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i

NASA/TM—2012–217456

Analytical Round Robin for Elastic-Plastic

Analysis of Surface Cracked Plates: 

Phase I Results

D.N. Wells and P.A. Allen

Marshall Space Flight Center, Huntsville, Alabama

March 2012

National Aeronautics and

Space Administration

Marshall Space Flight Center • Huntsville, Alabama 35812
ii

Available from:

NASA Center for AeroSpace Information

7115 Standard Drive

Hanover, MD 21076 –1320

443–757–5802

This report is also available in electronic form at

<https://www2.sti.nasa.gov/login/wt/>

Acknowledgments

The authors would like to acknowledge the support of the ASTM Committee on Fatigue and Fracture (E08), 

with a special note to James Joyce (U.S. Naval Academy, Retired), Richard Link (U.S. Naval Academy), and Robert 

Tregoning (U.S. Nuclear Regulatory Commission) for their support during the development of the revised surface 

crack testing standard and encouraging this analytical round robin to go forward. As always, our mentor Robert H. 

Dodds Jr. (University of Illinois at Urbana-Champaign) has provided a continuous stream of support and guidance. 

Clearly, the most important part of any round robin study is the group of participants graciously lending their time 

and effort to the undertaking. We would like to thank each of them sincerely: Steven Altstadt at Stress Engineering Services, Mark Messner at the University of Illinois at Urbana-Champaign with the support of Robert Dodds, 

Ramesh Mageshwaren, and Marc Scibetta at SCK CEN (Belgian Nuclear Research Center), Matt McCutcheon at 

NASA Marshall Space Flight Center (MSFC) (Propulsion Structures), Dawn Phillips and Brian Steeve at MSFC 

(Vehicle Structures), Howard Rathbun at the U.S. Nuclear Regulatory Commission, Guowu Shen at CANMET 

(Canada Centre for Mineral and Energy Technology), Greg Thorwald and Chris Tipple at Quest Integrity Group, 

Igor Varfolomeev at the Fraunhofer-Institut fuer Werkstoffmechanik IWM (Germany), Xin Wang at Carleton University (Canada), Stefan Bickelmaier and Michael Windisch at MT Aerospace AG (Germany), and Yoshiki Yamada 

at NASA Glenn Research Center. Our work on surface cracks at MSFC would not be possible without the support 

of Vann Bradford, Tommy Stewart, and Damon Cleghorn. We greatly appreciate their efforts in helping us move 

the new surface crack test standard forward.
iii

TABLE OF CONTENTS

1. INTRODUCTION ............................................................................................................. 1

1.1 Motivation for the Inter-Laboratory Study ................................................................... 1

1.2 Scope and Objectives for Phase I of the Inter-Laboratory Study .................................. 1

2. PROBLEM STATEMENT, PHASE I ................................................................................ 3

3. PARTICIPANT OVERVIEW ............................................................................................. 5

4. RESULTS AND DISCUSSION ......................................................................................... 8

4.1 Tying Analysis to Experiment ....................................................................................... 10

4.2 J-Integral Results .......................................................................................................... 19

4.3 Assessing Sources of Variability ................................................................................... 30

5. CONCLUSIONS ................................................................................................................ 68

6. RECOMMENDED PRACTICES ..................................................................................... 69

APPENDIX A—PHASE I PROBLEM STATEMENT AND TABULAR DATA ................. 71

APPENDIX B—EVALUATION OF INTER-LABORATORY STUDY RESULTS

ASSUMING A NORMAL DISTRIBUTION ............................................. 75

REFERENCES ....................................................................................................................... 79
iv

LIST OF FIGURES

 1. Surface crack test specimen configuration ................................................................ 3

 2. ILS specimen configured for testing .......................................................................... 9

 3. ILS specimen fracture surface with tearing location indicated .................................. 10

 4. Experimental force versus CMOD response ............................................................. 11

 5a. Force versus CMOD for all ILS participants ............................................................ 12

 5b. Force versus CMOD for all ILS participants at end of test ....................................... 13

 6. Deviation in predicted CMOD relative to lab-5 ........................................................ 14

 7. Elastic stiffness for each model relative to current ±2.5% limits ................................ 15

 8a. Overall force versus CMOD with “corner bracket” validity limit .............................. 16

 8b. Force versus CMOD, zoomed to “corner bracket” validity limit .............................. 17

 9a. Overall force versus CMOD with CMOD-based validity limit ................................. 18

 9b. Force versus CMOD, zoomed to CMOD-based validity limit .................................. 19

10a. J-integral at φ=17° versus force ................................................................................ 20

10b. J-integral versus φ at 200 kN ..................................................................................... 21

10c. J-integral versus φ at 252 kN ..................................................................................... 22

10d. J-integral versus φ at 289 kN ..................................................................................... 23

10e. J-integral versus φ at 200 kN normalized by maximum J-integral value .................... 24

10f. J-integral versus φ at 252 kN normalized by maximum J-integral value .................... 25

10g. J-integral versus φ at 289 kN normalized by maximum J-integral value .................... 26

11. J-integral versus CMOD at φ=17° ........................................................................... 29
v

LIST OF FIGURES (Continued)

12. J-integral versus CMOD at φ=17° with CMOD-based evaluation criteria ............... 30

13. J-integral at φ=17° versus analysis code ................................................................... 33

14. Typical 1⁄4 symmetric surface crack finite element mesh ............................................ 34

15. J-integral at φ=17° versus model size parameter from equation (1) .......................... 35

16a. Variation in elastic CMOD stiffness versus element integration order ...................... 36

16b. J-integral at φ=17° versus element integration order ................................................ 37

17. Crack tip mesh with undeformed geometry on left and deformed on right 

for (a) collapsed elements with tied nodes, (b) collapsed elements with untied 

nodes, and (c) keyhole pattern .................................................................................. 38

18. J-Integral at φ=17° versus crack tip mesh geometry ................................................. 39

19. Illustration of typical J-integral results showing degrees of path dependency ........... 40

20. Illustration of dimension used to characterize domain size ....................................... 41

21. J-integral at φ=17° versus normalized domain size .................................................. 42

22. J-integral at φ=17° versus method used to evaluate J from multiple domains ........... 43

23. J-integral at φ=17° versus model boundary condition .............................................. 44

24a. Curved crack front with skewed elements relative to the crack front normal, 

a common source of oscillating J-integral solutions ................................................. 45

24b. Curved crack front with elements aligned with the crack front normal ..................... 46

25. Oscillating J-integral solution affecting the lab-8 solution ........................................ 46

26. J-integral at φ=17° versus use of true or engineering stress-strain material 

properties .................................................................................................................. 48

27. J-integral at φ=17° versus use of small-strain or finite-strain formulation ................ 49

28. Stress-strain inputs in engineering stress-strain space ............................................... 50
vi

LIST OF FIGURES (Continued)

29a. Engineering stress versus plastic strain ..................................................................... 51

29b. Engineering stress versus plastic strain illustrating range of proportional limits ....... 52

30. J-integral at φ=17° versus proportional limit in constitutive inputs .......................... 53

31. Area under the engineering stress-plastic strain curve as function of stress .............. 54

32a. Engineering stress-strain curves used in the proportional limit substudy .................. 55

32b. All variations in proportional limit occur prior to standard 0.2% offset 

strain yield definition ................................................................................................ 56

33. Area under the engineering stress-plastic strain curve as function of stress 

for the proportional limit substudy ........................................................................... 57

34. Images (a) through (j) illustrate the effects of proportional limit on the resulting 

J-integral value as the model is loaded through the ILS load regime and beyond ..... 58

35. Bounding models at the tearing CMOD for the CMOD-matching substudy ............ 65

36. Comparison of scaled material property inputs for the CMOD-matching 

substudy ................................................................................................................... 66

37. Bounding models at the deformation-limited CMOD for the CMODmatching substudy .................................................................................................... 67

38. Probability plot for normal distribution of J-integral values ..................................... 75

39. Probability plot for normal distribution of elastic stiffness of modeled 

CMOD response ....................................................................................................... 76

40. Probability plot for normal distribution of the number of nodes used 

to model the crack front ........................................................................................... 77

41. Probability plot for normal distribution of the model size parameter 

from equation (1) ...................................................................................................... 77
vii

LIST OF TABLES

 1. List of ILS participants and affiliation ........................................................................ 5

 2. Summary of J-integral results ...................................................................................... 27

 3. Summary of ILS model parameters for all participants ............................................... 31

 4. Summary of results from CMOD-matching substudy ................................................. 64
viii

LIST OF ACRONYMS

ASTM American Society for Testing and Materials

CMOD crack mouth opening displacement

CTOD crack tip opening displacement

DCPD direct current potential drop

FEA finite element analysis

ILS inter-laboratory study

MSFC Marshall Space Flight Center

MSP model size parameter

TM Technical Memorandum
ix

NOMENCLATURE

2c surface crack length (see fig. 1)

a crack depth (see fig. 1)

B specimen thickness (see fig. 1)

E Young’s modulus of elasticity

K stress intensity factor

L specimen height (see fig. 1)

Ma deformation state characterized by crack length, i.e., Ma=rφaσys/Jφ

rφa and rφb specimen characteristic length used to evaluate the deformation state 

(see inset in fig. 34a)

W specimen width (see fig. 1)

m mean of population (number of nodes)

n Poisson’s ratio

s standard deviation of the population (number of nodes)

sys material yield strength

l local crack extension (see fig. 3)
x
1

TECHNICAL MEMORANDUM

ANALYTICAL ROUND ROBIN FOR ELASTIC-PLASTIC ANALYSIS OF SURFACE 

CRACKED PLATES: PHASE I RESULTS

1. INTRODUCTION

The ASTM Committee on Fatigue and Fracture (E08) Task Group on Fracture Toughness 

of Surface Cracks (E08.07.03) is developing a new material test standard for the assessment of 

surface crack toughness in the linear-elastic and the elastic-plastic regime. While evaluation of 

the linear-elastic stress intensity factor, K, is well accepted for surface crack geometries through 

the Newman-Raju equations1,2 or other existing tabulations,3 the evaluation of elastic-plastic 

J-integral values requires the user to perform an analytical assessment of the experiment using 

methods that cannot be easily contained or conveyed within a test standard. The feasibility of 

allowing this latitude in assessment methodology in a testing standard requires validation through 

an inter-laboratory study (ILS). This Technical Memorandum (TM) provides an overview of the 

problem statement and results of the first phase of the study.

1.1 Motivation for the Inter-Laboratory Study

Existing mechanical testing standards for fracture mechanics applications, such as ASTM 

E1820,4 include within the document all the necessary equations and supporting data to assess the 

experiment directly, producing a “standard” result. In the case of a surface crack in a flat plate, no 

openly available method exists for accurately evaluating the J-integral under elastic-plastic conditions along the full perimeter of a surface crack that can feasibly be reduced to equation form for 

inclusion into a test standard. However, the ability to assess this problem using methods of analysis 

such as finite elements has progressed considerably in recent years and may now be a suitable substitute for assessment of experimental results in lieu of fixed relations within the document. Codes 

defining standard practice for structural evaluation of defects, such as the API Recommended 

Practice 579, Fitness-for-Service,3 allow for such external analysis methods. This study is designed 

to evaluate the feasibility of an external analysis method for experimental data assessment.

1.2 Scope and Objectives for Phase I of the Inter-Laboratory Study

This ILS is planned to develop in phases based on need and observations made as the study 

progresses. Additional phases are likely to investigate different crack shapes, bending as opposed 

to tension loading, and different materials. This first phase approached the ILS concept differently 

than most experimentally based studies—the analysis methodology was not specified to participants and key experimental results were withheld, as explained below. This made the study “blind” 
2

in the sense that the participants did not have the experimental force and displacement information available to them to validate their model, or to potentially alter their model to converge on the 

experimental result, and they were free to approach the analysis using any method of choice. This 

approach allowed this first phase of the ILS to serve as a current measure of the state of the art for 

elastic-plastic fracture mechanics analysis. An open study of this nature regarding elastic-plastic 

analysis methods has not been performed for some time.5,6

Given this approach, the ILS phase I results are representative of the surface crack test 

standard providing essentially no guidance regarding the method of test analysis; however, analysis 

guidelines are to be provided within the testing standard to maximize the likelihood that users will, 

to the extent possible, reach a “standardized” answer. These guidelines are expected to derive from 

three sources: an assessment of the causes of variability within the ILS results, a compilation of 

the best generally accepted modeling practices, and the commonality of methods chosen by the ILS 

participants. 
3

2. PROBLEM STATEMENT, PHASE I

The initial effort, described as phase I—problem A, was an analysis of a surface crack in 

tension. The basic geometry and definition of parameters for a generic surface crack specimen 

are shown in figure 1. The complete problem statement as it was provided to each participant 

is included in appendix A along with the necessary tabular data for the reader to independently 

analyze the problem. The material properties for the problem were provided to all participants 

in the form of engineering stress versus strain data from a tensile test. The elastic properties 

(E=74.46 GPa and ν=0.33) were defined to remove that potential source of variability from 

the study. The choice of constitutive models and treatment of the stress-strain data was left to 

the discretion of the participants.

CMOD

W

A 2c

L/2

L

B

A

φ

B

cco

a

Section A-A

Typical Surface-Crack Specimen

(Grip Details Omitted)

a = crack

2c = crack length

B = plate thickness Back

Surface

Fatigue

Precrack

2c

 Figure 1. Surface crack test specimen configuration. 
4

From their analysis, each participant was requested to provide three primary results, selected 

based on the nature of results required by the test method. Participants were requested to run their 

elastic-plastic analysis with sufficient quantity of load steps to report the following data:

(1) Force versus crack mouth opening displacement (CMOD) throughout the test.

(2) J-integral versus parametric angle location (φ) for a set of specific force values.

(3) J-integral versus force throughout the test for a given parametric angle location (φ).
5

3. PARTICIPANT OVERVIEW

This phase of the study had 15 participants covering a range of industries, academia, and 

countries. There was also intent to include participants with a broad range of expertise in the study. 

The user community for the test standard is potentially broad; therefore, our intent was not to have 

only “experts” in the field participating. A study with only expert participants would not be properly representative. It could lead to overly optimistic results and a lack of comprehensive analytical 

recommendations for users of the test standard.

A list of the participants is included in table 1. All participants have been assigned a random designation from lab-1 to lab-15. All results remain anonymous, though each participant was 

informed of their lab identification.

Table 1. List of ILS participants and affiliation.

Name Affiliation

Phillip Allen (lab-1) MSFC Materials Laboratory

Steven Altstadt Stress Engineering Services

Mark Messner/Bob Dodds University of Illinois at Urbana-Champaign

Ramesh Mageshwaran SCK CEN, Belgium

Matt McCutcheon MSFC Propulsion Structures

Dawn Phillips MSFC Vehicle Structures

Howard J. Rathbun U.S. Nuclear Regulatory Commission

Guowu Shen MTL/CANMET, Canada

Brian Steeve MSFC Vehicle Structures

Greg Thorwald Quest Integrity Group

Chris Tipple Quest Integrity Group

Igor Varfolomeev Fraunhofer-Institut fuer Werkstoffmechanik IWM

Xin Wang Carleton University, Canada

Stephan Bickelmaier MT Aerospace AG, Germany

Yoshiki Yamada NASA Glenn Research Center

After receiving all contributions, the authors sent a survey to each participant to document 

the input parameters for each of the analyses. The questions in the survey were not exhaustive, 

but were selected with insight into expected sources of variation in these assessments. To minimize interpretation issues in the answers, a list of likely answers was provided; however, the survey 

always allowed the participant to provide any response along with explanatory comments. The 

survey requested the following information, with the standardized responses provided appearing 

in {brackets}:
6

(1) General

(a) Principle background of analyst {materials, structures, other}

(b) Method of analysis used {finite elements, SINTAP, other}

(c) What code was used {ABAQUS, ANSYS, WARP3D, other}

(2) Model specifics

(a) Total number of elements

(b) Total number of nodes

(c) Number of nodes along the crack front

(d) Type of element {linear hex (8-node), quadratic hex, (20+ node), other}

(e) Element integration order {full, reduced}

(f) Strain assumption {small strain, finite strain}

(3) Crack tip modeling

(a) Crack tip elements/geometry {collapsed hex-untied nodes, collapsed hex-tied nodes, 

collapsed hex-tied nodes with 1⁄4 point mid-side node, key-hole or finite radius, no special crack tip 

meshing, other}

(4) Boundary conditions

(a) Grip end modeling {simple rectangular flat plate, modeled grip end details 

(provide comments)}

(i) If grip end details modeled, description of boundary condition is required

(ii) If rectangular flat plate {uniform displacement, uniform pressure stress, other}

(b) Number of symmetry planes {none, 1, 2}

(5) Material modeling (actual stress-strain inputs requested)

(a) Type of plasticity {incremental, nonlinear elastic/deformation}

(b) Engineering or true stress and strain {engineering, true} 

(c) Stress-strain curve representation {bilinear, multilinear table, linear + power law, 

power law, other}

(6) J-integral calculations

(a) Method {domain integral, crack tip opening displacement (CTOD), other}

(b) If domain integral was used

(i) Number of domains

(ii) Method to get J from multiple domains {average of all, average of all but first, 

average of converged, average of subset (e.g., last 4 of 10), maximum value of all domains, other}

(iii) Dimension of outermost domain.

An additional short survey was provided if the submitted analysis was not based on finite 

element methods. While a few participants did submit supplementary analysis that was not finite 

element based, these analyses were provided for reference purposes with the finite element results 

being the primary submittal to the study. All results included in the study were based on finite 

element models.
7

Each lab’s submission was reviewed upon receipt. If there appeared to be any form of basic 

issue with the provided data, the authors contacted the participant for clarification. There were 

a few cases of errors created during the copy and paste process to move data from finite element 

code output to the requested Excel spreadsheet summaries. These corrections were allowed in order 

to avoid unwarranted variation in the results. In each case, the simple errors corrected in this process would have been easily noticed had the study not been blind to the experimental results.
8

4. RESULTS AND DISCUSSION

As a basis for discussing the ILS results, it is helpful to present the experimental results 

upon which this phase of the ILS is based. The participants were not aware of these results at 

the time of their assessment.

The experiment consisted of a 2219-T8 aluminum specimen of the dimensions shown in 

appendix A. The photograph of the specimen during test (see fig. 2) shows the ring gauge used for 

CMOD as well as input current and probe wires for direct current potential drop (DCPD) measurements. Changes in the DCPD signal assisted as a nonquantitative indicator of crack extension 

during the test. The DCPD probe spacing was ≈1 mm using local platinum probe wires welded 

near the right-hand 2c location of the precrack. The outer probe wires visible in the photograph 

provided a reference signal to normalize the DCPD measurement. The specimen was precracked 

in tension, instrumented, and then loaded monotonically until the combination of CMOD and 

DCPD signals indicated sufficient change to suggest local tearing had occurred. The specimen 

was partially unloaded and the extent of the crack tearing was marked using a sodium hydroxide 

solution. The solution was rinsed from the crack and the crack dried while the partial load was 

maintained. The specimen was then subjected to fatigue marking cycles at ≈60% of the tearing load 

to help accentuate the ductile tearing fracture surface. Finally, the specimen was monotonically 

loaded to failure using the same instrumentation.
9

 Figure 2. ILS specimen configured for testing.

The proposed surface crack test method relies on a multispecimen technique to narrow in 

on appropriate amounts of crack tearing for test assessment. The fracture surface for the ILS test, 

annotated in figure 3, reveals the local (maximum) crack extension (l) was 0.5 mm, at a parametric 

angle of 17°. This information, combined with the tearing force of 251.8 kN and the corresponding 

CMOD, constitute the experimental test result. The intent of the proposed surface crack standard 

is to define the crack tip conditions at the initiation of ductile tearing. Therefore, the analysis 

requested by the ILS was to estimate the J-integral at the parametric angle φ=17° at the tearing 

force of 252 kN, and to predict the force versus CMOD response throughout the test. Though 

experimental results cannot be compared beyond this force level, the requested analysis was 

extended to a force of 289 kN in order to provide insight into analysis variability at higher 

deformation levels.
10

a

B

φ

2c

Figure 3. ILS specimen fracture surface with tearing location indicated.

4.1 Tying Analysis to Experiment

The only physical tie between the analysis and experiment is the force versus CMOD 

response. Based on this metric, the proposed test standard requires agreement between analysis and 

experiment in two ways: (1) the slope of the force-CMOD elastic response and (2) the final analytical force-CMOD prediction. The limits of acceptable agreement for these metrics are still under 

review and studies such as this ILS will help determine the required bounds. 

The force-CMOD response from the experiment is shown in figure 4. The nonlinearity 

in the test record and the permanent CMOD offset of ≈0.015 mm are indicators of elastic-plastic 

conditions in the specimen. The force-CMOD predictions from the 15 ILS labs are plotted along 

with the experimental record in figures 5a and 5b. Consistent agreement is found in the elastic 

slope with only lab-2 predicting a slightly stiffer response. The predicted nonlinearity of the forceCMOD response due to plasticity is also reasonably consistent up to the tearing force of 252 kN. 

The deviation in CMOD prediction increases significantly at force levels above 252 kN. Figure 6 

illustrates the variation in CMOD predictions relative to the arbitrarily chosen lab-5 result. Three 

response families are visible in the result: lab-2 stands unique; labs -6, -14, and -15 share a slightly 

stiffer response; and the remaining labs are grouped as a general family largely within the ±1 

standard deviation lines shown. The appropriateness of applying normal distribution statistics 

to various ILS data is discussed in appendix B.
11

300

250

200

150

100

50

0

0 0.02 0.04 0.06 0.08 0.1 1.12

CMOD (mm)

Force (kN)

Experiment

Figure 4. Experimental force versus CMOD response.
12

300

250

200

150

100

50

0

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

CMOD (mm)

Force (kN)

Experiment

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 5a. Force versus CMOD for all ILS participants.
13

Experiment

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

295

290

285

280

275

270

265

260

255

250

245

0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17

CMOD (mm)

Force (kN)

Figure 5b. Force versus CMOD for all ILS participants at end of test.
14

0.015

0.01

0.005

0

–0.005

–0.01

–0.015

–0.02

–0.025

–0.03

0 50 100 150 200 250 300

Deviation CMOD, Lab-5 as Basis (mm)

Force (kN)

+1 Std Dev

–1 Std Dev

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 6. Deviation in predicted CMOD relative to lab-5.

To maintain acceptable bounds on the analytical assessment of the experiment, the forceCMOD traces must agree using reasonable metrics. Currently, two metrics are proposed. The first 

is a simple evaluation of the elastic stiffness of the force-CMOD relationship. This evaluation 

provides a broad check on numerous basic inputs to the analysis, including the specimen and crack 

geometry, the elastic stiffness in the material constitutive model, and to a lesser degree, the modeled 

boundary conditions. Many fundamental analysis mistakes can be screened with this evaluation. 

Currently, the test standard’s allowable range for the analytical prediction is ±2.5% of the experimental stiffness. Figure 7 shows the modeled stiffness for each of the labs relative to the experiment 

along with the ±2.5% bounding lines. The stiffness was determined by a regression of force-CMOD 

data between 30 and 150 kN. Thirteen of 15 labs matched the experimental stiffness within these 

limits with their blind analysis. It is important to note that in this phase of the ILS, the elastic modulus of the material (E=74.46 GPa) was provided directly as part of the problem statement. The 

inevitable variability of selecting an elastic modulus from literature, handbooks, or test data is not 

reflected in this part of the study. The most direct means of correcting the analytically predicted 

specimen stiffness is through changing the model’s elastic modulus. Additional guidance will be 

required to appropriately limit the liberties taken with the modulus in order to match experimental
15

Exp 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

2,500

2,550

2,600

2,650

2,700

2,750

2,800

2,850

2,900

Elastic Stiffness (kN/m

m)

Lab

Stiffness

Experiment

2.5%

–2.5%

Figure 7. Elastic stiffness for each model relative to current ±2.5% limits.

results. A need to alter the modulus value in the analysis by more than a few percent to match the 

experiment is an indicator that some other aspect of the analysis or experimental data is in error.

The second proposed metric to match the analysis results to the test results concerns the 

maximum force and CMOD recorded during the test. This is the point in the test when the specimen is unloaded after the initiation of tearing. It is unlikely the test and analysis records will match 

exactly, so there is a potential to evaluate the test result based on the analysis matching either the 

test force or the test CMOD. The current draft of the test standard proposes test evaluation when 

the analysis reaches either the maximum experimental force or CMOD, whichever is first. This 

evaluation method is illustrated in figures 8a and 8b. The corner bracket shown at the end of the 

experimental record represents a limit on force, allowing up to –10% error in CMOD prediction, or 

a limit on CMOD allowing up to a –10% error in force prediction. All ILS analyses reflect a model 

stiffer than the experiment regarding the response to plastic deformation. This is an expected result 

from two perspectives:
16

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

0

50

100

150

200

250

300

CMOD (mm)

Force (kN)

Experiment

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 8a. Overall force versus CMOD with “corner bracket” validity limit.
17

0.09 0.095 0.1 0.105 0.11 0.115 0.12

220

225

230

235

240

245

250

255

260

265

270

CMOD (mm)

Force (kN)

Experiment

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 8b. Force versus CMOD, zoomed to “corner bracket” validity limit.

(1) The finite element methodology discretizes the geometry and enforces an assumed 

displacement field within each of the elements, creating a model response stiffer than reality. 

(2) The nonlinearity in the experimental force-CMOD trace incorporates plasticity as well 

as geometric changes such as localized tearing; therefore, the test record nonlinearity may have 

plastic CMOD offset and elastic compliance changes due to crack extension superimposed. 

The crack geometry in the analysis does not change—creating a stiffer analytical response. 

It is important that the test record not have excessive compliance change due to crack extension 

(tearing) in order for this assessment methodology to apply; i.e., the crack shape after tearing 

should not be appreciably different from the precrack shape. The proposed test standard sets limits 

on the amount of crack tearing allowed.

More clearly visible in figure 8b, 11 of 15 analyses satisfy the criteria of matching CMOD 

within 10% at the maximum test force of 252 kN. The stiffer results of labs -2, -6, -14, and -15 

underpredict the CMOD by slightly more than 10%. Note that lesser values of CMOD at a given 

force correspond to a lower value of the J-integral. Setting limits using the force-CMOD corner 
18

bracket ensures a conservative assessment of the experiment. The maximum J-integral evaluated 

from the test would result from an exact match of the experimental record—hitting the corner 

of the bracket. Any deviations from the experimental force CMOD record are truncated by this 

method and will reduce the resulting J value. Anticipating the use of this methodology, this ILS 

problem statement requested reporting of J-integral values at specific force levels.

Tests that result in tearing under predominantly linear-elastic conditions are well characterized by either force or CMOD. Tests extending into the elastic-plastic regime behave differently, 

with the value of J increasing as a strong exponential of the force and varying more directly proportional to increases in crack opening displacements, characterized by the CTOD and, as related 

mostly by geometry, the CMOD. Therefore, as plastic effects begin to dominate at the crack front, 

the CMOD becomes a better behaved and more reliable indicator of the crack tip conditions than 

applied force. Based on these arguments, a CMOD-based criterion for comparing test and analysis records may be a more accurate way to determine J. Figures 9a and 9b illustrate the use of this 

criterion. The error bar represents a ±5% allowable error in force prediction at the experimental 

CMOD. The same subset of results is valid by this evaluation as was valid in the case of the corner 

bracket method. While potentially more accurate, this approach allows evaluation of the test result

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

50

0

100

150

200

250

300

CMOD (mm)

Force (kN)

Experiment

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 9a. Overall force versus CMOD with CMOD-based validity limit.
19

0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 0.13

220

230

240

250

260

270

280

CMOD (mm)

Force (kN)

Experiment

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 9b. Force versus CMOD, zoomed to CMOD-based validity limit.

at a force higher than that actually achieved in the experiment. The result is that an exact match 

to the experimental record represents the mean of allowable evaluations, not the maximum. Since 

analyses without crack extension are biased towards a stiffer response (higher force for given 

CMOD), this method will be biased toward producing higher J-integral values. This method is 

likely to produce more consistent test evaluations when the force-CMOD record is highly nonlinear 

due to plasticity and has achieved a near zero slope at tearing. However, most tests producing this 

type of force-CMOD record are not likely within the currently established limits of deformation for 

the elastic-plastic regime established by the proposed test standard. Further review of the effects of 

these evaluation methods follow during discussion of J-integral results and the influence of constitutive modeling choices.

4.2 J-Integral Results

The reported J-integral values from each of the ILS labs are presented in figures 10a 

through 10g. The surface crack is a three-dimensional crack case, which results in a strong variation in J around the crack perimeter. Ideally, the experimental results will indicate where along the 

perimeter the maximum amount of crack extension occurred. In the case of this ILS problem, the 

location of maximum tearing occurred at a parametric angle, φ=17°. In analysis of this test, the 
20

ILS labs reported J values at each analytical load step from zero to 289 kN at this φ=17° location. 

Additionally, the J values were requested at discrete forces (200, 252, and 289 kN) to be reported 

along the crack perimeter for 0° ≤φ≤90°.* Figure 10a presents the J-integral versus force results 

for all labs at φ=17°. The variability across labs is not extensive up to the test force of 252 kN. At 

higher forces, the variability increases, but not exorbitantly, considering the blind nature of the 

study. The values of J at φ=17° are tabulated along with their average, range, and standard deviation in table 2 to assist with quantitative comparison.

0 50 100 150 200 250 300

0

5

10

15

20

25

30

35

Force (kN)

J-Integral (kJ/m2

)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 10a. J-integral at φ=17° versus force.

—————

*Some participants cautioned against the use of their J results for φ ≤ 5° due to the strong gradient created by the loss of the crack singularity at the free surface. The proposed surface crack test 

standard prohibits evaluation of J in this range of φ; therefore, these values are not significant to 

the study.
21

20

18

14

16

12

8

10

6

6

4 φ a

co 2occ

2

Parametric Angle, (deg)

J-Integral (kJ/m2

)

0 15 30 45 60 75 90

 a = Crack Depth

2c = Crack Length

Back B = Plate Thickness

Surface

Fatigue

Precrack B

φ

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 10b. J-integral versus φ at 200 kN.
22

20

18

16

12

14

10

8

4

0 15 30 45 60 75 90

6

J-Integral (kJ/m2

)

φ a

co 2occ

 a = Crack Depth

2c = Crack Length

Back B = Plate Thickness

Surface

Fatigue

Precrack B

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Parametric Angle, φ (deg)

Figure 10c. J-integral versus φ at 252 kN.
23

40

35

30

25

20

15

10

5

0 15 30 45 60 75 90

J-Integral (kJ/m2

)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

φ a

co 2occ

 a = Crack Depth

2c = Crack Length

Back B = Plate Thickness

Surface

Fatigue

Precrack B

Parametric Angle, φ (deg)

Figure 10d. J-integral versus φ at 289 kN.
24

1.1

1

0.9

0.7

0.8

0.6

0.5

0 3

0.4

0 15 30 45 60 75 90

ao

co

2co

φ a

co 2occ

 a = Crack Depth

2c = Crack Length

Back B = Plate Thickness

Surface

Fatigue

Precrack B

J-Integral, Normalized by Maximum J Value

Parametric Angle, φ (deg)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 10e. J-integral versus φ at 200 kN normalized by maximum J-integral value.
25

r

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

φ a

co 2occ

 a = Crack Depth

2c = Crack Length

Back B = Plate Thickness

Surface

Fatigue

Precrack B

0 15 30 45 60 75 90

Parametric Angle, φ (deg)

1.1

1

0.9

0.7

0.8

0.6

0.5

0 3

0.4

J-Integral, Normalized by Maximum J Value

Figure 10f. J-integral versus φ at 252 kN normalized by maximum J-integral value.
26

r

co

1.1

1

0.9

0.7

0.8

0.6

0.5

0 3

0.4

0 15 30 45 60 75 90

J-Integral, Normalized by Maximum J Value

Parametric Angle, φ (deg)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

φ a

co 2occ

 a = Crack Depth

2c = Crack Length

Back B = Plate Thickness

Surface

Fatigue

Precrack B

Figure 10g. J-integral versus φ at 289 kN normalized by maximum J-integral value.
27

Table 2. Summary of J-integral results.

J-Integral at ø= 17° (kJ/m2)

Assessment

Condition 200 kN 252 kN

CMOD at

Crack Tearing 289 kN

CMOD at

0.15 mm

Max 10.05 18.42 20.94 34.59 32.51

Avg 9.58 16.95 19.35 29.66 30.28

Min 8.65 14.92 17.02 24.74 26.20

Std Dev 0.34 0.82 0.83 3.15 1.49

CMOD

(mm)

J at ø= 17°

(kJ/m2)

Lab-2 0.0956 15.9

Lab-6 0.1009 16.7

Lab-15 0.1009 16.5

Lab-14 0.1012 16.4

Lab-8 0.1034 14.9

Lab-9 0.1034 17.1

Lab-4 0.1040 17.2

Lab-12 0.1043 16.9

Lab-3 0.1048 17.3

Lab-11 0.1055 17.0

Lab-13 0.1057 17.3

Lab-5 0.1064 17.4

Lab-1 0.1066 17.8

Lab-7 0.1068 17.4

Lab-10 0.1089 18.4

For evaluation of the surface crack J at a particular angle φ, variation can be considered 

in terms of the general magnitude of J as well as the distribution of J as a function of φ around 

the perimeter of the crack. Figures 10b–10d show the computed J values around the crack at the 

three discrete force values, 200, 252, and 289 kN, respectively. The progression of the J versus φ

distributions with increasing load further illustrates the increasing variability shown in figure 10a. 

All but two labs (lab-2 and lab-8) have computed a response that appears to vary principally in 

magnitude of J with a consistent response of J as a function of φ . To better evaluate this effect, 

figures 10e–10g reflect the same data normalized by each lab’s maximum J value. These figures 

illustrate a magnitude-independent evaluation of the φ-dependence of J. The near-surface loss of 

singularity is consistently predicted in all solutions except lab-2, which utilized a different analytical method. For all other solutions, the predicted φ location of maximum J is consistent despite no 

standardization of crack mesh refinement for this study. The difference visible in the lab-8 solution 

for 30°<φ<90° is of more interest since the analytical method used here is the same as the other 

consistent solutions (domain integral). The cause of the lab-8 difference is discussed in further 

detail in section 4.3.4. Overall, these results strongly support the ability of current analytical techniques to reliably compute the φ-dependence of J. 

Max 18.42

Avg 17.38

Min 16.85

Std Dev 0.44
28

Figure 11 illustrates the J versus CMOD response for each of the labs. The discrete force 

values are identified along each trajectory. In the linear elastic regime, J varies quadratically with 

CMOD. Once the linear elastic regime has been exceeded, the response between J and CMOD 

is mostly linear, as follows from the relation of J proportional to CTOD. When compared to the 

highly sensitive response between force and J in the elastic-plastic regime (see fig. 10a), there is an 

advantage to the use of CMOD as the criterion to anchor analysis to experiment. Figure 12 shows 

the same data with the experimental CMOD at 252 kN and the trajectory lines removed. Valid 

solutions at a force of 252 kN with predicted CMOD within 10% of the experiment provide a solution range from 14.9 to 18.4 kJ/m2. Symbols plotted along the experimental CMOD line represent 

J values interpolated from the provided solutions at this CMOD. In this case, valid solutions within 

5% of the 252 kN force (see fig. 9b) provide a range of J values from 17.3 to 20.1 kJ/m2. Use of the 

CMOD criterion reduces the variability in the solutions relative to the force criterion, but for most 

analyses, this will correspond to analysis at a force higher than the experiment and therefore higher 

J values. As deformation levels increase, the use of the CMOD criteria becomes more effective at 

containing variability. A set of J values is estimated at a CMOD of 0.15 mm by interpolating (or 

extrapolating) each lab’s result. Again, the represented variability is reduced by approximately half. 

Note that including the lab-8 result nearly doubles the variability in the solution to the experiment 

at 252 kN, 19% with lab-8 versus 8.5% without.
29

0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18

0

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

CMOD (mm)

J-Integral (kJ/m2

)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5 

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 11. J-integral versus CMOD at φ=17°.
30

0.06 0.08 0.1 0.12 0.14 0.16 0.18

5

10

15

20

25

30

35

Experimental CMOD @ 252 kN

90% Experimental CMOD @ 252 kN

J @ 289 kN J @ 252 kN J @ 200 kN

CMOD (mm)

J-Integral (kJ/m2

)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

J at Exp. CMOD

J at CMOD=0.15 mm

Figure 12. J-integral versus CMOD at φ=17° with CMOD-based evaluation criteria.

4.3 Assessing Sources of Variability

To address various potential contributions to variability, the analysis methodology of each 

lab was collected through the survey previously discussed. Table 3 presents a summary of the 

reported parameters utilized in each model. The parameters are assessed independently by observing the reported J values at φ=17° as a function of each parameter. This provides a rough guide 

to likely causes of variation, but cannot be fully conclusive because the parameters are not always 

fully independent, i.e., other influences which happen to correlate may produce trends in data 

which are not actually due to the observed parameter. The following section steps through a series 

of observations from table 3.

 
31

Tables 3. Summary of ILS model parameters for all participants.

Lab ID

Analyst

Background

Model Size Formulation

Code No. Elem. No. Nodes CF Nodes

CF

Density Model Size

Elem. 

Type Integration Strain

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Structures

Structures

Structures

Structures

Structures

WARP3D

ADINA

ABAQUS

WARP3D

WARP3D

4,780

10,416

12,348

25,195

9,624

22,716

46,859

53,459

111,228

45,222

49

24

41

181

145

–0.37

–0.91

–0.54

2.48

1.70

–1.65

–1.37

–0.78

4.20

1.18

Quad

Quad

Quad

Quad

Quad

Reduced

Full

Full

Reduced

Reduced

Small

Small

Small

Finite

Small

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Materials

Structures

Structures

Structures

Structures

ANSYS

WARP3D

ABAQUS

ANSYS

ABAQUS

22,742

14,316

11,032

20,479

2,248

90,547

64,281

50,585

70,249

16,805

101

42

41

73

37

0.75

–0.52

–0.54

0.15

–0.63

1.77

–0.39

–0.88

0.48

–2.11

Quad

Quad

Quad

Quad

Quad

Reduced

Reduced

Full

Reduced

Reduced

Finite

Small

Finite

Small

Small

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Structures

Structures

Structures

Structures

Structures

ABAQUS

ABAQUS

ABAQUS

ABAQUS

ABAQUS

13,999

9,055

100,000

21,648

6,584

62,699

41,718

110,000

91,297

30,102

61

25

25

97

49

–0.11

–0.89

–0.89

0.67

–0.37

–0.04

–1.52

0.79

1.71

–1.40

Quad

Quad

Linear

Quad

Quad

Reduced

Reduced

Full

Full

Reduced

Small

Finite

Small

Small

Finite

Lab ID

Crack Tip

Mesh

Specimen

Grip

Boundary

Condition

Constitutive Model Formulation

Plasticity Eng/True SS-Curve J Method

No. 

Dom.

Avg. 

Method

Dom. 

Size

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

C-UT

C-UT

C-T_1/4

KH

C-UT

Simple

Simple

Simple

Simple

Simple

Stress

Stress

Stress

Stress

Stress

Incr

NLE

Incr

Incr

Incr

Eng

True

True

True

True

Table

Eq

Table

Table

Table

Domain

VCE

Domain

Domain

Domain

10

–

8

16

6

Maximum

–

Drop 2

Maximum

Drop 1

0.89

0.00

2.20

2.12

0.74

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

C-UT

C-UT

C-UT

C-T_1/4

C-UT

Simple

Simple

Simple

Simple

Simple

Disp

Stress

F w/C

Stress

Stress

Incr

Incr

Incr

Incr

Incr

True

Eng

True

True

Eng

Table

Table

Table

Table

Table

Domain

Domain

Domain

Domain

Domain

10

5

10

10

5

Converged

Drop 1

Maximum

Subset

(5 only)

Drop 1

0.63

0.60

0.50

1.08

1.01

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

KH

C-UT

C-UT

C-T

KH

Simple

Details

Details

Simple

Simple

Stress

RS

F w/C

Stress

Disp

Incr

Incr

Incr

Incr

Incr

True

True

Eng

True

True

Table

Table

Table

Table

Table

Domain

Domain

Domain

Domain

Domain

5

10

5

10

10

Drop 0

Subset 

(3–6)

Maximum

Converged

Maximum

0.47

0.94

2.26

0.82

0.75
32

Tables 3. Summary of ILS model parameters for all participant (Continued).

CF Nodes = Number of nodal locations along the crack perimeter

Quad = Element with quadratic shape functions

Linear = Element with linear shape functions

Stress = Uniform pressure stress

Disp = Uniform displacement condition

RS = Rigid surface contact for pin(s)

F w/C = Force with coupling

CF Density = See equation (1)

Model Size = See equation (1)

Incr = Incremental plasticity

NLE = Non-linear elasticity

Full = Quadrature integrates stiffness exactly for undistorted 

elements

Reduced = Quadrature at a lower order than full

Eng = Engineering stress and strain material properties

True = True stress and strain material properties

True = True stress and strain properties used small-strain formulation

Small = Nodal geometry is not updated due to strain

Finite = Nodal geometry is updated due to strain

Table = Stress strain values supplied as a look-up table for interpolation

Eq = Stress-strain values determined by an equation

C-UT = Collapsed hex, untied nodes

C-T = Collapsed hex, tied nodes, standard mid-side nodes

C-T_1/4 = Collapsed Hex, tied nodes, 1/4 point mid-side nodes

KH = Key-hole mesh

Simple = Rectangular flat plate

Details = Some grip detail added using contact, coupling, or other 

methods

Domain = Domain integral method used

VCE = Virtual crack extension (other than domain integral)

Maximum = J reported is the maximum of all domain results

Drop X = The first X domain results were not included in the average

Converged = Value or average was taken from domains with unchanging 

value

Subset = The domain value was selected from a single or group of domains

Dom. Size = Radial size of outer domain divided by avg. domain size

(1.01 mm)

Though not plotted, it is of interest that the background of all but one reported participant 

was “structures.” This could have a bearing on the ability of typical mechanical test labs to have the 

experience and capability to execute the analysis reliably. Most labs participating in the ILS were 

engineering organizations that include a damage tolerance assessment capability. Many material 

test organizations may not have adequate depth in structural analysis to reliably execute the assessment of surface crack tests to the new standard. This issue can readily be addressed by these organizations’ use of third party analysis capabilities within the commercial industry.

4.3.1 Codes and Modeling Choices

Four analysis codes were represented in the study: ABAQUS,7 WARP3D,8 ANSYS,9 and 

ADINA.10 Figure 13 reports the J results at three force levels by lab for each of the codes. As 

expected, there are no visible trends based on analysis code; therefore, no restrictions on analysis 

code are likely in the test standard.
33

ABAQUS WARP3D ANSYS ADINA

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

J-Integral (kJ/m2

)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 13. J-integral at φ=17° versus analysis code.

A common concern for converged finite element analysis (FEA) is mesh refinement. To 

address this issue in a simple, yet quantitative manner, the authors requested the number of nodes 

and elements in the model as well as the number of nodal locations along the crack front perimeter. This cannot fully define the quality of a mesh in that it does not address the efficient use of 

elements or the quality of the element geometry, but it does allow a very basic assessment to see if 

the results are strongly model size dependent. Figure 14 shows a typical 1⁄4 symmetric surface crack 

finite element mesh and the associated mesh refinement near the crack tip and in the far field. To 

develop a parameter to represent model size, the number of nodes in each model was normalized to 

a standard normal space by subtracting the mean (μ) number of nodes and dividing by the standard deviation (σ) of the number of nodes. To balance this parameter somewhat with a parameter 

representing the degree of refinement at the crack tip, the number of crack front nodes was similarly normalized and the two values summed. The model size parameter (MSP) is shown in equation form below:

MSP = (Nmodel − μmodel)

σ model

+ (Ncrack front − μcrack front )

σ crack front

. (1)
34

Symmetry

Plane

Symmetry

Plane

Figure 14. Typical 1⁄4 symmetric surface crack finite element mesh.

For the average model, the MSP will be zero. Negative values represent models smaller than 

average and positive values represent larger models. Lab-11 represents an “average size” model 

using this parameter. Figure 15 illustrates the influence of model size on the predicted J values. 

There are no distinct trends with model size. All models in the study were sufficiently refined to 

produce reliable J values. Based on the efficiency of computing resources and the robustness of the 

domain integral method for calculating J, this result is not unexpected. 
35

–4 –3 –2 –1 0 1 2 3 4 5 6

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

J-Integral (kJ/m2

)

Model Size Parameter

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 15. J-integral at φ=17° versus model size parameter from equation (1).

The MSP is a rough gauge of the refinement of the continuum discretization in the finite 

element models. The ability of the model to reproduce the displacements in the experiment is 

a function of this elemental discretization and the assumed shape of the displacement field within 

these elements. The need to discretize and assume displacement field shapes makes the typical finite 

element model stiffer than reality. Because the objective of this analysis is to reproduce the force 

and displacement results of the experiment as faithfully as possible, it is desirable to minimize the 

overprediction of stiffness. A common means to this end is to utilize an element formulation with 

“reduced integration.” Generally, reduced integration is defined as choosing a quadrature for element integration that is one order lower than the function used to estimate the displacement field. 

In this process, the higher order terms are removed from the integration of the element stiffness 

matrix and the element stiffness is slightly reduced. This has the desirable effect of offsetting, to 

some degree, the increased stiffness associated with the discretization process. A further benefit 

of reduced element integration vis-à-vis model stiffness is the elimination of “plasticity locking,” 

a manifestation of artificial stiffening in distorted, fully integrated, quadratic isoparametric elements due to their inability to properly represent the incompressibility assumption of the underlying plasticity theory.11 The use of reduced integration comes at the potential cost of enabling 

zero-energy deformation modes into the solution. This is most likely to occur around the crack tip 
36

and must be guarded against. The effect of using full versus reduced integration elements in the 

ILS problem is illustrated in figures 16a and 16b. In figure 16a, a weak trend is visible toward a less 

stiff elastic response with reduced integration elements. Note that this response may also be a function of mesh refinement. The trends in the J solutions with element integration are even weaker, as 

shown in figure 16b. Based on these observations, it is not likely that element integration order will 

have a substantial influence on J values for models with sufficient element density. If zero-energy 

deformation modes are avoided, the use of reduced integration is helpful at tempering the overly 

stiff response of the finite element solution and may assist in developing an acceptable match to the 

experimental force versus CMOD response.

Full Reduced

2,550

2,600

2,650

2,700

2,750

2,800

2,850

2,900

Elastic Stiffness (kN/mm)

Experiment

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 16a. Variation in elastic CMOD stiffness versus element integration order.
37

Full Reduced

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

J-Integral (kJ/m2)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 16b. J-integral at φ=17° versus element integration order.

Another aspect of fracture mechanics modeling open to wide interpretation is the choice 

of meshing technique at the tip of the crack. A wide variety of practices exist that were developed 

with specific methodologies in mind, such as reproducing a linear-elastic singularity, or providing 

accurate crack face opening profiles for the displacement method of determining K, or perhaps getting the CTOD accurately modeled. In the ILS submittals, there were three different techniques of 

crack tip modeling represented:

(1) Collapsed brick elements with tied nodes at the crack tip and mid-side nodes relocated 

to the quarter point.

(2) Collapsed brick elements with duplicate, untied nodes at the tip.

(3) A key-hole mesh pattern.

Figure 17 illustrates the three crack tips along with their deformed shapes. Figure 18 shows 

the resulting J values sorted relative to the crack tip meshing practice with no discernable influence 

on the results. None of the ILS submittals utilized a J evaluation scheme that is sensitive to the 

resolution of the near-crack tip fields. All were based on the virtual crack extension method, with 
38

14 out of 15 labs using the domain integral approach for the implementation. These results illustrate precisely why the domain integral approach has become the de facto standard for such calculations. The fact that the most accurate J evaluations actually come from domains remote from the 

crack tip without need to accurately resolve these stress and strain fields is powerful. Other fracture 

mechanics assessments, such as resolving the CTOD or evaluating the constraint condition in the 

crack tip fields, require more care be taken in the meshing of the crack tip elements. For purposes 

of J-integral analysis in the elastic-plastic regime, the use of collapsed brick elements with duplicate 

nodes which can allow the crack tip to open under load is the recommended method for assessments related to the surface crack standard. (The method using tied crack tip nodes with quarter 

point mid-side nodes was developed to best represent the linear elastic fracture mechanics singularity and the solution for K and is not recommended for elastic-plastic analysis. The key-hole and 

initial radius methods are useful for properly resolving crack tip stress and strain fields under very 

high deformations. This is not required for an accurate evaluation of the J-integral if the domain 

integral method is used.)

(b)

(a)

(c)

Figure 17. Crack tip mesh with undeformed geometry on left and deformed on right 

for (a) collapsed elements with tied nodes, (b) collapsed elements with untied 

nodes, and (c) keyhole pattern.
39

Collapsed Untied Collapsed 1⁄4 pt, Tied Key-hole

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

J-Integral (kJ/m2)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 18. J-Integral at φ=17° versus crack tip mesh geometry. 

4.3.2 The Domain Integral

When using the domain integral method, the development of a crack tip mesh also involves 

establishing the orderly element patterns that make up the volume domains. Most finite element 

codes that support the domain integral calculation also have tools to assist in the generation of 

domains. Each domain is defined by a volumetric region encompassing the crack tip from one 

crack face to the other (or to a symmetry plane). Generally, there are multiple concentric domains 

which ideally render the same value based on the path-independent nature of the J-integral. When 

modeling elastic-plastic conditions, maintaining a true path-independent behavior requires the use 

of a constitutive model that does not violate the fundamental assumptions of the conservation 

integral—a conservative strain energy field. This is done using a hyperelastic (nonlinear elastic) 

constitutive model, referred to as deformation plasticity in much of the fracture mechanics literature.12 These constitutive models are not universally available and are often limited in the manner 

in which they represent material flow behavior, such as allowing only power law representations 

of the stress-strain curve. The use of incremental plasticity is much more common, versatile, and 

representative of metallic plasticity; however, the nature of irreversible deformation and the local 

unloading and stress rearrangement that occurs violates the conservation integral assumptions 

and creates path dependence in the domain integral calculation. This can largely be avoided if 
40

the domains are sufficiently remote to pass through material that remains predominantly elastic. Unfortunately, the most common method of domain generation starts directly at the crack 

tip; and, for most elastic-plastic analysis, this ensures that some of the near-tip domains will be 

substantially within the crack tip plastic zone. In incremental plasticity, the proper J value is that 

measured in the far-field, away from the influences of plasticity. J-integral values calculated within 

domains influenced by plasticity will be lower than the far-field J value. Determining the proper J 

value from a domain integral analysis requires evaluation of the path dependency of the domain 

results. The physical size of the domains relative to the state of deformation in the model (size of 

the plastic zone) will influence in the degree of path independence achieved. Figure 19 notionally 

illustrates a typical loss of path dependence as plasticity effects overwhelm the domains. The highest value in the most remote domain will be the best evaluation of the J-integral.13

0 2 4 6 8 10 12 14 16 18 20

0

0.2

0.4

0.6

0.8

1

Domain Number

J/J Far Field

Increasing Path Dependence

Due to Incremental Plasticity

as Deformation Increases, i.e.,

as Plastic Zone Size Increases

Best Far-Field Value

Figure 19. Illustration of typical J-integral results showing degrees of path dependency.
41

To determine if the size and treatment of the domain results is a source of variability for 

the ILS problem, the authors requested each lab using the domain integral to report the physical 

size (radius) of the outermost domain, as illustrated in figure 20, as well as a description of how 

path dependence was evaluated, i.e., how an answer was chosen from the multiple domain results. 

Figure 21 shows resulting J values versus the domain size normalized by the average domain size, 

which was 1.01 mm. For the ILS problem, the J results do not correlate with domain size, indicating that the domains used were sufficiently large to accommodate the deformation without large 

error, or that path dependence in domain results was properly handled. The method of handling 

the domain path dependence was grouped into three categories based on the survey results: 

(1) all but the first domain was averaged, (2) some smaller subset of the domain values was averaged, or (3) only the fully converged or maximum value was reported. Figure 22 illustrates the 

effect of these choices on the J-integral results. If these effects were significant in the models, 

a trend toward higher J values for the converged or maximum domain result would be likely when 

compared to results involving averaging over many domains. In fact, the family of lower J values 

at 289 kN each reported using converged/maximum domain value. This indicates other factors 

are present to cause these variations.

Radial Distance to Outermost Domain

Figure 20. Illustration of dimension used to characterize domain size.
42

0 0.5 1 1.5 2 2.5 3

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

J-Integral (kJ/m2

)

Domain Size/Average Domain Size (1.01 mm)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 21. J-integral at φ=17° versus normalized domain size.
43

No Domains All But First Subset Converged/Max

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

J-Integral (kJ/m2

)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 22. J-integral at φ=17° versus method used to evaluate J from multiple domains.

4.3.3 Boundary Conditions

Three basic types of model boundary conditions were reported by ILS participants: remote 

uniform pressure stress, remote uniform displacement, and a few with various forms of contact or 

the use of the ABAQUS force with coupling approach (*COUPLE). Figure 23 reports the J results 

versus these three forms of boundary conditions, again without any significant trends. The surface 

crack test standard limits the specimen geometry, controlling the height (L) and surface length of 

the crack (2c) relative to the specimen width (W) so that 2c ≤W/5 and L≥2W (see fig. 1). These 

limitations are in place to help minimize (but not eliminate) the influence of boundary conditions 

in both experiment and analysis. The family of solutions with lower J values actually spans all three 

types of boundary conditions. For a direct comparison, the lab-1 model was run with displacement 

boundary conditions to compare against the original solution using stress boundary conditions. 

The maximum difference in J was 2% between the two boundary conditions, indicating the variation in boundary conditions is not a significant source of the variation in the ILS problem.
44

Stress Displacement Force w/c or Contact

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

J-Integral (kJ/m2

)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 23. J-integral at φ=17° versus model boundary condition.

4.3.4 Variability Due to Element Skewing 

Lab-8 kindly collaborated to uncover the source of variability in that solution. As discussed 

below, it is the only solution with significant differences not well explained by the constitutive 

input. In this case, a review of figures 6–9 illustrate that the predicted deflections from the model 

are reasonable, producing a force versus CMOD response that matches the experiment well and 

agrees with many other solutions. The series of graphs in figure 10 illustrates a slightly lower 

J-integral solution and a somewhat different φ-dependence in the solution. Figure 11 is the strongest clue that there may be issues with the J-integral solution. If the CMOD has been well predicted, the J-integral solution should follow the clear trend from the other analyses. This solution 

tends to stand alone.
45

The most unique aspect of the lab-8 submittal is that the model geometry was created from 

a previously existing surface crack mesh and warped to meet the geometry of the ILS problem. 

Scaling fracture mechanics finite element meshes to meet new needs has long been a common 

practice; however, this is most successful for subtle changes in two-dimensional crack geometries. 

The curved front of the surface crack makes any but the smallest of mesh distortions problematic. 

In this case, the result was skewed elements along the crack front at a strong angle to the local 

crack front normal. Figures 24a and 24b illustrate an example of the skewed crack front element 

shape, though the mesh in these figures is not from the ILS problem. Despite previous statements 

regarding the robustness of the domain integral method with respect to mesh quality, this aspect 

of mesh integrity is important. The lab-8 solution has a strong oscillation in the J-integral solution 

from corner to mid-side nodes, shown in figure 25. These oscillations are frequently observed when 

the domain definition includes elements with such distortion. It is important to maintain elements 

along the crack front as perpendicular to the crack front as possible. 

Figure 24a. Curved crack front with skewed elements relative to the crack front normal, 

a common source of oscillating J-integral solutions.
46

Figure 24b. Curved crack front with elements aligned with the crack front normal.

0 15 30 45 60 75 90

10

15

20

25

30

35

Parametric Angle, (deg) 

J-Integral (kJ/m2

)

J

Primary

Midside

φ

Figure 25. Oscillating J-integral solution affecting the lab-8 solution.
47

4.3.5 Constitutive Modeling

For modeling elastic-plastic behavior, the choices made regarding the constitutive properties 

of the material often have the largest impact on model results compared to other modeling choices. 

As expected, this follows through to the ILS problem as well. Each lab was provided with a spreadsheet containing a set of engineering stress-strain data, very much as it would appear from a test 

lab reporting a tensile test. See appendix A. Each participant was free to represent this material 

property in their model in any fashion they chose. In this case, the ILS problem was actually simplified because the value of elastic modulus (and Poisson’s ratio) was provided in addition to the 

stress-strain data. This should have assisted with the evaluation of the stress-strain curve and likely 

minimized some of the variability in its representation. Fourteen of the 15 labs used a table of 

stress-strain data pairs to represent the stress-strain curve. Lab-2 fit a linear-plus-power law equation to the data for use with a nonlinear elastic representation of the curve. Beyond these consistencies, there was a wide variety in choices. The survey recorded the use of true stress-strain versus 

engineering stress-strain as well as the fidelity of the curve representation regarding proportional 

limit/yield point selection and hardening behavior. Though not constitutive in nature, the use of 

small strain assumptions (nodal geometry definition does not change) versus a finite strain formulation (nodal geometry updates due to displacements) was also documented since it relates to the 

constitutive inputs.

Figures 26 and 27 illustrate the J results relative to the choice of true versus engineering 

stress-strain and the use of small strain versus finite strain formulations, respectively. These choices 

are related in that one should generally follow the other. If small strain assumptions are used, the 

engineering stress-strain curve is appropriate, while if a finite strain formulation is used, the true 

stress-strain curve is proper. Based on the survey responses, 6 of the 15 labs used a true stressstrain input with small strain assumptions. In figures 26 and 27, these cases are indicated by a small 

arrow. J-integral evaluation is generally insensitive to the choice of small versus finite strains, except 

for an increase in path dependency with the finite strain formulation.13 However, the transformation of the stress-strain curve into true stress-strain space elevates the curve and tends to suppress 

the resulting J values as plasticity becomes a large contributor. This trend is visible in figure 26.
48

True Stress-Strain Eng. Stress-Strain

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

←←←

←

←

← Indicates

Incompatible

Use of True Strain

With Small Strain

Analysis.

J-Integral (kJ/m2

)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 26. J-integral at φ=17° versus use of true or engineering stress-strain

material properties.
49

Finite Strain Small Strain

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

←←←

←

←

← Indicates

Incompatible

Use of True Strain

With Small Strain

Analysis.

J-Integral (kJ/m2

)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 27. J-integral at φ=17° versus use of small-strain or finite-strain formulation.

The changes in the early post-yield portions of the stress-strain curve due to the true stressstrain transformation are small, yet they noticeably affect the J-integral results reported at the 

specified force values. Based on this observation, a closer evaluation of each of the lab’s stressstrain inputs is warranted. To provide a consistent basis for comparison, the provided data were 

converted into a consistent format to eliminate differences due to units, plastic versus total strain, 

and the use of true stress-strain versus engineering stress-strain. Figure 28 shows the provided data 

along with the inputs of all labs using engineering stress and strain. In this larger view, the consistency across labs appears good, likely well within the variability one would expect from typical 

material property scatter. To isolate when each analysis would begin to accumulate plastic strain, 

the curves are plotted as engineering stress versus plastic strain in figures 29a and 29b. In these 

figures, a range of proportional limit is visible from approximately 315 to 385 MPa, represented 

by star-shaped symbols. This range is due in part to a lack of a standardized treatment for rendering experimental tensile data into constitutive inputs for an analysis code. For example, a likely 

explanation for the higher proportional limits chosen is the use of a 0.2% offset definition of yield 

strength that is standard in engineering practice, versus setting the proportional limit when the first 

substantial plastic strain occurs. The J results are plotted versus the proportional limit used in each 

lab’s analysis in figure 30. With the exception of the lab-8 result, the choice of proportional limit 

effectively isolates each of the results that provide lower J results at 289 kN. This follows clearly 
50

since the higher proportional limit delays the onset of plastic contributions to J because the plastic 

area under the stress-strain curve for a given stress level is reduced. 

0 0.02 0.04 0.06 0.08 0.1 0.12

0

50

100

150

200

250

300

350

400

450

500

Engineering Strain

Engineering Stress (MPa)

Provided Data

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 28. Stress-strain inputs in engineering stress-strain space.
51

0 0.02 0.04 0.06 0.08 0.1 0.12

300

320

340

360

380

400

420

440

460

480

500

Plastic Strain

Engineering Stress (MPa)

Provided Data

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 29a. Engineering stress versus plastic strain.
52

0 0.5 1 1.5 2 2.5 3

x 10–3

310

320

330

340

350

360

370

380

390

400

Plastic Strain

Engineering Stress (MPa)

Provided Data

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 29b. Engineering stress versus plastic strain illustrating range

of proportional limits.
53

300 320 340 360 380 400

5

10

15

20

25

30

35

J @ 289 kN J @ 252 kN J @ 200 kN

J-Integral (kJ/m2

)

Proportional Limit (MPa)

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 30. J-integral at φ=17° versus proportional limit in constitutive inputs.

Figure 31 illustrates the area under the stress versus plastic strain curve. In the figure, the 

area is normalized relative to the area under the provided stress-strain curve and plotted versus 

stress. Thus, inputs exactly matching the provided data produce a normalized area of unity. The 

curves with higher proportional limits (labs -6, -14, and -15) do not begin accumulating plastic area 

until the local stresses exceed 380 MPa. Also clear from this plot is that though the curve fit chosen 

by lab-2 has a slightly higher than average proportional limit, it accumulates plastic area quicker 

than average in the range of 360 to 400 MPa, and the overall result is that the lab-2 estimate 

of J at 289 kN is typical of the larger population. Of further interest is the deviation among labs 

in normalized area under the plastic strain curve converges as the stress-strain state increases. This 

suggests that variability in the analysis due to the precision of the proportional limit in the constitutive input should decrease as deformation increases.
54

300 320 340 360 380 400 420 440 460

–0.5

0

0.5

1

1.5

2

2.5

3

Normalized Area Under σ−εpl Curve

Stress (MPa)

Provided Data

Lab-1

Lab-2

Lab-3

Lab-4

Lab-5

Lab-6

Lab-7

Lab-8

Lab-9

Lab-10

Lab-11

Lab-12

Lab-13

Lab-14

Lab-15

Figure 31. Area under the engineering stress-plastic strain curve as function of stress.

To directly study the effects of proportional limit variation in the constitutive inputs, 

the lab-1 model was run under four scenarios, where only the proportional limit was altered. 

Figures 32a and 32b illustrate the stress-strain curves used in the assessment. The differences are 

subtle, with the proportional limit changing from 276 to 379 MPa, yet the curve remaining identical to the provided input data after a plastic strain of 0.002, the typical engineering yield point. 

The smallest proportional limit value of 276 MPa was purposefully chosen below the actual proportional limit of the test data on the linear portion of the stress-strain curve to illustrate no issues 

arise with a low proportional limit as long as the curve is properly followed. An area evaluation 

similar to that previously reviewed for the participant inputs is shown in figure 33. In this case, the 

normalizing curve is the one with the lowest proportional limit (276 MPa). The delay in accumulation in plastic area in the early region of the curve is clear as the proportional limit increases. These 

plastic area differences quickly dissipate once the stress-strain curves converge and large amounts 

of plastic area develop.
55

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

0

50

100

150

200

250

300

350

400

450

500

Engineering Strain

Engineering Stress (MPa)

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

0.2% Offset Line

Figure 32a. Engineering stress-strain curves used in the proportional limit substudy.
56

450

500

400

375

400

350

325

350

300 275

300

Sys = 276 MPa

Sys = 317 MPa

Sys = 345MPa

0.01 0.02 0.03 0.04 0.05

250

0 0.5 1 1.5 2 2.5 × 10–3

250

Plastic Strain

Plastic Strain

Sys 345

Sys = 379 MPa

0.2% Offset Line

Engineering Stress (MPa)

Figure 32b. All variations in proportional limit occur prior to standard 0.2% offset 

strain yield definition.
57

300 320 340 360 380 400 420 440 460

–1

–0.5

0

0.5

1

1.5

2

Stress (MPa)

Normalized area is the area under each curve

divided by the area under the 276 MPa curve for

a given stress level.

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

Normalize Area Under – pl Curve

ε

σ

Figure 33. Area under the engineering stress-plastic strain curve as function

of stress for the proportional limit substudy.

Figures 34a through 34j present the results from a series of load steps from each of the four 

models in this substudy superimposed, each illustrating both the force versus CMOD response and 

the J(φ) results around the crack perimeter at the indicated force. The force values of interest in the 

ILS are highlighted along the force axis and the deformation level as defined in the test standard 

is shown for crack length and ligament at φ=17°, rφa and rφb in figure 34a. The load step producing the J(φ) result is indicated along the force-CMOD trace. In figure 34a, at step 11, the response is 

predominantly linear-elastic and the four results are nearly identical. Figure 34b, at step 12, corresponds closely with the 200 kN force of interest in the ILS and very little deviation in J(φ) is present 

among the models, particularly at φ=17°. As the analysis progresses through steps 13–15 (figs. 34c–

34e), separation in the J(φ) results becomes clear and is ordered based on the proportional limit. 

Step 15 corresponds closely to the 252 kN force where results were requested in the ILS. Note 

that the variability in the proportional limit leads to approximately 6% variation in J(φ=17°). At 

steps 16 and 17 in figures 34f and 34g, respectively, the two models with lower proportional limits 

have converged, yet the overall variation in J(φ) has increased across the models, accounting for 

almost a 20% deviation in J(φ=17°). At step 18, which is slightly above the 289 kN force value 

reported in the ILS, the lower proportional limit models are further converging, leaving the model 
58

with the 379 MPa proportional limit lagging as a separate family (fig. 34h). At even higher levels 

of deformation, the early influence of the proportional limit dissipates. Figures 34i and 34j illustrate that the variation in J has decreased to approximately 4% for J(φ =17°). The model violates 

the current deformation limit on crack size in the test standard (Ma ≥25) between steps 19 and 20.

0 0.1 0.2 0.3 0.4 0.5

0.7 0

50

100

150

200

250

300

350

Force (kN)

Sys = 276 MPa Sys = 317 MPa Sys = 345 Mpa Sys = 379 MPa

Step 11

Force vs CMOD

0 10 20 30 40 50 60 70 80 90

6

7

8

9

J-Integral (kJ/m2)

J-Integral vs for Step 11

Deformation Measures at = 17° φ

Crack, Ma= 290

Ligament, Mb= 1,246

CMOD (mm)

5.5

6.5

7.5

8.5

r

φ

φ

φ

b

a

B

r

φ a

m

2c

φ (deg)

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

(a)

a ys

a

r M J

φ σ = b ys

b

r M J

φ

φ

σ σ = Jφ J

Figure 34. Images (a) through (j) illustrate the effects of proportional limit on the resulting 

J-integral value as the model is loaded through the ILS load regime and beyond.
59

0 0.1 0.2 0.3 0.4 0.5

0 0.6 0.7

CMOD (mm)

Force (kN)

Sys = 276 MPa Sys = 317 MPa Sys = 345 Mpa Sys = 379 MPa

Step 12

Force vs CMOD

7

8

9

10

J-Integral (kJ/m2)

J-Integral vs for Step 12

Deformation Measures at = 17°

φ

φ

φ

Crack, Ma= 238

Ligament, Mb= 1,024

50

100

150

200

250

300

350

0 10 20 30 40 50 60 70 80 90

(deg)

7.5

8.5

9.5

10.5

a ys

a

r M J

φ σ = b ys

b

r M J

φ

φ

σ σ = Jφ J

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

(b)

0 0.1 0.2 0.3 0.4 0.5

0.6 0.7

Force (kN)

Sys = 276 MPa Sys = 317 MPa Sys = 345 Mpa Sys = 379 MPa

Step 13

Force vs CMOD

0 10 20 30 40 50 60 70 80 90

8

8.5

9

9.5

10

10.5

11

11.5

12

12.5

J-Integral (kJ/m2)

Deformation Measures at = 17° φ

Crack, Ma= 198

Ligament, Mb= 850

a ys

a

r M J

φ σ = b ys

b

r M J

φ

φ

σ =

0

50

100

150

200

250

300

350

CMOD (mm)

J-Integral vs φ for Step 13

φ (deg)

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

Jφ J

(c)

Figure 34. Images (a) through (j) illustrate the effects of proportional limit on the resulting 

J-integral value as the model is loaded through the ILS load regime and beyond 

(Continued).
60

0 0.1 0.2 0.3 0.4 0.5

0.6 0.7 0

50

100

150

200

250

300

350

CMOD (mm)

Force (kN)

Sys = 276 MPa Sys = 317 MPa Sys = 345 Mpa Sys = 379 MPa

Step 14

Force vs CMOD

0 10 20 30 40 50 60 70 80 90

9

10

11

12

13

14

15

(deg)

J-Integral (kJ/m

J-Integral vs for Step 14

Deformation Measures at φ = 17° 

φ

φ

Crack, Ma= 164

Ligament, Mb= 707

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

(d)

a ys

a

r M φ σ = b ys

b

r M φ

φ

σ = Jφ J

0 0.1 0.2 0.3 0.4 0.5

0.6 0.7 0

50

100

150

200

250

300

350

CMOD (mm)

Force (kN)

Sys = 276 MPa Sys = 317 MPa Sys = 345 Mpa Sys = 379 MPa

Step 15

Force vs CMOD

0 10 20 30 40 50 60 70 80 90

10

11

12

13

14

15

16

17

18

(deg)

J-Integral (kJ/m2)

J-Integral vs for Step 15

Deformation Measures at φ

φ

φ

= 17

Crack, Ma= 136

Ligament, Mb= 584

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

a ys

a

r M J

φ σ = b ys

b

r M J

φ

φ

σ = Jφ J

(e)

Figure 34. Images (a) through (j) illustrate the effects of proportional limit on the resulting 

J-integral value as the model is loaded through the ILS load regime and beyond 

(Continued).
61

0 0.1 0.2 0.3 0.4 0.5

0.6 0.7 0

50

100

150

200

250

300

350

CMOD (mm)

Force (kN)

Sys = 276 MPa Sys = 317 MPa Sys = 345 Mpa Sys = 379 MPa

Step 16

Force vs CMOD

0 10 20 30 40 50 60 70 80 90

12

14

16

18

20

22

24

J-Integral (kJ/m2)

J-Integral vs for Step 16

Deformation Measures at = 17° φ

φ

Crack, Ma= 106

Ligament, Mb= 455

φ (deg)

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

a ys

a

r M J

φ σ = b ys

b

r M J

φ

φ

σ = Jφ J

(f)

0 0.1 0.2 0.3 0.4 0.5

0.6 0.7 0

50

100

150

200

250

300

350

CMOD (mm)

Force (kN)

Sys = 276 MPa Sys = 317 MPa Sys = 345 Mpa Sys = 379 MPa

Step 17

Force vs CMOD

0 10 20 30 40 50 60 70 80 90

14

16

18

20

22

24

26

28

30

 (deg)

J-Integral (kJ/m2)

J-Integral vs for Step 17

Deformation Measures at = 17° φ

φ

φ

Crack, Ma= 79

Ligament, Mb= 342

a ys

a

r M J

φ σ = b ys

b

r M J

φ

φ

σ = Jφ J

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

(g)

Figure 34. Images (a) through (j) illustrate the effects of proportional limit on the resulting 

J-integral value as the model is loaded through the ILS load regime and beyond 

(Continued).
62

0 0.1 0.2 0.3 0.4 0.5

0.6 0.7 0

50

100

150

200

250

300

350

CMOD (mm)

Force (kN)

Sys = 276 MPa Sys = 317 MPa Sys = 345 Mpa Sys = 379 MPa

Step 18

0 10 20 30 40 50 60 70 80 90

15

20

25

30

35

40

45

Deformation Measures at = 17° φ

Crack, Ma= 57

Ligament, Mb= 245 J-Integral (kJ/m2)

a ys

a

r M J

φ σ = b ys

b

r M J

φ

φ

σ = Jφ J

J-Integral vs φ for Step 18

φ (deg)

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

Force vs CMOD

(h)

0 0.1 0.2 0.3 0.4 0.5

0.6 0.7 0

50

100

150

200

250

300

350

CMOD (mm)

Force (kN)

Sys = 276 MPa Sys = 317 MPa Sys = 345 Mpa Sys = 379 MPa

Step 19

0 10 20 30 40 50 60 70 80 90

35

40

45

50

55

60

65

70

75

80

 (deg)

J-Integral (kJ/m2)

Deformation Measures at =17° φ

φ

Crack, Ma= 31

Ligament, Mb= 134

Force vs CMOD

J-Integral vs φ for Step 19

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

(i)

a ys

a

r M J

φ σ = b ys

b

r M J

φ

φ

σ σ = Jφ J

Figure 34. Images (a) through (j) illustrate the effects of proportional limit on the resulting 

J-integral value as the model is loaded through the ILS load regime and beyond 

(Continued).
63

0 0.1 0.2 0.3 0.4 0.5

0.6 0.7 0

50

100

150

200

250

300

350

CMOD (mm)

Force (kN)

Sys = 276 MPa Sys = 317 MPa Sys = 345 Mpa Sys = 379 MPa

Step 20

0 10 20 30 40 50 60 70 80 90

70

80

90

100

110

120

130

140

150

160

170

J-Integral (kJ/m2)

Deformation Measures at = 17° φ

φ

φ (deg)

Crack, Ma= 15

Ligament, Mb= 64

Sys = 276 MPa

Sys = 317 MPa

Sys = 345 MPa

Sys = 379 MPa

Force vs CMOD

J-Integral vs for Step 20

(j)

a ys

a

r M J

φ σ = b ys

b

r M J

φ

φ

σ σ = Jφ J

Figure 34. Images (a) through (j) illustrate the effects of proportional limit on the resulting 

J-integral value as the model is loaded through the ILS load regime and beyond 

(Continued).

The sequence of plots showing J-integral and force versus CMOD results in figure 34 illustrates the variability in J-integral assessment due to constitutive input when the criterion used 

to match the experiment is enforced at the far-field boundary conditions, in this case, matching 

the experimentally measured force to the reaction force in the model. Similar J-integral variability is observed if the experimental displacement at the end of the specimen is used to match 

displacement-controlled boundary conditions on the model. The constitutive model must convey 

these far-field conditions through the specimen (or structure) to the crack, resulting in considerable sensitivity once nonlinear conditions prevail. These constitutive sensitivities require significant 

consideration for typical structural defect assessments, when local knowledge of crack conditions 

are not known and remote boundary conditions are (at best) all that are available. In the case of 

surface crack toughness experiments in the laboratory, the presence of the CMOD gauge provides 

an option to enforce the tie from experiment to analysis based on conditions local to the crack. As 

hinted during previous discussion of figures 11 and 12, the CMOD provides a more direct and less 

sensitive indicator of crack tip conditions. Moreover, tying experiment and analysis using CMOD 

should lessen constitutive sensitivities because of the local nature of the criterion.

To evaluate the benefit of using a CMOD-based criterion to reduce the sensitivity of constitutive inputs to the J-integral results, the authors conducted a second substudy using the lab-1 

model. In this substudy, the constitutive inputs were altered to target a ±5% deviation in predicted 
64

force when the model’s CMOD matches the experimental CMOD. The constitutive inputs were 

modified using a simple multiplicative factor on the stress values of the stress-strain curve. A factor of 0.96 applied to the stress inputs resulted in a very close match to the force and CMOD at 

the initiation of tearing in the experiment. This result is designated “analysis A” and “material A” 

in table 4 and figure 35, respectively. Further iteration developed material inputs B and C to target 

the ±5% bounds for prediction of force on the specimen at the tearing CMOD. As tabulated in 

table 4, the resulting range in J-integral prediction is only ±3.2% when interpolated at the tearing 

CMOD. The modified constitutive inputs are illustrated in figure 36. In stark comparison to the 

previous substudy, which allowed zero deviation in the 0.2% offset yield strength, bounding assessments B and C incorporate a 57 MPa (14%) difference in the 0.2% offset yield strength, and yet, 

by matching the CMOD, the results maintain very small errors in the J-integral. 

Table 4. Summary of results from CMOD-matching substudy.

CMOD-Matched Analysis at Tearing CMOD

Analysis

Factor

on Stress*

0.2% Offset Yield

Strength (Mpa)

Force Difference**

(%)

J-Integral

(kJ/m2)

J Difference***

(%)

Lab-1†

A

B

C

1.000

0.960

1.040

0.893

386

370

401

344

2.92

0.01

5.03

–4.94

19.8

19.4

20.0

18.8

1.67

0

3.17

–3.23

CMOD-Matched Analysis at Deformation-Limited CMOD

Analysis

Factor

on Stress*

0.2% Offset Yield

Strength (Mpa)

Force Difference‡

(%)

J-Integral

(kJ/m2)

J Difference***

(%)

A-2

D

E

0.960

1.010

0.910

370

390

351

0

5.02

–5.17

159

167

152

0

4.88

–4.70

*

**

***

†

‡

Multiplication factor applied to the material stress inputs of the Lab-1 analysis.

Percent difference in predicted force at tearing CMOD.

Percent difference in J at φ=17° relative to analysis “A”.

Material properties as used in Lab-1 results.

Percent difference in predicted force at CMOD from analysis A-2.
65

300

Experiment

250

Lab-1 Result

Material A

Material B

Material C

200

100

150

Force (kN) 

50

0 0.02 0.04 0.06 0.08 0.1

0.12 0.14 0.16 0.18 0

0.1 0.105 0.11 0.115 0.12 0.125 0.13

CMOD (mm)

CMOD (mm)

Force (kN)

280

260

270

250

230

240

220

Figure 35. Bounding models at the tearing CMOD for the CMOD-matching substudy.
66

0 0.01 0.02 0.03 0.04 0.05 0.06

250

300

350

400

450

500

Plastic Strain

Engineering Stress (MPa)

Material, Lab-1

Material A

Material B

Material C

Material D

Material E

0.2% Offset Line

Figure 36. Comparison of scaled material property inputs for the CMOD-matching 

substudy.

There is only a small amount of plasticity in the round robin experiment at tearing. 

To evaluate the same sensitivities at higher levels of plasticity, the analysis using material A 

was run to the maximum allowed deformation in the proposed surface crack standard, resulting in analysis A-2 in table 4. Using this analysis as a baseline, materials D and E (fig. 36) 

were developed to target the allowable ±5% force range at this deformation-limited CMOD, 

as illustrated in figure 37. Interpolating J-integral values from analyses D and E at the final 

CMOD of analysis A-2, the resulting range in J is approximately ±4.8%. For comparison, 

the original tearing CMOD bracket is also shown in figure 37. Note that the factors on stress 

input for materials D and E are smaller than those for materials B and C, resulting in a range 

of 39 MPa for offset yield strength.

Though the issue of constitutive modeling sensitivity remains for most elastic-plastic 

assessment of defects in structures, the advantage of measuring the CMOD during surface 

crack toughness testing allows the proposed test method to mitigate much of this sensitivity 

during the analytical assessment of the experiment, thus exacting knowledge of the material’s 

stress-strain response is not necessary to reasonably evaluate the toughness result.
67

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

0

50

100

150

200

250

300

350

CMOD (mm)

Force (kN)

Material A

Material D

Material E

Figure 37. Bounding models at the deformation-limited CMOD for the CMODmatching substudy.
68

5. CONCLUSIONS

Regarding the overall state of the art for elastic-plastic J-integral analysis, the results from 

phase I of the ILS are encouraging. Providing a request for analysis with no further instruction 

regarding methodology has yielded results with surprisingly little variation. Many of the commonly assumed sources of error in finite element analysis appear less significant than they once 

were, when computing resources for mesh building and equation solving were at a higher premium. 

General modeling choices such as analysis code, model size (mesh density), crack tip meshing, or 

boundary conditions were not found to be sources of significant variability. This finding is made 

possible by the almost ubiquitous standardization toward the finite element implementation of the 

domain integral method for numerical estimation of the J-integral. The method has been adopted 

by numerous commercial and open source codes and provides a robust and reliable method to 

evaluate elastic-plastic J-integral values, as long as the domains are defined by elements of reasonable quality.

 

Because the ILS called for results at specified force values, the most pressing source of variability in the J-integral assessment is introduced through the constitutive model. The ILS has illustrated, for assessments controlled only by far-field boundary conditions, the representation of the 

material’s stress-strain relation can influence elastic-plastic J results more strongly than most other 

choices left to the analyst, within commonly accepted modeling practices. In fact, the first substudy 

which evaluated proportional limit effects in force control would argue that constitutive model 

sensitivity is likely inherent to the process despite best efforts of the analyst since the degree of 

material property variability reflected in the study is reasonably representative of the expected variability in flow properties within any given piece of structural metal. The vast majority of structural 

assessments for J will be based on applied forces or a resulting stress field where the local displacements (CMOD) are unknown. In this case, the substudy illustrates the importance of representing 

the stress-strain curve faithfully throughout. For most J-integral evaluations related to structural 

integrity, other structural factors of safety are likely to limit stresses such that the stress-strain 

conditions in the vicinity of the crack span the range through the proportional limit and out only 

modestly onto the tangent modulus of the curve. Analysis in this region of the stress-strain curve is 

highly prone to introducing variability into the estimated elastic-plastic values of J.

In the controlled case of surface crack toughness testing, the advantage of having the 

CMOD data is significant. The second substudy illustrates that the extreme sensitivity to constitutive input can be largely avoided by using the CMOD as the basis for evaluating the experimental 

results. The resulting error in the J-integral will be bounded by the percent error in the predicted 

force at a given CMOD. The proposed surface crack standard will be updated to utilize only a 

CMOD-based assessment of surface crack tests in the elastic-plastic regime.
69

6. RECOMMENDED PRACTICES

In the development of a standardized test procedure, the objective is to provide all necessary 

controls to allow users to achieve a consistent test result. In the case of the elastic-plastic surface 

crack test, the approach used by other test standards to contain all needed relations to evaluate 

the test record is not currently feasible. The variations possible due to nonlinear material behavior 

add sufficient complexity that an external, stand-alone analysis is required. This ILS has provided 

strong evidence that such a method is feasible without the introduction of excess variability, even 

without any significant guidance regarding the analysis methodology. However, the ILS has also 

provided clear insight into what common practices are in use, such that they could be standardized without undue burden on the user of the standard. This allows the test result to be standardized to the extent possible by providing specifics regarding what analysis practices are considered 

“standard.” The following list of recommended practices is proposed for inclusion in the surface 

crack test standard. These are a combination of accepted best practices and observations garnered 

through this phase of the ILS. They are not to be construed as the only acceptable way to perform elastic-plastic J-integral analysis of surface cracks, but rather as a set of rules that help users 

achieve a more standardized answer when evaluating test records according to the test standard:

(1) The analysis shall be performed using the finite element method.

(2) The model shall consist of three-dimensional elements with quadratic shape functions 

and utilizing reduced integration. Crack tip deformations should be monitored for spurious zero 

energy deformation modes.

(a) Full integration may be used for elements near the crack tip to avoid spurious zero 

energy deformation modes.

(3) The domain integral method shall be used for J-integral calculations.

(a) The domains should be as large as possible without compromising mesh integrity.

(b) The domains should consist of elements without excessive skew and should be 

normal to the local crack front within 30°. 

(c) A minimum of five domains shall be used to monitor path dependence.

(d) All domain solutions shall be checked for path dependence, reporting only the highest fully converged value from the outermost domain(s). Unconverged domains indicate the physical size of the domain is too small for the deformation state in the model.

(e) All domain solutions shall be checked for oscillatory J-integral results along the 

crack front. Solutions with oscillations >5% shall not be used.

(f) Small strain assumptions shall be used, such that the nodal geometry is not updated 

due to displacements.
70

(4) The following procedures shall be followed to develop constitutive model inputs:

 (a) Tensile test data from ASTM E8 or equivalent is required to establish the stressstrain curve. Tensile specimens should come from the same material and metallurgical orientation 

used for surface crack testing. Multiple tests are preferable to evaluate a typical response.

 (b) The finite element constitutive model shall follow Mises plasticity (incremental 

plasticity).

 (c) Stress-strain data shall be input as a table look-up, developed as follows:

 (i) Elastic modulus values shall come from either handbook values or dedicated 

modulus testing per ASTM E111 or equivalent. Do not rely on tensile test results for accurate 

elastic modulus values.

 (ii) Use engineering stress-strain values to be compatible with the small strain 

assumption in the element formulation.

 (iii) Separate plastic strain from experimental stress-strain data by subtracting the 

elastic strain based on the best fit modulus to the actual tensile test response (not the handbook 

value). If required, the quality of the linear fit should be biased toward the proportional limit 

region. Define the proportional limit when the plastic strain value is consistently >0.0001. 

 (iv) Develop the input table as required by the analysis code (stress versus total 

strain or plastic strain). If the FEA code requires total strain input, then use the elastic modulus 

value (4)(c)(i) to calculate the elastic strain for a given stress value and sum the elastic and plastic 

strain components to calculate total strain. The table should have a sufficient number of entries to 

accurately define the proportional limit, rollover, and tangent modulus characteristics of the stressstrain curve. The material model should be linear-elastic up to the proportional limit.
71

ASTM Committee on Fatigue and Fracture

E08.07.03 Task Group on Surface Cracks

Analytical Round Robin on Elastic‐Plastic Analysis

of Surface Cracks in Flat Plates

Introduction: Introduction:  The task group on fracture fracture toughness toughness of surface surface cracked cracked plates is in the process process of

developing a  new material test standard for the assessment of surface crack toughness tests in the linear‐

elastic and the elastic‐plastic regime.  While evaluation of stress intensity factor, K, is well‐accepted for

surface crack geometries through the Newman‐Raju equations or other existing tabulations, the evaluation

of elastic‐plastic J‐Integral values requires the user to perform an analytical assessment of the experiment

using methods that  cannot be explicitly dictated by the test standard.   The feasibility of allowing this

latitude latitude in assessment assessment methodology methodology in a testing testing standard standard requires requires validation validation through through an inter‐laboratory laboratory

study.

Objective:  Determine the variability in J‐Integral values from analytical assessment of experimental surface

crack tests under elastic‐plastic conditions providing minimal guidance. Phase I of the study is conducted

“blind.”  No test results are provided to the participants at this time.

Phase I, Problem Problem A

Surface Cracked Plate in Tension

Requested Analysis: Elastic‐plastic analysis with sufficient quantity of

load steps to report the following data:

1.  Force versus CMOD from 0 to 289kN (65kip)

2.  J‐Integral versus φ, for at least the following force values

a. 200kN (45kip)  b. 251.8kN (56.6kip)  c. 289kN (65kip)

Request sufficient data to plot J vs φ

3.  J‐Integral versus Force from 0 to 289kN (65kip) at φ ≈ 17°

Any consistent system of units is fine, but ksi and inches would be helpful

for consolidating the data.   See the draft surface crack test standard for

definition of φ and CMOD for the surface crack specimen.

Material Properties:  Stress‐strain data shall be provided to all

participants in an Excel spreadsheet.  Interpretation of the data and type

of constitutive model is left to the discretion of the user.  For consistency,

all analysis should utilize the following elastic properties:

E = 74.46 GPa (10.8E3 ksi)

ν = 0.33

W = 88.82 (3.497)

(0.374)

Test Cross Section A‐A: Dimensions taken directly from test specimen A.  

Dimensions are mm(inches).  See Figure 3 for complete specimen geometry.

ν 0.33

Figure 2.  Experimental test

set‐up for specimen A.

B = 9.50

a

a = 6.17 (0.243) 2c 2c = 12.70 (0.500)

Figure 1.  Cross‐section A‐A and crack size definition.

Crack assumed centered with respect to W.

Questions?  Please email us:

Douglas.N.Wells@nasa.gov

Phillip.A.Allen@nasa.gov

APPENDIX A—PHASE I PROBLEM STATEMENT AND TABULAR DATA
72

Scope and Schedule: This is the first of at least two problems in Phase I of this Analytical Round Robin.  A

similar problem under 4‐pt bending should follow shortly.  Future phases of the round robin will be defined

as needed to develop the necessary minimum guidance for the surface crack test standard to sufficiently

control analytical variability, or to determine sources of variability observed in Phase I.  We request that

results of this first problem be returned by email (see “Questions?” box on other side) by July 11, 2011.  As

always, earlier would be very helpful.

THANK YOU for participating in this very helpful study!    Doug Wells and Phillip Allen, NASA MSFC.

127(5)

25.4(1) 25.4(1)

57.15(2.25)

mm(inches)

12.7(0.5)

R 381(15)

W

406.4(16)

B

R 47.6(1.875)

203.2(8) 203.2(8)

A A

114.3(4.5)

76.2(3)

30 23(1 19) TYP

114.3(4.5)

76.2(3)

18 29(0 72)

Figure 3.  Surface Crack Tension Specimen Geometry

18.29(0.72) 30.23(1.19) TYP

33.27(1.31)

63.5(2.5)

15.2(0.6)
73

Strain Stress (MPa) Strain Stress (MPa) Strain Stress (MPa) Strain Stress (MPa) Strain Stress (MPa) Strain Stress (MPa)

0.00000 0.0 0.00577 369.5 0.02034 425.7 0.03792 456.7 0.05704 470.2 0.07715 468.10.00016 11.6 0.00609 375.3 0.02089 426.8 0.03850 457.4 0.05762 470.6 0.07786 467.80.00031 23.3 0.00620 376.9 0.02147 427.9 0.03909 458.0 0.05823 471.0 0.07858 467.60.00047 34.9 0.00669 382.3 0.02203 429.5 0.03968 458.5 0.05839 471.2 0.07927 467.00.00062 46.5 0.00721 386.0 0.02260 430.9 0.04026 459.1 0.05886 470.8 0.07991 465.70.00078 58.2 0.00736 386.9 0.02318 432.2 0.04083 459.9 0.05950 471.1 0.08058 465.80.00094 69.8 0.00777 388.9 0.02373 432.9 0.04142 460.1 0.06013 471.2 0.08130 465.30.00109 81.4 0.00835 391.9 0.02430 434.4 0.04201 460.6 0.06076 471.6 0.08199 464.80.00125 93.1 0.00895 394.4 0.02486 435.7 0.04259 461.1 0.06140 471.5 0.08267 463.70.00141 104.7 0.00954 396.2 0.02544 436.5 0.04314 462.2 0.06205 471.5 0.08336 462.90.00156 116.3 0.01011 398.1 0.02600 437.4 0.04374 462.8 0.06264 471.5 0.08408 462.70.00172 128.0 0.01057 399.5 0.02642 438.8 0.04434 462.7 0.06327 471.6 0.08478 461.80.00187 139.6 0.01072 400.1 0.02656 438.7 0.04494 463.8 0.06377 472.2 0.08548 460.60.00203 151.2 0.01130 402.2 0.02712 439.9 0.04553 464.2 0.06393 471.7 0.08622 459.7

0.00219 162.9 0.01159 402.8 0.02769 441.2 0.04612 464.1 0.06457 471.8 0.08693 459.1

0.00234 174.5 0.01187 403.4 0.02826 442.0 0.04666 464.8 0.06522 471.7 0.08761 457.9

0.00250 186.1 0.01247 405.1 0.02883 443.1 0.04725 465.6 0.06588 471.9 0.08828 457.60.00266 197.8 0.01261 405.9 0.02936 443.7 0.04771 465.7 0.06654 471.8 0.08900 456.10.00281 209.4 0.01304 406.9 0.02991 445.2 0.04786 465.9 0.06719 471.5 0.08969 455.40.00297 221.1 0.01362 408.5 0.03049 446.0 0.04847 466.4 0.06785 471.7 0.09040 454.50.00312 232.7 0.01377 408.7 0.03104 446.5 0.04909 466.3 0.06852 471.9 0.09112 453.3

0.00328 244.3 0.01421 410.0 0.03158 448.1 0.04969 466.7 0.06917 471.4 0.09186 452.10.00344 256.0 0.01481 411.4 0.03216 448.8 0.05031 467.6 0.06983 471.4 0.09258 451.1

0.00359 267.6 0.01537 412.8 0.03273 449.4 0.05093 467.7 0.07050 471.5 0.09330 450.10.00375 279.2 0.01579 414.3 0.03328 450.4 0.05154 467.6 0.07117 471.2 0.09400 448.3

0.00391 290.9 0.01593 415.0 0.03385 451.4 0.05211 467.9 0.07179 470.0 0.09467 447.30.00406 302.5 0.01642 415.2 0.03439 451.6 0.05273 468.9 0.07238 470.6 0.09541 446.30.00422 314.1 0.01693 417.5 0.03496 452.4 0.05333 469.1 0.07308 470.8 0.09612 444.50.00428 317.0 0.01750 419.1 0.03556 453.8 0.05396 468.9 0.07376 470.4 0.09684 443.30.00456 326.1 0.01807 420.6 0.03613 454.3 0.05459 469.5 0.07441 469.7 0.09758 441.60.00481 338.3 0.01864 421.8 0.03672 455.1 0.05521 469.9 0.07510 469.4 0.09828 439.60.00509 350.5 0.01919 422.7 0.03701 455.2 0.05584 470.1 0.07578 469.2 0.09898 438.1

0.00540 361.1 0.01976 424.1 0.03731 455.8 0.05645 469.8 0.07647 468.8 0.09979 434.7

Table A1. Tabular data, full engineering stress-strain data provided to the ILS participants. 

Provided Data, Cont'd Provided Data Provided Data, Cont'd Provided Data, Cont'd Provided Data, Cont'd Provided Data, Cont'd
74

Table A2. Useful data for independent evaluation of the ILS problem. These data were not provided to ILS participants. 

Force CMOD Force CMOD Force CMOD Strain Stress

kN mm kN mm kN mm mm/mm MPa

0.0 0.00000 128.5 0.04674 0.0 0.00000 0.00426 317.0

1.2 0.00051 134.2 0.04877 14.5 0.00528 0.00540 361.1

3.6 0.00127 139.7 0.05105 28.9 0.01057 0.00609 375.4

6.8 0.00229 145.3 0.05334 43.4 0.01586 0.00721 386.0

10.6 0.00356 150.8 0.05563 57.8 0.02116 0.00835 391.8

14.8 0.00483 156.3 0.05791 72.3 0.02649 0.00954 396.2

19.2 0.00635 161.7 0.06020 86.7 0.03184 0.01057 399.6

23.9 0.00813 167.1 0.06248 101.2 0.03724 0.01159 402.9

28.8 0.00991 172.4 0.06477 115.7 0.04269 0.01247 405.1

33.9 0.01168 177.7 0.06706 130.1 0.04821 0.01362 408.5

39.2 0.01346 183.0 0.06934 144.6 0.05382 0.01481 411.4

44.2 0.01524 188.3 0.07188 159.0 0.05955 0.01578 414.3

49.6 0.01727 193.6 0.07442 173.5 0.06545 0.02600 437.4

54.9 0.01905 198.8 0.07696 187.9 0.07159 0.03613 454.4

60.5 0.02108 204.2 0.07976 202.4 0.07807 0.04666 464.8

65.9 0.02311 209.2 0.08255 216.9 0.08506 0.05645 469.8

71.5 0.02515 214.5 0.08560 231.3 0.09285 0.06377 472.2

77.1 0.02718 219.7 0.08865 245.8 0.10202

82.6 0.02921 224.9 0.09169 260.2 0.11470

88.5 0.03150 230.2 0.09550 274.7 0.13530

93.9 0.03353 235.3 0.09906 289.1 0.15986

99.9 0.03581 240.3 0.10312

105.7 0.03785 245.6 0.10770

111.7 0.04013 250.6 0.11227

117.3 0.04242 252.0 0.11354

123.1 0.04445

Experiment Experiment, Cont'd Lab-1 Analysis Lab-1 Constitutive Input
75

APPENDIX B—EVALUATION OF INTER-LABORATORY STUDY RESULTS 

ASSUMING A NORMAL DISTRIBUTION

In many cases, the ILS results have been reduced to report average and standard deviation 

information. In doing so, the results are assumed to be distributed normally. To evaluate the quality 

of that assumption, many of the parameters in the study have been plotted on normal probability 

axes to evaluate the degree to which they follow the normal distribution. The J-integral results at 

φ=17° are plotted in figure 38 for each of the specified force levels. The majority of responses at 

the 200 and 252 kN force levels are described well with the linear fit in this space. The lower values 

of J fall below what would be expected from the normal distribution fit. At 289 kN, the trends are 

stronger, with the family of J ≈ 25 kJ/m2 standing alone and influencing the fit. The slope of the 

result subset with J ≈30 kJ/m2 looks to be a better match to the slope at the previous force levels 

and indicates that this subset of models likely better fits the normal distribution. In this case, the 

extreme values in the ILS result population for J are not perfectly fit to the normal distribution and 

will cause errors in the standard deviation estimates; however, the use of these statistics within this 

TM are still useful for indicating the relative variability in the result set.

Probability Plot, Normal, J( =17) at 200, 252, and 289 kN φ

0.75

0.9

0.95.

0.25

0.5

Probability

200 kN

252 kN

289 kN

5 10 15 20 25 30 35

0.05

0.1

J-Integral (kJ/m2)

Figure 38. Probability plot for normal distribution of J-integral values.
76

Figure 39 illustrates the distribution of data regarding the elastic stiffness of the ILS models. This figure can be compared with figure 7. Again, the central values of stiffness reasonably 

reflect a normal distribution, but the slightly less stiff models are more so than the normal distribution would predict. The result from lab-2 stands unique in this view as well, with stiffness well 

outside the expected upper bound predicted by a normal fit.

0.9

0.95

Probability Plot for Normal Distribution

0.5

0.75

Stiffness

0.1

0.25

2,550 2,600 2,650 2,700 2,750 2,800 2,850 2,900

0.05

CMOD Stiffness (kN/mm)

Probability

Figure 39. Probability plot for normal distribution of elastic stiffness of modeled 

CMOD response.

Figures 40 and 41 relate to the use of the model size statistics. In figure 40, the size of the 

model as measured by the number of crack front nodes is plotted in normal space. In this case, 

the quality of fit to the normal distribution is not very good. This raises some question regarding 

the rigorous validity of the MSP that uses the average and standard deviation of the model nodal 

counts as shown in equation (1). The MSP sums two values that have been transformed into a standard normal space under assumptions that the values are normally distributed. The result in figure 41 shows that the central set of models characterized by –1<MSP<2 are somewhat normally 

distributed, but relative to those, the smallest models are a bit larger than predicted. As in the case 

of the J-integral results, the distributions are not rigorously normal, but sufficiently so to conclude 

that normal statistics derived from the results are sufficient for relative comparison within the context of this study.
77

Probability Plot for Normal Distribution

0.75

0.9

0.95

0.25

0.5

Probability

CF Node Count

20 40 60 80 100 120 140 160 180 200

0.05

0.1

Number of Crack Front Nodes

Figure 40. Probability plot for normal distribution of the number of nodes used 

to model the crack front.

0.9

0.95

Probability Plot for Normal Distribution

0.5

0.75

r

MSP

0.1

0.25

–3 –2 –1 0 1 2 3 4 5

0.05

Model Size Parameter (MSP)

Probability

Figure 41. Probability plot for normal distribution of the model size

parameter from equation (1).
78
79

REFERENCES

 1. Newman, J.C.; and Raju, I.S.: “Stress-intensity Factor Equations for Cracks in ThreeDimensional Finite Bodies Subjected to Tension and Bending Loads,” Computational 

Methods in the Mechanics of Fracture, Vol. 2, pp. 311–334, 1986.

 2. Newman, J.C.; Reuter, W.G.; and Aveline, C.R.J.: “Stress and Fracture Analyses of SemiElliptical Surface Cracks,” ASTM Special Technical Publication, STP1360, pp. 403–423, 2000.

 3. API A, API Recommended Practice 579-1/ASME FFS-1, Fitness-For-Service, 2nd Ed., 

Ed: American Petroleum Institute, 2007.

 4. E1820-09, Standard Test Method for Measurement of Fracture Toughness, American Society 

for Testing and Materials, Philadelphia, PA, 2011.

 5. Bleackley, M.H.; and Luxmoore, A.R.: “Comparison of Finite Element Solutions With 

Analytical and Experimental Data for Elastic-Plastic Cracked Problems,” International Journal 

of Fracture, Vol. 22, pp. 15–39, 1983.

 6. Larrson, L.H.: “A Calculational Round Robin in Elastic-Plastic Fracture Mechanics,” 

International Journal of Pressure Vessels and Piping, Vol. 11, pp. 207–228, 1983.

 7. ABAQUS., Dassault Systèmes, <http://www.simulia.com/products/unified_fea.html>, 

Verified January 11, 2011.

 8. WARP3D, Static and Dynamic Nonlinear Analysis of Fracture in Solids, Dodds, R.H., 

University of Illinois at Urbana-Champaign, <http://code.google.com/p/warp3d/>, 

Verified January 11, 2011.

 9. ANSYS, ANSYS, Inc., <http://www.ansys.com/>, Verified January 11, 2011.

10. ADINA, ADINA R&D, Inc., <http://www.adina.com/index.shtml>, Verified January 11, 

2011.

11. Dodds, R.H.: “Effects of Reduced Integration on the 2-D Quadratic Isoparametric Element 

in Plane Strain Plasticity,” International Journal of Fracture, Vol. 19, pp. R75–R82, 1982.

12. Rice, J.R.: “A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks,” Journal of Applied Mechanics, Vol. 35, pp. 379–386, 1968.

13. Brocks, W.; and Scheider, I.: “Reliable J-Values; Numerical Aspects of the Path-Dependence 

of the J-integral in Incremental Plasticity,” MP Materialprüfung, Vol. 45, p. 264, 2003.
80

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Analytical Round Robin for Elastic-Plastic Analysis of Surface

Cracked Plates: Phase I Results

D.N. Wells and P.A. Allen

George C. Marshall Space Flight Center

Huntsville, AL 35812

National Aeronautics and Space Administration

Washington, DC 20546–0001

Unclassified-Unlimited

Subject Category 39

Availability: NASA CASI (443–757–5802)

Prepared by the Damage Tolerance Assessment Branch, Materials and Processes Laboratory, 

Engineering Directorate

M–1332

Technical Memorandum

NASA/TM—2012–217456

surface crack, finite element, elastic-plastic fracture, J-integral, round robin

01–03–2012

UU 94

NASA

U U U

An analytical round robin for the elastic-plastic analysis of surface cracks in flat plates was conducted with 15 participants. Experimental results 

from a surface crack tension test in 2219-T8 aluminum plate provided the basis for the inter-laboratory study (ILS). The study proceeded in 

a “blind” fashion given that the analysis methodology was not specified to the participants, and key experimental results were withheld. This 

approach allowed the ILS to serve as a current measure of the state of the art for elastic-plastic fracture mechanics analysis. The analytical results 

and the associated methodologies were collected for comparison, and sources of variability were studied and isolated. The results of the study 

revealed that the J-integral analysis methodology using the domain integral method is robust, providing reliable J-integral values without being 

overly sensitive to modeling details. General modeling choices such as analysis code, model size (mesh density), crack tip meshing, or boundary 

conditions, were not found to be sources of significant variability. For analyses controlled only by far-field boundary conditions, the greatest 

source of variability in the J-integral assessment is introduced through the constitutive model. This variability can be substantially reduced by 

using crack mouth opening displacements to anchor the assessment. Conclusions provide recommendations for analysis standardization.

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