Modelling Crack Growth in Additively Manufactured Inconel 718 and Inconel 625

Abstract

This paper first examines crack growth in a range of tests on additively manufactured (AM) and conventionally manufactured Inconel 718. It is shown that whereas when the crack growth rate (da/dN) is plotted as a function of the range of the stress intensity factor (ΔK), the crack growth curves exhibit considerable scatter/variability, when da/dN is expressed in terms of the Schwalbe crack driving force (Δκ), then each of the 33 different curves essentially collapse onto a single curve. This relationship appears to hold over approximately six orders of magnitude in da/dN. The same phenomenon also appears to hold for 20 room temperature tests on both conventionally and additively manufactured Inconel 625. Given that the 53 studies examined in this paper were taken from a wide cross section of research studies it would appear that the variability in the da/dN and ΔK curves can (to a first approximation) be accounted for by allowing for the variability in the fatigue threshold and the cyclic fracture toughness terms in the Schwalbe crack driving force. As such, the materials science community is challenged to address the fundamental science underpinning this observation.

1. Introduction

Durability and damage tolerance (DADT) analyses are essential for the design and through-life sustainment of both aircraft and reusable space vehicles, as described in the United States Air Force (USAF) MIL-STD-1530D [1], United States (US) Joint Services Structural Integrity Guidelines JSSG2006 [2], and National Aeronautics and Space Administration (NASA) NASA-HDBK-5010 [3]. Furthermore, as delineated in USAF Structures Bulletin EZ-19-01 [4] a damage tolerance and durability-based approach is also mandated for the airworthiness certification of additively manufactured airframes and limited life aerospace parts. Indeed, MIL-STD-1530D [1] mandates that these analyses must be based on linear elastic fracture mechanics. At this stage, it should be noted that USAF Structures Bulletin EZ-19-01 [4] states that the most difficult challenge is to establish an accurate fracture mechanics-based prediction of structural performance and that, as highlighted in the United States Joint Services Structural Integrity Guidelines JSSG2006 [2], there must be no yielding at 115% design limit load, where the design limit load is the maximum load that will be seen in an operational airframe. Furthermore, as first outlined in [5] and discussed in more detail in [6,7,8], the mandated durability analysis must use the small crack growth curve and must be consistent with the building block approach mandated in MIL-STD-1530D and discussed in JSSG2006. This latter statement means that the same crack growth equation used to describe the growth of small cracks in laboratory test specimens must be able to predict small crack growth under operation flight loads. In other words, there can be no disposable constants that are used to fit the measured experimental/operational crack growth history. As such, the equation governing the growth of small naturally occurring cracks in AM materials becomes particularly significant. The ASTM E647-13a [9] definition of small and short cracks is given in Appendix A.

In this context, it is now known that the Hartman–Schijve variant of the Nasgro crack growth equation can often be used to represent the growth of both long and short cracks in AM materials [7,8,10,11,12,13,14,15,16,17,18,19,20,21,22,23], cold spray additively manufactured (CSAM) parts [8,24,25], and to compute the durability of cold spray repairs to conventionally built aluminium alloys parts [26].

Important features of this formulation are that it has been shown [7,8,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] to:

(1)

Account for the effect of the build direction on crack growth;

(2)

Account for the large scatter seen in the da/dN versus ΔK curves associated with different AM processes;

(3)

Be able to capture the growth of both long and small cracks in AM parts;

(4)

Be able to predict the growth of cracks that arise due to the interaction between rough surfaces and surface breaking cracks;

(5)

Be able to predict the growth of cracks that nucleate and grow from surface breaking porosity;

(6)

Be able to compute the growth of cracks in cold spray repairs to corrosion damage.

Applications of this formulation to a range of conventionally manufactured materials are given in [6,27,28,29,30,31,32,33]. Its applicability to model crack growth in polymers, adhesives, composites, and nano-composites is discussed in [8,34,35,36,37,38,39,40,41,42,43,44,45,46]. An illustration of its ability to model crack growth in plasma sprayed refractory materials is given in [47].

The Hartman–Schijve crack growth equation can be expressed [7] as:

da/dN = D[Δκ]^p

(1)

where N is the number of cycles, a is the crack length/depth, p and D are material constants, and Δκ is Schwalbe’s crack tip driving force [48], viz:

Δκ= [(∆K − ∆K_thr)/(√(1 − K_max/A))]

(2)

Here, K_max and K_min are the maximum and minimum values of stress intensity factors in a given fatigue cycle, ∆K = (K_max − K_min), ∆K_thr is the fatigue threshold, which is defined such that when ∆K = ∆K_thr, the crack growth rate (da/dN) becomes identically zero, and A is the cyclic fracture toughness. As explained in [6,7,8], the terms ∆K_thr and A are best interpreted as parameters that are chosen so as to fit the measured da/dN versus ∆K data. (A summary of alternative equations that are commonly used to compute the growth of long cracks is given in [49]).

An intriguing feature of this formulation is that it is now known [8] that the variability in the da/dN versus ∆K curves associated with AM Ti-6Al-4V is captured by allowing for the variability in the two parameters ∆K_thr and A, and that D and p would appear to be true material parameters. By this, it is meant that for AM Ti-6Al-4V, the parameters D and p would appear to be independent of the build process and the orientation of the crack to the build direction. As such, this discovery simplifies our understanding of the key parameters governing crack growth in AM Ti-6Al-4V. As a result, when da/dN is plotted against ∆κ the 53 curves, which were obtained by a number of independent researchers, it essentially fell onto a single curve [8]. Furthermore, as discussed in [8], since there are only two variable parameters, namely ∆K_thr and A, this means that the worst case (mean-3σ, where is the variance) da/dN versus ∆K curve, that is mandated in NASA-HDBK-5010, can be easily determined. As such the purpose of the present paper is to address the question: Does this phenomenon also hold for both Inconel 718 and Inconel 625?

For the sake of comparison,

Table 1. Values of p and D associated with a number of AM materials, data from refs [16,17,18].

Material and Reference	D	p
Ti-6Al-4V, from [16,18]	2.79 × 10−10	2.12
316L Stainless Steel, from [16,25]	1.49 × 10−10	2.12
AerMet 100, from [16]	1.49 × 10−10	2.12
Inconel 625, from [17]	2.79 × 10−10	1.99
17-4 PH Stainless Steel, from [17]	4.46 × 10−10	1.83

presents the values of p and D associated with several AM materials.

2. Materials and Methods

Progress in science involves the interplay between theory and observation. In this context, it has been mentioned that for AM Ti-6Al-4V, it would appear that the variability in the long crack da/dN versus ΔK curves is essentially controlled by just two material parameters, namely ∆K_thr and A. The question thus arises: Is this true for other AM materials.

To address this question, it is necessary to focus on materials for which there are sufficient crack growth data to enable a reasonable conclusion to be reached. As a result, this paper studies crack growth in:

(i)

Thirty-three tests on both conventionally built and AM Inconel 718 specimen tests, including the NASA round robin study into additively manufactured Inconel 718 built using selective laser melt (SLM) [50].

(ii)

Twenty tests on both AM and conventionally built Inconel 625.

In each case, the crack growth data were taken from the open literature, and the da/dN versus ΔK curves were determined as per the ASTM E647 fatigue test standard [9]. In contrast, for these materials, the parameters D and p would appear to be true material constants.

This study reveals that, to a first approximation, the variability in the da/dN versus ΔK curves associated with each of the materials examined can be approximated using Equations (1) and (2) and allowing for the variability in the terms ∆K_thr and A. This finding simplifies our understanding of the role of the various terms in the Hartman–Schijve crack growth equation.

Given that this finding mirrors that for AM Ti-6Al-4V and for a range of other AM materials [16,17,18], as well as for a range of cold spray additively manufactured materials [24,25], the materials science community is challenged to address the fundamental science underpinning this observation.

We have previously explained that durability analysis is the key to the airworthiness certification of limited life AM parts, and that such analyses need to use the associated worst case small crack da/dN versus ΔK curve. In this context, [8] notes that this worst-case curve can often be estimated from the long crack curve by setting the fatigue threshold term in Equation (2) to a small value typically in the range 0.1 ≤ ΔK_thr ≤ 0.3. This suggestion is supported by the results given in [14,15,19,21,24] for the growth of small cracks in a range of AM materials. Unfortunately, a detailed search of the open literature did not reveal the necessary small crack da/dN versus ΔK curves for either Inconel 718 or Inconel 625. As a result, the methodology delineated in [8] is used to hypothesise the necessary worst-case curves. In so doing, this paper presents a means for these predictions to be independently (and scientifically) validated by researchers in the field. However, it should be noted that, as illustrated in [18], researchers should be aware of the need to determine sufficient crack growth curves that the variability in the curves can be adequately captured and the worst possible crack growth curve determined.

3. Crack Growth in Conventionally and Additively Manufactured Inconel 718

The NASA Round Robin study into crack growth in Selective Laser Melt (SLM) built Inconel 718 [50] presented both the R = 0.1 and the R = 0.7 da/dN versus ΔK curves. These curves, which were obtained by a number of laboratories, are shown in Figure 1, Figure 2 and Figure 3. The identifiers used in Figure 1, Figure 2 and Figure 3 and Table 2 to identify the various laboratories that took part in this study are those used in [50]. Figure 1, Figure 2 and Figure 3 also include the following da/dN versus ΔK curves:

(1)

The room temperature R = 0.1, 0.4, 0.7, and 0.8 crack growth curves given in the Nasgro database. The Specimen ID’s (descriptors) used for these tests in the Nasgro database are: Q3LA12AB01A4, Q3LA12AB01A3, Q3LA12AB01A2, and Q3LA12AB01A1, respectively.

(2)

The R = 0.1 crack growth curve given by Konecna et al. [51] for SLM Inconel 718;

(3)

The R = 0.1 and 0.7 crack growth curves given by Newman and Yamada [52] for conventionally manufactured Inconel 718;

(4)

The R = 0.1 and 0.7 crack growth curves given by Yadollahi et al. [53] for laser bed powder fusion (LPBF) built Inconel 718 specimens.

(5)

The R = 0.1 and (high R ratio) K_max fatigue crack growth curves given by Ostergaard et al. [54] for LPBF Inconel 718. These curves are for specimens both with and without HIPing. Following the notation used in [54], specimens that were subjected to both solution and duplex aging are given the suffix S-DA. Specimens that have been HIPed include the term HIP in their descriptor. Details of these two post-build treatments are given in [54].

(6)

The R = 0.1 crack growth curve given by Kim et al. [55] for LPBF Inconel 718;

(7)

The R = 0.1 crack growth curve given by Yu et al. [56] for laser-directed energy deposition (LDED) built Inconel 718 in the as-deposited condition.

(8)

The R = 0.5, 0.1, −1.0, and −2 crack growth curves given by Paluskiewicz et al. [57] for conventionally manufactured Inconel 718 tested at 100 °C.

Figure 1. The high R ratio da/dN versus ΔK curves for tests on both conventionally manufactured and AM Inconel 718. Here it should be noted that the curves with the prefix “NASA SLM” are data from ref. [50].

Figure 1. The high R ratio da/dN versus ΔK curves for tests on both conventionally manufactured and AM Inconel 718. Here it should be noted that the curves with the prefix “NASA SLM” are data from ref. [50].

Figure 2. The R = 0.1 and 0.4 da/dN versus ΔK curves for tests on conventionally manufactured Inconel 718 and the various R = 0.1 and 0.4 da/dN versus ΔK curves for AM Inconel 718 together with the curves given by Paluskiewicz [58] for conventionally manufactured Inconel 718 tested at 100 °C and at a range of R ratios. As per Figure 1, curves with the prefix “NASA SLM” are data from ref. [50].

Figure 2. The R = 0.1 and 0.4 da/dN versus ΔK curves for tests on conventionally manufactured Inconel 718 and the various R = 0.1 and 0.4 da/dN versus ΔK curves for AM Inconel 718 together with the curves given by Paluskiewicz [58] for conventionally manufactured Inconel 718 tested at 100 °C and at a range of R ratios. As per Figure 1, curves with the prefix “NASA SLM” are data from ref. [50].

Figure 3. A combined plot of the 28 da/dN versus ΔK curves shown in Figure 1 and Figure 2. This plot covers a range of build processes and R ratios.

Figure 3. A combined plot of the 28 da/dN versus ΔK curves shown in Figure 1 and Figure 2. This plot covers a range of build processes and R ratios.

An interesting feature of Figure 2 is the large variability in the R = 0.1 da/dN versus ΔK curves at low values of da/dN and the relatively low variability at higher values of da/dN.

Low Temperature Crack Growth in Conventionally Manufactured Inconel 718

Since Inconel 718 is also widely used in space-related applications, let us next consider the effect of low temperatures on the da/dN versus ΔK curves associated with conventionally manufactured Inconel 718. To this end, Figure 4 presents a few selected da/dN versus ΔK curves taken from Figure 3 together with the following datasets that were extracted from the Nasgro database, viz:

(1)

Specimen test ID: Q3LB11AA07A2. This is a middle crack tension (M(T)) specimen cut from a 12.7 mm thick sheet and tested at R = 0.6 and 195 °K.

(2)

Specimen test ID: Q3LB11AA07A1. This is a middle crack tension specimen (M(T)) cut from a 12.7 mm thick sheet and tested at R = 0.33 and 195 °K.

(3)

Specimen test ID: Q3LB24GB04A1. This is a 25.4 mm thick compact tension (CT) specimen cut from a forging and tested at R = 0.1 and 76 °K.

(4)

Specimen test ID: Q3LC10LA04A1. This is a 7.62 mm thick compact tension (CT) specimen cut from a plate and tested at R = 0.1 and 77.6 °K.

(5)

Specimen test ID: Q3LB24GC02A1. This is a 25.4 mm thick compact tension (CT) specimen cut from a forging and tested at R = 0.1 and 4 °K.

Examining Figure 4, we see that, to a first approximation, low temperatures appear to improve the fatigue performance of Inconel 718.

Having presented a range of da/dN versus ΔK crack growth curves associated both conventionally manufactured and AM Inconel 718, the next section examines the question:

Do these curves simplify if da/dN is expressed as a function of Δκ, and can, as has been shown [8] to be the case for AM Ti-6Al-4V, the variability in the da/dN versus ΔK curves be captured by allowing for the variability in the two parameters ∆K_thr and A?

4. Modelling the Variability in Crack Growth in Conventionally Manufactured Inconel 718 and Additively Manufactured Inconel 718

Figure 5 reveals that, to a first approximation, when da/dN is plotted against Δκ then the various (28) curves shown in Figure 3, that cover both conventionally built and AM Inconel 718 tested at a range of R ratios, all (essentially) collapse onto a single master curve, viz:

da/dN = 1.2 × 10⁻¹⁰ × [Δκ]^2.12

(3)

The values of the parameters ∆K_thr and A used in Figure 5, which were determined using the “Automated Total Least Squares” methodology described in [58], are listed in

Table 2. Values of the parameters used in Figure 5 and Figure 6.

Test Descriptor 1	ΔKthr (MPa √m)	A (MPa √m)	Coefficient of Determination (R2)
R = 0.1
++ NASA SLM M1-0253 * [50]	2.6	160	0.85
NASA SLM MFSC [50]	6.2	155	0.83
NASA SLM Lab B [50]	6.1	240	0.81
NASA SLM Lab C [50]	7.8	180	0.99
Nasgro data base, room temperature test, Nasgro specimen ID: Q3LA12AB01A4	6.0	170
Konecna et al. [51]	1.8	140
Yadollahi et al. [53] (L-PBF)	7.8	150
Newman et al. [52], conventionally manufactured	8.7	175
Kim et al. [55] for LPBF	4.0	180
Yu et al. [56] LED tested in the as-built condition	1.8	118
R = 0.4
Nasgro data base, room temperature test, Specimen ID: Q3LA12AB01A3	3.5	170
R = 0.7
NASA SLM M1-200 [50]	3.0	155	0.80
NASA SLM MFSC [50]	3.0	155	0.83
NASA SLM Lab B [50]	2.9	240	0.90
NASA SLM Lab C [50]	4.1	180	0.80
Newman et al. [52], conventionally manufactured	3.0	175
Nasgro data base, room temperature test, Nasgro specimen ID: Q3LA12AB01A2	2.95	170
Yadollahi et al. [53] (L-PBF)	3.0	150
Yu et al. [55] LED tested in the as-built condition	1.8	118
Nasgro data base R = 0.8, room temperature test, Nasgro specimen ID Q3LA12AB01A1	4.0	170
Paluskiewicz et al. [57], conventionally built Inconel 718 tested at 100 °C (373 °K) at a range of R ratios
R = 0.5	2.0	70 **
R = 0.1	6.0	70
R = −1.0	9.0	70
R = −2.0	4.0	70
Ostergaard et al. [56], LPBF built Inconel 718
Kmax = 36 MPa √m, SD-A with the crack in the XZ direction	1.8	150
Kmax = 36 MPa √m, HIP with the crack in the ZX direction	2.3	150
R = 0.1, SD-A with the crack in the XZ direction	3.9	150
R = 0.1, HIP with the crack in the XZ direction	6.7	150
R = 0.1, HIP with the crack in the ZX direction	7.0	150
Nasgro Low temperature tests on conventionally built Inconel 718
12.7 mm thick M(T) specimen tested at R = 0.6 and 195 °K, Nasgro Specimen ID: Q3LB11AA07A2.	7.0	250
12.7 mm thick M(T) specimen tested at R = 0.33 and 195 °K, Nasgro specimen ID: Q3LB11AA07A2.	13.0	250
25.4 mm thick CT specimen tested at R = 0.1 and 76 °K, Nasgro specimen test ID: Q3LB24GB04A1.	17	140
7.62 mm thick CT specimen tested at R = 0.1 and 76 °K, Nasgro specimen test ID: Q3LC10LA04A1.	3.9	150
25.4 mm thick CT specimen tested at R = 0.1 and 4 °K, Nasgro specimen test ID: Q3LB24GC02A1.	18	200

¹ Unless stated the specimen tests were compact tension (CT) specimens and the tests were performed in accordance with the fatigue test standard ASTM E647-13a. * Unless stated the specimens were tested at room temperature. ** These test specimens would appear to have a reduced value of A. It is not clear why this is the case. ⁺⁺ The prefix “NASA SLM” denotes data from the NASA round robin study into crack growth in Inconel 718 [50].

. Furthermore, as can be seen in Figure 5, this da/dN versus Δκ relationship appears to hold over approximately six orders of magnitude in da/dN, and holds regardless of whether the material was fabricated conventionally, using SLM, L-PBF, or LDED. It also appears to hold regardless of the R ratio. We also see that for SLM, L-PBF, and LDED Inconel 718, the value of the exponent p in Equation (1) is approximately 2, and hence is similar to the values obtained for the AM materials listed in Table 1. Interestingly, examining Figure 5, we see that the da/dN versus Δκ curve associated with SLM, L-PBF, and LDED built Inconel 718 is similar to the da/dN versus Δκ curves associated with additively manufactured Ti-6Al-4V, 17-4Ph steel, and Inconel 625.

Figure 6 reveals that, for the low temperature curves shown in Figure 4, when da/dN is plotted against Δκ, these curves also collapse onto Equation (3). The values of the parameters ∆K_thr and A used in Figure 6 are also listed in Table 2. Consequently, not only do these curves simplify if da/dN is expressed as a function of Δκ, but (to a first approximation) the variability in the curves can be captured by allowing for the variability in the two parameters ∆K_thr and A. To compliment Figure 6, Appendix B presents a selection of cases where the measured and computed da/dN is plotted against ΔK curves are compared. The computed da/dN is plotted against ΔK curves obtained using Equation (3) and the values of ∆K_thr and A listed in Table 2.

5. Modelling the Variability in Crack Growth in Conventionally Manufactured and Additively Manufactured Inconel 625

Section 3 noted that the da/dN versus Δκ relationship determined in [14] for a limited number of tests on AM Inconel 625 was similar to that of Inconel 718. Let us further investigate this observation by analysing the da/dN versus ΔK curves presented in [59,60,61,62] for tests on Inconel 625 specimens built using both SLM and LPBF where the cracks lay at various degrees to the build direction. Two SLM specimen tests were analysed in [59,62]. The R ratios in the tests discussed in [59] was 0.1. In Ref. [62], the SLM specimens were tested at R ratios of 0.1, 0.5, and 0.7. The SLM specimens in [59,62] were heat treated, but not HIPed. In Ref. [59], specimens with the crack in different directions, which were labelled SLM SR-0 or SLM-SR-90, were also tested. The SR notation indicates that the specimens were stress relieved, and the 0/90 notation indicates the direction of the crack relative to the build direction. The LPBF specimens tested in [60], at R ratios of 0.1, were either HIPed or stress relieved (SR). The LPBF specimens tested in [60] were assigned the designations P₀, P₁, P₂, and P₃ depending on the laser deposition parameters, [6] for details. In Ref. [60], tests were performed on specimens with the crack at either 0 and 90 degrees to the build direction The LBPF specimens tested in [61], which had an R ratio of 0.5, were labelled V2–V6 and tests were performed on specimens with the crack at 0 and 90 degrees to the build direction.

Figure 7 presents the R = 0.1 crack growth curves for both the AM specimens and the wrought material tested in [58]. Figure 8 presents the R = 0.5 and 0.7 curves given for this material. Figure 9 presents all of these curves on a single plot.

The resultant 20 different da/dN versus ΔK curves are given in Figure 9. The corresponding da/dN versus Δκ curves are given in Figure 10 together with the governing equation published in a prior paper for this material [17], viz:

da/dN = 2.79 × 10⁻¹⁰ (Δκ)^1.99

(4)

Figure 10 reveals that, regardless of the AM process, the R ratio, or the angle between the crack and the build direction, when da/dN is plotted as a function of Δκ, then the 20 different curves essentially collapse onto a single master curve that can be approximated by Equation (4). As previously, the values of ΔK_thr and A used in Figure 10 were determined using the “Automated Total Least Squares” methodology described in [59]. The values of the parameters associated with these 20 tests are given in

Table 3. Values of the parameters A and ΔK_thr used in Figure 10.

Test, Reference and Descriptor	ΔKthr (MPa √m)	A (MPa √m)
SLM [59]
SLM SR 0, R = 0.1	10.9	90
SLM SR 90, R = 0.1	6.4	140
SLM SR 45, R = 0.1	8.3	140
Wrought, R = 0.1	7.0	90
LPBF [60]
LPBF P0-0 HIP, R = 0.1	10.1	90
LPBF P0-90 HIP, R = 0.1	9.5	76
LPBF P2-90 HIP, R = 0.1	8.9	70
LPBF P3-0 HIP, R = 0.1	8.8	74
LPBF P3-90 HIP, R = 0.1	8.5	88
LPBF P3-0 SR, R = 0.1	7.3	71
SLM [62]
SLM ZX, R= 0.1	9.8	100
SLM YZ, R = 0.1	8.0	90
SLM ZX, R= 0.5	6.4	100
SLM YZ, R = 0.5	5.9	90
SLM ZX, R= 0.7	5.5	100
LPBF [61]
LBPF V2 0, R = 0.5	6.2	210
LBPF V4 0, R = 0.5	6.4	190
LBPF V5 0, R = 0.5	6.2	180
LBPF V5 90, R = 0.5	6.7	145
LBPF V6 90, R = 0.5	7.4	125

.

To compliment Figure 10, Appendix B presents a selection of cases where the measured and computed da/dN plotted against ΔK curves are compared. The computed da/dN plotted against ΔK curves are obtained using Equation (4) and the values of ∆K_thr and A listed in Table 3.

As such, this study supports the conclusion given above for crack growth in Inconel 718 and Ti-6Al-4V, namely that, to a first approximation, crack growth in these 20 tests would appear to be controlled by the two independent parameters A and ΔK_thr.

6. The Small Crack Growth Hypothesis

Appendix X3 of the fatigue test standard ASTM E647-13a [9] comments: It is unclear if a measurable threshold exists for small naturally occurring cracks in operational structures. The papers [6,7,12,14,15,20,21,23,26,28,29,30,31,32,63] subsequently built on this statement to illustrate that, for both conventionally and AM metals, Equations (1) and (2) with the values of D, p, and A determined from tests on long cracks and the fatigue threshold set to a small value, typically in the range 0.1 ≤ ΔK_thr ≤ 0.3 MPa √m, were often able to accurately compute the growth of small cracks under both constant amplitude and operational flight loads.

As a result, it is hypothesised that, in the absence of significant residual stresses, an upper bound estimate of the growth of small cracks in Inconel 718, that is required by NASA-HDBK-5010, may be able to be estimated using Equation (3) with the fatigue threshold term ΔK_thr set to 0.1, viz:

da/dN = 1.2 × 10⁻¹⁰ × [(∆K − 0.1)/(√(1 − K_max/A))]^2.12

(5)

It is similarly hypothesised that, in the absence of significant residual stresses, an upper bound estimate of the growth of small cracks in Inconel 625 may be able to be estimated using Equation (4) with the fatigue threshold term ΔK_thr set to 0.1, viz:

da/dN = 2.79 × 10⁻¹⁰ × [(∆K − 0.1)/(√(1 − K_max/A))]^1.99

(6)

The values of A in Equations (5) and (6) would need to be taken from tests on long cracks in specimens built using the same processes and post build treatments as those used to build the small crack test specimens. However, it should be stressed that further work is needed to validate these hypotheses.

7. Conclusions

Whereas when da/dN is expressed as a function of the range of ΔK, the crack growth curves associated with 33 tests on both conventionally and additively manufactured Inconel 718 exhibit considerable scatter/variability, when da/dN is expressed in terms of Δκ, each of these 33 different curves essentially collapse onto a single curve regardless of the manufacturing process. Furthermore, to a first approximation, the resultant relationship between da/dN and Δκ appears to hold over approximately six orders of magnitude in da/dN.

It would thus appear that accounting for the variability in the terms ∆K_thr and A in the Hartman–Schijve crack growth equation has the potential to account for both the variability in crack growth and the R ratio effect seen in crack growth tests on both AM and conventionally built Inconel 718. Furthermore, for tests on conventionally manufactured Inconel 718, this relationship appears to hold for tests performed at temperatures that range from 373 °K to cryogenic temperatures.

This finding suggests that since there are only two variable parameters, namely ∆K_thr and A, this formulation has the potential to determine the worst case (mean-3σ) da/dN versus ∆K curve mandated in NASA-HDBK-5010.

The same phenomenon also appears to hold for 20 room temperature tests on both conventionally and additively manufactured Inconel 625, albeit over approximately six orders of magnitude in da/dN.

We also found that the da/dN versus Δκ curve associated with SLM, L-PBF, and LDED built Inconel 718 is similar to the da/dN versus Δκ curves associated with additively manufactured Ti-6Al-4V, 17-4Ph steel and Inconel 625.

Unfortunately, the authors could not find a correlation between the values ∆K_thr and A determined in these studies and the corresponding build processes.

Furthermore, given that:

(1)

the 53 studies examined in this paper were taken from a wide cross section of researchers;

(2)

that the conclusions reached in this study mirror those stated in [8] for the assessment of 53 independent studies into crack growth in AM Ti-6Al-4V, which were also taken from a wide cross section of researchers, namely that the variability in the da/dN and ΔK curves can (to a first approximation) be accounted for by allowing for the variability in the terms ∆K_thr and A;

(3)

the materials science community is challenged to address the fundamental science underpinning this observation.

Prior studies have shown how for AM Ti-6Al-4V, AM 316L stainless steel, and wire arc additively manufactured (WAAM) 18Ni 250 Maraging steel, the mandated worst case small crack da/dN versus ΔK curve can be obtained from a knowledge of the da/dN versus Δκ curves by setting the term ∆K_thr to a small value (0.1 MPa √m). (To the best of the authors’ knowledge, no other such predictive capability exists for AM materials.)

As a result, the present paper has, for the first time, taken the opportunity to predict the necessary worst case da/dN versus ΔK curves for both AM Inconel 718 and AM Inconel 625, and in the spirit of the scientific method, the materials science community is also challenged to measure these curves and thereby confirm or disprove these predictions.

The ability to predict the worst-case small crack da/dN versus ΔK curve for these two materials, when taken together with the prior studies mentioned above, would further support the generality of this approach and have the potential to significantly simplify the durability analysis/certification.