March 2012 

NASA/CR-2012-217347 (Corrected Copy)

Composite Structures 

Damage Tolerance Analysis Methodologies 

James B. Chang, Vinay K. Goyal, John C. Klug and Jacob I. Rome 

The Aerospace Corporation, El Segundo, California 
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March 2012 

NASA/CR-2012-217347 (Corrected Copy)

Composite Structures 

Damage Tolerance Analysis Methodologies 

James B. Chang, Vinay K. Goyal, John C. Klug and Jacob I. Rome 

The Aerospace Corporation, El Segundo, California 
Available from: 

NASA Center for AeroSpace Information 

7115 Standard Drive 

Hanover, MD 21076-1320 

443-757-5802 

Acknowledgments 

The continuous encouragement from Dr. Ivatury Raju of National Aeronautics and Space 

Administration (NASA) Engineering and Safety Center (NESC) is highly appreciated. 

The administrative support from Mr. Allan Cohen of Aerospace Civil and Commercial 

Group is acknowledged. 

The use of trademarks or names of manufacturers in the report is for accurate reporting and does not 

constitute an official endorsement, either expressed or implied, of such products or manufacturers by the 

National Aeronautics and Space Administration. 
ERRATA

NASA/CR‒2012-217347

Issue Date: 06/04/2014

Cover, Title Page, and Report Documentation Page: Author name spelling corrected from Vanay 

to Vinay.
iii 

Foreword 

This report presents the results of a literature review as part of the development of composite 

hardware fracture control guidelines funded by NASA Engineering and Safety Center (NESC) 

under contract NNL04AA09B. J. B. Chang is the principal investigator. 

The objectives of the overall development tasks are to provide a broad information and database 

to the designers, analysts, and testing personnel who are engaged in space flight hardware 

production. 

The literature review concentrated on the state-of-the-art damage tolerance analysis methods that 

have the potential to be used in the damage tolerance demonstration for fracture-critical 

composite hardware in manned and unmanned spaceflight systems. 
 iv

Table of Contents 

1.0 Introduction ............................................................................................................................................ 1

2.0 Residual Strength Prediction Methodologies ......................................................................................... 2

2.1 Surface Cuts and Scratches .............................................................................................................. 2

2.1.1 Closed Form Solutions for Surface Cuts and Scratches .................................................... 2

2.1.2 Finite Element Analysis (FEA) Methods for Surface Cuts and Scratches ........................ 7

2.2 Delaminations ................................................................................................................................ 12

2.2.1 Closed Form Solutions for Delaminations ...................................................................... 12

2.2.2 FEA Methods for Delaminations .................................................................................... 15

2.3 Impact Damage .............................................................................................................................. 31

2.3.1 Prediction of Burst Strength After Impact (BAI) for Composite Overwrapped 

Pressure Vessel (COPV) ................................................................................................. 31

2.3.2 Predicting Damage after Low-Velocity Impact .............................................................. 34

3.0 Damage Tolerance Life Prediction Methodologies .............................................................................. 37

3.1 Delamination Growth in Cyclic Loading ....................................................................................... 37

3.1.1 Closed Form Solution for Delamination Growth ............................................................ 37

3.1.2 Delamination Growth Using FEA ................................................................................... 38

3.2 Damage Growth at Notch Tip ........................................................................................................ 40

3.2.1 Damage Growth in Monotonic Loading ......................................................................... 41

3.2.2 Damage Growth Model Applied to [90/0]s Laminates ................................................... 42

3.2.3 Damage Growth in [90i/0i]s Laminates .......................................................................... 43

3.2.4 Fatigue Damage Growth in [90i/0j]s Laminates.............................................................. 43

3.2.5 Damage Growth in [90/0]ns Laminates ........................................................................... 43

3.2.6 Damage Growth in [90/+45/-45/0]s Laminates ............................................................... 44

4.0 Damage Tolerance Analysis Codes ...................................................................................................... 45

4.1 ADAM-C ....................................................................................................................................... 45

4.2 PFA for Composites ....................................................................................................................... 47

5.0 References ............................................................................................................................................ 47

5.1 Government Documents ................................................................................................................ 47

5.2 Government References ................................................................................................................. 47

5.3 Non-Government References ......................................................................................................... 48

Appendix A: ADAM-C Computer Code ................................................................................................... 54

Appendix B: PFA Code for Composites .................................................................................................... 64
 v

List of Figures 

Figure 1. Comparison of curve shapes for notched strength prediction theories in small notch 

range. ...................................................................................................................................... 6

Figure 2. Comparison of curve shapes for notched strength prediction theories in large crack 

notch. ...................................................................................................................................... 7

Figure 3. Finite element mesh for /a = 1. ............................................................................................. 8

Figure 4. Schematic of typical subcritical damage in a [90/0]s laminate. .............................................. 9

Figure 5. Non-local damage theory predictions versus tested strength. .............................................. 10

Figure 6. 2-D eight-noded isoparametric quadrilateral element. ......................................................... 11

Figure 7. Mode I, II and mixed-mode loading of DCB specimens. .................................................... 13

Figure 8. Schematic load-displacement curve for a delamination toughness measurement. ............... 14

Figure 9. Thin-film model – three configurations. .............................................................................. 15

Figure 10. Finite element nodes near crack tip. ..................................................................................... 16

Figure 11. Inner and outer contours, which form a closed contour around the crack tip when 

connected by Γ+ and Γ- (Anderson, 1991). ........................................................................... 17

Figure 12. Isoparametric elements that are commonly used in 2-D and three-dimensional (3-D) 

crack problems. .................................................................................................................... 18

Figure 13. Degeneration of a quadrilateral element into a triangle at the crack tip. .............................. 19

Figure 14. Load-deflection response of DCB test. ................................................................................ 20

Figure 15 Crack-tip element and local loading. ................................................................................... 21

Figure 16. Cross-sectional view of laminate containing two symmetrically located delaminations 

that is subjected to compressive loading. ............................................................................. 23

Figure 17. Post-buckled configuration. ................................................................................................. 23

Figure 18. Crack-tip element and loading for delamination buckling problem. .................................... 24

Figure 19. Comparison of total ERR for a [02/90/02]3s laminate. .......................................................... 24

Figure 20. Laminate with a free edge delamination. ............................................................................. 25

Figure 21. (a)Transverse stress distribution for an uncracked mid-plane symmetric laminate 

subjected to uniform axial extension in the X1 direction. (b) Crack-tip element. 

Superposition of problem (b) onto (a) gives solution to cracked laminate. ......................... 25

Figure 22. ERR versus hygroscopic loading for a [45/0/–45/90]s T300/5208 laminate with edge 

delaminations at both (– 45/90). ........................................................................................... 27

Figure 23. Configuration of embedded delamination. ........................................................................... 27

Figure 24. Progressive delamination fronts for the [(90/0)s] laminate. ................................................. 29

Figure 25. Strain-energy release rate distributions for the [(90/0)s] laminate. ...................................... 29

Figure 26. Schematic of axial splitting and delamination in a [(90/0)]s laminate. ................................ 30

Figure 27. Local finite element mesh. ................................................................................................... 31

Figure 28. FEM of cylindrical COPV under impact.............................................................................. 31

Figure 29. Basic impact damage mechanism of laminates subjected to 2-D line-loading impact. 

Top: delamination induced by inner shear cracks. Bottom: delamination induced by 

surface bending crack. .......................................................................................................... 36

Figure 30. Schematics of the delamination growth mechanism due to a shear crack and a bending 

crack, respectively, in a laminated composite subjected to point-nose impact. ................... 36

Figure 31. Cyclic growth rates (dL/dN) in constant-amplitude compression-compression loading as 

a function of L (solid lines – experimental data, dashed lines – analytical prediction). ...... 37

Figure 32. Strain ERRs (GI and GII) as a function of applied load (εx in percent) and delamination 

length (L) for delamination located between plies 3 and 4. ................................................. 38
 vi

Figure 33. FEM procedure. ................................................................................................................... 39

Figure 34. Strain ERRs (GII) as a function of applied load (εx in %) and delamination length (L) for 

delamination located between plies 1 and 2. ........................................................................ 40

Figure 35. Strain ERRs as a function of applied load (εx in percent) and delamination length (L) 

for delamination between plies 2 and 3. ............................................................................... 40

Figure 36. Schematic of the notch tip damage pattern in a cross-ply specimen. ................................... 41

Figure 37. Normalized split length for a [90/0]S laminate as a function of load cycles showing the 

effect of varying peak stress. The solid lines represent the prediction of Equation (6.9). ... 42

Figure 38. Normalized split length as a function of load cycles for [90i/0i]ns laminates. The solid 

lines represent Equation (3.11). ........................................................................................... 43

Figure 39. Normalized split length as a function of load cycles for [90/0]ns laminates. The solid 

lines represent Equation (3.13). ........................................................................................... 44

Figure 40. Normalized split length as a function of load cycles in [90/+45/–45/0]s laminates. The 

solid lines represent Equation (3.14) in an integrated form. ................................................ 45

Figure 41. ADAM-C program flow chart. ............................................................................................. 45

Figure A-1 ADAM-C methodology. ...................................................................................................... 56

Figure A-2. Point stress model parametric study. From Reference A.8. ................................................. 58

Figure A-3. Hole data correlation. From Reference A.8. ........................................................................ 59

Figure A-4. Crack data correlation. From Reference A.13. ..................................................................... 60

Figure A-5. Point stress model correlation for plates with holes. ............................................................ 60

Figure A-6. IFM correlation for plates with cracks. The data points below the curve indicate 

conservative prediction. ....................................................................................................... 61

Figure A-7. Frequency correlation (hole data). Less than 1.0 indicates conservative prediction. ........... 61

Figure A-8. Frequency correlation (crack data). Less than 1.0 indicates conservative prediction. ......... 62

Figure B-1. Modeling flowchart. ............................................................................................................. 65

Figure B-2. Flowchart of the progressive damage implementation using ABAQUSTM. ......................... 66

Figure B.3. Failure modes integrated into a user subroutine package..................................................... 66

Figure B-4. Interface element consists of a continuous distribution of breakable springs. ..................... 67

Figure B-5. Interface element coupled with a cohesive zone model. ...................................................... 69

Figure B-6. Typical nonlinear shear constitutive law. ............................................................................. 69

Figure B-7. Potential fracture plane in the 3-D failure criterion for matrix-cracking. ............................ 71

Figure B-8. Test configurations for the characterization of fracture properties. ..................................... 73

Figure B-9. Test configurations for the characterization of strength properties. ..................................... 73

Figure B-10. Predictions by PFA of the DCB compare well to experimental data and analytical 

solutions. .............................................................................................................................. 77

Figure B-11. Predictions by PFA for the ELS compare well to analytical solutions. ............................... 77

Figure B-12. Predictions by PFA for the ENF compare well to analytical solutions. ............................... 77

Figure B-13. Predictions by PFA for the FRMM compare well to analytical solutions. .......................... 78

Figure B-14. Predictions by PFA for MMB compare well to analytical solutions. .................................. 78

Figure B-15. Predictions by PFA for ECT compare well to testing. ......................................................... 79

Figure B-16. Damage predicted by PFA compared well by scans of the failed coupon. .......................... 79

Figure B-17. PFA predicted load-displacement response for DCB and ELS accurately. ......................... 79

Figure B-18. Predicted catastrophic failure of the off-axis test configurations. ........................................ 80

Figure B-19. Failure envelope predictions compared to experimental data for compression tests. .......... 80

Figure B-20. Failure response compared well with the experimental response. ....................................... 80

Figure B-21. Flat panel loaded in shear. .................................................................................................... 81

Figure B-22. Curved panel loaded in compression. .................................................................................. 82

Figure B-23. Structural response of the flap panel loaded in shear. .......................................................... 82
 vii

Figure B-24. Structural response of the curved panel loaded in compression for d/W = 0.4. ................... 82

Figure B-25. Structural response of the curved panel loaded in compression for d/W = 0.2. ................... 83

Figure B-26. Visual indications of delaminations correspond to delamination prediction locations. ....... 83

Figure B-27. Sandwich structure impacted by a free-falling mass. ........................................................... 84

Figure B-28. Strain response at bottom facesheet of the sandwich structure. ........................................... 84

Figure B-29. Delamination predicted by analysis compared well with the test. ....................................... 84

Figure B-30. J-section of a circular fuselage frame subject to a radial load. ............................................. 85

Figure B-31. Structural response of the J-beam subject to radial loads for the prediction and testing. .... 85

Figure B-32. Failure and response of the single lap joint test configuration. ............................................ 86

Figure B-33. Failure and response of a double cantilever beam test. ........................................................ 87

Figure B-34. Predicted crack path versus that observed in testing. ........................................................... 87

Figure B-35. Failure and response of a crack lap shear test. ..................................................................... 88

Figure B-36. Single-sided bearing and bypass load (top) and double sided bearing and bypass load 

(bottom). ............................................................................................................................... 88

Figure B-37. Joint strength failure of various specimen configurations. .................................................. 89

Figure B-38. Strain comparison for filled-hole tension test. ..................................................................... 89

Figure B-39. Strain comparison for single-sided bearing testing. ............................................................. 89

Figure B-40. A typical COPV with boundary conditions.......................................................................... 91

Figure B-41. Delamination location for this example. .............................................................................. 91

Figure B-42. Hypothetical fiber defect near delamination. ....................................................................... 92

Figure B-43. Direction of delamination growth in this example. .............................................................. 92

Figure B-44. Predicted delamination growth for low and high fracture toughness values........................ 93

List of Tables 

Table 1. Predicted Damage Modes for Each Ply ................................................................................... 32

Table 2. Calculated Ply Material Effectiveness Factors ........................................................................ 33

Table 3. Comparison of Burst Strengths for Cylindrical COPV ........................................................... 34

Table 4. Assessment of Analytical Methods ......................................................................................... 46

Table A-1. Summary of Literature Survey Results ................................................................................... 55

Table A-2. Assessment of Analytical Methods ......................................................................................... 57

Table B-1. Basic Material Properties Determined from Coupon Testing ................................................. 72

Table B-2. Summary Table for Fracture Test Configurations Analyzed Using Finite Elements .............. 75
 viii

Acronyms 

1-D One-dimensional 

2-D Two-dimensional 

3-D Three-dimensional 

ADAM-C Aerospace Damage Analysis Methodology for Composites 

ASTM American Society for Testing and Materials 

BAI Burst Strength after Impact 

CMH Composite Material Handbook 

COPV Composite Overwrapped Pressure Vessel 

COSAP COPV Stress Analysis Program 

da/dN crack growth rates 

DCB Double Cantilever Beam 

DF Material Degradation Factor 

ECT End Crack Torsion 

ED Edge Delamination 

ELS End Load Split 

ENF End-Notched Flexure 

ERR Energy Release Rate 

FCI Fracture Critical Item 

FEA Finite Element Analysis 

FEM Finite Element Model 

FRMM Fixed Ratio Mixed Mode 

Gr/Ep Graphite/Epoxy 

IFM Inherent Flaw Model 

LaRC Langley Research Center 

LEFM Linear Elastic Fracture Mechanics 

MIL-HDBK Military Handbook 

ML Mar-Lin Model 

MMB Mixed-Mode Bending 

NASA National Aeronautics and Space Administration 

NESC NASA Engineering and Safety Center 

NESC NASA Engineering and Safety Center 

PFA Progressive Failure Analysis 

PS Point Strain Model 

PSM Point Stress Model 

SSP Space Station Program 

UEL User Element Subroutine 

VCCT Virtual Crack Slosure Technique 

WN Whitney and Nuismer 

WN-PSM Whitney and Nuismer Point Stress Model 

WWFE World-Wide Failure Exercise 
1.0 Introduction 

The National Aeronautics and Space Administration (NASA) requires that all payload that used 

the Space Shuttle and all equipment installed in the International Space Station comply with the 

fracture control requirements specified in NASA-STD-5003, Fracture Control Requirements for 

Payloads Using the Space Shuttle and SSP 30558B, Fracture Control Requirements for Space 

Station. In a general fracture control standard, NASA-STD-5007, General Fracture Control 

Requirements for Manned Spaceflight Systems, fracture critical items (FCIs) shall be 

demonstrated to have adequate damage tolerance capabilities, i.e., to have adequate “safe-life.” 

This requirement is applicable to all FCIs independent of materials. 

NASA-STD-5007 has been replaced by NASA-STD-5019, Fracture Control Requirements for 

Spaceflight Hardware. In this standard, fracture control of the composite and bonded structures 

specified in MSFC-RQMT-3479, Fracture Control Requirements for Composite and Bonded 

Vehicle and Payload Structures, shall be applied to all fracture critical composite hardware. The 

detail requirements specified in these standards are shown in NASA-STD-5019. 

In fracture control for hardware items made of metal, the primary failure mode that should be 

considered is the crack propagation leading to catastrophic failure. Cracks or crack-like defects 

are the “damage” to be assessed. Damage tolerance demonstration consists of life and residual 

strength predictions that are obtained by performing fatigue crack growth analysis and static 

fracture strength analysis. Linear elastic fracture mechanics (LEFM) principle is the base for 

such analyses. The primary parameter used in LEFM is the crack-tip stress intensity factor, 

which is a function of the crack size, crack geometry and applied loading conditions. Crack 

growth analysis computer codes used in the aerospace industry, including CRKGRO (Chang 

et al., 1981) and FLAGRO (Hu, 1980), all contain various K-solutions and a large database of 

fracture properties (Kc) and crack growth rates (da/dN). The capabilities of crack-growth life 

predictions using LEFM have been shown to be adequate when correlated with experimental 

results (Chang, 1981). NASGRO (JSC-22267B, Fatigue Crack Growth Computer Program 

NASGRO®, 2001), a derivative code from FLAGRO, is widely used for space flight hardware. 

In fracture control of composite hardware, it is generally agreed that analytical methods that can 

be used for damage tolerance demonstration of composite structures are not as mature as the 

metallic counterparts. It is known that tensile failures of composite structures are primarily K

associated with fiber breaking. On the other hand, compression and shear failures of composite 

structures are often associated with matrix cracking and local delamination. Furthermore, a 

commonly used composite material system such as graphite/epoxy (Gr/Ep) is very brittle and 

highly susceptible to impact damage. Therefore, the evaluations of damage tolerance capabilities 

of composite hardware are much more complicated than their metallic counterparts. 

To meet fracture control requirements levied in NASA-STD-5003 and SSP 30558B, it is 

common practice in the spacecraft industry to perform acceptance proof tests on all flight 

composite structures at a 1.2 × limit load level. For many applications, it has been shown to be a 

viable option to meet the requirement set forth in those standards. However, unless there is 

reasonable compensation for the effect of environment, proof tests are required to be conducted 

at the service environment. In some cases, to perform a proof test at a specific environment is 

both costly and time consuming. For example, a composite cryogenic tank adapter used in an 

upper stage vehicle should be proof tested at cryogenic temperature. However, due to facility 
 2

limitations and other technical concerns, it may be decided to conduct a damage tolerance test 

program instead (Sollars, 1987). Furthermore, to avoid a high proof load causing matrix cracking 

and fiber breakage, there is a limit for a proof test level for composite structural items. NASASTD-5003 specifies that the proof test level should be limited to less than 80 percent of ultimate 

strength. For a marginal design, the proof load should be ≤1.1 × limit load; in this case, the 

“1.2 × limit load” requirement cannot be met. Hence, damage tolerance testing is the only viable 

option if damage tolerance analysis is not acceptable by most fracture-control authorities. 

The aircraft industry has traditionally used the “building block” approach to demonstrate the 

damage tolerance capability of fracture critical airframe structures (Whitehead and Dee, 1983). 

In the building block approach, one would run different level tests in concert with damage 

tolerance analysis. This approach is currently being adopted by the space industry for the same 

purpose (Konno et al., 2001). Because there are many impact threat events, such as tool drop, 

runway debris, hail, lighting strikes, etc. in the operation of an aircraft, impact damage is the 

predominant “defect” to be considered. Compressive strength after impact is used as the 

allowable for composite wing and other critical airframe structural designs. Many attempts have 

been made by investigators to predict residual strength for those structures (Rhoades et al., 1978; 

Byers, 1980; Smith and Wilson, 1985; Horton et al., 1988; Madan, 1988). Another important 

defect that needs to be evaluated is delamination. The growth of delamination in compressioncompression fatigue spectrum applied structures such as upper wing covers was the focus of 

research in the 1970s and 1980s (Rybicki and Kanninen, 1977; Chai and Babcock, 1985). From 

this research, a few growth models have been proposed. 

For spaceflight composite structures such as satellite trusses and platforms and launch vehicle 

fairings, delaminations are very critical defects since these structures are also subjected to 

compression loads. For other composite hardware items such as overwrapped pressure vessels 

and pressurized structures, impact damage and other types of mechanical damage including 

surface cuts and scratches are potential damages that need to be considered. Damage tolerance 

testing on some of the above-mentioned composite hardware could be very costly. Thus, the use 

of damage-tolerance analysis technologies to demonstrate the damage tolerance ability of 

composite FCIs should be encouraged. Accordingly, this state-of-the-art review was conducted; 

the results are presented in this document. 

2.0 Residual Strength Prediction Methodologies 

Damage tolerance analysis methods reviewed in this task cover two categories: residual strength 

analysis and “safe-life” analysis. Defects included are surface cuts and/or scratches, delamination 

and impact damage. The residual strength analysis methods are presented in this section while 

the life prediction methods are discussed in Section 3. 

2.1 Surface Cuts and Scratches 

2.1.1 Closed Form Solutions for Surface Cuts and Scratches 

For laminated composite structures, it has been observed by many investigators that defects such 

as cuts, scratches and impact damage will in general not grow under tensile cyclic loading. In 

these cases, damage tolerance assessments require the consideration of the residual strength 
 3

instead of “safe-life.” A few methods and models have been developed for predicting the 

residual strength of these types of damage. 

• LEFM (Classic) Method 

• Inherent Flaw Model 

• Point Stress Model 

• Point Strain Model 

• Mar-Lin (ML) Model 

2.1.1.1 Brief Description of Closed Form Solutions 

LEFM (Classic) Method 

The LEFM method uses the crack-tip stress intensity factor, K, to characterize the crack behavior 

in an isotropic body. This method has been routinely used to predict the residual strength of 

cracked metallic spaceflight structures since the 1970s. For a through-the-thickness cut in a 

quasi-isotropic composite laminate plate subjected to tensile load, attempts have been made to 

use the same parameter, K. In this case, the cut is treated as a crack. 

Based on the LEFM method, the strain in the fiber direction at a distance r directly ahead of a cut 

(crack) tip can be written in the following infinite series (Poe, 1983). 

 (2.1) 

where 

 (2.2) 

and 

 (2.3) 

In Equations (2.2) and (2.3), Kc is the critical stress intensity factor (or fracture toughness) of the 

composite laminate, x and y are Cartesian coordinates with x perpendicular to the cut; E is 

modulus of elasticity, ν is Poisson’s ratio, and α is the angle that the fiber makes to the x-axis 

(perpendicular to the crack). 

Inherent Flaw Model 

The inherent flaw model (IFM) was originally developed by Waddoups et al. (1971) (Aronsson, 

1986) for predicting the residual strength of notched laminates with infinite width under tensile 

load. It has been extended to finite width by introducing a finite-width correction factor. For a 

composite laminate with a notch (crack) of length 2a and inherent damage zone size, co, 

symmetrically on both sides of the crack tip, the notched tensile strength is given by 

 (2.4) 

( ) ( ) 1 0 2 1 Q2 r Or − ε π = +

Q E = c x K ξ

( )( ) 2 2

xy yx x y ξ =− + 1 E E sin cos νν α α

IFM σ N

IFM o o N o

c

Y ac

σ

σ = +
 4

where σo is the unnotched tensile strength and Y is a finite-width correction factor. 

While the material is considered homogeneous, orthotropic and linear elastic, the following 

relation-ship between and the critical strain-energy release rate (fracture toughness), Gc, is 

obtained (Aronsson, 1986) from the equation: 

 (2.5) 

where 

 (2.6) 

In Equation (2.6), Aij are the in-plane laminate stiffnesses as determined from laminate plate 

theory (Ashton and Whitney, 1970). A linear relationship between co and Gc may further be 

obtained from 

 (2.7) 

Point Stress Model 

The Point Stress Model (PSM) was derived by Whitney and Nuismer (WN) who considered the 

exact elasticity solution of the normal stress ahead of a notch. By approximating the exact 

solution of the normal stress ahead of the notch (crack) with the asymptotic solution, the model 

can be extended to notched laminates with finite geometry. This is justified if the notch length is 

sufficiently large compared to the size of the damage zone size do (Agarwal and Giare, 1982; 

Zweben, 1973). 

For a tension loaded laminate with a notch length 2a, an approximate expression for the 

unnotched tensile strength according to the Whitney and Nuismer Point Stress Model (WNPSM) is given by (Aronsson, 1986): 

 (2.8) 

where do is the characteristic length defined in the PSM (Whitney and Nuismer, 1974; Nuismer 

and Whitney, 1975). 

The finite-width correction factor, Y, used in the calculations was obtained from (Brown, 1970): 

 (2.9) 

IFM σ N

IFM o N 2

o

c

a Y 1

C

σ

σ

π σ

=

+

G

11 22 22 12 66

11 11

AA A 2A A C

2 A 2A

  +

= +  

 

o c 2

o

C

c

πσ = G

PSC o o N

2d

Y a

σ

σ =

234

a aa a a Y 1 09 1 73 8 2 14 2 14 6

w ww w w .. . . .      =− + − +          
 5

This factor was derived originally for an isotropic material and is used for the quasi-isotropic 

laminates considered here. 

The WN-PSM (Nuismer and Whitney, 1975; Whitney and Nuismer, 1974), predicts failure when 

the stress at a characteristic dimension, d1, ahead of the crack tip equals or exceeds σo. The 

residual notched strength is given by 

 (2.10) 

The two parameters in this model, σo and d1, are usually determined by tests. 

PSM 

The PSM proposed by Poe and Sova (Poe, 1983; Poe and Sova, 1980), may also be formulated 

with a characteristic dimension, d2. This model predicts failure when the strain at a distance 

ahead of the notch (crack) tip equals or exceeds the fiber failure strain. The notched failure stress 

is given by 

 (2.11) 

where ξ is a functional that depends on elastic constants and the orientation of the principal load 

carrying plies. The characteristic dimension relates to a material toughness parameter, which was 

found to be relatively independent of lay-up. The two parameters that must be determined for 

this model are the fiber failure strain and d2. 

ML Model 

The Mar-Lin (ML) model (Lin and Mar, 1977; Mar and Lin, 1977) allows the singularity, n, in 

the stress intensity factor to be a value other than square-root. The notched failure stress is given 

by 

 (2.12) 

where Hc is the composite fracture toughness. In general, Hc and the exponent n are the two 

parameters that must be determined by test. In studies, the exponent, n, was related to the 

theoretical singularity of a crack in the matrix, with the tip at the fiber/matrix interface. In this 

case, the singularity is a function of the ratio of fiber and matrix shear moduli and Poisson’s 

ratios. Using this method, the singularities for a range of typical fiber/matrix combinations were 

determined to be between 0.25 and 0.35. 

2.1.1.2 Comparison of Various Closed Form Models 

The following discussions are taken from CMH-17, Composite Materials Handbook Volume 3. 

In this reference, predictions for both small notch (2a ~ 1.2 in. (3 cm)) and large notch (2a up to 

2

n o 1

a 1

a d

σ σ ∞   = −     +

o n 2

2

a 1

2d

σ

σ

ξ

∞ =

+

( )

c n n

H

2a

σ∞ =
 6

20 in. (0.5 m)) sizes are compared. The former notch sizes are characteristic of much of the data 

collected for composites to date. Four models are compared; LEFM (Classical), WN (point 

stress), point strain (PS), and ML. As a baseline for comparing changes in notch length predicted 

by the four models, curves are generated based on average experimental results (finite width 

corrected) for the IM6/937A tape material with W/2a = 4 and a lay-up of [+45/90/–45/0/+30/–

30/0/–45/90/+45]. This will ensure that all theories agree for at least one crack length. 

Figure 1 shows a comparison of the four models for small crack notches. Only a small difference 

is seen between PS and WN models. A close examination of the LEFM and ML curves indicates 

that the singularity has a significant effect on curve shape. For notch lengths less than the 

baseline point, ML predictions are less than those of Classical. For notch lengths greater than the 

baseline point, the opposite is true, and models tend to segregate based on singularity, i.e., WN, 

PS and Classical yield nearly the same predictions. 

Figure 2 shows that singularity dramatically affects differences between predictions in the large 

notch length range. The ratio of notched strength predictions for models with the same order of 

singularity becomes a constant. For example, WN and LEFM become functionally equivalent, 

and the relationship 

 (2.13) 

will yield a value for KIC such that the two models compare exactly for large notches. 

Figure 1. Comparison of curve shapes for notched strength prediction theories in small notch range. 

IC o 1 K =σ π2 d
 7

Figure 2. Comparison of curve shapes for notched strength prediction theories in large crack notch. 

2.1.2 Finite Element Analysis (FEA) Methods for Surface Cuts and Scratches 

The FEA method provides flexibility and accuracy for multiple configurations. 

2.1.2.1 Conventional FEA Methods 

Two conventional FEA methods exist to account for the effects of damage progression on load 

redistribution in finite element models (FEM). Progressive damage methods that degrade various 

stiffness properties of individual elements as specified failure criteria are met (Dopker et al., 

1995) have shown some success in modeling damage growth in specimen configurations. The 

magnitude of the calculations, however, provides a significant obstacle to incorporating them 

into the complex models required for stiffened structures. 

Notched Strength Modeling Using FEA 

In order to achieve a truly general notched strength model, the physical basis of failure must be 

considered, and an extensive study of subcritical damage is an essential element of this process. 

To this end, a notched strength model was developed by Kortschot and Beaumont (1991) based 

on experimental results that illustrate the dependence of notched strength on subcritical damage. 

The model was finite element based and used to determine the modified notch tip stress 

distributions. The failure criterion was complemented by theoretical predictions of damage 

growth. 

The approach of this method is to calculate the stress distribution near the notch tip for a FEM 

that includes a subcritical damage region. Using stress results from the FEA, the stress 

concentration factor, Kt, can be determined. The failure criterion is derived from curve fitting the 

finite element results to test data. 

An example of this method uses a one-quarter model of the specimen using appropriate boundary 

conditions, as shown in Figure 3. Two-dimensional (2-D), eight-noded quadrilateral plane stress 

elements were used. Two layers of elements were superimposed with one layer of elements 
 8

given the properties of a 0o ply and the other given the properties of a 90o ply. Corresponding 

elements in the two layers shared all nodes and hence deformed identically except in the area 

representing the delamination region where the 90o ply was disconnected from the 0o ply. In 

addition, the elements of the 90o ply were continuous across the split in the 0o ply, as illustrated 

in Figure 4. Although this geometry is quite complicated, it is a direct representation of the 

observed subcritical damage in [90/0]s laminates. 

The FEM was constructed to yield information about the tensile stress distribution in the 0o ply 

near the notch tip. The model produced stress contour maps that displayed finite maximum 

stresses for all values of /a ≠ 0, where  is the split length and a is the crack length (V-notch). 

The stress concentration factor, Kt, was determined as the maximum stress in the 0o ply divided 

by the remote stress on the laminate for each /a ratio. The relationship between Kt and /a was 

approximated very closely by the empirical function 

 Kt = 8.16(/a)-0.284 (2.14) 

Figure 3. Finite element mesh for /a = 1. 
 9

Figure 4. Schematic of typical subcritical damage in a [90/0]s laminate. 

In order to derive a failure criterion, that is, to discover what aspect of the 0o ply stress field is 

constant at failure, the terminal stress distributions for a variety of specimens must be compared. 

To do this in a simplified way, the peak stress in the 0o ply immediately prior to failure, σ0p, was 

calculated for all specimens. A value for terminal /a was obtained from a radiograph of the 

failed specimen and substituted into Equation (2.14). The remote stress at failure, σ∞f, was then 

multiplied by Kt to obtain σ0p. The predicted failure stress can be expressed as 

 (2.15) 

where 

Kt = 8.16(/a)-0.284, 

σ∞f = remote failure stress of the laminate; and 

σ0f = failure stress of 0o ply. 

Strain Softening Models 

Strain softening models (Llorca and Elices, 1992; Mazars and Bazant, 1988) have been 

successfully used to simulate the fracture of concrete and other materials by providing the ability 

to capture the global load redistribution that occurs as the notch-tip region is softened by damage 

formation, without the computational concerns of detailed progressive damage models. A range 

of softening laws has been proposed. However, determination of the strain softening material 

laws through trial and error requires a large number of tests, and can be computationally 

intensive. Basham et al. have presented an approach to determine these laws using energy 

methods and relatively few tests (Basham et al., 1993). 

This method is a generalized continuum approach that is compatible with complex FEMs 

required to properly approximate structural configurations. The approach allows the capture of 

0

t

predicted f

f ( )

σ

σ ∞ = K
 10

load redistribution due to local damage formation and growth. One approach is the use of 

nonlinear springs that undergo reductions in bending stiffness as damage occurs. The models can 

be calibrated using small-notch test results, and then extended to large-notch configurations. 

Modeling and calibrating the stiffness reductions are of concern for most structural 

configurations, where out-of-plane loading, load eccentricities, and bending loads are common 

(Kongshavn and Poursartip, 1999). 

Non-local Damage Theory Using Strain Softening 

Nahan and Kennedy have used the damage induced strain softening approach for degradation of 

laminate properties to simulate crack growth in laminate plates (Nahan and Kennedy, 2000). 

This approach defines bulk plate damage as a linear variation of orthotropic damage through the 

plate thickness. A damage term is defined that can range from zero, being a no damage 

condition, to one, which corresponds to complete material failure. A bulk laminate stress-strain 

softening relationship is derived and incorporated into a FEM of the laminate. The FEA will 

generate a simulated load history, in which the damage can be observed to develop at the notchtip and propagate to eventual failure. The theory performs well in predicting tension fracture 

over a wide range of notch sizes as shown in Figure 5. 

Figure 5. Non-local damage theory predictions versus tested strength. 

Predicting Crack Growth Using Micropolar Strain Softening 

In another study, Kennedy used micropolar strain softening for predicting damage growth in 

composite plates with different notch geometries (Kennedy, 1999). The micropolar approach 

differs from classical elasticity theory in that the deformation of the body is described by a 

displacement vector and an independent microrotation vector (Eringen, 1968). The theory 

prevents singularities due to localization of deformation in strain softening materials under 

simple stress states. 

An eight-node, isoparametric quadrilateral element, as shown in Figure 6, was developed for 

carrying out the plain strain micropolar elasticity analysis. The element is similar to that used in 

classical elasticity theory, except the addition of a rotation degree of freedom, φI, at each node. 

The theory was used to predict damage growth in a rectangular laminate plate with notches 

consisting of a circular hole, an elliptical hole, and a sharp crack. In all cases, the results showed 

that the specimen size had a relatively weak effect on the failure load predicted, which indicates 

that the usefulness of this model is limited to materials with this characteristic. 
 11

Figure 6. 2-D eight-noded isoparametric quadrilateral element. 

Finite Element Solution Issues 

Finite element solutions for problems involving strain softening material laws involve a number 

of complexities not often associated with static structural analysis. Specifically, singular stiffness 

matrices are encountered when material failure is occurring in a sufficiently large area and/or 

when the structure is buckling. A solver with a variety of robust, nonlinear solution algorithms 

that is capable of modeling strain-softening response for orthotropic materials should be selected. 

Arc-length methods, such as Riks (1979), have proven useful in dealing with snap-through 

stability problems, and were useful in initial efforts to model tension loaded strain softening 

problems. 

Solving the problem dynamically minimizes a number of numerical difficulties because damping 

and inertial forces can smooth out system response and greatly reduce numerical noise in the 

solution process. As the maximum load is reached, local failure occurs, thus accelerating parts of 

the system, and the numerical integration in time can be stopped when a minimum acceleration 

related to system failure has been achieved. This has proven to be a very accurate failure 

criterion for compression-loaded structural systems (Dopker et al., 1995). 

Element Size and Formulation 

Strain-softening laws and the finite element size are interrelated, due to the effect of element size 

on notch-tip strain distribution. A great amount of mesh sensitivity has been found when strain is 

incorporated into FEAs based on classical plasticity theory. The sensitivity occurs because as the 

mesh is refined, there is a tendency for the damage zone to localize to a zero volume, which can 

lead to the physically impossible condition of structural failure with zero energy dissipation. It 

has been found that larger elements (i.e., ≥ 0.20 in. (0.5 cm)) result in less-severe, but broader, 

stress concentrations and in strain-softening curves with steeper unloading segments (Scholz 

et al., 1997). Element size, therefore, is another parameter in the strain softening approach that 

must be determined. Fortunately, damage in composite materials typically localizes on a 

relatively large scale, e.g. relative to plastic yielding at a crack tip in metal. Element sizes 

required to accurately predict notch-length and finite-width effects in compression are typically 

larger than those required for tension. 

Element formulation and strain softening laws are also interrelated. Limited studies of 4-, 8-, and 

9-noded shell elements found that higher order elements lead to higher fracture strengths and 

large damage zones (Dopker et al., 1994). 
 12

2.2 Delaminations 

Delamination, often referred to as interlaminar fracture, is one of the few instances where 

fracture mechanics formalism is applicable to fiber-reinforced composites on a global scale 

(Anderson, 1991). A delamination surface can be treated as a crack, where the resistance of the 

material to the propagation of the delamination is the critical strain-energy release rate (fracture 

toughness), Gc. Since the delamination typically is confined to the matrix material, e.g., between 

plies, continuum theory is applicable, and the delamination growth is self similar, i.e., keeping 

the same aspect ratio. For composite structures under compressive loads, one of the most 

predominant failure modes is the growth of delamination. This type of “crack” has been studied 

extensively by the aircraft industry since the aircraft wing and other control surfaces are 

generally loaded in both tension and compression. 

2.2.1 Closed Form Solutions for Delaminations 

Closed form solutions are often used in the determination of the critical strain-energy release 

rates, Gc, of a composite laminate specimen. Typical specimens include the double cantilever 

beam (DCB) specimen, the end-notched flexure (ENF) specimen and the edge delamination 

(ED) specimen. An advantage of the DCB specimen configuration is that it permits 

measurements of Mode I, Mode II, or mixed-mode fracture toughness (see Figure 7). The ENF 

specimen has essentially the same geometry as the DCB specimen, but the latter is loaded in 

three-point bending, which imposes Mode II displacements of the crack faces. The ED specimen 

simulates the conditions in an actual structure – tensile stresses normal to the ply are highest at 

the free edge; thus, delamination zones often initiate at the edges of a panel. 

For pure Mode I loading, elastic beam theory leads to the following expression for energy release 

rate (ERR): 

 (2.16) 

where B is the beam’s thickness, and 

 (2.17) 

2 2

I

I

P a

B E I

G =

3

I

I

2P a E I

3 = ⋅ Δ
 13

Figure 7. Mode I, II and mixed-mode loading of DCB specimens. 

Assuming linear beam theory, the corresponding relationship for Mode II is given by 

 (2.18) 

Mixed-mode loading conditions can be achieved by unequal tensile loading of the upper and 

lower portions of the specimens. The applied loads can be resolved into Mode I and Mode II 

components as follows: 

 (2.19a) 

 (2.19b) 

where PU and PL are the upper and lower loads, respectively. The components of G can be 

computed by inserting PI and PII into Equations (2.16), (2.17) and (2.18) above. 

2 2

II

II

3P a

4 B E I

G =

P P I L =

U L

II

P P

P

2

− =
 14

Linear beam theory may result in erroneous estimates of G, particularly when the specimen 

displacements are large. In this case, the area method provides an alternative measure of G. 

Figure 8 schematically illustrates a typical load-displacement curve, where the specimen is 

periodically unloaded. The loading portion of the curve is typically nonlinear, but the unloading 

curve is usually linear and passes through the origin. G can be estimated from the incremental 

area inside the load-displacement curve, divided by the change in crack area: 

 (2.20) 

The Mode I and Mode II components of G can be inferred from the PI – ΔI and PII – ΔII curves, 

respectively. 

(Chai, 1982). The one-dimensional (1-D) thin film model is shown below. 

Figure 8. Schematic load-displacement curve for a delamination toughness measurement. 

2.2.1.1 Thin-Film Models 

Three delamination models derived by Chai in the early 1980s include the 1-D Thin-Film Model, 

Thick-Column Model and the 2-D Elliptical Model (Chai, 1982). The 1-D thin-film model is 

shown below. 

Consider the three stages in thin-film delamination and buckling as shown in Figure 9. The 

delaminated film of thickness h and length  is part of an infinitely thick body characterized by 

Young’s Modulus, E, and Poisson’s ratio, ν. The strain-energy release rate, G, can be expressed 

as 

 (2.21) 

where ε0 is the compressed strain and εcr is the critical buckling strain. 

U

B a

Δ

Δ

G =

( ) ( )( )

2

0 cr 0 cr

Eh 1

3

2

ν

εε ε ε

−

G = −−
 15

Figure 9. Thin-film model – three configurations. 

The thin-film model provides a simple means of predicting delamination growth, but it has very 

limited application due to its simplifying assumptions such as infinite thickness and through the 

thickness delamination. 

2.2.2 FEA Methods for Delaminations 

The FEA method provides flexibility and accuracy for multiple configurations. Analyses 

methods have been presented that utilize conventional continuum elements. Other methods use 

singular elements that contains singularity to predict delaminations. 

2.2.2.1 Conventional FEA for Delaminations 

Analyses that use conventional continuum elements, such as, quadrilateral and triangular shell 

elements or hexagonal and pentagonal solid elements, are referred to as conventional methods. 

One such method for predicting delaminations in a composite structure using conventional 

elements is the Virtual Crack Closure Technique (VCCT), first presented by Rybicki and 

Kanninen in a classic paper (Rybicki and Kanninen, 1977). 

Calculation of Stress Intensity Factor by FEA 

In their paper in 1977, Rybicki and Kanninen presented a method for evaluating stress intensity 

factors using a modified crack closure integral and a constant strain finite element analysis. 

The method uses Irwin’s crack closure integral. The assumption is that if a crack extends by a 

small amount Δc, the energy absorbed in the process is equal to the work required to close the 

crack to its original length (Irwin, 1958). The following is the mathematical interpretation 

 (2.22) 

( ) ( )

( ) ( )

c

y 0 0

c

xy 0 0

1 lim r 0 r dr

2

1 lim r 0 u r dr

2

c

c

c ,v,

c

c, ,

c

σ π

τ π

Δ

Δ →

Δ

Δ →

= Δ−

Δ

+ Δ−

Δ





G
 16

where G is the energy release rate, σy and τxy are the stresses near the crack tip, u and v are the 

relative sliding and opening displacements between points on the crack faces, and Δc is the crack 

extension at the crack tip. The above equation can also be expressed as G = GI + GII. 

Taking the limit and using the stress intensity factors KI and KII gives 

 and (2.23-2.24) 

where β = 1 for plane stress and 1 – ν2 for plane strain and E is Young’s modulus. 

Rybicki used constant strain elements near the crack tip, and found a good match comparing 

with the other results, using more complex methods. The finite element approach for evaluating 

GI and GII from Equation (2.22) is based on the nodal forces and displacements as shown in 

Figure 10. 

In equation form, the work required to close the crack an amount Δc is 

 and (2.25-2.26) 

where the value of Fc and Tc are the y- and x-forces, respectively, that are required to hold nodes 

c and d together. 

Figure 10. Finite element nodes near crack tip. 

The energy domain integral methodology (Shih et al., 1986; Moran and Shih, 1987) is a general 

framework for numerical analysis of the J-integral, energy release rate. This approach is 

extremely versatile, as it can be applied to both quasistatic and dynamic problems with elastic, 

plastic, or viscoplastic material responses, as well as thermal loading. Moreover, the domain 

integral formulation is relatively simple to implement numerically, and it is very efficient. This 

approach is very similar to the VCCT. 

2

I

I E = β K G

2

II

II E = β K G

I ccd ( ) 0

1 lim F

c 2

v v

Δ → c = ⋅− Δ

G II c c d ( ) 0

1 lim T u u

Δ →c 2 c = ⋅− Δ

G

24

l1 Δc l2

m n o p

g

h

i j k l

y,v

x,u

Element

4

Element

3

Element

2

Element

1 c

d

a

b

l1

e

f
 17

Taking account of plastic strain, thermal strain, body forces, and crack face tractions leads to the 

following general expression for J in two dimensions: 

p p j ij j

ij 1i ij ij i

1 i 11 1 1 A

u u q w w F q dA

x x xx x x *

ε Θ σ δ σ ασ

      ∂ ∂ ∂   ∂ ∂∂

= − + −+ −       ∂ ∂ ∂∂ ∂ ∂        

 J (2.27) 

where α is the coefficient of thermal expansion, Θ is the temperature relative to ambient, the 

super-script p denotes the plastic strain, and q is an arbitrary but smooth function that is equal to 

unity on Γo and zero on Γ1 (Figure 11). 

Inertia can be taken into account by incorporating T, the kinetic energy density, into the 

group of terms that are multiplied by q. For a linear or nonlinear elastic material under 

quasistatic conditions, in the absence of body forces, thermal strains, and crack face tractions, 

Equation (2.27) reduces to 

j

ij 1i

1 i A

u q w dA

x x *

σ δ

  ∂ ∂ = −   ∂ ∂    J (2.28) 

Figure 11. Inner and outer contours, which form a closed contour around the crack tip when connected by 

Γ+ and Γ- (Anderson, 1991). 

Equation (2.28) is equivalent to Rice’s path-independent J-integral. When the sum of the 

additional terms in the more general expression (Equation (2.27)) is nonzero, J is path 

dependent. The variable, q, can be interpreted as a normalized virtual displacement. The q 

function is merely a mathematical device that enables the generation of an area integral, which is 

better suited to numerical calculations. 

Implementing J-Integral into FEA 

The domain integral approach can be implemented into finite elements with some modifications 

(Shih et al., 1986; Dodds and Vargas, 1988). In 2-D problems, one must define the area over 

which the integration is to be performed. The inner contour, Γo is often taken as the crack tip, in 

which case A* corresponds to the area inside of Γ1. The boundary of Γ1 should coincide with 

element boundaries. An analogous situation applies in three dimensions, where it is necessary to 

j

2j

1

u

q d

x

Γ Γ

σ Γ

+ − +

∂ − ∂ 
 18

define the volume of integration. The latter situation is somewhat more complicated, however, 

since J(η) is usually evaluated at a number of locations along the crack front. 

In the absence of thermal strains, path-dependent plastic strains, and body forces within the 

integration volume or area, the J integral is expressed as follows: 

 (2.29) 

where the spatial derivatives of q are given by 

 (2.30) 

and m is the number of Gaussian points per element, wp are w are weighting factors, n is the 

number of nodes per element, qi are the nodal values of qi, ξi are the parametric coordinates for 

the element, and NI are the element shape functions. The quantities within { }p are evaluated at 

the Gaussian points. Note that the integration over crack faces is only necessary when there are 

nonzero tractions. 

Finite Element Mesh Design 

Although many commercial codes have automatic mesh generation capabilities, producing a 

properly designed FEM always requires some human intervention. Crack problems, in particular, 

require skill on the part of the user. Figure 12 shows several common element types for crack 

problems. At the crack tip, four-sided elements (in 2-D problems) are often degenerated down to 

triangles, as Figure 13 illustrates. An analogous situation occurs in three dimensions, where a 

brick element is degenerated to a wedge. 

Figure 12. Isoparametric elements that are commonly used in 2-D and three-dimensional (3-D) crack 

problems. 

* *

m jj j

ij 1i p 2j

A or V p 1 crack faces 1 ik 1

p

ux u q w w qw

xx x

J det σδ σ ξ =

           ∂∂ ∂ ∂ =− −          ∂ ∂∂ ∂             

n 2 or 3

I k

i kj I1k1

q N

x x

ξ

ξ = =

∂ ∂ ∂ = ∂ ∂∂  
 19

Figure 13. Degeneration of a quadrilateral element into a triangle at the crack tip. 

Using the Strain Softening Method to Perform Progressive Delamination Analysis 

Using the strain-softening method, Davila et al., have developed a method to simulate 

progressive debonding or delamination using decohesion elements (Davila et al., 2001). The 

onset of damage and the growth of delamination can be simulated without previous knowledge 

about the location, size, and direction of propagation of the delaminations. The approach has 

been found to be well suited to nonlinear progressive failure analyses where both ply damage 

and delaminations are present. 

The approach consists of placing interfacial decohesion elements between composite layers. A 

decohesion failure criterion that combines aspects of strength-based analysis and fracture 

mechanics is used to simulate delamination by softening the element. As the delamination grows, 

the tractions across the interface will grow to a maximum value, then decrease and eventually 

vanish when complete decohesion occurs. The work of normal and tangential separation can be 

related to the critical values of ERR. 

Davila et al. developed an 8-node decohesion element and implemented it in ABAQUSTM to 

perform the analyses (de Moura et al., 2000; 1997). The decohesion element models the interface 

between sublaminates or between two bonded components. The element consists of a zerothickness volumetric element in which the kinematics of the elements that are being connected 

on the top and bottom faces are correctly modeled. The element represents damage by a material 

response using a cohesive zone ahead of the crack tip to predict the delamination growth. The 

interface elements have been formulated to deal with mixed-mode delamination growth 

problems. 

An example comparing this approach with a beam solution and test data uses a DCB to 

determine interlaminar fracture toughness in Mode I (GIC). A Gr/Ep specimen made of a 

unidirectional fiber-reinforced laminate containing a thin insert at the mid-plane near the loaded 

end is chosen. The specimen was 15 cm (6 in.) long and 2 cm (0.8 in.) wide with an initial crack 

length of 5.5 cm (2.2 in.). A plot of the reaction force as a function of the applied end 

displacement, d, is shown in Figure 14. The beam solution was developed by Mi and Crisfield 

(Mi and Crisfield, 1996) for isotropic adhered materials and used plane stress assumptions. After 

the initiation of delamination, fiber bridging in the test specimen caused a small drift in the 

response compared to the FEM and analytical results. The results of the decohesion method were 

in good agreement with the test data. 
 20

Figure 14. Load-deflection response of DCB test. 

A parametric study revealed that the accuracy of the prediction was reduced when the element 

sizes in the region of strain softening were greater than a prescribed value. Poor results were 

obtained when the element size was greater than 1.25 mm (0.5 in.). 

2.2.2.2 FEA With Singular Element Formulation 

In contrast to the conventional delamination prediction approach presented in the previous 

section, the following approach uses singular elements to predict delamination growth along a 

crack tip. 

Davidson developed a crack-tip element methodology for predicting delamination growth in 

laminated composite structures (Davidson, 1998). First, the toughness versus mode mix relation 

was determined experimentally. The definition of mode mix sought was the one that, when 

applied to all experimental results, produced a single-valued toughness versus mode mix curve. 

The mode mix definition was found from the results of a series of delamination toughness tests, 

and obtained within the construct of a crack-tip element analysis. This approach allowed for the 

possibility that the singular field-based decomposition was valid; it also allowed for the 

possibility that an alternative definition was valid. This alternative definition was based on 

parameters that uniquely defined the loading at the crack tip but which were insensitive to the 

details of the near-tip damage. The second component of the methodology consisted of analyzing 

the structure of interest and assessing delamination growth. This was also done using the crack 

tip element analysis and incorporated the experimentally determined definition of mode mix. 

One of the attractions of this approach is that it removes the need for detailed 2-D and 3-D FEAs. 

Total Energy Release Rate 

Consider the crack tip element of Figure 15 that represents a 3-D portion of the crack tip region 

in a general interfacial fracture problem. The crack lies locally in the x-y plane at constant z, the 

element is in a state of plane stress or plane strain with respect to the y coordinate direction. The 

crack front may be straight or curved with the length of the element in the y direction assumed to 

be sufficiently short such that there is no significant variation in the loading and in the 

orientation of the edges of the element with respect to y. It is assumed that the lengths of the 

cracked and uncracked regions comprising the element are large with respect to their thicknesses 

but are sufficiently small such that geometric nonlinearities are negligible. 
 21

Figure 15. Crack-tip element and local loading. 

Classical plate theory is used to predict the overall deformations and strain energies of the 

element (Davidson et al., 1995; Schapery and Davidson, 1990; Hu, 1995). It has been shown 

(Davidson et al., 1995; Schapery and Davidson, 1990) that the loading on the crack-tip element, 

which produces the stress singularity can be fully characterized in terms of a concentrated crack 

tip force, Nc, and moment, Mc. The concentrated crack-tip force and moment are found by 

enforcing the condition that the displacements of the upper and lower plates be compatible along 

the crack plane over –b < x < 0. The ERR of the crack-tip element is obtained through a 

modified virtual crack closure method and may be expressed in terms of Nc and Mc, rather than 

the four independent quantities N1, N2, M1, and M2. This gives 

 (2.31) 

where 

 (2.32) 

 (2.33) 

The concentrated crack-tip force and moment are given by 

 (2.34) 

and 

 (2.35) 

( ) 2 2

1 c 2 c 12 c c

1 c N c M 2 c c N M sin

2

G = ++ Γ

12

1 2

c sin

c c

Γ =

2 2

11 22

1 1 2 11 2 2

212

2 2

22 11

12 1 2

Dt Dt c A A Bt B t

4 4

c D +D

Dt Dt c BB

2 2

′ ′ =+ + − + + ′′′ ′

= ′ ′

′ ′ = − −− ′ ′

N N aN aM c 1 11 12 =− + +

1 1 11 1 12 1

c 1 21 22

Nt a t a t MM aN aM

22 2

   =− + − + −      
 22

where 

 (2.36) 

and t1 and t2 are the thicknesses of plates 1 and 2 as shown in Figure 15. 

Mode Decomposition – Classical Definition 

Mode decomposition is performed to define the ERR in terms of its Mode I and Mode II 

components. The Mode I and Mode II ERRs are given by 

 (2.37) 

and 

 (2.38) 

where is defined as 

 (2.39) 

In the above equations, is a fixed dimension, regardless of the body being analyzed, whereas L

is a characteristic dimension that scales with the body. Equation (2.39) ensures that the correct 

mode mix will be predicted for those cases where an oscillatory singularity exists. That is, Ω is 

determined for a specific crack-tip element geometry and is based on the characteristic 

dimension L. The scaling of given by Equation (2.39) ensures that the same Ω will predict 

the correct mode mix for arbitrary loadings of a geometrically similar element with different 

absolute dimensions (Davidson et al., 1995). 

Example Problems in Two Dimensions 

Instability-Related Delamination Growth

Davidson presented the prediction of instability-related delamination growth (Davidson, 1998) 

using a global cylindrical buckling analysis along with a local crack tip element analysis 

(Davidson and Krafchak, 1993). Figure 16 shows a cross-sectional view of a midplane 

symmetric laminate containing two through-width delaminations located symmetrically with 

respect to the laminate’s midplane. 

1 2

11 1 1

1 2

12 1

1 2

21 1 1

1 2

22 1 1

A t a AA B B

2

A t a EB D

2

B t a BA D

2

B t a BB D D

2

E

  = +− ′ ′    

  =+ − ′    

  = +− ′    

  = +− ′ ′    

( ) 2

I c1 c2

1 N c sin M c cos

2

Ω ΩΓ ˆ ˆ −

=− + +     G

( ) 2

II c 1 c 2

1 N c sin M c cos

2

Ω ΩΓ ˆ ˆ −= ++     G

Ωˆ

( ) ln ˆ ˆ L

L ΩΩε = +

ˆ

L

Ωˆ
 23

A cylindrical buckling analysis is performed on the laminate model (Davidson and Krafchak, 

1993; Yin, 1988). The post-buckled laminate appears as shown in Figure 17, with all internal 

force and moment resultants shown in their positive conventions (the actual in-plane forces are 

compressive in this problem and therefore the reverse of those shown in the figure). 

Figure 16. Cross-sectional view of laminate containing two symmetrically located delaminations that is 

subjected to compressive loading. 

Figure 17. Post-buckled configuration. 

Figure 18 shows the crack-tip element and local loading. Minor modifications to the crack-tip 

element formulation were made to enforce the symmetry constraints. The loads, N, N1, and N2, 

and moment, M1, acting on the crack-tip element are given by 

 (2.40) 

Substituting these loads into the crack-tip element equations provides an analytical methodology 

for determining mixed-mode ERRs as a function of the applied far-field loading. 

Figure 19 presents a comparison for total ERR between the linear crack-tip element analysis and 

a geometrically nonlinear finite element analysis (Davidson and Krafchak, 1993). A [02/90/02]3s

Gr/Ep laminate with delaminations at the interfaces between the 5th and 6th and 25th and 26th plies 

was modeled in both cases. The laminate is assumed to be in a state of plane strain. The 

horizontal axis, denoted as applied compressive strain, is the deformation of one of the loaded 

ends of the laminate divided by the laminate half-length. The value of used was 14.6o and 

was found by a linear finite element analysis of the crack-tip element geometry. Because the 

delamination growth occurs at a 0/0 interface, the stress singularity obeys a classical inversesquare root relation. 

B

D DD x 1x 2 12 1 x x

N N N N N N N M M N W

2

= = =+ = −

Ωˆ
 24

Figure 18. Crack-tip element and loading for delamination buckling problem. 

Figure 19. Comparison of total ERR for a [02/90/02]3s laminate. 

Using this method, only a single, linear finite element analysis need be performed to obtain Ω; 

thereafter, all results are generated analytically. For certain specific problems, no finite element 

analyses are required, as the value of Ω may be taken directly from Davidson et al. (1995). 

Edge Delamination

Consider a mid-plane symmetric laminate with a single delamination at its free edge, as shown 

in Figure 20. The term “sublaminates” will be used to refer to the regions bounded by the 

delamination and the laminate free surfaces. It is assumed that the delamination length, a, is large 

compared to the thickness of either of the sublaminates and that the strains and curvatures in the 

laminate and sublaminates are given by 

 (2.41) 

First, consider the classical laminated plate theory solution to an uncracked laminate under a 

specified loading. A transverse stress distribution is obtained for which the resultant transverse 

force, N2, and moment, M2, vanish along those faces of the laminate which are defined by a 

1o o 6o 1 o 6 εεη = == = 0 0 χχ χ
 25

normal vector in the positive or negative x2 direction. Assuming a crack is introduced at an 

arbitrary interface, the resultant transverse forces and moments along the free edge of each of the 

sublaminates, as well as the shear and normal stresses along the crack plane, must vanish. To 

achieve this, superpose onto the solution to the uncracked laminate the solution to the problem 

where the sublaminates are loaded by transverse forces and moments, which are equal and 

opposite to those which are given by the uncracked solution. This superposition problem is 

shown in Figure 21 (with all forces and moments shown in their positive conventions). 

Figure 20. Laminate with a free edge delamination. 

Figure 21. (a)Transverse stress distribution for an uncracked mid-plane symmetric laminate subjected to 

uniform axial extension in the X1 direction. (b) Crack-tip element. Superposition of problem (b) onto 

(a) gives solution to cracked laminate. 
 26

Since the ERR for the uncracked laminate (Figure 21(a)) is zero, the ERR for the cracked 

laminate is simply that from the “crack-tip element” of Figure 21(b). The ERR and fracture mode 

ratio for this element may be obtained through a modified virtual crack closure method 

(Davidson, 1995). The equation for G is derived as in Equation (2.31) above. The subscripted 

quantities A', B', and D' are defined such that, for sublaminate 1 or 2 

 (2.42) 

 (2.43) 

where 

For those cases where crack growth is between orthotropic layers and the crack tip stress field 

exhibits an inverse square root singularity, GI and GII may each be expressed as Equations (2.37) 

and (2.38) where = Ω. 

The value of GI and GII will never be less than zero. However, when the bracketed portion of GI

is less than zero, a negative Mode I stress intensity factor is predicted and the Mode I ERR 

component is actually a “closing mode.” This implies a crack face interference and the present 

solution, which does not enforce crack face contact constraints, is invalid for that particular 

loading case. 

The value of Ω is independent of the loading and may be found from numerical or other results 

of one special loading case. This may be done, for example, by FEA of the crack-tip element or 

taken from a generalized plane strain FEA of the free edge delamination problem. Also, the 

value of the total ERR is independent of Ω; that is, the sum of GI and GII will always give the 

same result as G, regardless of the value of Ω. 

As an example, consider the determination of the ERR in a [45/0/–45/90]s T300/5208 laminate 

with delaminations at both –45/90 interfaces (Davidson, 1994a and 1995). The laminate is 

subjected to a uniform strain in the global x1 direction. The value of the ERR for various 

hygroscopic loading cases are presented in Figure 22 along with the finite element results from 

O’Brien et al. (1986). The results are found to match almost exactly. 

(i) (i) (i)

2o 2 2 A N B M , i 1 2 i i ε =+ = ′ ′ ,

(i) (i) (i)

222 B N D M , i 1 2 i i χ =+ = ′ ′ ,

(1)

2o 2

(1)

2 2

is the midplane strain in the x direction in sublaminate 1

is the curvature in the x direction in sublaminate 1, etc.

ε

χ

=

=

Ωˆ
 27

Figure 22. ERR versus hygroscopic loading for a [45/0/–45/90]s T300/5208 laminate with edge 

delaminations at both (– 45/90). 

2.2.2.3 Progressive Delamination Growth 

A self-contained finite element program developed for progressive delamination growth 

technique allows non self-similar delamination progression. The approach is demonstrated on the 

post-buckling delamination growth of a thin composite sublaminate from a thicker sublaminate 

structure shown in Figure 23. In order to save computational costs, a double shell model is 

pursued using Mindlin shell theory. Efficiency and accuracy of this approach has been 

demonstrated in comparison to a 3-D model. 

 

Figure 23. Configuration of embedded delamination. 

The conditions for delamination growth are as follows: (1) the delamination exists prior to 

loading, (2) the growth is along the delamination interface plane, (3) stable growth occurs by 

increasing the boundary load, and (4) the computed distribution of strain energy release rate will 

exceed the critical value at the section of the delamination front that is propagating. 

The modified crack closure technique (Rybicki and Kanninen, 1977; Jih and Sun, 1990) is used 

to calculate the strain energy release rate. In this technique, the strain energy released during 

crack extension is assumed to be equal to the work needed to close the opened surfaces. 

Generally, if the relationship between the nodal force and relative displacement is nonlinear, the 
 28

closure operation must be performed incrementally (Cheong and Sun, 1992). However, it was 

discovered that even while using large deflection theory, the relations between the nodal force 

(moment) and relative displacement (rotation) are nearly linear at the delamination front. Thus, 

one-step closure is adopted in this study. This condition must be verified for each problem as 

there may be problems where the linear relation does not persist. 

In the crack closure method, the virtual crack extension Δa is usually taken to be small compared 

to the crack length. Thus, the nodal displacements (crack opening displacements) at the original 

crack tip after the extension can be approximated by the nodal displacements behind the crack tip 

before the extension. This reduces the analysis to one step instead of two. This procedure is valid 

if the elements in front of and directly behind the delamination crack front are similar. The 

conditions for similarity require that the size of each element both in front of and directly behind 

the crack front must be equal, the length of the elements normal to the delamination front must 

be Δa, and ideally, the two sides of the elements should be made perpendicular to the 

delamination front. 

A program, FECM209, developed at Purdue University is used to simulate delamination 

propagation. For simplicity, a total strain-energy release rate, G, is chosen for the approach. The 

procedure for progressive delamination growth is as follows: 

1. Calculate the linear stiffness matrix. 

2. Apply the in-plane load incrementally by a specified amount Δε0 until the final load is 

reached. 

3. Compute the nonlinear stiffness matrix based on the current stress field and 

displacements. 

4. Solve the displacement and stress fields. 

5. Find the converged solution or return to Step 3 if the solution is not convergent. 

6. Evaluate the strain-energy release rate along the delamination front. 

7. Check the distribution of the strain-energy release rate to find the corner nodes that 

exceed the critical value. If none exists, return to Step 2. 

8. Move the nodes in the outward normal direction to the delamination front where the 

critical strain-energy release rate is exceeded and generate a new mesh. 

9. Compute the linear stiffness matrix for the current mesh. 

10. Go to Step 3. 

In Step 6 the local value of G is computed at every corner node. When the strain-energy release 

rate at a node exceeds Gcr, the node will propagate in the local outward normal directions. The 

amount of propagation is proportional to the difference in the strain energy release rate and Gcr. 

During remeshing, the ratios of element sizes along the delamination front remain constant. In 

addition, the number of elements along the x and y axes remains constant. A large change in 

element sizing could lead to poor results. 

One problem with changing the mesh during loading is that the stiffness matrix must be updated 

accordingly. In Step 9, the linear stiffness matrix is updated based on the current mesh, but the 

nonlinear stiffness matrix must be calculated based on current displacements and stress fields. 

Since the location of the nodes has changed, the displacements and stresses at the new location 

should be interpolated from those at the previous locations. However, since the movements of 
 29

the nodes are very small each time the mesh is updated, the displacements and stresses at the 

previous location can be used as an approximation for the current location. 

Most commonly used commercial codes do not allow the movement of nodes in a mesh during 

the loading procedure. Consequently, after the mesh is updated, the loading procedure would 

have to be restarted. In other words, the whole loading process has to be restarted for each 

movement of the delamination front. This is computationally very time-consuming for 

simulating delamination growth. 

Post-buckling Delamination Growth Predictions

The following are results from simulations of post-buckling embedded and edge delaminations 

using the above technique. An example of an embedded circular delamination is shown in 

Figure 24. An initially circular delamination front with a radius of 15 mm (0.6 in.) is 

embedded in a 100 mm (3.9 in.) square panel. The critical strain-energy release rate 

Gcr = 200 J/m2 (13.7 ft-lb/ft2)is chosen for the analysis. The panel is loaded in the x-direction 

under uniform displacement. 

Overlapping of the upper and lower sublaminates often occurs over some range at the 

delamination front. This overlapping can be constrained by contact elements. 

The growth contours and corresponding normalized strain-energy release rate distributions for 

the [(90/0)s] sublaminate are shown in Figures 24 and 25, respectively. The upper sublaminate is 

four plies while the lower sublaminate is twenty-eight plies. Transverse deflection of the lower 

sublaminate is constrained. The growth initiates perpendicular to the loading direction and is 

stable for the first four loading levels (ε1 – ε4). Curves labeled ε5 and ε6 have become neutrally 

stable since the loading level is not increased and the distribution of the strain-energy release rate 

in the growth-region remains near Gcr. The last curve labeled ε7 is unstable because G has 

increased to 208 J/m2

 (13.8 ft-lb/ft2) while the loading level remained constant. 

Figure 24. Progressive delamination fronts for the 

[(90/0)s] laminate. 

Figure 25. Strain-energy release rate distributions 

for the [(90/0)s] laminate. 

The distribution plot of G for the ideal ellipse (with the aspect ratio of contour ε6) has a very 

large oscillation about the Gcr line while the distribution at ε6 is constant. In fact, the maximum 
 30

value is more than twice the value of Gcr. Thus, the propagating delamination cannot be modeled 

as an elliptical crack. 

It has been found that the initiation and subsequent stable delamination growth is very sensitive 

to the stacking sequence of the upper sublaminate. Lower bending stiffness parallel to the 

loading direction leads to higher predictions of strain-energy release rate and delamination 

growth at lower load levels. Un-symmetric sublaminates may have strong extensional-bending 

coupling that effectively reduces the bending stiffness leading to higher strain energy release 

rates. 

The analysis neglects matrix cracking that can occur in the post-buckled sublaminate as observed 

by Whitcomb (1988). Matrix cracking can change the stiffness and may affect the stability of the 

growth. 

A similar approach has been used to simulate the damage at the tip of a notch in cross-ply 

laminates. A schematic of the failure mode is shown in Figures 26 and 27. An axial split forms in 

the 0o-ply from a matrix crack and a delamination forms between the 0o and 90o layers. The 

fractures at the split and along the delamination front are modeled independently. This model has 

been used to investigate the size of the damage zone under various degrees of ply clustering or 

repeated plies. Larger damage zones are predicted and observed for increased ply clustering. 

Laminate strength approaches the laminate net section strength in the with large-ply clustering 

and damage zones. 

See “Efficient Modeling of Postbuckling Delamination Growth in Composite Laminates Using 

Plate Elements” by Klug et al. (1996) for more detailed descriptions of the analysis results. 

Figure 26. Schematic of axial splitting and delamination in a [(90/0)]s laminate. 
 31

Figure 27. Local finite element mesh. 

2.3 Impact Damage 

2.3.1 Prediction of Burst Strength After Impact (BAI) for Composite Overwrapped Pressure 

Vessel (COPV) 

The prediction of BAI for COPVs is one of most challenging tasks in COPV fracture control. 

Chen et al. (1998 and 1999) developed a procedure that can be used in the BAI predictions for 

COPVs under impact. A 3-D FEM was constructed with multi-layered, laminated, composite 

shell elements for the cylindrical COPV and the analysis was performed with ABAQUSTM 

(Vaidya et al., 1998). The model consisted of two portions. In the cylindrical portion, where the 

low energy impact occurs, the liner thickness and the composite overwrap lay-up were used to 

obtain individual ply stress/strain responses. In the hemispherical ends, where the laminate 

material thickness and fiber orientation change continuously, integrated properties for the 

overwrap were used for simplicity. Due to symmetry, only half of the vessel was modeled, as 

shown in Figure 28. The vessel was fully constrained on one end while the other end was 

constrained only in the radial and tangential directions to simulate the constraint condition in the 

tests. 

Figure 28. FEM of cylindrical COPV under impact. 
 32

The impactor mass was specified through the use of a point mass element, and the impactor 

geometry was specified using a rigid surface element. Surface definitions were specified for both 

the impactor and the COPV to ensure proper surface interaction during impact. The analysis was 

carried out in two steps. In the first step, an internal pressure was applied to the shell structure, 

and its static response was determined. In the second step, the impactor was assigned an initial 

velocity, and the resulting transient dynamic response of the COPV to the impact was evaluated 

using the implicit time integration procedure provided in ABAQUSTM. 

2.3.1.1 Impact Damage Prediction 

Since fiber breakage is more detrimental to the tensile fracture strength of composite laminates 

than either delamination or transverse matrix cracking, different modes of impact damage must 

be distinguished in order to accurately predict the BAI of an impact damaged COPV. A modified 

maximum strain failure theory was used to predict both the extent and the mode of composite 

damage. The ply normal strains, in the fiber and transverse directions, and shear strains at each 

integration point were calculated and compared with their respective ultimate strains to 

determine whether any damage occurred in each ply. The three distinctive damage modes were: 

damage in the fiber direction (Mode I), damage in the transverse direction (Mode II), and 

damage in shear (Mode S). The extent of damage was determined by examining the strains in all 

the finite elements adjacent to the impact location. 

A FEM of the cylindrical COPV is shown in Figure 28. The predicted damage mode and extent 

of damage obtained from the FEA results are summarized in Table 1, where the inner ply is 

shown at the top of the table and outer ply is shown at the bottom. The damage modes for the 

two elements adjacent to the impact location are separated by a dash, and a “0” indicates that 

there is no damage. A blank space in the table indicates that neither element was damaged and a 

“2s” means that both Mode II and Mode S damages were predicted. It is observed that the 

damage concentrates on the 72.5o plies and the 27.7o plies. For the 15 and 20 ft-lb (20 and 

27 joules) impacts, the inner 19.2o ply was also damaged. 

Table 1. Predicted Damage Modes for Each Ply 

Laminate 

Ply 

Impact Energy 

5 ft-lb 

(7 joule) 

10 ft-lb 

(14 joule) 

15 ft-lb 

(20 joule) 

20 ft-lb

(27 joule)

19.2o 0-S 0-S 

-19.2o 

85.7o 

-85.7o 

27.7o 1-0 1-S 

-27.7o S-0 S-1 S-1 

72.5o S-2 2S-2 2S-2 

-72.5o 2-0 2-2S 2-2S 2-2S 

85.8o 

-85.8o 

To account for the effect of damage on laminate strength, empirical material degradation factors 

(DF) were used to reduce the stiffness of damaged plies in the subsequent strength calculations. 
 33

The selection of the DF was based on the following empirical rules. For undamaged plies, the DF 

was zero. For plies with damage, the DF is as follows: 

DF = 1.0 if damaged in Mode I 

DF = 0.8 if damaged in Mode I or IIS 

DF = 0.6 if damaged in Mode S 

Mode IIS is considered to be no worse than Mode II and therefore the same DF was assigned to 

these two modes. In this way, the material effectiveness can be obtained for each ply by 

considering the damage mode and the damage extent, as 

 (2.44) 

where EF is the ply material effectiveness factor and summation is performed over the two 

elements surrounding the impact location. 

For the COPV studied, the results of the material effectiveness calculation of each ply are 

presented in Table 2. It is observed that the material effectiveness in one of the 72.5o plies was 

reduced to 60 percent under the 5 ft-lb (7 joule) impact. Similarly, the 72.5o plies and the 27.7o 

plies were all reduced to 20 percent effective, and the inner 19.2o ply was reduced to 70 percent 

effective under the 20 ft-lb (27 joule) impact. 

Table 2. Calculated Ply Material Effectiveness Factors 

Laminate 

Ply 

Impact Energy 

5 ft-lb

(7 joule)

10 ft-lb 

(14 joule)

15 ft-lb

(20 joule)

20 ft-lb

(27 joule)

19.2o - - 0.7 0.7 

-19.2o - - - - 

85.7o - - - - 

-85.7o - - - - 

27.7o - - 0.5 0.2 

-27.7o - 0.7 0.2 0.2 

72.5o - 0.3 0.2 0.2 

-72.5o 0.6 0.2 0.2 0.2 

85.8o - - - - 

-85.8o - - - - 

2.3.1.2 Post Impact Burst Strength Predictions 

It was observed in the impact-testing program that debonding between the aluminum liner and 

the composite laminate occurred upon impact. Therefore, the aluminum liner and composite 

laminate were assumed to be un-bonded in the analysis and their contributions to the BAI of the 

COPV were calculated separately. The BAI of the COPV is the sum of the two component 

strengths. Once the impact damage was determined, the BAI of the COPV laminate was 

determined by a progressive failure analysis procedure. As the aluminum liner has yielded long 

before the laminate fails, the burst strength contribution by the aluminum liner, p, was 

conservatively estimated with a closed-form formula 

 (2.45) 

DF EF 1 0

2 = − . 

y s t

p

r =
 34

where r is the radius of the pressure vessel; sy and t are the aluminum liner’s yield strength and 

thickness, respectively. The yield strength of the aluminum liner was assumed to be 42,000 psi 

(290 MPa). 

The burst strength contributed by the undamaged laminate was predicted to be 10,170 psi 

(70 MPa). The contribution of the aluminum liner was approximately 510 psi (3.5 MPa). The 

total predicted burst strength was therefore 10,680 psi (73.6 MPa), which was in close agreement 

with the average test value of 10,700 psi (73.8 MPa). Table 3 compares the predicted burst 

strengths with test data for the cylindrical COPV with damage caused by impact at various 

energy levels. The degradation in the burst strength was predicted to range from 7 percent to 

29 percent for impact energies of 5 to 20 ft-lb (7 to 27 joules), respectively. One of the 15 ft-lb 

(20 joule) BAls does not appear to be reasonable because it indicates a greater residual burst 

strength than that of the 10 ft-lb test (14 joule). Ignoring that particular test data, one can see that 

the predicted degradation in burst strength was a little larger than the test data for the 10 ft-lb 

(14 joule) case and fairly close to the test data for all other cases. (See “Impact Damage Effects 

on Gr/Ep Composite Overwrapped Pressure Vessels” by Chen et al. (1999) for more detailed 

descriptions of the test results.) 

Table 3. Comparison of Burst Strengths for Cylindrical COPV 

Impact 

Energy 

Test Results Analytical Results 

Burst Pressure Degradation Burst Pressure Degradation 

0 ft-lb 

(0 joule) 

10,700 psi 

(73.7 MPa) - 10,680 psi (73.6 MPa) - 

5 ft-lb 

(7 joule) 

9,800 psi 

(67.6 MPa) 8.4% 9,920 psi (68.4 MPa) 7.1% 

10 ft-lb 

(14 joule) 

8,884 psi 

(61.2 MPa) 17.0% 8,220 psi (56.7 MPa) 23.0% 

15 ft-lb 

(20 joule) 

8,246 psi 

(56.9 MPa) 22.9% 7,960 psi (54.9 MPa) 25.5% 

15 ft-lb 

(20 joule) 

8,377 psi 

(57.8 MPa) 21.7% 7,960 psi (54.9 MPa) 25.5% 

15 ft-lb 

(20 joule) 

9,257 psi 

(63.8 MPa) 13.5% 7,960 psi (54.9 MPa) 25.5% 

20 ft-lb 

(27 joule) 

7,681 psi 

(53.0 MPa) 28.2% 7,590 psi (52.3 MPa) 28.9% 

2.3.2 Predicting Damage after Low-Velocity Impact 

Choi and Chang developed a method for predicting the impact damage of laminated composite 

plates resulting from both line-nose and point-nose impacts (ABAQUSTM User’s Manual, 1992). 

The method consists of a stress analysis for determining the stress distributions inside the 

laminates during impact and a failure analysis for predicting the initiation and extent of the 

impact damage. The authors used a transient dynamic stress analysis code developed by Wu and 

Chang (Choi and Chang, 1992). The stress analysis was based on a 3-D linear elastic theory that 

assumes the materials in each layer to be homogeneous and orthotropic. Eight-node brick 

elements incorporating incompatible modes were used in the FEA. Elements with incompatible 

modes were chosen because of the improved accuracy in calculating bending stiffnesses and 

interlaminar shear stresses. The method was found to correspond well with experiments in 

predicting the location and extent of impact damage. 
 35

The procedure for determining the extent of the impact damage was as follows: 

1. Calculate the transient dynamic stresses within each layer as a function of time. 

2. Apply the matrix failure criterion for predicting the critical matrix cracks in each layer for 

determination of the extent of delaminations. 

3. If matrix cracking is predicted in a layer of the laminate, then apply the delamination 

criterion subsequently in the upper and lower layer of the interface during the entire 

period of impact. 

The procedure should be repeated at the other layers during impact for determining any 

additional matrix cracking and delaminations. The final size of each delamination is determined 

by the area within which the stress components satisfy the delamination failure criterion during 

the entire duration of the impact. No material degradation was considered in the model, and the 

model does not take into account delamination interaction during impact. 

Two failure criteria were proposed in the study to predict the initiation of critical matrix cracks 

and estimate the extent of delamination. 

2.3.2.1 Critical Matrix Cracking Criterion 

The failure criterion proposed for matrix cracking can be expressed as (Wu and Chang, 1989) 

 (2.46) 

where x and y are local coordinates of the nth layer parallel and normal to the fiber directions, 

respectively, and z is the out-of-plane direction. Si is the in situ interlaminar shear strength within 

the laminate under consideration and Yt and Yc are the in situ ply transverse tensile and 

compressive strengths, respectively, as defined by Choi and Chang (ABAQUSTM user’s manual, 

1992). 

Whenever the calculated average stresses in any one of the plies in the laminate first satisfy the 

criterion (i.e., eM = 1) during impact, the initial impact damage is predicted. It was assumed that 

the matrix crack would propagate throughout the thickness of the ply group that contained the 

cracked ply. A delamination can be immediately induced from the location of the matrix crack 

along the interfaces of the ply group. Because additional matrix cracking can occur in other 

layers as time increases, the criterion should be continuously applied to the neighboring layers. 

2.3.2.2 Impact-Induced Delamination Criterion 

Once the crack matrix criterion is applied and a critical matrix crack is predicted within a 

laminate, a delamination is initiated from the crack. A semi-empirical model was developed to 

estimate the extent of delaminations in the composites. The model includes the following two 

types of critical cracks initiating delaminations resulting from the impact: the shear crack 

generated within the laminates, and the bending crack produced from the bottom surface of the 

laminates as shown in Figures 29 and 30. 

2 2 n n yy yz 2

n n M

i

e

Y S

      + =

  

σ σ

M

M

n n t yy

n n yy

e 1 Failure

e 1 No Failure

Y Y if 0

Y Y if 0 c

 ≥ 

 <

 = ≥ 



 = < 

σ

σ
 36

It was reasoned that delamination growth due to low-velocity impact could occur only when two 

sequential conditions were met by the following: 

1. One of the ply groups directly above or below the concerned interface failed due to 

matrix cracking. 

2. The combined stresses governing the delamination growth mechanisms through the 

thicknesses of the upper and lower ply groups of the interface reaches a critical value. 

This is the impact-induced delamination growth criterion. 

Figure 29. Basic impact damage mechanism of laminates subjected to 2-D line-loading impact. Top: 

delamination induced by inner shear cracks. Bottom: delamination induced by surface bending crack. 

 

Figure 30. Schematics of the delamination growth mechanism due to a shear crack and a bending crack, 

respectively, in a laminated composite subjected to point-nose impact. 

The second condition can be written in equation form as 

 (2.47) 

where Da is an empirical constant which must be determined experimentally. The constant has 

been found to be insensitive to ply orientation and thickness of the laminate, and primarily 

dependent on the material system used. The superscripts n and n + 1 correspond to the upper and 

lower plies of the nth interface, respectively. and are the averaged interlaminar and in2 2 2 n n1 n 1

yz yy xz 2

a D n n1 n1

i i

D e

SSY

σ σ σ + +

+ +

            ++ =           

D

D

n1 n1

yy

n1 n1

c yy

e 1 Failure

e 1 No Failure

Y Y if 0

Y Y if 0

t σ

σ

+ +

+ +

 ≥ 

 <

 = ≥ 



 = < 

σ yz σyy
 37

plane transverse stresses within the nth and n + 1th ply, respectively. is the averaged 

interlaminar longitudinal stress within the n + 1th ply. 

3.0 Damage Tolerance Life Prediction Methodologies 

This section presents life prediction methods for damage growth under fatigue-type loading. 

3.1 Delamination Growth in Cyclic Loading 

3.1.1 Closed Form Solution for Delamination Growth 

A delamination growth criteria that incorporates Mode II was investigated (dL/dN = f(GII)). A 

different delamination growth-rate criterion has been proposed that includes the effects of both 

ΔGII and GIImax. The following expression was found to provide best fit of the experimental data 

for a wide range of stress ratios: 

 (3.1) 

This equation is similar to the approaches used to characterize crack growth rates in metals 

for different stress ratios (Broek, 1986), and for R = ∞ it reduces to Equation (3.1) since 

GIImax = ΔGII. A comparison of the analytical and experimental cyclic growth rates for three 

different R values is shown in Figure 31. Again, good agreement in the cyclic growth rates is 

observed for larger delaminations for which Mode II drives damage growth. 

Figure 31. Cyclic growth rates (dL/dN) in constant-amplitude compression-compression loading as a 

function of L (solid lines – experimental data, dashed lines – analytical prediction). 

A 3-D FEM would be required for a full understanding of parameters that influence delamination 

growth. However, the simplified modeling approach given by Han et al. (1999) can provide 

useful information about the major forces that drive delamination growth. For a delamination 

that spans only a few surface plies the results suggest that Mode II is the dominant factor that 

governs delamination growth with minor contributions from Mode I for smaller delamination 

lengths. If the delamination is nested deep in the thickness of the laminate, the influence of 

Mode I component becomes dominant and cannot be neglected (see Figure 32). However, the 

σ xz

2 n

II

II max 2

IIc

dL

m

dN

    Δ   Δ       =  

 

 

G G

G

G
 38

maximum value of GI (Whitcomb, 1981). Furthermore, large scatter in reported critical strain 

ERRs (Shyprykevich, 1997) and uncertainties in loading spectrum and damage characteristics 

could lead to substantially different delamination growth predictions. 

Figure 32. Strain ERRs (GI and GII) as a function of applied load (εx in percent) and delamination length 

(L) for delamination located between plies 3 and 4. 

3.1.2 Delamination Growth Using FEA 

While a number of researchers have investigated delamination growth using the concepts of 

LEFM (Whitcomb, 1989; 1990; 1992; O’Brien et al., 1986; Davidson, 1994b; Kardomateas 

et al., 1994; Russell and Street, 1989; and Tratt, 1991), the question of characterizing 

delamination growth under fatigue is still not well understood. Typically, delamination growth is 

governed by strain ERRs GI, GII, and GIII, which in the Han et al. study were determined by 

FEA (Han et al., 1999). 

Initial impact damage usually has a circular shape with multiple delaminations throughout the 

thickness. However Han argued that damage growth in fatigue more closely resembles a single 

delamination propagation front, rather than the growth of embedded multiple delaminations, as 

shown in Figure 33. Thus, an idealized plane-strain FEM was used to determine Mode I and 

Mode II components of the strain ERR. 
 39

Figure 33. FEM procedure. 

When subjected to a compressive load, the thin sublaminate buckles causing high interlaminar 

stresses at the delamination tip causing delamination growth. Hence, the influence of buckling 

and postbuckling of the sublaminate on delamination growth can be studied using a 

geometrically nonlinear static analysis. To initiate out-of-plane displacements an initial 

imperfection was assumed 

 (3.2) 

where the maximum initial imperfection (wmax) is chosen to be 10 percent of the delaminated ply 

thickness. Strain ERRs along the delamination front were determined using a modified crack 

closure technique (Rybicki and Kanninen 1977), and the individual strain ERRs, associated with 

the two modes of fracture, were (Mukherjee et al., 1994). 

 (3.3) 

3.1.2.1 Comparison with Experimental Results 

Experimental results indicate that once the damage propagated, it resembled a single 

delamination growth located one or two plies below the laminate surface. To gain a physical 

understanding of the mechanisms contributing to the delamination propagation, the correlation 

between the cyclic growth rates and the strain ERRs (G) was investigated. The GI and GII values 

of Figures 34 and 35 were calculated at different strain levels for delaminations located at one 

ply and two plies beneath the surface, respectively. 

In Figure 34, for a through width delamination, GI values of a single ply are very small due to the 

low bending stiffness of the delaminated ply. As the thickness of the delaminated ply increases, 

GI also increases (Whitcomb, 1981)as shown in Figure 35. However, the influence of Mode I 

max

x

w cos

2a

  π ⋅ =    

() ( )

() ( )

a

I yy y a 0

0

a

II xy x a 0

0

1 lim x u x a

2 a

1 lim x u x a

2 a

σ

σ

Δ

Δ →

Δ

Δ →

= Δ +Δ

Δ

= Δ +Δ

Δ





G

G
 40

decreases rapidly as crack length increases, and it does not contribute to the delamination growth 

once the delamination reaches ~18 mm (0.7 inches). On the other hand, values of GII remain 

relatively constant over the range of delamination lengths investigated; therefore, a constant 

value of GII can be assumed for a specific value of applied strain. However, after the initial 

buckling of the delaminated plies, the crack tip actually closes due to the compressive forces that 

develop near the crack tip. These compressive forces might decrease the calculated values for 

GII, thus leading to the decay of cyclic growth rate. 

Figure 34. Strain ERRs (GII) as a function of applied load (εx in %) and delamination length (L) for 

delamination located between plies 1 and 2. 

 a) Mode I. b) Mode II. 

Figure 35. Strain ERRs as a function of applied load (εx in percent) and delamination length (L) for 

delamination between plies 2 and 3. 

3.2 Damage Growth at Notch Tip 

In a study to predict damage growth at a notch tip of a center-notched carbon fiber/epoxy 

laminates, Spearing et al. modeled the damage as a series of interacting matrix cracks in various 

forms: splitting, delamination and transverse play cracking (Spearing et al., 1992). The basis of 

the growth model is the fatigue crack growth law known as Paris’ law (Hertzberg and Manson, 

1980). 

 (3.4) 

where da/dN is the crack growth rate, ΔK is the stress intensity factor range and λ1 and m are 

empirical constants. 

( )m 1

da λ

dN

= ΔK
 41

Fatigue damage at notches can be characterized by the split length, l, at the notch tip. It has been 

observed that splits and delaminations grow in combination at the notch tip. As the split length 

increases, the associated delamination area grows with a dependence on the square of the split 

length (Figure 36) which implies an increasing resistance to further crack advance. Because of 

this, Equation (3.5) can be written as 

 (3.5) 

where l is the split length at the notch tip, ΔG is the driving force, Gc the current toughness (as 

measured in monotonic loading), and λ3 is a constant. 

Figure 36. Schematic of the notch tip damage pattern in a cross-ply specimen. 

3.2.1 Damage Growth in Monotonic Loading 

The model that Spearing et al. (1992) apply to damage is based on a damage growth model for 

matrix cracks in monotonic loading. Ignoring transverse ply cracking and the effect of residual 

stresses, damage at the notch in a [90/0]s laminate will grow by the simultaneous formation of 

splits in the 0o plies and triangular-shaped delamination zones at the [90/0] interfaces. 

Spearing et al. used a modified FEM developed by Kortschot and Beaumont in order to calculate 

∂C/∂l, the ratio of specimen compliance to an increment of split growth (Kortschot and 

Beaumont, 1991). The Kortschot and Beaumont model consists of two superimposed identical 

layers of 2-D plane stress elements. The two layers share the same nodes everywhere except in 

the delaminated region, where separate nodes are generated in the 90o ply. This model was 

further discussed in section 2.1.2.1. The model was first developed for edge-notched specimens. 

In order to use this model for center-notched specimens, some modifications were made. The 

m 2

3 c

dl λ

dN

Δ  =    

G

G
 42

relationship between the value of ∂C/∂l obtained from the FEM (∂C/∂l|FE) and those for the 

actual [90/0]s specimen is given by the scaling equation 

 (3.6) 

where W is width and t is the thickness of the split. 

This can be extended to all laminates of the form [90i/0i]ns in which there is an equal total 

thickness of 90o and 0o plies. The scaling equation, in this case, becomes 

 (3.7) 

The method can be extended to laminates with unequal thicknesses of 0o and 90o plies by altering 

the relative ply thicknesses of the layers of the finite element mesh. 

3.2.2 Damage Growth Model Applied to [90/0]s Laminates 

For an initial length, l0, the split length after N cycles of constant load amplitude is given by 

 (3.8) 

where Gs and Gd are the energies absorbed per unit area of split and delamination, respectively, 

and ΔG is given by 

 (3.9) 

where P is the range of applied load. 

Figure 37 shows a comparison of experimental data with the results from Equation (3.9) for a 

[90/0]s laminate cycled at three different peak stresses. Good agreement is shown with the 

experimental data. 

Figure 37. Normalized split length for a [90/0]S laminate as a function of load cycles showing the effect 

of varying peak stress. The solid lines represent the prediction of Equation (6.9). 

( ) ()

( )( )

FE FE

FE

C C W 2 2t

l W 2 2t l

∂ ∂ = ∂ ∂

( )()

( )( )

FE FE

FE

C C W 2 int

l W 2 2int l

∂ ∂ = ∂ ∂

( ) ( )( )( )

2m2 ( ) m 2 m2 2

d s d0

d

1 m2 l λ tanα N t l tanα

tanα 2

+     + + = Δ × ++         G G GG

G

( )2 P C

2t l

Δ ∂ Δ =

∂

G
 43

3.2.3 Damage Growth in [90i/0i]s Laminates 

For laminates containing plies of equal thicknesses, ΔG increases linearly with ply thickness; 

i × t. For the [90/0]s case, there are only two interfaces at which delaminations can grow. In this 

case, the ratio ΔG/Gc increases as i increases; so the fatigue law becomes 

 (3.10) 

Good agreement has also been found with experimental data for this case as is shown in 

Figure 38. 

Figure 38. Normalized split length as a function of load cycles for [90i/0i]ns laminates. The solid lines 

represent Equation (3.11). 

3.2.4 Fatigue Damage Growth in [90i/0j]s Laminates 

Laminates with plies of unequal thicknesses can be modeled in a similar way. The relative ply 

thicknesses of the layers in the finite element meshes were adjusted and the resulting values of 

∂C/∂l|FE were scaled appropriately. In this case, the fatigue law is 

 (3.11) 

From this law, it is shown that for laminates with j > i more rapid damage progression occurs 

than in [90/0]s laminates. 

3.2.5 Damage Growth in [90/0]ns Laminates 

In the case of [90/0]ns laminates, ply thickness is constant and equal for the 90o and 0GII laminae; 

only the number of 90/0 interfaces varies. Through half the laminate thickness, 2n – 1 interfaces 

( ) ( )( )( ) ()

m 2

2

FE FE

FE

s d

1 C

σ W 2 2it W 2 2t

dl 2 l

dN it ltanα

∞

  ∂ Δ   ∂ = +

 

G G

( ) ( )( ) ( )( )( )

m 2

2

FE FE

FE

s d

1 C

σ W2 i jt W2 i jt dl 2 l

dN it ltanα

∞

  ∂ Δ+ +   ∂ = +

 

G G
 44

exist at which delaminations can propagate. Since the split area is n × t × l, and ∂C/∂l scales with 

the number of plies as shown in Equation (3.8), the fatigue law becomes 

 (3.12) 

As the number of plies increases, the model predicts that the rate of damage propagation 

decreases slightly which causes the model to underestimate the damage growth (Figure 39). This 

effect was theorized to be due to unequal damage growth through the thickness of the laminate. 

Figure 39. Normalized split length as a function of load cycles for [90/0]ns laminates. The solid lines 

represent Equation (3.13). 

3.2.6 Damage Growth in [90/+45/-45/0]s Laminates 

For these laminates, it was assumed that damage is controlled by splitting in the 0o plies and by 

delamination at the interface between the 0o ply and the innermost –45o plies. In this case, the 

finite element representation provides values for ∂C/∂l that are independent of the split length, 

which is different from the cross-ply laminates discussed previously. The fatigue law was 

derived for this configuration in a similar manner to that of the [90/0]s laminate, yielding 

 (3.13) 

where k is the dimension defining the intersection of the delamination with the center line of the 

notch. 

A good correlation between the experimental data and the fatigue law were also found in this 

case, as shown in Figure 40. 

( ) ( )( )( ) ()

( )

m 2

2

FE FE

FE

s d

1 C

σ W 2 2nt W 2 2t

dl 2 l λ

dN nt 2n 1 ltanα

∞

  ∂ Δ   ∂ = + −

 

G G

( )

m 2

2

s d

C 12 P

dl l λ

dN l k

t

2

   ∂   Δ      ∂ =      +   +     

G G
 45

Figure 40. Normalized split length as a function of load cycles in [90/+45/–45/0]s laminates. The solid 

lines represent Equation (3.14) in an integrated form. 

4.0 Damage Tolerance Analysis Codes 

Two computer codes have been developed at The Aerospace Corporation (Aerospace) for 

predicting residual strength of composite structures containing defects. The first such code is 

Aerospace Damage Analysis Methodology for Composites (ADAM-C). The other code is 

Progressive Failure Analysis (PFA). The following sections briefly introduce these two codes. 

Detailed discussions are presented in Appendices A and B. 

4.1 ADAM-C 

In a research and development project conducted by Aerospace in the 1980s, a few analytical 

models were selected for inclusion in a static residual strength prediction software code, 

ADAM-C (Chang et al., 1985); see Figure 41 for program flow. Refer to Appendix A for a more 

detailed explanation of the study. Table 4 contains a summary of some of the analytical models 

reviewed. 

Figure 41. ADAM-C program flow chart. 
 46

Table 4. Assessment of Analytical Methods 

Loading and Damage 

Type Analytical Method Advantages Disadvantages

Point stress failure model 

Notched fracture stress 

model 

Damage zone model

Simple calculation, adequate database 

Simple calculation 

Provides more accurate 

predictions

Stress intensity zone size is 

independent of hole sizes 

Extensive testing, limited 

database 

Lengthy calculation, not enough 

database

Inherent flaw model 

Strain energy density factor method 

Finite element approach 

Damage zone model

Simple calculation, adequate database 

Provides more accurate 

predictions 

Provides adequate accuracy for complex 

geometries 

Provides more accurate 

predictions

Characteristic zone size is independent of crack sizes 

Very limited database 

Length calculation 

Length calculation

Finite element method 

Edge delamination model

Provides adequate accuracy for complex 

geometries 

Simple calculation

Length calculation 

Limited application

Finite element method 

Thin-film 1-D model

Provides adequate accuracy for complex 

geometries 

Simple calculation

Length calculation 

Limited application

Finite element method 

Thin-film 2-D delamination 

model

Provides adequate accuracy for complex 

geometries 

Simple calculation

Length calculation 

Prediction accuracy needs to be 

improved

The IFM developed by Waddoups et al. was selected for use in predicting the residual strength 

of cracked composite laminates (Waddoups et al., 1971). The PSM developed by Whitney and 

Nuismer was selected for use with laminates containing holes and cutouts (Whitney and 

Nuismer, 1974). These models require simple calculations and have the advantage of an 

extensive test database for the more common composite systems such as graphite/epoxy (Gr/Ep). 

Details of these methods can be found in section 5.1.1.1 of Whitney and Nuismer, 1974. 

For composites with delaminations at free edges, the edge delamination model (O’Brien, 1982) 

of O’Brien was selected. Through-the-width or embedded delaminations are characterized by 

Chai’s thin film model (Chai et al. 1981). These two models have two common features in that 

they avoid lengthy finite element calculations, and use the strain ERR G to characterize 

delamination growth. 
 47

4.2 PFA for Composites 

In recent years, the research and development for the PFA of composites at Aerospace has 

integrated and extended the PFA developed at NASA Langley Research Center (LaRC) in the 

late 1990s and early 2000s. The PFA is implemented in a software tool that integrates with 

existing commercial FEM software. The tool simulates the growth and propagation of failure 

modes in a composite and it is constantly adapted for new classes of problems and extended to 

incorporate new physics. The key features of the PFA code include: (1) delamination simulation 

by using interface elements; (2) Prediction of in-plane and out-of-plane nonlinear shear damage; 

(3) Prediction of matrix-cracks based on a modified version of Puck’s criterion (Puck and 

Schurmann, 1998); and (4) A fiber failure criterion based on the work of Chang and Chang 

(Chang, and Chang, 1987). 

The PFA code utilizes concepts from damage mechanics to degrade the material properties of the 

composite appropriately. The code also handles contact in the event of delamination closure 

following growth. Similar to ADAM-C, the code was developed to be as generic as possible and 

applicable to a large range of problems with adequate accuracy and conservatism. The accuracy 

of the code, however, relies on developing a complete mechanical, strength, and fracture 

database that can be used directly as input to the code. Appendix B discusses the progressive 

failure philosophy, the damage models in the PFA code, the characterization testing that is 

required to define the damage model, and validation of the PFA code. The PFA code is also 

demonstrated for few examples pertaining to pressure vessels. 

5.0 References 

5.1 Government Documents 

NASA-STD-5003 Fracture Control Requirements for Payloads Using the Space 

Shuttle, 1996. 

NASA-STD-5007 General Fracture Control Requirements for Manned Spaceflight 

Systems, 2000. 

NASA-STD-5019 Fracture Control Requirements for Spaceflight Hardware, 2008. 

MSFC-RQMT-3479 Fracture Control Requirements for Composite and Bonded 

Vehicle and Payload Structures, 2006. 

JSC-22267B Fatigue Crack Growth Computer Program NASGRO®, Version 

3.0, November, 2001. 

SSP 30558B Fracture Control Requirements for Space Station, 1994. 

CMH-17 (aka MIL-HDBK-17) Composite Materials Handbook, Polymer Matrix, Vol. I-V. 

5.2 Government References 

The references listed below are in alphabetical order by author. 

Byers, B. A. (1980): Behavior of Damaged Graphite/Epoxy Laminates Under Compression Loading. 

NASA-CR-159293, August 1980. 
 48

Davidson, B. D. (1998): A Predictive Methodology for Delamination Growth in Laminated Composites, 

Part I: Theoretical Development and Preliminary Experimental Results. FAA Report No. 

DOT/FAA/AR-97/87, April 1998. 

Dopker, B.; et al. (1995): “Composite Structural Analysis Supporting Affordable Manufacturing and 

Maintenance.” Paper presented at the Sixth NASA/DOD/ARPA Advanced Composite Technology 

Conference, Anaheim, California; August 1995. Also published in conference publication 

NASA CP-3326, 1996. 

Han, H. T.; Mitrovic, M.; and Turkgenc, O. (1999): The Effect of Loading Parameters on Fatigue of 

Composite Laminates: Part III. FAA Report No. DOT/FAA/AR-99/22, June 1999. 

Konno, K.; Woytach, J; and Tyburski, T. E. (2001): Flywheel Energy Storage System (FESS) Rotor 

Certification Plan. NASA Report No. 69011-PLAN-311, Rev. A. 

Madan, R. C. (1988): Composite Transport Wing Technology Development. NASA-CR-178409, 

February 1988. 

Poe, C. C., Jr.; and Sova, J. A. (1980): Fracture Toughness of Boron/Aluminum Laminates with Various 

Proportions of 0° and ±45° Plies. NASA-TP-1707. 

Scholz, D.; et al. (1997): Advanced Technology Composite Fuselage – Materials and Processes. 

NASA-CR-4731, April 1997. 

Smith, P. J.; and Wilson, R. D. (1985): Damage Tolerant Composite Wing Panels for Transport Aircraft. 

NASA CR-3951, December 1985. 

Sollars, T. A. (1987): “Shuttle/Centaur G-Prime Composite Adapters Damage Tolerance/Repair Test 

Program.” AIAA Paper 87-0792, 28th Structures, Structural Dynamics and Materials Conference, 

Monterey, California, April 6–8, 1987. 

Whitcomb, J. D. (1988): Instability-Related Delamination Growth of Embedded and Edge 

Delaminations. NASA TM-100655, August 1988. 

5.3 Non-Government References 

The references listed below are in alphabetical order by author. 

ABAQUS User’s Manual (1992): Pawtucket, Rhode Island: Hibbitt, Karlson and Sorensen, Inc. 

Agarwal, B. D.; and Giare, G. S. (1982): “Influence of the Properties of a Matrix Material on the Fracture 

Toughness of Short-Fiber Composites.” Materials Science and Engineering, Vol. 52, pp. 139-145. 

Anderson, T. L. (1991): Fracture Mechanics: Fundamentals and Applications. Boca Raton, Florida, 

CRC Press. 

Aronsson, C-G. (1986): “Tensile Fracture of Laminates With Cracks.” Journal of Composite Materials, 

Vol. 20, No. 3, pp. 287-307, May 1986. 

Ashton, J. E.; and Whitney, J. M. (1970): Theory of Laminate Plates. Stamford, Technomic Publishing 

Co. 

Basham, K. D.; Chong, K. P.; and Boresi, A. P. (1993): “A New Method to Compute Size Independent 

Fracture Toughness Values for Brittle Materials,” Engineering Fracture Mechanics, Vol. 46, No. 3, pp. 

357-363. 

Broek, D. (1986): Elementary Engineering Fracture Mechanics. M. Nijhoff Publishers. 
 49

Brown, W. F. (1970): “Tentative Method of Test for Plane Strain Fracture Toughness of Metallic 

Materials.” Review of Developments in Plane Strain Fracture Toughness Testing, American Society for 

Testing and Materials, ASTM STP 463, American Society for Testing and Materials, 1970, pp. 249-269. 

Chai, H.; Babcock, C. D. (1985): “Two-Dimensional Modeling of Compressive Failure in Delami-nated 

Laminates.” Journal of Composite Materials, Vol. 19, No. 1, pp. 67-98. 

Chai, H.; Babcock, C. A.; and Knauss, W. G. (1981b). “One Dimensional Modeling of Failure in 

Laminated Plates by Delamination Buckling.” International Journal of Solids and Structures, Vol. 17, 

No. 11, pp. 1069-1083. 

Chai, H. (1982): The Growth of Impact Damage in Compressively Loaded Laminates. Unpublished Ph.D. 

Thesis, March 1982. Pasadena: California Institute of Technology (Caltech). 

Chang, F. K.; and Chang, K. Y. (1987): “A Progressive Damage Model for Laminated Composites 

Containing Stress Concentrations.” Journal of Composite Materials, Vol. 32, pp. 834-855. 

Chang, J. B. (1981): “Round-Robin Crack Growth Prediction on Center-Cracked Tension Specimens 

under Random Spectrum Loading.” Methods and Models for Predicting Fatige Crack Growth under 

Random Loading, ASTM-STP-748, J. B. Chang and C. M. Hudson, Eds., American Society for Testing 

and Materials, pp. 3-40 

Chang, J. B.; Rogers, J. C.; and Kamrath, A. R. (1985): Damage Tolerance Analysis Methodology for 

Composite Structures. From Aerospace Sponsored Research (ASR) Summary Report, Aero-space Report 

No. ATR-85(8498)-2, The Aerospace Corporation. 

Chang, J. B.; Szamossi, M.; and Liu, K. W. (1981): A User’s Manual for a Detailed Level Fatigue 

Crack-Growth Analysis Commuter Code – The CRKGRO Program. AFWAL-TR-81-3093. 

Chen, H. C.; Chiu, S. T.; and Chang, J. B. (1999): “Impact Damage Effects on Gr/Ep Composite Overwrapped Pressure Vessels.” Paper presented at the Proceedings of the 40th AIAA/ASME/ASCE/AHS/ASC 

Structures, Structural Dynamics, Materials Conference and Exhibit, AIAA Paper No. 99-1321. Vol. 2, 

April 12–15, St. Louis, Missouri. 

Chen, H. C.; Chiu, S. T.; and Chang, J. B. (1998): Residual Burst Strength Evaluation of Composite 

Overwrapped Pressure Vessels After Impact. JANNAF NDES/RNTS/S & MBS Joint Meeting, Salt Lake 

City, Utah, March 1998. 

Cheong, S. K.; and Sun, C. T. (1992): “Analysis of Postbuckled Delaminations Using Incremental Crack 

Closure Integral.” Proceedings of the Second International Symposium on Composite Structures and 

Materials, Peking Press, Beijing: PRC, pp. 519-524. 

Choi, H. Y.; and Chang, F. K. (1992): “A Model for Predicting Damage in Graphite/Epoxy Laminated 

Composites Resulting From Low-Velocity Point Impact.” Journal of Composite Materials, Vol. 26, 

No. 14, pp. 2134-2169. 

Davidson, B. D.; and Krafchak, T. M. (1993): “Analysis of Instability-Related Delamination Growth 

Using a Crack Tip Element,” AIAA Journal, Vol. 31, No. 11, pp. 2130-2136. 

Davidson, B. D. (1994a): “Energy Release Rate Determination for Edge Delamination Under Combined 

In-Plane, Bending and Hygrothermal Loading. Part I – Delamination at a Single Inter-face.” Journal of 

Composite Materials, Vol. 28, No. 11, pp. 1009-1031. 

Davidson, B. D. (1994b): Prediction of Delamination Growth in Laminated Structures. Paper presented 

at the Proceedings of the 1994 International Mechanical Engineering Congress and Exposition: Failure 

Mechanics in Advanced Polymeric Composites. Chicago, Illinois, USA, AMD-Vol. 196, pp. 43-65. 
 50

Davidson, B. D. (1995): “Prediction of Energy Release Rate for Edge Delamination Using a Crack Tip 

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 51

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 54

Appendix A: ADAM-C Computer Code 

A-1 ADAM-C Methodology 

Aerospace developed a computer software, ADAM-C, in the mid-1980s (Ref. A.1) with the 

intention to predict residual strength for composite structures containing defects such as cuts, 

holes and delaminations. A detailed discussions of the models incorporated in ADAM-C are 

presented as follows: 

Four basic failure modes determine the damage tolerance of composite materials: matrix 

cracking, fiber breakage, interfacial debonding, and interply delamination. All failure modes may 

be traced to localized flaws and defects that are introduced into the material through many 

potential sources, e.g., manufacturing and assembly procedures, impact damage, and 

hygrothermal environment. 

The literature survey results indicate that the growth behavior and failure modes of all flaws are 

strongly dependent on such parameters as laminate strength, stiffness, ply orientation, stacking 

sequence, fiber volume fractions, load application, and environmental conditions. These 

parameters must be integrated into a consistent methodology before full exploitation of 

composite materials can be achieved with confidence. 

A survey of aircraft industry practices revealed that the damage tolerance capability of critical 

composite hardware is often demonstrated by conducting safe-life tests on coupons, or small 

structural components, with preconceived and prefabricated flaws. This approach, however, may 

require an extensive test program when several potential laminates with different layups are 

being considered. Inasmuch as each laminate has unique properties, the test results for one layup 

are not directly applicable to the other candidate laminates. 

Several investigators have developed analytical models that predict laminate response to a 

variety of damage types common to aircraft structures. The environmental and structural 

requirements of aircraft and spacecraft, however, are quite different. Spacecraft components 

generally tend to be more highly stressed than their aircraft counterparts; hence, the applicability 

of these models to space hardware needs further evaluation. 

It is common practice in the spacecraft industry to perform acceptance proof tests on all primary 

composite structures, an approach that is both costly and time consuming. In some cases, such as 

the cryogenic temperature environment, only limited testing is possible because of the difficulty 

in simulating environmental conditions. Thus, the need for a reliable damage tolerance 

methodology that could minimize the number of these expensive tests is apparent. 

As shown in Table A-1 survey results have yielded two broad categories that encompass most 

combinations of flaw type and loading condition. The category I models concentrate on throughthe-thickness flaws such as holes, cutouts, and cracks, whereas the category II models are 

restricted to delamination, which is the predominant failure mode unique to composite structures. 
 55

Table A-1. Summary of Literature Survey Results 

Category Method/Model Description Developer Ref. 

No.

I

Inherent flaw model Uses linear fracture mechanics parameter for predicting 

residual strength of cracked body.

Waddoup et 

al., 1971

A. 2

Point stress failure 

model

Uses laminate theory to calculate stress concentration 

factor for predicting residual strength of holes and 

cutouts.

Whitney and 

Nuismer, 1974

A. 3

Finite element 

method

Uses finite element analysis for predicting residual 

strength of laminate containing cracks.

Wang, 1974 A. 4

Notched fracture 

stress model

Relates fracture stress to composite fracture toughness; 

for cracked body residual strength calculation.

Lin, 1977 A. 5

Strain energy density factor methodUses strain energy density factor to predict residual 

strength of cracked composite structures. 

Sih, 1978 A. 6

Strain failure 

criterion

Relates critical fiber strain to a general fracture toughness 

parameter; for cracked body residual strength prediction.

Poe and 

Sova, 1980

A. 7

Damage zone 

model

Uses Dugdale approach to represent damage; for predicting strength of holes, cutouts, and cracks.Backlund, 

1981

A. 8

II

Finite element 

method

Uses 3-D finite element to determine interlaminar 

stresses; for delamination growth prediction.

Ratwani, et 

al., 1979

A. 9

1-D thin-film model Uses laminate buckling theory for through-the-thickness 

delamination prediction.

Chai et al., 

1981

A.10

Edge-delamination 

model

Uses rule-of-mixture for onset of edge delamination 

prediction.

O’Brien, 1982 A.11

2-D thin-film model Uses laminate buckling theory for embedded delamination prediction.Shyprykevich 

et al., 1984 

A.12

The analytical methods in each category of Table A-1 have been evaluated. The advantages and 

disadvantages of each approach are summarized in Table A-2. In general, none of the existing 

models provides reliable results at reasonable cost. This is particularly true for category II 

models. The finite element approach is ideally suited for analyzing composite structures that 

contain delaminations, but computer costs are often prohibitive. 

The following selection criteria were established to distinguish which of the available models 

should form the basis of the required methodology: 

Models shall be generic in nature. 

• Lengthy calculations shall be avoided. 

• Models shall provide predictions with adequate accuracy and conservatism for space 

hardware application. 

• Models shall use the existing composite material database. 

• Methodology shall be readily adaptable to existing Aerospace computer systems. 

On the basis of these criteria, the IFM developed by Waddoups et al. (Ref. A.2) was selected for 

use in predicting the residual strength of cracked composite laminates. The PSM developed by 

Whitney and Nuismer (Ref. A.3) was selected for use with laminates containing holes and 

cutouts. These models require simple calculations and have the advantage of an extensive test 

database for the more common material systems such as fiberglass/epoxy and Gr/Ep. 

For composites with through-the-width or embedded delaminations are characterized by the thin 

film model of Chai et al.(Ref. A.10). For delaminations at free edges, the edge delamination 

model of O’Brien (Ref. A.11) was selected. These two models have two common features: they 

avoid lengthy finite element calculations, and they use the strain-energy release rate G to 

characterize delamination growth. 
 56

The previously mentioned models form the basis for an Aerospace-developed damage tolerance 

software package called ADAM-C, which was the primary product of the Aerospace Sponsored 

R program. ADAM-C architecture is illustrated in Figure A-1. Material properties, laminate, and 

flaw geometries are input into preprocessor modules (GENLAM and RANKHO), which generate 

the laminate constitutive relations and the stress concentration factors for certain flaw shapes. 

These codes were acquired from the Air Force Materials Laboratory and linked to the remaining 

modules such that unnecessary programming effort was avoided. 

Figure A-1. ADAM-C methodology. 

Once the stiffness matrices and stress concentration factors have been calculated, control is 

transferred to the appropriate module, depending on the type of flaw selected. The program 

addresses holes, cracks, and elliptical cutouts. In each case, a residual strength prediction is the 

final product. The safe-life and residual buckling strength of a composite laminate that has a 

delamination may also be obtained by running ADAM-C. 

ADAM-C is available on both the VAX and IBM-PC systems. Each model is independent and 

modular, thus facilitating any future enhancements. The program is menu-driven and userfriendly. The material database contains many commonly used graphite and glass/epoxy systems, 

but this list can be updated by users at any time by simply running an interactive program called 

MATRPR (Ref. A.13). Thus, any new material systems can be quickly accommodated. 

The following list summarizes the current ADAM-C output: 

Laminated plate theory stiffness matrices 

• Stress concentration factors due to holes or cutouts at arbitrary angles of inclination; 

• Static residual strength of cracked panels; 

• Static residual strength of panels with holes and cutouts; 

• 1-D and 2-D delamination growth with residual buckling strength; 

• Edge delamination growth and residual strength. 
 57

Table A-2. Assessment of Analytical Methods 

Loading and Damage 

Type Analytical Method Advantages Disadvantages

Point stress failure 

model 

Notched fracture stress 

model 

Damage zone model

Simple calculation, adequate database 

Simple calculation 

Provides more accurate 

predictions

Stress intensity zone size is 

independent of hole sizes 

Extensive testing, limited 

database 

Length calculation, not enough 

database

Inherent flaw model 

Strain energy density 

factor method 

Finite element approach 

Damage zone model

Simple calculation, adequate database 

Provides more accurate 

predictions 

Provides adequate accuracy 

for complex geometries 

Provides more accurate 

predictions

Characteristic zone size is independent of crack sizes 

Very limited database 

Length calculation 

Length calculation

Finite element method 

Edge delamination 

model

Provides adequate accuracy 

for complex geometries 

Simple calculation

Length calculation involved 

Limited application

Finite element method 

Thin-film 1-D model

Provides adequate accuracy 

for complex geometries 

Simple calculation

Length calculation involved 

Limited application

Finite element method 

Thin-film 2-D 

delamination model

Provides adequate accuracy 

for complex geometries 

Simple calculation

Length calculation involved 

Prediction accuracy needs to be 

improved

Certain refinements were incorporated into the selected ADAM-C modules. In the PSM, for 

example, a finite width correction factor that accounts for the interaction between the hole and 

the free edge of the plate was added. The importance of the correction function is illustrated in 

Figure A-2. Experimental residual strengths for a Gr/Ep laminate are compared with theoretical 

values obtained for different combinations of damage zone sizes and finite width correction 

factors. It is apparent that only the curve that represents the combined effect of the damage zone 

and the finite width correction factor correlates most closely with the experimental data. Thus, 

the incorporation of an appropriate correction function is vital to the residual strength prediction 

of the model. 
 58

Figure A-2. Point stress model parametric study. From Reference A.8. 

The IFM is based on the linear elastic fracture mechanics model for metals containing cracks, but 

was modified to include the damage zone size. This expression depends on such parameters as 

the crack length, damage zone size, unnotched laminate tensile strength, finite width correction 

function, and most importantly, the laminate fracture toughness Kq. 

Unlike metals, there are no standards for the definition and experimental determination of Kq for 

composite materials, i.e., the fracture toughness is open to interpretation. Therefore, the 

calculation of Kq by analytical techniques is an important topic, as we have already established 

that experimental determination of the important parameters of all candidate laminates in a given 

structure is not economically feasible. At the present time, our methodology uses two approaches 

to calculate the fracture toughness because no single model adequately covers the wide range of 

laminate behavior. The first approach assumes a simple relationship between the fracture 

toughness and the unnotched tensile property of a laminate, i.e., Kq, is one-half of the unnotched 

strength (in English units). The second approach uses an analytical prediction of Kq proposed by 

Poe (Ref. A.14), who attempted to develop a general fracture toughness parameter that depends 

on the lamination constants as well as certain assumptions about the importance of load carrying 

plies. For conciseness, the first Kq approximation is abbreviated as IFM-A and the second as 

IFM-B. 

Results of the delamination studies indicate that the strain-energy release rate G is a preferred 

parameter from the standpoint of delamination growth. An approach based on G avoids the 

necessity and expense of calculating stress distributions near flaw boundaries. Often, these 

stresses become singular and thus of no practical use in predicting stable delamination growth. 

Delamination growth requires work that may be supplied by external forces during delamination 

or by the decrease in the strain energy of the system or both. For thin film models, energy is 

supplied by the change in strain energy resulting from delamination extension. If the energy loss 

is greater than the work required to create a unit area of delaminated composite, growth will 

occur. The delamination growth is assumed to follow a power law based on the strain-energy 

release rate raised to some exponent. This approach has been incorporated into ADAM-C. 
 59

In order to substantiate the predictive capabilities of the ADAM-C program, an extensive 

correlation study was performed. Test data found in the open literature were correlated with 

predictions by means of the PSM and IFM modules. Although appreciable data are available on 

laminates with holes, cutouts, and cracks, information about delamination growth under cyclic 

loads is very sparse. Our future experimental work will seek to fill this data gap, and correlations 

will be made with the ADAM-C delamination growth modules. 

In Figures A-3 and A-4, the predictive accuracy of the PSM and IFM is correlated with some 

typical experimental data found in the literature. All of the data, both holes and cracks, are 

compared with three models: PSM, IFM-A, and IFM-B. The purpose of this approach is to 

investigate the extent to which the hole and crack models are interchangeable and thus develop 

some insight into the nature of these flaws in composite structures. Briefly, the results indicate 

that, for cracked structures, the IFM-B approach is preferred, whereas for composites with holes, 

the PSM provides the desired conservatism. Laminated structures have such a wide range of 

behavior that no single model can be completely endorsed, but it is believed that these models 

have the potential to be extended to more general types of loading conditions. 

Figure A-3. Hole data correlation. From Reference A.8. 

The correlation between the PSM and typical data for laminates with holes is shown in 

Figure A-3. The selected database represents a wide range of laminate configurations and 

sources within the open literature. It should be noted that both axes have been normalized with 

respect to unflawed laminate strengths such that all data can be plotted as percentages of unity. 

Perfect correlation between theory and experiment is achieved along the darkened line with a 

slope of 1. The data points below the curve indicated conservative predictions. In general, the 

PSM provides adequately conservative predictions over most hole sizes of practical interest. 

In Figure A-4, IFM-A and B are correlated to some of the existing cracked laminate data. Again, 

the data have been normalized and the six symbols represent the three data sources and their 

comparisons with the two approaches for determining K. The scatter in the residual strength 

predictions is greater for cracks than for holes. 
 60

Figure A-4. Crack data correlation. From Reference A.13. 

Further insight into each model is provided in Figures A-5 and A-6, where frequency of 

occurrence is plotted versus the ratio of the predicted-to-experimental residual strength values for 

the PSM and IFM calculations. A perfect model would always duplicate an experimental result, 

and hence the strength ratio (X-axis) would always be 1; there would be no distribution. 

Theoretical models are seldom perfect, however, and a reasonable basis for comparison is to 

cross-plot all distributions on the same figure. A good model should peak near X = 1 and should 

have a low-frequency response in the unconservative regime, i.e., X > 1. 

Figure A-5. Point stress model correlation for plates with holes. 
 61

Figure A-6. IFM correlation for plates with cracks. The data points below the curve indicate conservative 

prediction. 

In Figure A-7, the peak PSM response is in the range of 0.9 to 1.1. The IFM models also perform 

well, but are more frequently unconservative than is the PSM. Therefore, these findings indicate 

that the PSM is the preferred ADAM-C module for analyzing laminated structures containing 

holes. 

Figure A-7. Frequency correlation (hole data). Less than 1.0 indicates conservative prediction. 

In Figure A-8, the three models are cross-plotted with respect to the cracked laminate data. The 

PSM is slightly unconservative. IFM-A yields a relatively flat response with both very 

conservative and unconservative answers. IFM-B results are sometimes too conservative but 

seldom unconservative. The broad scatter indicated in Figure A-8 suggests that it is difficult for a 

single approach to cover the residual strength variations of cracked laminate data. 
 62

Figure A-8. Frequency correlation (crack data). Less than 1.0 indicates conservative prediction. 

In summary, this project addresses the need for a reliable and cost-effective damage tolerance 

methodology for composite spacecraft structures. During the project, a literature survey was 

conducted, model selection and enhancements were made, software was developed, and 

comparisons were made between experimental and analytical results. These tasks resulted in the 

development and implementation of the ADAM-C code. This work is the first step in an effort to 

characterize the damage tolerance capabilities of composite structures in space environment. 

Now that the basic methodology is in place, the process of final refinement begins. As stated 

earlier, the aircraft and aerospace structural requirements can be quite different. Generally 

speaking, aircraft structures operate in a low-stress, high-cycle environment, whereas the reverse 

is often true for spacecraft components. The latter condition is more severe, but the existing test 

data were generated for aircraft applications in the low-cycle, high-stress regime. Therefore, an 

accurate correlation and refinement of ADAM-C will require further experimental testing under 

conditions relevant to spacecraft structures. 

A damage tolerance testing program is planned with specimens to be constructed from 

commonly used composite material systems. Flaws and defects will be prefabricated in the 

specimens to represent a variety of damage conditions. The results of these tests will be 

correlated with ADAM-C, and appropriate refinements to the methodology will be incorporated. 

With this approach, the goal of developing reliable damage tolerance analysis techniques with 

respect to spacecraft structures will be achieved. 

A-2 References 

A.1 Chang, J. B.; Rogers, J. C.; and Kamrath, A. R. (1985): Damage Tolerance Analysis 

Methodology for Composite Structures. From Aerospace Sponsored Research (ASR) Summary 

Report, Aerospace Report No. ATR-85(8498)-2, The Aerospace Corporation. 

A.2 Waddoups, M. E.; Eisenmann, J. R.; and Kaminski, B. E. (1971): “Macroscopic Fracture 

Mechanics of Advanced Composite Materials.” Journal of Composite Materials, Vol. 5, 

October 1971, pp. 446-455. 
 63

A.3 Whitney, J. M.; and Nuismer, R. J. (1974): “Stress Fracture Criteria for Laminated Composites 

Containing Stress Concentrations.” Journal of Composite Materials, Vol. 8, pp. 253-265. 

A.4 Wang, S. S. (1974): Fracture of Graphite Fiber Reinforced Composites. Unpublished Sc. D. 

Thesis, 1974, Boston: Massachusetts Institute of Technology. 

A.5 Lin, K. Y. (1977): Fracture of Filamentary Composite Materials, Ph. D. Thesis, 1977, 

Massachusetts Institute of Technology. 

A.6 Sih, G. C. (1978): “Mechanics of Fracture: Linear Response.” Paper presented at the 

Proceedings of the First International Conference on Numerical Methods in Fracture 

Mechanics, Swansea, United Kingdom. 

A.7 Poe, C. C.; and Sova, J. A. (1980): Fracture Toughness of Boron/Aluminum Laminates with

Various Proportions of 0 and +/-45 Degree Plies. NASA TP-1707, November 1980. 

A.8 Backlund, J. (1981): “Fracture Analysis of Notched Composites.” Computational Structures. 

Vol. 13. 

A.9 Ratwani, M. M.; and Kan. H. P. (1979): Compression Fatigue Analysis of Fiber Composites.

Report No. NADC-78049-60, Warminster, Pennsylvania: Naval Air Development Center. 

A.10 Chai, H.; Babcock, C. D.; and Knause, W. G. (1981): “One Dimensional Modeling of Failure in 

Laminated Plates by Delamination Buckling.” International Journal of Solids and Structures, 

Vol. 17, No. 1069. 

A.11 O’Brien, T. K. (1982): “Characterization of Delamination Onset and Growth in a Composite 

Laminate.” Damage in Composite materials, ASTM STP 775, K. L. Reifsnider, Ed., American 

Society for Testing and Materials, 1982, pp. 140-167. 

A.12 Shyprykevich, P.; Flanagan, G.; and Sharp, D. (1984): Damage Tolerance of Composites, Bell 

Helicopter Report No. 699-089-088. Ft. Worth, Texas. 

A.13 Tsai, S. (1985): Composites Design – 1985. Think Composites, Dayton, Ohio. 

A.14 Poe, C. C. (1984): Fracture Toughness of Fibrous Composite Materials. NASA TP-2370, 

November 1984. 
 64

Appendix B: PFA Code for Composites 

The research and development for the PFA of composites at Aerospace has integrated and 

extended the PFA developed at NASA LaRC in the late 1990s and early 2000s (Ref. B-1). The 

PFA is implemented in a software tool that integrates with existing commercial FEM software. 

The tool simulates the growth and propagation of failure modes in a composite and it is 

constantly adapted for new classes of problems and extended to incorporate new physics. The 

key features of the PFA code include: (1) Delamination simulation by using interface elements; 

(2) Prediction of in-plane and out-of-plane nonlinear shear damage; (3) Prediction of matrixcracks based on a modified version of Puck’s criterion (Ref. B-2, B-3); and (4) A fiber failure 

criterion based on the work of Chang (Ref. B-4). The PFA code utilizes concepts from damage 

mechanics to degrade the material properties of the composite appropriately. The code also 

handles contact in the event of delamination closure following growth. Similar to ADAM-C, the 

code was developed to be as generic as possible and applicable to a large range of problems with 

adequate accuracy and conservatism. The accuracy of the code, however, relies on developing a 

complete mechanical, strength, and fracture database that can be used directly as input to the 

code. The next subsections will discuss the progressive failure philosophy, the damage models in 

the PFA code, the characterization testing that is required to define the damage model, and 

validation of the PFA code. The PFA code is also demonstrated for few examples pertaining 

pressure vessels. 

B-1 Progressive Failure Philosophy 

Refer to Figure B-1 for the following discussion on progressive damage philosophy and its 

relation to validation. The goal is to develop a reliable and practical analytical tool capable of 

simulating the response of composite joints and predicting their structural margins. This effort 

can be broadly divided into three sections: coupon testing, component testing and analysis. 

Coupon testing is essential in the damage modeling of composites since it is used to determine 

material properties, fracture properties and failure criteria. Coupon tests can also give insight into 

the progression of damage and response of the structure based upon observations and 

measurements. From component testing, the second leg in the model development, the structure 

can be evaluated on a macro-scale to evaluate margins. 

However, insight into the progression of failure is more difficult since failure may be unstable 

and the testing cost is considerably higher for flight components than it is for coupons. Finally, 

analysis is conducted to validate the material model. Since the coupon tests are simpler than the 

flight component, it follows that the analytical model will first be used to predict the response of 

the coupon tests. A FEM is created to reflect the experimental set-up; material properties are 

carried over from coupon testing or found in the literature, as appropriate, and care is taken to 

ensure that the loads and boundary conditions reflect the actual test configuration. The next step 

in the process is to compare predictions with test observations – including the measured strain, 

delamination formation and growth, shear damage, matrix cracking or fiber failure – in order to 

correlate the relevant material parameters. This is especially important in the modeling of 

damage, whereby friction coefficients and even strength the finite element results and the test 

data, the model can be used to analyze a subscale structural component and compared to 

experimental results. If large differences exist between the finite element results and the test data, 

then the model needs to be re-examined. Typically, the material, boundary conditions, and 
 65

failure criteria would be revised at the coupon level. This refinement continues until the subscale 

analysis shows good agreement with the subscale test data. The analytical tool is considered to 

be reliable once the model can accurately predict the response of both the coupon and component 

tests. 

Figure B-1 illustrates the modeling technique and adjust for specific applications. Both coupon 

and component testing are required, along with corresponding finite element analyses. 

Figure B-1. Modeling flowchart. 

B-2 Progressive Failure Methodology 

A progressive failure methodology is developed that is an extension of the methodologies found 

in Pifko and Kushner (Ref. B-5) and Sleight et al. (Ref. B-6]). The methodology requires a 

nonlinear solver to establish equilibrium because both geometric and material nonlinearity are 

considered. The geometric nonlinearity is due to large strain and large rotation kinematics. The 

nonlinear material behavior is due to the degradation of the material properties of the plies and 

the material in between the plies to simulate intralaminar and interlaminar damage mechanisms. 

An incremental iterative approach is adopted for the nonlinear finite element analysis, and the 

Newton-Raphson method is used to trace the loading path of the structure with a displacementcontrol analysis. A line search algorithm is only activated to provide an initial approximation to 

the Newton-Raphson method, or to minimize a large residual force. The progressive failure 

methodology complies with two requirements: (i) optimum size load steps to achieve solution 

convergence while representing accurately local load redistribution, and (ii) a fine mesh in 

regions that undergo material degradation to represent proper stress gradients at the boundary of 

the damaged zones. 

Damage parameters used in the analysis can be divided in two parts: one relates to the failure 

prediction at the lamina level, and second relates to the prediction of damage initiation at the 

laminate level, which can then progress and lead to ultimate failure. The progressive damage 

implementation is depicted in Figure B-2. The damage evolution equations are solved at the 

Gaussian integration points within each material ply, thus explicitly accounting for damage 

variation through the thickness and in the plane of each material ply. The flowchart is colorcoded as follows: Green represents the stage at which the user interfaces directly with the finite 

element software ABAQUSTM (Ref. B-7); Blue represents the user material subroutine UMAT; 

and Peach represents the ABAQUSTM standard nonlinear incremental procedure. First, the 

Goal

Reliable 

Analytical Tool

Improve 

Model 

Fidelity

No

Material & Loads 

Modeling

Material 

Characterization

Coupon Testing

Failure 

Criteria

Finite Element of 

Coupon Test

Margins 

Test

Flight 

Component 

Testing

Component Finite 

Element Model

Margins 

Model

No

Yes

Margins 

Agree?

FEM & Testing Agree?

Yes

Goal

Reliable 

Analytical Tool

Improve 

Model 

Fidelity

No

Material & Loads 

Modeling

Material 

Characterization

Coupon Testing

Failure 

Criteria

Finite Element of 

Coupon Test

Margins 

Test

Flight 

Component 

Testing

Component Finite 

Element Model

Margins 

Model

No

Yes

Margins 

Agree?

FEM & Testing Agree?

Yes
 66

ABAQUSTM input file needs to be prepared in a typical fashion with additional parameters 

relating to the strength and fracture of the composite. Then, the analysis is initiated by applying 

an initial load to the FEM. A converged solution must be obtained because the structure is no 

longer in equilibrium due to differences between the internal loads and the external loads. The 

stress, strain, and stiffness matrix (Jacobian) are updated. At each load step, failure criteria are 

applied at each Gaussian point. If failure occurs, the damage state is obtained and the 

components of the stiffness matrix are reduced correspondingly. The components are reduced to 

a small value instead of zero, because a reduction to zero leads to numerical difficulties in the 

nonlinear procedure. Following the application of the damage tensor, the modified NewtonRaphson nonlinear iterative procedure is conducted to achieve equilibrium prior to the next load 

step. 

Figure B-2. Flowchart of the progressive damage implementation using ABAQUSTM. 

The progressive failure methodology was implemented in a user subroutine package compatible 

with ABAQUSTM finite element code, Figure B-3, because it has the ability to solve large 

models, has advanced material models and contact definitions, and it has a built-in post 

processing software. 

Figure B.3. Failure modes integrated into a user subroutine package. 

Converged

Solution?

Load

Pr+1 = Pr + ΔPr

ΔPr ≠ 0? Input

σi

, εi, Δεi

Output

σi+1, K, D

Update Stresses

εi+1 = εi + Δεi

, σi+1 = f(εi+1)

Update Stiffness Matrix

kl

ij

ε

σ K

∂

∂ =

Update Damage Tensor D

Equilibrium

Iterations

ABAQUS PreProcessing

ABAQUS 

PostProcessing

ABAQUS 

Processing 

Initial Load 

ΔPo, Po

Failure Load

Iterations

Exceeded?

Decrease 

ΔPr

No Yes

No

Yes

Yes

No

Converged

Solution?

Load

Pr+1 = Pr + ΔPr

ΔPr ≠ 0? Input

σi

, εi, Δεi

Output

σi+1, K, D

Update Stresses

εi+1 = εi + Δεi

, σi+1 = f(εi+1)

Update Stiffness Matrix

kl

ij

ε

σ K

∂

∂ =

Update Damage Tensor D

Update Stresses

εi+1 = εi + Δεi

, σi+1 = f(εi+1)

Update Stiffness Matrix

kl

ij

ε

σ K

∂

∂ =

Update Damage Tensor D

Equilibrium

Iterations

ABAQUS PreProcessing

ABAQUS 

PostProcessing

ABAQUS 

Processing 

Initial Load 

ΔPo, Po

Failure Load

Iterations

Exceeded?

Decrease 

ΔPr

No Yes

No

Yes

Yes

No

Model Subsystems Integration Benefits

Delamination 

model

Nonlinear 

shear model

Matrix-cracking 

model

Fiber failure

ABAQUS

FEM

Ability to model over 

30K DOF 

Advanced material models 

and contact definitions

Damage visualization using variety 

of post-processing software

Efficient nonlinear solvers 

with parallel processing

Developers maintain ABAQUS as the 

state of the art in structural analysis

Stress recovery from 

discrete composite layers

User Subroutine 

Package

Delamination 

model

Nonlinear 

shear model

Matrix-cracking 

model

Fiber failure

ABAQUS

FEM

Ability to model over 

500K DOF 

Advanced material models 

and contact definitions

Damage visualization using variety 

of post-processing software

Efficient nonlinear solvers 

with parallel processing

Developers maintain ABAQUS as the 

state of the art in structural analysis

Stress recovery from 

discrete composite layers

User Subroutine 

Package

Model Subsystems Integration Benefits

Delamination 

model

Nonlinear 

shear model

Matrix-cracking 

model

Fiber failure

ABAQUS

FEM

Ability to model over 

30K DOF 

Advanced material models 

and contact definitions

Damage visualization using variety 

of post-processing software

Efficient nonlinear solvers 

with parallel processing

Developers maintain ABAQUS as the 

state of the art in structural analysis

Stress recovery from 

discrete composite layers

User Subroutine 

Package

Delamination 

model

Nonlinear 

shear model

Matrix-cracking 

model

Fiber failure

ABAQUS

FEM

Ability to model over 

500K DOF 

Advanced material models 

and contact definitions

Damage visualization using variety 

of post-processing software

Efficient nonlinear solvers 

with parallel processing

Developers maintain ABAQUS as the 

state of the art in structural analysis

Stress recovery from 

discrete composite layers

User Subroutine 

Package

Delamination 

model

Nonlinear 

shear model

Matrix-cracking 

model

Fiber failure

ABAQUS

FEM

Ability to model over 

30K DOF 

Advanced material models 

and contact definitions

Damage visualization using variety 

of post-processing software

Efficient nonlinear solvers 

with parallel processing

Developers maintain ABAQUS as the 

state of the art in structural analysis

Stress recovery from 

discrete composite layers

User Subroutine 

Package

Delamination 

model

Nonlinear 

shear model

Matrix-cracking 

model

Fiber failure

ABAQUS

FEM

Ability to model over 

500K DOF 

Advanced material models 

and contact definitions

Damage visualization using variety 

of post-processing software

Efficient nonlinear solvers 

with parallel processing

Developers maintain ABAQUS as the 

state of the art in structural analysis

Stress recovery from 

discrete composite layers

User Subroutine 

Package
 67

B-3 Failure Modes and Degradation of Mechanical Properties in PFA 

A brief description of following failure modes considered in the progressive failure methodology 

is presented: (1) Delamination; (2) In-plane and out-of-plane nonlinear shear damage; 

(3) Matrix-cracking; and (4) Fiber failure. 

Delamination

A new computational fracture mechanics technology has been developed in recent years 

(Refs. B-8 and B-9) for the prediction of discrete delaminations in composite materials via the 

use of interface elements coupled with cohesive zone models. The interface element formulation 

consists of a nonlinear distribution of continuous springs, Figure B-4, with a cohesivedecohesion constitutive law that satisfies a strength based failure criterion and a fracture-based 

criterion. The interfacial constitutive law for the formulation of the interface element governs the 

deformations of these interface layers and determines whether delamination growth occurs or 

not. This unified strength-fracture approach allows the prediction of the initiation and growth of 

delamination. Interface elements are positioned at interfaces where the delamination may 

potentially grow. The prediction of multiple delaminations and non-self similar delamination 

growth is then obtained depending upon the natural response of the structure being analyzed. 

Figure B-4. Interface element consists of a continuous distribution of breakable springs. 

Prior versions to ABAQUSTM 6.5-6 did not have the interface finite element formulation to 

predict delamination, and thus the element was previously developed and implemented in 

ABAQUSTM using a user element subroutine (UEL). The drawback with the UEL approach is 

that ABAQUSTM does not support visualization of the damage distribution at the interface 

element level, so the shape and size of the delamination is not known, except for separations that 

are visible from the deformation plot. To overcome this visualization difficulty, an effort was 

undertaken to predict delamination using standard, but very thin, solid brick elements (C3D8I) 

available in the ABAQUSTM element library. The traction-separation law normally 

accompanying interface elements was transformed to a stress-strain law for an interphasic 

element of very small thickness and implemented in ABAQUSTM with a user material subroutine 

UMAT. Severe convergence issues surfaced occasionally, mainly triggered by excessive element 

distortion. The element distortion problem occurred more often under the initiation and growth 

of shear delamination. Moreover, the formulation of the C3D8I element was not meant to be 

used for the prediction of delamination. The most current version of ABAQUSTM has added the 

capability to predict delamination using interface elements. 

Undeformed

Deformed
 68

The interface element type used in this paper is the COH3D8, an eight-noded element, three 

displacement degrees of freedom per node, and linear interpolation. There are two convergence 

issues with the COH3D8. The first issue is that the convergence has been found to be excessively 

slow with the built-in bilinear constitutive law that accompanies this element formulation. The 

source of the slow convergence is the tangent stiffness discontinuity present in the bilinear 

constitutive law. The Newton-Raphson nonlinear solver enters into nonconvergent oscillating 

cycles of the residual force while attempting to converge to a solution. The second issue for slow 

convergence is related to numerical instabilities arising from stiffness matrices with negative 

eigenvalues and ill-conditioned stiffness matrices that often accompany the type of problems 

involving softening behavior. 

The first convergence issue is addressed by selecting a constitutive law with a smooth slope, 

Figure B-5. The constitutive law has an exponential form, and it naturally satisfies the following 

multi-axial quadratic stress criterion for the onset of delamination: 

 , 

 ; 

where T1 and T2 are the interfacial traction components associated with shear and S1 and S2 are 

the interfacial shear strengths. T3 is the normal interfacial traction component, S3t is the tensile 

interfacial normal strength, and S3c is the compressive interfacial normal strength. The quadratic 

failure criterion includes a ‘friction coefficient’ that incorporates the apparent increase in shear 

strength due to transverse compression. The constitutive law also naturally satisfies the following 

mixed-mode fracture criterion for the progression of delamination: 

 ; 

where the Mode I, II, and III energy release rates are GI, GII, and GIII, respectively; and the 

critical Mode I, II, and III energy release rates are GIc, GIIc, and GIIIc, respectively. Naturally 

means that the criteria do not need to be evaluated in the numerical algorithm, because the 

criteria and its corresponding failure degradation rule is built-in in the constitutive law. These 

constitutive equations are also thermo-mechanically consistent, and this is achieved by including 

a damage parameter to prevent the restoration of the previous cohesive state between the 

interfacial surfaces. The second convergence issue is resolved by setting negative diagonal terms 

in the material tangent stiffness matrix to zero. This modification does not affect the solution, but 

it improves convergence efficiency. The constitutive law implementation is conducted with a 

user material subroutine UMAT. The constitutive law requires definition of the critical energy 

release rate for Mode I, II, and III; and the shear and normal interlaminar strengths. 

2 2 2

3 1 2

3

3 12

1 0

t

T T T , T S SS

       + + =≥        

222

3 1 2

3

3 13 23

1 0

c

T T T , T S ST S T

          + + =< − −      η η

I II III

Ic IIc IIIc

++ = 1 GGG

GGG
 69

Figure B-5. Interface element coupled with a cohesive zone model. 

In-Plane and Out-of-Plane Nonlinear Shear

Composite laminates typically exhibit a nonlinear response when subjected to shear loads either 

in the plane of the composite or transverse to the plane of a thick composite. Material 

nonlinearity in the plane of the composite can be replicated during off-axis testing or Iosipescu 

testing. A typical nonlinear curve is shown in Figure B-6. Testing is generally used to extract the 

initial shear modulus and the shear strength. Hahn and Tsai (Ref. B-10) have proposed a cubic 

polynomial to represent the nonlinear in-plane shear stress-strain relationship. 

Figure B-6. Typical nonlinear shear constitutive law. 

The work at Aerospace has been extended to account for nonlinear shear through-the-thickness 

of the composite. The extension was conducted because there is extensive evidence presented in 

the literature indicating that the nonlinear shear model plays an important role in accurate 

predictions of the failure strength. Including this nonlinearity is of paramount importance to find 

the appropriate stress distribution and concentration at the bolt hole. Otherwise, premature 

predictions of in-plane matrix-cracks are predicted and may result on a gross underprediction of 

strength. Including the nonlinear shear behavior of the composite in the failure modeling relieves 

the in-plane shear stress concentration at the hole boundary and delays the prediction of matrixcracks. Although complex polynomial forms and sophisticated nonlinear models have been 

proposed, at Aerospace the following exponential constitutive law was developed that requires a 

minimum number of material parameters: 

 

 

T

Tc Gc

μ

c ε

d

T

Tc Gc

μ

c ε

d

T

Tc Gc

μ

c ε

d

T

Tc Gc

μ

c ε

d

y i i

ii i

i i

e 1 m1 1 n1

m 1 n1 ( )

     − − = +−      − −     

λ

σ σ

i

i i 1

i i

i i

e1 m1 1 ln i 12 13 23

m1 n1 , , ,, −   − − =− = =   − −  

φ

λ φ
 70

 . 

The material properties that need to be specified for the nonlinear shear constitutive law are the 

initial shear modules, ultimate shear strengths, ultimate shear strains, and the yield strengths. 

Matrix-cracking

The strength-based failure criteria developed by Hashin (Ref. B-11) based on experimental 

observations of unidirectional composites subjected to tensile loads, has been extended because 

it does not always fit experimental data as shown by numerous studies, especially in the case of 

matrix compression. The World-Wide Failure Exercise (WWFE) was conducted, whereby the 

leading theories for predicting failure in composite laminates were compared against 

experimental evidence in order to assess the qualitative and quantitative robustness and accuracy 

of each failure theory. The comparisons included biaxial strength envelopes for a range of 

unidirectional and multi-directional composites. The WWFE results indicated that even for 

simple layups and loadings, many of the failure criteria that have been proposed to-date fall short 

of predicting the failure strength accurately. Since the numerical predictions with the failure 

theory by Puck and Schurmann (Ref. B-3) showed good agreement with experimental results for 

the prediction of biaxial failure envelopes in the WWFE studies, their criterion is extended from 

in-plane only to a 3-D form that accounts for through-the-thickness failure. The failure theory 

follows the work of Hashin. Hashin’s failure criteria for matrix-cracks consists of a quadratic 

interaction of stress invariants, whereby the criteria is made independent of the stress parallel to 

the fiber based on the fact that any possible failure plane is parallel to the fibers. The failure 

criteria proposed by Puck and Schurmann is essentially an extension of Hashin’s postulate. 

The fracture plane is unknown and it must be determined. This is accomplished by maximizing 

the failure index with respect to the fracture plane angle. Since matrix-cracking occurs in a plane 

parallel to the fibers, the fracture plane oriented at an angle β with respect to the global 

coordinate system must be obtained. The tractions acting in a potential fracture plane are 

determined by force equilibrium as follows: 

 

where σijis the local state of stress that rotates with the lamina orientation. σ11 is the stress 

parallel to the fiber, σ33 is the stress in the stacking direction, σ12 is the in-plane shear stress, and 

σ13, σ23 are the transverse shear stresses. The matrix-cracking failure criterion for a tensile 

traction is postulated as 

 

u u i ii

iii yyy i ii

e nm

ε === ; ;

ε σε

σ ε

22 33 22 33

nn 23

22 33

nt 23

nl 12 13

cos(2 ) sin(2 ) 2 2

sin(2 ) cos(2 ) 2

cos( ) sin( )

+ −

=+ +

− =− +

= +

σ σσσ

σ β σ β

σ σ σ βσ β

σ σ βσ β

2 2 2

nn nl nt

l t

1

YSS

     

  ++ =          

σ σ σ
 71

and for a compressive traction as follows: 

 

The contribution of the compression traction has been eliminated but added to increase the shear 

strength in order to indicate that shear fracture is impeded in compression. In compression there 

is additional resistance to shear fracture. This effect is incorporated with a ‘friction coefficient’. 

Here, the denominators in this failure index can be regarded as effective shear strengths. The 

potential fracture plane as shown in Figure B-7 is obtained by maximizing the failure index. 

Once failure has been established, corresponding mechanical properties are degraded to emulate 

fracture. 

Figure B-7. Potential fracture plane in the 3-D failure criterion for matrix-cracking. 

Fiber-breakage

The failure criteria for the prediction of fiber failure utilizes critical strain and it is simply: 

ε11 < cε11, ε11 < 0; ε11 > tε11, ε11 > 0 

where the c

ε11 is the strain allowable in compression and tε11 is the strain allowable in tension. 

For fibers surrounding a hole, an area satisfying the failure criterion is used as the criteria. The 

need for a critical fiber area criterion in the case of a hole is made evident by stress 

concentrations that significantly exceed the fiber strain allowable, thus causing rapid damage 

growth at load levels that are not realistic. Utilizing the critical fiber area as a criterion delays the 

failure of the composite with a notch or hole. Otherwise, the strength could be under predicted by 

more than 50 percent. 

Degradation to Mechanical Properties

Now that the failure mechanisms with their respective failure criteria have been defined, the 

constitutive law for the composite material needs to be modified accordingly to simulate damage 

initiation and growth. The new constitutive law, as derived from damage mechanics is: 

 

where i = 1,2,3 correspond to the direction of the fiber, to the direction perpendicular to the fiber 

in the plane of the laminate, to the direction perpendicular to the fiber transverse to the laminate, 

respectively. 

Fiber breakage corresponds to d1 = 1 and the components of the damage elasticity tensor C1j,Cj1

become zero. The physical interpretation of the reduction in stiffness C13, C31, C21, C12 is that if 

2 2

nl nt

l nn t nn

1

S S

      + =   − −    

σ σ

ησ ησ

ij C C (1 d )(1 d ) = −− ij i j
 72

the fiber fails, there is no load transfer from the fiber to the matrix. From the engineering 

standpoint, this reduction represents that the Poisson ratio no longer affects the mechanical 

behavior of the composite. Similarly, for matrix cracking in the plane of the laminate and 

transverse to the laminate d2 = 1, and the components of the damage elasticity tensor C2j,Cj2

become zero. If matrix-cracking occurs, the shear stiffness for each shear plane is also degraded 

using the damage parameter d2. For delamination, the energy dissipated by this failure 

mechanism will be taken into account using interface elements, hence the damage variable d3

will be held as zero. 

B-4 Material Characterization for the PFA 

To properly simulate damage initiation and growth of different failure mechanisms and their 

interaction, the mechanical, strength, and fracture properties must be determined from coupon 

testing. Table B-1 shows basis material properties determined from coupon testing. The type of 

tests needed are simple tensile test, Iosipescu testing, and the open-hole strip testing. For fiber 

failure and matrix-cracking coupon testing can be used to fully characterize failure in large scale 

structures, Figures B-8 and B-9. From the tensile test, one can determine the Young modulus and 

the ultimate tensile strength in the fiber direction. The shear stiffness, the yield stress, and the 

ultimate shear strength in the plane of the laminate can be determined either from [−45/+45]ns

off-axis loading test, short-beam shear test, or torsion test. From the open-hole strip testing, one 

can determine the critical area for which neighboring fibers fail. 

The fracture properties to characterize delamination growth are obtained from fracture testing 

that are either an American Society for Testing and Materials (ASTM) standard or are in the 

process of being certified. The Mode I critical energy release rate is determined from the double 

cantilever beam. End loads are applied in opposite directions to peel the composite beam in to 

two sublaminates. The test is generally performed on unidirectional composites with the fibers 

oriented such that these are parallel to the length of the initial delamination. The experiment 

consists of a load that is applied at aluminum end blocks jointed to the DCB specimen by an 

adhesive. The test is performed under displacement-controlled tensile test machine. The 

conventional quasi-static tests are generally performed at constant displacement rates that can be 

chosen between 5 mm/min and 15 mm/min. 

Table B-1. Basic Material Properties Determined from Coupon Testing 

Test Coupon Stiffness Strength ASTM Standard 

Tensile E11 X D3039/3039M-95a 

Iosipescu Shear G12 σy, σu D5379/D5379M-98 

Open Hole Strip Af D5766/D5766M-95 

Test Coupon Stiffness Strength ASTM Standard 

Tensile E11 X D3039/3039M-95a 

Iosipescu Shear G12 σy, σu D5379/D5379M-98 

Open Hole Strip Af D5766/D5766M-95 
 73

Figure B-8. Test configurations for the characterization of fracture properties. 

Figure B-9. Test configurations for the characterization of strength properties. 

The initial delamination is achieved by incorporating a Teflon film at the midplane of the 

laminate layup prior to curing. There are issues pertaining the test specimen layup. With these 

test specimens, the delamination front may not be straight due to uneven distribution of the 

energy release rate across the width of the beam. This uneven distribution is caused by the 

anticlastic bending effect. Secondly, the main delamination may branch into multiple cracks that 

may follow the fiber-matrix interfaces. This situation may lead to Mode II loading. In addition, 

End Notch Flexure

End Load Split

End Crack Torsion

Mode II

Mode II

Mode III

Crack Lap Shear

80% Mode II

Fixed Ratio Mixed Mode

57% Mode II

Mixed Mode Bending

10% to 100% Mode II

Mixed Mode

Mixed Mode

Mixed Mode

ASTM

Double Cantilever Beam

Mode I

ASTM

End Notch Flexure

End Load Split

End Crack Torsion

Mode II

Mode II

Mode III

Crack Lap Shear

80% Mode II

Fixed Ratio Mixed Mode

57% Mode II

Mixed Mode Bending

10% to 100% Mode II

Mixed Mode

Mixed Mode

Mixed Mode

Crack Lap Shear

80% Mode II

Fixed Ratio Mixed Mode

57% Mode II

Mixed Mode Bending

10% to 100% Mode II

Mixed Mode

Mixed Mode

Mixed Mode

ASTM

Double Cantilever Beam

Mode I

ASTM

Shear Interlaminar Strength

Iosipescu Test Modified Arcan Test

DERA 

Tensile Interlaminar

Strength

Compression Test

Compression Interlaminar

Strength & η

Shear Interlaminar Strength

Iosipescu Test Modified Arcan Test

DERA 

Tensile Interlaminar

Strength

Compression Test

Compression Interlaminar

Strength & η
 74

the phenomenon of fiber bridging caused by fiber nesting in 0-degree composites can lead to 

variations of the critical energy release rate. 

Table B-2 summaries fracture test configurations. A complex fracture behavior involving fiber 

breakage, ply jumping, and fiber bridging often occurs when arbitrary orientation in ply angles 

are considered in the design of DCB test specimen. The delamination branches to interfaces 

away from the midplane, which leads to larger values of fracture toughness. They developed a 

DCB test method to suppress crack jumping and fiber bridging effects. To achieve pure Mode I 

delamination in a layup other than 0-degrees, the arms of the DCB specimen were designed such 

that these are balanced and symmetric to eliminate the stretching-shearing and stretching 

coupling effects. In addition, the layup of the angle ply laminate was designed to minimize the 

bending-twisting effects. The tests were designed to ensure there was no curvature or shear 

distortion of the laminate due to thermal stresses from curing. 

The critical energy release rate for Mode II is determined either from the ENF, end load split 

(ELS), mixed-mode bending (MMB) apparatus, or the four-point ENF. Only the ENF and the 

ELS will be discussed because these are the most commonly used in determining the fracture 

energies under Mode II loading. The ENF is simply supported at the ends with the load applied 

at the middle of the specimen. If the delamination length is approximately less than 35 percent of 

the total length, the test specimen undergoes delamination. 

The ELS test configuration is clamped at one end and a load is applied at the other end. For this 

test configuration, the delamination growth is unstable for short delamination lengths that are 

less than 55 percent of the length of the specimen. In both test configurations, Mode II loading is 

promoted by the relative sliding of the upper surface of the lower lamina with respect to the 

lower surface of the upper lamina. The relative sliding occurs due to the rotation of the upper and 

lower arms. The practical difficulties in measuring the Mode II fracture toughness with these test 

configurations is the choice of starter defect, the stability of the test, and frictional effects 

between the crack faces. Using a film insert as a starter defect leads to a larger fracture toughness 

than a test specimen with a pre-crack. The main problem encountered with the Mode II tests is 

that determining the stability of short cracks is complex since these can dynamically propagate. 

The critical energy release rate for Mode III can be determined from the end crack torsion 

(ECT), the split cantilever beam, and the crack rail shear. The split cantilever beam generates a 

substantial Mode II strain-energy release rate component near the free edges, thus invalidating 

the test. Extensive efforts undertaken by NASA LaRC to standardize the ECT test to characterize 

Mode III have been unsuccessful. The edge-crack torsion test configuration consists of a 

rectangular composite of a [90/(±45)n/(±45)n/90]s laminate with a rectangular pre-crack 

introduced by a non-adhesive film inserted at the midplane along one edge during manufacture. 

The sample is mounted in a special fixture that provides support at three points, and a transverse 

load is introduced at the fourth point by compressing a loading pin. Usually, the specimen and 

fixture assembly is loaded in a test machine with a constant stroke rate of 1.5 mm/min. This 

special fixture creates torsion along the length of the laminate, thus inducing Mode III 

delamination. 

The MMB test, crack lap shear, or fixed ratio mixed mode (FRMM) can be used to determine the 

failure envelope for a delamination under a mixed-mode condition. From all the tests discussed, 

the double cantilever beam and the MMB tests are ASTM standards, while the end notch flexure 

is a European Agency standard. Extensive efforts are underway to develop the other fracture test 
 75

configurations into ASTM standards fully. For the full characterization of the transverse strength 

properties including the nonlinearities required in the PFA, the Iosipescu test, Modified Arcan 

Test, and the off-axis compression tests can be used to determine the interlaminar shear strength, 

tensile strength, and the mixed-mode interaction. The benefit of the off-axis compression test is 

that it provides the means to determine the ‘friction coefficient’ required for the formulation of 

the interface elements. Typically, this ‘friction coefficient’ is 0.28 and it can be confirmed by 

testing. This ‘frictional parameter’ is essentially a phenomenological representation of physical 

events occurring at the microscopic level that are not well understood. Many researchers 

including Puck have treated this parameter as friction for matrix undergoing brittle fracture under 

compression. 

Table B-2. Summary Table for Fracture Test Configurations Analyzed Using Finite Elements 

Test Configuration Failure 

Modes Failure Description 

1. DCB 

Layups: [0]2

Mode I 

(1) Delamination front is curved. 

(2) For short cracks, delamination is dominated by 

strength, while for long cracks delamination is dominated 

by fracture energy. Unstable delamination growth occurs 

under load control. 

2a. ELS 

 

2b. ENF 

Mode II 

(1) Unstable delamination growth occurs either under 

displacement or load control if the crack length to 

specimen length ratio is 0.55 for the ELS and 0.35 for the 

ENF. 

(2) The energy dissipated due to friction is negligible 

compared to the fracture dissipation. However, frictional 

forces can cause temporary crack arrest. 

(3) For short cracks, a snapback algorithm is required to 

follow the load-displacement response.

3. ECT 

 Layups: [90/±453/ 453/90]s 

Mode III 

(1) Delamination is governed by GIII, although a GII

component does exist. Thus, delamination growth occurs 

non-self-similarly. 

(2) Proper determination of the transverse shear modulus 

G23 is critical for an accurate determination of the antiplane shear mode. 

(4) Unstable delamination growth occurs if the initial 

delamination is sufficiently small. 

4a. FRMM 

 Layups: [0]2

4b. MMB 

Mixed-mode 

failure 

(1) Delamination is governed by GI and GII. 

(2) The delamination growth is unstable under either load 

or displacement control if the crack length to specimen 

length ratio is 0.41 for the FRMM and 0.35 for the MMB. 

Snapback required to trace the structural response. 

(3) The Riks algorithm required to follow the loading 

path of the MMB, since the loads are applied 

proportionally. (5) 


 76

B-5 Validation of PFA for Coupon Test Configurations 

To demonstrate the capability to predict delamination and the irreversibility capability of the 

PFA code, steady-state delamination growth of unidirectional T300/977-2 carbon fiberreinforced epoxy laminate beams were simulated for quasi-static loading-unloading cycle of 

various fracture test configurations. The finite element results for Mode I double cantilever beam 

(Figure B-10), the Mode II ELS (Figures B-11), ENF (Figure B-12), and the mixed-mode tests 

(Figures B-13, Figure B-14) were in excellent agreement with either experimental data available 

in the literature or with linear elastic fracture mechanics analytical solutions. The PFA code was 

also validated with test data available for the ECT, a very unstable test configuration. The ECT 

test specimen is a G40-800/R6376 graphite epoxy composite laminate with a 

[90/(±45)3/(±45)3/90]s stacking sequence. The predicted failure load was within 5 percent of the 

test, Figure B-15, and the predicted damage distribution was in close agreement to the scans of 

the failed specimens, Figure B-16. Lastly, the finite element simulations of the DCB and the ELS 

made of AS4/PEEK were conducted and the numerical response was in close agreement with 

experiments, Figure B-17. 

The effect of through-the-thickness compressive/tensile loads on the strength and response of 

woven carbon/epoxy composite blocks loaded in an off-axis configuration with respect to the 

stack direction was investigated using the PFA code. This particular problem tests the ability of 

the code to consider the apparent increase in shear strength when a composite is subjected to 

through-the-thickness compressive stress. Published data suggests that the composite blocks fail 

primarily by delamination. Thus, to predict the strength of the blocks, decohesion elements were 

placed between adjacent lamina. This is an improvement over previous codes that relied on a 

simple quadratic strength-based failure criterion, thus, under-predicting the failure load. 

Compression simulations were conducted for 0-degree, 7.5-degree, 19-degree, 25-degree, and 

45-degree off-axis angles test configurations. Simulations were able to capture the catastrophic 

failure of the blocks, whereby the composite blocks crack into two pieces and the upper and the 

lower portions instantaneously rotate in opposite directions, Figure B-18. The failure event as 

observed from testing and predicted by analysis is dynamic and catastrophic. The predicted 

failure loads were in close agreement with the test data for all compression test configurations. 

The failure envelopes for the composite blocks were in close agreement with the testing 

conducted by Schubel and Sun’s failure criteria, Figure B-19. The PFA code also includes the 

nonlinear shear constitutive law, which enabled the prediction of the nonlinear structural 

response, Figure B-20. 
 77

Figure B-10. Predictions by PFA of the DCB compare well to experimental data and analytical solutions. 

Figure B-11. Predictions by PFA for the ELS compare well to analytical solutions. 

Figure B-12. Predictions by PFA for the ENF compare well to analytical solutions. 
 78

Figure B-13. Predictions by PFA for the FRMM compare well to analytical solutions. 

Figure B-14. Predictions by PFA for MMB compare well to analytical solutions. 
 79

Figure B-15. Predictions by PFA for ECT compare well to testing. 

Figure B-16. Damage predicted by PFA compared well by scans of the failed coupon. 

Figure B-17. PFA predicted load-displacement response for DCB and ELS accurately. 
 80

 

Figure B-18. Predicted catastrophic failure of the off-axis test configurations. 

Figure B-19. Failure envelope predictions compared to experimental data for compression tests. 

Figure B-20. Failure response compared well with the experimental response. 
 81

B-6 Subscale test panels 

The PFA code is now demonstrated to simulate the initiation and growth of intralaminar and 

interlaminar failure modes occurring in AS4-3503 Gr/Ep panels with centrally located circular 

cutouts when subject to compression or shear loads, Ref. B-1. The testing of these panels was 

conducted at NASA LaRC. The failure of these panels is of special interest, because buckling, 

interlaminar, and intralaminar failures interact into the postbuckling regime of the structural 

response. The flat composite panel, Figure B-21, consisted of 24 plies with a stacking sequence 

of [±45/02]3s and framed by a square fixture made of steel. Three corners of the frame were held 

by pin joints, while a displacement was specified in the remaining corner at 45 degrees with 

respect to the edge of the panel. 

Two curved composite panels one with diameter to width ratio of 0.4 and the other one with a 

ratio of 0.2, Figure B-22, were subject to axial compression. Both panels had a laminate stacking 

sequence of [±45/0/90]3s. The PFA code was necessary to predict the structural response of these 

panels reasonably, Figures B-23 thru B-25. The extent and location of delaminations predicted 

by the PFA code was confirmed, Figure B-26, by visual inspection of the failed test specimens. 

The delaminations both as observed from test and analysis occurred at the midplane of the flat 

panel loaded in shear and the curved panel with d/W = 0.4 loaded in compression. For the curved 

panel with d/W = 0.2, delamination occurred in multiple locations throughout the thickness of 

the panel. Specifically, the PFA code predicted delamination initiation and growth perpendicular 

to the loading direction in regions where large compressive loads were present. As simulations 

identified the shear panel collapsing, a rapid progression of delamination perpendicular to the 

loading axis was predicted, while delamination arrest was predicted parallel to the loading axis. 

Figure B-21. Flat panel loaded in shear. 
 82

Figure B-22. Curved panel loaded in compression. 

Figure B-23. Structural response of the flap panel loaded in shear. 

Figure B-24. Structural response of the curved panel loaded in compression for d/W = 0.4. 
83

Figure B-25. Structural response of the curved panel loaded in compression for d/W = 0.2. 

Figure B-26. Visual indications of delaminations correspond to delamination prediction locations. 

The PFA code also predicted delamination growth occurring perpendicular to the compressive 

loads for the two curved panels with different cutout sizes. Simulations indicated that the 

compression panel with smaller diameter cutout-to-width ratio failed with rapid progression of 

delamination from the edge of the cutout. For the compression panel with larger diameter cutoutto-width ratio, the predicted delamination growth was followed by delamination arrest. At this 

point, the numerical response indicates a residual strength, a finding consistent with the 

experimental data. After further loading, the predicted delamination growth leads to catastrophic 

failure. Thus, the PFA code was capable of predicting the rapid progression of delamination from 

the edge of the cutout at a location perpendicular to the loading axis including non-self-similar 

delamination growth, delamination arrest, and closure of delaminated faces. 

B-7 Impact Damage of Structures 

The PFA code was also used to determine the extent and shape of delaminations of a 

square sandwich panel with facesheets made from woven carbon fabric/epoxy laminates 

(AGP370-5H/3501-6S) and a closed-cell core made of PVC foam (Divinycell H250) subject to a 

low velocity impact caused by a free-falling blunt impactor, Figure B-27, Ref. B-12. The 

predicted delaminations were ringed shape, extended outwards from just the circular edge of the 

impactor, and were located in the midplane of the upper facesheet. The predictions compared 

well with ultrasonic scanning of the failed test specimen, Figure B-28. The predicted global 

response of the sandwich structure was in close agreement with the tests, Figure B-29. 
 84

Figure B-27. Sandwich structure impacted by a free-falling mass. 

Figure B-28. Strain response at bottom facesheet of the sandwich structure. 

Figure B-29. Delamination predicted by analysis compared well with the test. 

Two characteristics of the PFA code were instrumental in the good correlation that was achieved: 

it incorporates both the increase in interlaminar shear strength, Ref. B-13, and Mode II fracture 

toughness when the composite is subject to transverse compression loads. The simulation of this 

impact event demonstrated that considering other damage modes not included in the PFA code is 

important. A separate damage model for the foam core was necessary in order to properly 

simulate the extent and shape of delamination. The PFA code was also able to predict a decrease 

in load carrying capability caused by delamination growth, which was also observed in testing. 

The PFA code was validated with test data generated at NASA LaRC of a J-section circular 

fuselage frame made of textile perform 3M PR500 epoxy resin subject to radially inward load 

Drop mass

and impactor

Instrumented

sandwich panel

0.0%

0.2%

0.4%

0 1000 2000 3000 4000 5000 6000

Load (pounds)

Strain in bottom facesheet

Analysis

Low-velocity

A n a lys is T e s t
 85

representing a crash-type event, Figure B-30. The architecture of the textile composite is a 2×2, 

2-D triaxial braid with a yarn layup of [0 degree 18k/±64 degrees 6k] 39.7 percent axial. The 

PFA of the frame correlated reasonably well with the predicted failure modes of the composite, 

including the structural response of the structure, Figure B-31. The PFA code was numerically 

stable enough to predict the load-displacement response through several major failure events 

interacting with buckling. Since different failure modes, as predicted by analysis, occur 

simultaneously at different locations of the stiffener, the exact sequence from the model could 

not be extracted. However, in multiple locations kinking of axial tows led to matrix-crazing and 

eventually delamination. 

Figure B-30. J-section of a circular fuselage frame subject to a radial load. 

Figure B-31. Structural response of the J-beam subject to radial loads for the prediction and testing. 

B-8 Adhesively Bonded Joints 

The PFA code was validated against test data for several adhesively bonded joint configurations, 

Ref. B-14. The first validation was conducted with a single lap joint test configuration. The 

single lap joint test specimen, Figure B-32, was made of woven carbon epoxy fabric and epoxyacrylate adhesive. The PFA code predicted two cracks initiating from the reentrant corners and 

propagated towards the middle of the test specimen. The PFA predictions compared well with 

the structural response from the experiments. 
 86

Figure B-32. Failure and response of the single lap joint test configuration. 

The second validation example was that of an Aluminum 6061-T6 DCB with a thermoplastic 

acrylic polymer adhesive, Figure B-33. Interface elements were placed within the adhesive to 

provide multiple paths for crack propagation and allow for the prediction of hackle crack 

patterns. The PFA code predicted crack paths similar to those observed with tests and compared 

well with the structural response from the experiment. Through the PFA it was found that a small 

component of Mode II fracture turned the crack and contributed to temporary crack arrests also 

observed in testing. 
 87

Figure B-33. Failure and response of a double cantilever beam test. 

The third validation example was that of a crack lap shear test, Figure B-34. The crack initiates 

in the vicinity of the reentrantcorner, or, at the material discontinuity as predicted by PFA and 

observed in testing. The PFA indicates that the crack is mainly dominated by a Mode II loading. 

The predicted failure and response of the crack lap shear test configuration compared well with 

the testing, Figure B-35. 

Figure B-34. Predicted crack path versus that observed in testing. 
 88

Figure B-35. Failure and response of a crack lap shear test. 

B-9 Mechanically Fastened Joints 

The PFA code was further validated for problems involving composite mechanically fastened 

joints, Ref. B-15 and Ref. B-16. The composites were 128 plies [0/–45/90/45]16s quasi-isotropic 

panels and 0.75 inches thick. The analysis simulated bypass only test, double-sided bearing and 

bypass load test, and single-sided bearing and bypass load test, Figure B-36. These tests were 

performed at Aerospace’s Space Materials Laboratory. The model incorporated nonlinear 

material behavior, and progressive damage of composites. 

Figure B-36. Single-sided bearing and bypass load (top) and double sided bearing and bypass load 

(bottom). 

The orientation and density of matrix-cracks and fiber damage are represented through a damage 

tensor, an internal state tensor that is thermodynamically consistent within the framework of 

continuum damage mechanics. The damage tensor evolves as damage associated with different 

failure modes accumulates. As described earlier, matrix-cracks are predicted using a 3-D failure 

criteria, which is based on the stresses acting on a potential fracture plane. For use with the 

failure criteria, a new anisotropic shear constitutive law is postulated governing the transverse 

and the in-plane shear nonlinear mechanical behavior. 

The prediction of initiation and progression of multiple delaminations through the thickness of 

the composite is simulated via cohesive-decohesive constitutive law. Numerical simulations 

were in good agreement using filled hole tension, double sided bypass and bearing load, and 

single sided bypass and bearing test configurations. The predicted strength and the strain 

behavior was in good correlation with in-house experimental data, Figures B-37 thru B-39. 

Bypass load

Bearing load

Bypass load

Bearing load

Bearing load
 89

Figure B-37. Joint strength failure of various specimen configurations. 

Figure B-38. Strain comparison for filled-hole tension test. 

Figure B-39. Strain comparison for single-sided bearing testing. 

0

10

20

30

40

50

60

0 15 30 45 60 75

Bearing Stress (ksi)

Bypass Stress (ksi)

FHT Test FHT Model

DSJ Test DSJ Model

SSJ Test SSJ Model

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150

Total Load (kips)

Strain (%)

SG-1 SG-1

SG-2 SG-2

Test Model

SG-2

SG-4

0.5” 1.0”

1.75” 

Pbypass 

Pbearing 

SG-1 

SG-3 

0.0

0.2

0.4

0.6

0.8

1.0

0 50 100 150

Total Load (kips)

Strain (%)

SG-1 SG-1

SG-2 SG-2

SG-3 SG-3

SG-4 SG-4

Test Model

SG-2

SG-4

0.5” 1.0” 

1.75” 

Pbypass 

Pbearing 

SG-1 

SG-3 
 90

B-10 Modeling Delaminations in COPVs 

Delaminations occasionally occur in COPVs. The objective of this section is to describe a 

method used to evaluate these defects by using a pre-processor in conjunction with progressive 

failure analysis. An example is used to illustrate this approach for a typical COPV. COPVs store 

highly pressurized gases in launch vehicles and spacecraft. Typical COPV construction consists 

of high-strength fiber composites wrapped around a thin metal liner. The purpose of the liner is 

to act as a leakage barrier while the composite carries the most of the load from pressurization. 

This combination leads to lighter weight designs for the same pressure capability, which is 

valuable for space applications. 

Structural integrity and conformance to safety requirements demand accurate stress, fatigue and 

fracture mechanics analyses. Typically, netting analysis has been used as a design tool for 

COPVs, but is a highly simplified method that cannot always be used to evaluate the structural 

integrity of a discrepant COPV. Accurate analysis is particularly important since the 

consequences of a local defect propagating can lead to catastrophic failure and possibly complete 

mission loss. Aerospace has been on the forefront in the development of high-fidelity analysis of 

composites, including the specialized tools that are required to evaluate delaminations and 

debonds in COPVs. 

Aerospace continues development of the COPV Stress Analysis Program (COSAP), begun in the 

1990s under Air Force and NASA sponsorship. COSAP is primarily comprised of pre-processing 

code that is designed to work with the commercial finite element code ABAQUSTM. COSAP is 

flexible enough to model various vessel configurations, winding patterns and peculiar situations 

resulting from manufacturing considerations including delaminations. COSAP has been 

validated through experiments (Ref B-17). Although COSAP is designed for analysis of COPVs, 

it may also be used for analysis of other filament-wound vessels, such as composite solid rocket 

motor cases. 

COSAP is designed to handle complicated features, such as the continuously changing local 

wrap angle and thickness resulting from the filament winding process. COSAP is designed to 

automatically generate an axisymmetric FEM for a COPV, while calculating the effective 3-D 

composite material properties. The input file requires major dimensions, basic material 

properties and definition of the winding pattern. The dome geometry can be specified by the user 

directly or optimized by the pre-processor. Depending upon the winding method, geodesic or 

planar, COSAP will automatically determine the local wrap angle and thicknesses of composite 

layers based on the input data and then generate the finite element mesh. To limit the total 

number of elements, a group of composite layers, sublaminate, can be incorporated into a row of 

elements. 

COSAP generates a FEM suitable for analysis with ABAQUSTM. ABAQUSTM is used to perform 

a multi-step analysis that may include fabrication, sizing and burst cycles. Sizing, or autofrettage, 

is part of a typical manufacturing process, whereby the COPV is pressurized in order to yield the 

metal liner and as a result, the liner retains compressive residual stresses at lower operating 

pressure. This increases the number of cycles the tank can sustain before leaking. Upon 

completion of the multi-step analysis, the results are processed to evaluate burst pressure 

capability, fatigue life and the leak-before-burst failure mode. Prediction of the burst pressure is 

based upon progressive failure analysis described earlier, wherein for COPVs the ultimate failure 
 91

is usually fiber breakage. Preceding the global failure, delaminations may propagate into the boss 

region or into other regions of criticality, thereby leading to decreased fiber capability. 

The objective of this section is to demonstrate how COSAP or equivalent tools can be used to 

evaluate the burst strength reduction due to the presence of a delamination. A typical COPV 

with a 20-inch diameter and alternating hoop and helical plies with a total thickness of up to 

0.5 inches is analyzed, Figure B-40. In this example, the model development consisted of the 

following steps: 

The liner contour was found by digitizing the available drawings and using software to 

determine the exact shape; 

Ply thickness and initial ply angle at the tangent point were specified in the winding program, 

which then generated the remaining ply angle and thickness throughout; 

The modeling also considered the inclusion of proper ply drop off geometry and location; 

this data was obtained from a cross section of a constructed component; 

Candidate delamination locations were identified in critical ply interfaces, Figure B-41; 

With the above geometric information, COSAP was used to generate an axisymmetric FEM 

with cohesive elements located at candidate locations including the liner-to-overwrap 

interface. 

Figure B-40. A typical COPV with boundary conditions. 

Figure B-41. Delamination location for this example. 

The analysis was performed in ABAQUSTM and included elastic-plastic material properties of 

the metallic liner and contact modeling was enforced to prevent delaminated surfaces from 

penetrating. Three load steps were required to properly evaluate the COPV: 1) loaded to 

autofrettage pressure of 4000 psi; 2) unloaded to 0 psi; and 3) loaded to maximum effective 

operating pressure, 3000 psi. Progressive failure analysis was active during these analysis steps. 

To build confidence in the model, strain gage data compared well to the predicted strains in the 

model. This was also true for the COPV axial and radial growth during autofrettage. 

Two failure modes that could result from an initial delamination were evaluated. The first failure 

mode is decreased burst capability due to a hypothetical fiber defect occurring directly above the 

delamination, Figure 42. The concern was that the delamination growth could amplify the effects 

P

5” Delamination
 92

of local fiber damage. The fiber defect was modeled by substantially reducing the elastic 

modulus in the hoop direction. The analysis showed that while the fiber defect itself could 

substantially reduce the burst capability, the addition of a delamination near this hypothetical 

fiber damage reduced the strength by less than 5 percent. Based on previous experience the 

effects of delamination can be much larger than in this example, arising from multiple factors. 

Figure B-42. Hypothetical fiber defect near delamination. 

The second failure mode is delamination propagation towards the boss, Figure B-43. A 

delamination in this region could have uncertain effects due to the large external loads that 

exist in this region. Progressive failure analysis, as described earlier, was used to predict 

delamination growth. Fracture toughness properties were not available from test data, so values 

from 1.0–5.0 lbs/in were assumed. The analysis predicted delamination growth in the region of 

ply drop-offs, which can be explained by the high transverse shear stresses present in this region. 

Once the delamination grew outside of the drop-off region, the delamination growth rate slowed 

substantially. Further, results showed that this growth was insensitive to the fracture toughness 

chosen, Figure B-44. Again, based on previous experience, delamination growth characteristics 

are driven by multiple factors and this example should not be used for the assessment of any 

other structure, especially considering that delamination growth to the boss is plausible. 

Figure B-43. Direction of delamination growth in this example. 

Delamination

Fiber Defect

Growth
 93

Figure B-44. Predicted delamination growth for low and high fracture toughness values. 

In summary, an example was used to demonstrate how progressive failure analysis in 

conjunction with state-of-the-art pre-processing tools can be used to evaluate delaminations in 

COPVs. A limitation of this example is that not all failure modes were considered. While these 

tools can be used as part of an assessment of other failure modes, additional sophisticated 

analysis may be required. Particular to liner buckling, analysis tools need to be improved and 

validated through a test, non-destructive evaluation and analysis program. 

B-11 References 

B-1. Goyal, V. K.; Jaunky, N.; Johnson, E.; and Ambur, D.: “Intralaminar and Interlaminar 

Progressive Failure Analyses of Composite Panels with Circular Cutouts.” Composite 

Structures, 64, 2004, pp. 91–105. 

B-2. Goyal, V. K.; Rome, J. I.; and Klug, J. C.: “Progressive Failure Analysis for the Prediction of 

Bearing Strength in Composites.” 47th AIAA/ASME/ASCE/AHS/ASC Structures, Structural 

Dynamics, and Materials Conference, Newport, Rhode Island, 1–4 May 2006, 

AIAA-2006-2172. 

B-3. Puck, A.; and Schürmann, H. (1998): “Failure Analysis of FRP Laminates by Means of 

Physically Based Phenomenological Models.” Composite Science and Technology, Vol. 58, 

Issue 7, July 1998, pp. 1045-1067. 

B-4. Chang, F.; and Chang, K.: “A Progressive Damage Model for Laminate Composites Containing 

Stress Concentrations.” Journal of Composite Materials, Vol. 21, 1987, pp. 834-855. 

B-5. Pifko, A. B.; and Kushner, A. S.: Impact Analysis of Composite Aircraft Structures,

Computational Methods for Crashworthiness. Ahmed K. Noor and Huey D. Carden 

(Compilers), NASA CP-3223, 1993. 

B-6. Sleight, D. W.; Knight, N. F., Jr.; and Wang, J. T.: “Evaluation of a Progressive Failure 

Methodology for Laminate Composite Structures.” In Proceedings of the 

AIAA/ASME/ASCE/AHS/ASC 38th Structures, Structural Dynamics, and Materials. 

B-7. Anon.: ABAQUS User’s Manual, Vol. 1-3, Version 5.6. Hibbitt, Karlsson, and Sorensen, 

Pawtucket, RI, 1995. 

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 1000 2000 3000 4000 5000 6000 7000

Delamination Growth (inch)

Pressure (psi)

Low Gc

High Gc

0 2500 4500
 94

B-8. Goyal, V. K.; Johnson, E.; Davila, C.; and Jaunky, N.: “An Irreversible Constitutive Law for 

Modeling the Delamination Process Using Interface Elements.” Proceedings of the 

AIAA/ASME/ASCE/AHS/ASC 43rd Structures, Structural Dynamics, and Materials Conference, 

April 22–25, 2002,Paper AIAA-2002-1576, CD-ROM, 2002. 

B-9. Goyal, V. K.; and Klug, J.: “Interphasic Formulation for the Prediction of Delamination.” 45th 

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference, Palm 

Springs, California, Apr. 19–22, 2004, AIAA-2004-1845, 2004. 

B-10. Hahn, H.; and Tsai, S.: “Nonlinear Elastic Behaviour of Unidirectional Composite Laminates.” 

Journal of Composite Materials, 7, 1973, pp. 102-118. 

B-11. Hashin, Z.: 1980, “Failure Criteria for Unidirectional Fiber Composites.” Journal of Applied 

Mechanics, 47, 329-334. 

B-12. Rome, J. I.; Schubel, P. M.; and Goyal, V. K.: “Prediction of Low Velocity Impact Damage in a 

Composite Sandwich using a Progressive Failure Methodology.” Paper presented at the 22nd 

ASC Technical Conference. 

B-13. Goyal, V. K.; Schubel, P. M.; Rome, J. I.; and Keough, M. R.: “Enhancements to the Interfacial 

Element Formulation for the Prediction of Fracture.” Paper presented at the 48th AIAA 

Structures, Structural Dynamics, and Materials Conference. 

B-14. Goyal, V. K.; and Johnson, E. R.: “Predictive Strength-Fracture Model for Composite Bonded 

Joints.” Composite Structures, Volume 82, Issue 3, February 2008, Pages 434-446. 

B-15. Goyal, V. K.; Rome, J. I.; Klug, J. C.; Chang, D. J.; Katzman, H. A.; and Ross, F. D.: “Failure 

Modeling and Simulation of Composites Subjected to Bypass and Bearing Loads.” 47th 

AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 

AIAA 2006-2172, 1–4 May 2006. 

B-16. Rome, J. I.; and Goyal, V. K.: “Effects of Geometric and Material Parameters on the Strength of 

Composite Joints.” 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and 

Materials Conference, AIAA 2007-2281, 23–26 April 2007. 

B-17. Chiu, S. T.; Shu, J. C.; and Chang, J. B.: “Evaluation of Analysis Methods for Composite 

Overwrapped Pressure Vessels.” The 1994 JANNAF NDE and Structures & Mechanical 

Behavior Joint Meeting, Hill AFB, Utah, 24–28 October, 1994. 
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Composite Structures Damage Tolerance Analysis Methodologies

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Chang, James B.; Goyal, Vinay K.; Klug, John C.; Rome, Jacob I.

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NASA Langley Research Center

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Linear elastic fracture mechanics; Crack growth; International Space Station; Fracture critical items; Stress analysis

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