R. Craig McClung, Michael P. Enright, and Wuwei Liang

Southwest Research Institute, San Antonio, Texas

Integration of NASA-Developed Lifi ng Technology 

for PM Alloys into DARWIN®

NASA/CR—2011-216977

March 2011
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R. Craig McClung, Michael P. Enright, and Wuwei Liang

Southwest Research Institute, San Antonio, Texas

Integration of NASA-Developed Lifi ng Technology 

for PM Alloys into DARWIN®

NASA/CR—2011-216977

March 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135

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NASA/CR—2011-216977 iii 

TABLE OF CONTENTS 

Page 

1.0 INTRODUCTION .................................................................................................. 1 

1.1 Background ................................................................................................ 1 

1.2 Approach ................................................................................................... 2 

2.0 STATUS SUMMARIES ........................................................................................ 5 

2.1 Research in P/M Lifing Conducted at 

NASA Glenn Research Center .................................................................. 5 

2.1.1 Determination of Anomaly Distributions .......................................... 5 

2.1.2 Calculation of Fracture Probability .................................................. 9 

2.2 A Summary of Current DARWIN Capabilities 

Relevant to P/M Lifing .............................................................................. 16 

2.2.1 Determination of Anomaly Distributions ........................................ 17 

2.2.2 Calculation of Fracture Probability ................................................ 17 

3.0 ALGORITHM DEVELOPMENT .......................................................................... 25 

3.1 Algorithms for Multiple Anomaly Types .................................................... 25 

3.1.1 Introduction ................................................................................... 25 

3.1.2 Estimating Marginal Probability Densities from 

Component Failure Data ............................................................... 27 

3.1.3 Application to Gas Turbine Engine Materials ................................ 35 

3.2 Development of an Unfolding Algorithm for Estimating 

Spheroid Parameters from Sectioning Data Measurements .................... 42 

3.2.1 Introduction ................................................................................... 42 

3.2.2 The Stereological Equation ........................................................... 42 

3.2.3 Numerical Solution of the Stereological Equation ......................... 43 

3.2.4 Application Example ...................................................................... 44 

3.2.5 Discussion ..................................................................................... 49 

3.3 Algorithms for Modeling the Growth Rates of Small Fatigue Cracks ....... 50 

3.3.1 Introduction ................................................................................... 50 

3.3.2 Analysis Methods .......................................................................... 50 

3.3.3 Results .......................................................................................... 52 

3.3.4 Discussion ..................................................................................... 59 
NASA/CR—2011-216977 iv 

TABLE OF CONTENTS (cont.) 

Page 

4.0 PROGRAM PLAN FOR FUTURE RESEARCH ................................................. 61 

5.0 SUMMARY ......................................................................................................... 65 

6.0 BIBLIOGRAPHY ................................................................................................ 67 

LIST OF FIGURES 

Figure Page 

1. Crack formation computations are performed via a 

user-supplied crack formation module. ............................................................... 18 

2. The conditional probability of fracture failure PF d of 

multiple anomaly materials is dependent on the number 

of anomalies j at a specified location in the component. .................................... 22 

3. For multiple anomaly materials, both the probability of fracture 

given an anomaly PFd   i j and the anomaly occurrence probability 

P d  1 are dependent on the number of anomalies present. ................................. 22 

4. For components with multiple anomalies of multiple types at 

multiple locations, the probability of failure can be modeled 

using several nested series systems to represent the various 

failure events. ..................................................................................................... 24 

5. Analytical probability of failure values associated with single and 

multiple anomalies of a single type for the example component 

containing multiple anomaly types. ..................................................................... 28 

6. Analytical and simulated probability of failure of failure values for the 

example component containing two anomaly types were in close agreement, 

as expected. ....................................................................................................... 30 

7. CDF values for individual anomaly types estimated from censored 

failure data were significantly different from the analytical values. ..................... 31 

8. CDF values for individual anomaly types estimated using the combined 

Kaplan-Meier/Meeker-Escobar approach were in close agreement 

with the analytical values. ................................................................................... 34 

9. The transformed CDF values based on the Kaplan-Meier/Meeker-Escobar 

approach were in close agreement with the analytical parent distributions. ....... 35 

10. The marginal CDFs for each anomaly type were obtained from the 

Kaplan-Meier/Meeker-Escobar fit of the failure data. .......................................... 38
NASA/CR—2011-216977 v 

LIST OF FIGURES (cont.) 

Figure Page 

11. The parent CDFs were determined by transforming the 

marginal CDF values for each anomaly type. ..................................................... 39 

12. The influence of volume to surface area ratio on probability of failure 

and associated contribution of: (a) surface NMPs, (b) subsurface NMPs, 

and (c) surface pores. ......................................................................................... 40 

13. Bivariate histograms (stereograms) of anomaly size and shape parameter 

associated with anomalies in the fictitious material: (a) original population 

of spheroids, and (b) predicted population of spheroids based on sectioning 

data transformed using the Cruz-Orive unfolding algorithm. ............................. 46 

14. Influence of the number of section plane measurements on 

predicted spheroid size parameter values: (a) 10 × 10 bin grid, and 

(b) 5 × 5 bin grid. ................................................................................................ 47 

15. Influence of the number of section plane measurements on predicted 

spheroid shape parameter values: (a) 10 × 10 bin grid, and 

(b) 5 × 5 bin grid. ................................................................................................ 48 

16. Schematic Kitagawa diagram relating smooth specimen 

fatigue endurance limit (Δσe) and large-crack threshold (ΔKth). .......................... 51 

17. Comparison of small-crack and large-crack growth rate data for 

Ti-6Al-4V tested by Lenets et al. and Caton et al. at R = 0.1. ............................. 53 

18. Comparison of small-crack and large-crack growth rate data for 

Ti-6Al-4V tested by Lenets et al. and Caton et al. at R = 0.1, with 

small-crack data adjusted using El Haddad parameter. ..................................... 54 

19. Comparison of small-crack and large-crack growth rate data 

for IN-100 tested by Jha et al. at R = 0.05. ......................................................... 55 

20. Comparison of small-crack and large-crack growth rate data 

for IN-100 tested by Jha et al. at R = 0.05, with small-crack data 

adjusted using two estimates of the El Haddad parameter. ............................... 55 

21. Comparison of small-crack and large-crack growth rate data 

for U720 at R = -0.33. ......................................................................................... 57 

22. Comparison of small-crack and large-crack growth rate data for 

U720 at R = -0.33, |with small-crack data adjusted using El Haddad 

parameter. .......................................................................................................... 57 

23. Comparison of small-crack and large-crack growth rate data for 

U720 at R = -1. ................................................................................................... 58 

24. Comparison of small-crack and large-crack growth rate data for 

U720 at R = -1, with small-crack data adjusted using El Haddad 

parameter. .......................................................................................................... 58 
NASA/CR—2011-216977 vi 

LIST OF TABLES 

Table Page 

1. Anomaly Parameters for Example Component 

Containing Two Types of Anomalies ................................................................... 28 

2. Simulated Failure of Example Component with 

Two Types of Anomalies ..................................................................................... 29 

3. Simulated CDF Values for Example Component 

Containing Two Types of Anomalies ................................................................... 29 

4. Failure CDF for Anomaly Type A Using Kaplan-Meier Approach 

Combined with Meeker-Escobar Data Fit ........................................................... 33 

5. Failure CDF for Anomaly Type B Using Kaplan-Meier Approach 

Combined with Meeker-Escobar Data Fit ........................................................... 33 

6. Anomaly Types Associated with Nickel-Based Superalloy 

Test Specimens .................................................................................................. 35 

7. Component Failure CDF for Nickel-Based Superalloy Smooth Specimen 

Fatigue Failure Data with Applied Stress of 1100 MPa ....................................... 36 

8. Failure CDF for Surface NMPs Using Kaplan-Meier/Meeker-Escobar 

Approach ............................................................................................................ 37 

9. Failure CDF for Subsurface NMPs Using Kaplan-Meier/Meeker-Escobar 

Approach ............................................................................................................ 37 

10. Failure CDF for Surface Pores using Kaplan-Meier/Meeker-Escobar 

Approach ............................................................................................................ 38 

11. Influence of Component Volume to Surface Area Ratio on 

Number of Anomalies for Each Anomaly Type .................................................... 39 

12. Spheroid Parameters Associated with Application Example .............................. 45 
NASA/CR—2011-216977 1 

1.0 INTRODUCTION 

In recent years, Southwest Research Institute® (SwRI®) and NASA Glenn Research Center 

(GRC) have worked independently on the development of probabilistic lifing methods for 

materials used in gas turbine engine rotors. The two organizations have addressed different but 

complementary technical challenges. This final report summarizes a brief investigation into the 

current status of relevant technology at SwRI and GRC with a view towards a future integration 

of methods and models developed by GRC for probabilistic lifing of powder metallurgy (P/M) 

nickel turbine rotor alloys into the DARWIN® software developed by SwRI. 

1.1 Background 

Previous Activity at SwRI. Work at SwRI was originally motivated by incidents involving 

uncontained rotor fracture at Sioux City, Iowa, in 1989 and Pensacola, Florida, in 1996. As a 

result of these incidents, the Federal Aviation Administration (FAA) requested that the aircraft 

engine industry, through the Aerospace Industries Association (AIA) Rotor Integrity SubCommittee (RISC), review available techniques to determine whether a damage tolerance 

approach could be introduced to produce a reduction in the rate of uncontained rotor events. 

During the development of this probabilistic damage tolerance approach, it became apparent to 

RISC that the capabilities and effectiveness of the emerging technology could be significantly 

enhanced by further research and development. SwRI, in partnership with four major 

U.S. engine manufacturers and with guidance from RISC, proposed and was awarded a series of 

major FAA grants beginning in 1995 to address identified shortfalls in technology and data. 

One of the key outcomes of this work has been the probabilistic damage tolerance computer code 

DARWIN (Design Assessment of Reliability With INspection). DARWIN integrates finite 

element models and stress analysis results, fracture mechanics models, material anomaly data, 

probability of anomaly detection, and uncertain inspection schedules with a user-friendly 

graphical user interface (GUI) to determine the probability-of-fracture of a high-energy rotating 

component as a function of operating cycles with and without inspections. 

Initial DARWIN capabilities were focused on the hard alpha titanium problem, which had 

caused the Sioux City incident. Later DARWIN work focused on surface damage at machined 

circular holes, which had caused the Pensacola incident. With direct guidance from the FAA, 

DARWIN has been tailored to address specific advisory circulars issued by the FAA, including 

AC 33.14-1 for titanium hard alpha, AC 33.70-2 on circular holes in all materials, and the more 

general AC 33.70-1 for all engine life-limited parts. The FAA has indicated that DARWIN is an 

acceptable means of compliance with these advisory circulars, and DARWIN has been employed 

in the formal FAA certification process for numerous engines. 

Some elementary DARWIN capabilities have also been developed to address superalloys with 

high anomaly occurrence rates, including probabilistic three-dimensional (3D) descriptions of 

anomaly sizes and orientations, and provisions for users to integrate their own probabilistic crack 

formation life modules. However, this work has been limited due to other, more pressing FAA 

and RISC priorities, and additional effort on these topics is not planned under the current FAA 

grant. 
NASA/CR—2011-216977 2 

At the request of the FAA and the engine industry, SwRI began licensing and providing formal 

user support for DARWIN in 2000. Currently DARWIN is licensed by six aircraft engine 

companies and two foreign government laboratories. DARWIN is also available on a 

royalty-free basis to all U.S. government agencies. 

Previous Activity at NASA Glenn Research Center. For more than twelve years, GRC has been 

actively investigating and developing models for the fatigue behavior of P/M nickel-based alloys 

that are being increasingly used in high energy rotating components for gas turbine engines. 

These alloys have many advantages over conventionally cast and wrought alloys, but they suffer 

from the rare occurrence of non-metallic defects (ceramic inclusions) that arise from the powder 

atomization process. These inclusions can have a potentially large detrimental effect on the 

fatigue life of individual components. GRC activity has primarily focused on understanding and 

modeling the effects of ceramic inclusions on fatigue life, and on developing improved life 

prediction methods that could be applied to components. The work has resulted in the 

development of an extensive and unique data base that captures the relationships between 

inclusion size and shape distributions and the low cycle fatigue life for various temperature and 

stress regimes. Probabilistic models and algorithms have been developed that simulate the effect 

of inclusion occurrence rate, size distribution and material processing forging strains on fatigue 

life. Work is continuing to refine these models further and to begin their verification and 

validation. 

1.2 Approach 

This report summarizes a joint critical review of previous work by SwRI and GRC to identify 

specific GRC data and models that could be integrated into DARWIN, to identify new DARWIN 

functionality that must be developed to enable this integration, to identify additional data or 

models that are needed, and then to outline a program plan to accomplish the desired integration. 

Some initial work was also performed to begin developing some of the needed algorithms. 

This critical review began by SwRI and GRC each compiling and presenting to the other 

organization a detailed briefing describing the relevant background technology. 

SwRI summarized current DARWIN capabilities with special emphasis on multiple anomaly 

materials, while GRC summarized their existing data and models for P/M materials. SwRI then 

performed a detailed review and critique of this work, with a focus on determining how existing 

GRC data and models could fit into current DARWIN capabilities, as well as identifying 

apparent gaps in current DARWIN capabilities. 

This briefing and the first critique was conducted at a face-to-face initial technical exchange 

meeting conducted at Honeywell Aerospace in Phoenix, Arizona on December 9, 2008. 

The main attendees were Jack Telesman and Peter Bonacuse from GRC, Craig McClung and 

Michael Enright from SwRI, and Peter Kantzos (formerly at GRC) from Honeywell. Most of the 

meeting was devoted to a series of technical presentations that summarized previous work at 

GRC and SwRI applicable to P/M lifing. 
NASA/CR—2011-216977 3 

This technical exchange meeting launched efforts to develop formal written summaries of the 

existing data, models, and software, along with an initial assessment of future needs. 

These written summaries are provided in Section 2 of this final report. 

New algorithm development was conducted in three areas. First, a methodology for estimating 

anomaly distributions for materials in which multiple anomaly types are operative was developed 

and verified. Second, an unfolding algorithm to estimate the dimensions and orientations of 3D 

spheroids based on 2D sectioning measurements was implemented and investigated. Third, the 

behavior of small fatigue cracks in U720 tests conducted by GRC was investigated, and an 

existing model to predict this small-crack behavior from large-crack growth rates was evaluated. 

These activities are described in Section 3. 

One of the primary objectives of the current study was to identify technologies for assessment of 

P/M materials developed at GRC that could be transferred for use by gas turbine engine 

manufacturers. Based on the GRC status summary and the DARWIN capabilities summary, 

several technology gaps were identified related to (1) characterization and simulation of 3D 

anomalies based on data from different types of inspections, (2) understanding and analysis of 

anomaly deformation and breakup during the forging process, and (3) treatment of various 

factors that can influence fatigue crack growth lives. A program plan for the development and 

implementation of key algorithms and software modules in DARWIN to address the technology 

gaps was developed and is described in Section 4. 

It was originally envisioned that, during the course of this project, SwRI would interact with all 

four of the major U.S. aircraft engine manufacturers as well as other government agencies that 

are performing research in the subject areas. This communication and consultation would be 

designed to assess previous industry work in this area and to provide a formal overview of the 

technical exchange activities conducted during this year, as well as the status summaries and 

program plans. The manufacturers and the government agencies would be invited to review and 

comment on this information, and their interest in participating in future implementation efforts 

would be assessed. However, as the funding scenario for future efforts changed, NASA GRC 

directed that this communication and consultation effort be de-emphasized at the present time. 

Informal conversations were conducted with two of the four manufacturers on topics of specific 

interest. In addition, relevant public domain literature authored by staff from various aircraft 

engine manufacturers was critically reviewed. About twenty technical reports, journal articles, 

and conference papers related to fatigue life prediction of P/M alloys were included in this 

exercise. A formal written summary was not prepared. 
(This page intentionally left blank.) 
NASA/CR—2011-216977 5 

2.0 STATUS SUMMARIES 

2.1 Research in P/M Lifing Conducted at NASA Glenn Research Center 

For more than ten years, NASA Glenn Research Center has been actively investigating and 

developing models for the fatigue behavior of powder metallurgy (P/M) nickel-based alloys used 

in high energy rotating components for gas turbine engines. The purpose of this section is to 

provide an overview of the major investigations and findings at GRC. The technical details are 

not repeated here, but GRC publications and presentations that provide those details are 

thoroughly cited. This status summary is an important first step towards the development of a 

plan for further research and development activities that would be necessary to integrate the 

GRC technology into the DARWIN software. 

The GRC investigations focused on a single materials system—Udimet® 720—in both seeded 

and unseeded forms. Therefore, some of the specific conclusions of the GRC work may be 

limited to this specific material. However, it was the intent of GRC, and it is the intent of the 

current survey, to identify phenomena, relationships, and methods that are applicable to a broad 

range of P/M superalloys. 

The GRC seeded investigations included two different types of seeds. One set of artificial 

inclusions was made of crushed Alcoa T64 alumina, a common crucible material, selected to 

represent a hard, blocky Type I inclusion. These seeds were screened to different size ranges in 

different studies, including -140/+170 mesh (nominal diameter 122 m) and -80/+100 mesh. 

A second seed type was produced by crushing pre-baked Ram 90 alumina crucible paste, 

representative of soft/friable Type II inclusions. The Ram 90 seeds were screened to mesh sizes 

of either -270/+325 (nominal diameter 54 m) or -140/+170. The base material was production 

quality U720 powder screened to -270 mesh and processed conventionally. Seeds did not affect 

the resultant microstructure. Seeding rates were chosen to ensure an appropriate number of 

surface inclusions in fatigue test specimens. 

The GRC research can be organized into two broad categories. The first category involves the 

determination of the proper statistical descriptions of anomaly size, shape, and frequency. 

These anomaly distributions would then be used as input to the reliability calculations. 

The second category involves the calculation of the probability of fracture of a coupon or 

component, given that an anomaly is present. This determination involves some treatment of 

fatigue crack formation life and fatigue crack growth life, including the effects of various 

residual stresses. 

2.1.1 Determination of Anomaly Distributions

Because non-metallic inclusions and other metallurgical defects can have a significant 

deleterious effect on the fatigue performance of P/M superalloys, the “cleanliness” of a particular 

alloy or material lot is a critical characteristic. Kantzos et al. (2004) have correctly emphasized 

that both the size and number of inclusions must be considered to characterize cleanliness. 
NASA/CR—2011-216977 6 

Three different sources of data may be available to determine anomaly sizes and frequencies. 

The first is measurements of inclusions that have been completely separated from the matrix 

material, the second is measurements of implanted inclusions from sectioning of the material, 

and the third is measurements of implanted inclusions that initiated significant cracks in a fatigue 

test. Each is considered here in turn. 

Non-metallic inclusions in powder metallurgy alloys tend to be irregularly shaped although 

mostly convex. The GRC approach to characterizing defect sizes and shapes employed an 

idealized ellipsoid model to approximate the defect geometry, whether data resulted from HLS, 

metallography, or fractography. 

Calculation of Anomaly Distributions from Heavy Liquid Separation Data 

Several methods have been attempted to separate ceramic inclusions from a base metal powder, 

including water elutriation, froth flotation, and heavy liquid separation (Roth et al., 1994). 

The most successful of these methods has been heavy liquid separation (HLS), originally 

developed by Battelle for Wyman-Gordon, and developed further by GE and Wyman-Gordon for 

application to nickel-base powder alloys. In order to perform HLS analysis of anomaly sizes, the 

blended metal powder is mixed with a heavy liquid whose density is less than that of the 

superalloy powder but greater than the density of the inclusions. Low density ceramic inclusions 

float to the surface while higher density powder metal will sink. The ceramic inclusions can be 

separated from the solution and characterized (number and size) using standard visual 

examination techniques. A major concern with HLS is that the heavy liquid used 

(thallium malonate-formate solution) is highly toxic and requires careful handling procedures. 

The HLS method can generate extensive data on the size and shape of inclusions, and these data 

can easily be characterized in terms of standard statistical quantities. These inclusions are 

typically lying on a flat surface when inspected, and so it is easy to identify and measure the 

largest dimension of the inclusion, as well as the size orthogonal to this dimension in the plane of 

the flat surface. However, the third dimension of the inclusion (the out-of-plane dimension) is 

not so easily measured, unless the inclusion is manipulated to present the third dimension to the 

optical plane. 

In earlier GRC investigations of artificial seeds (Kantzos et al., 2003; Bonacuse et al., 2002), 

when no data were available for the third dimension, it was assumed that the relationship of the 

third dimension to the second dimension was the same as the relationship of the second 

dimension to the first dimension (i.e., the mean and standard deviation were a factor of 

0.667 times the size of the larger dimension), and it was further assumed that the correlation 

between the second and third dimensions was the same as between the first and second. 

However, GRC subsequently generated data for the third dimension by inclining the tape 

(to which the seeds were adhered) perpendicular to the original inspection direction 

(Bonacuse et al., 2006). The resulting data revealed that the assumed scaling of the third 

dimension (mean and standard deviation reduced by a factor of 0.667 relative to the second 

dimension) was remarkably accurate. It also revealed, however, that the correlation between the 

second and third dimensions was completely different than the first and second dimensions; the 
NASA/CR—2011-216977 7 

third dimension was, in fact, inversely correlated with the second dimension. This was 

rationalized by noting that the largest size the minimum dimension can achieve is limited by the 

largest hole in the sieve, and inclusions with smaller third dimensions are more likely to fall 

through the sieve. Therefore, inclusions with the largest second dimensions are more likely to 

have smaller third dimensions or they would not have made it through the sieve to begin with. 

The specific quantitative observations made by GRC about the relative dimensions of the 

anomalies are likely limited to the specific seeds used in this study. It is not clear if the 

qualitative observations are applicable to the broader class of inclusions in powder metal. 

Further data are needed to determine if this true. It also remains that additional effort would be 

required in practice to obtain quantitative data on the third dimension of inclusions using the 

HLS process. 

Another inherent limitation of the HLS approach to characterizing inclusion size distributions is 

that the inclusions are characterized at the powder stage, before the component is forged into its 

production shape. GRC (Kantzos et al., 2003) found that seed dimensions in forgings 

(as determined using metallographic sectioning) were significantly different from the original 

seed dimensions as characterized by HLS data, even after compensating for errors introduced by 

the sectioning method (discussed in the following section). The differences were attributed to 

seeds breaking up and the fragments becoming strung out along the extrusion direction during 

the forging process. The effective aspect ratios of the seeds in the forging were much larger than 

the aspect ratios of the initial seeds. Similar observations had been made for natural inclusions 

in an earlier study (Gabb et al., 2000). Kantzos (2006) indicated that Type I inclusions tend to 

break up more, while Type II inclusions tend to deform more, and that larger inclusions tended to 

break up more. 

Calculation of Anomaly Distributions from Sectioning Data 

Because seed or inclusion dimensions can be altered by the forging process, it may be preferable 

to characterize these dimensions with measurements taken from the final forged shape. This can 

be done with metallographic sectioning of the forging. Metallographic sections can be taken 

along different planes (hoop, radial, axial) in order to determine seed dimensions and shapes in 

each of these planes. 

However, this method also has inherent limitations, as Kantzos et al. (2003) has pointed out. 

One limitation is that the metallographic section will not, in general, pass through the maximum 

dimension of the defect, and so the resulting measurement will underestimate the actual size of 

the defect by an unknown amount. The distribution based on sectioning and the actual 

distribution tend to converge at the largest areas. Another limitation is that the same defect 

cannot be sectioned along three orthogonal planes, and so it is not possible to know the threedimensional aspect ratio of any single defect. 

Ghosn et al. (2003) presented a method to address the first limitation by “unfolding” the 

inclusion size distribution from the metallographic sectioning information. He adapted a 

published scheme to generate a 3D reconstruction of size distributions from 2D sections using an 

Expectation Maximization (EM) algorithm. The method is most commonly used for spherical 

bodies, but Ghosn also obtained equations for the special case of prolate spheroids of constant 
NASA/CR—2011-216977 8 

shape (a spheroid being an ellipsoid having two equal axes). Ghosn et al. demonstrated that the 

method worked successfully to predict the actual size distribution of spherical shot peen media 

and -80+100 mesh metal powder from 2D section data; the method was less successful with -270 

mesh metal powder. They also applied the same methods to the metallographic images of the 

largest seeds (fragmented particles) in forgings, documented in more detail by Kantzos et al. 

(2003). Evaluation of success was difficult because the “correct” answer was not known 

(the original seeds had severely broken up during forging). This method has not been 

demonstrated for general ellipsoids. 

As noted earlier, the forging process can cause significant fragmentation of defects 

(Kantzos et al., 2003). As a result, the “defect” observed in a metallographic image may in fact 

be a collection of neighboring defect fragments of varying size, proximity, and orientation. 

Therefore, GRC geometric models of defects in the forging employing an ellipsoidal form 

provide only an average shape reflecting the overall result of the forging process. 

The characteristic length of the major or minor axis of the “ellipse” in the sectioning plane 

generally represents the span of a collection of neighboring fragments, including some semiarbitrary criterion for the maximum distance between neighboring particles that should be 

considered constituents of the same “defect.” In this spirit, the major axis is the longest 

dimension of this particle collection, and the minor axis is the approximate width of the 

collection along a direction orthogonal to the major axis. 

This method requires metallographic sampling of relatively large areas in order to generate 

adequate data. The GRC studies of seeded material involved sampling 3500 mm2

 to 10,000 mm2

of surface area (Kantzos et al., 2004), but they noted that much larger areas would have to be 

sampled for unseeded material due to the significantly lower defect occurrence rates. 

Calculation of Anomaly Distributions from Fractography Data 

A third possibility (Kantzos et al., 2004) is to determine anomaly distributions from inclusion 

measurements on the fracture surface itself. These measurements correspond to defects that 

actually caused fracture, not the general population of all defects in the material. The total 

number of defects thus characterized would generally be much smaller, which would present 

additional statistical challenges. Dimensional information on other orthogonal planes is 

generally not available. 

Fractographic approaches do not give any real indication of how many inclusions were present in 

the tested volume. It is difficult to extrapolate from the extreme defect size distributions 

obtained from fatigue test coupons to those expected in a turbine rotor or a fleet of turbine rotors. 

Kantzos et al. (2004) noted from their seeded studies that the size distribution for crack-initiating 

defects was larger than the initial (HLS) distribution for their seeded material, since fatigue was 

an extreme process that tended to favor the larger defects in the population. The opposite was 

true for unseeded (relatively clean) material: the size distribution for the crack-initiating defects 

was smaller than the initial HLS size distribution. The unseeded material was so clean 

(inclusions were so rare) that a large inclusion was unlikely to be found in the coupon volume 

tested even though some large inclusions were present in the original powder. 
NASA/CR—2011-216977 9 

Comparative Remarks 

Kantzos et al. (2004) concluded that all three of these methods—HLS, quantitative 

metallography, and quantitative fractography—were useful and capable of quantifying powder 

cleanliness to various degrees. They suggested that a combination of all three methods was 

desirable, although they did not propose any methods for such a combination. The HLS method 

is the only method that can provide an accurate size distribution as well as the defect occurrence 

rate. However, the HLS distribution needs to be adjusted in order to take into account the extent 

of defect fragmentation during forging. 

Multiple Anomaly Types 

All of these observations and methods are based on the assumption that only one type of defect is 

causing fatigue failure and therefore needs to be characterized. The cited GRC research did not 

explicitly address the issue of multiple anomaly types. All work was focused on non-metallic 

inclusions (and artificial seeds that were intended to emulate these natural inclusions). 

The seeded specimens always failed from inclusions (seeds). 

Earlier GRC work on unseeded U720 did observe multiple failure modes, including some 

systematic effects that favored one failure mode over another. Gabb et al. (2000) performed 

unseeded U720 tests at 538C (1000F) and found that strain ratio R = 0 tests predominantly 

(but not always) failed from cracks at elongated clusters of inclusions, while R = -1 tests usually 

failed from cracks emanating from large sheared grains (“grain facet failures”) or at pores, with a 

few inclusion failures also occurring. The total strain range in the R = 0 tests (0.75%) was 

slightly smaller than the total strain range in the R = -1 tests (0.90%). No quantitative treatment 

of different failure modes was offered. 

Later GRC testing (Gabb et al., 2002) at 650C (1200F), three strain ratios, and a wider variety 

of strain ranges found that unseeded specimens failed at cracks initiated at two different types of 

inclusions as well as micropores. Micropore-initiated failures appeared to be rare. 

Initiation sites tended to be located at or near the surface in tests at high strain ranges and internal 

for lower strain ranges. Kantzos et al. (2004) reported that the unseeded fatigue test coupons 

failed from inherent inclusions as well as other flaws such as porosity and grain facets. 

Kantzos (2006) indicated tendencies toward pore failure at lower temperatures and higher strains, 

and towards inclusion failures at high temperatures and lower strains. 

2.1.2 Calculation of Fracture Probability

The second major part of the reliability determination is the calculation of the probability of 

fracture, given that a defect does exist. This part includes several steps, including the 

determination of the initial crack size due to the inclusion, the number of fatigue cycles required 

to form a growing crack at the defect, and the number of fatigue cycles required to grow this 

crack to failure. The fatigue life can also be influenced by residual stress effects and 

environmental effects. 
NASA/CR—2011-216977 10 

Determination of the Initial Crack Size 

The size and shape of the crack-initiating inclusion must be translated into the size and shape of 

the initial fatigue crack in order to perform a fatigue crack growth (FCG) calculation. 

Bonacuse et al. (2002) originally hypothesized that the inclusion with the largest cross-section 

orthogonal to the load would be the most likely to initiate a propagating fatigue crack. 

However, this assumption led to predicted distributions of initial crack area that were 

consistently smaller than what was revealed on the fracture surfaces. Bonacuse et al. (2006) 

subsequently proposed that the projected area of the inclusion on the plane orthogonal to the load 

might be a better predictor of the inclusion’s crack initiating potential, noting that other 

researchers had come to the same conclusion. 

Since the defects are being modeled as arbitrary ellipsoids, their orientation relative to the load 

plane is another important factor in determining the size and shape of the resulting fatigue crack. 

Bonacuse et al. (2002) reported that the extrusion and forging processes tend to orient the 

inclusions along the high strain directions. In forgings, sections taken from orthogonal directions 

seem to show a preferential orientation in the plane of the forging. Bonacuse et al. (2002) 

elaborated that there was a correlation between the magnitude of the processing strain and the 

degree of orientation: larger strains lead to more alignment. In addition, larger inclusions 

seemed to be more likely to orient themselves to the flow direction than smaller inclusions. 

Kantzos et al. (2003, 2004) provided a more detailed report, presenting detailed forging maps 

with data indicating that the orientation of the inclusions in the forging was not random, but was 

correlated with the direction and magnitude of the metal forming strains in the forging process. 

Both break-up and orientation effects were stronger in the transition and rim regions of their 

simple disc shapes, where effective forging strains were largest. They speculated that these 

effects would be more significant in large forgings where the material flow and deformations are 

appreciably larger than in the pancake forgings used in the GRC study. 

Detailed studies of these orientation correlations were not performed, so Bonacuse et al. (2006) 

optimized an assumed correlation to fit the available data. Combining this information with the 

projected area criterion mentioned earlier, Bonacuse et al. (2006) simulations showed reasonable 

success in predicting the distributions of crack-initiating defect sizes observed in fatigue tests. 

Gabb et al. (2000) had earlier observed some statistical correlations between smooth specimen 

fatigue life and original location of the specimen within a complex forging shape for unseeded 

material, although the correlation coefficients were relatively low. It is not clear if these 

correlations might have been due to defect issues (perhaps arising from different forging strains 

at different locations in the forging) or due to base microstructure issues (such as grain size). 

This question was not addressed in the Gabb et al. work. 

Most of the failure cracks in the GRC seeded studies initiated at inclusions on the specimen 

surface. Kantzos et al. (2004) reported that surface initiations frequently favored inclusions that 

were barely sectioned by the surface and whose bulk was mostly contained within the specimen. 

Internal initiations typically occurred at much larger inclusions than surface initiations. 
NASA/CR—2011-216977 11 

This was as expected, due to the much larger volume sampled in the interior compared to the 

surface, and perhaps also because much smaller inclusions in the interior might not initiate 

growing cracks. 

Given that defects are being modeled as arbitrarily oriented ellipsoids with three independent 

dimensions, and given that the intersection and/or projection of these ellipsoids with sectioning 

planes, load planes, and external surfaces defines significant area values with direct implications 

for life prediction, it is useful to be able to calculate these areas quickly and accurately. 

Bonacuse (2008) has derived a set of mathematical models for these parameters, including the 

intercepted area of a randomly sectioned ellipsoid, the dimensions and orientation of the 

intercepted ellipse, the area of a randomly oriented sectioned ellipse, the depth and width of a 

randomly oriented sectioned ellipse, and the projected area of a randomly oriented ellipsoid. 

Without these mathematical models, computationally expensive search algorithms would be 

required to compute these parameters, and this might not be feasible for probabilistic simulations 

of fatigue life and component reliability. 

Calculation of Crack Formation Life 

The nature and location of the initiating defect appears to have a significant impact on crack 

formation life. 

Early GRC studies (Gabb et al., 2000) with unseeded test coupons, where 75% of the cracks 

initiated sub-surface, concluded that most tests spent over 90% of life in the initiation phase 

(including both pores and inclusions). These estimates did not consider the possibility of slower 

FCG rates for embedded crack (in the absence of environmental effects), as discussed further 

below. This might mean that the formation life fraction was smaller than originally inferred. 

More recent GRC seeding studies (Kantzos et al., 2004), where most failures were initiated at the 

surface, observed little or no formation life for surface-connected inclusions. Crack formation at 

subsurface locations was not studied in detail in this work. In these seeded tests, the tendency 

for surface initiation was stronger at lower strain ranges and higher R ratios. 

Gabb et al. (2002) found that prior exposure at elevated temperatures for extended periods of 

time shifted the failure locations from internal to surface, due to environmental attack on the 

surface. An increase in total scatter was attributed to the competing failure modes: internal 

initiations at inclusions or large grains (longer lives), or surface initiations at an environmentaffected surface layer (shorter lives). Proper analysis required identifying and separating these 

two modes. 

GRC has not proposed any general physics-based models to predict crack formation life as a 

function of defect size and shape, temperature, strain range, and mean stress. Gabb et al. (2003) 

developed a non-linear regression model describing the relationship between total fatigue test life 

and a combination of test control parameters (strain range and strain ratio) and defect 

characteristics (mean log inclusion areas and inclusion area densities) for a specific set of test 

results. However, this model did not differentiate between crack formation life and crack growth 

life. 
NASA/CR—2011-216977 12 

Because crack formation life has not been isolated from crack growth life in GRC analyses, no 

explicit attention has been given to scatter in crack formation life. 

Detailed studies of specimen surfaces using a special rotational stage in the scanning electron 

microscope (Telesman et al., 2002, 2006) revealed that cracks were initiated at nearly all 

surface-breaking defects being monitored. However, not all initiated cracks grew to failure. 

The cracks that did grow to failure often formed at defects with mapped surface sizes that were 

actually smaller than the average of all monitored inclusions. This was interpreted as evidence 

of an “iceberg” effect: the most deleterious inclusions may be surface-breaking inclusions with 

much larger dimensions beneath the surface, as noted also by Kantzos et al. (2004). 

Calculation of Crack Growth Life 

The calculation of FCG life is a critical step in prediction of total fatigue life, particularly for 

situations where crack formation life is negligible. Because inclusions can be very small, the 

early growth of fatigue cracks formed at defects falls into the “small-crack” regime. One of the 

key issues is whether the growth of small cracks near defects follows the general trends of 

conventional long-crack growth behavior, or exhibits classical “small-crack” effects. 

Telesman et al. (2006, 2007) showed crack growth data that, at first, appeared to exhibit classical 

small-crack effects: increased scatter, accelerated growth rates relative to the extrapolated Paris 

line, and growth at nominal K values below the large-crack threshold. However, much of this 

behavior was characterized as “pop-in”: cracks appeared very quickly, but exhibited decreasing 

da/dN values and often arrested completely. No quantitative models were proposed by GRC for 

pop-in behavior, but machining residual stresses, crack closure, localized residual stresses due to 

coefficient of thermal expansion mismatch, or highly localized stress concentrations due to 

irregularly shaped inclusions (Kantzos et al., 2004) were offered as potential explanations. 

When Telesman et al. screened out this crack growth during the first 250 cycles, some of the 

apparent “small-crack” trends in the data vanished. The remaining data often agreed with the 

large-crack Paris line. However, some data still fell below the large-crack threshold, and some 

data (especially at R ~ 0) exhibited slightly faster growth than the Paris line. It was speculated 

that this might be due to reduced crack closure, since FCG rates were still bounded by high-R

large-crack trend lines. Further study of the data is needed to determine how well the smallcrack behavior is described by large-crack models. 

As noted earlier, Telesman et al. (2002, 2007) and Kantzos et al. (2004) reported that most 

monitored inclusions initiated cracks for all test conditions, but only a few cracks grew to failure. 

It is not clear why some cracks arrest and some grow. The relative fraction of active inclusions 

was higher at higher strain ranges and higher R-ratios.

Relatively little attention was given to large-crack growth, aside from the generation of basic 

property data. Telesman et al. (2006) used a Walker model to correlate 1200°F long-crack FCG 

data generated with Kb-bar (surface crack) specimens at four negative and three positive R-ratios. 

Different Walker exponents were used at positive vs. negative stress ratios. Data were only 

shown at R = 0.05, 0.5, -0.25, and -1. No attention was given to scatter in large crack growth 

rates, since this is not usually a significant source of variability in this problem. 
NASA/CR—2011-216977 13 

Bonacuse et al. (2003) reported a life prediction study based on simulations of inclusions with 

random size and shape distributions. Predicted seed maps (including orientation and intercepted 

surface areas) generally agreed with experimental observations. Fatigue life was calculated 

based on fatigue crack growth with no threshold or initiation considerations. The ten largest 

surface connected inclusions and all embedded inclusions in the simulated specimens were 

considered, with the lowest calculated life assumed to give the specimen life. Room temperature 

air FCG properties were used for embedded cracks, while elevated temperature air properties 

were used for surface cracks (see remarks below about environmental effects). The simulations 

exhibited mixed success in predicting observed behavior: conservative in some cases, 

non-conservative in others, and generally underestimating the observed life scatter. 

Predictions of FCG lives for the seeded tests (for both surface and internal cracks) based on large 

crack growth rate models (assuming negligible crack formation life, and taking the size of the 

defect where the failure crack originated as the initial crack size) met with mixed success in work 

reported by Telesman et al. (2006). Predictions were reasonably accurate in some cases 

(especially for some larger seeds at R = 0), but very conservative in other cases. 

These calculations did not address small-crack behavior (including pop-in, decreasing da/dN, or 

crack arrest), machining residual stresses, or environmental effects. These latter two effects are 

discussed in subsequent sections. 

Ghosn et al. (2005) documented some statistical analysis of the FCG data and used the analysis 

to perform a Monte Carlo simulation study. The number and size of the inclusions in each 

simulated bar was randomly generated from appropriate distributions. They lumped small-crack 

and large-crack data together and used simple linear Paris relationships to model the data. 

They developed empirical equations to describe the percentages of cracks with zero or negative 

growth as a function of the maximum stress. They also developed statistical expressions for the 

deviations from the Paris line, which were larger at smaller Walker-corrected K values 

(consistent with smaller crack sizes). They presented an error analysis to claim that 

measurement errors were not a significant source of uncertainty. The simulations were generally 

successful at reproducing the overall trends in the observed life behavior, although some details 

were not satisfactorily predicted. 

Residual Stress Effects 

GRC has given considerable attention to surface residual stress effects in P/M alloys, but has not 

studied bulk residual stress effects (arising, for example, from forging or heat treating). 

Effects of Residual Stress on Fatigue Life 

Telesman et al. (2007) observed surface crack initiation very early in low-cycle fatigue (LCF) 

life at 1200°F, R = 0, and various strain ranges, for both as-machined and one shot-peened 

condition. Since the significant compressive residual stress induced by the shot peening (SP) did 

not delay surface crack initiation, but did increase total life for some test conditions, they 

concluded that residual stress affects crack growth rates, but not crack formation lifetimes. 
NASA/CR—2011-216977 14 

Barrie et al. (2008) studied the effects of SP on fatigue behavior at 427C and 650C for multiple 

strain ranges from 0.6% to 1.2% and for strain ratios of -1, 0, and 0.5. An initial screening study 

of different SP conditions found that all SP conditions considered significantly improved life for 

the 54 μm seeded material, by shifting the failure from surface to interior. However, only SP to 

high intensity resulted in fatigue life improvement for the 122 μm seeded material, probably 

because the depth of the residual compressive stress was on the same order as the seed sizes. 

Further studies by Barrie et al. (2008) with the selected high intensity/low coverage SP 

conditions (8A/200%) found some improvements in fatigue life at low strain ranges and low 

strain ratios. However, no improvements were observed at Rε = 0.5 or larger strain ranges. 

The larger strain ranges were observed to cause significant residual stress relaxation during the 

test. No explanation was given for the lack of effect at high strain ratio, but this would be 

consistent with a FCG explanation since the high R values would counteract some of the effects 

of the compressive residual stress from a crack closure standpoint. Barrie et al. did not study 

initiation or growth phases separately. 

Relaxation of Residual Stresses 

Residual stresses induced by shot peening or other surface enhancement methods can relax 

during thermal exposure and/or fatigue cycling. 

Gabb et al. (2004) investigated thermal exposure effects in retired T700 disks manufactured from 

P/M alloy René 95. Residual stresses were measured in the as-received (full lifetime) disks, and 

then following thermal exposure at 593°C, 650°C, and 704°C for 1 hr, 24 hr, and 500 hr. 

Significant relaxation occurred. An empirical regression model was developed to describe the 

effects of T, t, initial cold work, and initial residual stress on stress relaxation and cold work 

relaxation. Stress relaxation was accentuated for combinations of high temperature with long 

time, as well as combinations of high initial compressive residual stress with high temperature, 

time, or initial cold work. The equations are not quantitatively predictive for other materials or 

conditions. Limited tensile specimen testing of mechanical stress relaxation with time and 

temperature found qualitatively similar trends, but no quantitative correlations were established. 

Telesman et al. (2007) presented limited data on measurements of shot peening (8A, 200%) 

residual stress on specimens before and after LCF testing at 1200°F (650C), R = 0, Δε = 0.8%. 

Significant residual stress relaxation occurred. Another limited data set showed nearly complete 

relaxation of low plasticity burnishing (LPB) residual stress following LCF testing at 1200°F, R

= 0.5, and Δε = 0.8%, although the conditions were cited as being “overly aggressive” to 

evaluate the effect of LPB on LCF life. One data set showed significant relaxation of LPB 

residual stress following 17 hrs at 1200°F (no fatigue cycling), suggesting that the relaxation 

observed in the LCF tests may have been primarily due to thermal mechanisms. LCF lifetimes 

with LPB were similar to untreated lifetimes. No models for residual stress relaxation were 

proposed. 

Kantzos (2006) has presented data showing the effect of LCF cycling at 1200°F, R = 0, 

Δε = 0.8% on SP residual stress for four different shot peening intensity-coverage combinations. 

All four conditions exhibited significant relaxation. No models were proposed. 
NASA/CR—2011-216977 15 

Machining Residual Stresses 

Telesman et al. (2002, 2006) speculated that low stress grind machining residual stresses 

(which are compressive on the surface) might be related to observed microcrack behavior 

(decreasing crack growth rates and crack arrest), although no specific mechanisms were 

proposed. Simple predictions (neglecting residual stresses) of surface crack growth rates at R = 0, 

Δε = 0.6% significantly (10×) overpredicted observed crack growth rates, but agreement was 

good at Δε = 0.8%. Telesman et al. (2002) noted that the higher strain ranges created enough 

plasticity to wash out the shallow machining residual stresses. Limited measurements of surface 

and near-surface residual stresses showed some relaxation in initial (machining-induced) residual 

stresses after only 250 LCF cycles. Limited test data were also presented showing that a single 

1% overload increased the FCG rate and decreased the fatigue life for specimens with large 

(T64) defects. It was speculated that this effect might be due to residual stress reduction, but 

also perhaps due to potential inclusion debonding. Single 1.2% overloads did not affect the 

fatigue life for specimens with smaller 54 μm inclusions. Another group of specimens was 

electropolished to remove residual stresses, but the electropolishing (EP) step also preferentially 

removed matrix material surrounding the inclusions. The EP step resulted in a significant 

reduction in LCF life (but with a smaller number of inclusions initiating cracks). Another series 

of tests with thermal exposure in vacuum to reduce residual stress also reduced the fatigue life. 

Environmental Effects 

As noted earlier, Gabb et al. (2002) observed that prior exposure of unseeded fatigue specimens 

to elevated temperatures (650°C to 705°C) for extended periods of time (100 hr to over 1000 hr) 

caused a significant reduction in total fatigue life (up to 70 percent) and increased the scatter in 

life compared to unexposed levels. Fatigue tests were conducted at 650°C. The differences were 

attributed to a shift in failure mode from subsurface to surface initiation due to environmental 

attack on the surface (i.e., a significant decrease in surface initiation life due to environmental 

effects, not related to surface inclusions). Unexposed fatigue tests at 704°C and a 650°C fatigue 

test with a 1 minute cyclic dwell period also exhibited surface failure modes consistent with the 

exposed tests. 

Telesman et al. (2006) noted that proper modeling of subsurface crack growth required crack 

growth models for vacuum conditions (in the absence of environmental effects). A survey of the 

literature found only isolated vacuum FCG data (Paris regime) at high temperature (600°C and 

700°C), but these data generally agreed with available room temperature air data (FCG rates 

were generally slower than elevated temperature air data). The possibility of using room 

temperature air data to model surface cracks was suggested. 
NASA/CR—2011-216977 16 

2.2 A Summary of Current DARWIN Capabilities Relevant to P/M Lifing 

SwRI has been actively developing technology and software to support probabilistic damage 

tolerance analysis of gas turbine engine rotors since 1995. One of the key outcomes of this work 

has been the DARWIN (Design Assessment of Reliability With INspection) computer code 

(Millwater et al., 2000; Leverant et al., 2004; McClung et al., 2004a; McClung et al., 2009). 

DARWIN integrates finite element models and stress analysis results, fracture mechanics models, 

material anomaly data, probability of anomaly detection, and uncertain inspection schedules with 

a user-friendly graphical user interface (GUI) to determine the probability-of-fracture of a 

high-energy rotating component as a function of operating cycles with and without inspections. 

The SwRI work was originally motivated by an uncontained rotor fracture incident at Sioux City, 

Iowa, in 1989. As a result of this incident, the Federal Aviation Administration (FAA) requested 

that the aircraft engine industry, through the Aerospace Industries Association (AIA) Rotor 

Integrity Sub-Committee (RISC), review available techniques to determine whether a damage 

tolerance approach could be introduced to produce a reduction in the rate of uncontained rotor 

events. During the development of this probabilistic damage tolerance approach, it became 

apparent to RISC that the capabilities and effectiveness of the emerging technology could be 

significantly enhanced by further research and development. SwRI, in partnership with four 

major U.S. engine manufacturers and with guidance from RISC, proposed and was awarded a 

series of FAA grants beginning in 1995 to address identified shortfalls in technology and data. 

Initial DARWIN capabilities were focused on the hard alpha titanium problem, which had 

caused the Sioux City incident. A general framework was established to address inherent 

material defects that could occur anywhere in an axisymmetric body, as described by a 2D finite 

element model. Another notable uncontained rotor incident in Pensacola, Florida, in 1996, was 

caused by surface damage at machined circular holes, and as a result, a second thrust of 

DARWIN development emerged. A second general framework was established to address 

induced material defects arising during manufacturing or maintenance processes, focusing on 

damage occurring at the component surface, and incorporating 3D finite element models. 

With direct guidance from the FAA and the OEM Steering Committee, DARWIN has been 

tailored to address specific advisory circulars issued by the FAA, including AC 33.14-1 for 

titanium hard alpha, AC 33.70-2 on circular holes in all materials, and the more general AC 

33.70-1 for damage tolerance of all engine life-limited parts. The FAA has indicated that 

DARWIN is an acceptable means of compliance with these advisory circulars, and DARWIN 

has been employed in the formal FAA certification process for numerous engines. 

The FAA also directed that some basic DARWIN capabilities be developed to address 

superalloys with high anomaly occurrence rates, including powder metallurgy alloys. 

Some capabilities have already been implemented in DARWIN, although this work has been 

limited due to other, more pressing FAA and RISC priorities. 

The purpose of this document is to summarize the existing capabilities in DARWIN that are 

relevant to the P/M lifing problem. The document follows the general outline of the summary of 

research in P/M lifing conducted at NASA Glenn Research Center. 
NASA/CR—2011-216977 17 

2.2.1 Determination of Anomaly Distributions

DARWIN currently has no capability to derive or generate anomaly distributions. A limited 

number of distributions are provided in a special DARWIN library for specific anomaly types: 

hard alpha titanium anomalies (as a function of material vintage and billet/forging inspection 

sensitivity), and surface damage at machined circular holes. The user has the option of providing 

additional anomaly distributions as files in predetermined formats that DARWIN can read. 

The user cannot modify these distributions inside DARWIN itself. 

For hard alpha titanium, the anomalies are assumed to have a spherical shape. The probability 

distribution of these anomalies is described in a two column format consisting of anomaly area 

and associated number of anomalies (exceedance format). For general materials such as P/M 

alloys, the anomalies are assumed to have an ellipsoidal shape. The probability distribution of 

these anomalies is based on the ellipsoid major axis (exceedance format), anomaly aspect ratios 

(normal or lognormal distributions), and three anomaly orientation angles (uniform or normal 

distributions). 

2.2.2 Calculation of Fracture Probability

Determination of the Initial Crack Size 

For titanium alloys with hard alpha anomalies, the initial crack dimensions are based on the 

anomaly area described in the anomaly distribution. The anomaly area is converted to crack 

dimensions assuming that the initial crack has a circular shape for embedded cracks, a halfcircular shape for surface cracks, and a quarter-circular shape for corner cracks. For general 

materials, the initial crack dimensions are based on either the intersection or projection of the 3D 

anomaly with the fracture plane. Similar as for titanium hard alpha, the anomaly dimensions are 

adjusted when the crack is located near a surface or corner of a component. In addition, an initial 

or “production” inspection can be applied to general materials to remove components with 

anomalies that are detected prior to placement in service (this inspection cannot be applied to 

titanium materials, because the anomaly distributions for these materials were calibrated to 

components that had already been placed in service). 

Calculation of Crack Formation Life 

DARWIN has no built-in models to calculate the cycles required to form a fatigue crack at a 

material anomaly. However, DARWIN does facilitate user-provided crack formation models. 

A Crack Formation Module is provided that allows users to link their own specific crack 

formation algorithms with DARWIN, as shown in Figure 1. The Crack Formation Module and 

associated data are developed and maintained by the individual user. During run time, DARWIN 

communicates with the Crack Formation Module to obtain the crack formation life associated 

with an anomaly having specified temperature and applied stress values. 
NASA/CR—2011-216977 18 

Figure 1. Crack formation computations are performed via a 

user-supplied crack formation module. 

The Crack Formation Module is executed as a separate program that receives input from 

DARWIN. The following data are currently passed from DARWIN to the Crack Formation 

Module: global coordinates (r, z, theta), distance to the nearest free surface of the fracture 

mechanics plate, size of anomaly (a, c) in crack plane and orientation of plane, six components 

of stress at the anomaly location for each load pair, relaxed stresses (if shakedown is performed 

in DARWIN), and temperature at the anomaly location for each stress pair. The maximum 

temperature of each stress pair is provided to the formation module, and the crack plane is 

limited to hoop, axial, and radial stress planes. These data are used by the Crack Formation 

Module for crack formation life prediction. The following data are passed from the Crack 

Formation Module to DARWIN: crack formation life for entire load history, and crack size at 

end of crack formation life (a, c). 

Linking two independent computer programs together is a nontrivial task, particularly when 

multiple platforms and associated operating systems must be supported. The approach used is to 

create a third program that is linked separately with the other two programs. A formation 

Application Program Interface (API) was developed for this purpose, in which DARWIN and the 

Formation API are compiled as stand-alone programs. The Crack Formation Module is also 

compiled as a stand-alone program. During run time, data is passed between DARWIN and the 

Crack Formation Module via the Formation API. This approach does not require the sharing of 

source code, nor does it require the creation of ASCII files to pass data. In the future, the 

interface between DARWIN and the Crack Formation Module can easily be expanded to include 

additional variables associated with enhanced crack formation life models. 

Calculation of Crack Growth Life 

DARWIN has a comprehensive capability for fatigue crack growth analysis. The Flight_Life 

fracture mechanics module contains libraries of stress intensity factor (SIF) solutions and 

multiple models for FCG rate calculations. SIF solutions include original univariant and/or 

bivariant weight function solutions for elliptical embedded cracks, semi-elliptical surface cracks, 
NASA/CR—2011-216977 19 

quarter-elliptical corner cracks, and straight through cracks in plates or at holes (Enright et al. 

2003; McClung et al., 2004b; Lee et al., 2008). Available crack growth equations include Paris, 

bilinear Paris, sigmoidal, hyperbolic sine, and the NASGRO 4.0 equation, as well as general 

capabilities to accommodate tabular data. Available mean stress models include Walker and 

closure models, as well as Walker interpolation. Interpolation of crack growth rates at different 

temperatures is also available. Scatter or uncertainty in calculated FCG rates is described by a 

life scatter factor (a log-normal distribution with a user-specified coefficient of variation) that is 

uniformly applied at all crack sizes. 

DARWIN currently has no explicit treatment of small-crack behavior, except that the standard 

NASGRO crack growth equation in DARWIN does include a size-dependent threshold term that 

reduces Kth at very small crack sizes, following the El Haddad functional form. 

Some investigations of alternative small-crack formulations are underway at SwRI in the current 

FAA grant, and this could potentially lead to a new small-crack model being implemented in 

DARWIN. 

Static and fatigue testing of coupons with either artificial or natural hard alpha defects gave 

results indicating the potential influence of local residual stresses on fracture behavior 

(McKeighan et al., 2001). Experimental measurements of the coefficient of thermal expansion 

of the hard alpha (titanium nitride) inclusion and the base titanium alloy indicated a large enough 

difference to induce significant local residual stresses under some conditions (Laz et al., 2003). 

Idealized models were developed to estimate these residual stress fields and their effect on crack 

initiation and growth behavior, and they agreed satisfactorily with the observed cracking. 

These models have not been implemented in DARWIN, however. 

Residual Stress Effects 

DARWIN currently has a basic capability to superimpose a user-provided tabular residual stress 

gradient with the 2D finite element model stress results. The resulting combined stress gradient 

is used to calculate K and R; the superposition of the residual stress does not change K but it 

does change R, which will change the predicted FCG rate. Special optimum point spacing 

algorithms facilitate the accurate and efficient superposition of stress gradients with significantly 

different length scales. 

This capability is currently limited to semi-elliptical surface cracks with univariant stress 

gradients in 2D axisymmetric models. Work is planned under the current FAA grant to extend 

the superposition capability to some additional crack types and also to 3D finite element models. 

DARWIN currently has no capability to calculate relaxation or redistribution of residual stresses 

due to thermal exposure, mechanical loading, fatigue cycling, or crack growth, and no facilities 

are provided to allow the user to update the residual stress inputs as cycling proceeds. 

A comprehensive literature review of this topic was conducted under one of the FAA grants and 

has been published (McClung, 2007). 
NASA/CR—2011-216977 20 

Environmental Effects 

DARWIN currently permits the user to supply different crack growth properties 

(even different crack growth equation types) for embedded cracks versus all other crack types 

(which are all surface-breaking). This was specifically designed to facilitate the specification of 

properties obtained under vacuum conditions for embedded cracks, while other crack types 

would use customary air properties. 

The earlier FAA grants led by SwRI also generated fatigue crack growth properties under 

appropriate vacuum conditions at relevant temperatures for several selected rotor alloys, both 

titanium and nickel-based superalloys (McClung et al., 1999). This effort included the 

generation of vacuum FCG properties for U720 using material provided by NASA GRC 

(material identical to that used in the GRC studies of P/M lifing). U720 data were generated 

using Kb-bar (surface crack) specimens at 800F for R = 0.05 and R = 0.5. These data have been 

provided to GRC, but they have not yet been published. 

DARWIN includes the ability to extract from finite element models not only the full stress tensor 

throughout the model, but also the temperature distributions at each time point. DARWIN also 

permits the user to provide different crack growth properties at different temperatures. 

During complex thermal-mechanical cycling, DARWIN currently calculates FCG rates using the 

properties at the temperature corresponding to the time point at which the maximum stress in that 

particular load pair (cycle) occurs, performing temperature interpolation as needed. Work is 

underway in the current FAA grant to implement new capabilities for time-dependent crack 

growth (e.g., fatigue cycling with dwell periods) as well as more advanced thermal-mechanical 

fatigue crack growth algorithms. 

Reliability Calculations 

Anomalies in gas turbine engine materials can lead to fracture. The fracture event is dependent 

on (1) the presence of an anomaly, and (2) the formation and growth of a crack that exceeds the 

fracture toughness of the material before the design life has been reached. The occurrence 

probability of an anomaly P(dj) can be measured by counting the number of anomalies of various 

sizes on the surface or within the volume of a component. P(dj) is typically modeled as a Poisson 

point process (Roth, 1998; Haldar and Mahadevan, 2000): 

    exp  !

j

i i

j i i

V

Pd V

j

   (1)

The likelihood of fracture failure can be estimated using fatigue crack growth modeling. 

Since this value is dependent on the presence of an anomaly of a specified initial size, it is 

commonly expressed as a conditional probability P Fd  , the probability of fracture given that an 

anomaly is present. P Fd   is dependent on a number of random variables related to the applied 

stress values and the fatigue nucleation and growth processes (Wu et al., 2002). 
NASA/CR—2011-216977 21 

The occurrence of an anomaly in a component is a relatively rare event for some materials, such 

as hard alpha in titanium (Aerospace Industries Association Rotor Integrity Subcommittee, 1997; 

Leverant et al., 2003). For these materials, the probability of more than one anomaly is assumed 

to be negligible (Wu et al., 2002). The probability of fracture at a specified location pi is 

therefore based on the occurrence of a single rare anomaly P d 1  and the probability of fracture at 

location i given that a single anomaly is present PFd  i 1 : 

pi i  PFd Pd  1 1    (2)

On the other hand, components made from some rotor-grade alloy materials 

(including P/M alloys) may contain hundreds or even thousands of anomalies. For these 

multiple anomaly materials, the conditional probability PF d  is dependent on the number of 

anomalies present, and is expressed as PFd  i j. Since cracks can form at the location of any 

anomaly, the failure of the component can be considered as a weakest-link (series) system 

consisting of the fracture failure associated with each discrete number of anomalies. 

For conservative risk estimates, PFd  i j is set to the maximum probability of fracture 

associated with any one of the j anomalies present. The resulting system model is expressed as 

(Enright and Huyse, 2006): 

  1 1   j

PFd PFd ij ij       (3)

The influence of the number of anomalies j on PFd  i j is illustrated conceptually in Figure 2. 

The anomaly occurrence probability is also influenced by the number of anomalies in a 

component. This relationship is reflected in the Poisson model of Eqn. (1). The dependence of 

PFd   i j

 and P j d on the number of anomalies j is shown conceptually in Figure 3. 

The members in Figure 3 represent the discrete number of anomalies that could be present in a 

given component which are events that have a mutually exclusive relationship (i.e., it is not 

possible for a selected component to have only two anomalies and only three anomalies 

simultaneously). i p is equal to the sum of the failure probabilities associated with each discrete 

number of anomalies (Hoyland and Rausand, 1994). To obtain i p , Eqns. (1) and (3) are 

substituted into Eqn. (2) for each member and summed over the total number of members: 

        1

1 1 exp !

j n j i i

i i j ii

j

V

p PFd V

j



 

             

 (4)
NASA/CR—2011-216977 22 

PFd  i j 

N 

j = 2

j = 1

j = n

PFd  i j 

N 

j = 2

j = 1

j = n

Figure 2. The conditional probability of fracture failure P  F d of 

multiple anomaly materials is dependent on the number of anomalies j 

at a specified location in the component. 

System PF di j Pd j

j 1

j  2

j n 

PFd  i j2 

j d 1

PDF P d j1  PDF

j d 2

PDF P d  j2

j d n

PDF PFd  i jn  P d jn 

PFd  i j1 

PDF

PDF

System PF di j Pd j

j 1

j  2

j n 

PFd  i j2 

j d 1

PDF P d j1  PDF

j d 2

PDF P d  j2

j d n

PDF PFd  i jn  P d jn 

PFd  i j1 

PDF

PDF

Figure 3. For multiple anomaly materials, both the probability of fracture 

given an anomaly PFd   i j and the anomaly occurrence probability P d  1

are dependent on the number of anomalies present. 
NASA/CR—2011-216977 23 

Eqn. (4) requires specification of the number of anomalies n. It is shown in Enright et al. (2005) 

that Eqn. (4) reduces to: 

pi ii i    1 exp  V PFd  1    (5)

Note that Eqn. (5) is based on the average number of anomalies present at location i ( i i V ). 

The probability of fracture is also dependent on the location of an anomaly within a component. 

The component is often discretized into a number of subregions called zones, and the probability 

of fracture is assessed for anomalies located within each zone (McClung et al., 2004a). 

Since component failure occurs when there is a failure within any zone, the component is 

modeled as a series system of zones with a probability of fracture that can be expressed as: 

1 2  

1

[ ]1 1

m

F mi

i

p PF F F p



        (6)

If multiple anomalies are present in each zone, pi can be estimated for each zone using Eqn. (5). 

Substitution of Eqn. (5) into Eqn. (6) yields the following expression for multiple anomalies at 

multiple locations (Enright and Huyse, 2006): 

  1

1

1 exp

m

F ii i

i

p V PFd



        (7)

Multiple anomaly types may also be present in one or more regions of a component. 

The multiple anomaly types can be modeled as additional members of the series system of zones. 

This is illustrated conceptually in Figure 4, where it is shown that components with multiple 

anomaly types and locations can be modeled using several nested series systems to represent the 

relationship among the various failure events. 
NASA/CR—2011-216977 24 

j 1

j  2

j  n

Anomaly

Location

Anomaly

Type

Number of

Anomalies

Pore

NMP

Other

j 1

j  2

j  n

Anomaly

Location

Anomaly

Type

Number of

Anomalies

Pore

NMP

Other

Figure 4. For components with multiple anomalies of multiple types at multiple locations, 

the probability of failure can be modeled using several nested series systems 

to represent the various failure events. 

Several sampling-based probabilistic analysis methods are used in DARWIN to predict the life 

of disks subjected to periodic inspection. Monte Carlo simulation provides accurate results 

(the accuracy is dependent on the failure probability, confidence interval, and number of random 

samples) but is relatively inefficient because the failure limit state must evaluated for each 

random sample using a fatigue crack growth algorithm. A method called a life approximation 

function (LAF) creates deterministic life and grown area arrays for a family of initial anomalies. 

During Monte Carlo simulation, the failure limit state is evaluated for each random sample using 

values interpolated from the deterministic arrays, thereby improving computational efficiency. 

Another method referred to as importance sampling (IS) focuses analysis on the initial conditions 

(anomaly size and other random variables) that would result in lives shorter than the specified 

design life. This approach reduces the size of the analysis region and is significantly more 

efficient than Monte Carlo simulation. However, it is currently limited to 2D anomalies and 

does not provide treatment for crack formation life. 

Previous DARWIN versions required the user to specify the dimensions and orientation of the 

fracture mechanics model (rectangular plate) that is superimposed on the finite element model 

geometry. To reduce the bias introduced by individual users, an automated fracture model called 

“auto-plate” was recently developed that automatically identifies the optimum dimensions and 

orientation of the fracture mechanics model at a user-specified location. A new feature to 

display life contours for 2D axisymmetric finite element models was also recently developed that 

automatically places an anomaly at each of the nodes in the finite element model, generates a 

fracture model at each anomaly location, and computes the fatigue crack growth (FCG) lifetime 

at each location. The resulting family of calculated life results is displayed in the GUI using 

conventional contouring methods. Future development will be focused on automatic generation 

of fracture risk contours based on the recently developed auto-plate and life contours capabilities. 
NASA/CR—2011-216977 25 

3.0 ALGORITHM DEVELOPMENT 

The critical reviews of accomplishments and existing capabilities in the GRC methods/models 

and the SwRI DARWIN software identified a number of significant technology gaps. 

Addressing all of these technology gaps was always understood to be well beyond the scope of 

this initial effort. Instead, the primary intended outcome of this initial effort was the 

development of a program plan for further research, and that program plan is provided in the 

later Section 4.0. However, it was possible to perform some new algorithm development under 

the current project, and the results of these activities are documented in the following 

subsections. 

New algorithm development was conducted in three areas. First, a methodology for estimating 

anomaly distributions for materials in which multiple anomaly types are operative was developed 

and verified. Second, an unfolding algorithm to estimate the dimensions and orientations of 3D 

spheroids based on 2D sectioning measurements was implemented and investigated. Third, the 

behavior of small fatigue cracks in U720 tests conducted by GRC was investigated, and an 

existing model to predict this small-crack behavior from large-crack growth rates was evaluated. 

3.1 Algorithms for Multiple Anomaly Types 

3.1.1 Introduction

As noted earlier in the Status Summary section, all of the GRC work developing methods to 

determine anomaly distributions was based on the assumption that only one type of defect is 

causing fatigue failure. However, other GRC investigations with unseeded U720 specimens did 

observe multiple failure modes due to multiple anomaly types, including inclusions and pores. 

Therefore, in order to address this technology gap, additional work was undertaken under the 

current project to develop methods for generating anomaly distributions when multiple anomaly 

types are operative. 

Prior to the current project, SwRI performed work for the Air Force Research Laboratory 

(AFRL) investigating multiple fatigue failure modes in a representative nickel-based superalloy, 

IN-100 (McClung and Enright, 2009). As part of this work for AFRL, SwRI developed a 

preliminary method to treat multiple anomaly types. However, this preliminary method had 

several limitations. Under the current project, SwRI conducted additional research to develop a 

more rigorous treatment, using the IN-100 data from the AFRL effort to demonstrate the new 

method. That new work is described in this section. 

The probability of failure of a component with multiple anomaly types can be predicted using 

established system reliability methods provided that the failure probabilities associated with 

individual anomaly types are known (Thoft-Christensen and Murotsu, 1986; Hoyland and 

Rausand, 1994; Enright and Frangopol, 1998; Melchers, 1999; Enright and Huyse, 2006). If test 

data are available for components containing a single anomaly of a single type, they can be used 

to construct probability density function distributions (PDFs) describing the likelihood of fatigue 

failure versus flight cycles. These PDFs can be transformed to predict PDFs of components 

containing more than one anomaly of a single type by considering each of the anomalies as a 
NASA/CR—2011-216977 26 

single member of a series system of anomalies. If a material contains more than one type of 

anomaly, it is usually impractical (if not impossible) to prepare test specimens with a single type 

of naturally occurring anomaly. When the test specimens contain multiple types of anomalies, it 

may be difficult to accurately estimate the failure probabilities associated with the individual 

anomaly types, even when component failures can be traced to individual anomaly types. This is 

due to the potential overlap among failure of different anomaly types that is not captured by 

system-level testing, the so-called problem of competing risks. 

One of the primary challenges associated with modeling materials with multiple types of 

anomalies is that the test specimens typically contain multiple types of anomalies. If the lives 

associated with individual anomaly types cannot be characterized from the test data, then it 

becomes impossible to develop nucleation and growth models for each anomaly type that are 

required for the system reliability models. A review of failure data for a nickel-based superalloy 

provided by the Air Force Research Laboratory (Jha et al., 2008) revealed that the overlap 

among the failure modes for various anomaly types and locations can be significant for some 

engine materials (the extent of this overlap is presented later in this section). Since all of the life 

models are eventually based on test specimen data, and test specimens for this class of materials 

commonly contain multiple types of anomalies, it became essential to identify and demonstrate a 

method to accurately quantify the lives associated with individual anomaly types. 

If the failures of different anomaly types are independent and sufficient failure data are available, 

the marginal cumulative distribution functions (CDFs) associated with them can be estimated 

using nonparametric statistical models such as Kaplan-Meier (1958), Nelson (1972), and 

Aalan (1978), among others. The Kaplan-Meier approach is most commonly used. It is based 

on the widely accepted theorem of conditional probability (Ang and Tang, 1975), and is even 

included in some statistical software. When derived in maximum likelihood estimator format, it 

can be expressed as an asymptotic normal distribution with quantifiable confidence bounds 

(Hoyland and Rausand, 1994). It is restricted to systems with independent failure modes, an 

assumption that may be difficult to test in practice (Pepe and Mori, 1993). However, this may 

become less of an issue as new methods emerge for assessing the dependence among failure 

modes (e.g., Crowder, 1997). Some researchers have noted that in some cases it may lead to 

overly conservative risk estimates when the survivability estimates are converted to CDF 

predictions (Gaynor et al., 1993). 

It is well known that correlation among failure modes can have a substantial influence on system 

reliability predictions (Thoft-Christensen and Murotsu, 1986; Enright and Frangopol, 1998; 

Melchers, 1999). Researchers have reported that failure mode correlation also can have a 

substantial influence on survival function estimates for the problem of competing risks 

(Peterson, 1976; Crowder, 1991). To quantify the extent of this influence, several models have 

been developed that establish the upper and lower bounds associated with the range of 

correlation among the failure modes (Peterson, 1976; Slud and Rubinstein, 1983; Klein and 

Moeschberger, 1988; Zheng and Klein, 1995). In addition, many new models are emerging for 

the prediction of the marginal failure densities that are not restricted to independent failure 

modes (Moeschberger and David, 1974; Moeschberger and Klein, 1995; Chen et al., 2008; 

Cai and Prentice, 1995). Specialized models have also been developed to provide treatment of 
NASA/CR—2011-216977 27 

competing risks for sequential failures (Dignam et al., 2007) and parallel systems (Tan, 2007) as 

well. 

Once the failure probabilities associated with the individual anomaly types are known, they can 

be used to calibrate probabilistic models for each anomaly type. The resulting calibrated 

probabilistic models can then be used to predict the behavior of components with different 

numbers and types of anomalies. 

3.1.2 Estimating Marginal Probability Densities from 

Component Failure Data

The conditional probability of failure PF di 1  associated with a single anomaly can be estimated 

from test specimens containing single anomalies. The probability of failure associated with 

multiple anomalies pi can then be obtained by inserting PFd  i 1  in Eqn. (5). However, if the test 

specimens contain multiple anomalies, the results are provided in terms of pi. 

Reliability predictions based on these test results are only valid for components with exactly the 

same number of anomalies associated with the test specimens. To estimate risk for different 

numbers of anomalies, the test results can be converted to conditional probability format by 

solving Eqn. (5) for P  F di 1 and applying the values of pi and i i V associated with the test: 

  1   1 ln 1 i i

i i

PFd p V

   (8)

The conditional probability density associated with PFd  i 1  is referred to herein as a 

“parent distribution” because it represents life values without the influence of multiple anomalies. 

The mean life value associated with the parent distribution may be significantly greater than the 

mean of the measured life values of specimens containing multiple anomalies, particularly when 

the number of anomalies is large. 

For components with more than one type of anomaly, the above procedure can be performed 

separately for test specimens each containing a single anomaly type. If test data are available 

only for specimens containing multiple anomalies of multiple types, then the test results are an 

expression of pF (Eqn. (7)) rather than pi (Eqn. (5)). In this situation, each component test 

represents the entire system of multiple anomaly types/locations as well as the multiple 

anomalies subsystems associated with each. If the failure of components with multiple 

anomalies can be traced to specific anomaly types, then the pi values can be estimated using the 

Kaplan-Meier (1958) method. The approach is illustrated in the following example. 

Consider the fictitious component indicated in Table 1 with two types of anomalies (types A and 

B) that can form growing cracks which can lead to fracture failure of the component. The 

component has a volume of 1 mm3

 and contains an average of 10 and 100 anomalies of type A

and B, respectively. For each anomaly type, the total fatigue life given the presence of a single 

anomaly of a single type (parent life) is modeled as a lognormal random variable with the main 

descriptors (median and coefficient of variation (COV)) indicated in Table 1. 
NASA/CR—2011-216977 28 

Table 1. Anomaly Parameters for Example Component 

Containing Two Types of Anomalies 

Anomaly 

Type 

Anomaly Rate 

(1/mm3) 

Total Life Given a Single Anomaly 

of a Single Type (Parent Life) 

Median COV Dist 

A 10 1000 0.3 Lognormal 

B 100 2000 0.5 Lognormal 

The analytical probability of failure cumulative distribution functions (CDFs) associated with the 

life distributions of the parent total life and multiple anomalies of a single type are shown in 

Figure 5. Also shown is the probability of failure of the example component containing multiple 

anomalies of type A and B, computed analytically using Eqn. (7). This figure illustrates the risk 

prediction approach of components with multiple anomaly types when the parent lives are 

known. 

Total Life, cycles

0 500 1000 1500 2000 2500 3000

Cumulative Probability

0.0

0.2

0.4

0.6

0.8

1.0

Anomaly A, Parent

Anomaly B, Parent

Anomaly A, V = 10

Anomaly B, V = 100

Component (AUB) 

Figure 5. Analytical probability of failure values associated with 

single and multiple anomalies of a single type for the 

example component containing multiple anomaly types. 

The probability of failure of the example component containing multiple anomaly types can also 

be obtained using numerical simulation. 1000 simulated specimens containing multiple 

anomalies of each anomaly type were obtained using Monte Carlo simulation, indicated in 

Table 2. The failure time of each component (the minimum life associated with anomaly types A

and B) was recorded along with the anomaly type responsible for the failure. 
NASA/CR—2011-216977 29 

Table 2. Simulated Failure of Example Component with Two Types of Anomalies 

Simulated 

Specimen 

Cycles to Failure Failure Data 

A B min(A,B) A B AUB

1 607 648 607 607 — 607 

2 492 501 492 492 — 492 

3 633 566 566 — 566 566 

4 488 720 488 488 — 488 

5 483 642 483 483 — 483 

7 803 664 664 - 664 664 

8 488 508 488 488 — 488 

— — — — — — — 

998 567 534 534 - 534 534 

999 619 663 619 619 — 619 

1000 818 602 602 — 602 602 

The CDF for the simulated component can be obtained by arranging the component failure times 

in ascending order and assigning cumulative failure probabilities F(t) to each time value. 

For uncensored failure data, F(t) is typically estimated as a/(b+1) (Ang and Tang, 1975) or 

(a-0.5)/b (Meeker and Escobar, 1998), where i is the rank order of the specimens and k is the 

total number of specimens. Using this approach (with F(t) = a/(b+1)), CDF values were 

obtained as indicated in Table 3 and shown in Figure 6. Also shown in Figure 6 is the CDF 

obtained analytically using Eqn. (7), which is in close agreement with the simulated values, as 

expected. 

Table 3. Simulated CDF Values for Example Component 

Containing Two Types of Anomalies 

k Total Life 

(cycles) F(t) 

1 247 0.0010 

2 279 0.0020 

3 284 0.0030 

4 290 0.0040 

5 292 0.0050 

— — — 

997 747 0.9960 

998 756 0.9970 

999 773 0.9980 

1000 815 0.9990 
NASA/CR—2011-216977 30 

Total Life, cycles

400 600 800 1000

Cumulative Probability

0.0

0.2

0.4

0.6

0.8

1.0

Component (Analytical AUB) 

Simulated Component (1000 MC Samples)

Figure 6. Analytical and simulated probability of failure of failure values for the example 

component containing two anomaly types were in close agreement, as expected. 

But suppose instead that the 1000 samples were obtained from component fatigue tests. In this 

situation, the CDFs for the individual anomaly types and their associated parent distributions 

would be unknown, and would have to be estimated from the failed specimen data. 

Assuming that the failure of each specimen could be traced to a specific anomaly type, the data 

could then be used to estimate the CDF associated with each anomaly type. However, since the 

data were not obtained from specimens of a single anomaly type, special care must be taken 

when fitting the data to CDFs. 

A common approach for estimating the CDF of a single anomaly type from test data containing 

multiple anomaly types is to consider only the failure times that can be traced to the individual 

anomaly type and constructing a CDF using the previously described procedure for uncensored 

data. The CDFs obtained using this approach are shown in Figure 7 for anomaly types A and B. 

The analytical CDF values for anomaly types A and B are also shown in Figure 7 for comparison 

purposes. The CDF curves based on a fit of only the failure times that can be traced to the 

individual anomaly types are significantly different from the analytical values. Furthermore, 

when these CDF curves are used to predict the CDF of the component (using Eqn. (6)), the 

results are not in agreement with either the analytical or simulated results of the component 

(from Figure 6). 
NASA/CR—2011-216977 31 

Total Life, cycles

200 400 600 800 1000

Cumulative Probability

0.0

0.2

0.4

0.6

0.8

1.0

Anomaly A (V = 10), Analytical

Anomaly B (V = 100), Analytical

Component (AUB), Analytical 

Anomaly A, Simulated Data, Uncond. Fit

Anomaly B, Simulated Data, Uncond. Fit

AUB, Simulated Data, Uncond. Fit 

Figure 7. CDF values for individual anomaly types estimated from censored failure data 

were significantly different from the analytical values. 

The reason for this difference is that some of the life values are censored by the testing process. 

A given component fails when the minimum lifetime associated with any anomaly is reached, 

and the lifetimes associated with all other anomaly types in the component are not recorded 

(they are censored). The CDFs cannot be obtained using this approach because many of the data 

are missing. 

Fortunately, statistical methods are well-established for treatment of competing risks 

(Kaplan and Meier, 1958; Nelson, 1972; Aalan, 1978; Moeschberger and David, 1974; 

Moeschberger and Klein, 1995; Chen et al., 2008; Cai and Prentice, 1995) that can be applied to 

this problem. If the failure times associated with the different anomaly types are treated as 

independent random variables, then the Kaplan-Meier method (1958) can be used to estimate the 

CDFs associated with each anomaly type. In this method, the lifetime is discretized into k

intervals, and the probability of survival within an interval Pk for an anomaly type is based on the 

ratio of the number of components that survive over the interval to the number of components 

that were available at the beginning of the interval: 

k k

k

k

n f P

n

  (9)

A conditional probability argument is made to establish the relationship among the survival 

probabilities associated with all of the non-overlapping intervals. After the first time interval, 

some of the components may fail. During the second time interval, only the components that 
NASA/CR—2011-216977 32 

have not failed during the first interval are tested. In other words, to survive the second time 

interval, it is implied that the component has also survived the first time interval: 

P  I PI I PI PI I 2 1 2 1 21          (10)

Combining Eqns. (9) and (10), and recognizing that P2 is the conditional probability that a 

component survives interval 2 given that it has also survived interval 1 ( PI I  2 1  ): 

  11 2 2

1 2 12

1 2

nf nf PI I PP

n n

               (11)

It follows that the probability of survival over m number of time intervals is given by: 

  1

m

k k

k k

n f P t  n

         (12)

and the associated probability of failure is: 

  1

1 1

m

k k

k k

n f Ft Pt  n

          (13)

One potential drawback of the Kaplan-Meier method is that the probability of failure values are 

expressed in a step function format, similar to the integrated form of a histogram. The resulting 

curve contains discontinuities at each of the failure times and therefore does not satisfy the 

requirements of a true CDF. To estimate the continuous CDF, Meeker and Escobar (1998) 

recommend setting the CDF values at each step equal to the average of the two values associated 

with each step change: 

     1

1

2

Ft Ft Ft   k k      (14)

The application of the Kaplan-Meier approach is illustrated in Tables 4 and 5 for the simulated 

failure data associated with anomaly types A and B, respectively. Similar to the approach 

previously presented for uncensored data, the component failure times were arranged in 

ascending order, and the time intervals were selected so that only a single failure occurred during 

a given interval. However, for a given anomaly type, CDF values were assigned to all of the 

component failure times for both anomaly types. In Table 4, the first failure associated with 

anomaly A did not occur until interval 24. The CDF value obtained using the Kaplan-Meier 

method (indicated as “K-M” in Table 4) experienced a step change at interval 24, and remained 

unchanged until the next failure of anomaly A at interval 31, and so on. In Table 5, the first 

failure associated with anomaly B occurred at the first interval, and the K-M CDF experienced a 

step change over every interval until the first failure associated with anomaly A. The continuous 

CDF estimate obtained using Eqn. (14) is also indicated in Tables 4 and 5 as the “M-E” CDF. 
NASA/CR—2011-216977 33 

Table 4. Failure CDF for Anomaly Type A Using Kaplan-Meier Approach 

Combined with Meeker-Escobar Data Fit 

Total Life k nk fk pk Pk K-M F(t) M-E F(t) 

0 0 1000 0 1.000 1.000 0.000 0.0000 

247 1 1000 0 1.000 1.000 0.000 0.0000 

279 2 999 0 1.000 1.000 0.000 0.0000 

— — — — — — — — 

340 23 978 0 1.000 1.000 0.000 0.0000 

345 24 977 1 0.999 0.999 0.001 0.0005 

348 25 976 0 1.000 0.999 0.001 0.0010 

— — — — — — — — 

358 30 971 0 1.000 0.999 0.001 0.0010 

360 31 970 1 0.999 0.998 0.002 0.0015 

360 32 969 0 1.000 0.998 0.002 0.0021 

— — — — — — — — 

747 997 4 1 0.750 0.075 0.925 0.9122 

756 998 3 0 1.000 0.075 0.925 0.9247 

773 999 2 0 1.000 0.075 0.925 0.9247 

815 1000 1 1 0.000 0.000 1.000 0.9624 

Table 5. Failure CDF for Anomaly Type B Using Kaplan-Meier Approach 

Combined with Meeker-Escobar Data Fit 

Total Life k nk fk pk Pk K-M F(t) M-E F(t) 

0 0 1000 0 1.000 1.000 0.000 0.0000 

247 1 1000 1 0.999 0.999 0.001 0.0005 

279 2 999 1 0.999 0.998 0.002 0.0015 

284 3 998 1 0.999 0.997 0.003 0.0025 

290 4 997 1 0.999 0.996 0.004 0.0035 

— — — — — — — — 

453 221 780 1 0.999 0.824 0.176 0.1752 

453 222 779 0 1.000 0.824 0.176 0.1757 

454 223 778 0 1.000 0.824 0.176 0.1757 

— — — — — — — — 

746 996 5 0 1.000 0.040 0.960 0.9601 

747 997 4 0 1.000 0.040 0.960 0.9601 

756 998 3 1 0.667 0.027 0.973 0.9668 

773 999 2 1 0.500 0.013 0.987 0.9801 

815 1000 1 0 1.000 0.013 0.987 0.9867 
NASA/CR—2011-216977 34 

The CDFs estimated using the Kaplan-Meier approach combined with the Meeker-Escobar 

adjustment are shown in Figure 8 for anomaly types A and B. Also shown is the analytical 

probability of failure of components containing multiple anomalies of only type A or type B. 

It can be observed that the CDF values for individual anomaly types estimated using the KaplanMeier approach combined with the Meeker-Escobar data fit were in close agreement with the 

analytical values. The CDF for a component with anomalies A and B (estimated using Eqn. (6)) 

is also shown in Figure 8. It is also in close agreement with the analytical component values. 

Total Life, cycles

200 400 600 800 1000

Cumulative Probability

0.0

0.2

0.4

0.6

0.8

1.0

Anomaly A (V = 10), Analytical

Anomaly B (V = 100), Analytical

Component (AUB), Analytical 

Anomaly A, Simulated Data, K-M Fit

Anomaly B, Simulated Data, K-M Fit

AUB, Simulated Data, K-M Fit 

Figure 8. CDF values for individual anomaly types estimated using the combined 

Kaplan-Meier/Meeker-Escobar approach were in close agreement 

with the analytical values. 

Once the marginal CDF values for anomaly types A and B have been estimated using the KaplanMeier/Meeker-Escobar approach, they can be transformed to the associated parent distributions 

using Eqn. (8). The resulting CDF values are shown in Fig. 9. The analytical parent 

distributions are also shown in Fig. 9, where it can be observed that they are both in close 

agreement with the values estimated from the transformed failure data. This illustrates how the 

approach is used to obtain the parent distributions which are ultimately used for risk prediction 

of components with different numbers of anomalies of multiple types. 
NASA/CR—2011-216977 35 

Total Life, cycles

0 500 1000 1500 2000

Cumulative Probability

0.0

0.2

0.4

0.6

0.8

1.0 Anomaly A, Parent

Anomaly B, Parent

Anomaly A, Transformed K-M Fit

Anomaly B, Transformed K-M Fit

Figure 9. The transformed CDF values based on the Kaplan-Meier/Meeker-Escobar 

approach were in close agreement with the analytical parent distributions. 

3.1.3 Application to Gas Turbine Engine Materials

The approach described in Section 3.1.2 was applied to reliability prediction of a nickel-based 

superalloy based on data provided by the AFRL (AFRL, also reported in Jha et al., 2008). 

The material contained non-metallic particles (NMPs) and pores with anomaly occurrence rates 

indicated in Table 6. AFRL had previously performed fatigue tests on cylindrical 

smooth specimens (diameter = 5 mm and gage length = 15 mm) at four stress levels 

(1200 MPa, 1150 MPa, 1100 MPa, and 1000 MPa). All tests were conducted at a stress ratio of 

R = 0.05 and a temperature of 650C. The test results at 1100 MPa revealed failures that could be 

traced to several anomaly types/locations and were selected for use in the reliability studies. 

As indicated in Table 7, NMP failures were identified at both the surface and subsurface of the 

specimens at this stress level, whereas pore failures occurred only on the surface. 

Table 6. Anomaly Types Associated with Nickel-Based Superalloy Test Specimens 

Anomaly Occurrence Rate Volume Average Number 

Type Location per mm3 per mm2 mm3 Total Effective 

NMPs Surface 10 0.17 4.0 40.2 3.4 

NMPs Subsurface 10 0.17 290.5 2905.1 250.3 

Pores Surface 2445 19 1.8 4475.0 306.9 
NASA/CR—2011-216977 36 

Table 7. Component Failure CDF for Nickel-Based Superalloy Smooth Specimen 

Fatigue Failure Data with Applied Stress of 1100 MPa 

k Type Location Total Life F(t) 

0 — — 0 0 

1 NMP Surface 1736 0.0625 

2 NMP Surface 2210 0.1250 

3 NMP Surface 2309 0.1875 

4 NMP Surface 3237 0.2500 

5 Pore Surface 9446 0.3125 

6 Pore Surface 13188 0.3750 

7 NMP Subsurface 18405 0.4375 

8 Pore Surface 18525 0.5000 

9 NMP Subsurface 21936 0.5625 

10 NMP Subsurface 25081 0.6250 

11 NMP Subsurface 26417 0.6875 

12 NMP Subsurface 31736 0.7500 

13 Pore Surface 45337 0.8125 

14 NMP Subsurface 65021 0.8750 

15 NMP Subsurface 76339 0.9375 

The volume of material associated with the surface anomalies was based on a thin annulus 

(often referred to as an “onion skin”) at which the number of anomalies associated with the 

volumetric occurrence rate (Table 6) matched the number of anomalies associated with the 

surface occurrence rate. The subsurface volume was based on the remaining volume 

(i.e., the total volume minus the surface volume). The average number of anomalies for each 

anomaly type and location was computed as the product of the volume and the volumetric 

occurrence rate, indicated in Table 6. Since many of the anomalies were too small to initiate 

growing cracks, an effective number of anomalies that could lead to failure was estimated based 

on a minimum threshold equivalent anomaly diameter (40 m for NMPs, and 21 m for pores). 

The CDFs for each of the three anomaly type/location combinations associated with the material 

were estimated using the combined Kaplan-Meier/Meeker-Escobar approach described in 

Section 3.1.2. The results for surface NMPs are indicated in Table 8, where the failures all 

occurred at relatively short life values compared to the other anomaly types/locations. The 

results for subsurface NMPs and surface pores are indicated in Tables 9 and 10, respectively. 

The marginal CDFs indicated in Tables 8-10 are shown in Figure 10. Also shown in Figure 10 is 

the CDF for all anomalies (indicated in Table 7) which was obtained using the approach for 

uncensored data described previously in Section 3.1.2. 
NASA/CR—2011-216977 37 

Table 8. Failure CDF for Surface NMPs Using Kaplan-Meier/Meeker-Escobar Approach 

Total Life k nk fk pk Pk K-M F(t) M-E F(t) 

0 0 15 0 1.000 1.000 0.000 0.000 

1736 1 15 1 0.933 0.933 0.067 0.033 

2210 2 14 1 0.929 0.867 0.133 0.100 

2309 3 13 1 0.923 0.800 0.200 0.167 

3237 4 12 1 0.917 0.733 0.267 0.233 

9446 5 11 0 1.000 0.733 0.267 0.267 

13188 6 10 0 1.000 0.733 0.267 0.267 

18405 7 9 0 1.000 0.733 0.267 0.267 

18525 8 8 0 1.000 0.733 0.267 0.267 

21936 9 7 0 1.000 0.733 0.267 0.267 

25081 10 6 0 1.000 0.733 0.267 0.267 

26417 11 5 0 1.000 0.733 0.267 0.267 

31736 12 4 0 1.000 0.733 0.267 0.267 

45337 13 3 0 1.000 0.733 0.267 0.267 

65021 14 2 0 1.000 0.733 0.267 0.267 

76339 15 1 0 1.000 0.733 0.267 0.267 

Table 9. Failure CDF for Subsurface NMPs Using 

Kaplan-Meier/Meeker-Escobar Approach 

Total Life k nk fk pk Pk K-M F(t) M-E F(t) 

0 0 15 0 1.000 1.000 0.000 0.000 

1736 1 15 0 1.000 1.000 0.000 0.000 

2210 2 14 0 1.000 1.000 0.000 0.000 

2309 3 13 0 1.000 1.000 0.000 0.000 

3237 4 12 0 1.000 1.000 0.000 0.000 

9446 5 11 0 1.000 1.000 0.000 0.000 

13188 6 10 0 1.000 1.000 0.000 0.000 

18405 7 9 1 0.889 0.889 0.111 0.056 

18525 8 8 0 1.000 0.889 0.111 0.111 

21936 9 7 1 0.857 0.762 0.238 0.175 

25081 10 6 1 0.833 0.635 0.365 0.302 

26417 11 5 1 0.800 0.508 0.492 0.429 

31736 12 4 1 0.750 0.381 0.619 0.556 

45337 13 3 0 1.000 0.381 0.619 0.619 

65021 14 2 1 0.500 0.190 0.810 0.714 

76339 15 1 1 0.000 0.000 1.000 0.905 
NASA/CR—2011-216977 38 

Table 10. Failure CDF for Surface Pores using Kaplan-Meier/Meeker-Escobar Approach 

Total Life k nk fk pk Pk K-M F(t) M-E F(t) 

0 0 15 0 1.000 1.000 0.000 0.000 

1736 1 15 0 1.000 1.000 0.000 0.000 

2210 2 14 0 1.000 1.000 0.000 0.000 

2309 3 13 0 1.000 1.000 0.000 0.000 

3237 4 12 0 1.000 1.000 0.000 0.000 

9446 5 11 1 0.909 0.909 0.091 0.045 

13188 6 10 1 0.900 0.818 0.182 0.136 

18405 7 9 0 1.000 0.818 0.182 0.182 

18525 8 8 1 0.875 0.716 0.284 0.233 

21936 9 7 0 1.000 0.716 0.284 0.284 

25081 10 6 0 1.000 0.716 0.284 0.284 

26417 11 5 0 1.000 0.716 0.284 0.284 

31736 12 4 0 1.000 0.716 0.284 0.284 

45337 13 3 1 0.667 0.477 0.523 0.403 

65021 14 2 0 1.000 0.477 0.523 0.523 

76339 15 1 0 1.000 0.477 0.523 0.523 

Total Life, Cycles

103 104 105

Cumulative Probability

0.0

0.2

0.4

0.6

0.8

1.0

All Anomalies

Surface NMP

Subsurface NMP

Surface Pore

Figure 10. The marginal CDFs for each anomaly type were obtained 

from the Kaplan-Meier/Meeker-Escobar fit of the failure data. 
NASA/CR—2011-216977 39 

The parent distributions were obtained by applying the effective number of anomalies and the 

F(t) values to Eqn. (8) at each life value. The transformed CDFs are shown in Figure 11, where 

it can be observed that the surface NMPs have a significant influence on component reliability 

when single anomalies are considered. 

Total Life, cycles

103 104 105

Cumulative Probability

0.00

0.02

0.04

0.06

0.08

0.10

Surface NMP, Parent

Subsurface NMP, Parent

Surface Pore, Parent

Figure 11. The parent CDFs were determined by transforming the 

marginal CDF values for each anomaly type. 

The parent CDFs shown in Figure 11 can be used to make reliability predictions for components 

with different numbers of anomalies. Compared to realistic engine components such as rotors or 

disks, the smooth specimens have a relatively small volume to surface area ratio. To study this 

effect, reliability predictions were made for a range of volume to surface area ratio values 

indicated in Table 11 (the volume to area ratio of the test specimens was assigned a value of 1.0). 

The results are shown in Figure 12, where it can be observed that component failure probabilities 

at early lives were dominated by surface NMPs for all of the volume to surface area ratios 

considered. The surface pores and subsurface NMPs had the most influence near the middle and 

end of life, where the component CDF values were already very large. 

Table 11. Influence of Component Volume to Surface Area Ratio on 

Number of Anomalies for Each Anomaly Type 

Volume to Area Ratio Diameter 

(mm) 

Number of Active Anomalies 

Surface NMP Subsurface NMP Surface Pore 

1 5 3 250 306 

2 10 7 1008 613 

5 25 17 6326 1532 

10 50 35 25338 3065 
NASA/CR—2011-216977 40 

Total Life, cycles

103 104 105

Cumulative Probability

0.0

0.2

0.4

0.6

0.8

1.0

Vol/Surf Area = 1.0

2.0

5.0

10.0

All Anomalies

Surface NMP

(a) 

Total Life, cycles

103 104 105

Cumulative Probability

0.0

0.2

0.4

0.6

0.8

1.0 Vol/Surf Area = 1.0

2.0

5.0

10.0

All Anomalies

Subsurface NMP

(b) 

Figure 12. The influence of volume to surface area ratio on 

probability of failure and associated contribution of: 

(a) surface NMPs, (b) subsurface NMPs, and (c) surface pores. 
NASA/CR—2011-216977 41 

Total Life, cycles

103 104 105

Cumulative Probability

0.0

0.2

0.4

0.6

0.8

1.0 Vol/Surf Area = 1.0

2.0

5.0

10.0

All Anomalies

Surface Pore

(c) 

Figure 12 (cont). The influence of volume to surface area ratio on 

probability of failure and associated contribution of: 

(a) surface NMPs, (b) subsurface NMPs, and (c) surface pores. 

As indicated in Table 11, the number of subsurface NMPs becomes very large as the volume to 

area ratio is increased. For a volume/area value of 10, there are substantially more subsurface 

NMP anomalies compared to the other types present. This suggests that these anomaly types 

should play more of a role in component failures as the volume to area ratio is increased. This is 

confirmed in Figure 12(b) where it can be observed that the slope of the CDF associated with 

subsurface NMPs becomes considerably steeper as the volume to area ratio increases from 

1 to 10. With such a large number of subsurface NMPs available at relatively high volume to 

area ratios, one might intuitively expect subsurface NMPs to also play a role in component 

failures at relatively low life values. However, Figure 12(a) shows that the surface NMPs are 

much more likely to fail at the low life values because the slope of the CDF of surface NMPs 

increases significantly with an increase in the volume to surface area ratio. The probabilistic 

model predicts that subsurface NMPs and surface pores will fail only after the surface pores have 

been exhausted. So although the subsurface NMPs are more plentiful than the other anomaly 

types, there are still enough surface NMPs available to substantially influence the early stages of 

component life. 

For this material, the dominant failure mechanism was not significantly influenced by changes in 

the volume to area ratio, which suggests that the smooth specimen test results could be directly 

used to predict the failure mechanisms of full scale engine components. Further research is 

required to investigate the extent of this relationship. 
NASA/CR—2011-216977 42 

3.2 Development of an Unfolding Algorithm for Estimating Spheroid 

Parameters from Sectioning Data Measurements

3.2.1 Introduction

As part of the OEM engagement task, SwRI learned that at least one of the OEMs uses only 

sectioning data to characterize anomalies in powder materials. Several GRC studies focused on 

this method, and an unfolding algorithm was developed by Ghosn et al. (2003) that was 

successful for some anomalies (-80+100 mesh metal powder) but was less successful for others 

(-270 mesh metal powder). Since one of the ultimate objectives of this project was to transfer 

the technologies developed at GRC into DARWIN, an unfolding algorithm was developed and 

implemented in software at SwRI to investigate the accuracy of the unfolding method. To verify 

the algorithm, a fictitious material containing a known distribution of spheroidal anomalies was 

simulated through sectioning and unfolding to reveal the potential computational errors 

associated with this approach. 

Wicksell (1925, 1926) investigated the problem of estimating the size-shape distributions of 

spheres and ellipsoids from the data of the circles or ellipses observed in planar sections. 

Cruz-Orive (1976, 1977) proved that the ellipsoidal problem cannot be solved universally. 

The equation for the general stereological problem has unique solutions only for particular 

families of ellipsoids called spheroids, which are biaxial ellipsoids. The spheroids may have 

random size and shape, not necessarily independent from each other, but they must all be either 

prolate or oblate for a given model. For prolate spheroids (elongated “football” shape) the major 

axis is the rotator axis, whereas the minor axis is the rotator axis for oblate spheroids 

(flattened “pancake” shape). 

3.2.2 The Stereological Equation`

The general problem of predicting the size-shape distribution from the measurements of 

section profiles is described by the stereological equation (Cruz-Orive 1976, 1977; 

Ohser and Mücklich 2000). Let the size parameters U and S be the major caliper parameter of a 

spheroid and its planar section profile, respectively. Let V be the shape parameter of a typical 

spheroid and T be the random shape parameter of a random section profile. Let FV(u,v) and 

FA(s,t) denote the bivariate size-shape distribution functions of the spheroids and their elliptical 

section profiles, respectively, where FV(u,v)=P(U≤u,V≤v) and FA(s,t)=P(S≤s,T≤t). 

The probability of both S>s and T>t is given by: 

P(S s,T t) 1 F ( ,t) F (s,1) F (s,t)     A   A  A (15) 

The stereological equation describes the relationship among the distribution functions of the 

spheroids and elliptical section profiles of prolate or oblate spheroids whose principal major and 

minor semi-axes are a and b, respectively: 

[1 ( , ) ( ,1) ( )] ( ) ( , ), 1 N F t F s F s,t N p u,v, s,t dF u v A A A A V s t V         (16) 
NASA/CR—2011-216977 43 

where 0<s and 0<t≤1. The shape parameter is defined as v = 1 – (b/a)

2

. The bi-dimensional 

size-shape variables (u,v) for prolate and oblate spheroids are (b,v) and (a,v), respectively. 

Similarly, the size-shape parameter (s,t) for elliptical profiles is defined as (a, 1 – (b/a)

2

 ), where 

a and b are the principal major and minor semi-axes of the elliptical profiles, respectively 

(Cruz-Orive, 1976). 

3.2.3 Numerical Solution of the Stereological Equation

The unfolding algorithm proposed by Cruz-Orive is described in this section. There is a close 

correspondence between linear integral Eqn. (2) which specify linear relations among 

distribution functions and linear equations which specify analogous relations among vectors of 

relative frequencies. The stereological integral Eqn. (2) can be transformed into a linear equation 

system: 

NA = PNVQ (17) 

where NV and NA are vectors of relative frequencies of the spheroids and their elliptical profiles, 

respectively. P and Q are transformation matrices corresponding to the kernel function p(u,v,s,t) 

in Eqn. (2). Suppose the range (0, B) of size component b (or a) is divided into s bins of equal 

width B/s, and the range (0,1) of the shape component is divided into k bins of equal width 1/k. 

The rectangular domain of variable (b,v) or (a,v) is divided into a grid comprising s  k “cells”. 

The relative frequencies of spheroids per unit volume are: 

NV  NV i, j;i  1,s, j 1,k, (18) 

where N/V  NV, note that V V

k

j

s

i

  N i j  N  

( , )

1 1

. The relative frequencies of elliptical profiles per 

unit area are: 

NA  NA , ; 1,s, 1,k, (19) 

where NA(, )  NA.. Clearly, A A

s k

  N  N  

( , )

1 1

   

. Solving Eqn. (3) yields: 

N    i, j p N , q ,i ,s, j ,k  j

A

S

i

k

j

ia

V 1 1 1        

 

 

  , (20) 

where p

i , qj are the elements of matrices P-1 and Q-1, respectively. P is an upper triangular 

matrix that is size dependent, whereas Q is a lower triangular matrix that is shape dependent. 

The elements of matrices P and Q are given as follows: 
NASA/CR—2011-216977 44 

 (21) 

For populations consisting of prolate spheroids, we have, 

 (22) 

where: 

and: 

 (23) 

where: 

3.2.4 Application Example

A Fortran program based on the Cruz-Orive unfolding algorithm was implemented and 

illustrated for a cubic section of a fictitious material containing oblate spheroidal anomalies. 

The spheroid dimensions (major axis a and aspect ratio b/a) and orientation angles were modeled 

as independent uniformly distributed random variables with the values indicated in Table 12. 

Using Monte Carlo simulation, spheroids with random values of a and b/a were generated and 

placed at random 3D locations within a cubic section of the fictitious material. To compute the 

relative frequency of anomalies per unit volume NV of a given size and aspect ratio, the full 

ranges of a and b/a values were discretized into s and k bins, respectively. Two discretization 

schemes were considered: 5 × 5 and 10 × 10. 

To simulate the measurements that would be obtained by the sectioning process, horizontal 

section cuts were applied at various heights within the cube, and the relative frequency of 

anomalies per unit area NA was obtained for the size and shape parameters associated with each 

anomaly that appeared on the section plane. The values obtained from the section plane were 

converted using the Cruz-Orive unfolding algorithm to predict the size and shape distribution of 

the spheroids. 
NASA/CR—2011-216977 45 

Table 12. Spheroid Parameters Associated with Application Example 

Variable Description Lower Bound Median Upper Bound Distribution

a major axis 0.0 5.0 10.0 Uniform 

b/a aspect ratio 0.0 0.5 1.0 Uniform 

1, 2 , 3 orient. angles 0.0  2 Uniform 

Figure 13 shows the 3D histogram plots of the relative frequencies of the actual population of 

spheroids and the predicted population based on measurements of their elliptical profiles on 

98 planar sections. The predicted results have reasonable agreement with the actual distributions. 

Computational accuracy improves with the increasing number of sections, as expected, due to 

the increased total number of Monte Carlo samples associated with the additional sections. 

The influence of the number of section plane measurements on predicted spheroid size parameter 

and shape parameter values is shown in Figures 14 and 15, respectively for a 10 × 10 grid and a 

5 × 5 grid. For both parameters, predictions were significantly improved when the number of 

sections was increased from 1 to 10, but increasing the number of sections by another order of 

magnitude (i.e., from 10 to 100) had only a marginal improvement on prediction accuracy. 

This suggests that there may be an optimum number of sections that provides a good balance 

between computational accuracy and the number of sections (and associated costs). 

Generally, the errors associated with the stereological unfolding algorithm come from two 

primary competing sources: statistical error (i.e., not enough Monte Carlo samples) and 

discretization (i.e., bin size) error. Discretization error decreases with increasing number of bins, 

while statistical error increases with increasing number of bins. The bin size should not be too 

large or too small. There is an optimal discretization parameter that minimizes the total error 

(Ohser and Sandau, 2000). In practical experiments, the number of sections is limited and data 

noise often dominates, so the discretization error is generally insignificant compared with 

statistical error. For example, the accuracies of the results using 5 × 5 bins are comparable to 

those with 10 × 10 bins, if not better. Therefore, the focus should be on reducing statistical error 

in practical applications. It is important to have a sufficient number of samples in each bin to 

have small statistical errors. Statistical error is smaller if measurements are done on more planar 

sections, or if the material has a higher density of spheroid centers and the spheroids are larger. 

However, there is a maximum limit on the density of spheroids for the Cruz-Orive stereological 

unfolding algorithm because this algorithm assumes that the spatial distribution of spheroids is 

uniform and dilute. In practice, “dilution” is satisfactory whenever the volume fraction of the 

spheroids to the total volume is less than about 10% (Cruz-Orive, 1977). This condition is 

required to assume that the set of spheroid centers approximately forms a “Poisson ensemble” in 

three dimensions. 
NASA/CR—2011-216977 46 

(a) 

(b) 

Figure 13. Bivariate histograms (stereograms) of anomaly size and shape parameter 

associated with anomalies in the fictitious material: (a) original population of spheroids, 

and (b) predicted population of spheroids based on sectioning data transformed 

using the Cruz-Orive unfolding algorithm. 
NASA/CR—2011-216977 47 

(a) 

(b) 

Figure 14. Influence of the number of section plane measurements on predicted 

spheroid size parameter values: (a) 10 × 10 bin grid, and (b) 5 × 5 bin grid. 
NASA/CR—2011-216977 48 

(a) 

(b) 

Figure 15. Influence of the number of section plane measurements on predicted spheroid 

shape parameter values: (a) 10 × 10 bin grid, and (b) 5 × 5 bin grid. 
NASA/CR—2011-216977 49 

3.2.5 Discussion

As mentioned in Section 3.2.1, it currently is impossible to accurately predict the shape and 

orientation parameters of a general ellipsoid based on measurements obtained from 2D 

sectioning data. If data are available in another form, such as HLS, then it may be possible to 

predict general ellipsoid parameters by combining the data using Bayesian updating or a similar 

method. On the other hand, if data are only available from 2D sectioning, then a general 

ellipsoid model is probably not an appropriate one because it simply cannot be fully 

characterized. In this situation, the anomaly model should be limited to either the prolate or 

oblate spheroid model. 

In this study, an unfolding algorithm was implemented and verified that can be used to estimate 

the dimensions and orientations of 3D spheroids based on 2D sectioning measurements. It was 

shown that the accuracy of the predicted spheroid model is dependent on the number of sections 

and the discretization of the mesh used to characterize the data. In section 2.1.1, it was 

mentioned that the unfolding algorithm developed by Ghosn et al. (2003) performed well for 

some mesh sizes and not so well for others. This difference may possibly have been due to the 

number of sections or the discretization bin width. 

In some instances, the number of sections may be a fixed quantity (e.g., historical data). In this 

case, the unfolding algorithm could be used to quantify confidence bounds on fracture risk 

associated with the uncertainty in the spheroid model for the fixed number of sections. For new 

design, the number of sections is probably limited by the cost to obtain them. In this situation, 

the unfolding algorithm could be used to identify the minimum number of sections that would be 

required to meet a fracture risk reliability target that accounts for the uncertainty associated with 

the number of sections. In addition, since DARWIN includes a capability to quantify the 

influences of manufacturing and service inspections on fracture risk, it could be used to extend 

the cost tradeoff to include the cost of inspection. This would allow the engineer to identify the 

best combination of testing (sectioning) and inspections to minimize overall life-cycle cost. 

The unfolding algorithm used in this study was limited to spheroid dimensions and orientations 

that were modeled as independent random variables. GRC studies have shown that anomaly 

dimensions are strongly correlated variables, so this influence would need to be considered 

before implementing an unfolding algorithm in DARWIN. 
NASA/CR—2011-216977 50 

3.3 Algorithms for Modeling the Growth Rates of Small Fatigue Cracks 

3.3.1 Introduction

The tendency for small fatigue cracks to grow at rates faster than anticipated from large-crack 

data trends and/or to grow at stress intensity factor ranges below the usual large-crack threshold 

has long been noted, and a variety of reasons have been proposed (McClung et al., 1996; 

Davidson et al., 2003). 

GRC measured the growth rates of small fatigue cracks in Udimet 720 and compared the results 

with large-crack behavior (Telesman et al., 2006). Their initial comparison suggested that 

classic small-crack behavior was observed, including high initial crack growth rates and large 

scatter. Many cracks exhibited decreasing crack growth rates and some arrested completely. 

Few small cracks grew to become dominant cracks with increasing crack growth rates. 

However, further examination of the data revealed that much of the characteristic “small crack” 

behavior was associated with “pop-in” behavior during the first 250 cycles. When data from this 

initial crack formation and growth period were removed, the remaining data often fell at or 

below the extrapolated Paris line, and some of the classic “small crack” behavior was no longer 

apparent. 

In research (currently unpublished) funded by the FAA, the current authors have recently 

surveyed small-crack behavior in a variety of common gas turbine engine rotor materials. 

Small-crack data were collected from the literature and (when possible) from related private 

communications with the experimentalists who generated the data. The specific rotor materials 

included Ti-6Al-4V, Ti-6Al-2Zr-4Sn-6Mo, IMI685, IN-100, and Astroloy. Classic small-crack 

effects were observed in every case. 

The pragmatic engineering challenge is not only to predict when this “anomalous” small-crack 

behavior will occur, but to predict the actual growth rates of small fatigue cracks on the basis of 

large-crack data and other appropriate parameters (but without actually having to generate 

fatigue crack growth (FCG) rate data for small cracks, since this is typically an expensive effort). 

The FAA-funded study critically evaluated one simple engineering model that can be used to 

correlate small-crack behavior with corresponding large-crack behavior. 

A brief study was conducted in the present project to extend this survey of small-crack behavior 

to the GRC U720 data, and to evaluate the same engineering model using the U720 data. 

3.3.2 Analysis Methods

Two types of crack growth data analysis methods require delineation. The first method is that 

used to calculate the average crack growth rate, da/dN, from measured crack length vs. cycles 

data. In this study, a direct secant method that calculates the slope between each two adjacent 

(a, N) pairs was used. The second method is that used to calculate the stress intensity factor 

range from the crack size and applied stress range. The U720 small-crack data were generated 

with small cylindrical specimens under uniform remote axial tension. The surface crack depth 

was estimated to be equal to half of the measured surface length, and the Forman-Shivakumar 

(1986) solution for a thumbnail crack in a solid cylinder (also available in NASGRO as the SC07 
NASA/CR—2011-216977 51 

solution) was used to calculate the stress intensity factor. This solution was used for consistency 

with the other data analysis performed in the FAA-funded effort. 

A wide variety of different analysis methods have been proposed to calculate an enhanced crack 

driving force for small cracks and hence to predict their accelerated growth rates compared to 

large cracks with the same nominal driving force. The purpose of this small study was to 

investigate one simple method that has shown some promise in earlier studies. The method 

was originally proposed by El Haddad et al. (1979) based on the so-called Kitagawa 

(Kitagawa and Takahashi, 1976) diagram construction, Figure 16. The Kitagawa diagram 

integrates the traditional large crack fracture mechanics threshold, Kth, with the traditional 

smooth specimen endurance limit, σe. The region on the Kitagawa diagram both above the 

horizontal endurance limit line and above the sloping threshold line is “unsafe” because fatigue 

failure is predicted to occur. The region below both lines is “safe” because fatigue failure is 

predicted not to occur. The intersection of the two lines, defined as a0, is given by the equation: 

2

0

1



















 

e

th

F

K

a

  (24) 

where, for consistency with the smooth specimen endurance limit data, F = 0.67 is the geometry 

correction factor in the K solution for the semi-circular surface crack in the smooth specimen. 

Figure 16. Schematic Kitagawa diagram relating smooth specimen 

fatigue endurance limit (Δσe) and large-crack threshold (ΔKth). 

e

2

1























e

th

o F

K

a

 

F a

K

S th



  
NASA/CR—2011-216977 52 

The Kitagawa diagram indicates that cracks smaller than a0 must be able to grow at nominal 

stress intensity factor ranges that are less than the large crack threshold, since smooth specimens 

fail by the initiation and growth of microcracks to failure in this regime. El Haddad suggested 

that small cracks could grow below the large-crack threshold because the effective driving force 

of the small cracks was larger than expected from traditional large-crack fracture mechanics. 

He proposed that the actual “equivalent” driving force could be calculated by replacing the 

physical crack size a with the sum (a + a0) according to the relationship: 

  0 Keq  F(a)S  a  a (25) 

Here F(a) is the crack shape and specimen/component geometry correction factor for the 

geometry of interest; a0 is not included in the F(a) calculation. Note that the a0 contribution is 

negligible for large values of a, but becomes increasingly significant for smaller values of a. 

Although the El Haddad (EH) concept has been criticized for its apparent lack of physical 

justification, several researchers have independently derived similar formulations based on 

detailed micromechanical considerations, and these have been shown to be numerically identical 

to the EH formulation under typical conditions. Chan (1999) related a0 to dislocation pileups 

and mode II shear cracks, and found satisfactory agreement with small-crack data from titanium 

aluminides. Tanaka et al. (1981) formulated the problem in terms of a crack-tip slip band 

propagating across one or more grain boundaries, and demonstrated reasonable agreement 

between his formulation and a variety of experimental data, mostly for steels and aluminum 

alloys. 

3.3.3 Results

The ability of the EH small-crack model to correlate small-crack and large-crack data is first 

illustrated by showing some of the data evaluated in the FAA-funded study. 

Lenets et al. (2000, 2002) at Honeywell and Caton et al. (unpublished) at the AFRL tested 

Ti-6Al-4V from a common material source. The Ti-6Al-4V was provided in a solution-treated 

and over-aged (STOA) condition. The microstructure consisted of 60 volume percent primary 

alpha phase and 40 volume percent lamellar transformed beta phase. The grain structure was 

equiaxed and uniform with an average alpha grain size around 10 m (0.0004 in.). 

Both sets of small-crack tests were conducted on smooth cylindrical specimens at room 

temperature. Lenets et al. and Caton et al. both conducted tests at R = 0.1 and R = 0.5, and 

Caton et al. also conducted tests at R = -1. The small surface cracks in the Caton et al. tests were 

initiated at focused ion beam (FIB) micro-notches roughly 30 – 40 m wide, 15 – 20 m deep, 

and 3 – 4 m high. The small surface cracks in the Lenets et al. tests were initiated naturally. 

Crack growth was monitored in both laboratories with standard cellulose acetate replication 

methods. Large-crack data had been generated at the same stress ratios with the same material 

using both compact tension and surface-crack tension specimens (Sheldon et al., 1999; 

Boyce and Ritchie, 2001). 
NASA/CR—2011-216977 53 

Comparisons of the small-crack and large-crack growth rate data for the HCF program 

Ti-6Al-4V are shown in Figure 17 for stress ratio R = 0.1. The data exhibited a pronounced 

small-crack effect at the smallest crack sizes, growing faster than the corresponding large-crack 

trends. 

Ti-6Al-4V

R = 0.1

K (ksiin)

1 10 100

da/dN (in/cyc)

10-9

10-8

10-7

10-6

10-5

10-4

Long Cracks

AFRL Small Cracks

Honeywell Small Cracks

Figure 17. Comparison of small-crack and large-crack growth rate data for 

Ti-6Al-4V tested by Lenets et al. and Caton et al. at R = 0.1. 

The EH small-crack parameter a0 was calculated to be a0 = 0.0021 in. (53 m) from Eqn 1 based 

on available large-crack threshold and smooth-specimen endurance limit data for the same 

material. The El Haddad-corrected values of K were then calculated from Eqn 2 for all data. 

The results are shown for R = 0.1 in Figure 18. Following the EH correction, the small-crack 

data agree very closely with the large-crack data trends. Similar correlations (not shown here) 

were successful at R = 0.5 and R = -1, although the available data were more limited. 

Jha et al. (2008) performed large-crack and small-crack tests at 650C (1200F) with the nickelbased superalloy IN-100. The material was fine-grained, subsolvus, and powder-processed with 

a median -grain size about 4 m (0.00016 in.). The yield strength at the test temperature was 

about 1100 MPa (160 ksi), the UTS was about 1379 MPa (200 ksi), and the % elongation at 

fracture was about 20%. 

Small-crack tests were performed at R = 0.05 using mostly cylindrical smooth specimens under 

uniform tension (one test was conducted with a square bar under bending). Some cracks 

originated naturally at non-metallic particles on the surface, while other cracks initiated naturally 

at surface pores. Crack length measurements were performed with acetate replicas at room 

temperature during interruptions in the test. Large crack da/dN vs. K data were generated with 

C(T) specimens. 
NASA/CR—2011-216977 54 

Ti-6Al-4V

R = 0.1

K (ksiin)

1 10 100

da/dN (in/cyc)

10-9

10-8

10-7

10-6

10-5

10-4

Long Cracks

AFRL Small Cracks (EH)

Honeywell Small Cracks (EH)

Figure 18. Comparison of small-crack and large-crack growth rate data for Ti-6Al-4V 

tested by Lenets et al. and Caton et al. at R = 0.1, with small-crack data adjusted 

using El Haddad parameter. 

Two approaches were used to estimate the EH parameter, one based on Eqn. (24) and one based 

on scaling the grain size. The large-crack threshold was measured to be 12.3 MPam from a 

single reported large-crack test. Estimating the “endurance limit” was difficult because 

smooth-specimen fatigue tests (R = 0.05) were only conducted at maximum stresses of 

1000 MPa, 1100 MPa, 1150 MPa, and 1200 MPa. At the lowest stress level tested, failures were 

observed at lives ranging from 60,000 cycles to nearly 1,000,000 cycles. The earlier failures 

were mostly associated with crack initiation at subsurface non-metallic particles. The longer 

lives (300,000 to 950,000 cycles) were mostly associated with crack initiation at surface pores, 

which is thought to be more relevant to the EH estimate. Based on extrapolation of the available 

failure data for surface pore cracks, the endurance limit was crudely estimated to be about 

σe = 900 MPa. This estimate and the measured large-crack threshold led to an estimated EH 

parameter a0 = 132 m (0.0052 in.). An alternative value was calculated by scaling the median 

-grain size: six times this dimension is 24 m (about 0.001 in.). The earlier studies of smallcrack behavior in titanium rotor alloys had found six-times-grain-size to be a reasonable estimate 

of the EH parameter. 

Comparisons of the large-crack and small-crack data are shown in Figure 19, and comparisons of 

large-crack data and EH-corrected small-crack data are shown in Figure 20. The EH estimate 

based on Eqn. (24) over-predicted the small-crack effect (compared to the available large-crack 

data), while the EH estimate based on scaling the median -grain size slightly 

under-predicted the small-crack effect. 
NASA/CR—2011-216977 55 

IN100

R = 0.05

K (ksiin)

1 10 100

da/dN (in/cyc)

10-9

10-8

10-7

10-6

10-5

10-4

167 ksi (Pore)

167 ksi (NMP)

174 ksi (Pore)

Long Cracks

Figure 19. Comparison of small-crack and large-crack growth rate data 

for IN-100 tested by Jha et al. at R = 0.05. 

IN100

R = 0.05

K (ksiin)

1 10 100

da/dN (in/cyc)

10-9

10-8

10-7

10-6

10-5

10-4

167 ksi -EH (Pore)

167 ksi -EH (NMP)

174 ksi -EH (Pore)

EH (6x grain size)

EH (6x grain size)

EH (6x grain size)

Long Cracks

Figure 20. Comparison of small-crack and large-crack growth rate data for IN-100 

tested by Jha et al. at R = 0.05, with small-crack data adjusted using two estimates 

of the El Haddad parameter. 
NASA/CR—2011-216977 56 

Consider now the GRC U720 data. The GRC small-crack tests were performed at 1200F 

(650°C) on cylindrical, axially-loaded fatigue specimens with a 0.40-inch (10.2 mm) diameter. 

All cracks were initiated naturally at artificially-seeded ceramic inclusions with average seed 

diameters of 0.0021 in. (54 m) or 0.0048 in. (122 m). Crack size measurements were 

performed in a special rotational stage in a scanning electron microscope. Fatigue cycling at 

20 cpm in ambient laboratory air was periodically interrupted and the specimen transferred to the 

SEM stage to map and track surface crack growth. The fatigue cycling was initially performed 

under strain control at 20 cpm (0.33 Hz) with strain ratios of R = 0 or -1 under strain ranges of 

0.6% or 0.8%. Following an initial cycling period during which maximum and minimum loads 

stabilized, testing continued to failure under load control at a frequency of 5 Hz. 

The corresponding small-crack load ratios were approximately R = -0.33 or R = -1. 

The maximum stresses in the small crack tests varied from 154 ksi to 163 ksi for R = -0.33 and 

from 107 ksi to 109 ksi for R = -1. 

GRC provided an extensive data base of small-crack results to SwRI, and only a representative 

selection of these tests were included in this study. Specifically, R = 0 specimens included were 

5776-C-L17, 5776-J-L22, 5776-D-L3, and 5776-H-L21. The R = -1 specimens included were 

5776-C-L2, 5776-D-L18, 5776-H-L14, and 5776-J-L5. Furthermore, data from the first 

250 cycles of fatigue testing (the so-called pop-in period) were excluded from this study. 

Large-crack tests were performed for comparison under load control at 1200F with so-called Kb 

bar (surface crack) specimens at R = -0.25 or -1. Comparisons of large-crack and small-crack 

data (with and without EH corrections) are shown in Figures 21 and 22 for R = -0.33 and 

Figures 23 and 24 for R = -1. Since endurance limit data were not available, the EH parameter 

was again estimated by scaling the average grain size (8 m) by a factor of six. 

Even after excluding the data from the pop-in period, the remaining small-crack data still 

exhibited classical small-crack effects, growing at K values smaller than the large-crack 

threshold, and growing at rates faster than large-crack trends. The EH correction was generally 

successful in correlating large-crack and small-crack data at R = -0.33. At R = -1, the EH 

correction under-corrected the small-crack data relative to the large-crack data, although it did 

successfully consolidate the small-crack data. Unfortunately, only near-threshold small-crack 

data were available. 
NASA/CR—2011-216977 57 

NASA U720

R = -0.33 (Strain Control R



K (ksiin)

1 10 100

da/dN (in/cyc)

10-9

10-8

10-7

10-6

10-5

10-4

Long Cracks (R = -0.25)

Small Cracks

Figure 21. Comparison of small-crack and large-crack growth rate data 

for U720 at R = -0.33. 

NASA U720

R = -0.33 (Strain Control R



K (ksiin)

1 10 100

da/dN (in/cyc)

10-9

10-8

10-7

10-6

10-5

10-4

Long Cracks (R = -0.25)

Small Cracks (EH)

Figure 22. Comparison of small-crack and large-crack growth rate data for U720 

at R = -0.33, with small-crack data adjusted using El Haddad parameter. 
NASA/CR—2011-216977 58 

NASA U720

R = -1.0 (Strain Control R



K (ksiin)

1 10 100

da/dN (in/cyc)

10-9

10-8

10-7

10-6

10-5

10-4

Long Cracks (R = -1.0)

Small Cracks

Figure 23. Comparison of small-crack and large-crack growth rate data for 

U720 at R = -1. 

NASA U720

R = -1.0 (Strain Control R



K (ksiin)

1 10 100

da/dN (in/cyc)

10-9

10-8

10-7

10-6

10-5

10-4

Long Cracks (R = -1.0)

Small Cracks (EH)

Figure 24. Comparison of small-crack and large-crack growth rate data for 

U720 at R = -1, with small-crack data adjusted using El Haddad parameter. 
NASA/CR—2011-216977 59 

3.3.4 Discussion

There are several practical difficulties associated with the El Haddad approach when a0 is 

estimated from Eqn. (24). For example, it is sometimes difficult to define an “endurance limit” 

for some materials. Studies during the last two decades in the high cycle fatigue and very high 

cycle fatigue regimes have indicated that many materials do not exhibit a distinct plateau in 

fatigue strength at very long lives, but that the fatigue strength may continue to decrease with 

further cycling. A stated “endurance limit” may merely reflect the limits imposed on fatigue test 

duration. A shorter test duration may lead to an overestimate of the endurance limit, which will 

give an underestimate of the EH parameter, and potentially an underestimate of small-crack 

growth rates. 

In addition, it is sometimes observed that the failure mode for smooth specimen fatigue tests may 

shift from surface initiation to subsurface initiation at very long lifetimes. Since Eqn. (24) 

requires some assumption about the crack geometry of interest, a shift from a surface crack to an 

embedded crack will require a change in the formulation (and may introduce a discontinuity or 

conflict between data derived from different failure modes). Eqn (24) as written above is 

applicable only to endurance limits associated with surface failure modes. The failure origins 

associated with high cycle fatigue specimens are not always reported. If the endurance limit for 

surface crack behavior is estimated to be too low because it is influenced by subsurface failures 

at longer lives, then the EH parameter may be estimated to be too large, thereby overestimating 

small-crack growth rates. 

Furthermore, fatigue failure behavior in some materials may be controlled by populations of 

inclusions, and the life-limiting crack in a specimen may be associated with a particularly large 

defect. Any effect of this defect on the growth (or non-growth) of microcracks (associated, for 

example, with local stress concentrations or residual stresses) is not reflected in the formulation 

behind Eqn. (24). In some cases, the defect may be sufficiently large that the initial resulting 

microcrack is larger than a0, thereby raising additional questions about the formulation. If the 

intended goal of the EH small-crack formulation is to describe crack growth behavior in normal 

matrix material, then properties derived from defect-dominating behavior may be inappropriate. 

However, if the cracks of interest in the material always arise from such defects, then the 

formulation may end up being self-consistent. 

A different set of difficulties are associated with characterization of the large-crack threshold, the 

other key input into Eqn. (24). It is now known that the apparent large-crack threshold can be 

influenced both by test technique (e.g., traditional load-shedding methods) and specimen 

geometry (large-crack thresholds generated with C(T) specimens are sometimes observed to be 

significantly higher than with M(T) specimens). The most common error appears to be an 

overestimation of Kth based on C(T) tests at R  0 using traditional load-shedding methods. 

Underestimation of Kth appears to be less likely. An overestimate of Kth will contribute to an 

overestimate of a0 and an overestimate of small-crack growth rates. 

Another complication associated with the large-crack threshold is that other environmental 

factors may influence the apparent value of this threshold (for example, if corrosion product 

builds up on the crack faces and causes premature arrest due to oxide-induced crack closure). 
NASA/CR—2011-216977 60 

There may also be environmental influences on small surface cracks in service, but there may not 

be adequate similitude between the threshold test configuration and the small-crack 

configuration for the threshold data to be transferable to the EH calculation. Again, an 

overestimate of Kth would lead to an overestimate of small-crack growth rates by the EH model. 

Given these difficulties associated with obtaining appropriate input data to estimate the EH 

parameter, it is encouraging that some success was observed using microstructurally-motivated 

estimates of a0. It may also be practical to use totally empirical estimates of a0 in some cases, as 

was also done here in a few cases. However, the more empirical the estimate, the greater is the 

need for adequate small-crack growth rate data to validate the estimate. 

The physical meaning of the EH length parameter also deserves some further discussion. 

El Haddad originally considered a0 to be an intrinsic crack length in the microstructure. Such an 

interpretation is troublesome, however, since a0 is on the order of several grain diameters. 

The FCG threshold value for small cracks is known to increase with crack length, approaching 

the long-crack value when the crack becomes large. A more reasonable and physically appealing 

interpretation for a0 is simply that it describes the length scale where the FCG threshold of the 

small crack approaches that of the large-crack threshold. 
NASA/CR—2011-216977 61 

4.0 PROGRAM PLAN FOR FUTURE RESEARCH 

One of the primary objectives of the current study was to identify technologies for assessment of 

P/M materials developed at GRC that could be transferred for use by gas turbine engine 

manufacturers. Based on the GRC status summary and the DARWIN capabilities summary, 

several technology gaps were identified related to (1) characterization and simulation of 3D 

anomalies based on data from different types of inspections, (2) understanding and analysis of 

anomaly deformation and breakup during the forging process, and (3) treatment of various 

factors that can influence fatigue crack growth lives. A program plan for the development and 

implementation of key algorithms and software modules in DARWIN to address the technology 

gaps has been developed and is described in this section. 

Pre-Processing Module for 3D Anomalies. Over the past decade, GRC pioneered a number of 

studies that focused on methods for measuring 3D anomalies in P/M materials anomalies. 

Several methods were identified for anomaly characterization, including direct measurement of 

the three orthogonal axes of anomalies recovered using the heavy liquid separation process 

(HLS), and indirect measurement of these dimensions based on data from sectioning and 

fractography. 3D anomalies are simulated in DARWIN using six independent random variables 

based on three orientation angles and the dimensions of the primary axis and two aspect ratios 

for the remaining two orthogonal axes. A software module is needed to convert the data from 

anomaly measurements into the format that is used in DARWIN. For HLS data, the module 

would convert anomaly major and minor axis measurements into anomaly primary axis and 

aspect ratios. For data from sectioning or fractography, the module would convert the data from 

2D measurements into 3D anomaly dimensions/orientations using the unfolding algorithm 

described previously. Since the number of sections probably will be limited, DARWIN would 

need to be enhanced to quantify confidence bounds on fracture risk associated with the 

uncertainty in the spheroid model for the limited number of sections. Note that HLS 

measurements do not provide any information regarding the orientation or breakup of anomalies 

due to the forging process. On the other hand, 3D ellipsoids cannot be fully characterized based 

on data from sectioning or fractography. A Bayesian updating algorithm is therefore envisioned 

that would combine data from the various test data sources into a common set of variables for 

use in numerical simulations. DARWIN would also be enhanced to display fracture risk 

confidence bounds reduction associated with use of combined data from the various test data 

sources. 

Pre-Processing Module for Failure Data from Multiple Anomaly Types. As mentioned in 

Section 3.1, one of the challenges associated with modeling P/M materials is that the materials—

and the test specimens (particularly unseeded specimens)—typically contain multiple types of 

anomalies that can cause fatigue failure. Under this project, a method was developed for 

estimating the marginal CDFs associated with individual anomaly types, using an approach 

based on the nonparametric statistical model of Kaplan-Meier (1958) combined with the data 

smoothing algorithm proposed by Meeker and Escobar (1998). This method was documented in 

Section 3.1. Although the approach has been demonstrated using real failure data, some effort 

would be required to implement the algorithm in DARWIN. It is envisioned that this would be 

implemented in a separate module that would be used to preprocess the raw failure data into a 

form that could be used directly in DARWIN. 
NASA/CR—2011-216977 62 

Probabilistic Simulation of 3D Anomalies. The current version of DARWIN is limited to 

simulation of uncorrelated random variables. However, GRC studies suggest that the anomaly 

dimensions could be highly correlated random variables, and anomaly orientations could be 

correlated to additional random variables related to forging strain. DARWIN enhancements 

would be required to model these relationships among the various anomaly dimensions and 

orientations. One of the key challenges with the use of correlated random variables to represent 

anomaly dimensions is that it may be difficult to visualize the relationships among the crack 

dimensions in the fracture plane. Further DARWIN enhancements would be required to provide 

GUI visualization of 3D anomalies at user-specified fracture planes. 

Identification of Life-Limiting Anomaly Orientation. To estimate the fracture risk of disk 

materials, the analyst is typically faced with identifying the minimum life location in a 

component. Since the minimum life location is dependent on a number of factors, such as the 

orientation of the fracture plane, the projected area of the anomaly on the fracture plane, and 

local constraint changes associated with crack transitions, it may be difficult to identify. 

Fortunately, DARWIN enhancements such as life contours and risk contours are under 

development, and these will assist the user with identifying the location once a fracture plane has 

been selected. However, the user will still need to identify the orientation of the fracture plane, 

and there is no guarantee that it will be conservative. Anomaly orientation may also influence 

the life-limiting location within a component. An algorithm is therefore needed in DARWIN 

that will automatically identify the life-limiting anomaly orientation. 

Small-Crack Modeling. Small-crack behavior has been identified as being potentially significant 

for accurate lifing of P/M materials. The current DARWIN software does not include any 

special treatment of small-crack behavior, either in terms of accelerated growth rates 

(from a deterministic perspective) or in terms of greater scatter in possible growth rates 

(from a probabilistic perspective). As noted in Section 3.3, SwRI has been investigating 

a simple model for deterministic small-crack growth rate behavior under FAA funding 

(a model which showed some promise for P/M U720), and it is possible that this particular 

model might be implemented in the FAA project. Investigation/development/implementation of 

alternative models would require additional funding. The current DARWIN model for life 

scatter is independent of crack size, and there are no current plans in the FAA project to develop 

and implement alternative probabilistic models for scatter in small-crack growth rates. 

Therefore, this would require additional work. 

Analysis of Forging Effects. Previous GRC studies with P/M materials have indicated that the 

size, orientation, and behavior of inherent anomalies may be significantly influenced by the 

forging process. Several of the aircraft engine manufacturers have expressed interest in exploring 

this phenomenon, and the U.S. Air Force has recently funded research programs related to the 

topic. SwRI will participate in two studies, one a Small Business Innovative Research (SBIR) 

project in partnership with Scientific Forming Technologies Corporation (SFTC), and the other 

as a subcontractor to Pratt & Whitney (P&W). The SFTC study will focus on linking the SFTC 

DEFORM software with DARWIN, with the initial phase focused on the transfer of forging 

residual stress data from DEFORM to DARWIN. Future phases, if funded, would consider 

additional variables such as anomaly size and orientation as well as microstructural information. 

The SwRI effort for the P&W study will also use DEFORM data to model the influence of the 
NASA/CR—2011-216977 63 

forging process on anomaly orientation. Since the USAF has funded only very specific features, 

it is anticipated that further funding may be required to combine the various features related to 

forging effects into a comprehensive toolset that will be of use to the general DARWIN user. 

Residual Stress Issues. A number of residual stress issues are relevant to P/M lifing. As noted in 

the earlier Status Summary (Section 2.2), DARWIN currently has a basic capability to 

superimpose a user-provided residual stress gradient with 2D finite element model stress results 

for a single crack geometry. Some additional work is planned under the current FAA grant to 

extend this capability to additional crack geometries and additional analysis modes. 

Further effort would be required to address a complete set of crack geometries and analysis 

modes. As noted in the previous paragraph, SwRI is currently working with SFTC to develop a 

new DARWIN capability to read DEFORM results files for forging (bulk) residual stresses and 

superimpose these with DARWIN service stresses, and this effort should be completed if the 

SBIR funding continues. Additional work remains to develop and implement predictive models 

for local residual stresses (e.g., machining residual stresses, or residual stresses arising from 

differential thermal expansion of inclusions) and for the significant relaxation and redistribution 

of residual stresses that can occur in service due to both thermal and mechanical mechanisms. 

Other Issues. The development of DARWIN has been funded primarily by the FAA, and the 

existing features reflect the priorities of the FAA, RISC, and the Industry Steering Committee. 

There are a number of issues that influence fracture risk that are not addressed in the current 

project, and due to competing FAA priorities, probably will not be completed in future FAA 

projects. For example, an approach is needed to model anomalies that intersect the free surface 

of the component. In the current version of DARWIN, the free surface is assumed to intersect 

the midpoint of the anomaly, but an approach is needed to model intersections at various 

anomaly locations to simulate the volume of material that is removed by the machining process. 

 

Modeling of the crack formation life is another critical area of need for DARWIN. Currently, 

DARWIN provides no internal models for predicting the formation/nucleation/initiation life. 

It instead provides a software link to a separate user-supplied crack formation module that 

provides only limited treatment of the crack formation life. Additional work is needed to 

develop general crack formation modeling capabilities, or to expand the existing software link to 

include other important variables that the user may wish to consider in their own crack formation 

module. 

The implementation of formal algorithms in DARWIN necessarily requires validation of the 

engineering models, which is commonly achieved by comparison to actual test data. 

Verification of the software algorithms is also required to confirm that the engineering models 

are implemented in the intended form, and a standardized procedure has been developed at SwRI 

to accomplish this task. It is essential to include verification and validation (V&V) activities in 

any future task funded to enhance DARWIN for P/M lifing. GRC has generated extensive test 

data on the behavior of inclusions in P/M materials and the resulting effect on fatigue life, and 

these data should prove invaluable in future V&V activities. 
(This page intentionally left blank.) 
NASA/CR—2011-216977 65 

5.0 SUMMARY 

Southwest Research Institute and NASA Glenn Research Center have worked independently in 

recent years on the development of probabilistic lifing methods for gas turbine engine rotor 

materials. This final report summarizes a brief investigation into the current status of relevant 

technology at SwRI and GRC with a view towards a future integration of methods and models 

developed by GRC for probabilistic lifing of powder metallurgy (P/M) nickel rotor alloys into 

the DARWIN software developed by SwRI. The current status of GRC data and models and the 

DARWIN software was reviewed in detail, and significant accomplishments/capabilities and 

technology gaps were identified. New algorithm development was conducted in three 

areas. First, a rigorous methodology for estimating anomaly distributions for materials in which 

multiple anomaly types are operative was developed and verified. Second, an unfolding 

algorithm that can be used to estimate the dimensions and orientations of 3D spheroids based on 

2D sectioning measurements was implemented and investigated. Third, the behavior of small 

fatigue cracks in U720 tests conducted by GRC was investigated, and an existing model to 

predict small-crack behavior from large-crack growth rates was evaluated. Finally, a program 

plan was outlined for potential future research, identifying the most significant opportunities to 

address technology gaps and to develop needed new DARWIN capabilities. 
(This page intentionally left blank.) 
NASA/CR—2011-216977 67 

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NASA/CR—2011-216977 69 

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NASA/CR—2011-216977 70 

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NASA/CR—2011-216977 71 

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NASA/CR—2011-216977 73 

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14. ABSTRACT

In recent years, Southwest Research Institute (SwRI) and NASA Glenn Research Center (GRC) have worked independently on the 

development of probabilistic life prediction methods for materials used in gas turbine engine rotors. The two organizations have addressed 

different but complementary technical challenges. This report summarizes a brief investigation into the current status of the relevant 

technology at SwRI and GRC with a view towards a future integration of methods and models developed by GRC for probabilistic lifing of 

powder metallurgy (P/M) nickel turbine rotor alloys into the DARWIN (Darwin Corporation) software developed by SwRI.

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Fatigue; Probabilistic life prediction; P/M disk superalloy; Fatigue crack growth 

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