curve presented in [42] and the corresponding curve predicted using Equation (1) with *D* = 1.86 x 10−9, and *n* = 2, Δ*Kthr* = 0.6 MPa √m and *A* = 111 MPa √m is given in Figure 4. Figure 4 reveals that there is an excellent agreement between the computed and the measured curve presented in [42]. The next section will use these values of *D, n*, Δ*Kthr*, and *A* to compute crack growth in the DST Helicopter Challenge. Having determined the crack growth equation for AA7075-T7351 a comparison between the *R* = 0.8 AA7075-T73 short crack da/dN versus ∆K curve presented in [44] and the corresponding curve predicted using Equation (1) with *D* = 1.86 × 10−<sup>9</sup> , and *<sup>n</sup>* <sup>=</sup> 2, <sup>∆</sup>*K*thr <sup>=</sup> 0.6 MPa <sup>√</sup> m and *<sup>A</sup>* <sup>=</sup> 111 MPa <sup>√</sup> m is given in Figure 4. Figure 4 reveals that there is an excellent agreement between the computed and the measured curve presented in [44]. The next section will use these values of *D, n*, ∆*K*thr, and *A* to compute crack growth in the DST Helicopter Challenge. Having determined the crack growth equation for AA7075-T7351 a comparison between the *R* = 0.8 AA7075-T73 short crack da/dN versus ΔK curve presented in [42] and the corresponding curve predicted using Equation (1) with *D* = 1.86 x 10−9, and *n* = 2, Δ*Kthr* = 0.6 MPa √m and *A* = 111 MPa √m is given in Figure 4. Figure 4 reveals that there is an excellent agreement between the computed and the measured curve presented in [42]. The next section will use these values of *D, n*, Δ*Kthr*, and *A* to compute crack growth in the DST Helicopter Challenge.

**Figure 4.** Comparison with the small/short crack growth curve given in [42]. **Figure 4.** Comparison with the small/short crack growth curve given in [42]. **Figure 4.** Comparison with the small/short crack growth curve given in [44].

#### **4. Computing Crack Growth in the DST Small Crack Helicopter Round Robin Challenge** helicopter. The crack growth data and details of the specimen and the various helicopter flight load spectra were made publicly available via the DST ASSIST initiative and are available at [11,12].

**4. Computing Crack Growth in the DST Small Crack Helicopter Round Robin Challenge** 

The focus of problem proposed in the (DST) Group's small crack Helicopter Round Robin Challenge was to compute the growth of small cracks in 8.4 mm thick AA7075-T7351 specimens under a range of simplified helicopter flight load spectra [11,12]. The baseline spectrum, which is described in [45], was obtained from a flight strain survey conducted on a US Army H-60 Black Hawk helicopter. The crack growth data and details of the specimen and the various helicopter flight load spectra were made publicly available via the DST ASSIST initiative and are available at [11,12]. The load sequences provided by DST as part of the ASSIST Round Robin were termed IRF-E14, IRF-E15, and IRF-E16. These spectra are simplified/reduced versions of the baseline spectrum, where different numbers of small amplitude cycles have been removed. Sequences termed CSL090SSXX, which are truncated versions of the IRF-E16 spectrum, were also provided. The CSL090SSXX spectra had: a) an additional 90% of the smallest cycles removed, and b) the mid-range cycles were scaled by XX%.

described in [45], was obtained from a flight strain survey conducted on a US Army H-60 Black Hawk

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The focus of problem proposed in the (DST) Group's small crack Helicopter Round Robin Challenge was to compute the growth of small cracks in 8.4 mm thick AA7075-T7351 specimens

The load sequences provided by DST as part of the ASSIST Round Robin were termed IRF-E14, IRF-E15, and IRF-E16. These spectra are simplified/reduced versions of the baseline spectrum, where different numbers of small amplitude cycles have been removed. Sequences termed CSL090SSXX, which are truncated versions of the IRF-E16 spectrum, were also provided. The CSL090SSXX spectra had: (a) an additional 90% of the smallest cycles removed, and (b) the mid-range cycles were scaled by XX%. A plan view of the test specimens used by DST in the ASSIST test program [11] is shown in Figure 5. The number of turning points in each of the spectra used in this test study are given in Table 2. The surface of the specimen was etched to promote organic crack nucleation, using a solution of Hydrofluoric acid (1%), Nitric acid (50%), and water (49%). Further details of the test specimen and the spectra are given in [11,12].

A plan view of the test specimens used by DST in the ASSIST test program [11] is shown in Figure 5. The number of turning points in each of the spectra used in this test study are given in Table 2. The surface of the specimen was etched to promote organic crack nucleation, using a solution of Hydrofluoric acid (1%), Nitric acid (50%), and water (49%). Further details of the test specimen and the spectra are given in [11,12]. **Table 2.** The number of turning points in each spectrum. **IRF-E15 IRF-E15 IRF-E16 CSL090SS00 CSL090SS05 CSL090SS15 CSL090SS20** 82,839 248,255 649,666 64,958 64,958 64,958 64,958

**Figure 5.** Geometry of the 8.4 mm thick test specimen. (units are in mm).

**Figure 5.** Geometry of the 8.4 mm thick test specimen. (units are in mm). **Table 2.** The number of turning points in each spectrum.


#### growth histories associated with the Round Robin tests subjected to the following spectra: IRF-E14, IRF-E15, IRF-E16, CSL090SS00, CSL090SS05, CSL090SS15, and CSL090SS20. The analysis was *Short Cracks in 7075-T7351*

performed using both *A* = 40 MPa √m, and *A* = 111 MPa √m. The value of *A* = 40 MPa √m was investigated since prior DST constant amplitude tests [41] had yielded values of *A* ≈ 32 MPa √m for twenty four mm thick AA7075-T7351 specimens, and *A* ≈ 40 MPa √m for three mm thick AA7075- T7351 specimens. The value of *A* = 111 MPa √m was investigated since it is associated with the short crack tests reported in [42]. As per the requirements enunciated in the ASSIST challenge [11], the initial crack size was taken to be a centrally located 0.01 mm semi-circular surface crack. The stress intensity factors were computed using the methodology outlined in [46]. Comparisons between the measured and computed crack growth histories are given in Figures 6–12, where the computed crack depth histories are labelled "Computed Δ*Kthr* = 0.6 A = XX", where XX is either 40 or 111 depending on what value of *A* was used in the analysis. Here it should be noted that, as shown in Figures 6–12, each spectrum test program had a number of repeated tests. Figures 6–12 reveal that there is little Equation (1), with *D* = 1.86 × 10−<sup>9</sup> , *<sup>n</sup>* <sup>=</sup> 2, and <sup>∆</sup>*K*thr <sup>=</sup> 0.6 MPa <sup>√</sup> m, was used to predict the crack growth histories associated with the Round Robin tests subjected to the following spectra: IRF-E14, IRF-E15, IRF-E16, CSL090SS00, CSL090SS05, CSL090SS15, and CSL090SS20. The analysis was performed using both *<sup>A</sup>* <sup>=</sup> 40 MPa <sup>√</sup> m, and *<sup>A</sup>* <sup>=</sup> 111 MPa <sup>√</sup> m. The value of *<sup>A</sup>* <sup>=</sup> 40 MPa <sup>√</sup> m was investigated since prior DST constant amplitude tests [41] had yielded values of *<sup>A</sup>* <sup>≈</sup> 32 MPa <sup>√</sup> m for twenty four mm thick AA7075-T7351 specimens, and *<sup>A</sup>* <sup>≈</sup> 40 MPa <sup>√</sup> m for three mm thick AA7075-T7351 specimens. The value of *<sup>A</sup>* <sup>=</sup> 111 MPa <sup>√</sup> m was investigated since it is associated with the short crack tests reported in [44]. As per the requirements enunciated in the ASSIST challenge [11], the initial crack size was taken to be a centrally located 0.01 mm semi-circular surface crack. The stress intensity factors were computed using the methodology outlined in [46]. Comparisons between the measured and computed crack growth histories are given in Figures 6–12, where the computed crack depth histories are labelled "Computed ∆*K*thr = 0.6 *A* = XX", where XX is either 40 or 111 depending on what value of *A* was used in the analysis. Here it should be noted that, as shown in Figures 6–12, each spectrum test program had a number of repeated tests. Figures 6–12 reveal that there is little difference between the crack growth histories computed using *<sup>A</sup>* <sup>=</sup> 40 MPa <sup>√</sup> m or *<sup>A</sup>* <sup>=</sup> 111 MPa <sup>√</sup> m. This is because the

IRF-E15.

IRF-E15.

majority of the life is consumed in growing to a depth of 1 mm. We also see that there is reasonable agreement between the measured and predicted crack growth curves. difference between the crack growth histories computed using *A* = 40 MPa √m or *A* = 111 MPa √m. This is because the majority of the life is consumed in growing to a depth of 1 mm. We also see that there is reasonable agreement between the measured and predicted crack growth curves. 10

there is reasonable agreement between the measured and predicted crack growth curves.

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This is because the majority of the life is consumed in growing to a depth of 1 mm. We also see that

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**Figure 6.** The measured and computed crack depth histories for the helicopter flight load spectrum **Figure 6.** The measured and computed crack depth histories for the helicopter flight load spectrum IRF-E14. IRF-E14.

**Figure 7.** The measured and computed crack depth histories for the helicopter flight load spectrum **Figure 7.** The measured and computed crack depth histories for the helicopter flight load spectrum **Figure 7.** The measured and computed crack depth histories for the helicopter flight load spectrum IRF-E15.

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CLS0900SS00.

CLS0900SS00.

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**Figure 8.** The measured and computed crack depth histories for the helicopter flight load spectrum **Figure 8.** The measured and computed crack depth histories for the helicopter flight load spectrum IRF-E16. IRF-E16.

**Figure 9.** The measured and computed crack depth histories for the helicopter flight load spectrum **Load Blocks Figure 9.** The measured and computed crack depth histories for the helicopter flight load spectrum **Figure 9.** The measured and computed crack depth histories for the helicopter flight load spectrum CLS0900SS00.

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CLS0900SS15.

CLS0900SS15.

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**Figure 10.** The measured and computed crack depth histories for the helicopter flight load spectrum **Figure 10.** The measured and computed crack depth histories for the helicopter flight load spectrum CLS0900SS05. CLS0900SS05.

**Figure 11.** The measured and computed crack depth histories for the helicopter flight load spectrum **Load Blocks Figure 11.** The measured and computed crack depth histories for the helicopter flight load spectrum **Figure 11.** The measured and computed crack depth histories for the helicopter flight load spectrum CLS0900SS15.

**Figure 12.** The measured and computed crack depth histories for the helicopter flight load spectrum **Figure 12.** The measured and computed crack depth histories for the helicopter flight load spectrum CLS0900SS20.

#### CLS0900SS20. **5. Material Variability**

**5. Material Variability**  The variability in crack growth that can arise from a fatigue test was first highlighted by a paper by Virkler et al. [47] This study presented the results of more than sixty constant amplitude R = 0.2 tests on identical 2024-T3 panels which had a constant initial half crack length of 9 mm (see Figure 13). Figure 2 illustrates the variability in crack growth seen in tests on long cracks tested under an operational flight load spectra. Unfortunately, for small cracks the variability in the crack depth histories can be significantly greater than that seen in the long crack curves shown in Figures 2 and 13 [16,48,49]. (The effect of (local) material variability on the growth of small cracks is compounded by the fact that the size and shape of the initial material discontinuity is variable, and generally cannot be tightly controlled [49–51].) The variability in the crack depth history associated with spectra CSL090SS15 is a good example of this, see Figure 11 that presents the variability in the crack depth The variability in crack growth that can arise from a fatigue test was first highlighted by a paper by Virkler et al. [47] This study presented the results of more than sixty constant amplitude R = 0.2 tests on identical 2024-T3 panels which had a constant initial half crack length of 9 mm (see Figure 13). Figure 2 illustrates the variability in crack growth seen in tests on long cracks tested under an operational flight load spectra. Unfortunately, for small cracks the variability in the crack depth histories can be significantly greater than that seen in the long crack curves shown in Figures 2 and 13 [16,48,49]. (The effect of (local) material variability on the growth of small cracks is compounded by the fact that the size and shape of the initial material discontinuity is variable, and generally cannot be tightly controlled [49–51].) The variability in the crack depth history associated with spectra CSL090SS15 is a good example of this, see Figure 11 that presents the variability in the crack depth histories associated with six different cracks.

histories associated with six different cracks. This raises the question: How much greater would the variability in the crack growth histories This raises the question: How much greater would the variability in the crack growth histories shown in Figures 6–12 have been if significantly more tests been performed?

shown in Figures 6–12 have been if significantly more tests been performed? Whilst it is not possible to definitively answer this question, it may be possible to shed some light on the problem space by investigating the effect of small changes in the fatigue threshold on the computed crack growth histories. Given that more than 90% of the Black Hawk flight load spectrum consists of small amplitude loads [45], and that the variability in the growth of small cracks can often be captured by allowing for variability in the term Δ*Kthr* (see Section 3.1 and [2,15,16,18,38,44]) it is anticipated that a small change in Δ*Kthr* should result in a much greater change in the crack growth history. To investigate this hypothesis we reanalysed the various test spectra using Δ*Kthr* = 0.5 MPa √m and *A* = 111 MPa √m. The resultant crack depth histories are also shown in Figures 6–12 where they are labelled "Computed Δ*Kthr* = 0.5 A = 111". Here we see that, as expected, when values of Δ*Kthr* = 0.5 MPa √m and *A* = 111 MPa √m are used there is a significant reduction in the computed fatigue lives, when compared to the lives computed using Δ*Kthr* = 0.6 MPa √m and *A* = 111 MPa √m, and that the computed fatigue lives are now conservative. Bearing in mind that for small cracks growing under combat, maritime, and civil aircraft flight load spectra, it has been shown that the variability in the crack growth histories is captured by allowing for (relatively small) changes in Δ*Kthr*—this Whilst it is not possible to definitively answer this question, it may be possible to shed some light on the problem space by investigating the effect of small changes in the fatigue threshold on the computed crack growth histories. Given that more than 90% of the Black Hawk flight load spectrum consists of small amplitude loads [45], and that the variability in the growth of small cracks can often be captured by allowing for variability in the term ∆*K*thr (see Section 3.1 and [2,15,16,18,38,43]) it is anticipated that a small change in ∆*K*thr should result in a much greater change in the crack growth history. To investigate this hypothesis we reanalysed the various test spectra using <sup>∆</sup>*K*thr <sup>=</sup> 0.5 MPa <sup>√</sup> m and *<sup>A</sup>* <sup>=</sup> 111 MPa <sup>√</sup> m. The resultant crack depth histories are also shown in Figures 6–12 where they are labelled "Computed ∆*K*thr = 0.5 *A* = 111". Here we see that, as expected, when values of <sup>∆</sup>*K*thr <sup>=</sup> 0.5 MPa <sup>√</sup> m and *<sup>A</sup>* <sup>=</sup> 111 MPa <sup>√</sup> m are used there is a significant reduction in the computed fatigue lives, when compared to the lives computed using <sup>∆</sup>*K*thr <sup>=</sup> 0.6 MPa <sup>√</sup> m and *<sup>A</sup>* <sup>=</sup> 111 MPa <sup>√</sup> m, and that the computed fatigue lives are now conservative. Bearing in mind that for small cracks growing under combat, maritime, and civil aircraft flight load spectra, it has been shown that the variability in the crack growth histories is captured by allowing for (relatively small) changes in ∆*K*thr—this appears to suggest that in order to evaluate the effect of simplifying a spectrum, so as to reduce test

variability".

time, on the fastest possible (lead) crack a statistically significant number of tests should be performed. This requirement is highlighted in Section 3.2.19.1 of the US Joint Services Structural Guidelines [9] that states: "The allowable structural properties shall include all applicable statistical variability". time, on the fastest possible (lead) crack a statistically significant number of tests should be performed. This requirement is highlighted in Section 3.2.19.1 of the US Joint Services Structural Guidelines [9] that states: "The allowable structural properties shall include all applicable statistical

appears to suggest that in order to evaluate the effect of simplifying a spectrum, so as to reduce test

**Figure 13.** The variability in the crack length histories reported in [46]. **Figure 13.** The variability in the crack length histories reported in [46].

To further investigate the variability (scatter) seen in the ASSIST tests let us examine the data associated with test spectra IRF-15 and CSL090SS15, which had information on the largest number of cracks (six). The mean lives, standard deviation (σ), and the projected worst case (mean-3σ) lives are given in Table 3. Here we see that the standard deviation in the test lives is a significant proportion of the mean life. It should be noted that whilst the mean-3σ life and the standard deviation calculations are based on small sample statistics, they nevertheless indicate the need for data on the growth of a greater number of cracks, i.e., additional testing. To further investigate the variability (scatter) seen in the ASSIST tests let us examine the data associated with test spectra IRF-15 and CSL090SS15, which had information on the largest number of cracks (six). The mean lives, standard deviation (σ), and the projected worst case (mean-3σ) lives are given in Table 3. Here we see that the standard deviation in the test lives is a significant proportion of the mean life. It should be noted that whilst the mean-3σ life and the standard deviation calculations are based on small sample statistics, they nevertheless indicate the need for data on the growth of a greater number of cracks, i.e., additional testing.

**Table 3.** The values of *A* and ∆*K*thr and *A* used in Figure 14. **Table 3.** The values of *A* and ∆*K*thr and *A* used in Figure 14.


*A* = 40 MPa √m

Computed Δ*Kth*r = 0.3 MPa √m, *A* = 111 MPa √m 9.7 13.2 It has been suggested [18] that for small cracks in aluminium alloys the threshold term lies in the range 0.1 < Δ*Kthr* < 0.3. Consequently, for spectra IRF-15 and CSL090SS15 the analysis was repeated using Δ*Kthr* = 0.3 MPa √m, and either *A* = 40 MPa √m, or *A* = 111 MPa √m. Comparisons between the measured and computed crack depth histories for these spectra are shown in Figures 14 and 15. The (computed) number of cycles to failure obtained using *A* = 40 MPa √m and *A* = 111 MPa √m are given It has been suggested [18] that for small cracks in aluminium alloys the threshold term lies in the range 0.1 < ∆*K*thr < 0.3. Consequently, for spectra IRF-15 and CSL090SS15 the analysis was repeated using <sup>∆</sup>*K*thr <sup>=</sup> 0.3 MPa <sup>√</sup> m, and either *<sup>A</sup>* <sup>=</sup> 40 MPa <sup>√</sup> m, or *<sup>A</sup>* <sup>=</sup> 111 MPa <sup>√</sup> m. Comparisons between the measured and computed crack depth histories for these spectra are shown in Figures 14 and 15. The (computed) number of cycles to failure obtained using *<sup>A</sup>* <sup>=</sup> 40 MPa <sup>√</sup> m and *<sup>A</sup>* <sup>=</sup> 111 MPa <sup>√</sup> m are given in Table 3. Here we see that Table 3 and Figures 14 and 15 reveal that the computed crack depth history is a weak function of the cyclic fracture toughness. We also see that when using <sup>∆</sup>*K*thr <sup>=</sup> 0.3 MPa <sup>√</sup> m the resultant computed crack depth histories have a near exponential

flight load spectra IRF-E15.

shape. Furthermore, the computed lives to failure are more conservative than the "Mean-3σ" lives as determined from the "limited" number of tests. It is thus suggested that in the absence of a statistically significant number of small crack tests the crack depth curve calculated using the value <sup>∆</sup>*K*thr <sup>=</sup> 0.3 MPa <sup>√</sup> m may be a reasonable first estimate for this "worst case" curve. MPa √m the resultant computed crack depth histories have a near exponential shape. Furthermore, the computed lives to failure are more conservative than the "Mean-3σ" lives as determined from the "limited" number of tests. It is thus suggested that in the absence of a statistically significant number of small crack tests the crack depth curve calculated using the value Δ*Kthr* = 0.3 MPa √m may be a reasonable first estimate for this "worst case" curve. history is a weak function of the cyclic fracture toughness. We also see that when using Δ*Kthr* = 0.3 MPa √m the resultant computed crack depth histories have a near exponential shape. Furthermore, the computed lives to failure are more conservative than the "Mean-3σ" lives as determined from the "limited" number of tests. It is thus suggested that in the absence of a statistically significant number of small crack tests the crack depth curve calculated using the value Δ*Kthr* = 0.3 MPa √m may be a

*Materials* **2020**, *13*, x FOR PEER REVIEW 12 of 16

history is a weak function of the cyclic fracture toughness. We also see that when using Δ*Kthr* = 0.3

in Table 3. Here we see that Table 3 and Figures 14 and 15 reveal that the computed crack depth

**Figure 14.** The effect of different toughness on the measured and computed curves for the helicopter flight load spectra IRF-E15. **Figure 14.** The effect of different toughness on the measured and computed curves for the helicopter flight load spectra IRF-E15. **Figure 14.** The effect of different toughness on the measured and computed curves for the helicopter

flight load spectra CSL090SS15. **Figure 15.** The effect of different toughness on the measured and computed curves for the helicopter flight load spectra CSL090SS15. **Figure 15.** The effect of different toughness on the measured and computed curves for the helicopter flight load spectra CSL090SS15.

### *On the Shape of the Crack Depth History Curves*

In the previous section, we remarked that using <sup>∆</sup>*K*thr <sup>=</sup> 0.3 MPa <sup>√</sup> m gave crack depth histories that had a near exponential shape. It was also suggested that, in the absence of a statistically significant number of small crack tests, the crack depth curve computed using <sup>∆</sup>*K*thr <sup>=</sup> 0.3 MPa <sup>√</sup> m may be a reasonable first estimate for the worst-case crack depth history. In this context, it should be noted that the USAF Durability Design Handbook [52] explains that the growth of small "lead" cracks, i.e., the fastest growing cracks in an airframe or a component [53,54], in military aircraft is generally exponential. Indeed, this exponential crack growth model is contained within the USAF approach to assessing the risk of failure [55]. This feature, i.e., the exponential growth of lead cracks growing under flight load spectra, was independently validated in [49,56] and is discussed in more detail in [2,49]. However, examining Figures 6–12, we see that the crack growth histories are not exponential. This observation raises an additional question, viz:

If significantly more tests had been performed would the "worst-case" crack depth versus load blocks curves have been (approximately) exponential?

### **6. Conclusions**

The assessment of the economic lives of operational metallic helicopter airframes requires a durability analysis in which the EIDS are sub mm, typically 0.254 mm. Unfortunately, whereas several studies into the ability of crack growth models to perform a damage tolerance analysis of helicopter components subjected to a representative flight load spectrum have been performed few, if any, studies can be found on the ability of crack growth models to perform a valid durability assessment of a component subjected to an operational helicopter flight load spectrum. In this context, the present study has found that the HS equation is able to reasonably accurately compute the growth of small naturally occurring cracks in AA7075-T7351 under several simplified/reduced Black Hawk flight load spectra. This suggests that the HS equation may have the potential to address the question of how to simplify measured spectra in order to reduce the time and complexity of full-scale helicopter fatigue tests.

It is also suggested that, given the inherent variability seen in small crack growth, any round robin test on small cracks, and any test program performed to the effect of spectrum truncation on the growth of small cracks should involve a statistically significant number of tests.

**Author Contributions:** Conceptualization and methodology—R.J.; analysis of the FALSTAFF loading test case—P.H.; development and validation of the computer software and the helicopter test analysis—D.P.; review, editing, and supervision of D.P. at Monash—R.K.S.R. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Acknowledgments:** The authors wish to acknowledge Madeline Burchill at DST for the development of the ASSIST Round Robin Helicopter Program, and Beau Krieg at DST for providing the associated crack growth data. The authors would also like to acknowledge the assistance of Weiping HU at DST who provided the data on the growth of cracks in 7075-T7351 under a FALSTAFF spectrum.

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

### **References**


© 2020 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* 

### *Article* **Fatigue and Crack Growth under Constant- and Variable-Amplitude Loading in 9310 Steel Using "Rainflow-on-the-Fly" Methodology Fatigue and Crack Growth under Constant- and Variable-Amplitude Loading in 9310 Steel Using "Rainflow-On-The-Fly" Methodology**

**James C. Newman, Jr. James C. Newman, Jr.** 

> Department of Aerospace Engineering, Mississippi State University, Mississippi, MS 39762, USA; newmanjr@ae.msstate.edu; Tel.: +1-(901)-734-6642 Department of Aerospace Engineering, Mississippi State University, Mississippi State, MS 39762, USA; newmanjr@ae.msstate.edu; Tel.: +1-(901)-734-6642

**Abstract:** Fatigue of materials, like alloys, is basically fatigue-crack growth in small cracks nucleating and growing from micro-structural features, such as inclusions and voids, or at micro-machining marks, and large cracks growing to failure. Thus, the traditional fatigue-crack nucleation stage (*N<sup>i</sup>* ) is basically the growth in microcracks (initial flaw sizes of 1 to 30 µm growing to about 250 µm) in metal alloys. Fatigue and crack-growth tests were conducted on a 9310 steel under laboratory air and room temperature conditions. Large-crack-growth-rate data were obtained from compact, C(T), specimens over a wide range in rates from threshold to fracture for load ratios (*R*) of 0.1 to 0.95. New test procedures based on compression pre-cracking were used in the near-threshold regime because the current ASTM test method (load shedding) has been shown to cause load-history effects with elevated thresholds and slower rates than steady-state behavior under constant-amplitude loading. High load-ratio (*R*) data were used to approximate small-crack-growth-rate behavior. A crack-closure model, FASTRAN, was used to develop the baseline crack-growth-rate curve. Fatigue tests were conducted on single-edge-notch-bend, SEN(B), specimens under both constant-amplitude and a Cold-Turbistan+ spectrum loading. Under spectrum loading, the model used a "Rainflow-on-the-Fly" subroutine to account for crack-growth damage. Test results were compared to fatigue-life calculations made under constant-amplitude loading to establish the initial microstructural flaw size and predictions made under spectrum loading from the FASTRAN code using the same microstructural, semi-circular, surface-flaw size (6-µm). Thus, the model is a unified fatigue approach, from crack nucleation (small-crack growth) and large-crack growth to failure using fracture mechanics principles. The model was validated for both fatigue and crack-growth predictions. In general, predictions agreed well with the test data. **Abstract:** Fatigue of materials, like alloys, is basically fatigue-crack growth in small cracks nucleating and growing from micro-structural features, such as inclusions and voids, or at micro-machining marks, and large cracks growing to failure. Thus, the traditional fatigue-crack nucleation stage (*Ni*) is basically the growth in microcracks (initial flaw sizes of 1 to 30 μm growing to about 250 μm) in metal alloys. Fatigue and crack-growth tests were conducted on a 9310 steel under laboratory air and room temperature conditions. Large-crack-growth-rate data were obtained from compact, C(T), specimens over a wide range in rates from threshold to fracture for load ratios (*R*) of 0.1 to 0.95. New test procedures based on compression pre-cracking were used in the near-threshold regime because the current ASTM test method (load shedding) has been shown to cause load-history effects with elevated thresholds and slower rates than steady-state behavior under constant-amplitude loading. High load-ratio (*R*) data were used to approximate small-crack-growth-rate behavior. A crack-closure model, FASTRAN, was used to develop the baseline crack-growth-rate curve. Fatigue tests were conducted on single-edge-notch-bend, SEN(B), specimens under both constant-amplitude and a Cold-Turbistan+ spectrum loading. Under spectrum loading, the model used a "Rainflow-on-the-Fly" subroutine to account for crack-growth damage. Test results were compared to fatigue-life calculations made under constant-amplitude loading to establish the initial microstructural flaw size and predictions made under spectrum loading from the FASTRAN code using the same micro-structural, semi-circular, surface-flaw size (6-μm). Thus, the model is a unified fatigue approach, from crack nucleation (small-crack growth) and large-crack growth to failure using fracture mechanics principles. The model was validated for both fatigue and crack-growth predictions. In general, predictions agreed well with the test data.

**Keywords:** fatigue; crack growth; metallic materials; plasticity; crack closure; spectrum loading **Keywords:** fatigue; crack growth; metallic materials; plasticity; crack closure; spectrum loading

#### **1. Introduction 1. Introduction**

This article is dedicated to Dr. Wolf Elber and his remarkable achievements. This article is dedicated to Dr. Wolf Elber and his remarkable achievements.

On 12 January 2019, Wolf took off on his final flight. His passion was to soar in his glider over the Blue Ridge Mountains in southwest Virginia, USA. With his passing,

**Citation:** Newman, J.C., Jr. Fatigue and Crack Growth under Constantand Variable-Amplitude Loading in 9310 Steel Using and Crack Growth under Constantand Variable-Amplitude Loading in 9310 Steel Using "Rainflow-On-The-Fly" Methodology. *Metals* **2021**, *11*,

**Citation:** Newman, J.C. Jr. Fatigue

"Rainflow-on-the-Fly" Methodology. *Metals* **2021**, *11*, 807. https://doi.org /10.3390/met11050807 x. https://doi.org/10.3390/xxxxx Academic Editor: Filippo Berto

Academic Editor: Filippo Berto Received: 17 April 2021 Accepted:13 May 2021

Received: 17 April 2021 Accepted: 13 May 2021 Published: 15 May 2021 Published: **Publisher's Note:** MDPI stays neutral with regard to jurisdictional

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. claims in published maps and institutional affiliations. **Copyright:** © 2021 by the authors.

**Copyright:** © 2021 by the author. 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 (https:// creativecommons.org/licenses/by/ 4.0/). tribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

the fatigue and fracture mechanics community lost a great pioneer. Dr. Elber had a distinguished career at the NASA Langley Research Center, and later as head of the U.S. Army Research Laboratory at Langley. His professional accolades are immortalized by his discovery of the "plasticity induced crack closure" phenomenon [1,2]. Thank you, Wolf, for your friendship and brilliant mind over the years. We will greatly miss you.

Fatigue of metallic materials is divided into several phases: crack nucleation, smalland large-crack growth, and fracture [3]. Crack nucleation is controlled by local stress and strain concentrations and is associated with cyclic slip-band formation from dislocation movement leading to intrusions and extrusions [4,5]. Although cyclic slip may be necessary in pure metals, the presence of inclusions, voids, or machining marks in metal alloys greatly affects the crack-nucleation process. (Herein, small-crack growth includes microstructurally and physically small cracks.) The small-crack growth regime is the growth in cracks from inclusions, voids, or machining marks, ranging from 1 to 30 µm in depth [6]. Schijve [7] has shown that, for polished surfaces of pure metals and commercial alloys, the formation of a small crack of about 100 µm in size can consume 60 to 80% of the fatigue life. This is the reason that there is so much interest in the growth behavior of small cracks. Large-crack growth and failure are regions where fracture-mechanics parameters have been successful in correlating and predicting fatigue-crack growth and fracture. In the past three decades, fracture-mechanics concepts have also been successful in predicting the growth in small cracks under constant-amplitude and spectrum-loading using crack-closure theory [6].

The engineered metallic materials are inhomogeneous and anisotropic when viewed on a microscopic scale. For example, these materials are composed of an aggregate of small grains, inclusion particles or voids. These inclusion particles are of different chemical compositions to the bulk material, such as silicate or alumina inclusions in steels. Because of their nonuniform microstructure, local stresses may be concentrated at these locations and may cause the initiation of fatigue cracks. Crack initiation is primarily a surface phenomenon because: (1) local stresses are usually highest at the surface, (2) an inclusion particle of the same size has a higher stress concentration at the surface than in the interior, (3) the surface is subjected to adverse environmental conditions, and (4) the surfaces are susceptible to inadvertent damage. The growth in "naturally" initiated cracks in commercial aluminum alloys has been investigated by Bowles and Schijve [8], Morris et al. [9] and Kung and Fine [10]. In some cases, small cracks initiated at inclusions and the Stage I period of crack growth were eliminated [6]. This tendency toward inclusion initiation rather than slip-band (Stage I) cracking was found to depend on stress level and inclusion content [10]. Similarly, defects (such as tool marks, scratches and burrs) from manufacturing and service-induced events will also promote initiation and Stage II crack growth [6].

During the last three decades, test and analysis programs on "small-crack" behavior have shown that majority of the fatigue life is consumed by small-crack growth from a micro-structural feature for a variety of metal alloys [11–14]. The smallest measured flaw sizes using plastic-replica methods ranged from 10 to 30 µm for aluminum alloys (2024-T3; 7075-T6), aluminum–lithium alloy (2090-T8ED41) and 4340 steel; and the crack-propagation life was about 90 percent of the fatigue life for constant-amplitude and spectrum loading. Thus, a large portion of nucleation life (*N<sup>i</sup>* ) in classical fatigue is small-crack growth, from a micro-structural feature to about 250 µm for these materials. The exception was the Ti-6Al-4V alloy [13], where the smallest measured flaw sizes were about 20 µm and crack-growth lives were about 50% of the fatigue life. However, small-crack analyses of similar Ti-6Al-4V specimens machined from two engine discs [15,16] showed that the initial flaw sizes from 2 to 20 µm predicted the scatter band in fatigue life on open-hole fatigue tests quite well. Therefore, the crack-growth approach provides a unified theory for the determination of fatigue lives for these materials. However, for pure- and single-crystal materials, nucleation cycles are required to transport dislocations at critical locations, develop slip bands, and cracks.

Fatigue-crack growth under variable-amplitude and spectrum loading is composed of complex crack-shielding mechanisms (plasticity, roughness and fretting debris) and damage accumulation due to cyclic plastic deformations around a crack front in metallic materials. Typically, "rainflow" methods [17] are applied to variable-amplitude loading to develop a load sequence that is used to compute damage accumulation and life because damage relations are, generally, a non-linear function of the crack-driving parameters. However, crack-tip damage is only a function of the current loading and load history (material memory). Loading in the future has no bearing on the "current" damage. The life-prediction code, FASTRAN [18], has a "rainflow-on-the-fly" methodology [19] to compute damage as the cyclic load history is applied, and there was no need to reorder the spectra. Herein, several variable-amplitude loading sequences were developed to test and to validate the "rainflow-on-the-fly" subroutine. Fatigue-crack growth under variable-amplitude and spectrum loading is composed of complex crack-shielding mechanisms (plasticity, roughness and fretting debris) and damage accumulation due to cyclic plastic deformations around a crack front in metallic materials. Typically, "rainflow" methods [17] are applied to variable-amplitude loading to develop a load sequence that is used to compute damage accumulation and life because damage relations are, generally, a non-linear function of the crack-driving parameters. However, crack-tip damage is only a function of the current loading and load history (material memory). Loading in the future has no bearing on the "current" damage. The lifeprediction code, FASTRAN [18], has a "rainflow-on-the-fly" methodology [19] to compute damage as the cyclic load history is applied, and there was no need to reorder the spectra. Herein, several variable-amplitude loading sequences were developed to test and to validate the "rainflow-on-the-fly" subroutine.

crystal materials, nucleation cycles are required to transport dislocations at critical loca-

*Metals* **2021**, *11*, x FOR PEER REVIEW 3 of 19

tions, develop slip bands, and cracks.

The paper presents the results of large-crack-growth-rate tests conducted on compact, C(T), specimens made of 9310 steel [20] over a wide range of constant-amplitude loading to establish the baseline crack-growth-rate curve for fatigue and crack-growth analyses. Compression pre-cracking methods [21–26] were used to generate test data in the nearthreshold regime because the ASTM E-647 test method [27,28] using the load-shedding procedure was shown to cause load-history effects and slower crack-growth rates than steady-state behavior [25,26]. Both compression pre-cracking constant-amplitude (CPCA) and load-reduction (CPLR) methods were used. A crack-closure analysis was used to collapse the rate data from C(T) specimens into a narrow band over many orders of magnitude in rates using a plane-strain constraint factor for low rates and modeled a constraint-loss regime to plane-stress behavior at high rates. For steels, small- and largecrack data (without load-history effects) tend to agree well [29]. Thus, the high-R large-crack data in the near-threshold regime is a good estimate for small-crack behavior, as proposed by Herman et al. [30]. A Two-Parameter Fracture Criterion [31] was used to characterize the fracture behavior. Fatigue tests were conducted on 9310 steel single-edge-notch-bend, SEN(B), specimens [32] under both constant-amplitude and a modified Cold-Turbistan [33] spectrum loading. The test results were compared to the life calculations or predictions made using the FASTRAN code. The paper presents the results of large-crack-growth-rate tests conducted on compact, C(T), specimens made of 9310 steel [20] over a wide range of constant-amplitude loading to establish the baseline crack-growth-rate curve for fatigue and crack-growth analyses. Compression pre-cracking methods [21–26] were used to generate test data in the near-threshold regime because the ASTM E-647 test method [27,28] using the loadshedding procedure was shown to cause load-history effects and slower crack-growth rates than steady-state behavior [25,26]. Both compression pre-cracking constant-amplitude (CPCA) and load-reduction (CPLR) methods were used. A crack-closure analysis was used to collapse the rate data from C(T) specimens into a narrow band over many orders of magnitude in rates using a plane-strain constraint factor for low rates and modeled a constraint-loss regime to plane-stress behavior at high rates. For steels, small- and large-crack data (without load-history effects) tend to agree well [29]. Thus, the high-R large-crack data in the near-threshold regime is a good estimate for small-crack behavior, as proposed by Herman et al. [30]. A Two-Parameter Fracture Criterion [31] was used to characterize the fracture behavior. Fatigue tests were conducted on 9310 steel single-edgenotch-bend, SEN(B), specimens [32] under both constant-amplitude and a modified Cold-Turbistan [33] spectrum loading. The test results were compared to the life calculations or predictions made using the FASTRAN code.

#### **2. Material and Specimen Configurations 2. Material and Specimen Configurations**

The Boeing Company (Seattle, WA, USA) provided a 9310 steel rod (150 mm diameter by 950 mm length) to Mississippi State University. C(T), SEN(B), and tensile specimens (*B* = 6.35 mm) were machined in the longitudinal direction (cracks perpendicular to longitudinal direction) and heat-treated by special procedures [20]. The yield stress, *σys*, was 980 MPa, ultimate tensile strength, *σu*, was 1250 MPa, and modulus of elasticity was *E* = 208.6 GPa. Figure 1 shows C(T) and SEN(B) specimens with back-face strain (BFS) gauges used to monitor crack growth in the C(T) specimens and crack nucleation and growth in the SEN(B) specimen. The Boeing Company (Seattle, WA, USA) provided a 9310 steel rod (150 mm diameter by 950 mm length) to Mississippi State University. C(T), SEN(B), and tensile specimens (*B* = 6.35 mm) were machined in the longitudinal direction (cracks perpendicular to longitudinal direction) and heat-treated by special procedures [20]. The yield stress, *σys*, was 980 MPa, ultimate tensile strength, *σu*, was 1250 MPa, and modulus of elasticity was *E* = 208.6 GPa. Figure 1 shows C(T) and SEN(B) specimens with back-face strain (BFS) gauges used to monitor crack growth in the C(T) specimens and crack nucleation and growth in the SEN(B) specimen.

**Figure 1. Figure 1.** Fatigue-crack-growth and fatigue specimens tested and analyzed. Fatigue-crack-growth and fatigue specimens tested and analyzed.

#### **3. "Rainflow-on-the-Fly" Methodology** Application of minimum stress, *S5*, caused the crack surfaces to close, and rainflow-on-

Δ

crack-growth rate (*dc/dN*) per loading amplitude, which gives

*c1* is a function of

Δ

and Δ

is Δ*c =* Δ*c1 +* Δ

*= f(S4 − So) −*

**3.". Rainflow-on-the-Fly" Methodology** 

*Metals* **2021**, *11*, x FOR PEER REVIEW 4 of 19

Since the Paris–Elber non-linear power law is used to compute crack-growth damage in the FASTRAN life-prediction code [18], a "rainflow-on-the-fly" subroutine was developed to compute crack growth under variable-amplitude loading. Damage calculations are illustrated in Figure 2 and they are based on calculations of stress amplitudes above the crack-opening stress, *So*. Damage only occurs during the loading amplitude but unloading may change crack-tip deformations and affect the subsequent damage during the next loading amplitude. Crack-opening stress, *So*, is calculated at the minimum stress, *S1*, and ∆*c<sup>1</sup>* is a function of ∆*Seff = S<sup>2</sup>* − *So*. The ∆Seff value is used to compute ∆*Keff* and then the crack-growth rate (*dc/dN*) per loading amplitude, which gives ∆*c1*. The next unloading to S<sup>3</sup> did not close the crack, and the next maximum loading was to *S4*. Here, damage is ∆*c<sup>2</sup> = f(S<sup>4</sup>* − *So)* − ∆*c<sup>1</sup> + f(S<sup>2</sup>* − *S3)*, which captures the larger damage due to *S4*. The total damage is ∆*c* = ∆*c<sup>1</sup> +* ∆*c2*. Again, each stress range is used to calculate an effective stress-intensity factor and then the corresponding crack extension from the crack-growthrate relation. Application of minimum stress, *S5*, caused the crack surfaces to close, and rainflow-on-the-fly logic was reset. Total damage, ∆*c*, captures the essence of rainflow logic and applies damage in the proper sequence using a cycle-by-cycle calculation. (Note that a cycle is defined as any minimum–maximum–minimum stress.) the-fly logic was reset. Total damage, Δ*c*, captures the essence of rainflow logic and applies damage in the proper sequence using a cycle-by-cycle calculation. (Note that a cycle is defined as any minimum–maximum–minimum stress.) Herein, several specially designed spectra were developed to exercise the FASTRAN code and calculations are shown for these spectra. Each spectrum had sequences of stress ranges that would require "rainflow" logic to calculate the proper crack extension. Comparisons are made between: (1) linear damage, (2) rainflow logic and (3) FASTRAN. Linear damage is using only the stress range to calculate crack extension and ignoring load interactions. Rainflow logic is a manual calculation of crack extension using the basic rainflow concept principles to compute the proper crack extension. FASTRAN uses the "rainflow-on-the-fly" subroutine to remember stress history and to compute the proper damage as the crack-opening stress in FASTRAN was held constant, as specified, and the crack-growth relation was a simple Paris–Elber relation as ⁄ = 5 × 10ିଵ (∆)ଷ (1) where ∆ = ∆√. The spectra were applied to an infinite plate under remote uniform stress, *S*, with an initial crack half-length (*ci*) of 5 mm.

Since the Paris–Elber non-linear power law is used to compute crack-growth damage in the FASTRAN life-prediction code [18], a "rainflow-on-the-fly" subroutine was developed to compute crack growth under variable-amplitude loading. Damage calculations are illustrated in Figure 2 and they are based on calculations of stress amplitudes above the crack-opening stress, *So.* Damage only occurs during the loading amplitude but unloading may change crack-tip deformations and affect the subsequent damage during the next loading amplitude. Crack-opening stress, *So*, is calculated at the minimum stress, *S1*,

*Seff = S2 − So*. The ΔSeff value is used to compute

*c1 + f(S2 − S3)*, which captures the larger damage due to *S4*. The total damage

*c2*. Again, each stress range is used to calculate an effective stress-intensity

S3 did not close the crack, and the next maximum loading was to *S4*. Here, damage is

factor and then the corresponding crack extension from the crack-growth-rate relation.

Δ

*c1*. The next unloading to

Δ

*Keff* and then the

Δ*c2*

**Figure 2.** Damage calculations using crack-closure theory under variable-amplitude loading. **Figure 2.** Damage calculations using crack-closure theory under variable-amplitude loading.

Spectrum A is a Christmas-Tree type loading sequence and is shown in Figure 3a and was designed to severely test the Rainflow-on-the-Fly option in FASTRAN. Without Rainflow analyses, the difference in crack-growth lives for the sequence would be several orders-of-magnitude in error using linear damage [19]. One block of loading is defined from point A to B, and the sequence was repeated until the final crack length was reached or Herein, several specially designed spectra were developed to exercise the FASTRAN code and calculations are shown for these spectra. Each spectrum had sequences of stress ranges that would require "rainflow" logic to calculate the proper crack extension. Comparisons are made between: (1) linear damage, (2) rainflow logic and (3) FASTRAN. Linear damage is using only the stress range to calculate crack extension and ignoring load interactions. Rainflow logic is a manual calculation of crack extension using the basic rainflow concept principles to compute the proper crack extension. FASTRAN uses the "rainflow-on-the-fly" subroutine to remember stress history and to compute the proper damage as the crack-opening stress in FASTRAN was held constant, as specified, and the crack-growth relation was a simple Paris–Elber relation as

$$d\mathbf{c}/dN = \mathbf{5} \times 10^{-10} (\Delta \mathbf{K}\_{eff})^3 \tag{1}$$

where <sup>∆</sup>*Ke f f* <sup>=</sup> <sup>∆</sup>*Se f f* <sup>√</sup> *πc*. The spectra were applied to an infinite plate under remote uniform stress, *S*, with an initial crack half-length (*c<sup>i</sup>* ) of 5 mm.

Spectrum A is a Christmas-Tree type loading sequence and is shown in Figure 3a and was designed to severely test the Rainflow-on-the-Fly option in FASTRAN. Without Rainflow analyses, the difference in crack-growth lives for the sequence would be several orders-of-magnitude in error using linear damage [19]. One block of loading is defined from point A to B, and the sequence was repeated until the final crack length was reached or the specimen failed. The analytical crack-closure model in FASTRAN was turned off by intentionally setting the crack-opening stress, *So*, to a constant value of 50 MPa. application.) FASTRAN with the Rainflow subroutine gives the solid (black) symbols that exactly match what the Rainflow method predicted.

the specimen failed. The analytical crack-closure model in FASTRAN was turned off by

only for "linear damage" (lines with a solid blue symbol) and for loadings that do not require Rainflow methods. Diamond symbols show what a Rainflow method would give in crack extension. (Note that a cycle is defined as any minimum to maximum to minimum loading, and crack-growth damage only occurs during loading. The unloading part of the cycle causes reverse plastic deformations that may affect damage during the next load

Figure 3b shows crack-growth increments calculated for each loading amplitude. If

Δ

Δ

*Seff = Smax − Smin*. These equations apply

*Seff = Smax − So*. If

intentionally setting the crack-opening stress, *So*, to a constant value of 50 MPa.

the crack-opening stress is greater than the minimum stress, *Smin*, then

quence A.

*Metals* **2021**, *11*, x FOR PEER REVIEW 5 of 19

*Smin* is greater than the crack-opening stress, then

**Figure 3.** (**a**) Loading sequence A. (**b**) Calculated crack extension per applied cycle for load se-**Figure 3.** (**a**) Loading sequence A. (**b**) Calculated crack extension per applied cycle for load sequence A.

The second spectrum (B) was designed after some of the European standard spectra with various stress amplitudes that have a number of constant-amplitude cycles, and the spectrum had a number of cycles that go from maximum–minimum–maximum or minimum–maximum–minimum loading. Some cycles had a Christmas-Tree type loading that was interrupted with other stress amplitudes. Again, this spectrum does require rainflow logic to calculate the correct damage. The first case (B1) had a constant crack-opening stress (50 MPa) for the complete spectrum, as shown in Figure 4a. The calculated crack extension using linear damage, Figure 3b shows crack-growth increments calculated for each loading amplitude. If the crack-opening stress is greater than the minimum stress, *Smin*, then ∆*Seff* = *Smax* − *So*. If *Smin* is greater than the crack-opening stress, then ∆*Seff = Smax* − *Smin*. These equations apply only for "linear damage" (lines with a solid blue symbol) and for loadings that do not require Rainflow methods. Diamond symbols show what a Rainflow method would give in crack extension. (Note that a cycle is defined as any minimum to maximum to minimum loading, and crack-growth damage only occurs during loading. The unloading part of the cycle causes reverse plastic deformations that may affect damage during the next load application.) FASTRAN with the Rainflow subroutine gives the solid (black) symbols that exactly match what the Rainflow method predicted.

The second spectrum (B) was designed after some of the European standard spectra with various stress amplitudes that have a number of constant-amplitude cycles, and the spectrum had a number of cycles that go from maximum–minimum–maximum or minimum–maximum–minimum loading. Some cycles had a Christmas-Tree type loading that was interrupted with other stress amplitudes. Again, this spectrum does require rainflow logic to calculate the correct damage.

The first case (B1) had a constant crack-opening stress (50 MPa) for the complete spectrum, as shown in Figure 4a. The calculated crack extension using linear damage, rainflow and FASTRAN are shown in Figure 4b. Out of the 57 cycles, 18 cycles required

quence B1.

rainflow logic to compute the correct crack extension. FASTRAN agreed with the Rainflow predicted crack extensions. rainflow logic to compute the correct crack extension. FASTRAN agreed with the Rainflow predicted crack extensions.

rainflow and FASTRAN are shown in Figure 4b. Out of the 57 cycles, 18 cycles required

**Figure 4.** (**a**) Loading sequence B1. (**b**) Calculated crack extension per applied cycle for load se-**Figure 4.** (**a**) Loading sequence B1. (**b**) Calculated crack extension per applied cycle for load sequence B1.

In the FASTRAN code, crack-opening stresses will change as a function of stress history. Thus, to simulate a changing crack-opening stress, the second case (B2) had a constant crack-opening stress of 50 MPa for the first 22 cycles and then *So* = 90 MPa for the remainder of loading, as shown in Figure 5a. Of course, the first 22 cycles have the same behavior as Case B1, but now cycles 23 to 34 do not require rainflow logic (*Smin < So*) and crack extensions were computed directly from the Paris–Elber relation (Equation (1)). These crack-extension comparisons are shown in Figure 5b. However, the crack extensions during cycles 35 to 37 were much less than in Case B1 due to the higher crack-open-In the FASTRAN code, crack-opening stresses will change as a function of stress history. Thus, to simulate a changing crack-opening stress, the second case (B2) had a constant crack-opening stress of 50 MPa for the first 22 cycles and then *S<sup>o</sup>* = 90 MPa for the remainder of loading, as shown in Figure 5a. Of course, the first 22 cycles have the same behavior as Case B1, but now cycles 23 to 34 do not require rainflow logic (*Smin < So*) and crack extensions were computed directly from the Paris–Elber relation (Equation (1)). These crack-extension comparisons are shown in Figure 5b. However, the crack extensions during cycles 35 to 37 were much less than in Case B1 due to the higher crack-opening stress and lower ∆*Keff* values.

ing stress and lower Δ*Keff* values. Figure 6 shows the calculated crack-opening stresses from FASTRAN (*α* = 2) using the full crack-closure model for load sequence B using the cycle-by-cycle option. The results for Block 1 are shown as the dashed (red) lines. The crack-opening stresses are calculated at a minimum applied stress and remain constant until the crack closes again at the next minimum applied stress. In many cycles, the crack-tip region was open (*Smin > So*) during the applied stress amplitudes. The model starts with a "zero" crack-opening stress Figure 6 shows the calculated crack-opening stresses from FASTRAN (*α* = 2) using the full crack-closure model for load sequence B using the cycle-by-cycle option. The results for Block 1 are shown as the dashed (red) lines. The crack-opening stresses are calculated at a minimum applied stress and remain constant until the crack closes again at the next minimum applied stress. In many cycles, the crack-tip region was open (*Smin > So*) during the applied stress amplitudes. The model starts with a "zero" crack-opening stress and the opening stresses increase as the crack grows and leaves plastically deformed material in the wake of the crack tip.

terial in the wake of the crack tip.

**Figure 5.** (**a**) Loading sequence B. (**b**) Calculated crack extension per applied cycle for load sequence B2. **Figure 5.** (**a**) Loading sequence B. (**b**) Calculated crack extension per applied cycle for load sequence B2.

and the opening stresses increase as the crack grows and leaves plastically deformed ma-

**Figure 6. Figure 6.**  Loading sequence B and calculated crack-opening stresses. Loading sequence B and calculated crack-opening stresses.

Calculated crack-opening-stress history for Block 400, at about 110,000 cycles (halflife), are shown as solid (blue) lines. Here, the cycle numbers are the same cycles as in Block 1 (first sequence). The calculated crack-opening stresses where in agreement over most of the sequence, except the only major difference was in cycles 1 to 5, where the crack surfaces were fully open. However, the crack extension during the first five cycles in Block 1 was

a factor of 8 larger than that during the 400th block because of the higher crack-opening stress. Most of the damage from cycles 1 to 5 was from the loading on cycle 1. 1 was a factor of 8 larger than that during the 400th block because of the higher crackopening stress. Most of the damage from cycles 1 to 5 was from the loading on cycle 1.

Calculated crack-opening-stress history for Block 400, at about 110,000 cycles (halflife), are shown as solid (blue) lines. Here, the cycle numbers are the same cycles as in Block 1 (first sequence). The calculated crack-opening stresses where in agreement over most of the sequence, except the only major difference was in cycles 1 to 5, where the crack surfaces were fully open. However, the crack extension during the first five cycles in Block

#### **4. Fatigue-Crack-Growth and Fracture Tests on 9310 Steel 4. Fatigue-Crack-Growth and Fracture Tests on 9310 Steel**

*Metals* **2021**, *11*, x FOR PEER REVIEW 9 of 19

Compact, C(T), specimens were used to generate the ∆*K* against rate (*dc/dN*) data on the 9310 steel at room temperature and 20 Hertz [20] over a wide range in stress ratios (*R* = 0.1 to 0.95). These ∆*K*-rate data are shown in Figure 7. Tests were conducted from near threshold to fracture. A BFS gauge was used to monitor crack growth and to measure crackopening loads using the compliance-offset method. In the low-rate regime, compression pre-cracking constant amplitude (CPCA) and load reduction (CPLR) methods were used to generate the ∆*K*-rate data. In general, the test frequency was dropped to about 5 to 8 Hertz as the cracks were grown to failure. The data show the normal spread with the R value but shows more spread in the threshold and fracture regions. In the fracture region, the spread is due to the fracture being controlled by the maximum stress-intensity factor and would not collapse on a ∆*K*-rate plot. For example, the stress-intensity-factor range at failure, ∆*K<sup>c</sup> = KIe (1* − *R)*, where *KIe* is the elastic fracture toughness. Thus, high *R* tests would fracture at a lower ∆*K<sup>c</sup>* value than low *R* tests. The spread in the threshold region is suspected to be due to the load-history effects caused by the load-reduction procedure used in the CPLR method. Load-reduction tests, as specified in ASTM E-647 [28], have been shown [25,26] to induce more spread in ∆*K* in the threshold region than the mid-rate region. The spread has been associated with a rise in the crack-closure behavior during load-reduction tests [34,35]. Compact, C(T), specimens were used to generate the Δ*K* against rate (*dc/dN*) data on the 9310 steel at room temperature and 20 Hertz [20] over a wide range in stress ratios (*R* = 0.1 to 0.95). These Δ*K*-rate data are shown in Figure 7. Tests were conducted from near threshold to fracture. A BFS gauge was used to monitor crack growth and to measure crack-opening loads using the compliance-offset method. In the low-rate regime, compression pre-cracking constant amplitude (CPCA) and load reduction (CPLR) methods were used to generate the Δ*K*-rate data. In general, the test frequency was dropped to about 5 to 8 Hertz as the cracks were grown to failure. The data show the normal spread with the R value but shows more spread in the threshold and fracture regions. In the fracture region, the spread is due to the fracture being controlled by the maximum stressintensity factor and would not collapse on a Δ*K*-rate plot. For example, the stress-intensity-factor range at failure, Δ*Kc = KIe (1 − R)*, where *KIe* is the elastic fracture toughness. Thus, high *R* tests would fracture at a lower Δ*Kc* value than low *R* tests. The spread in the threshold region is suspected to be due to the load-history effects caused by the loadreduction procedure used in the CPLR method. Load-reduction tests, as specified in ASTM E-647 [28], have been shown [25,26] to induce more spread in Δ*K* in the threshold region than the mid-rate region. The spread has been associated with a rise in the crackclosure behavior during load-reduction tests [34,35].

**Figure 7.** Stress-intensity-factor range against rate for 9310 steel over range in stress ratio (R). **Figure 7.** Stress-intensity-factor range against rate for 9310 steel over range in stress ratio (R).

To evaluate the fracture toughness of the material, only the fatigue-crack-growth tests were available. In most tests, the cracks in the C(T) specimens were grown to failure under cyclic loading. Here, the final recorded crack length and the maximum fatigue load were used to calculate *KIe* (elastic stress-intensity factor at failure), and these results are To evaluate the fracture toughness of the material, only the fatigue-crack-growth tests were available. In most tests, the cracks in the C(T) specimens were grown to failure under cyclic loading. Here, the final recorded crack length and the maximum fatigue load were used to calculate *KIe* (elastic stress-intensity factor at failure), and these results are shown in Figure 8, as solid (blue) circular symbols. Most tests failed at large *c<sup>i</sup> /w* ratios (>0.7). For comparison, similar test results conducted on a 4340 steel [36] are shown as open circular symbols. Here, one 4340 steel specimen was fatigue cracked to a lower crack-length-to-width ratio and statically pulled to failure (solid square symbol).

**Figure 8.** Elastic stress-intensity factor at failure on 9310 and 4340 steel C(T) specimens. **Figure 8.** Elastic stress-intensity factor at failure on 9310 and 4340 steel C(T) specimens.

The solid curve in Figure 8 was based on the Two-Parameter Fracture Criterion (TPFC). A fracture criterion was derived [37] that accounted for the elastic-plastic behavior of a cracked material based on notch-strength analysis. The criterion was based on expressing the elastic stress-intensity factor at failure in terms of net-section stress as The solid curve in Figure 8 was based on the Two-Parameter Fracture Criterion (TPFC). A fracture criterion was derived [37] that accounted for the elastic-plastic behavior of a cracked material based on notch-strength analysis. The criterion was based on expressing the elastic stress-intensity factor at failure in terms of net-section stress as

shown in Figure 8, as solid (blue) circular symbols. Most tests failed at large *ci/w* ratios (>0.7). For comparison, similar test results conducted on a 4340 steel [36] are shown as open circular symbols. Here, one 4340 steel specimen was fatigue cracked to a lower crack-

length-to-width ratio and statically pulled to failure (solid square symbol).

$$K\_{I\varepsilon} = P\_f / \sqrt{BW} \text{ F} = \mathbb{S}\_n \sqrt{\pi c} \text{ F}\_n \tag{2}$$

where *Fn* is the usual boundary-correction factor based on net-section stress. Equations for *Sn* and *Fn* are given in reference 37 for the C(T) specimen. The fracture criterion is where *F<sup>n</sup>* is the usual boundary-correction factor based on net-section stress. Equations for *S<sup>n</sup>* and *F<sup>n</sup>* are given in [37] for the C(T) specimen. The fracture criterion is

$$K\_F = K\_{I\varepsilon} / \Phi \tag{3}$$

(5)

(8)

*f or S<sup>n</sup>* ≤ *σys*

$$\Phi = 1 - m \, S\_{11}/S\_1 \tag{4}$$

$$\mathbf{Q} = \left(\sigma\_{\rm H}(\mathbf{S}\_{\rm t})\right)\left(1 - \mathbf{w}\,\mathbf{S}\_{\rm t1}/\mathbf{S}\_{\rm t1}\right) \tag{5}$$

region or hinge on the net section based on the ultimate tensile strength, *σu*. For the compact specimen, *Su* is a function of load eccentricity and is 1.62*σu* for c/w = 0.5 [37]. The fracture parameters, *KF* and *m*, are assumed to be constant in the same sense as the ultimate tensile strength; that is, the parameters may vary with material thickness, state of stress, temperature, and rate of loading. If *m* is equal to zero in Equations (3) and (4), then *KF* is equal to the elastic stress-intensity factor at failure, and the equation represents the behavior of low-toughness (brittle) materials under plane-strain behavior. If *m* is equal to unity, the equation represents fracture behavior of high-toughness materials (plane-stress fracture). For given fracture-toughness parameters, *KF*, and *m*, the elastic stress-intensity factor where *K<sup>F</sup>* and m are the two material fracture parameters. The stress *S<sup>u</sup>* (ultimate value of elastic net-section stress) was computed from the load required to produce a fully plastic region or hinge on the net section based on the ultimate tensile strength, *σu*. For the compact specimen, *S<sup>u</sup>* is a function of load eccentricity and is 1.62*σ<sup>u</sup>* for c/w = 0.5 [37]. The fracture parameters, *K<sup>F</sup>* and *m*, are assumed to be constant in the same sense as the ultimate tensile strength; that is, the parameters may vary with material thickness, state of stress, temperature, and rate of loading. If *m* is equal to zero in Equations (3) and (4), then *K<sup>F</sup>* is equal to the elastic stress-intensity factor at failure, and the equation represents the behavior of low-toughness (brittle) materials under plane-strain behavior. If *m* is equal to unity, the equation represents fracture behavior of high-toughness materials (plane-stress fracture).

at failure is ூ = ி/(1 + 2/௬௦) ≤ ௬௦ (6) For given fracture-toughness parameters, *KF*, and *m*, the elastic stress-intensity factor at failure is

$$\mathcal{K}\_{l\varepsilon} = \mathcal{K}\_{\mathcal{F}} / \left(1 + 2m\gamma / \sigma\_{\rm ys}\right) \tag{6}$$

$$K\_{\rm IF} = \left\{ \sqrt{\left(\mathbf{w}\gamma\right)^2 + 2\gamma S\_{\rm r}} - \mathbf{w}\gamma \right\} \sqrt{\mathbf{x}} \cdot \mathbf{F\_{\rm F}} \tag{7}$$

$$X\_{k^\*} = S\_{k^\*} \sqrt{\pi \alpha} F\_k$$

where *γ* = *KFσys*/ - 2 *S<sup>u</sup>* √ *πc F<sup>n</sup>* . In 1973, a relationship between *KF/E* and *m* [37] was found for many materials and crack configurations. Herein, the relationship was used to help select an m value. As the 9310 steel exhibited a high toughness, an *m*-value of 0.5 was selected. The corresponding *<sup>K</sup><sup>F</sup>* value was 500 MPa-<sup>√</sup> m to fit the 9310 steel fracture data. Solid curve in Figure 8 was calculated from the TPFC for the 76.2 mm wide compact specimens. These calculations show how the initial crack length affects the elastic fracture toughness. In addition, the variation in specimen width (not shown) would greatly affect the elastic fracture toughness, in that larger width specimens produce large *KIe* values.

### **5. Fatigue-Crack-Growth and Crack-Closure Analyses**

*A*1 

Crack-opening-stress equations for constant-amplitude loading were developed from an early analytical crack-closure model (*So*) calculations [38]. As the number of elements within the plastic-zone region in the model was increased to 20 [18] and the crack-growth increment was modelled on a cycle-by-cyclic basis, new *S<sup>o</sup>* equations were made for a single crack in a very wide plate under uniform remote applied stress, *S*. The new set of equations was developed to fit the results from the revised closure model and, again, gave *S<sup>o</sup>* as a function of stress ratio (*R*), maximum stress level (*Smax*/σo) and the constraint factor (*α*). The new equations are

$$S\_o / S\_{\text{max}} = A\_0 + A\_1 R + A\_2 R^2 + A\_3 R^3 \tag{9}$$

$$S\_4/S\_{\rm MLIC} = Aq + AqR$$

where *R* = *Smin/Smax*, *Smax* < 0.8 *σo*, and *Smin* > −*σo*. The A<sup>i</sup> coefficients are functions of *α* and *Smax/σ<sup>o</sup>* and are given by

$$A\_0 = \left(0.9453 - 0.514 \, a + 0.1355 \, a^2 - 0.0133 \, a^3\right) \left[\cos(\beta)\right]^{(0.8 \, a - 0.1)} \tag{11}$$

$$\beta = \pi S\_{\text{max}} / \left(2a\sigma\_o\right)$$

$$= 0.5719 - 0.1726 \text{ } \alpha + 0.019 \text{ } a^2 \Big) S\_{\text{max}} / \sigma\_o \tag{12}$$

$$A\_2 = 0.975 - A\_0 - A\_1 - A\_3 \tag{13}$$

$$A\_3 = 2A\_0 + A\_1 - 1\tag{14}$$

(10)

*f or R* ≥ 0

A crack-closure analysis was then performed on the fatigue-crack growth (∆*K*-rate) data from C(T) specimens in Figure 7 to determine the ∆*Keff*-rate relation. The *K*-analogy concept [18] was used to calculate the crack-opening stresses (or loads) for C(T) specimens from the above equations. The ∆*Keff*-rate data are shown in Figure 9. Selection of the lower constraint factor, 2.5, was found to reasonably collapse the ∆*K*-rate data into an almost unique relation.

In the threshold region, the lower R tests exhibited a rise in crack-opening loads as the ∆*K* level was reduced in a load-reduction test. Even the CPLR method showed a load-history effect, but not as much as the current ASTM procedure [28]. The upper constraint factor, 1.15, and constraint-loss range was selected to help fit spectrum crackgrowth tests [20]. The lower vertical dashed line at (∆*Keff*)th is the estimated threshold for the steel [39], and the upper vertical dashed line at (∆*Keff*)<sup>T</sup> is the location of constraint loss from plane-strain to plane-stress behavior [40]. The solid (blue) lines with circular (yellow) symbols shows the baseline crack-growth-rate curve for FASTRAN. Fatigue-crack-growth, fracture and tensile properties for the 6.35-mm thick 9310 steel are given in Table 1.

**Figure 9.** Effective stress-intensity factor range against rate for 9310 steel C(T) specimens. **Figure 9.** Effective stress-intensity factor range against rate for 9310 steel C(T) specimens.

**Table 1.** Effective stress-intensity factor range against rate, fracture and tensile properties for 9310 steel (*B* = 6.35 mm). **Table 1.** Effective stress-intensity factor range against rate, fracture and tensile properties for 9310 steel (*B* = 6.35 mm).


#### **6. Fatigue Behavior of Notched Specimens 6. Fatigue Behavior of Notched Specimens**

tor [42,43] is

For most fatigue-crack-growth analyses, linear-elastic analyses have been found to be adequate. The linear-elastic effective stress-intensity factor range developed by Elber [2] is given by For most fatigue-crack-growth analyses, linear-elastic analyses have been found to be adequate. The linear-elastic effective stress-intensity factor range developed by Elber [2] is given by

$$
\Delta \mathcal{K}\_{eff} = (\mathcal{S}\_{\max} - \mathcal{S}\_o) \sqrt{\pi c} \,\mathrm{F}(\mathcal{c}/\mathcal{w}) \tag{15}
$$

Δ = (௫ − ) √ (/) (14) where *Smax* is maximum stress, *So* is crack-opening stress, and *F*(*c*/*w*) is the boundary correction factor. However, for high-stress intensity factors and low-cycle fatigue conditions, linear-elastic analyses are inadequate and nonlinear crack-growth parameters are needed. To account for plasticity, a portion of the Dugdale [41] cyclic-plastic-zone size (*ω*) has been added to the crack length, *c*. The cyclic-plastic-zone-corrected effective stress-intensity facwhere *Smax* is maximum stress, *S<sup>o</sup>* is crack-opening stress, and *F*(*c*/*w*) is the boundary correction factor. However, for high-stress intensity factors and low-cycle fatigue conditions, linear-elastic analyses are inadequate and nonlinear crack-growth parameters are needed. To account for plasticity, a portion of the Dugdale [41] cyclic-plastic-zone size (*ω*) has been added to the crack length, *c*. The cyclic-plastic-zone-corrected effective stress-intensity factor [42,43] is

$$\left(\left(\Delta K\_p\right)\_{eff} = \left(\mathcal{S}\_{\max} - \mathcal{S}\_o\right)\sqrt{\pi d}\ F(d/w) \tag{16}$$

(Δ) = (௫ − ) √ (/) (15) where *d* = *c* + *ω*/4 and *F*(*d*/*w*) is the cyclic-plastic-zone-corrected boundary-correction facwhere *d* = *c* + *ω*/4 and *F*(*d*/*w*) is the cyclic-plastic-zone-corrected boundary-correction factor. The cyclic plastic zone is given by

$$
\mathcal{O} = \left(1 - R\_{eff}\right)^2 \rho/4\tag{17}
$$

 = (1 − )ଶ /4 (16) where *Reff* = *So/Smax* and plastic-zone size (*ρ*) for a crack in a large plate [41] is

ρ

$$\rho = \mathcal{c}\left\{\text{sec}[\pi\text{S}\_{\text{max}}/(2a\sigma\_o)] - 1\right\} \tag{18}$$

where *Reff* = *So/Smax* and plastic-zone size (

where flow stress, *σo*, is multiplied by the constraint factor (*α*). Herein, the cyclic-plasticzone corrected effective stress-intensity factor range (Equation (17)) will be used in the fatigue-life predictions.

The FASTRAN life-prediction code [18] was used to model crack growth from an initial micro-structural flaw size to failure and the crack-growth relation used is

$$\text{dc}/d\text{N} = \mathbb{C}\_{1i}[\left(\Delta \mathcal{K}\_p\right)\_{eff}]^{\mathbb{C}\_{2i}} \left\{ 1 - \left[\Delta \mathcal{K}\_o/\left(\Delta \mathcal{K}\_p\right)\_{eff}\right]^p \right\} / \left[1 - \left(\mathcal{K}\_{\text{max}}/\mathcal{K}\_{Ie}\right)^q\right] \tag{19}$$

where *C1i* and *C2i* are coefficient and exponent for each linear segment (*i* = 1 to n), respectively. The (∆*Kp*)*eff* is cyclic-plastic-zone corrected effective stress-intensity factor, ∆*K<sup>o</sup>* is effective threshold, *Kmax* is maximum stress-intensity factor, *KIe* is elastic fracture toughness (which is, generally, a function of crack length, specimen width, and specimen type), *p* and *q* are constants selected to fit test data in either the threshold or fracture regimes, respectively. Herein, no threshold was modeled and ∆*K<sup>o</sup>* was set equal to zero; thus, p was not needed. Near-threshold behavior was modeled with the multi-linear equation (independent of *R*). Fracture was modeled using the TPFC (*K<sup>F</sup>* and *m*) [31,37].

Fatigue tests were conducted on SEN(B) specimens under: (1) constant-amplitude loading (*R* = 0.1) and (2) Cold-Turbistan+ loading. The semi-circular edge notch was chemically polished. The Cold-Turbistan+ spectrum was obtained from [33], adding a constant load to make the overall *R* = 0.1 (compression loads not allowed on SEN(B) specimens). Figure 10 shows the constant-amplitude tests plotting *σmax* (notch root elastic stress) against cycles to failure (*N<sup>f</sup>* ). The square symbols are single-edge-notch tension, SEN(T), specimens [44], whereas the solid circular symbols are from the current study. A trial-and-error procedure was used to find the 6-µm semi-circular surface flaw at the center of the notch to fit the fatigue data. Herein, the 6-µm flaw is considered an equivalent initial flaw size (EIFS) to fit the S-N behavior under constant-amplitude loading. The solid curve shows calculated lives that used available crack-growth-rate data (rates <sup>≥</sup> <sup>4</sup> <sup>×</sup> <sup>10</sup>−<sup>11</sup> m/cycle) for *σmax* ≥ 980 MPa. The dashed (blue) curve used the estimated ∆Keff-rate relation for rates below 4 <sup>×</sup> <sup>10</sup>−<sup>11</sup> m/cycle (see Figure 9). On Figure 10, the horizontal dashed (black) line is where <sup>∆</sup>*Keff* = (∆*Keff*)th = 2.3 MPa<sup>√</sup> m. Upper and lower bound calculations were made with 4- and 10-µm, respectively.

Figure 11 shows the test data (solid circular symbols) under the Cold-Turbistan+ spectrum. The open symbols are the retest of the two runout tests, but at higher applied notch elastic stress. FASTRAN predictions using the same 6-µm surface flaw fell at the lower bound of the test data using the baseline curve from Figure 9 (Table 1). If a threshold of 2.3 MPa√ m (no crack growth) was selected, then the cycles to failure were near the upper bound of the test data. Selecting a ∆*Keff*-rate relation between the vertical dashed line and the baseline (blue) curve below 10−<sup>10</sup> m/cycle, would have given a more accurate predicted fatigue strength. These results indicate that the selection of the baseline curve in Figure 9 for rates below 10−<sup>10</sup> m/cycle is very important. In addition, in the fatigue endurance-limit region, there could also be some build-up of fretting oxide debris on the small-crack surfaces, which could increase the fatigue life by elevating the crack-opening loads. Further study is required to include fretting debris on the crack surfaces in the FASTRAN model.

To study the crack-closure behavior and life predictions under the Cold-Turbistan+ spectrum, the predicted crack-opening-stress ratios are shown in Figure 12 for a small portion of the spectrum (first 80 cycles). Here, the FASTRAN code used the full crackclosure model with a constraint factor of 2.5. Calculated crack-opening stresses began at the minimum stress in the spectrum and the crack-opening stress increases as the crack grows and leaves plastically deformed material in the wake of the crack tip. Calculations were also made at Block 35 (about half-life) and show that the crack-opening stresses are, generally, higher than Block 1. These results are somewhat surprising, in that the predicted crack-opening stresses were generally at the minimum applied stress values for the larger cyclic amplitudes. This would imply that the loading cycle would be fully

effective in growing the crack. Thus, the Cold-Turbistan+ spectrum would be classified as an "accelerating" spectra. the accuracy of fatigue-crack-growth calculations under spectrum loading would be greatly improved with "Rainflow-on-the-Fly" methodology.

larger cyclic amplitudes. This would imply that the loading cycle would be fully effective in growing the crack. Thus, the Cold-Turbistan+ spectrum would be classified as an "ac-

These results are important, in that Rainflow analyses for fatigue-crack growth are a function of load history. Fortunately, the current Rainflow analyses, used in the literature, would be conservative without considering crack-closure load-history effects. However,

*Metals* **2021**, *11*, x FOR PEER REVIEW 14 of 19

celerating" spectra.

**Figure 10. Figure 10.** Measured and calculated stress-life behavior under constant-amplitude loading. Measured and calculated stress-life behavior under constant-amplitude loading.

**Figure 11.** Measured and predicted stress-life behavior under Cold-Turbistan+ spectrum loading. **Figure 11.** Measured and predicted stress-life behavior under Cold-Turbistan+ spectrum loading.

**Figure 12.** Part of Cold Turbistan+ spectrum loading and calculated crack-opening-stress ratios. **Nomenclature Figure 12.** Part of Cold Turbistan+ spectrum loading and calculated crack-opening-stress ratios.

transport dislocations at critical locations, develop slip bands, and cracks.

Δ

damage, unless the method was updated during crack-growth history.

nucleating at micro-structural features, such as inclusions and voids, or at micro-machining marks, and large cracks growing to failure. Thus, the traditional fatigue-crack nucleation stage (*Ni*) is basically the growth in microcracks (initial flaw sizes of 1 to 30 μm growing to about 250 μm) in a variety of metal alloys. Large-crack growth and failure are regions where fracture-mechanic parameters were successful in correlating and predicting fatigue-crack growth and fracture. In the last three decades, fracture-mechanics concepts have also been successful in predicting "fatigue" (growth of small cracks) under constantamplitude and spectrum loading using crack-closure theory. Therefore, the crack-growth approach provides a unified theory for the determination of fatigue lives for metal alloys. However, for pure- and single-crystal materials, there are nucleation cycles required to

Tests were conducted on compact, C(T), and single-edge-notch-bend, SEN(B), specimens made of a 9310 steel (*B* = 6.35 mm) under laboratory air and room temperature conditions. The C(T) crack-growth specimens were tested over a wide range in stress ratios (*R* = 0.1 to 0.95) and crack-growth rates from threshold to fracture. A crack-closure model

Fatigue tests were conducted on the SEN(B) specimens under constant-amplitude

lation was used to calculate or predict fatigue behavior on the SEN(B) specimens using the crack-closure model and small-crack theory. The constant-amplitude fatigue tests were used to determine an initial semi-circular surface flaw size (6-μm) to fit the test data. The 6-μm flaw is considered an equivalent initial flaw size (EIFS). Using the same initial flaw size enabled the FASTRAN code to predict the fatigue behavior under the Cold-Turbistan+ spectrum loading quite well. Rainflow-on-the-Fly methodology was validated on a complex spectrum loading and indicated that the calculated damage was a function of load history and that the usual Rainflow methods would not capture correct crack-growth

*K*-rate data onto an almost unique

Δ

Δ

*Keff-*rate crack-growth re-

*Keff-*rate relation

**7. Concluding Remarks** 

(FASTRAN) was used to collapse the

over more than four orders-of-magnitude in rates.

**Funding:** This research received no external funding

**Conflicts of Interest:** The author declares no conflict of interest.

loading (*R* = 0.1) and a Cold-Turbistan+ spectrum loading. The

These results are important, in that Rainflow analyses for fatigue-crack growth are a function of load history. Fortunately, the current Rainflow analyses, used in the literature, would be conservative without considering crack-closure load-history effects. However, the accuracy of fatigue-crack-growth calculations under spectrum loading would be greatly improved with "Rainflow-on-the-Fly" methodology.

### **7. Concluding Remarks**

Fatigue of materials, like alloys, is basically the fatigue-crack growth in small cracks nucleating at micro-structural features, such as inclusions and voids, or at micro-machining marks, and large cracks growing to failure. Thus, the traditional fatigue-crack nucleation stage (*N<sup>i</sup>* ) is basically the growth in microcracks (initial flaw sizes of 1 to 30 µm growing to about 250 µm) in a variety of metal alloys. Large-crack growth and failure are regions where fracture-mechanic parameters were successful in correlating and predicting fatiguecrack growth and fracture. In the last three decades, fracture-mechanics concepts have also been successful in predicting "fatigue" (growth of small cracks) under constant-amplitude and spectrum loading using crack-closure theory. Therefore, the crack-growth approach provides a unified theory for the determination of fatigue lives for metal alloys. However, for pure- and single-crystal materials, there are nucleation cycles required to transport dislocations at critical locations, develop slip bands, and cracks.

Tests were conducted on compact, C(T), and single-edge-notch-bend, SEN(B), specimens made of a 9310 steel (*B* = 6.35 mm) under laboratory air and room temperature conditions. The C(T) crack-growth specimens were tested over a wide range in stress ratios (*R* = 0.1 to 0.95) and crack-growth rates from threshold to fracture. A crack-closure model (FASTRAN) was used to collapse the ∆*K*-rate data onto an almost unique ∆*Keff*-rate relation over more than four orders-of-magnitude in rates.

Fatigue tests were conducted on the SEN(B) specimens under constant-amplitude loading (*R* = 0.1) and a Cold-Turbistan+ spectrum loading. The ∆*Keff*-rate crack-growth relation was used to calculate or predict fatigue behavior on the SEN(B) specimens using the crack-closure model and small-crack theory. The constant-amplitude fatigue tests were used to determine an initial semi-circular surface flaw size (6-µm) to fit the test data. The 6-µm flaw is considered an equivalent initial flaw size (EIFS). Using the same initial flaw size enabled the FASTRAN code to predict the fatigue behavior under the Cold-Turbistan+ spectrum loading quite well. Rainflow-on-the-Fly methodology was validated on a complex spectrum loading and indicated that the calculated damage was a function of load history and that the usual Rainflow methods would not capture correct crack-growth damage, unless the method was updated during crack-growth history.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.

### **Nomenclature**



### **Abbreviations**


### **References**


## *Article* **Computational Failure Analysis under Overloading**

**Slobodanka Boljanovi´c 1,\* and Andrea Carpinteri <sup>2</sup>**


**Abstract:** The aim of this research work is to shed more light on performance-based design through a computational framework that assesses the residual strength of damaged plate-type configurations under overloading. Novel expressions are generated to analyze the power of crack-like stress raisers coupled with retardation effects. Analytical outcomes show that careful consideration of the overload location and crack size can be quite effective in improving safety design and failure mode estimation.

**Keywords:** analytical framework; fatigue crack; residual strength; retardation effect

### **1. Introduction**

The well-known role of surface flaws in compromising bearing capacity during service operations, characterized by variable amplitude loading where overload and underload often exist, constantly warns that such stress raisers can cause sudden hazards in large moving systems.

An understanding of the load interaction effects on the fatigue crack extension, resulting from changes in the cyclic load level, is desirable to ensure the safety integrity and full functioning of structural components exposed to dynamic load environments with/without mixed loading modes, by employing reliable computational frameworks.

From the point of view of fracture mechanics, the driving mode interactions due to overload may be considered through several mechanisms such as residual stress [1], crack closure [2], crack tip blunting [3], strain hardening [4], crack branching [5] and reversed yielding [6].

The interest in the overload phenomenon, even if it has been researched for a long time, has become more pronounced in order to meet increasingly stringent damage tolerancebased demands for modern systems, combining relevant mechanisms and/or concepts. Thus, Elber [2] suggested that the concept of crack opening stress can be applied to generate the driving interactions in the vicinity of the crack tip due to overload. Budiansky and Hutchinson [7] employed the strip-yield hypothesis, generally attributed to the work of Dugdale [8] and Barenblatt [9], to model the plasticity-induced fatigue crack closure. Ohji et al. [10] introduced an incremental plasticity model incorporating kinematic hardening and crack growth simulation, extending the fatigue flaw in each stress cycle by a prescribed length that was equal to the finite element mesh size. Willenborg et al. [11] and Wheeler [12] suggested that residual compressive stresses can retard post-overload crack growth, and they developed their fracture mechanics models. Later, Suresh [5] discussed that retardation can persist even when the post-overload has traversed through the predicted zone of residual compressive stresses. Fleck [13] and Wang et al. [14] performed the finite element analysis to generate a plasticity-induced crack closure in two-dimensional configurations under plain-strain conditions.

Through a stability analysis under overload, Sander and Richard [15] employed the strip yield model and the modified generalized Willenborg model, implemented in the NASGRO [16] software package. Pavlou et al. [17] developed a fracture mechanics model

**Citation:** Boljanovi´c, S.; Carpinteri, A. Computational Failure Analysis under Overloading. *Metals* **2021**, *11*, 1509. https://doi.org/10.3390/ met11101509

Academic Editor: George A. Pantazopoulos

Received: 2 August 2021 Accepted: 15 September 2021 Published: 23 September 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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 (https:// creativecommons.org/licenses/by/ 4.0/).

based on strain-hardening fatigue mechanisms in the plastic zone in order to explore the influence of yield stress changes within the overload plastic zone on the fatigue crack growth rate. Huang et al. [18] introduced a modified Wheeler model incorporating an improved fatigue crack growth rate solution and the equivalent stress-intensity concept. [20]. Harmain [21] proposed several modifications to Wheeler's crack growth idea involving an effective stress intensity factor based on Elber's crack closure concept [2], the relationship between overload ratio and the Wheeler's exponent, and the fatigue crack growth

NASGRO [16] software package. Pavlou et al. [17] developed a fracture mechanics model based on strain-hardening fatigue mechanisms in the plastic zone in order to explore the influence of yield stress changes within the overload plastic zone on the fatigue crack growth rate. Huang et al. [18] introduced a modified Wheeler model incorporating an improved fatigue crack growth rate solution and the equivalent stress-intensity concept.

Further, Mohanty et al. [19] took into account Frost and Dugdale's fracture mechanics concept and the modified two-parameter crack growth model proposed by Jones et al.

*Metals* **2021**, *11*, x FOR PEER REVIEW 2 of 11

Further, Mohanty et al. [19] took into account Frost and Dugdale's fracture mechanics concept and the modified two-parameter crack growth model proposed by Jones et al. [20]. Harmain [21] proposed several modifications to Wheeler's crack growth idea involving an effective stress intensity factor based on Elber's crack closure concept [2], the relationship between overload ratio and the Wheeler's exponent, and the fatigue crack growth rate calculation. Boljanovi´c and Maksimovi´c [22] examined the fatigue performance of a through-crack under overloading using the Kujawski crack growth concept [23] together with the Wheeler retardation concept. Further, Boljanovi´c et al. [24] analyzed the stability of the same stress raiser, linking the Zhan et al. concept [25] and the Wheeler concept [12] together with relevant solutions suggested by Richard et al. [26] for assessing the stress intensity factor under mixed mode loading with overload. rate calculation. Boljanović and Maksimović [22] examined the fatigue performance of a through-crack under overloading using the Kujawski crack growth concept [23] together with the Wheeler retardation concept. Further, Boljanović et al. [24] analyzed the stability of the same stress raiser, linking the Zhan et al. concept [25] and the Wheeler concept [12] together with relevant solutions suggested by Richard et al. [26] for assessing the stress intensity factor under mixed mode loading with overload. In the present work, the trends of fatigue strength degradation are evaluated taking into account the overload effect. In order to identify and characterize the potential features of such a phenomenon, an analytical framework is proposed, and a detailed body of evidence is provided for evaluating detrimental effects of a crack-like through flaw coupled

In the present work, the trends of fatigue strength degradation are evaluated taking into account the overload effect. In order to identify and characterize the potential features of such a phenomenon, an analytical framework is proposed, and a detailed body of evidence is provided for evaluating detrimental effects of a crack-like through flaw coupled with retardation effect due to overloading. Solutions from relevant literature demonstrate that this computational tool is able to provide a physical interpretation of the overload mechanism governing the propagation of the fatigue flaw so that it is possible to generate stress intensities and residual life through a consistent set of fracture-mechanics parameters. with retardation effect due to overloading. Solutions from relevant literature demonstrate that this computational tool is able to provide a physical interpretation of the overload mechanism governing the propagation of the fatigue flaw so that it is possible to generate stress intensities and residual life through a consistent set of fracture-mechanics parameters. **2. Driving Mode Analysis under Cyclic Loading** 

#### **2. Driving Mode Analysis under Cyclic Loading** Achieving high safety performances necessarily requires a detailed analysis of the

Achieving high safety performances necessarily requires a detailed analysis of the fatigue-critical hot spots in large moving systems through relevant fracture mechanicsbased crack growth concepts [19,24,27,28]. Hence, the driving mode progression due to crack-like edge flaw (Figure 1) is evaluated by employing the Huang–Moan crack growth concept [29], expressed as follows: fatigue-critical hot spots in large moving systems through relevant fracture mechanicsbased crack growth concepts [19,24,27,28]. Hence, the driving mode progression due to crack-like edge flaw (Figure 1) is evaluated by employing the Huang–Moan crack growth concept [29], expressed as follows:

$$\frac{da}{dN} = \mathcal{C}(M\,\Delta K)^m \tag{1}$$

where *da*/*dN* and *a* are the crack growth rate and crack length, respectively, ∆*K* is the stress intensity factor range, *C* and *m* are material parameters experimentally obtained, and *N* is the number of loading cycles. where *da*/*dN* and *a* are the crack growth rate and crack length, respectively, ∆*K* is the stress intensity factor range, *C* and *m* are material parameters experimentally obtained, and *N* is the number of loading cycles.

**Figure 1.** Geometry of the plate with edge crack-like flaw subjected to cyclic loading with overload (unit: mm). **Figure 1.** Geometry of the plate with edge crack-like flaw subjected to cyclic loading with overload (unit: mm).

Further, relevant interactions between in-service loading profiles and environmental effects are theoretically examined by means of the following fracture mechanics parame-Further, relevant interactions between in-service loading profiles and environmental effects are theoretically examined by means of the following fracture mechanics parameter [29]:

ter [29]:

$$M = \begin{cases} \left(1 - R\right)^{-\beta\_1} & -5 \le R < 0\\ \left(1 - R\right)^{-\beta} & 0 \le R < 0.5\\ \left(1.05 - 1.4R + 0.6R^2\right)^{-\beta} & 0.5 \le R < 1 \end{cases} \tag{2}$$

where *R* is the stress ratio, *β*<sup>1</sup> and *β* are material parameters, experimentally obtained in the case of one negative and two positive cyclic loading domains [29], respectively.

Through a safety-relevant analysis herein presented, after integration crack growth rate (Equation (1)) from initial *a<sup>0</sup>* to final *a<sup>f</sup>* crack length, the residual life can be evaluated as follows:

$$N = \int\_{a\_0}^{a\_f} \frac{da}{\mathcal{C}(M\,\Delta K)^m} \tag{3}$$

In order to make correct durability decisions and have well-coordinated fatigue responses to potential hazard events, a reliable assessment of residual bearing capacity and severity of plate-type system damage is needed. According to damage tolerance requirements, nonlinear driving force interactions in the vicinity of the crack tip are herein explored through the stress intensity range expressed as follows:

$$
\Delta \mathcal{K} = \frac{\Delta P}{w \, t} \, f\_{\mathcal{I}} \left( \frac{a}{w} \right) \sqrt{\pi \, a} \tag{4}
$$

where ∆*P* is applied force range, *w* and *t* are width and thickness of the plate, respectively.

Further, in order to quantify the effect of an edge crack-like through flaw and geometry of considered plate, the following fracture mechanics-based parameter is employed [19].

$$f\_I\left(\frac{a}{w}\right) = 1.12 - 0.231\left(\frac{a}{w}\right) + 10.55\left(\frac{a}{w}\right)^2 - 21.72\left(\frac{a}{w}\right)^3 + 30.39\left(\frac{a}{w}\right)^4 \tag{5}$$

### **3. Crack Growth Evolution Taking into Account the Retardation Effect**

Variable amplitude cyclic-load environment may often have a significant impact on the strength capacity of large moving systems due to the appearance of overload, underload, and overload followed by underload, characterized by retardation, acceleration, and reduced retardation. Thus, if overload is applied, a relevant plastic zone of residual compressive stresses is generated in the vicinity of the crack tip. Such a nonlinear stress state field causes retardation in crack propagation by reducing the exiting crack tip driving force. In this context, fatigue analysis is herein performed employing the fracture mechanicsbased concepts [29] combined with the retardation concept [12], i.e.,

$$\frac{da}{dN} = \mathcal{C}\_{pi}\mathcal{C}(M\Delta K)^m\tag{6}$$

where *Cpi* is the fatigue parameter related to the overload effect.

Complex interactions in the zone where compressive stress states exist are theoretically examined using the following fracture mechanics solution proposed by Wheeler [12]:

$$\mathcal{C}\_{pi} = \left\{ \begin{array}{c} \left( \frac{r\_{pi}}{a\_{ol} + r\_{po} - a\_i} \right)^p \\ 1 \end{array} \; ; \; a\_i + r\_{pi} \le a\_{ol} + r\_{po} \\ \text{or} \\ a\_i + r\_{pi} \ge a\_{ol} + r\_{po} \end{array} \tag{7}$$

where *rpi* and *a<sup>i</sup>* are the current plastic zone in the *i*th cyclic and corresponding crack size, and *rpo* and *aol* represent the overload plastic zone size and appropriate overload crack size, respectively, and *p* is the retardation exponent experimentally obtained.

Since the power of the overload effect decreases when the current plastic zone approaches the overload plastic zone, the relevant sizes of current plastic zone and monotonic overload plastic zone are computed [30,31] as follows:

$$r\_{pi} = \frac{1}{\pi} \left(\frac{\Delta K}{2\sigma\_{ys}}\right)^2 r\_{po} = \frac{1}{\pi} \left(\frac{K\_{ol}}{\sigma\_{ys}}\right)^2 \tag{8}$$

where ∆*K* and *Kol* are the relevant stress intensity factor range and the stress intensity factor generated by an overload, respectively, and *σys* is the yield strength of the material.

Furthermore, fatigue-based overload evaluations are herein performed in terms of the number of loading cycles, integrating the proposed solution for crack growth rate in Equation (6), i.e.,

$$N = \int\_{a\_0}^{a\_f} \frac{da}{\mathbb{C}\_{pi}\mathbb{C}(\Delta \mathbb{K})^m} \tag{9}$$

where *a<sup>0</sup>* and *a<sup>f</sup>* are initial and final crack length, respectively.

With the need to reduce design life cyclic time and costs with ever more complex large moving systems, the possibility of assessing the failure strength against fatigueinduced loading interactions through advanced computational strategies is attractive. Thus, a damage tolerance-based analytical framework is herein developed, in which Euler's algorithm is implemented for generating fatigue resistance under overloading, as is discussed through several case study applications in the next section.

### **4. Residual Strength Design Using Developed Computational Framework**

### *4.1. Fatigue Evaluations under Cyclic Loading with Overload*

The failure performance design presented here tackles the crack-like edge stress raiser (Figure 1) under cyclic loading with overload. Through the life evaluations for the plate made of 6061 T6 aluminum alloy, the following material and loading parameters are employed: *<sup>C</sup>* = 6 <sup>×</sup> <sup>10</sup>−<sup>11</sup> , *m* = 3.2, *β* = 0.7, *λ* = 1, *p* = 0.21, *σys* = 296 MPa, *Pmax* = 11,772 N with *R* = 0. Relevant geometrical sizes are characterized by *w* = 50 mm, *t* = 3 mm, *L* = 180 mm, and an initial crack length equal to *a0*= 6 mm is examined. By adopting that the crack length of a single overload is equal to *aol* = 7.5 mm, driving mode interactions are theoretically examined in the case of three overload stress ratios (*Rol* = 1.42, 1.67, and 1.88) which are shown in Table 1.

**Table 1.** Evaluated number of loading cycles and applied force ranges with corresponding stress ratios under overloading.


The fatigue vulnerability analysis performed via a novel computational framework evaluates the driving mode caused by through-crack and the residual life through Equations (1)–(5) coupled with Equations (6)–(9), respectively. Safety outcomes shown in Figure 2a generate the stress intensities in the vicinity of the crack tip caused by overload (*Rol* = 1.42), whereas the number of loading cycles versus crack length is examined in Figures 2b and 3, and Table 1 in the case of three different overload stress ratios (*Rol* = 1.42, 1.67, and 1.88). Further experimental outcomes discussed by Kumar [32] are employed to estimate the predictive capability of generated plate lives.

mate the predictive capability of generated plate lives.

*Metals* **2021**, *11*, x FOR PEER REVIEW 5 of 11

**Figure 2.** Fatigue resistance analysis: (**a**) *KI* versus *a*, *Rol* = 1.42 and (**b**) *a* versus *N*, *Rol* = 1.42, calculated curves from the present work, experiments reported by Kumar [32]. **Figure 2.** Fatigue resistance analysis: (**a**) *K<sup>I</sup>* versus *a*, *Rol* = 1.42 and (**b**) *a* versus *N*, *Rol* = 1.42, calculated curves from the present work, experiments reported by Kumar [32]. curves from the present work, experiments reported by Kumar [32].

**Figure 3.** Fatigue resistance analysis: (**a**) *a* versus *N*, *Rol* = 1.67 and (**b**) *a* versus *N*, *Rol* = 1.88, calculated **Figure 3.** Fatigue resistance analysis: (**a**) *a* versus *N*, *Rol* = 1.67 and (**b**) *a* versus *N*, *Rol* = 1.88, calculated curves from the present work, experiments reported by Kumar [32]. **Figure 3.** Fatigue resistance analysis: (**a**) *a* versus *N*, *Rol* = 1.67 and (**b**) *a* versus *N*, *Rol* = 1.88, calculated curves from the present work, experiments reported by Kumar [32].

curves from the present work, experiments reported by Kumar [32].

Through different comparisons presented in this section, it can be inferred that the developed computational framework provides conservative estimates for plates with edge through-crack subjected to overload. In addition, the main contribution of the fatigue resistance analysis presented is that it can be stated that, for the considered plate with through-crack (characterized by *t* = 3 mm), the higher the overload ratio and applied over-Through different comparisons presented in this section, it can be inferred that the developed computational framework provides conservative estimates for plates with edge through-crack subjected to overload. In addition, the main contribution of the fatigue resistance analysis presented is that it can be stated that, for the considered plate with through-crack (characterized by *t* = 3 mm), the higher the overload ratio and applied overload force range, the more pronounced the impact of retardation effects under cyclic load-Through different comparisons presented in this section, it can be inferred that the developed computational framework provides conservative estimates for plates with edge through-crack subjected to overload. In addition, the main contribution of the fatigue resistance analysis presented is that it can be stated that, for the considered plate with through-crack (characterized by *t* = 3 mm), the higher the overload ratio and applied overload force range, the more pronounced the impact of retardation effects under cyclic loading (see Table 1). Since complex interactions between the effects of through-crack

load force range, the more pronounced the impact of retardation effects under cyclic load-

ing (see Table 1). Since complex interactions between the effects of through-crack and the

to failure.

and the stress ratio effect coupled with the effects of thin plate thickness are adequately evaluated via fatigue life (with respect to experiments [32]), the computational strategy herein proposed can significantly contribute to further improvements in safety design and optimization of large moving systems. *4.2. The Effect of Overload Crack Length and the Stress Ratio Effect on the Fatigue Strength*  In this section, the strength performance under cyclic loading with single overload is

### *4.2. The Effect of Overload Crack Length and the Stress Ratio Effect on the Fatigue Strength* designed for plates made of 6061 T6 aluminum alloy (*w* = 50 mm, *t* = 4 mm, *L* = 200 mm, Figure 1). Fatigue stability is explored in the case of overload crack length equal to *aol* = 8.5

posed can significantly contribute to further improvements in safety design and optimi-

*Metals* **2021**, *11*, x FOR PEER REVIEW 6 of 11

zation of large moving systems.

In this section, the strength performance under cyclic loading with single overload is designed for plates made of 6061 T6 aluminum alloy (*w* = 50 mm, *t* = 4 mm, *L* = 200 mm, Figure 1). Fatigue stability is explored in the case of overload crack length equal to *aol* = 8.5 mm, 14.5 mm, and 24.5 mm assuming the following overload and cyclic loading parameters: *Pmaxol* = 22,150 N, *Rol* = 1.75, *Pmax* = 15,500 N, *R* = 0.2. The initial crack length is characterized by *a<sup>0</sup>* = 5.5 mm. mm, 14.5 mm, and 24.5 mm assuming the following overload and cyclic loading parameters: *Pmaxol* = 22,150 N, *Rol* = 1.75, *Pmax* = 15,500 N, *R* = 0.2. The initial crack length is characterized by *a0* = 5.5 mm. Damage tolerance-based assessments in terms of the residual life generated via novel analytical solutions (Equations (2)–(5)) are shown in Figure 4a for the three different sin-

Damage tolerance-based assessments in terms of the residual life generated via novel analytical solutions (Equations (2)–(5)) are shown in Figure 4a for the three different single-overload conditions considered. Furthermore, the interactions between the effect of through-crack and overload effect are theoretically examined for plates (*w* = 50 mm, *t* = 3.5 mm, *a<sup>0</sup>* = 4.5 mm) subjected to three different values of stress ratio, i.e., *R* = 0.1, 0.3, and 0.5, respectively, as is shown in Figure 4b. Note that relevant force range and maximum overload force are equal to *Pmax* = 17,200 N, *Pmaxol*= 24,420 N, *Rol* = 1.95, and *ao*<sup>l</sup> = 10.5 mm, whereas material parameters are the same as those adopted in the previous section. gle-overload conditions considered. Furthermore, the interactions between the effect of through-crack and overload effect are theoretically examined for plates (*w* = 50 mm, *t* = 3.5 mm, *a0* = 4.5 mm) subjected to three different values of stress ratio, i.e., *R* = 0.1, 0.3, and 0.5, respectively, as is shown in Figure 4b. Note that relevant force range and maximum overload force are equal to *Pmax* = 17,200 N, *Pmaxol*= 24,420 N, *Rol* = 1.95, and *ao*l = 10.5 mm, whereas material parameters are the same as those adopted in the previous section.

**Figure 4.** Fatigue resistance analysis: (**a**) *a* versus *N* (1 − *aol* = 8.5 mm, 2 − *aol* = 14.5 mm, 3 − *aol* = 24.5 mm) and (**b**) *a* versus *N* (1 − *R* = 0.1, 2 − *R*= 0.3, 3 − *R* = 0.5), calculated curves from the present work. **Figure 4.** Fatigue resistance analysis: (**a**) *a* versus *N* (1 − *aol* = 8.5 mm, 2 − *aol* = 14.5 mm, 3 − *aol* = 24.5 mm) and (**b**) *a* versus *N* (1 − *R* = 0.1, 2 − *R*= 0.3, 3 − *R* = 0.5), calculated curves from the present work.

By analyzing relevant theoretical outcomes shown in Figure 4, it can be concluded that the number of loading cycles decreased from 49,450 to 45,180 due to the increase in overload crack length from 8.5 mm to 14.5 mm. Furthermore, if the stress ratio increases from 0.1 to 0.3, the number of loading cycles increases from 27,820 to 36,790. Evidently, an increase in overload crack length leads to a decrease in the retardation effect due to overloading. In addition, increasing the value of stress ratio under cyclic loading can contribute to increasing the impact of overload leading to an additional increase in residual life By analyzing relevant theoretical outcomes shown in Figure 4, it can be concluded that the number of loading cycles decreased from 49,450 to 45,180 due to the increase in overload crack length from 8.5 mm to 14.5 mm. Furthermore, if the stress ratio increases from 0.1 to 0.3, the number of loading cycles increases from 27,820 to 36,790. Evidently, an increase in overload crack length leads to a decrease in the retardation effect due to overloading. In addition, increasing the value of stress ratio under cyclic loading can contribute to increasing the impact of overload leading to an additional increase in residual life to failure.

mixed mode flaws (Figure 5). The fatigue life is assessed here in the case of plate made of 2024 T3 aluminum alloy (*w* = 52 mm, *t* = 6.5 mm *σys* = 324 MPa, *E* = 73.1 GPa, *β* = 0.7, *p* =

*4.3. Fatigue Evaluations under Mixed Mode Loading with Overload* 

### *4.3. Fatigue Evaluations under Mixed Mode Loading with Overload Metals* **2021**, *11*, x FOR PEER REVIEW 7 of 11

Finally, the performance design carries out the driving mode progression due to the mixed mode flaws (Figure 5). The fatigue life is assessed here in the case of plate made of 2024 T3 aluminum alloy (*w* = 52 mm, *t* = 6.5 mm *σys* = 324 MPa, *E* = 73.1 GPa, *β* = 0.7, *<sup>p</sup>* = 0.21, *<sup>C</sup>* = 5 <sup>×</sup> <sup>10</sup>−<sup>11</sup> , *m* = 3.05). Further, maximum force *Pmax* = 7197 N (with stress ratio *R* = 0.1) and relevant maximum overload force *Pmaxol* = 17,993 N (with overload stress ratio *R*ol = 0.1 applied at crack length *aol* = 20.4 mm) are adopted in order to analyze the loading angle effect of crack-like flaw, whose initial length is equal to *a<sup>0</sup>* = 17.75 mm. 0.21, *C* = 5 × 10−11, *m* = 3.05). Further, maximum force *Pmax* = 7197 N (with stress ratio *R* = 0.1) and relevant maximum overload force *Pmaxol* = 17,993 N (with overload stress ratio *R*ol = 0.1 applied at crack length *aol* = 20.4 mm) are adopted in order to analyze the loading angle effect of crack-like flaw, whose initial length is equal to *a0* = 17.75 mm.

**Figure 5.** Geometry of the plate with edge crack-like flaw subjected to mixed-mode loading with overload. All dimensions are in millimeters. **Figure 5.** Geometry of the plate with edge crack-like flaw subjected to mixed-mode loading with overload. All dimensions are in millimeters.

In-service large moving systems often face a complex fatigue environment which can cause the formation of mixed-mode and/or multi-axial flaws. From the point of view of fracture mechanics, the detrimental effects of such randomly oriented quantities, represented as manufacturing and in-service flaws/defects, should be analyzed through the equivalent stress intensity factor [33]. Thus, the stress state in the vicinity of the crack tip, In-service large moving systems often face a complex fatigue environment which can cause the formation of mixed-mode and/or multi-axial flaws. From the point of view of fracture mechanics, the detrimental effects of such randomly oriented quantities, represented as manufacturing and in-service flaws/defects, should be analyzed through the equivalent stress intensity factor [33]. Thus, the stress state in the vicinity of the crack tip, where combined loading modes exist (see Figure 5), may be evaluated as follows:

$$
\Delta K\_{eq} = \left(\Delta K\_I^4 + 8\Delta K\_{II}^4\right)^{0.25} \tag{10}
$$

( )0.25 <sup>4</sup> <sup>4</sup> <sup>Δ</sup>*Keq* <sup>=</sup> <sup>Δ</sup>*KI* <sup>+</sup>8Δ*KII* (10) Fatigue-induced interactions caused by local mode I and mode II load environment are herein assessed via relevant stress intensity factors [34], i.e.,

$$
\Delta K\_I = \frac{\Delta P}{w \cdot t} \cos \phi \, f\_I \left(\frac{a}{w}\right) \sqrt{\pi \, a} \tag{11}
$$

$$
\Delta K\_{II} = \frac{\Delta P}{w \cdot t} \sin \phi \, f\_I \left(\frac{a}{w}\right) \sqrt{\pi \, a} \tag{12}
$$

Through the failure resistance design, the crack growth rate and residual life are evaluated employing Equations (1)–(3) and Equations (6)–(8) together with Equations (9)–(12) for three different loading profiles i.e., *φ* = 18°, 36°, and 54°, respectively. Figure 6a presents the equivalent stress intensity factors evaluated in the case of three different loading angles, whereas the residual strength in terms of the number of loading cycles is plotted

*w wt* where ∆*P* is applied force ranges, *a* represents the crack length, and *w* and *t* are the width and thickness of the plate, respectively.

*a w <sup>a</sup> <sup>f</sup> wt <sup>P</sup> KII* φ *<sup>I</sup>* π <sup>Δ</sup> <sup>Δ</sup> <sup>=</sup> sin (12) where ∆*P* is applied force ranges, *a* represents the crack length, and *w* and *t* are the width and thickness of the plate, respectively. Through the failure resistance design, the crack growth rate and residual life are evaluated employing Equations (1)–(3) and Equations (6)–(8) together with Equations (9)–(12) for three different loading profiles i.e., *φ* = 18◦ , 36◦ , and 54◦ , respectively. Figure 6a presents the equivalent stress intensity factors evaluated in the case of three different loading angles, whereas the residual strength in terms of the number of loading cycles is plotted in Figures 6b and 7a,b.

in Figures 6b and 7a,b.

*Metals* **2021**, *11*, x FOR PEER REVIEW 8 of 11

**Figure 6.** Fatigue resistance analysis: (**a**) *Keq* versus *a* (1 − *φ* = 18°, 2 − *φ* = 36°, 3 − *φ* = 54°), calculated curves from the present work, and (**b**) *a* versus *N* (*φ* = 18°), 1—calculated curve from the present work, 2—calculated curve reported by Mohanty et al. [19]. **Figure 6.** Fatigue resistance analysis: (**a**) *Keq* versus *a* (1 − *φ* = 18◦ , 2 − *φ* = 36◦ , 3 − *φ* = 54◦ ), calculated curves from the present work, and (**b**) *a* versus *N* (*φ* = 18◦ ), 1—calculated curve from the present work, 2—calculated curve reported by Mohanty et al. [19]. **Figure 6.** Fatigue resistance analysis: (**a**) *Keq* versus *a* (1 − *φ* = 18°, 2 − *φ* = 36°, 3 − *φ* = 54°), calculated curves from the present work, and (**b**) *a* versus *N* (*φ* = 18°), 1—calculated curve from the present work, 2—calculated curve reported by Mohanty et al. [19].

**Figure 7.** Fatigue resistance analysis: (**a**) *a* versus *N* (*φ* = 36°), 1—calculated curve from the present work, 2—calculated curve from Reference [19] and (**b**) *a* versus *N* (*φ* = 54°), calculated curve from **Figure 7.** Fatigue resistance analysis: (**a**) *a* versus *N* (*φ* = 36°), 1—calculated curve from the present work, 2—calculated curve from Reference [19] and (**b**) *a* versus *N* (*φ* = 54°), calculated curve from the present work, experiments reported by Mohanty et al. [19]. **Figure 7.** Fatigue resistance analysis: (**a**) *a* versus *N* (*φ* = 36◦ ), 1—calculated curve from the present work, 2—calculated curve from Reference [19] and (**b**) *a* versus *N* (*φ* = 54◦ ), calculated curve from the present work, experiments reported by Mohanty et al. [19].

the present work, experiments reported by Mohanty et al. [19].

Further, literature-based experimental/theoretical outcomes reported by Mohanty et al. [19] are examined for the same mixed mode configurations in order to assess the pre-Further, literature-based experimental/theoretical outcomes reported by Mohanty et al. [19] are examined for the same mixed mode configurations in order to assess the predictive capability of the generated computational framework. From Figures 6 and 7, it can Further, literature-based experimental/theoretical outcomes reported by Mohanty et al. [19] are examined for the same mixed mode configurations in order to assess the predictive capability of the generated computational framework. From Figures 6 and 7, it can be

those discussed by Boljanović et al. [24] and Mohanty et al. [19], as is shown in Table 2. Note that relevant calculations from [24] represent those in which stress intensities due to

Note that relevant calculations from [24] represent those in which stress intensities due to

the fact that mixed modes were analyzed using the Richard et al. [26] concept.

the fact that mixed modes were analyzed using the Richard et al. [26] concept.

inferred that relevant experimental data and theoretical outcomes are in quite good agreement. Further, fatigue lives evaluated through this research work are compared with those discussed by Boljanovi´c et al. [24] and Mohanty et al. [19], as is shown in Table 2. Note that relevant calculations from [24] represent those in which stress intensities due to the fact that mixed modes were analyzed using the Richard et al. [26] concept.

**Table 2.** Evaluated number of loading cycles and loading angles for relevant mixed mode conditions.


Comparisons in Table 2 indicate that relevant concepts discussed through this paper and those proposed within [24] adequately generate interactions between the stress raiser effects and mixed mode effects under overloading, where the framework previously developed provides a more conservative trend of estimates with respect to fatigue outcomes reported in [19]. It is evident that theoretical outcomes represent high performance/quality estimates.

Further, it should be noted that, during mandatory large moving systems inspections/controls in the case of loading angles *φ* = 36◦ and 54◦ , a computational framework herein developed can have an important role since it generates residual life values closer to relevant experimental/theoretical outcomes [19] than the ones discussed in [24].

Moreover, the safety analysis demonstrates that retardation effect due to overloading is pronounced for loading angle *φ* = 18◦ , while its impact in the case of *φ* = 36◦ and 54◦ decreased by about 10% if the edge crack-like flaw is subjected to mixed (I and II) loading modes.

### **5. Conclusions**

Due to heavy usage, large moving systems usually work and deteriorate in a variable amplitude dynamic environment, where deleterious factors caused by localized flaws can seriously compromise their bearing capacities. Thus, reliable and target-oriented assessments of fatigue degradation under overload, according to damage tolerance requirements, are very important. A challenging and also interesting task is to monitor/control the failure strength through the residual life evaluations simultaneously and independently in the case of axial loading and mixed mode loading, taking into account the load interaction effect. In this context, the present research work proposes a novel analytical framework for analyzing interactions between the effects of crack-like through flaws and overload effect.

In order to explore crack growth retardation, a two-parameter driving force concept is combined with the fracture mechanics concept proposed by Wheeler. Further, relevant fatigue life solutions are established which are essential to the reliable control of throughflaw configurations under axial and mixed mode loading. Several case study applications are given to demonstrate the fatigue assessments under targeted cyclic load profiles and to verify the developed computational tool taking into account the effect of overload stress ratio and the loading angle effects due to mixed modes. In addition, a significant contribution of this research is that it supplies notable information about overload mechanisms and sheds a light on improving the safe-integrity performance design of failure-critical aeronautical systems.

**Author Contributions:** Conceptualizations, S.B. and A.C.; methodology, S.B. and A.C.; software, S.B.; validation, S.B. and A.C. writing-original draft preparation, S.B.; writing-review and editing S.B. and A.C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Data Availability Statement:** Not available.

**Acknowledgments:** The present scientific research was supported by the Serbian Ministry of Education, Science and Technological Development through the Mathematical Institute of the Serbian Academy of Sciences and Arts, Belgrade and the COST Association, Brussels, Belgium within the Action CA 18203, which is gratefully acknowledge.

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

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