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Article

On the Link between Plastic Wake Induced Crack Closure and the Fatigue Threshold

by
Rhys Jones
1,2,*,
Andrew Ang
1,
Nam D. Phan
3 and
Michael Nicholas
4
1
ARC Industrial Transformation Training Centre on Surface Engineering for Advanced Materials, Faculty of Science, Engineering and Technology, Swinburne University of Technology, John Street, Hawthorn, VIC 3122, Australia
2
Centre of Expertise for Structural Mechanics, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, VIC 3800, Australia
3
Structures Division, Naval Air Systems Command, Patuxent River, MD 20670, USA
4
US Army Research Laboratory, U.S. Army Combat Capabilities Development Command Weapons and Materials Research Directorate, Aberdeen Proving Ground, Aberdeen, MD 21005, USA
*
Author to whom correspondence should be addressed.
Metals 2024, 14(5), 523; https://doi.org/10.3390/met14050523
Submission received: 4 April 2024 / Revised: 23 April 2024 / Accepted: 27 April 2024 / Published: 29 April 2024
(This article belongs to the Special Issue Advances in Fatigue and Fracture of Metals and Alloys and Their Applications)
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Versions Notes

Abstract

:
This purpose of this paper is to examine the relationship between crack growth equations based on Elber’s original plastic wake induced crack closure concept and the fatigue threshold as defined by the American Society for Testing and Materials (ASTM) fatigue test standard ASTM E647-15el. It is shown that, for a number of conventionally manufactured metals, the function U(R), where R is the ratio of the minimum to maximum applied remote stress, that is used to relate the stress intensity factor ΔK to the effective stress intensity factor ΔKeff is inversely proportional to the fatigue threshold ΔKth(R). This finding also results in a simple closed form equation that relates the crack opening stress intensity factor Ko(R) to ΔK, Kmax, and the fatigue threshold terms ΔKth(R) and ΔKeff,th. It is also shown that plotting da/dN as function of ΔK/ΔKth(R) would appear to have the potential to help to identify the key fracture mechanics parameters that characterise the effect of test temperature on crack growth. As such, for conventionally manufactured metals, plotting da/dN as function of ΔK/ΔKth(R) would appear to be a useful addition to the tools available to assess the fracture mechanics parameters affecting crack growth.

1. Introduction

Whilst this paper focuses on fracture mechanics based tools for assessing crack growth in aerospace structures, it should be noted that the study of crack growth is central to a wide crossection of industries, viz: rail, nuclear power, offshore oil and gas, etc. Indeed, to put things into perspective, ref. [1] states that “in the United States of America (USA), there are more trips per day over structurally deficient bridges than there are McDonald’s hamburgers eaten in the entire USA”.
The certification requiremets for military aircraft are delineated in MIL-STD-1530D [2]. The guidelines for the airworthiness certification of additively manufactured (AM) and cold spray additively manufactured (CSAM) parts are gven in United Stes Air Force Structures Bulletin EZ-SB-19-01 [3]. Both MIL-STD-1530D and EZ-SB-19-01 mandate that all durability and damage tolerance (DADT) analyses be performed using linear elastic fracture mechanics (LEFM). Indeed, EZ-SB-19-01 stated that the accurate prediction of the DADT of an AM part is perhaps the greatest challenge facting the airworthiness acceptance of AM parts. Consequently, in this paper, we will confine our focus on fatigue crack growth studies that relate da/dN to ΔK. Here a is the crack length, N is the number of fatigue cycles, and ∆K = Kmax − Kmin, where Kmax and Kmin are the maximum and minimum values of the crack tip stress intensity factor (K) in a load cycle.
As a result, the DADT assessment of conventionall manufactured, AM and CSAM parts must be based on the appropriate da/dN versus ΔK curve. However, it is now known [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20] that the variability in the da/dN versus ΔK curves associated with the growth of long cracks in more than one hundred and twenty independent tests performed on a range of additively manufactured (AM) materialscan often be accounted for by expressing the crack growth rate (da/dN) as a function of Δκ, where Δκ is the Schwalbe crack driving force [21], and by allowing for the effect of the manufacturing process on just two parameters, namely the cyclic fatigue threshold and the fracture toughness. Here Δκ is Schwalbe’s crack driving force which is defined as:
∆κ = (∆K − ∆Kthr)/√(1 − Kmax/A)
The terms ∆Kthr and A in Equation (1) are the fatigue threshold and the apparent cyclic fracture toughness respectively. The additively manufactured materials for which this observation has been found to hold include: Ti-6Al-4V [6,7,11,12], Inconel 718 [5], Inconel 625 [5], 316L steel [6,17,20], 304L steel [20], Aermet 100 steel [6], 17-4 Ph steel [15], Scalmalloy [12], an aluminium-scandium-magnesium alloy [4], 18Ni 250 Maraging steel [8,19]. This observation also appears to hold for a range of conventionally manufactured materials [22,23,24,25,26,27,28,29], a range of CSAM materials [20] and for plasma sprayed metals and alloys [30].
Figure 5 can often be predicted by setting the fatigue threshold term in Equation (1) to a small value, typically in the range 0.1 ≤ ΔKthr ≤ 0.3 MPa √m.
(i)
The multi-axial fatigue life of AM parts can often be computed using the same formulation [13,18];
(ii)
The variability in the growth of long cracks in conventionally manufactured metals can also often be captured by allowing for the variability associated with the fatigue threshold [22].
Indeed, variants of this approach have also been shown to be able to model delamination growth in composites as well as cohesive crack growth in adhesives and nano-composites, see [31,32,33,34,35,36,37].
Many legacy fixed and rotary wing aircraft make extensive use of 7000 and/or 2000 series aluminium alloys. Consequently, the recent finding [38] that Boeing Intelligence and Weapon Systems laser powder bed fusion (LPBF) Scalmalloy has a damage tolerance that is superior to that of the conventionally manufactured aluminium alloy 7075-T6 has highlighted the potential of Scalmalloy to be used to print limited life load bearing parts. However, as previously noted USAF Structures Bulletin EZ-19-01 [3] states that the accurate prediction of its durability and damage tolerance (DADT) is one of the most challenging issues facing the acceptance of AM parts. These various findings/observations raise the question: Can we link the observations delineated above, namely that the fatigue threshold would appear to be a key parameter in characterising crack growth in AM materials and cold spray repairs, to the crack closure concept that is commonly used to assess the damage tolerance of conventionally built aerospace parts?
In this context it should be noted that Paris et al. [39] were the first to suggest that, since [40,41] had shown that the stress-intensity factor K, uniquely characterises the near tip stress field, then the rate of fatigue crack growth should be a function of ΔK and Kmax [39,42,43]. However, when expressing da/dN as a function of ΔK it is found that the resultant da/dN versus ΔK curves can become dependent on the R ratio. (Here R is defined as R = Kmin/Kmax.)
Frost [44] subsequently revealed, for long cracks, the existence of a fatigue threshold (ΔKth(R)), which can be a function of the R ratio, below which a crack will not grow. (The ASTM fatigue test standard ASTM E647-15el [45] defines ΔKth as the value of ΔKth at a crack growth rate da/dN of 10−10 m/cycle.) Elber [46,47] subsequently introduced the idea that the R ratio dependency of the da/dN versus ΔK curves could be accounted for by introducing what was termed an “effective stress intensity factor” (ΔKeff) that accounts for plastic wake induced crack closure. (Here it should be noted that, as a result of the stress singularity at a crack tip the linear elastic fracture mechanics solution suggests that the region surrounding the crack will see an infinite stress. In reality, this region will yield. As a result, during fatigue crack growth the crack will grow through this yielded material and in the process the material is unloaded. This process results in compressive stresses behind the crack and the possibility that the crack faces will close. It is this process that is commonly refered to as “plastic wake induced crack closure. At this point, it should be mentioned that, whilst, as discussed in [48,49,50,51,52,53], there are other forms of crack closure and other formulations that attempt to account for this effect, this paper only addresses Elber’s [46,47] original formulation and not the larger cross-section of formulations that are summarised in [52].
As outlined in [48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64] variants of this approach are now widely used to model the growth of long cracks in conventionally manufactured metals. However, it should also be noted that it has recently been suggested [65] that R ratio effects on the growth of long cracks can be interpreted as a reflection of the effect of the environment on the crack tip region rather than crack closure per se.
The literature also contains a number of other crack growth equations that are based on the hypothesis that da/dN should be a function of how much ΔK exceeds its threshold value [4,21,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80]. As such the purpose of the present paper is to attempt to investigate a possible relationship between ΔKeff, as originally defined by Elber [46,47], and the fatigue threshold term ΔKth. This would provide a link between crack growth equations that are based on Elber’s plastic wake induced crack closure concept and those that are based on the assumption that the crack growth rate should be a function of how much ΔK exceeds its threshold value.
It should be stressed that the findings presented in this paper do not constitute a proof that the effect of crack closure effects can always be interpreted as being reflected by its effect on the fatigue threshold. Indeed, this study is specifically confined to those materials where Elber’s original formulation would appear to be a reasonable first approximation.
We also show that plotting da/dN as function of ΔK/ΔKth(R) would appear to have the potential to help identify the fracture mechanics parameters that characterise the effect of the test temperature on crack growth.

2. Materials and Methods

In order to achieve the goals stated above the authors have examined papers that are available in the open literature. Of the more than one hundred articles examined ninety eight are in peer reviewed Journals, five are available on the North American Space Administration (NASA) website, and one is available on the US Federal Aviation Administration (FAA) website. In those cases where the reference is not a Journal paper that is available in the open literature the web address of the reference is also given.

2.1. Choice of Materials

A focus of this paper is to examine if this phenomenon holds for a range of aerospace aluminium, steel, and titanium alloys. (By this we mean if plotting da/dN as a function of ΔK/ΔKth(R) collapses the various R ratio dependent da/dN versus ΔK curves onto a single master curve). In this context, it should be recalled that [81] had previously illustrated this phenomenon for a range of older aerospace aluminium alloys, namely 2024-T3, 7075-T651, 7050-T7451, 6013-T651 and 2324-T39. Consequently, we opted to study the more modern alloys 7085-T7452 [82], which is widely used on the F-35, and AA2524-T3 [83], which is a replacement for 2024-T3. On the other hand, given the role that [49,50] played in the development of the discipline, we thought it appropriate to examine if this phenomenon also held for the aluminium alloy 7055 that was studied in [49,50]. Similarly, crack growth in the aluminium alloy 7049 [84], which is widely used in military aircraft, is also examined. Furthermore, since [81] had not investigated stainless steels we opted to study 304L stainless steel. Similarly, we chose to study the Titanium alloy Ti622 since it is being considered for the next generation of supersonic airliners [85].
In the case of the choice of an engine material there were two obvious choices, viz: Inconel 718 and Inconel 625. However, test data at both a range of R ratio’s and temperatures was needed. As a result the super alloy GH4169 [86], which is the Chinese version of Inconel 718, was chosen.
Noting that, as previously mentioned, [81] revealed that in the case SLM Inconel 625 plotting da/dN as a function of ΔK/ΔKth(R) collapses the various R ratio dependent da/dN versus ΔK curves onto a single master curve, we opted to study a quite different AM material, namely Laser powder bed fusion (LPBF) built Hastelloy X [87]. (The SLM Inconel 625 sata analysed in [81] was taken from [88]).
The examples given [89,90], were chosen since they illustrated the temperature effects can sometimes merely reflect the effect of temperature on the fatigue threshold. In the case of [89] this example was chosen since it related to 304L stainless steel, which is widely used in aerospace. On the other hand [90] was chosen to investigate a quite different class of metals, namely medium entroy Cantor alloy CrCoNi.

2.2. The Relationship between Plasticity Induced Crack Closure and the Fatigue Threshold

Examining the above references revealed that Elber [46,47] was the first to introduce a function U(R), which he [47] suggested was independent of both ΔK and Kmax, that was used to define a term ΔKeff, viz:
ΔKeff = U(R) ΔK
such that, for long cracks that experience plastic wake induced crack closure, the resultant da/dN versus ΔKeff curves associated with each individual R ratio dependent da/dN versus ΔK curve all fell onto a single curve regardless of the R ratio.
Several possible forms of this (scaling) function U(R) are discussed in [50,51,52,53,54,55,56,59]. Here it should be noted that in [47] the function U was only a function of R. (A brief discussion of this statement is given in Appendix A) In such cases it thus follows from Equation (2) and from the facts that
(a)
if as per Elber [47] U(R) is only a function of R rather than of both ΔK and Kmax;
(b)
and, if as per [47] for a given value of R the function U(R) has a fixed (constant) value;
then if each R ratio dependent da/dN versus ΔK curve is to collapse onto a single da/dN versus ΔKeff curve by merely multiplying (each value of) ΔK by the function U(R), then Equation (2) can be rewritten in the form:
ΔKeff = (ΔKeff,th/ΔKth(R)) ΔK
Here ΔKeff,th is the value of ΔKeff at a crack growth rate da/dN of 10−10 m/cycle, and ΔKth(R) is the value ΔK, for a given R ratio dependent crack growth curve, at a crack growth rate da/dN of 10−10 m/cycle respectively. A graphical explanation of this, for two arbitrary R ratios, which we have termed R = R1 and R = R2, is shown in Figure 1. (Equation (3) forces the various (scaled) R ratio dependent da/dN versus ΔK curves to coincide at a crack growth rate da/dN = 10−10 m/cycle). It thus follows that, for long cracks where crack growth is consistent with Elber’s plastic wake induced crack closure equation, the function U(R) can be expressed as:
U(R) = ΔKeff,th/ΔKth(R)
In other words ΔKeff, as defined by Elber [46,47], should be linearly proportional to ΔK/ΔKth(R).
Elber [46,47] also introduced a function Ko(R), which is a function of the R-ratio and which he defined as the value of the stress intensity factor at which a crack opens, such that
U(R) = (Kmax − Ko(R))/(Kmax − Kmin)
As a result it is common to plot da/dN as a function of (Kmax − Ko(R)). However, as explained in [49,50], it is sometimes best to plot da/dN as a function of ΔK2/π0, where ΔK2/π0 = (Kmax − 2Ko(R)/π). This formulation, which [49,50,51] termed the ΔK2/π0 approach, is discussed further in Appendix A.
Equations (4) and (5) enable us to determine a simple (closed form) equation that relates the crack opening stress intensity factor Ko(R) to ΔK, Kmax and ΔKth(R), viz:
Ko(R) = Kmax (1 − (1 − R) ΔKeff,th/ΔKth(R))
which, when expressed in terms of the energy release rate G becomes
√Go(R) = √G max (1 − (1 − R) Δ√Geff,th/Δ√Gth(R))
It follows from the above discussion that whenever Elber’s approach to plastic wake-induced crack closure, i.e., Equation (2), holds, then when da/dN is plotted against ΔK/ΔKth(R) the various “normalised” curves should also all fall onto a single curve. However, it is important to note that, as shown in [61,91,92,93,94], the use of the ASTM load-reducing test protocol to determine the fatigue crack thresholds can (sometimes) result in erroneous values of ΔKth(R). Hence, Equation (4) may not (always) hold for data sets obtained using the American Society for Testing and Materials (ASTM) ASTM E647-15el [45] load-reducing test protocol, see Appendix B. As such, the use of the equation:
Ko(R) = Kmax (1 − (1 − R) ΔK’eff,th /ΔK’th(R))
where ΔK’eff,th and ΔK’th(R) are the values of ΔKeff and ΔK(R) at a crack growth rate of da/dN = 10−8 m/cycle, may sometimes be more appropriate.
Furthermore, as illustrated in [95,96,97] there are a range of metals that whilst they have an R ratio dependency in the near threshold region, see minimal R ratio effects outside of this region and hence see little (if any) crack closure in the Paris region. As a result, the concept of crack closure and the concept discussed in this paper does not hold for these materials.
Similarly, Appendix X3 (contained) in the fatigue test standard ASTM E647-15el [45] notes that it is not clear if a fatigue threshold exists for small naturally occurring cracks. This finding has significant implications when assessing the durability of metallic airframes since it means that the crack growth equation needed to assess the durability can differ from that associated with the closure free crack growth curve, see [19] for more details. A more detailed discussion on the state of the art can be found in [61,98,99].
To the best of the authors knowledge, the observation that, for long cracks, when da/dN was plotted against ΔK/ΔKth(R) the resultant R ratio dependent curves collapsed onto a single curve was first presented in [100]. This paper studied crack growth in: 17-4PH steam turbine blade steel tested in air at 90 °C at R ratio’s that ranged from −1 to 0.9. A range of additional examples, viz: 7050-T7451, 20-23-T39, 2024-T3, 6013-T651, 7075-T6511, 7055-T6511, a cold-rolled metastable Austenitic stainless steel, and an additively manufactured Inconel 625 material built using selective laser melt (SLM) tested with the crack at different orientations to the build direction, were subsequently presented in [81].
An interesting, and possibly related, feature of [81]was that it was shown that for studies into delamination growth in double cantilever beam (DCB) composite laminates the da/dN versus Δ√G curves, where G is the energy release rate, were a strong function of the level of pre-cracking. However, when da/dN was plotted as a function of Δ√G/Δ√Gth these different curves essentially collapsed onto a single curve. (The different da/dN versus Δ√G curves are as a result of the retardation due to the different levels of fibre bridging that are associated with the different levels of pre-cracking).

3. Illustrative Examples

To further illustrate this phenomenon, i.e., that when da/dN is plotted against ΔK/ΔKth(R) the resultant “normalised” curves can collapse onto a single curve, let us consider crack growth in the aluminium alloy AA7085-T7452 which is extensively used in the Lockheed F-35 Joint Strike Fighter (JSF). The R = 0.1 and 0.8 da/dN versus ΔK curves given in [82] for this material are reproduced in Figure 2. These curves were obtained from tests on specimens with an array of surface beaking defects, The corresponding da/dN versus ΔK/ΔKth curves are shown in Figure 3. Here we see that when normalised in this fashion the R = 0.1 and 0.8 da/dN versus ΔK/ΔKth curves do indeed collapse onto a single curve. The values of ΔKth used in Figure 3 are given in Table 1.
To continue this study, let us next consider the R = 0.1 and 0.5 da/dN versus ΔK curves given in the NASA study [85] into crack growth in the titanium super alloy Ti-6A1-2Zr-2Cr-2Sn-2Mo (Ti-6-2-2-2-2). These da/dN versus ΔK curves, which were obtained using ASTM E447 standard eccentrically-loaded single edge notch tensile (ESE(T)) specimens, are shown in Figure 4.
In this instance, estimated values of ΔKth were given in [85]. These values of ΔKth are reproduced in Table 2. These values of ΔKth can now be used to plot da/dN against ΔK/ΔKth(R). The corresponding “normalised plot” is shown in Figure 5, where we also see that when plotted in this fashion the R = 0.1 and 0.5 curves again essentially collapse.
These observations, when taken together with Equation (4) and the additional examples given in [81,100] and in and Appendix A and Appendix C, support the conclusion reached above that crack growth rate in conventionally manufactured metals would often appear to be primarily controlled by the fatigue threshold.
The observation that for conventionally manufactured metals the R ratio effect on crack growth is, to a first approximation, often reflected in the change of the fatigue threshold ΔKth, is also consistent with the hypothesis first proposed by Hartman and Schijve [66], and also by a number of other subsequent authors [61,67,68,69,70,71,72,73,74,75,76,77,78,79,80], that the crack growth rate (da/dN) should be a function of how much ΔK exceeds its threshold value (ΔKth). This hypothesis, i.e., that the crack growth rate (da/dN) should be a function of how much ΔK exceeds its threshold value, would appear to be supported by the data presented in [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,81] for crack growth in conventionally manufactured, AM and CSAM metals. However, as illustrated in [5,6,7,11,20], when when using the Hartman-Schijve crack growth equation to model crack growth in AM and CSAM metals it would appear that allowance must also be made for the variability in the fracture toughness that arises due to the manufacturing process used to build the part and the anisotropic nature of AM and CSAM materials.

4. The Potential to Help Clarify Seemingly Anomalous Behaviour

The paper by Jones et al. [81] also reported that plotting da/dN as a function of ΔK/ΔKth appeared to have the ability to help understand what, at first glance, appeared to be anomalous da/dN versus ΔK curves. To further illustrate this let us consider the da/dN versus Kmax curves given in [83] for crack growth in underaged (UA) high strength 7049 aluminium alloy tested in air at R = 0, −1, −2, −3. These curves, which were obtained using ASTM E647 standard middle tension (MT) specimens,
Here it should be noted that ASTM E647-15el [45] states that for R ≤ 0 when plotting the da/dN versus ΔK curve the term ΔK can be approximated by Kmax. As such it is unusual to see that whilst the R = −1, −2, −3 curves lie on a single curve the R = 0 curve lies on a quite different curve, see Figure 6. However, when these curves are plotted with da/dN expressed as a function of Kmax/Kmax,th, where Kmax,th is the value of Kmax at a crack growth rate (da/dN) of 10−10 m/cycle, then the two different curves collapse onto a single curve, see Figure 7. The values of Kmax,th used in Figure 7 are given in Table 3. As such it would appear that, from a mechanics perspective, the reason for the difference shown in Figure 6 is due to the different fatigue threshold that arose when testing at negative R ratio’s.

5. The Potential to Help Clarify Temperature Effects

To continue this study let us next examine the R = 0.1 and 0.7 da/dN versus ΔK curves presented in [86] for the growth of cracks in the Ni-based supper alloy GH4169 tested at room temperature (RT) and also at 550 °C. (GH4169 is a Chinese developed superalloy that has similar mechanical properties to Inconel 718 (IN718) [86]). The room temperature and 550 °C R = 0.1 and 0.7 da/dN versus ΔK curves are shown in Figure 8. Here we see that the da/dN versus ΔK curves are a function of both the test temperature and the R ratio.
Since Figure 8 does not contain crack growth data beneath a crack growth rate of approximately 10−8 m/cycle the various curves were normalised by dividing ΔK by the estimated value of ΔK at a crack growth rate of da/dN = 10−8 m/cycle, which we will define as ΔK′th. The resultant normalised plots are shown in Figure 9. Interestingly Figure 9 reveals that, when normalised in this fashion, both the R ratio and the temperature dependency effects shown in Figure 8 essentially vanish. In other words, for this material, the temperature and R ratio effects are (to a first approximation) reflected in their effect on a single parameter. Other examples that illustrates how expressing da/dN as a function of ΔK/ΔK′th can sometimes help when evaluating the effect of temperature on crack growth are given in Appendix C. The values of ΔK′th used in Figure 9 are given in Table 4.

6. The Potential to Help Clarify Crack Growth in AM Hastelloy X

We have previously noted that, as previously mentioned, [81] revealed that in the case SLM Inconel 625 plotting da/dN as a function of ΔK/ΔKth(R) collapses the various R ratio dependent da/dN versus ΔK curves onto a single master curve. Consequently. To further investigate the potential for this approach to assit in understanding crack growth in additively manufactured materials we analysed the (room temperature) R = 0.1, da/dN versus ΔK curves associated with laser bed powder fusion (LPBF) built ASTM standard CT tests on Hastelloy X where the crack was at either 0° or 90° degrees to the build direction. This paper presented the R = 0.1 crack growth curves associated with two different post-build treatments, viz: solution heat treatment (SHT); and hot isostatic pressing (HIP), with the crack at either 0 degress or 90 degress to the build direction.. The resultant the da/dN versus ΔK curves given in [87] are shown in Figure 10.
The corresponding da/dN versus ΔK/ΔK’th curves are shown in Figure 11. Figure 11 again appears to suggest that when normalised in this fashion the effect of anisotropy essentially vanishes. The values of ΔK’th used in Figure 11 are given in Table 5.

7. Brief Summary

Table 6 contains a brief summary of the finding seen in this study. Here we see that in each case expressing da/dN as a function of ΔK/ΔK’th appears to have collapsed the various R ratio and/or temperature depemdency seen when da/dN vwas expressed as a function of ΔK.

8. Conclusions

This paper has shown that, whenever Elber’s crack closure based approach to modelling crack growth in conventionally manufactured metals is valid, the function U(R) that is used to relate the stress intensity factor ΔK to the effective stress intensity factor ΔKeff, so as to account for plastic wake induced crack closure, would appear to be (to a first approximation) inversely proportional to the fatigue threshold. It thus follows that, in such instances, from a fracture mechanics perspective crack growth in conventionally manufactured metals would often appear to be primarily controlled by the fatigue threshold.
This finding also results in a simple closed-form equation that relates the crack opening stress intensity factor Ko(R), that was originally introduced by Elber, to ΔK, Kmax, the fatigue thresholds ΔKth(R) and ΔKeff,th.
At this point it should be restated that this paper does not constitute a proof that R ratio effects can always be interpreted as being reflected by its effect on the fatigue threshold. Indeed, there are numerous examples where whilst there are clear R ratio effects in the near threshold region this R ratio dependency essentially vanishes in the Paris region. Consequently, the discussion presented in this paper is not applicable for such materials. Indeed, the present paper is specifically confined to those materials where Elber’s original formulation, by this we mean Equation (2), is a reasonable first approximation. That said, there would appear to be numerous examples where the simplification proposed in this paper would appear to be a reasonable first approximation.
It is also shown that plotting da/dN as function of against ΔK/ΔKth(R) would appear to have the potential to help identify the key fracture mechanics parameters that characterise the effect of the test temperature on crack growth. However, it should be stressed that this may not always be the case. Nevertheless, replotting the da/dN versus ΔK(R) curve with da/dN plotted as function of against ΔK/ΔKth(R) would appear to be a useful additional approach to help to identify the (fracture mechanics based) parameters affecting crack growth.
In the case of the annealed SLM Inconel 625 tests and the various R = 0.1 tests on LPBF Hastelloy X specimens that are examined in this study it would appear that plotting da/dN versus ΔK/ΔKth may also help to identify the fracture mechanics parameters that characterise the effect of the build direction on crack growth in AM materials. However, it should be stressed that further work is needed to investigate if this hypothesis holds for other AM materials.

Author Contributions

Conceptualization, N.D.P. and M.N.; methodology, R.J.; software, R.J.; validation, A.A.; formal analysis, R.J. and A.A.; resources, A.A.; data curation, R.J.; writing—original draft preparation, A.A.; writing—review and editing, R.J., M.N. and N.D.P.; project administration, R.J. and M.N.; funding acquisition, A.A. and N.D.P. All authors have read and agreed to the published version of the manuscript.

Funding

Andrew Ang and Rhys Jones would like to acknowledge funding provided by the US Army International Technology Center, Indo-Pacific (ITC-IPAC), Tokyo, Contract No. FA520923C0010.

Data Availability Statement

The data presented in this study are available on request from the corresponding author. The data are not publicly available due to the ongoing nature of the study.

Acknowledgments

The findings and conclusions or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the ITC-IPAC.

Conflicts of Interest

The authors declare no conflict of interest.

Nomenclature

Material and R ratioΔKth MPa √m
acrack length
AMadditive manufacturing
ASTMAmerican Society for Testing and Materials (ASTM)
da/dNthe increment in the crack length per cycle
DADTdurability and damage tolerance
Gthe energy release rate
Gmaxthe maximun value of G in a load cycle
Gminthe minimun value of G in a load cycle
Δ √ G range of √ G in a fatigue cycle, as defined below
Δ √ G = √ G m a x − K √ G m i n
Δ √ G t h the value of Δ √ G at a crack growth rate (da/dN) of 10−10 m/cycle
Δ √ G e f f the crack closure corrected value of Δ √ G
Δ √ G e f f , t h the value of Δ √ G e f f at a crack growth rate (da/dN) of 10−10 m/cycle
Go(R)the value of G at which the crack first opens
Kstress-intensity factor
K m a x the maximun value of K in a load cycle
K m i n the minimun value of K in a load cycle
ΔKrange of the applied stress-intensity factor in the fatigue cycle, as defined below
ΔK = Kmax − Kmin
ΔKththe value of ΔK at a crack growth rate (da/dN) of 10−10 m/cycle
ΔK’ththe value of ΔK at a low crack growth rate, typically da/dN = 10−8 m/cycle
ΔKeffThe crack closure corrected value of ΔK that is defined as
ΔKeff = U(R) ΔK
ΔKeff,ththe value of ΔKeff at a crack growth rate (da/dN) of 10−10 m/cycle
Ko(R)the value of K at which the crack first opens
ΔK2/π0,an alternative formulation for the crack closure corrected value of ΔK, namely
ΔK2/π0 = (Kmax − 2Ko(R)/π)
Δκthe Schwalbe crack driving force
LBPFlaser bed powder fusion, an additive manufacturing process
LEFMlinear-elastic fracture-mechanics
Rthe R ratio, defined as R = Kmin/Kmax
SLMSelective laser melting, an additive manuring process
U(R)Elber’s crack closue function that relates ΔK to ΔKeff, viz:
U(R) = (Kmax − Ko(R))/(Kmax − Kmin)
USAUnited States of America
USAFUnited States Air Force

Appendix A

Whilst, as shown in [52,53,55], Elber’s [47] statement that for plasticity induced crack closure the function U is essentially only a function of R is supported by numerous studies there are also studies that suggest otherwise, see [48,49,50,51,52]. Unfortunately, as illustrated in [91.95] and in Appendix B, the load reducing test protocol described in the ASTM Fatigue Test Standard ASTM E647-15el [45] can sometimes lead to erroneous da/dN versus ΔK curves. The problem of assessing Elber’s the validity of original hypothesis is further complicated by:
(i)
The variability in the measured da/dN versus ΔK curves that is known to be associated with repeat tests [22,101];
(ii)
The level of experimental error that is often associated with the crack growth rates, particularly at low crack growth rates. The experimental data presented in [52] is a good example of the latter;
(iii)
The need, as delineated by Paris et al. [50], for precise computer controlled load-displacement data.
Consequently, it is sometimes difficult to make a statement with respect to the voracity or otherwise of Elber’s hypothesis that U is essentially only a function of the R ratio. However, in light of the data given in [52,53,55], which suggest that to a first approximation U is often essentially only a function of R, and given that [81,84] have shown that there are a number of examples where the experimental data reveals that U can often be reasonably well approximated as being inversely proportional to ΔKth, the present paper suggests that Elber’s statement may often represent a reasonable first approximation.
Nevertheless, it also should be emphasised that the findings outlined in the present paper, that expressing da/dN as function of ΔK/ΔKth(R) would appear to have the potential to help identify the fracture mechanics based parameters that characterise crack growth, are (in theory) specifically confined to those materials where Elber’s original formulation, i.e., Equation (2), is a reasonable first approximation.
Let us now return to the question of crack closure and the R ratio dependency of the da/dN versus ΔK curves. References [49,50] gave a number of examples where whilst expressing da/dN as a function of (Kmax − Ko(R)) helped collapse the various curves in region where da/dN was greater than approximately 10−8 m/cycle this was not the case for the region where da/dN was less than approximately 10−8 m/cycle. Whilst [49,50] revealed an improvement could be achieved by representing da/dN as a function of both Kmax and (Kmax − Ko(R)) they reported that the best result was achieved by plotting da/dN as a function of ΔK2/π0.
In this context, it should be noted that reference [81] illustrated that the various R ratio dependent crack growth curves given in [49,50] for the aluminium alloys 6013-T651 and 2324-T39 collapsed if da/dN was plotted as a function of ΔK/ΔKth. However, [49,50] also presented data for the aluminium alloy 7055. Neither the heat treatment or the nature of the specimen geometry was given in [49,50]. The R = 0.75, 0.5, 0.3, and −1 da/dN versus ΔK curves given in [49,50] for this alloy are shown in Figure A1. Since not all curves have crack growth data at 10−10 m/cycle the various curves were normalised by dividing ΔK by its estimated value at a crack growth rate of da/dN = 10−8 m/cycle, which we will define as ΔK’th. The resultant normalised plots are shown in Figure A2 where we see that, to a first approximation, the various (normalised) R ratio dependent curves also collapse. Consequently, for this material, the R ratio effects shown in Figure A1 would appear to be (to a first approximation) reflected by their effect on the fatigue threshold.
Table A1. The values of ΔK′th used in Figure A1.
Table A1. The values of ΔK′th used in Figure A1.
R RatioΔK′th MPa √m
R = 0.753.0
R = 0.53.5
R = 0.34.3
R = 0.15.0
R = −110.2
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Figure A1. The R = 0.75, 0.5, 0.3, and −1 da/dN versus ΔK curves for the aluminium alloy 7055.
Figure A1. The R = 0.75, 0.5, 0.3, and −1 da/dN versus ΔK curves for the aluminium alloy 7055.
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Figure A2. The R = 0.75, 0.5, 0.3, 0.1 and −15 da/dN versus ΔK/ΔK’th curves for the aluminium alloy 7055.
Figure A2. The R = 0.75, 0.5, 0.3, 0.1 and −15 da/dN versus ΔK/ΔK’th curves for the aluminium alloy 7055.
Metals 14 00523 g0a2
That the purpose of this particular example is not to suggest that whenever the function U would appear to be a function of both Kmax and R the curves will nevertheless collapse if da/dN was plotted as a function of ΔK/ΔKth. However, as illustrated above, there are cases when it will. As such, evaluationg if this concept holds would appear to be a useful addition to the tools available to assess the fracture mechanics parameters affecting crack growth.

Appendix B

It is now known [61,91,92,93,94] that the ASTM load reducing test protocol can results in da/dN versus ΔK curves that are dependent on the specimen test geometry. Furthermore, it is also known [101] that, for a given specimen geometry, even if both the loads and the starting crack size are tightly controlled the da/dN versus ΔK curve is not necessarily unique. (This observation is supported by the test data discussed in [22,102]). As a result of these and related observations, the NASA Fracture Control Handbook NASA-HDBK-5010 [103] mandates the use of the worst case da/dN versus ΔK curve.
These observations complicate our assessment of the suggestion that, to the best of the author’s knowledge was first delineated in [84], that for long cracks the function U(R) could be a function of both R and Kmax. Consequently, the present paper is focused on those data sets that would appear to conform to Equation (2).
At this point, it should also be noted that the present paper does not propose to advocate either for or against the use of crack closure based crack growth equations. It’s purpose is to:
(i)
Examine the relationship between Elber’s original equation, i.e., Equation (2), and those equations that are based on the assumption that crack growth is a function of how much ΔK exceeds the fatigue crack growth threshold;
(ii)
Present a new tool to help assess the fracture mechanics parameters affecting crack growth.

Appendix C. Additional Examples

To further illustrate this phenomenon let us consider the R = 0. 1, 0.2, 0.3, and 0.5 da/dN versus ΔK curves presented in [83] for the aluminium alloy 2524-T3. These curves, which were obtained using ASTM E647-15el standard compact tension (CT) specimens, are shown in Figure A3. Since Figure A3 does not contain crack growth data beneath 10−8 m/cycle the various curves were normalised by dividing ΔK by the estimated value of at a crack growth rate of da/dN = 10−8 m/cycle, which we will define as ΔK’th. The resultant normalised plots are shown in Figure A4 where we see that, to a first approximation, the various (normalised) R ratio curves also collapse. The values of ΔK’th(R) used in Figure A4 are listed in Table A2. Consequently, it would appear that, to a first approximation, the R ratio effects shown in Figure A3 are reflected by their effect on the fatigue threshold.
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Figure A3. The R =0.1, 0.2, 0.3 and 0.5 da/dN versus ΔK curves for the aluminium alloy AA2524-T3.
Figure A3. The R =0.1, 0.2, 0.3 and 0.5 da/dN versus ΔK curves for the aluminium alloy AA2524-T3.
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Figure A4. The R = 0.1, 0.2, 0.3 and 0.5 da/dN versus ΔK/ΔK’th curves for the aluminium alloy AA2524-T3.
Figure A4. The R = 0.1, 0.2, 0.3 and 0.5 da/dN versus ΔK/ΔK’th curves for the aluminium alloy AA2524-T3.
Metals 14 00523 g0a4
Table A2. The values of ΔK′th used in Figure A4.
Table A2. The values of ΔK′th used in Figure A4.
R Ratio ΔK′th MPa √m
R = 0.16.6
R = 0.26.3
R = 0.35.9
R = 0.55.6
To continue this study let us next consider the R = 0.05 and 0.5 da/dN versus ΔK curves presented in [89] for crack growth in 304L stainless steel tested at cryogenic temperatures (77K) and the R = 0.05 da/dN versus ΔK curve presented in [89] for crack growth in 304L stainless steel at room temperature (RT). These curves, which were obtained using ASTM E647 standard compact tension (CT) specimens, are reproduced in Figure A5.
Figure A5 reveals that the da/dN versus ΔK curves are functions of both the test temperature and the R ratio. Figure A6 reveals that when da/dN is plotted as function of ΔK/ΔKth(R) then, to a first approximation, these curves also essentially collapse onto a single curve. In other words, for this material, the R ratio and temperature dependencies seen in Figure A5 are (to a first approximation) reflected by their effect on a single parameter. The values of ΔKth(R) used in Figure A6 are listed in Table A3.
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Figure A5. The da/dN versus ΔK curves for 304L stainless steel tested at room temperature and 77 K.
Figure A5. The da/dN versus ΔK curves for 304L stainless steel tested at room temperature and 77 K.
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Figure A6. The da/dN versus ΔK/ΔK′th curves for 304L stainless steel tested at room temperature and 77K.
Figure A6. The da/dN versus ΔK/ΔK′th curves for 304L stainless steel tested at room temperature and 77K.
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Table A3. The values of ΔKth used in Figure A6.
Table A3. The values of ΔKth used in Figure A6.
Material, R Ratio and Test TemperatureΔKth MPa √m
304L, R = 0.5, 77 K5.5
304L, R = 0.05, 77 K8.0
304L, R = 0.05, RT4.0
To further illustrate the potential advantages of plotting da/dN as function of ΔK/ΔKth(R) let us examine the R = 0.1 da/dN versus ΔK curves presented in [90]. for the 7 µm (fine grained) medium entropy alloy (MEA) CrCoNi tested at 77 K, 189 K and 293 K (room temperature), see Figure A7. These curves were obtained using disk-shaped compact-tension DC(T) specimens. The da/dN versus ΔK/ΔKth(R) curves associated with these test temperatures are shown in Figure A8. Here we again see that, to a first approximation, when plotted in this fashion the effect of the test temperature essentially vanishes. In other words, for this R ratio, the temperature dependencies seen in Figure A5 are (to a first approximation) reflected by their effect on a single parameter. The values of ΔKth(R) used in Figure A8 are given in Table A4.
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Figure A7. The R = 0.1 da/dN versus ΔK curves for 7 µm (fine grained) medium entropy alloy (MEA) CrCoNi tested at 77 K, 189 K and 293 K (room temperature).
Figure A7. The R = 0.1 da/dN versus ΔK curves for 7 µm (fine grained) medium entropy alloy (MEA) CrCoNi tested at 77 K, 189 K and 293 K (room temperature).
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Metals 14 00523 g0a8
Figure A8. The R = 0.1 da/dN versus ΔK/ΔKth curves for the MEA CrCoNi tested at 77 K, 189 K and 293 K.
Figure A8. The R = 0.1 da/dN versus ΔK/ΔKth curves for the MEA CrCoNi tested at 77 K, 189 K and 293 K.
Metals 14 00523 g0a8
Table A4. The values of ΔKth used in Figure A8.
Table A4. The values of ΔKth used in Figure A8.
Test TemperatureΔKth MPa √m
77 K8.1
189 K7.0
293 K (RT)6.2

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Figure 1. A graphical presentation of Elber’s crack closure concept.
Figure 1. A graphical presentation of Elber’s crack closure concept.
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Figure 2. The da/dN versus ΔK curves for the aluminium alloy AA7085-T7452.
Figure 2. The da/dN versus ΔK curves for the aluminium alloy AA7085-T7452.
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Figure 3. The da/dN versus ΔK/ΔKth curves for the aluminium alloy AA7085-T7452.
Figure 3. The da/dN versus ΔK/ΔKth curves for the aluminium alloy AA7085-T7452.
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Figure 4. The R = 0.1 and 0.5 da/dN versus ΔK curves for the titanium alloy Ti-6-2-2-2-2.
Figure 4. The R = 0.1 and 0.5 da/dN versus ΔK curves for the titanium alloy Ti-6-2-2-2-2.
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Figure 5. The da/dN versus ΔK/ΔKth curves for the titanium alloy Ti-6-2-2-2-2.
Figure 5. The da/dN versus ΔK/ΔKth curves for the titanium alloy Ti-6-2-2-2-2.
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Figure 6. The da/dN versus Kmax curves for AA7049.
Figure 6. The da/dN versus Kmax curves for AA7049.
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Figure 7. The da/dN versus Kmax/Kmax,th curves for the underaged 7049 tests.
Figure 7. The da/dN versus Kmax/Kmax,th curves for the underaged 7049 tests.
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Figure 8. The 550◦C and RT R = 0.1 and 0.7 da/dN versus ΔK curves the super alloy GH4169.
Figure 8. The 550◦C and RT R = 0.1 and 0.7 da/dN versus ΔK curves the super alloy GH4169.
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Figure 9. The 550 °C and RT R = 0.1 and 0.7 da/dN versus ΔK/ΔK’th curves shown in Figure 10 and Figure 11 for the super alloy GH4169.
Figure 9. The 550 °C and RT R = 0.1 and 0.7 da/dN versus ΔK/ΔK’th curves shown in Figure 10 and Figure 11 for the super alloy GH4169.
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Figure 10. The da/dN versus ΔK curves for LPBF Hastelloy X.
Figure 10. The da/dN versus ΔK curves for LPBF Hastelloy X.
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Figure 11. The da/dN versus ΔK/ΔK’th curves for LPBF Hastelloy X.
Figure 11. The da/dN versus ΔK/ΔK’th curves for LPBF Hastelloy X.
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Table 1. The values of ΔKth used in Figure 3.
Table 1. The values of ΔKth used in Figure 3.
Material and R RatioΔKth MPa √m
7085-T7452, R = 0.10.75
7085-T7452, R = 0.81.02
Table 2. The values of ΔKth used in Figure 4.
Table 2. The values of ΔKth used in Figure 4.
Material and R RatioΔKth MPa √m
Ti-6-2-2-2-2, R = 0.52.22
Ti-6-2-2-2-2, R = 0.13.21
Table 3. The values of Kmax,th used in Figure 7.
Table 3. The values of Kmax,th used in Figure 7.
R RatioKmax,th (MPa √m)
R = −1, −2, −3 3.9
R = 0.05.0
Table 4. The values of ΔK’th used in Figure 9.
Table 4. The values of ΔK’th used in Figure 9.
Material, R Ratio and Test TemperatureΔK′th MPa √m
Room temperature tests
GH4169, R = 0.117.0
GH4169, R = 0.713.2
Tests at 550 °C
GH4169, R = 0.1 13.8
GH4169, R = 0.710.5
Table 5. The values of ΔK′th used in Figure 11.
Table 5. The values of ΔK′th used in Figure 11.
Material, Build Direction and R Ratio ΔK′th MPa √m
LPBF Hastelloy X, HIP-09.4
LPBF Hastelloy X, HIP-010.1
LPBF Hastelloy X, SHT-012.5
LPBF Hastelloy X, SHT-9011.0
Table 6. Summary Table.
Table 6. Summary Table.
Material, Test Conditions, and ReferenceOutcome
7085-T7452 (an aluminum alloy), specimen contained an array of surface notched and was tested at R = 0.1 and 0.8, [82].Expressing da/dN as a function of ΔK/ΔK’th essentially collapsed the R ratio dependent da/dN versus ΔK curves onto a single curve.
7055 (an aluminium alloy), tested at R = 0.75, 0.5, 0.3, and −1, the specimen geometry was not specified, [49.50].ibid.
2524-T3 (an alumium alloy), ASTM E647 ESE(T) specimens that were tested at R = 0.1, 0.2, 0.3, and 0.5 [83].ibid.
7049 (aluminium alloy), tested at R = 0, −1, −2 and −4, [84].Expressing da/dN as a function of ΔK/ΔK’th essentially removed the anomalous behaviour seen in the R ratio dependent da/dN versus Kmax curves.
The titanium alloy Ti-6-2-2-2-2, an ASTM E647 ESE(T) specimens that were tested at R = 0.1 and 0.5, [85].The R ratio depemdency essentially vanished when da/dN was plotted as a function of ΔK/ΔK’th.
GH4169 (a Chinese super alloy that has mechanical properties similar to Inconel 718), ASTM E647 CT specimens that were tested at R = 0.1 and 0.7 at both RT and 550 °C, [86].Both the R ratio and temperature dependent effects essentially vanishwd when da/dN was plotted as a function of ΔK/ΔK’th.
AM (LPBF) Hastelloy X, an ASTM E647 CT specimens that were either solution heat treated (SHT) or subjected to hot isostatic pressing (HIP) and tested at R = 0.1, [87].The anisotropy seen in the da/dN versus ΔK curves essentially vanished when da/dN was plotted as a function of ΔK/ΔK’th.
304L stainless steel, ASTM E647 CT specimens tested at cryogenic temperatures and at two different R = 0.05 and 0.5 and at RT with R = 0.05, [89].Both the temperature and R ratio dependency essentially vanished when da/dN was plotted as a function of ΔK/ΔK’th.
The medium entropy alloy CrCoNi, an ASTM E647 CT specimens that were tested with R = 0.1 and at 77K, 189K and 293K (room temperature), [90].The effect of the test temperature essentially vanished when da/dN was plotted as a function of ΔK/ΔK’th.
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Jones, R.; Ang, A.; Phan, N.D.; Nicholas, M. On the Link between Plastic Wake Induced Crack Closure and the Fatigue Threshold. Metals 2024, 14, 523. https://doi.org/10.3390/met14050523

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Jones R, Ang A, Phan ND, Nicholas M. On the Link between Plastic Wake Induced Crack Closure and the Fatigue Threshold. Metals. 2024; 14(5):523. https://doi.org/10.3390/met14050523

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Jones, Rhys, Andrew Ang, Nam D. Phan, and Michael Nicholas. 2024. "On the Link between Plastic Wake Induced Crack Closure and the Fatigue Threshold" Metals 14, no. 5: 523. https://doi.org/10.3390/met14050523

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