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AN ABSTRACT OF A THESIS

J-INTEGRAL FINITE ELEMENT ANALYSISOF SEMI-ELLIPTICAL SURFACE

CRACKS IN FLAT PLATESWITH TENSILE

LOADING

Eric N. Quillen

Master of Science in Mechanical Engineering

Linear elastic fracture mechanics (LEFM) is used when response to the loadis elastic, and the fracture is brittle. For LEFM, the K-factor is the most commonlyused fracture criterion. However, high temperatures and limited high stress cycles be-fore component replacement are factors that can cause significant plastic deformationand a ductile failure. In these cases, an elastic-plastic fracture mechanics (EPFM)approach is required. The J-integral is commonly used as an EPFM fracture param-eter.

The primary goal of this research was to develop three-dimensional finite el-ement analysis (FEA) J-integral data for surface crack specimen geometries andcompare to existing solutions. The finite element models were analyzed as elas-tic, and fully plastic using ABAQUS. The J-integral data were used to find the loadindependent variable, h1 for comparison purposes.

There were two other goals in this research. The second goal was to examinethe effect of various finite element modelling parameters including mesh density, ele-ment type, symmetry, and specimen size effects, on the resulting J-integral. The thirdgoal was to perform elastic-plastic finite element analyses that utilize a stress vs. plas-tic strain table based on a power law hardening material behavior. The elastic-plasticand fully plastic results were compared.

For the most part, the current data compared well with the data published byother researchers. The elastic results compared more favorably than the fully plasticand elastic-plastic data. For both the elastic and plastic analyses, the finite elementmodels (FEMs) produced sudden increases in the K-factor and J-integral at the freesurface and/or depth. The plastic FEMs also exhibited an anomaly in the J-integralat the third and fourth angles from the surface. The anomaly could be taken as ajump at the third angle or a dip at the fourth angle, depending on how the data weretrended. The third angle varied with the model geometry (2.71◦ to 11.24◦).

J-INTEGRAL FINITE ELEMENT ANALYSIS

OF SEMI-ELLIPTICAL SURFACE

CRACKS IN FLAT PLATES

WITH TENSILE

LOADING

A Thesis

Presented to

the Faculty of the Graduate School

Tennessee Technological University

by

Eric N. Quillen

In Partial Fulfillment

of the Requirements for the Degree

MASTER OF SCIENCE

Mechanical Engineering

May 2005

STATEMENT OF PERMISSION TO USE

In presenting this thesis in partial fulfillment of the requirements for a Master

of Science degree at Tennessee Technological University, I agree that the University

Library shall make it available to borrowers under rules of the Library. Brief quota-

tions from this thesis are allowable without special permission, provided that accurate

acknowledgment of the source is made.

Permission for extensive quotation from or reproduction of this thesis may be

granted by my major professor when the proposed use of the material is for scholarly

purposes. Any copying or use of the material in this thesis for financial gain shall not

be allowed without my written permission.

Signature

Date

iii

DEDICATION

This thesis is dedicated to my wife Julie, whose encouragement has been critical

in the completion of my graduate degree and the composition of this thesis.

iv

ACKNOWLEDGMENTS

I would like to thank the following people for their help with this work: Dr.

Chris Wilson, Dr. Phillip Allen, Mike Renfro, Krishna Natarajan, and Richard

Gregory. I would also like to thank my employer, Fleetguard, Inc., and cowork-

ers. Without their cooperation, it would not have been possible for me to perform

this research.

v

TABLE OF CONTENTS

Page

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

Chapter

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

1.1 Fracture Mechanics . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Overview of Research . . . . . . . . . . . . . . . . . . . . . . 2

2. TECHNICAL BACKGROUND . . . . . . . . . . . . . . . . . . . . . . 4

2.1 J-Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 EPRI Estimation Scheme . . . . . . . . . . . . . . . . . . . . 8

2.3 Reference Stress Method . . . . . . . . . . . . . . . . . . . . 17

3. RESEARCH PROCEDURE . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 Finite Element Modeling . . . . . . . . . . . . . . . . . . . . 19

3.1.1 Mesh Generation . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1.1 mesh3d scp . . . . . . . . . . . . . . . . . . . . . . . 21

3.1.1.2 FEA-Crack . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Analysis Procedure . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 J-Integral Convergence . . . . . . . . . . . . . . . . . . . . . 26

vi

vii

Chapter Page

3.3.1 Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3.2 Fully Plastic Zone . . . . . . . . . . . . . . . . . . . . . 27

3.4 Comparison to Other Work . . . . . . . . . . . . . . . . . . . 31

3.4.1 Kirk and Dodds . . . . . . . . . . . . . . . . . . . . . . . 31

3.4.2 McClung et al. [15] . . . . . . . . . . . . . . . . . . . . . 35

3.4.3 Lei [17] . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4.4 Nasgro Computer Program . . . . . . . . . . . . . . . . 38

3.5 Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Finite Size Effects . . . . . . . . . . . . . . . . . . . . . . . . 40

3.7 Material Properties . . . . . . . . . . . . . . . . . . . . . . . 41

3.7.1 Deformation Plasticity . . . . . . . . . . . . . . . . . . . 41

3.7.2 Incremental Plasticity . . . . . . . . . . . . . . . . . . . 43

4. RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.1 Fully Plastic Zone . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 Kirk and Dodds Incremental Plasticity . . . . . . . . . . . . 49

4.3 McClung and Lei Comparisons . . . . . . . . . . . . . . . . . 50

4.3.1 Elastic Analysis . . . . . . . . . . . . . . . . . . . . . . . 51

4.3.2 Fully Plastic Analysis . . . . . . . . . . . . . . . . . . . 67

4.3.3 Incremental Elastic-Plastic Analysis . . . . . . . . . . . . 86

4.4 Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . 91

4.5 Size Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

viii

Chapter Page

4.5.1 Height Effects . . . . . . . . . . . . . . . . . . . . . . . . 93

4.5.2 Width Effects . . . . . . . . . . . . . . . . . . . . . . . . 95

5. CONCLUSIONS AND RECOMMENDATIONS . . . . . . . . . . . . . 104

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5.2 Recommendations . . . . . . . . . . . . . . . . . . . . . . . . 106

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

APPENDICES

A: INSTRUCTIONS FOR MESH3D SCP MODIFICATIONS . . . . . . . . . 113

B: COARSE VERSUS REFINED MESHES FOR K-FACTORS . . . . . . . 115

C: COARSE VS. REFINED MESHES FOR FULLY PLASTIC MODELS . . 120

D: HEIGHT EFFECTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

E: K-FACTOR RESULTS FOR COARSE MESHES . . . . . . . . . . . . . . 137

F: FULLY PLASTIC RESULTS FOR COARSE MESHES . . . . . . . . . . . 152

G: INCREMENTAL PLASTICITY TABLES . . . . . . . . . . . . . . . . . . 163

VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

LIST OF TABLES

Table Page

2.1 McClung et al. h1 values in tension, n = 15 [15] . . . . . . . . . . . . . 14

2.2 McClung et al. h1 values in tension, n = 10 [15] . . . . . . . . . . . . . 14

2.3 McClung et al. h1 values in tension, n = 5 [15] . . . . . . . . . . . . . . 15

2.4 Lei h1 values in tension, n = 5 [17] . . . . . . . . . . . . . . . . . . . . 16

2.5 Lei h1 values in tension, n = 10 [17] . . . . . . . . . . . . . . . . . . . . 16

3.1 Number of nodes and elements in the duplication of the Kirk and Dodds[23] geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2 Incremental plasticity values for the Kirk and Dodds models . . . . . . 33

3.3 McClung et al. fully plastic geometries . . . . . . . . . . . . . . . . . . 36

3.4 Geometries for Nasgro comparison and width effect investigation . . . . 39

3.5 Number of crack front nodes in the coarse and refined meshes . . . . . 40

3.6 Stress vs. plastic strain data at n = 15, used for ABAQUS models . . . 47



4.1 Comparison of FEM results to Kirk and Dodds values . . . . . . . . . . 50

4.2 Surface and depth phenomenon for K-factors . . . . . . . . . . . . . . 56

4.3 Maximum percent differences between Newman-Raju and FEM solutions(quarter symmetry) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4 Maximum percent differences between McClung et al. [15] and FEMsolutions (quarter symmetry) . . . . . . . . . . . . . . . . . . . . . . 82

ix

x

Table Page

4.5 Maximum percent differences between McClung et al. [15] and Lei [17]solutions (quarter symmetry) . . . . . . . . . . . . . . . . . . . . . . 83

4.6 Model 1 (a/t=0.2 and a/c=0.2): h1 values at different heights . . . . . 94

4.7 Comparison of Nasgro and FEM results for n = 15 . . . . . . . . . . . 95

4.8 Comparison of Nasgro and FEM results for n = 10 . . . . . . . . . . . 96

4.9 Comparison of Nasgro and FEM results for n = 5 . . . . . . . . . . . . 96

D.1 Model 4 (a/t=0.5 and a/c=0.2): h1 values for at different heights (Part 1) 134

D.2 Model 4 (a/t=0.5 and a/c=0.2): h1 values for at different heights (Part 2) 135

D.3 Model 5 (a/t=0.5 and a/c=0.6): h1 values for at different heights . . . 135

D.4 Model 9 (a/t=0.8 and a/c=1.0): h1 values for at different heights . . . 136

E.5 Model 1 (a/t=0.2, a/c=0.2): K-Factor data from ABAQUS (Part 1) . 138


E.7 Model 2 (a/t=0.2, a/c=0.6): K-Factor data from ABAQUS . . . . . . 140








xi

Table Page





F.19 Model 1 (a/t=0.2, a/c=0.2): h1 data from ABAQUS . . . . . . . . . . 153



F.22 Model 4 (a/t=0.5, a/c=0.2): h1 data from ABAQUS (Part 1) . . . . . 156








G.30 Stress vs. strain data at n = 15, based on Equation 3.13 . . . . . . . . 164

G.31 Stress vs. strain data at n = 10, based on Equation 3.13 . . . . . . . . 165

G.32 Stress vs. strain data at n = 5, based on Equation 3.13 . . . . . . . . . 166

G.33 Stress vs. plastic strain data at n = 15, used for ABAQUS models . . . 167


xii

Table Page


LIST OF FIGURES

Figure Page

2.1 Contour around a crack tip [4] . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 EPRI J-Integral estimation scheme [4] . . . . . . . . . . . . . . . . . . 9

2.3 Sample of finite element mesh used by McClung et al. [15] . . . . . . . 12

2.4 Close up of the finite element mesh around the crack front used byMcClung et al. [15] . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1 Degeneration of elements around crack tip [4] . . . . . . . . . . . . . . 20

3.2 Plastic singularity element [4] . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Zones created in the mesh by mesh3d scp [20] . . . . . . . . . . . . . . 22

3.4 Mesh created using FEA-Crack . . . . . . . . . . . . . . . . . . . . . . 23

3.5 Close up of mesh from Figure 3.4 created using FEA-Crack . . . . . . . 23

3.6 Contours (semi-circular rings) around the crack tip . . . . . . . . . . . 24

3.7 Coordinate scheme for mapping crack face angles . . . . . . . . . . . . 26

3.8 Fully plastic element set consisting of the elements around the crack tip 28

3.9 Fully plastic element set consisting of part of layer 1 . . . . . . . . . . . 29

3.10 Fully plastic element set consisting of layer 1 . . . . . . . . . . . . . . . 29

3.11 Fully plastic element set consisting of partial layers 1 and 2 . . . . . . . 30

3.12 Geometries used by Kirk and Dodds for estimating the J-Integral [23] . 32

3.13 Stress vs. strain curve for Kirk and Dodds elastic-plastic models [23] . . 34

3.14 Refined mesh along the crack front . . . . . . . . . . . . . . . . . . . . 41

xiii

xiv

Figure Page

3.15 Effect of n on the stress vs. strain curve using a Ramberg-Osgood model 42

3.16 Intersection of Ramberg-Osgood curves at σo . . . . . . . . . . . . . . . 44

3.17 Elastic, modified elastic, and Ramberg-Osgood stress vs. strain curves forn = 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.1 Model 1 (a/t=0.2, a/c=0.2): Normalized K factor vs. angle along crackfront . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51









4.10 Elastic singularity element [4] . . . . . . . . . . . . . . . . . . . . . . . 58

4.11 Model 1 (a/t=0.2, a/c=0.2): Normalized K-factor vs. angle along crackfront for untied and tied nodes . . . . . . . . . . . . . . . . . . . . . 58

xv

Figure Page

4.12 Model 8 (a/t=0.8, a/c=0.6): Normalized K-factor vs. angle along crackfront for untied and tied nodes . . . . . . . . . . . . . . . . . . . . . 59

4.13 Model 1 (a/t = 0.2, a/c = 0.2): Reduced vs. full integration elements . 60

4.14 Model 8 (a/t = 0.8, a/c = 0.6): Reduced vs. full integration elements . 61

4.15 K-factor results from FEA-Crack Validation Manual [26] . . . . . . . . 63

4.16 FEM mesh for a flat plate with no symmetry exploited [26] . . . . . . . 63

4.17 FEM mesh for a flat plate with half symmetry . . . . . . . . . . . . . . 64

4.18 Model 6 (a/t = 0.5, a/c = 1.0): K-Factor results for half symmetry model 64

4.19 Model 8 (a/t = 0.8, a/c = 0.6): K-Factor results for half symmetry model 65

4.20 Model 1 (a/t=0.2, a/c=0.2): h1 vs. angle along the crack front . . . . . 67











xvi

Figure Page

















4.47 Model 6 (a/t=0.5, a/c=1.0): h1 results for half symmetry model at n = 15 84

4.48 Model 8 (a/t=0.8, a/c=0.6): h1 results for half symmetry model at n = 15 85

4.49 Model 1 (a/t=0.2, a/c=0.2): h1 vs. angle for fully plastic andelastic-plastic models at n=5 . . . . . . . . . . . . . . . . . . . . . . 87

xvii

Figure Page






4.55 Elastic, Ramberg-Osgood, modified elastic, and modifiedRamberg-Osgood stress vs. strain curves for n = 10 . . . . . . . . . . 90

4.56 Model 1 (a/t = 0.2, a/c = 0.2): Normalized K-factor vs. angle alongcrack front . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.57 Model 1 (a/t = 0.2, a/c = 0.2): h1 vs. angle along the crack front . . . 92

4.58 Model 1 (a/t = 0.2, a/c = 0.2): h1 vs. c/w for Nasgro and FEM at n = 15 97








xviii

Figure Page





B.1 Model 1 (a/t=0.2, a/c=0.2): Normalized K factor vs. angle along crackfront . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116








C.9 Model 1: h1 vs. angle along the crack front . . . . . . . . . . . . . . . . 121




xix

Figure Page





















LIST OF SYMBOLS

Symbol Description

a Crack depthaeff Effective crack length, includes plastic zoneb Uncracked ligament lengthc Half crack lengthds Increment of length along the contourG Strain energy release rateh1 Dimensionless parameter used to calculate Jpl

h2 Dimensionless parameter used to calculate CTODh3 Dimensionless parameter used to calculate δp

n Strain hardening exponentnj Unit vector components normal to Γr Crack tip radiusrc Radius of projected circlet Specimen thicknessw Half specimen widthx1 Distance along x-axis for projected circlex2 Distance along x-axis for projected circley1 Distance along y-axis for projected circley2 Distance along y-axis for projected circleA Crack areaCTOD Crack tip opening displacementE Young’s ModulusIn Integration constantJ Elastic-plastic fracture parameterJel Elastic portion of the J-integralJpl Plastic portion of the J-integralJtotal Sum of Jel and Jpl

K Stress intensity factorKnorm Normalized K-factorP Applied loadPo Limit loadTi Traction vectorui Displacement vectorW Specimen widthα Dimensionless Ramberg-Osgood material constantβ Plasticity constraint factorδp Load line displacement

xx

xxi

Symbol Description

ε Strainεij Strain tensorεo Yield strainεref Reference strainΓ ContourΠ Potential Energyµ Reference stress factorν Poisson’s ratioω Strain energy densityσ Stressσij Stress tensorσo Yield stressσref Reference stressθ Angle of crack tipEPFM Elastic plastic fracture mechanicsEPRI Electric Power Research InstituteFEA Finite element analysisFEM Finite element modelLEFM Linear elastic fracture mechanicsODB Output data base

CHAPTER 1

INTRODUCTION

1.1 Fracture Mechanics

Fracture mechanics is the study of the effects of flaws in materials under load.

Modern fracture mechanics was originated by Griffith [1] in the 1920’s when he suc-

cessfully showed that fracture in glass occurs when the strain energy resulting from

crack growth is greater than the surface energy. In 1948, Irwin [2] extended Griffith’s

strain energy release rate, G, to include metals by accounting for the energy absorbed

during plastic material flow around the flaw. By 1960, the fundamental principles of

linear elastic fracture mechanics (LEFM) were in place ([3, 4], for example).

LEFM is used to predict material failure when response to the load is elastic

and the fracture response is brittle. LEFM uses the strain energy release rate G or

the stress intensity factor K as a fracture criterion. K solutions for many geometries

have been calculated in the past and are widely available [5]. However, the design

parameters for some components violate the assumptions of LEFM. For example,

high temperatures and limited high stress cycles before component replacement are

factors that can cause significant plastic deformation and a ductile failure. In these

cases, where the LEFM approach is not valid, an elastic-plastic fracture mechanics

(EPFM) approach is required.

EPFM had its beginnings in 1961, when Wells [6] noticed that initially sharp

cracks in high toughness materials were blunted by plastic deformation. Wells pro-

posed that the distance between the crack faces at the deformed tip be used as a

1

2

measure of fracture toughness. The stretch between the crack faces at the blunted

tip is known as the crack tip opening displacement (CTOD).

In 1968 Rice [7] developed another EPFM parameter called the J-integral by

idealizing the elastic-plastic deformation around the crack tip to be nonlinear elastic.

The J-integral was shown to be equivalent to G for linear elastic deformation and to

the crack tip opening displacement for elastic-plastic deformation. During the same

year, Hutchinson [8], Rice, and Rosengren [9] showed that J was also a nonlinear

stress intensity parameter. The J-integral can be used as an elastic-plastic or fully

plastic crack growth fracture parameter, much like K is used as an elastic fracture

parameter.

The J-integral can be calculated using several experimental and analytical

techniques. The analytical techniques include the Electric Power Research Institute

(EPRI) estimation scheme, the reference stress method, and finite element methods.

It should be noted that many of the analytical techniques that do not directly require

finite element methods were established using finite element analysis.

1.2 Overview of Research

There are three goals in this research. The primary goal is to develop three-

dimensional finite element analysis (FEA) J-integral results using ABAQUS. These

results will be compared to existing solutions. The second goal is to investigate

the effect of various finite element modelling parameters on the resulting J-integral.

These parameters include mesh density, element type, symmetry, and specimen size

effects. The third goal is to compare incremental plasticity FEAs that utilize a stress

vs. plastic strain table based on a power law hardening material with the deformation

plasticity solution for a power law material. This comparison will be made in an

3

attempt to see if the fully plastic results using a deformation plasticity model can be

approached by a series of increasing loads in an incremental plasticity model.

The finite element models (FEMs) used in this research were three-dimensional

flat plates with surface cracks. The plates contained various surface crack, height, and

width geometries. Because of the dual symmetry, only one quarter of each plate was

modeled. Meshes from two different mesh generation programs were used: mesh 3d

(Faleskog, 1996) and FEA-Crack from Structural Reliability Technology.

CHAPTER 2

TECHNICAL BACKGROUND

In this chapter the J-integral and different J-integral calculation methods will

be examined. The chapter begins with a discussion of the theory and mathematical

foundation of the J-integral. Next, two methods for calculating the J-integral are

discussed: the EPRI Estimation Scheme and the reference stress method. Both of

these methods can be implemented using “hand calculations” without an extensive

fracture mechanics background. In addition, both of these methods are incorporated

into Nasgro, a fracture mechanics and fatigue crack growth program. Finally, the

FEA method is used in this research, but a review is not included here. There are

many excellent texts on the subject of FEA (for example Cook et al. [10]).

2.1 J-Integral

Rice [7] developed J as a path-independent contour integral by idealizing

elastic-plastic deformation to be the same as nonlinear elastic material behavior. In

the arbitrary path around a crack tip (Figure 2.1),

J =

∫Γ

(ωdy − Ti

∂ui

∂xds

), (2.1)

where ω is the strain energy density, Ti are components of the traction vector, ui

are the displacement vector components, and ds is an increment of length along the

4

5

Figure 2.1 Contour around a crack tip [4]

contour(Γ). The strain energy density and the traction vector components are

ω =

εij∫0

σijdεij (2.2)

and

Ti = σijnj, (2.3)

where σij is the stress tensor, εij is the strain tensor, and nj are unit vector components

normal to Γ.

In idealizing elastic-plastic behavior to be the same as nonlinear elastic material

behavior, Rice assumed that the material stress versus strain curve followed a power

law relationship. The Ramberg-Osgood equation is commonly used to describe the

stress and total strain data for this type of material response:

ε

εo

=σ

σo

+ α

(σ

σo

)n

, (2.4)

6

where ε is the total material strain, σo is the reference stress (normally defined as the

yield strength, but not necessarily the same as the 0.2% offset yield strength), εo is

the strain at the reference stress and is defined by εo = σo/E. There are two other

material constants in Equation 2.4. The first of these, α, is a dimensionless constant,

and the second, n, is the strain hardening exponent (n ≥ 1).

The J-dominated elastic-plastic stress field contains a singularity of order

r−1

n+1 . For the elastic case (n = 1), this singularity reduces to r−12 in agreement

with the K-dominated field of LEFM. The following two equations were derived by

Hutchinson [8], Rice and Rosengren [9] and are called the HRR singularity. The HRR

singularity describes the actual stresses and strains near the crack tip and within the

plastic zone as

σij = σo

(EJ

ασ2oInr

) 1n+1 ∗

σij (n, θ) (2.5)

and

εij =ασo

E

(EJ

ασ2oInr

) nn+1 ∗

εij (n, θ) , (2.6)

where In is an integration constant depending on n, r is the crack tip radius, θ is

the angle at a point around the contour, and∗

σij and∗

εij are functions of n and θ.

Equations 2.5 and 2.6 are important because the J-integral determines the stress

amplitude within the plastic zone. This fact establishes J as a fracture parameter

under conditions of plastic deformation.

7

Rice [7] also showed that the J-integral is equivalent to the energy release rate

in a nonlinear elastic material containing a crack:

J = −dΠ

dA(2.7)

where Π is the potential energy and A is the area of the crack. For linear elastic

deformation:

Jel = G =K2

E ′ (2.8)

where, for plane strain

E ′ =E

(1− ν2), (2.9)

and, for plane stress

E ′ = E. (2.10)

Care should be taken when using the energy release rate with elastic-plastic

or fully plastic deformation. In an elastic material, the potential energy is released

as the crack grows. In an elastic-plastic material, a large amount of strain energy is

used in forming a plastically deformed region around the crack tip. This energy will

not be recovered when the crack grows, or when the specimen is unloaded [4].

8

2.2 EPRI Estimation Scheme

The elastic-plastic and fully plastic J-integral estimation scheme presented by

EPRI [11] is derived from the work of Shih [12] and Hutchinson [13]. The purpose

of this work was to devise a simple handbook-style procedure for calculating the J-

integral. This goal was made possible by compiling nondimensional functions in table

form that could be used to calculate J directly. The nondimensional functions were

based on FEA results using Ramberg-Osgood materials.

The EPRI procedure computes a total J by summing the elastic and plastic

J ’s for various 2D geometries. This is expressed as

Jtotal = Jel + Jpl (2.11)

where Jtotal is the total J , Jel is the elastic portion, and Jpl is the plastic portion. For

small loads, Jel is much larger than Jpl. For large loads with significant deformation,

Jpl dominates. This situation is shown graphically in Figure 2.2. As discussed previ-

ously, elastic-plastic behavior is idealized to follow a nonlinear elastic path along the

stress versus strain curve.

In the EPRI estimation scheme, Jel is calculated utilizing an adjusted crack

length (aeff ) to compensate for the strain hardening around the crack tip and is

expressed as

Jel = G =K2(aeff )

E ′ , (2.12)

9

Figure 2.2 EPRI J-Integral estimation scheme [4]

where K is the stress intensity factor as a function of aeff . The adjusted crack length

is given by

aeff = a +1

1 + (P/Po)2

1

βπ

(n− 1

n + 1

) (KI

σo

)2

, (2.13)

where a is the half crack length, P is the applied load, Po is the limit load per unit

thickness, β = 2 for plane stress and β = 6 for plane strain, n is the strain hardening

exponent specific to the material, KI is the elastic stress intensity factor, and σo is

the reference stress (typically the yield strength).

10

The fully plastic equations for Jpl, crack mouth opening displacement (CTOD),

and load line displacement (δp), applicable for most specimen geometries are

Jpl = αεoσobh1

( a

W, n

) (P

Po

)n+1

, (2.14)

CTOD = αεoah2

( a

W, n

) (P

Po

)n

, (2.15)

and

δp = αεoah3

( a

W, n

) (P

Po

)n

, (2.16)

where α and n are a material constants, b is the uncracked ligament length, W is

the specimen width, and a is the crack length. h1, h2, and h3 are dimensionless

parameters that are a function of geometry and the hardening exponent n.

The center-cracked and single-edge-notched specimen geometries have a dif-

ferent form for Jpl. This form reduces the effect of the crack length to width ratio on

the value of h1, and is

Jpl = αεoσoba

wh1

( a

w, n

) (P

Po

)n+1

, (2.17)

where, for a center-cracked specimen, a is the half crack length and w is the half

width. Po is the reference or limit load, and is typically the load at which net cross

section yielding occurs. For center-cracked plate in tension,

Po = 4cσo

/√3 for plane strain, (2.18)

11

and

Po = 2cσo for plane stress. (2.19)

For a single-edge-crack in tension,

Po = 1.455ηcσo for plane strain, (2.20)

and

Po = 1.072ηcσo for plane stress. (2.21)

The EPRI handbook includes tabulations of h1, h2, and h3 for various n values

and geometries. These values were calculated using results from a finite element pro-

gram called INFEM [11]. INFEM was developed for the specific purpose of analyzing

fully plastic cracks and utilizes incompressible elements in the model formulation.

Further details of the finite element formulation have been published by Needleman

and Shih [14].

In 1999 McClung, Chell, Lee, and Orient [15] extended the original EPRI work

to include fully plastic J solutions for 3D geometries. This work was performed using

3D finite element models. The meshes for these models were constructed using eight-

noded brick elements in ANSYS 5.0. A typical mesh is shown in Figure 2.3. A close

up view of the crack front may be seen in Figure 2.4.

12

Figure 2.3 Sample of finite element mesh used by McClung et al. [15]

Figure 2.4 Close up of the finite element mesh around the crack front used byMcClung et al. [15]

13

Although the meshes were created in ANSYS, ABAQUS was used to perform

the analysis of the finite element models. The version of ABAQUS used for this work

was only capable of performing an incremental plasticity analysis. An EPRI-type

scheme was used to separate the elastic and plastic J values. The fully plastic values

for h1 were then calculated using

h1 =Jpl

ασoεot(

σσo

)n+1 . (2.22)

A combination of three different a/t (0.2, 0.5, 0.8) and a/c (0.2, 0.6, and 1.0)

ratios were tabulated. The specimen geometry ratios were kept constant for all models

at h/c = 4 and c/w = 0.25. The values of h1 were calculated for strain hardening

exponents of n = 5, 10, and 15, and can be found in Tables 2.1, 2.2, and 2.3.

In 2004 Lei [17] duplicated part of the work performed by McClung et al. [15]

by performing elastic and elastic-plastic finite element analyses for plates containing

semi-elliptical surface cracks under tension. The models contained surface cracks with

the same a/t and a/c ratios used by McClung et al. [15]. For the elastic analysis,

Jel results were generated and converted into K using Equation 2.8. These K results

were then compared with Newman-Raju stress-intensity factor calculations [18]. The

elastic-plastic results for strain hardening values of n = 5 and n = 10 were presented

in terms of h1. These h1 results are reproduced in Tables 2.4 and 2.5 and compare

well with McClung et al. for most geometries. The comparison with McClung et

al. and the current results are presented in more detail in Chapter 4.

14

Tab

le2.

1M

cClu

ng

etal

.h

1va

lues

inte

nsi

on,n

=15

[15]

a/t

a/c

0◦

9◦

18◦

27◦

36◦

45◦

54◦

63◦

72◦

81◦

90◦

0.20

0.20

0.22

30.

370

0.60

80.

821

1.00

11.

148

1.31

01.

447

1.56

01.

623

1.64

40.

200.

600.

356

0.46

50.

622

0.69

80.

774

0.82

30.

875

0.91

50.

948

0.97

10.

981

0.20

1.00

0.38

90.

503

0.62

80.

638

0.65

90.

653

0.65

70.

657

0.65

30.

646

0.64

60.

500.

204.

085

7.61

511

.602

14.4

8817

.057

18.7

9820

.228

21.4

3422

.129

22.2

1222

.309

0.50

0.60

3.33

64.

808

6.56

47.

048

7.69

77.

939

8.00

08.

021

8.01

97.

922

7.88

10.

501.

002.

774

3.73

84.

750

4.75

94.

932

4.89

14.

816

4.61

34.

407

4.24

34.

198

0.80

0.20

37.6

0963

.511

82.4

0491

.460

99.1

9892

.725

90.0

9788

.292

89.5

4895

.447

98.9

410.

800.

6017

.660

25.8

9032

.760

30.1

7237

.828

34.0

0231

.546

28.9

7228

.224

29.0

9530

.806

0.80

1.00

12.6

6717

.231

20.8

8219

.281

23.0

2921

.124

18.0

0516

.467

14.5

5715

.003

15.5

33

Tab

le2.

2M

cClu

ng

etal

.h

1va

lues

inte

nsi

on,n

=10

[15]

a/t

a/c

0◦

9◦

18◦

27◦

36◦

45◦

54◦

63◦

72◦

81◦

90◦

0.20

0.20

0.19

80.

320

0.52

30.

703

0.86

30.

996

1.13

31.

250

1.34

51.

398

1.41

60.

200.

600.

324

0.41

60.

544

0.60

40.

671

0.71

50.

759

0.79

20.

820

0.83

90.

847

0.20

1.00

0.35

80.

450

0.55

00.

553

0.57

10.

565

0.56

90.

566

0.56

20.

557

0.55

60.

500.

202.

539

4.51

26.

957

8.84

110

.665

11.9

7913

.048

13.9

5314

.546

14.7

1214

.811

0.50

0.60

2.31

93.

205

4.26

44.

561

4.96

75.

128

5.18

95.

209

5.23

15.

200

5.18

60.

501.

002.

007

2.59

93.

210

3.17

93.

272

3.21

83.

168

3.04

02.

921

2.82

72.

804

0.80

0.20

17.7

3129

.512

39.5

5043

.774

49.6

0046

.576

44.8

5443

.844

43.7

0645

.805

47.4

960.

800.

609.

688

13.6

8516

.725

15.8

5019

.174

17.3

1815

.896

14.7

7614

.068

14.3

2314

.800

0.80

1.00

7.24

29.

472

11.0

7710

.311

11.8

9811

.108

9.23

98.

398

7.53

67.

533

7.62

5

15

Tab

le2.

3M

cClu

ng

etal

.h

1va

lues

inte

nsi

on,n

=5

[15]

a/t

a/c

0◦

9◦

18◦

27◦

36◦

45◦

54◦

63◦

72◦

81◦

90◦

0.20

0.20

0.16

40.

252

0.40

70.

544

0.67

60.

789

0.89

70.

988

1.06

21.

103

1.11

70.

200.

600.

286

0.35

20.

441

0.48

00.

533

0.57

00.

605

0.63

10.

652

0.66

60.

672

0.20

1.00

0.32

10.

383

0.44

60.

440

0.45

20.

446

0.44

70.

442

0.43

90.

435

0.43

50.

500.

201.

325

2.13

93.

357

4.38

45.

480

6.37

17.

136

7.76

48.

222

8.42

88.

516

0.50

0.60

1.50

21.

916

2.41

22.

548

2.76

42.

860

2.91

72.

938

2.96

82.

973

2.97

60.

501.

001.

377

1.65

81.

931

1.86

71.

894

1.83

91.

800

1.73

01.

677

1.63

71.

630

0.80

0.20

7.22

411

.273

15.7

4318

.150

21.8

7021

.460

20.6

3220

.051

18.9

6018

.993

19.3

690.

800.

604.

983

6.44

97.

582

7.38

98.

421

7.75

07.

034

6.69

56.

266

6.11

46.

178

0.80

1.00

3.91

04.

728

5.25

14.

849

5.14

24.

775

4.08

03.

734

3.39

73.

285

3.27

0

16

Tab

le2.

4Lei

h1

valu

esin

tensi

on,n

=5

[17]

a/t

a/c

0◦

9◦

18◦

27◦

36◦

45◦

54◦

63◦

72◦

81◦

90◦

0.2

0.2

0.17

90.

3003

0.45

720.

5949

0.72

230.

8389

0.95

561.

053

1.13

21.

177

1.19

60.

20.

60.

3151

0.39

580.

4897

0.52

360.

5777

0.61

340.

6517

0.67

780.

7007

0.71

130.

7177

0.2

10.

3575

0.43

680.

5011

0.48

70.

5004

0.48

950.

4912

0.48

580.

4863

0.48

250.

4839

0.5

0.2

1.34

32.

327

3.49

54.

524

5.45

56.

265

7.03

97.

616

8.08

98.

338

8.46

60.

50.

61.

564

2.03

2.52

42.

633

2.82

92.

897

2.97

42.

982.

993

2.97

22.

981

0.5

11.

441.

782.

042

1.95

1.97

1.87

81.

837

1.76

21.

722

1.67

61.

672

0.8

0.2

6.72

311

.78

17.2

519

.57

21.2

21.4

321

.16

20.3

519

.57

18.9

218

.86

0.8

0.6

5.38

86.

938

8.31

78.

145

8.18

57.

656

7.17

86.

633

6.39

66.

263

6.29

0.8

14.

119

5.07

55.

678

5.18

75.

014

4.48

24.

106

3.70

13.

505

3.39

63.

406

Tab

le2.

5Lei

h1

valu

esin

tensi

on,n

=10

[17]

a/t

a/c

0◦

9◦

18◦

27◦

36◦

45◦

54◦

63◦

72◦

81◦

90◦

0.2

0.2

0.21

690.

3749

0.57

140.

7474

0.90

661.

046

1.18

61.

302

1.39

71.

451

1.47

50.

20.

60.

3554

0.45

080.

5774

0.63

30.

7033

0.75

180.

7987

0.83

380.

8627

0.87

860.

8862

0.2

10.

3995

0.50

090.

5964

0.59

950.

6222

0.61

780.

6211

0.61

90.

6193

0.61

720.

618

0.5

0.2

2.53

34.

723

6.90

78.

859

10.3

411

.45

12.4

513

.09

13.6

613

.91

14.1

10.

50.

62.

379

3.25

44.

285

4.58

84.

954

5.07

15.

164

5.14

25.

122

5.07

5.07

70.

51

2.05

52.

682

3.26

3.25

33.

341

3.23

13.

148

3.01

22.

907

2.81

82.

799

0.8

0.2

16.0

129

.51

43.2

746

.42

47.1

745

.94

45.7

145

.34

45.1

144

.93

45.3

0.8

0.6

11.0

815

.38

19.3

19.1

319

.01

17.4

316

.15

15.2

315

.31

15.4

815

.67

0.8

17.

925

10.6

812

.74

12.1

311

.77

10.4

59.

369

8.43

98.

159

8.23

8.40

8

17

2.3 Reference Stress Method

As discussed previously, the EPRI J estimation scheme assumes that the mate-

rial has a power law stress-strain curve. There are many materials that do not exhibit

this type of response. In 1984 Ainsworth [19] devised a method for calculating J that

did not depend on the material’s behavior following a power law. This approach is

called the reference stress method. The reference stress is defined as

σref =

(P

Po

)σo (2.23)

where P is the applied load, Po is the same limit load defined previously in the EPRI

research [11], and σo is the yield strength.

The reference strain, εref , is defined as the uniaxial strain corresponding to

σref . By inserting σref and εref into the Ramberg-Osgood equation 2.4, it can be

modified to the following form:

εref

εo

=σref

σo

+ α

(σref

σo

)n

. (2.24)

Using Equations 2.23 and 2.24, Equation 2.14 can be altered to the form

Jpl = σrefbh1

(εref −

σrefεo

σo

). (2.25)

Equation 2.25 still contains the variable h1, a function of n - same h1 used in the EPRI

equations discussed in the previous section. Ainsworth’s approach was to choose Po

in such a way that the dependence of h1 on n was minimized. For certain values of

18

Po, he found that h1 was relatively constant for n ≤20. As a result,

h1∼= h1

( a

w, 1

)(2.26)

where h1 is the average h1 for a range of n’s and h1

(aw, 1

)is the h1 for n equal to one.

The fully plastic solution at n = 1 is identical to the elastic solution using a Poisson’s

ratio of υ = 0.5,

µK2 (a) = bh1

( a

w, 1

)σ2

ref (2.27)

where µ=1 for plane stress and µ=0.75 for plane strain. By substituting Equation

2.27 and using the conditions that establish Equation 2.26, the Jpl expression becomes

Jpl =µKI

E

(Eεref

σref

− 1

). (2.28)

The previously discussed McClung et al. [15] finite element results were used

to develop another reference stress method. This reference stress algorithm is used

within Nasgro. Nasgro is a crack propagation and fracture mechanics program devel-

oped by NASA and the Southwest Research Institute.

CHAPTER 3

RESEARCH PROCEDURE

In this chapter, the technical approach used for this thesis is presented. The

chapter begins with a discussion of the finite element modeling including mesh gen-

eration. Next, the analysis procedure for the FEMs is discussed. Then, the work

duplicated by other researchers is reviewed, and any material properties or model

parameters specific to a geometry set are looked at as well. This duplication of other

researchers’ work was to validate the methodology used by ensuring that the J-integral

analysis could be performed properly. The chapter concludes with a discussion of the

general material properties used.

3.1 Finite Element Modeling

The finite element analysis program ABAQUS was used to calculate the K-

factors and J-integrals for a variety of specimen geometries. The models were created

with quarter symmetry to reduce the number of nodes and elements (hence, the

computational time) of each model.

Unless otherwise specified, the FEMs consisted of reduced integration, 20-

noded brick elements specified as C3D20R within ABAQUS. Reduced integration

elements are recommended in the ABAQUS User Manuals [21] for plastic and large

strain elastic models. Full integration elements tend to be overly stiff and the results

may oscillate. A reduced integration element has a softening effect on the stiffness

that improves the finite element results.

The elements around the crack tip were also of type C3D20R. However, the

elements were modified by collapsing the brick element into a wedge (Figure 3.1).

19

20

When the elements were degenerated, the mid-side nodes were not moved, and the

collapsed nodes were left untied (Figure 3.2). This allows for movement of the nodes

as the element is deformed and produces a 1/r strain singularity, which duplicates

the actual crack tip strain field in the plastic zone [4].

Figure 3.1 Degeneration of elements around crack tip [4]

Figure 3.2 Plastic singularity element [4]

21

3.1.1 Mesh Generation

Two different programs were used to generate finite element meshes. The

first, called mesh3d scp [20] by Faleskog, is available as freeware. Many early finite

element meshes in this work were generated with mesh3d scp. However, this program

has serious limitations. Therefore, a second mesh generation program, FEA-Crack,

was also used. This software is commercially available from Structural Reliability

Technology, Colorado.

3.1.1.1 mesh3d scp. The mesh generation program mesh3d scp generates

a one-quarter model of a surface cracked plate. The program assumes that both the

geometry and the load possess planes of symmetry. This program divides the model

into three zones, as shown in Figure 3.3. The element density in each zone is altered by

changing variables in the mesh3d scp input file. The node and element numbering in

each zone is controlled such that the application of boundary conditions and external

loads is simplified. The meshes used to investigate the fully plastic volume and

location were created using mesh3d scp (Figures 3.8 - 3.11).

The program mesh3d scp requires an iterative approach. The set of input

variables for the program input file are changed, the program generates a mesh, the

mesh is plotted and then examined graphically. This process is repeated until a

satisfactory mesh by appearance is created. This program is capable of generating

good meshes for some geometries. However, this program does not work well for other

specimen geometries. For these geometries, mesh3d scp was found to produce a bad

mesh, no mesh, or, in the worst cases, a mesh with errors.

This program was originally written to generate meshes for an earlier version

of ABAQUS. This makes it necessary to modify the ABAQUS input files created by

22

Figure 3.3 Zones created in the mesh by mesh3d scp [20]

mesh3d scp to make them compatible with recent releases of ABAQUS(V6.5). The

file modifications used for the models in this thesis are listed in Appendix A.

3.1.1.2 FEA-Crack. The second mesh generation program utilized for this

research is called FEA-Crack. FEA-Crack is more robust than mesh3d scp and does

not require the same iterative approach on the user’s part. The mesh density in the

area around the crack can be controlled by adjusting the program settings. Also, the

generated model may be viewed immediately, and required changes to the ABAQUS

input file are minimal. A mesh created using FEA-Crack is shown in Figures 3.4 and

3.5.

23

Figure 3.4 Mesh created using FEA-Crack

Figure 3.5 Close up of mesh from Figure 3.4 created using FEA-Crack

24

3.2 Analysis Procedure

Each FEM analyzed for this research contained 5 contours around the crack

tip, as seen in Figure 3.6. The results for the first contour are generally considered

to be less accurate than the other contours because of numerical inaccuracy [21]. For

this reason, the K-factor and J-integral data from all of the contours, except the first,

were averaged [17]. These average K-factor and J-integral were used for all further

calculations and comparisons.

The FEMs contained multiple node sets along the crack front. A node set

is a group of nodes that have been associated as a group within ABAQUS. The

number of node sets depended on the physical size of the crack front. Each of these

particular node sets contain a number of nodes with the same coordinates. In the

untied condition, one node in each node set is constrained so that it can move in only

Figure 3.6 Contours (semi-circular rings) around the crack tip

25

one or two directions (it stays on the plane of symmetry). The direction of constraint

depends on the symmetry plane. These constrained nodes are listed in another node

set called “crack front nodes,” which will be significant later. The other nodes in

each node set are not constrained.

ABAQUS generates values for the K-factor and J-integral at each of the node

sets along the crack front. An Excel macro was written to allow for examination of

the variation of the K-factor and J-integral values generated along the crack front.

The program was written to calculate the angle, as projected onto a circle, at each

crack front node. The macro first finds and records the constrained nodes found in

the node set “crack front nodes,” which is located in the ABAQUS input file. The

coordinates for each of these crack front nodes are then retrieved from the input file.

The crack coordinates are then mapped onto a circle, as shown in Figure 3.7. The

equation for the projection circle is shown below as

x22 + y2

2 = r2c . (3.1)

Two facts should be noted from Figure 3.7. First, y1 is equal to y2. Second, the

circle radius, rc, is equal to the crack depth, a. Both of the previous statements are

valid as long as a/c ≤ 1, which is the case for this research. Using this information,

Equation 3.1 can now be rearranged into the form

x2 =√

a2 − y21. (3.2)

Once x2 is known, the angle, θ, may be calculated using

θ = tan−1

(x2

y1

). (3.3)

26

Figure 3.7 Coordinate scheme for mapping crack face angles

With θ known, the variation of the K-factor and J-integral values can be mapped

along the crack front contour.

3.3 J-Integral Convergence

Two quantities were initially tested to ensure that the fully plastic FEM results

had converged. The first quantity was load. The second involved the fully plastic

zone specified for the FEMs.

3.3.1 Load

The applied load in the FEMs was adjusted until the resulting J-integral values

did not change with an increase in load. The final load step was also examined for

each model to ensure that the entire load was not applied. In cases where the entire

27

specified load was applied, the load was increased, and the FEM was analyzed again.

This ensured that the specified element set became fully plastic. The fully plastic

option in ABAQUS utilizes a Ramberg-Osgood material model and ends the analysis

when the observed strain for the selected element set exceeds the offset yield strain

by ten times, assuming the load or maximum number of increments have not been

reached. Also, to ensure sufficient steps in the model, the loads were set such that at

least 33% of the specified load was applied to the model.

3.3.2 Fully Plastic Zone

The volume and location effect of the specified fully plastic element set was

examined for two reasons. First, it was necessary to determine how much of the

specimen must become fully plastic before the J-integral converged. The second

reason was to simplify the model generation. The two mesh generation programs used

in this research, mesh3d scp and FEA-Crack, established convenient, but different,

elements sets for use as fully plastic.

The fully plastic results were generated using the *FULLY PLASTIC command

within ABAQUS. This command requires the specification of an element set which

is monitored for the fully plastic condition discussed previously. Several fully plastic

element sets, or zones, were tested and the results compared. The fully plastic element

sets used in this research are defined as follows:

• LayerCR - Contains elements around the crack tip, (Figure 3.8);

28

Figure 3.8 Fully plastic element set consisting of the elements around the cracktip

• Partial Layer 1 - Contains elements in the first layer of the model, but does

not contain the elements closest to the crack tip, (Figure 3.9);

• Layer 1 - Contains the elements in the ligament plus the elements found in

LayerCR, (Figure 3.10);

• Layer 2 - Contains elements in the first and second layers of the model, but

does not contain the elements closest to the crack tip (Figure 3.11).

29

Figure 3.9 Fully plastic element set consisting of part of layer 1

Figure 3.10 Fully plastic element set consisting of layer 1

30

Figure 3.11 Fully plastic element set consisting of partial layers 1 and 2

31

3.4 Comparison to Other Work

A series of models with different crack ratios and specimen sizes were generated.

These models contained geometric parameters (e. g. a/t, a/c, etc.) identical to those

used by other researchers. The current results were compared to previous work with

the intent of validating the FEMs and methods used for this research.

3.4.1 Kirk and Dodds

FEMs were generated with the same geometries and material properties used

by Kirk and Dodds in 1992 [23]. These geometries are shown in Figure 3.12. The

mesh generation program mesh3d scp was used to generate models for all three cracks

defined by Kirk and Dodds. The models consisted of 20-noded brick elements with

reduced integration. The number of nodes and elements in each model is listed in

Table 3.1.

Table 3.1 Number of nodes and elements in the duplication of the Kirk and Dodds[23] geometries

Crack 1 Crack 2 Crack 3Nodes 16,597 12,227 12,227

Elements 3562 2593 2593

32

Figure 3.12 Geometries used by Kirk and Dodds for estimating the J-Integral [23]

33

These FEMs were analyzed to find Jtotal using an elastic-plastic analysis.

ABAQUS utilizes an incremental plasticity model for this type of analysis, and re-

quires a table of true stress versus plastic strain. The material properties for these

models were derived from Figure 3.13 and are listed below:

• E = 3.00× 104 kpsi

• ν = 0.3

• Tangent Modulus = 3.57× 102 kpsi

• Initial Yield = 80 kpsi.

These properties were used to calculate the total and elastic strains at the yield stress

and an arbitrary stress, selected to be much higher than the applied stress. This

arbitrarily large stress was used as an input because ABAQUS does not explicitly

allow the tangent modulus to be given. The plastic strains required by ABAQUS

were found by subtracting the total and elastic strains. Table 3.2 shows the calculated

strains.

Table 3.2 Incremental plasticity values for the Kirk and Dodds models

σ, kpsi total strain elastic strain plastic strain80 2.67E-03 2.67E-03 0.00E+00200 3.36E-01 6.67E-03 3.29E-01

34

Figure 3.13 Stress vs. strain curve for Kirk and Dodds elastic-plastic models [23]

35

3.4.2 McClung et al. [15]

The mesh generation program FEA-Crack was used to generate models for all

nine geometries defined in the research performed by McClung et al. (Table 3.3). Two

sets of models were generated. The first set contained a coarse mesh. The second set

utilized a more refined mesh around the crack front. The McClung et al. geometries

were analyzed as elastic, fully plastic and incrementally plastic models. The elastic

and fully plastic analyses were performed using both the coarse and refined meshes.

The incrementally plastic models were analyzed using only the coarse meshes.

In the elastic FEM analysis, the K factor was found in two ways. First,

ABAQUS was used to calculate K directly. Second, ABAQUS was used to find

the elastic J , and then Equation 2.8 was used to calculate K. These results were

compared to K factors calculated using equations from Newman and Raju [24]. The

Newman-Raju solution is given in Equations 3.4 - 3.9.

KI = σ

√π

(a

Q

) [M1 + M2

(a

t

)2

+ M3

(a

t

)4]

gfθfw, (3.4)

Q = 1 + 1.464(a

c

)1.65

, (3.5)

36

Tab

le3.

3M

cClu

ng

etal

.fu

lly

pla

stic

geom

etries

Model1

Model2

Model3

Model4

Model5

Model6

Model7

Model8

Model9

a/t

0.2

0.2

0.2

0.5

0.5

0.5

0.8

0.8

0.8

a/c

0.2

0.6

10.

20.

61

0.2

0.6

1h/c

44

44

44

44

4c/

w0.

250.

250.

250.

250.

250.

250.

250.

250.

25t

11

11

11

11

1a

0.2

0.2

0.2

0.5

0.5

0.5

0.8

0.8

0.8

c1

0.33

0.2

2.5

0.83

0.5

41.

330.

8w

41.

330.

810

3.33

216

5.33

3.2

h4

1.33

0.8

103.

332

165.

333.

2

37

M1 = 1.13− 0.09(

ac

),

M2 = −0.54 + 0.89

0.2+(ac )

,

M3 = 0.5− 10.65+a

c+ 14

(1− a

c

)24,

(3.6)

g = 1 +

[0.1 + 0.35

(a

t

)2]

(1− sin θ)2 , (3.7)

fθ =

[(a

c

)2

cos2 θ + sin2 θ

]1/4

, (3.8)

fw =

[sec

(πc

2w

√a

t

)]1/2

, (3.9)

where KI is the K factor at a given angle, σ is the applied stress, a is the crack depth,

Q is factor applicable for ac≤ 1, c is the half crack width, t is the specimen thickness,

θ is the angle, as previously defined in Figure 3.7, along the crack front, and w is the

half specimen width.

3.4.3 Lei [17]

In 2004, Lei performed elastic and elastic-plastic J analyses on models with

the same crack geometries used by McClung et al. [15]. He also maintained a spec-

imen geometry ratio of c/w = 0.25. However, Lei deviated from the McClung et

38

al. geometries by fixing the ratio h/w at four to one instead of one to one. Lei also

fixed c, therefore fixing w and h, and varied a and t.

Lei used ABAQUS to perform the analyses on his models. He used the *CON-

TOUR INTEGRAL command within ABAQUS to generate J-integral results for

fifteen contours around the crack tip. The averages of these contours, excluding the

first, were presented. Lei found that the deviation of data from any one contour is

less than 5% of the average value.

Lei used consistent material properties in his analyses. The properties for the

elastic analyses were set at E = 500 MPa and ν = 0.3. The elastic-plastic analyses

used the Ramberg-Osgood stress-strain relationship (Equation 2.4), where σo = 1.0

MPa, α = 1, and n = 5 and 10. For all analyses, Lei used the Mises yield criterion

and small strain isotropic hardening.

3.4.4 Nasgro Computer Program

Current FEM results were compared with the results produced using the crack

propagation and fracture mechanics section of Nasgro. Nasgro is a fracture mechanics

and fatigue crack growth program developed by NASA and the Southwest Research

Institue. The same Ramberg-Osgood material properties used for the McClung ge-

ometries were duplicated for this comparison. The different geometries analyzed using

Nasgro are shown in Table 3.4.

39

Table 3.4 Geometries for Nasgro comparison and width effect investigation

Model a a/t c c/w w1 0.2 0.2 1.0 0.25 4.001a 0.2 0.2 1.0 0.50 2.001b 0.2 0.2 1.0 0.67 1.493 0.2 0.2 0.2 0.25 0.803a 0.2 0.2 0.2 0.50 0.403b 0.2 0.2 0.2 0.67 0.304 0.5 0.5 2.5 0.25 10.04a 0.5 0.5 2.5 0.33 7.584b 0.5 0.5 2.5 0.40 6.256 0.5 0.5 0.5 0.25 2.006a 0.5 0.5 0.5 0.33 1.526b 0.5 0.5 0.5 0.40 1.25

40

3.5 Mesh Refinement

Two sets of finite element models were constructed using the McClung et

al. geometries [15] found in Table 3.3. The first set contained a coarse mesh refinement

along the crack front. The coarse mesh refinement along the crack front can be seen

in Figure 3.5. The second set of models had three times more elements around the

crack front (Figure 3.14). Table 3.5 shows the number of crack front nodes in the

coarse and refined meshes.

3.6 Finite Size Effects

FEMs were generated to test the effect of specimen height and width on the

J-integral. The a/t ratios of 0.2 and 0.5, and the a/c ratios of 0.2 and 1.0 were

used in this analysis. The height effect models utilized the crack ratios for Model 1

(a/t = 0.2, a/c = 0.2), Model 4 (a/t = 0.5, a/c = 0.2), and Model 9 (a/t = 0.8, a/c =

1.0). The width effect models utilized the same model geometries used in the Nasgro

J-comparison work (Table 3.4).

Table 3.5 Number of crack front nodes in the coarse and refined meshesModel a/t a/c Coarse Refined

1 0.2 0.2 31 912 0.2 0.6 17 493 0.2 1.0 17 494 0.5 0.2 45 1335 0.5 0.6 17 496 0.5 1.0 17 497 0.8 0.2 73 2658 0.8 0.6 31 919 0.8 1.0 17 49

41

Figure 3.14 Refined mesh along the crack front

3.7 Material Properties

The material properties, unless otherwise specified, were based on a structural

steel. These are the same material properties used Natarajan [22] for some FEMs

in his thesis work involving J-integral solutions. Two different yielding models were

used in this research. The first was the Ramberg-Osgood deformation plasticity

model. The second was an incremental plasticity method requiring a table of σ and

εpl. The elastic material properties for each model depended on the yielding scheme

used for the FEA.

3.7.1 Deformation Plasticity

The following material properties were used with the *Deformation Plasticity

command in ABAQUS:

42

• E = 30.0× 106 psi

• ν = 0.3

• σo = 40.0× 103 psi

• α = 0.5

• n = 5, 10, and 15

where E is Young’s modulus, ν is Poisson’s ratio, σo is yield or reference stress, α

is a dimensionless constant as described in Equation 2.4, and n is the hardening

exponent. The effect of n on the stress vs. strain curves modelled using the Ramberg-

Osgood equation is shown in Figure 3.15. Notice that the smaller n is, the greater

the hardening slope

Figure 3.15 Effect of n on the stress vs. strain curve using a Ramberg-Osgoodmodel

43

3.7.2 Incremental Plasticity

The incremental plasticity models, with the exception of the Kirk and Dodds

comparison work, were generated using the Ramberg-Osgood equation,

ε

εo

=σ

σ0

+ α

(σ

σ0

)n

, (3.10)

shown again for convenience. The material properties listed in the previous section

were used to generate the a new Young’s modulus and a table of stress vs. plastic

strain for use in ABAQUS. The Young’s modulus, E = 30.0 × 106 psi, used for the

fully plastic analyses was not used to derive the stress vs. plastic strain tables for

ABAQUS. It was replaced by a secant modulus,∗E, as shown in Equation 3.11:

∗E =

σ0

εo (1 + α). (3.11)

This value of Young’s modulus was selected because it intersects the Ramberg-Osgood

curve at the fully plastic reference stress, σo = 40.0× 103 psi (Figure 3.16).

44

Figure 3.16 Intersection of Ramberg-Osgood curves at σo

45

Using this scheme, the elastic strain, and therefore the J-integral, will be

underestimated at low stresses (Figure 3.17). But, for sufficiently high stresses, the

elastic strain becomes overwhelmed by the plastic strain, making the error negligible.

The reference strain can now be expressed as

εo =σ0∗E

. (3.12)

Figure 3.17 Elastic, modified elastic, and Ramberg-Osgood stress vs. strain curvesfor n = 10

46

Multiplying both sides of Equation 3.10 by εo and substituting Equation 3.12 yields:

ε =σ∗E

+ αεo

(σ

σ0

)n

. (3.13)

Equation 3.13 can be divided into the elastic and plastic strains as

εel =σ∗E

, (3.14)

and

εpl = αεo

(σ

σo

)n

. (3.15)

The plastic strains at different stresses were then calculated for use with the *PLAS-

TIC command in ABAQUS for incremental plasticity analyses.

In summary, elastic-plastic material properties used in this research are based

on a modified Young’s modulus. This modification makes it possible to generate

incremental plasticity models that exhibit the same yield stress for all n’s. The elastic

properties used for the incremental plasticity analyses are∗E = 20× 106 and ν = 0.3.

The stress vs. plastic strain values used with the *Plastic command in ABAQUS are

shown in Tables 3.6 - 3.8.

47

Table 3.6 Stress vs. plastic strain data at n = 15, used for ABAQUS models

Stress Plastic Strain40000 0.00066741200 0.00103942400 0.00159843600 0.00242844800 0.00364946000 0.00542547200 0.00798248400 0.01163349600 0.01679750800 0.02404252000 0.03412453200 0.04804954400 0.06714255600 0.09313956800 0.12830258000 0.17556159200 0.2386960400 0.32252561600 0.43323162800 0.57863364000 0.76861465200 1.0156


Stress Plastic Strain40000 0.044000 0.00172916248400 0.00448552800 0.01070651357200 0.02383796261600 0.05001680566000 0.09971217470400 0.19012333374800 0.34859793279200 0.61739149783600 1.060160459

48


Stress Plastic Strain40000 0.044000 0.00107448400 0.00172952800 0.00267257200 0.00398661600 0.00577466000 0.00815370400 0.01125874800 0.01524579200 0.02028883600 0.02658588000 0.03435892400 0.0438596800 0.055333101200 0.069105105600 0.085493110000 0.104851114400 0.127567118800 0.15406123200 0.184783127600 0.220223132000 0.260903136400 0.307384140800 0.360265145200 0.420186149600 0.487828154000 0.563913158400 0.649209162800 0.744528167200 0.850727171600 0.968713176000 1.099441

CHAPTER 4

RESULTS

This chapter begins with a discussion of the results for various fully plastic

element sets. Next, models generated for parameters used by Kirk and Dodds [23]

are compared to published results. McClung, Lei, and Newman-Raju data are then

compared to current FEM results. Finally, the effects of the specimen size on the

J-integral are examined, and the h1 values for various specimen widths are compared

with Nasgro results.

4.1 Fully Plastic Zone

The mesh generation program mesh3d scp was used to generate FEM’s for all

four of the fully plastic zones described in Chapter 3. The same mesh was used for

each model. Only the specified fully plastic element set was changed for the different

models. It was found that the J-integral was identical for all of the described zones.

Therefore, only Partial Layer 1 was used in later fully plastic models was used for

the FEA-Crack meshes, and LayerCR was used for any fully plastic meshes produced

with mesh scp.

4.2 Kirk and Dodds Incremental Plasticity

The results for models generated per the Kirk and Dodds geometries are shown

in Table 4.1. The results compared quite well to the published data. The maximum

difference between the current results and the published data was 2.9%. It should be

noted that this excellent agreement in results was obtained even though the meshes

49

50

Table 4.1 Comparison of FEM results to Kirk and Dodds values

Crack θ◦ J (in-lb) Kirk and Dodds % Differencefrom FEM J (in-lb)

1 30.9 0.749 0.732 2.31 90 0.892 0.867 2.92 30.9 2.055 2.014 2.02 90 0.892 0.867 2.933 30.9 2.077 2.046 1.53 90 3.207 3.173 1.7

used by Kirk and Dodds contained approximately 25% the number of nodes and

elements used in this research.

4.3 McClung and Lei Comparisons

Elastic, fully plastic, and incremental plasticity FEA results for the McClung

et al. geometries are presented in this section. The elastic results are compared to the

Newman-Raju [24] calculations, and graphical trends are noted in the comparison of

Lei’s [17] elastic results. The fully plastic data are compared to the tabular data of

McClung et al. [15] and Lei [17]. The effects of mesh refinement are discussed for

both the elastic and fully plastic FEMs. Finally the incremental plasticity and fully

plastic FEA results are compared.

51

Figure 4.1 Model 1 (a/t=0.2, a/c=0.2): Normalized K factor vs. angle along crackfront

4.3.1 Elastic Analysis

The K factors obtained from the FEM’s with the McClung geometries were

normalized using

Knorm =KI

σ√

π aQ

, (4.1)

from Newman and Raju [25]. The results of the elastic FEM models and the Newman

and Raju [24] calculations are presented in Figures 4.1-4.9.

52



53



54



55



56

Significant increases in the K-factor at the surface and/or depth were observed

in all of the models. Only Models 8 (a/t = 0.8, a/c = 0.6) and 9 (a/t = 0.8, a/c = 1.0)

do not have large increases in the K-factor at the free surface. Models 1, 4, and 7

(all with a/c = 0.2) are the only FEMs that do not contain the same K-factor spike

repeated in the depth (Table 4.2).

When the surface and depth spikes are disregarded, the ABAQUS results com-

pared very reasonably to the normalized K-factors calculated per the Newman and

Raju [24] equations. This favorable comparison occurred even though the mid-side

nodes were not moved to the quarter points, and the nodes along the crack tip were

left untied (two conditions which yield optimum accuracy in K-factor calculations

using FEMs). The largest observed error, approximately six percent, occurred with

Model 8. It should also be noted that the normalized K-factor results from the K

and elastic J models were very close. The elastic results are summarized in Table

4.3. This summary disregards the surface and depth results. There is no apparent

pattern to the differences.

The current FEA results were also compared visually to the graphical results

published by Lei [17]. Lei used a different normalizing scheme, resulting in different

Table 4.2 Surface and depth phenomenon for K-factors

Surface DepthModel a/t a/c Jump Jump

1 0.2 0.2 yes no2 0.2 0.6 yes yes3 0.2 1.0 yes yes4 0.5 0.2 yes no5 0.5 0.6 yes yes6 0.5 1.0 yes yes7 0.8 0.2 yes no8 0.8 0.6 no yes9 0.8 1.0 no yes

57

Table 4.3 Maximum percent differences between Newman-Raju and FEM solu-tions (quarter symmetry)

Max FEA K Max FEA K Max FEA KDirectly from Jel from tied nodes

Model a/t a/c (% diff.) (% diff.) (% diff.)1 0.2 0.2 -5.64 -5.38 -8.52 0.2 0.6 -2.28 -2.18 -3 0.2 1.0 3.11 2.88 -4 0.5 0.2 -6.40 -5.57 -5 0.5 0.6 -2.15 -1.96 -6 0.5 1.0 3.35 3.44 -7 0.8 0.2 4.52 4.53 -8 0.8 0.6 -7.07 -7.07 -7.179 0.8 1.0 2.8 3.03 -

scales on the y-axis, but the graphs had very similar shapes. Lei also showed some

models with the same spike at the surface that was experienced in this research.

However, the increase was not as significant. No sudden increases were observed at

the depth of his elastic models.

An investigation was performed to find the cause of the previously discussed

surface and depth K-factor spikes in the current FEMs. The first step was to explore

the potential error caused by not tying the crack tip nodes or moving the mid-side

nodes to the quarter points. Model 1 (a/t = 0.2, a/c = 0.2) and Model 8 (a/t = 0.8,

a/c = 0.6) meshes were recreated in FEA-Crack for elastic analysis only. These two

models were generated with an elastic singularity, 1/√

r, created by tying the crack

tip nodes and moving the mid-side nodes to the quarter points (Figure 4.10). The

results of the two elastic models are shown graphically in Figures 4.11 and 4.12. The

K-factors produced using these two FEM’s were almost identical to the previous

results for Model 1 and Model 8.

58

Figure 4.10 Elastic singularity element [4]

Figure 4.11 Model 1 (a/t=0.2, a/c=0.2): Normalized K-factor vs. angle alongcrack front for untied and tied nodes

59

Figure 4.12 Model 8 (a/t=0.8, a/c=0.6): Normalized K-factor vs. angle alongcrack front for untied and tied nodes

60

The second step in investigating the surface and depth K-factor spikes in-

volved changing from reduced integration to full integration elements, type C3D20R

to C3D20 in ABAQUS, for the FEMs. The results for Models 1 and 8 are shown in

Figures 4.13 and 4.14. As mentioned in the ABAQUS User’s Manuals [21], using a

full integration element type caused the K-factor results to oscillate. Full integration

elements did not correct the surface and depth deviations.

Figure 4.13 Model 1 (a/t = 0.2, a/c = 0.2): Reduced vs. full integration elements

61

Figure 4.14 Model 8 (a/t = 0.8, a/c = 0.6): Reduced vs. full integration elements

62

The default nonlinear solver for ABAQUS was also considered as a possible

source of error in the third attempt to reduce the surface and depth K-factor dis-

crepencies. Since the K-factor calculation is linear, the solver used within ABAQUS

was changed to a linear perturbation. Unfortunately, the *CONTOUR INTEGRAL

command used to output the K-factors will not function within a linear perturbation

step. Another attempt to force a linear solution was made by using the *STATIC

command to force the solution to be performed in one step. The surface and depth

results were not altered by this approach.

In a fourth attempt to solve the free surface and depth spikes, the FEA-

Crack Validation Manual [26] was examined. It was found that the FEA programs

WARP3D [27], ABAQUS, and ANSYS were used to produce K-factor calculations

for validating FEA-Crack meshes. The validation data were presented graphically as

K-factor vs. angle along the crack front (Figure 4.15 [26]). The K-factors produced

by these three FEA programs were virtually identical, except at the free surface. At

this location, the K-factor produced by ABAQUS was approximately 7.7% higher

than the other two FEA programs. It was also noted that there were no irregularities

presented in the depth.

Further examination of the FEA-Crack Validation Manual [26] revealed that

the surface crack results published in the validation manual were for full plates with no

symmetry conditions (Figure 4.16). To investigate boundary conditions in the depth

phenomenon, Model 6 (a/t = 0.5, a/c = 1.0) and Model 8 (a/t = 0.8, a/c = 0.6)

meshes were created with half symmetry (Figure 4.17). The results may be seen in

Figures 4.18 and 4.19.

63

Figure 4.15 K-factor results from FEA-Crack Validation Manual [26]

Figure 4.16 FEM mesh for a flat plate with no symmetry exploited [26]

64

Step: Step-1, Stress Analysis, Step # 1Increment 9: Step Time = 0.3394

3-D crack mesh model generated by FEA-CrackODB: model6-hs.odb ABAQUS/Standard 6.2-007 Sat Dec 18 11:58:57 CST 2004

1

2

3

Step: Step-1, Stress Analysis, Step # 1Increment 9: Step Time = 0.3394

3-D crack mesh model generated by FEA-CrackODB: model6-hs.odb ABAQUS/Standard 6.2-007 Sat Dec 18 11:58:57 CST 2004

Figure 4.17 FEM mesh for a flat plate with half symmetry

Figure 4.18 Model 6 (a/t = 0.5, a/c = 1.0): K-Factor results for half symmetrymodel

65

Figure 4.19 Model 8 (a/t = 0.8, a/c = 0.6): K-Factor results for half symmetrymodel

66

The half symmetry models produced with FEA-Crack did not exhibit a K-

factor spike in the depth. Therefore, it was deduced that the boundary conditions

specified in the ABAQUS input file could be an issue. Perhaps the solver within

ABAQUS is very sensitive to the boundary conditions. Problems with the boundary

conditions in the depth or at the free surface could account for the observed increases.

An ABAQUS input file for an elastic model was examined to investigate the

boundary conditions of nodes along the crack front. For the elastic models, it was

found that there were no coincident nodes along the crack front. The crack tip nodes

were shared by the surrounding elements. The depth node was constrained in the

directions perpendicular and parallel to the crack front. All of the other crack tip

nodes were constrained only in the direction perpendicular to the crack front. This

constraint scheme is logical given a quarter symmetry model with tied nodes along

the crack front.

A possible explanation for the surface deviation is that it is a result of the

surface elements in the 3-D model. The lateral surface is subjected to plane stress,

but the surface elements have some thickness in the direction normal to the free

surface. This is a plausible, but unproven, explanation for the source of the surface

phenomenon.

67

4.3.2 Fully Plastic Analysis

The fully plastic results from the McClung geometries are presented in the

following section. It should be noted that ABAQUS outputs Jtotal, but it is difficult

to actually quantify Jel for a Ramberg-Osgood material due to the initial nonlinearity

of the power law curve. But, for fully plastic results, Jel is only a very small fraction

of Jtotal. For this reason, the fully plastic Jtotal is (for all practical purposes) equal to

Jpl. The Jpl results from ABAQUS were used to calculate the h1 values using

h1 =Jpl

αε0σ0t (σ/σ0)(n+1)

. (4.2)

The h1 data from the current FEMs are plotted against the values from McClung et

al. [15] and Lei [17] in Figures 4.20-4.46.

Figure 4.20 Model 1 (a/t=0.2, a/c=0.2): h1 vs. angle along the crack front

68



69



70



71



72



73



74



75



76



77



78



79



80



81

The same spikes observed at the free surface and in the depth of the elastic

FEMs were also found in the fully plastic models. These discrepencies matched those

found for the elastic models (Table 4.2). As seen in the elastic results, the fully plastic

models, except Models 8 (a/t = 0.8, a/c = 0.6) and 9 (a/t = 0.8, a/c = 1.0), exhibited

noticeable increases at the free surface. Models 1, 4, and 7 (all with a/c = 0.2) were

the only models that did not also contain increases in the depth.

The fully plastic results from the current FEMs, McClung, and Lei did not

correlate as well as the elastic results compared with the Newman-Raju solutions.

The differences were estimated by visually examining the previous h1 plots (Figures

4.20-4.46). The comparison was visual because the angles for McClung et al. and Lei

results did not match the angles used for this research, and a method to force the

FEA-Crack or mesh3d scp mesh generation programs to place node sets at specific

angles was not known. As with the elastic analysis, the surface and depth results

were disregarded in determining the percent differences.

A visual examination of the h1 plots revealed some interesting findings. First,

the Lei results do not contain the same spikes in the h1 values at the surface and

depth that are observed in in the current FEMs. This is intriguing because the Lei

FEA work was performed using quarter symmetry flat plates in ABAQUS.

Second, the McClung et al. and current FEM results compared reasonably well

for some crack geometries but not for others. It was found that the differences between

the two sets of results increased with n. There were differences of approximately

thirty percent for some of the n = 15 models. Table 4.4 contains the largest percent

difference between the McClung et al. and current FEM results for each model.

It was observed that Models 2, 3, 5, 6, 8, and 9 (all with a/c = 0.6 or 1.0)

of the current FEMs, exhibited either spikes in h1 at the third angle or dips at the

82

fourth angle from the surface for for all three n’s. The uncertainty of a spike at the

third angle or a dip at the fourth is due to the fact that this judgement is relative to

how the data trend is pictured. The values in Table 4.4 are highlighted with the *

symbol when the largest difference from the McClung et al. results occurred at the

third angle. The difference between results did not tend to be as significant at the

fourth angle. An examination of the FEMs for these models revealed that third angle

for each model, except Model 8, was located at 11.24◦. The third angle in Model 8

occurrs at 8.09◦. Both the McClung et al. and Lei data exhibited the same behavior.

These results occurred at an angle of 18◦.

The third observation was that Lei [17] tended to correlate more closely with

the McClung et al. [15] results than the current FEMs. Unlike the current FEMs, the

Lei models did not exhibit a consistent increase in the difference as n increased. The

maximum percent difference between the two sets of results was twenty percent, ten

percent less than the difference of the current FEA results. Table 4.5 lists the largest

percent difference between the McClung et al. and Lei results for each model.

Table 4.4 Maximum percent differences between McClung et al. [15] and FEMsolutions (quarter symmetry)

n = 5 n = 10 n = 15Model a/t a/c (% diff.) (% diff.) (% diff.)

1 0.2 0.2 2.00 -5.66 -12.502 0.2 0.6 9.20 -9.26 -12.903 0.2 1.0 12.20∗ −16.67∗ 18.52∗

4 0.5 0.2 4.67 7.69 9.095 0.5 0.6 17.5∗ 15.71∗ 21.57∗

6 0.5 1.0 15.17∗ 25.93∗ 23.17∗

7 0.8 0.2 13.33 17.78 29.358 0.8 0.6 10.81 18.75 30.009 0.8 1.0 16.00∗ 23.30∗ 28.5∗

Note: Maximum differences at the third anglefrom the surface are marked by *.

83

Table 4.5 Maximum percent differences between McClung et al. [15] and Lei [17]solutions (quarter symmetry)

n = 5 n = 10Model a/t a/c (% diff.) (% diff.)

1 0.2 0.20 19.17 17.162 0.2 0.60 12.44 9.693 0.2 1.00 14.05 11.594 0.5 0.20 8.79 4.685 0.5 0.60 5.95 2.596 0.5 1.00 7.36 3.197 0.8 0.20 9.57 9.418 0.8 0.60 10.23 20.699 0.8 1.00 8.13 17.64

84

The J-integral spikes at the free surface and depth were also investigated for

the fully plastic FEMs. As discussed in the analysis of the elastic results, the free

surface phenomenon can be attributed, at least partially, to a plane stress issue.

Fully plastic models with half symmetry were also analyzed to ensure that the depth

phenomenon disappeared under conditions of severe deformation. Model 6 (a/t = 0.5,

a/c = 1.0) and Model 8 (a/t = 0.8, a/c = 0.6) geometries were again used. The half

symmetry models did not exhibit the jump in h1 in the depth (Figures 4.47 and 4.48).

Figure 4.47 Model 6 (a/t=0.5, a/c=1.0): h1 results for half symmetry model atn = 15

85

Figure 4.48 Model 8 (a/t=0.8, a/c=0.6): h1 results for half symmetry model atn = 15

86

An ABAQUS input file for an plastic model was also examined to investigate

the boundary conditions of nodes along the crack tip. As discussed previously, the

models with untied nodes contain multiple node sets along the crack front. The nodes

at each node set along the crack tip are coincident. In the depth node set, one node

was constrained in the directions perpendicular and parallel to the crack front. The

remaining nodes in the depth were only constrained in the direction perpendicular

to the crack front. The remaining node sets, including the surface, contained one

node that was constrained in the direction perpendicular to the crack front. All other

nodes in the remaining sets were left unconstrained. This constraint scheme is logical

given a quarter symmetry model with untied nodes along the crack front.

4.3.3 Incremental Elastic-Plastic Analysis

The elastic-plastic results are compared to the fully plastic results for Model

1 (a/t=0.2, a/c=0.2) and Model 2 (a/t=0.2, a/c=0.6) for n = 5, 10, and 15. These

results are shown graphically in Figures 4.49 - 4.54.

87

Figure 4.49 Model 1 (a/t=0.2, a/c=0.2): h1 vs. angle for fully plastic and elastic-plastic models at n=5


88



89



90

It was found that the h1 values of the elastic-plastic method varied with the

applied stress, but appeared to converge with the fully plastic results as the stress

was increased. The similarities between the elastic-plastic and fully plastic results

included the surface, depth, and interior angle anomalies discussed previously.

Some of the differences between the elastic-plastic and fully plastic results at

lower stresses may have been due to a mistake. The wrong stress versus plastic strain

tables were used in ABAQUS. The tables actually used for the elastic-plastic analyses

in this research are located in Appendix G. These plastic strain tables contain stresses

lower than the 40 ksi yield strength discussed in Chapter 3. This caused the utilized

stress-strain curve to be lower than the Ramberg-Osgood curve at lower stresses

(Figure 4.55). However, the difference due to this error is likely small for a fully

plastic condition.

Figure 4.55 Elastic, Ramberg-Osgood, modified elastic, and modified Ramberg-Osgood stress vs. strain curves for n = 10

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4.4 Mesh Refinement

The refined meshes made no notable difference in the normalized K factors

from the elastic analyses or the h1 values from the fully plastic analyses. A graphical

comparison of the Newman-Raju calculations, coarse mesh, and refined mesh for K-

factor analysis is shown in Figure 4.56. A graphical comparison of the McClung

results, coarse mesh, and refined mesh for the h1 analysis is shown in Figure 4.57.

The full set of results comparing the Newman-Raju calculations, the coarse meshes,

and the refined meshes are shown graphically in Appendix B. The full set of results

comparing the McClung work, the coarse meshes, and the refined meshes are shown

graphically in Appendix C.

Figure 4.56 Model 1 (a/t = 0.2, a/c = 0.2): Normalized K-factor vs. angle alongcrack front

92

Figure 4.57 Model 1 (a/t = 0.2, a/c = 0.2): h1 vs. angle along the crack front

It should be noted that the refined FEM mesh for Model 7 (a/t = 0.8, a/c =

0.2) would not run in ABAQUS. The Model 7 geometry generated the largest FEM in

this research, containing 12,140 elements and 56,142 nodes. A warning was generated

about the number of history requests for the ABAQUS output database (ODB) file,

but the terminal error was not explicitly mentioned by ABAQUS. Because the other

refined mesh models generated results that closely matched the coarse mesh models,

this exact cause of this issue was not pursued.

The same anomalies observed in the coarse mesh FEM’s were also present in

the refined mesh models. This includes the K-factor and J-integral spikes at the

free surface and depth. It should be noted that the mesh refinement did reduce the

magnitude of the surface and depth surges. The phenomenon at third and fourth

angles from the surface was also present. The values of the third and fourth angle in

the refined meshes were occurred at either 3.75◦ or 2.71◦.

93

Many of the refined mesh models contain noticeable oscillations in the h1 data

around the crack tip. This oscillation was most significant in Models 3 (a/t = 0.2,

a/c = 1.0), 4 (a/t = 0.5, a/c = 0.2), and 6 (a/t = 0.5, a/c = 1.0). For each of these

models, the magnitude of the oscillation became larger as n increased.

4.5 Size Effects

4.5.1 Height Effects

Finite element models with four different crack ratios, each at three different

heights, were analyzed to check for a height effect. Although small differences were

noticed (maximum difference of 5%), no appreciable height effect was observed. The

results for Model 1 (a/t=0.2, a/c=0.2) can be seen in Table 4.6. The results for

Models 4 (a/t=0.5, a/c=0.2), 5 (a/t=0.2, a/c=0.6), and 9 (a/t=0.8, a/c=1.0) may

be found in Appendix D.

94

Table 4.6 Model 1 (a/t=0.2 and a/c=0.2): h1 values at different heights

θ h/w = 1 h/w = 3 h/w = 50.00 0.51 0.51 0.513.04 0.20 0.20 0.206.10 0.28 0.28 0.289.00 0.34 0.33 0.3311.93 0.39 0.39 0.3814.60 0.44 0.44 0.4417.30 0.49 0.48 0.4819.73 0.54 0.53 0.5322.19 0.58 0.58 0.5824.25 0.61 0.61 0.6126.35 0.67 0.66 0.6628.23 0.69 0.68 0.6830.15 0.75 0.74 0.7431.88 0.76 0.76 0.7633.64 0.84 0.83 0.8335.24 0.83 0.82 0.8236.87 0.89 0.88 0.8841.60 1.00 0.99 0.9946.71 1.08 1.07 1.0750.74 1.16 1.14 1.1555.15 1.23 1.21 1.2158.76 1.28 1.26 1.2762.80 1.33 1.31 1.3166.12 1.37 1.35 1.3569.95 1.40 1.39 1.3973.03 1.43 1.41 1.4176.79 1.45 1.43 1.4479.57 1.47 1.45 1.4583.44 1.48 1.46 1.4685.36 1.49 1.47 1.4790.00 1.51 1.49 1.49

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4.5.2 Width Effects

The width effect results were varied. As expected, it was found that h1 in-

creased as c/w increased. However, a large amount of variation between the Nasgro

and FEM results was observed. These results are presented in Tables 4.7 - 4.9, and

graphically in Figures 4.58 - 4.69. Notice that Model 4a for a strain hardening value

of n = 15 does not have results posted. This FEM would not run in ABAQUS, but

the reason is not known.

Table 4.7 Comparison of Nasgro and FEM results for n = 15

Nasgro Nasgro FEM FEMModel a/t a/c c/w h1(a) h1(c) h1(a) h1(c) % diff(a) % diff(c)

1 0.2 0.2 0.25 1.80 0.48 1.58 0.35 -13.47 -36.631a 0.2 0.2 0.50 3.31 0.90 2.00 0.44 -65.65 -102.721b 0.2 0.2 0.67 5.13 1.42 2.43 0.56 -111.14 -154.313 0.2 1.0 0.25 0.65 0.76 0.68 0.64 4.40 -18.813a 0.2 1.0 0.50 1.26 1.52 1.55 1.52 18.49 0.263b 0.2 1.0 0.67 2.06 2.71 2.52 2.78 18.40 2.474 0.5 0.2 0.25 18.08 5.01 24.83 8.64 27.18 42.034a 0.5 0.2 0.33 31.01 8.75 - - - -4b 0.5 0.2 0.40 48.59 13.96 44.25 15.85 -9.81 11.976 0.5 1.0 0.25 4.25 5.54 4.58 5.05 7.21 -9.646a 0.5 1.0 0.33 7.33 9.81 6.32 7.10 -15.95 -38.106b 0.5 1.0 0.40 11.56 15.89 9.38 10.55 -23.28 -50.60

96



1 0.2 0.2 0.25 1.47 0.39 1.37 0.31 -7.28 -26.711a 0.2 0.2 0.5 2.2 0.6 1.6 0.36 -37.65 -66.191b 0.2 0.2 0.67 2.97 0.83 1.85 0.44 -60.76 -88.23 0.2 1.0 0.25 0.53 0.62 0.59 0.56 9.62 -11.483a 0.2 1.0 0.5 0.84 1.01 1.05 1.08 20.31 6.873b 0.2 1.0 0.67 1.18 1.56 1.6 1.84 26.03 15.184 0.5 0.2 0.25 10.8 2.99 15.66 4.88 31.07 38.74a 0.5 0.2 0.33 15.13 4.27 18.77 5.92 19.39 27.794b 0.5 0.2 0.4 20.68 5.94 22.59 7.12 8.46 16.586 0.5 1.0 0.25 2.53 3.3 2.99 3.38 15.32 2.396a 0.5 1.0 0.33 3.56 4.77 3.69 4.29 3.42 -11.166b 0.5 1.0 0.4 4.94 6.76 4.79 5.59 -3.18 -20.99



1 0.2 0.2 0.25 1.2 0.32 1.14 0.27 -5.59 -18.951a 0.2 0.2 0.5 1.46 0.4 1.2 0.28 -21.85 -41.751b 0.2 0.2 0.67 1.7 0.47 1.3 0.31 -30.56 -52.043 0.2 1.0 0.25 0.44 0.51 0.48 0.49 9.3 -4.033a 0.2 1.0 0.5 0.56 0.67 0.65 0.7 14.27 4.583b 0.2 1.0 0.67 0.68 0.9 0.85 1.04 20.04 13.84 0.5 0.2 0.25 6.45 1.79 8.86 2.32 27.25 22.974a 0.5 0.2 0.33 7.56 2.13 9.34 2.45 19.07 12.954b 0.5 0.2 0.4 8.79 2.53 10.19 2.68 13.75 5.776 0.5 1.0 0.25 1.51 1.97 1.73 2.06 12.62 4.436a 0.5 1.0 0.33 1.79 2.38 1.88 2.32 4.94 -2.796b 0.5 1.0 0.4 2.1 2.88 2.15 2.71 2.43 -6.38

97

Figure 4.58 Model 1 (a/t = 0.2, a/c = 0.2): h1 vs. c/w for Nasgro and FEM atn = 15


98



99



100



101



102



103

The differences between the current FEM and Nasgro results were varied.

Some geometries produced relatively close comparisons (less than 5%) between the

Nasgro and FEM results. Others demonstrated significant differences (greater than

100%). The differences are most likely a result of the reference stress algorithm used

within Nasgro. This is not to say that the Nasgro results are less correct than the

current FEM results, and no final judgement is made here.

It was also found that no two comparisons of FEM and Nasgro results demon-

strated exactly the same pattern. The specific observations are:

• Model 1 (a/t = 0.2, a/c = 0.2): The Nasgro results were higher for every

tested c/w at all n’s. The difference between the Nasgro and FEM results

increased with c/w for all n’s.

• Model 3 (a/t = 0.2, a/c = 1.0): The FEM results were higher for every


increased with c/w for all n’s.



increased with c/w for n = 10. The difference between the Nasgro and

FEM results decreased as c/w increased for n = 15 and n = 5.


tested c/w at n = 5. The FEM and Nasgro results cross for n = 15 and

n = 10. The difference between the Nasgro and FEM results increased

with c/w for n = 15. The difference between the Nasgro and FEM results

decreased as c/w increased for n = 10 and n = 5.

CHAPTER 5

CONCLUSIONS AND RECOMMENDATIONS

This chapter contains the conclusions and recommendations for this research.

Conclusions are stated for each of the stated goals of this work. The recommendations

for further work address questions that were not answered during this study.

5.1 Conclusions

The following are conclusions reached in this research for the primary goal

of comparing three-dimensional finite element analysis J-integral results produced in

ABAQUS with work completed by other researchers:

1. The K-factor and elastic J-integral results produced in ABAQUS for this

research compare well with the Newman and Raju [18] calculations and Lei

data, except as noted below.

• There is a spike in the K-factor at the surface for most analyzed

geometries. Models with a very deep crack (a/t = 0.8) have smaller

spikes. The elastic J-integral results presented by Lei [17] showed some

surface spikes, but not to same magnitude as found in this research.

• There is a sudden increase in the K-factor in the depth for geome-

tries that do not have highly elliptical (a/t = 0.2) cracks for quarter

symmetry models. There were no spikes observed for the depth in the

results presented by Lei.

2. The fully plastic J-integral results generated in this research compare rea-

sonably well, for some geometries, with work produced McClung et al. [15]

104

105

and Lei [17]. However, there are some issues with using ABAQUS to cal-

culate the J-integral.

• The same surface and depth issues found in the current elastic FEMs

are also present in the fully plastic models.

• The plastic FEMs from the current research, Lei, and McClung et

al. exhibit an anomaly at the third or fourth angle from the surface.

The value of the angle was dependent on the model and ranged from

2.71◦ to 18◦.

3. The quarter symmetry FEA results presented by McClung et al. and Lei

did not contain surface or depth spikes in the plastic J-integral, as found

in the current FEMs. Possible reasons for this finding are given below.

• Other researchers used a smoothing method.

• There was an issue with the quarter symmetry FEMs used in this

research, probably related to the boundary conditions.

4. FEM and Nasgro results compared well for some, but not all, specimen

geometries. There was no apparent pattern to the variation.

The second goal of this research was to investigate the effects of mesh density,

element type, symmetry, and specimen size effects on the J-integral. Based on this

research, the conclusions are made.

1. The schemes used by ABAQUS to calculate K and J are robust and do

not require a significant amount of mesh refinement to produce converged

results.

2. A refined mesh can result in oscillations of the calculated J-integral, espe-

cially at larger n values.

106

3. Reduced integration elements give more consistent results for K-factor and

J-integral calculations in ABAQUS. Full integration elements tend to os-

cillate.

4. Quarter symmetry models experience a spike in the depth. Half symmetry

models do not have the same issue.

5. Specimen height has no significant effect on the J-integral for uniform ten-

sile loading.

6. Specimen width has a very significant effect on the J-integral for uniform

tensile loading.

The third goal of this research was to compare elastic-plastic and fully plastic

results. For the elastic-plastic FEAs, the stress vs. plastic strain table was based

on the Ramberg-Osgood material properties. The elastic-plastic and fully plastic

results compared very well at sufficiently large stresses. The same surface and depth

phenomenon observed in the fully plastic FEMs were also present in the elastic-plastic

results.

5.2 Recommendations

The following recommendations for continuing this research are given.

1. Investigate the surface and depth phenomenon within ABAQUS to answer

these questions:

• Why does ABAQUS yield a spike in the K-factor and J-integral at

the free surface?

• How far into the depth does the surface phenomenon affect results?

• Why do quarter symmetry models have an increase in the depth?

107

2. Investigate the anomaly exhibited by some models at the third and fourth

angles from the surface.

3. Use the existing models to obtain results for corner cracks. This work can

be easily performed by changing the boundary conditions of the existing

models.

REFERENCES

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109

[1] Griffith, A.A., “The Phenomena of Rupture and Flow in Solids.” PhilosophicalTransactions, Series A, Vol. 221, 1920, pp. 163-198.

[2] Irwin, G.R., “Fracture Dynamics.” Fracturing of Metals, American Society forMetals, Cleveland, 1948, pp. 147-166.

[3] Irwin, G.R., “Analysis of Stresses and Strains near the End of a Crack Travers-ing a Plate.” Journal of Applied Mechanics, Vol. 24, 1957, pp. 361-364.

[4] Anderson, T. L., Fracture Mechanics Fundamentals and Applications, SecondEdition, CRC Press, FL, 1991.

[5] Tada, H, P.C. Paris, and G.R. Irwin, The Stress Analysis of Cracks Handbook,ASME Press, New York, 2000.

[6] Wells, A.A., “Unstable Crack Propagation in Metals: Cleavage and Fast Frac-ture.” Proceedings of the Crack Propagation Symposium, Vol. 1, Paper 84,Cranfield UK, 1961.

[7] Rice, J.R. “A Path Independent Integral and the Approximate Analysis ofStrain Concentraion by Notches and Cracks.” Journal of Applied Mechanics,Vol. 35, 1968, pp. 379-386.

[8] Hutchinson, J.W., “Singular Behavior at the End of a Tensile Crack Tip in aHardening Material.” Journal of the Mechanics and Physics of Solids, Vol. 16,1968, pp.13-31.

[9] Rice, J.R. and G.F. Rosengren, “Plane Strain Deformation Near a Crack Tipin a Power-Law Hardening Material.” Journal of the Mechanics and Physics ofSolids, Vol. 16, 1968, pp. 1-12.

[10] Cook, R.D., D.S. Malkus, M.E. Plesha, R.J. Witt, Concepts and Applicationsof Finite Element Analysis, Fourth Edition, John Wiley & Sons, Inc., 2002.

[11] Kumar, V., M.D. German, and C.F. Shih, “An Engineering Approach forElastic-Plastic Fracture Analysis,” Report NP-1931, Electric Power ResearchInstitute, 1981.

[12] Shih, C.F., “J-Integral Estimates for Strain Hardening Materials in AntiplaneShear Using Fully Plastic Solutions,” in Mechanics of Crack Growth, ASTMSpecial Publication 590, 1976, pp. 3-22.

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[13] Shih, C.F., and J.W. Hutchinson, “Fully Plastic Solutions and Large-ScaleYielding Estimates for Plane Stress Crack Problems,” Transactions of ASME,Journal of Engineering Materials and Technology, Series H, Vol.98, October1976, pp. 289-295.

[14] Needleman, A., and C.F. Shih, “A Finite Element Method for Plane Strain De-formations of Incompressible Solids.” Computer Methods in Applied Mechanicsand Engineering, 1978, pp. 223-240.

[15] McClung, R.C., G.G. Chell, Y.D. Lee, D.A. Russell, and G.E. Orient, 1999,“Developement of a Practical Methodology for Elastic-Plastic and Fully Plas-tic Fatigue Crack Growth,” National Aeronautics and Space Administration,Marshall Space Flight Center, Alabama, Contract NAS8-37828.

[16] ABAQUS Analysis User’s Manual Version 6.4, ABAQUS, Inc., 2003.

[17] Lei, Y., “J-integral and limit load analysis of semi-elliptical surface cracks inplates under tension.” International Journal of Pressure Vessels and Piping,2004, pp. 21-31.

[18] Newman, J.C. and Raju, I.S., “An empirical stress-intensity factor equation forthe surface crack.” Engineering Fracture Mechanics, 1981, pp. 185-192.

[19] Ainsworth, R.A., “The Assessment of Defects in Structures of Strain HardeningMaterials.” Engineering Fracture Mechanics, Vol. 19, 1984, p.633.

[20] Faleskog, Jonas, User’s Manual: mesh3d scp, 1996.

[21] ABAQUS User’s Manual, Version ???.

[22] Natarajan, Krishna R., Plastic J-Integral Solutions for Through Cracks Us-ing Finite Element Method, Master of Science Thesis, Tennessee TechnologicalUniversity, Cookeville, TN.

[23] Kirk, M.T. and R.H. Dodds, “Approximate Techniques of J Estimation Appli-cable to Part-Though Surface Cracks,” Engineering Fracture Mechanics, Vol.43, No. 1, 1992, pp. 123-136.

111

[24] Newman, J.C. and Raju, I.S., “Stress Intensity Factor Equations for Cracks inThree-Dimensional Finite Bodies”, Fracture Mechanics: Fourteenth Symposium- Volume 1: Theory and Analysis, ASTM STP 791, J. C. Lewis and G. Sines,Eds.,American Society for Testing and Materials, 1983, pp. I-238 - I-265.

[25] Newman, J.C. and Raju, I.S., “Analysis of Surface Cracks in Finite PlatesUnder Tension of Bending Loads,” NASA Technical Paper 1578, Dec. 1979.

[26] “3D Finite Element Software for Cracks, Version 2.5, Benchmarks and Valida-tion,” Structural Reliability Technology, Boulder, CO, July, 2003.

[27] Computational Fracture Mechanics Group at University of Illinois Urbana-Champaign, http://cern49.ce.uiuc.edu/cfm/warp3d.html, accessed 1-31-2005.

[28] Lloyd, W.R., and W.G. Reuter, “Evaluation of Elastic-Plastic Surface FlawBehavior and Related Parameters Using Surface Displacement Measurements,”Fracture Mechanics: Twenty-First Symposium, ASTM STP 1074, J.P. Gudas,J.A. Joyce, and E.M. Hackett, Eds., American Society for Testing and Materi-als, Philadelphia, 1990, pp. 322-336.

[29] Irwin, G.R., “Plastic Zone Near a Crack and Fracture Toughness,” SagamoreResearch Conference Proceedings, Vol. 4, 1961.

[30] Dodds, R. H. Jr. and David T. Read, “Experimental and Numerical Studies ofthe J-Integral for a Surface Flaw,” International Journal of Fracture, Vol. 43,1990, pp. 47-67.

[31] Read,D.T., “Applied J-Integral in HY-130 Tensile Panels and Implications forFitness for Service Assessment,” Report NBSIR 82-1670, National Bureau ofStandards, Boulder, CO, 1982.

[32] Yagawa, G., Y. Kitajima, H. Ueda, “Three-Dimensional Fully Plastic Solutionsfor Semi-elliptical Surface Cracks,” International Journal of Pressure Vesselsand Piping, Vol. 53, 1993, pp. 457-510.

APPENDICES

112

APPENDIX A: INSTRUCTIONS FOR MESH3D SCPMODIFICATIONS

113

114

1. Insure that the correct element type (8, 20, or 27-node) is specified in themesh3d scp input file. This is done by changing the “ETYP=??” field tothe appropriate value.

2. The mesh3d scp program creates several files when run. Three of thesecontain node and element data. The file names are described below:• test.015 - contains node and coordinate data• test.016 - contains element connectivities• test.017 - contains element connectivities

Use the following steps to modify these files for use with ABAQUS 6.5.(a) Combine test.016 and test.017 using cat test.016 test.017 > temp ele.inp.(b) At the top of temp ele.inp, add *ELEMENT, TYPE=C3D20R,ELSET=

ELEMALL.(c) Run abaqus free job=elements input=temp ele

3. Change the node file name from test.015 to nodes.inp.4. Insert the *NODE at the top of the nodes.inp file.5. Make the following changes to the test.inp file (input file for ABAQUS):

• There will be three lines referencing test.015 thru test.017. Deletethese lines.

• Add *INCLUDE, INPUT=nodes.inp• Add *INCLUDE, INPUT=elements.inp• At the bottom of the file, add comma’s to the spaces between the

contours in the *CONTOUR INTEGRAL command.• Many, but not all, of the commands between *MATERIAL and

*****STEP=1****** are not valid and should be replaced withthe appropriate commands.

APPENDIX B: COARSE VERSUS REFINED MESHES FORK-FACTORS

115

116

Figure B.1 Model 1 (a/t=0.2, a/c=0.2): Normalized K factor vs. angle alongcrack front


117



118



119



APPENDIX C: COARSE VS. REFINED MESHES FOR FULLYPLASTIC MODELS

120

121

Figure C.9 Model 1: h1 vs. angle along the crack front


122



123



124



125



126



127



128



129



130



131



132



APPENDIX D: HEIGHT EFFECTS

133

134

Table D.1 Model 4 (a/t=0.5 and a/c=0.2): h1 values for at different heights (Part1)

θ h/w = 1 h/w = 3 h/w = 50.00 33.58 33.42 33.493.04 16.67 16.65 16.686.10 23.62 23.62 23.679.00 27.80 27.79 27.8611.93 30.36 30.33 30.3914.60 34.61 34.55 34.6217.30 36.76 36.65 36.7319.73 40.42 40.27 40.3522.19 42.86 42.66 42.7424.25 45.91 45.64 45.7326.35 48.64 48.32 48.4028.23 51.13 50.77 50.8530.15 54.08 53.66 53.7531.88 55.60 55.12 55.2033.64 57.94 57.39 57.4835.24 59.64 59.05 59.1236.87 61.04 60.39 60.4639.40 63.83 63.09 63.1842.02 66.02 65.22 65.2844.32 67.30 66.41 66.4846.71 69.30 68.37 68.4548.83 69.82 68.81 68.8851.06 71.45 70.41 70.4853.05 71.79 70.69 70.7655.15 72.77 71.62 71.7057.05 73.12 71.93 71.9959.05 73.77 72.54 72.6260.87 74.00 72.76 72.8162.80 74.68 73.41 73.4964.55 74.51 73.18 73.2466.42 75.40 74.06 74.1368.12 74.83 73.51 73.5769.95 75.53 74.15 74.21

135

Table D.2 Model 4 (a/t=0.5 and a/c=0.2): h1 values for at different heights (Part2)

θ h/w = 1 h/w = 3 h/w = 571.60 75.13 73.76 73.8273.40 75.50 74.12 74.1875.00 75.22 73.83 73.9076.79 75.57 74.16 74.2178.34 75.26 73.85 73.9180.13 75.62 74.20 74.2681.62 75.28 73.84 73.9083.44 75.63 74.19 74.2684.81 75.33 73.88 73.9486.72 75.65 74.19 74.2587.68 75.33 73.87 73.9490.00 76.26 74.78 74.84

Table D.3 Model 5 (a/t=0.5 and a/c=0.6): h1 values for at different heights

θ h/w = 1 h/w = 3 h/w = 50.00 4.74 4.58 4.585.60 2.97 2.89 2.8811.25 4.43 4.32 4.3216.79 4.58 4.46 4.4622.50 5.10 4.97 4.9627.98 5.34 5.19 5.1833.75 5.62 5.45 5.4439.15 5.73 5.55 5.5445.00 5.85 5.67 5.6650.29 5.91 5.71 5.7056.25 6.00 5.79 5.7861.36 5.95 5.74 5.7367.50 6.02 5.79 5.7872.24 5.94 5.71 5.7078.75 5.97 5.74 5.7382.05 5.90 5.67 5.6690.00 6.51 6.26 6.25

136

Table D.4 Model 9 (a/t=0.8 and a/c=1.0): h1 values for at different heights

θ h/w = 1 h/w = 3 h/w = 50.00 7.27 6.95 6.945.60 9.35 8.99 8.9711.25 12.88 12.39 12.3716.79 12.19 11.81 11.7922.50 13.33 12.88 12.8627.98 12.87 12.42 12.4033.75 12.50 12.06 12.0439.15 11.76 11.33 11.3145.00 10.99 10.55 10.5350.29 10.37 10.01 9.9956.25 9.96 9.58 9.5661.36 9.53 9.16 9.1467.50 9.27 8.88 8.8672.24 9.27 8.83 8.8178.75 9.35 8.95 8.9282.05 9.22 8.78 8.7690.00 10.12 9.71 9.68

APPENDIX E: K-FACTOR RESULTS FOR COARSE MESHES

137

138

Tab

leE

.5M

odel

1(a

/t=

0.2,

a/c=

0.2)

:K

-Fac

tor

dat

afr

omA

BA

QU

S(P

art

1)

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

FE

AK

Dir

ectl

y%

diff

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

Tie

dN

odes

FE

AD

irec

tfr

omJ

elT

ied

Nodes

0.00

0.59

0.85

0.81

0.83

44.1

337

.15

40.9

33.

040.

600.

590.

590.

59-1

.16

-1.5

2-0

.74

6.10

0.62

0.58

0.59

0.56

-5.6

3-4

.86

-8.5

09.

000.

640.

610.

620.

62-4

.64

-4.5

0-4

.24

11.9

30.

680.

640.

640.

63-5

.64

-5.3

8-7

.29

14.6

00.

710.

670.

670.

68-5

.35

-5.2

2-5

.17

17.3

00.

750.

710.

710.

70-5

.30

-5.1

8-6

.30

19.7

30.

780.

740.

740.

74-5

.15

-5.0

5-5

.13

22.1

90.

800.

770.

770.

76-4

.83

-4.7

5-5

.62

24.2

50.

830.

790.

790.

79-4

.95

-4.8

6-4

.93

26.3

50.

850.

810.

810.

81-4

.49

-4.4

0-5

.10

28.2

30.

870.

830.

830.

83-4

.58

-4.4

9-4

.55

30.1

50.

890.

850.

850.

85-4

.09

-4.0

0-4

.65

31.8

80.

910.

870.

870.

87-4

.31

-4.2

3-4

.25

33.6

40.

920.

890.

890.

89-3

.43

-3.3

4-4

.10

35.2

40.

940.

900.

900.

90-4

.42

-4.3

2-4

.34

36.8

70.

950.

920.

930.

92-2

.95

-2.9

0-3

.26

41.6

00.

990.

970.

970.

97-2

.33

-2.2

0-2

.35

139

Tab

leE

.6M

odel

1(a

/t=

0.2,

a/c=

0.2)

:K

-Fac

tor

dat

afr

omA

BA

QU

S(P

art

2)

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

FE

AK

Dir

ectl

y%

diff

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

Tie

dN

odes

FE

AD

irec

tfr

omJ

elT

ied

Nodes

46.7

11.

031.

011.

011.

00-2

.44

-2.3

8-2

.60

50.7

41.

061.

041.

041.

04-2

.01

-1.9

1-2

.07

55.1

51.

081.

061.

071.

06-1

.85

-1.7

9-2

.05

58.7

61.

101.

091.

091.

08-1

.74

-1.6

6-1

.81

62.8

01.

121.

111.

111.

10-1

.63

-1.5

6-1

.82

66.1

21.

141.

121.

121.

12-1

.59

-1.5

2-1

.65

69.9

51.

151.

141.

141.

13-1

.51

-1.4

5-1

.70

73.0

31.

161.

151.

151.

14-1

.52

-1.4

5-1

.59

76.7

91.

171.

161.

161.

15-1

.48

-1.4

0-1

.63

79.5

71.

181.

161.

161.

16-1

.49

-1.4

2-1

.55

83.4

41.

181.

171.

171.

16-1

.46

-1.3

8-1

.61

85.3

61.

191.

171.

171.

17-1

.48

-1.4

1-1

.55

90.0

01.

191.

181.

181.

18-0

.90

-0.8

8-1

.05

140

Tab

leE

.7M

odel

2(a

/t=

0.2,

a/c=

0.6)

:K

-Fac

tor

dat

afr

omA

BA

QU

S

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

FE

AD

irec

tfr

omJ

el

0.00

0.96

1.25

1.21

30.5

226

.62

5.60

0.94

0.94

0.94

0.00

-0.3

011

.25

0.94

0.92

0.93

-1.7

4-1

.14

16.7

90.

940.

930.

93-1

.70

-1.5

922

.50

0.95

0.93

0.93

-2.2

5-1

.88

27.9

80.

960.

940.

95-1

.89

-1.7

633

.75

0.98

0.96

0.96

-2.0

4-1

.74

39.1

51.

000.

980.

98-1

.87

-1.7

445

.00

1.02

1.00

1.00

-1.9

1-1

.69

50.2

91.

031.

011.

01-1

.90

-1.7

756

.25

1.05

1.03

1.03

-1.9

4-1

.75

61.3

61.

071.

041.

05-2

.02

-1.9

067

.50

1.08

1.06

1.06

-2.0

6-1

.90

72.2

41.

091.

071.

07-2

.17

-2.0

678

.75

1.10

1.08

1.08

-2.2

1-2

.05

82.0

51.

101.

081.

08-2

.28

-2.1

890

.00

1.11

1.16

1.14

4.55

3.44

141

Tab

leE

.8M

odel

3(a

/t=

0.2,

a/c=

1.0)

:K

-Fac

tor

dat

afr

omA

BA

QU

S

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

FE

AD

irec

tfr

omJ

el

0.00

1.18

1.44

1.41

22.0

819

.76

5.60

1.15

1.19

1.19

3.11

2.88

11.2

51.

131.

161.

172.

402.

8116

.79

1.12

1.14

1.14

1.62

1.70

22.5

01.

101.

121.

121.

201.

5327

.98

1.09

1.10

1.10

1.22

1.33

33.7

51.

081.

091.

090.

981.

3139

.30

1.07

1.08

1.08

1.05

1.19

44.9

21.

071.

081.

080.

851.

1650

.54

1.06

1.07

1.07

0.83

0.99

56.1

51.

061.

071.

070.

650.

9461

.77

1.06

1.06

1.07

0.55

0.72

67.3

91.

061.

061.

060.

390.

6773

.00

1.06

1.06

1.06

0.25

0.44

78.6

21.

061.

061.

060.

170.

4484

.24

1.06

1.06

1.06

0.13

0.28

90.0

01.

061.

281.

2321

.22

16.2

3

142

Tab

leE

.9M

odel

4(a

/t=

0.5,

a/c=

0.2)

:K

-Fac

tor

dat

afr

omA

BA

QU

S(P

art

1)

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

FE

AD

irec

tfr

omJ

el

0.00

0.82

1.17

1.10

42.5

235

.03

3.04

0.82

0.80

0.80

-2.2

4-2

.56

6.10

0.84

0.78

0.79

-6.4

0-5

.57

9.00

0.87

0.83

0.83

-5.0

7-4

.91

11.9

30.

910.

860.

86-5

.70

-5.3

714

.60

0.95

0.90

0.90

-5.0

9-4

.95

17.3

00.

990.

940.

94-4

.74

-4.5

619

.73

1.02

0.98

0.98

-4.3

4-4

.21

22.1

91.

051.

011.

02-3

.78

-3.6

724

.25

1.08

1.04

1.04

-3.6

3-3

.51

26.3

51.

111.

081.

08-2

.96

-2.8

728

.23

1.13

1.10

1.10

-2.7

7-2

.67

30.1

51.

151.

131.

13-2

.19

-2.1

031

.88

1.17

1.15

1.15

-2.0

2-1

.92

33.6

41.

191.

181.

18-1

.36

-1.2

735

.24

1.21

1.19

1.19

-1.3

8-1

.28

36.8

71.

231.

221.

22-0

.79

-0.7

339

.40

1.25

1.25

1.25

-0.2

3-0

.12

42.0

21.

281.

281.

280.

020.

1144

.32

1.30

1.30

1.31

0.53

0.62

46.7

11.

321.

331.

330.

870.

9548

.83

1.34

1.35

1.35

1.18

1.25

51.0

61.

351.

371.

371.

491.

56

143

Tab

leE

.10

Model

4(a

/t=

0.5,

a/c=

0.2)

:K

-Fac

tor

dat

afr

omA

BA

QU

S(P

art

2)

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

FE

AD

irec

tfr

omJ

el

53.0

51.

371.

391.

391.

721.

7955

.15

1.38

1.41

1.41

1.97

2.03

57.0

51.

401.

431.

432.

152.

2159

.05

1.41

1.44

1.44

2.35

2.41

60.8

71.

421.

461.

462.

482.

5462

.80

1.43

1.47

1.47

2.66

2.71

64.5

51.

441.

481.

482.

762.

8266

.42

1.45

1.49

1.49

2.88

2.92

68.1

21.

461.

501.

502.

932.

9969

.95

1.47

1.51

1.51

3.03

3.08

71.6

01.

471.

521.

523.

103.

1573

.40

1.48

1.53

1.53

3.17

3.22

75.0

01.

491.

531.

533.

213.

2676

.79

1.49

1.54

1.54

3.26

3.31

78.3

41.

491.

541.

543.

293.

3380

.13

1.50

1.55

1.55

3.33

3.37

81.6

21.

501.

551.

553.

343.

3983

.44

1.50

1.56

1.56

3.37

3.41

84.8

11.

511.

561.

563.

383.

4286

.72

1.51

1.56

1.56

3.39

3.43

87.6

81.

511.

561.

563.

393.

4390

.00

1.51

1.57

1.57

3.76

3.79

144

Tab

leE

.11

Model

5(a

/t=

0.5,

a/c=

0.6)

:K

-Fac

tor

dat

afr

omA

BA

QU

S

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

FE

AD

irec

tfr

omJ

el

0.00

1.13

1.40

1.37

24.1

421

.54

5.60

1.10

1.12

1.12

2.15

1.82

11.2

51.

081.

091.

090.

481.

0716

.79

1.08

1.08

1.08

0.55

0.65

22.5

01.

081.

081.

080.

130.

4527

.98

1.08

1.09

1.09

0.40

0.49

33.7

51.

101.

101.

100.

280.

5039

.15

1.11

1.11

1.11

0.24

0.32

45.0

01.

131.

131.

130.

050.

2250

.29

1.15

1.14

1.14

-0.2

3-0

.16

56.2

51.

161.

161.

16-0

.52

-0.3

961

.36

1.18

1.17

1.17

-0.9

2-0

.86

67.5

01.

191.

181.

18-1

.24

-1.1

272

.24

1.20

1.19

1.19

-1.6

0-1

.56

78.7

51.

221.

191.

19-1

.83

-1.7

282

.05

1.22

1.19

1.20

-2.0

1-1

.96

90.0

01.

221.

261.

252.

651.

97

145

Tab

leE

.12

Model

6(a

/t=

0.5,

a/c=

1.0)

:K

-Fac

tor

dat

afr

omA

BA

QU

S

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

FE

AD

irec

tfr

omJ

el

0.00

1.31

1.52

1.50

16.0

914

.64

5.60

1.27

1.32

1.31

3.35

3.10

11.2

51.

241.

281.

283.

063.

4416

.79

1.21

1.24

1.24

2.53

2.63

22.5

01.

181.

211.

222.

482.

7227

.98

1.16

1.19

1.19

2.54

2.64

33.7

51.

151.

171.

182.

482.

7439

.15

1.13

1.16

1.16

2.44

2.55

45.0

01.

121.

151.

152.

242.

4950

.29

1.12

1.14

1.14

2.02

2.14

56.2

51.

111.

131.

131.

701.

9461

.36

1.11

1.12

1.12

1.36

1.49

67.5

01.

111.

121.

121.

021.

2672

.24

1.11

1.11

1.11

0.72

0.84

78.7

51.

111.

111.

110.

490.

7182

.05

1.11

1.11

1.11

0.36

0.45

90.0

01.

111.

271.

2415

.10

11.9

5

146

Tab

leE

.13

Model

7(a

/t=

0.8,

a/c=

0.2)

:K

-Fac

tor

dat

afr

omA

BA

QU

S(P

art

1)

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

FE

AD

irec

tfr

omJ

el

0.00

1.19

1.49

1.46

25.7

523

.29

2.44

1.17

1.23

1.22

4.52

4.07

4.88

1.18

1.19

1.20

0.62

1.29

7.20

1.21

1.22

1.22

0.90

0.98

9.53

1.25

1.24

1.25

-0.1

90.

0011

.64

1.28

1.28

1.28

-0.0

8-0

.03

13.7

71.

321.

321.

32-0

.06

0.01

15.6

81.

351.

351.

350.

010.

0317

.61

1.39

1.39

1.39

0.36

0.39

19.2

21.

411.

421.

420.

320.

3220

.84

1.44

1.45

1.45

0.94

0.95

22.3

11.

461.

481.

481.

021.

0023

.79

1.49

1.51

1.51

1.57

1.55

25.1

21.

511.

531.

531.

621.

6226

.48

1.53

1.56

1.56

2.26

2.26

27.7

11.

541.

581.

582.

212.

2028

.96

1.56

1.60

1.60

2.76

2.73

30.6

91.

581.

631.

633.

153.

1432

.46

1.61

1.66

1.66

3.30

3.27

34.0

51.

631.

691.

693.

663.

6435

.66

1.65

1.71

1.71

3.88

3.86

37.1

31.

661.

731.

734.

044.

0138

.62

1.68

1.75

1.75

4.22

4.18

147

Tab

leE

.14

Model

7(a

/t=

0.8,

a/c=

0.2)

:K

-Fac

tor

dat

afr

omA

BA

QU

S(P

art

2)

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

FE

AD

irec

tfr

omJ

el

40.0

01.

701.

771.

774.

284.

2441

.41

1.71

1.79

1.78

4.38

4.34

42.7

21.

721.

801.

804.

374.

3344

.05

1.74

1.81

1.81

4.41

4.37

45.2

91.

751.

831.

834.

334.

3046

.57

1.76

1.84

1.84

4.34

4.30

47.7

61.

781.

851.

854.

224.

1748

.99

1.79

1.86

1.86

4.12

4.07

50.1

41.

801.

871.

873.

923.

8651

.32

1.81

1.88

1.88

3.77

3.72

52.4

31.

821.

881.

883.

613.

5453

.58

1.83

1.89

1.89

3.41

3.36

54.6

61.

841.

901.

903.

183.

1255

.77

1.85

1.90

1.90

2.97

2.91

56.8

31.

861.

911.

902.

722.

6557

.91

1.86

1.91

1.91

2.47

2.41

58.9

41.

871.

911.

912.

192.

1260

.00

1.88

1.92

1.92

1.94

1.86

61.0

11.

891.

921.

921.

651.

5762

.05

1.90

1.92

1.92

1.39

1.30

63.0

31.

901.

921.

921.

091.

0164

.06

1.91

1.92

1.92

0.78

0.70

65.0

21.

921.

931.

920.

470.

3966

.03

1.92

1.93

1.92

0.19

0.10

66.9

81.

931.

931.

92-0

.09

-0.1

967

.98

1.93

1.93

1.93

-0.3

7-0

.48

148

Tab

leE

.15

Model

7(a

/t=

0.8,

a/c=

0.2)

:K

-Fac

tor

dat

afr

omA

BA

QU

S(P

art

3)

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

FE

AD

irec

tfr

omJ

el

68.9

11.

941.

931.

93-0

.68

-0.7

769

.89

1.95

1.93

1.93

-0.9

5-1

.05

70.8

21.

951.

931.

92-1

.22

-1.3

371

.79

1.96

1.93

1.92

-1.4

9-1

.59

72.7

01.

961.

931.

92-1

.75

-1.8

673

.67

1.96

1.93

1.92

-2.0

0-2

.11

74.5

71.

971.

921.

92-2

.25

-2.3

675

.52

1.97

1.92

1.92

-2.4

8-2

.59

76.4

11.

981.

921.

92-2

.70

-2.8

177

.36

1.98

1.92

1.92

-2.9

2-3

.03

78.2

41.

981.

921.

92-3

.13

-3.2

479

.19

1.99

1.92

1.92

-3.3

1-3

.43

80.0

61.

991.

921.

92-3

.47

-3.5

981

.01

1.99

1.92

1.92

-3.6

4-3

.75

81.8

61.

991.

921.

92-3

.78

-3.9

082

.82

1.99

1.92

1.91

-3.9

2-4

.04

83.6

62.

001.

921.

91-4

.03

-4.1

584

.62

2.00

1.92

1.91

-4.1

4-4

.26

85.4

32.

001.

921.

91-4

.22

-4.3

486

.42

2.00

1.91

1.91

-4.3

0-4

.42

87.1

72.

001.

911.

91-4

.34

-4.4

788

.21

2.00

1.91

1.91

-4.4

0-4

.51

88.7

32.

001.

911.

91-4

.41

-4.5

390

.00

2.00

1.92

1.91

-4.2

9-4

.42

149

Tab

leE

.16

Model

8(a

/t=

0.8,

a/c=

0.6)

:K

-Fac

tor

dat

afr

omA

BA

QU

S(P

art

1)

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

FE

AK

Dir

ectl

y%

diff

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

Tie

dN

odes

FE

AD

irec

tfr

omJ

elT

ied

Nodes

0.00

1.40

1.53

1.52

1.52

9.52

8.66

8.74

4.03

1.35

1.42

1.41

1.42

4.60

4.38

4.56

8.09

1.32

1.37

1.38

1.36

3.57

4.40

2.95

11.9

71.

291.

331.

341.

343.

193.

283.

2715

.92

1.27

1.31

1.31

1.30

2.69

3.10

2.07

19.5

51.

261.

301.

301.

302.

682.

782.

7823

.26

1.25

1.29

1.29

1.28

2.50

2.77

1.94

26.6

21.

251.

281.

281.

282.

392.

472.

4730

.09

1.25

1.28

1.28

1.27

2.26

2.43

1.75

33.0

01.

251.

271.

271.

271.

811.

901.

9136

.02

1.25

1.28

1.28

1.27

1.88

1.98

1.40

38.7

21.

261.

271.

281.

281.

331.

411.

4241

.52

1.26

1.28

1.28

1.27

1.23

1.29

0.79

44.0

21.

271.

281.

281.

280.

560.

640.

6446

.62

1.27

1.28

1.28

1.28

0.60

0.62

0.14

48.9

51.

281.

271.

271.

27-0

.45

-0.4

5-0

.48

150

Tab

leE

.17

Model

8(a

/t=

0.8,

a/c=

0.6)

:K

-Fac

tor

dat

afr

omA

BA

QU

S(P

art

2)

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

FE

AK

Dir

ectl

y%

diff

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

Tie

dN

odes

FE

AD

irec

tfr

omJ

elT

ied

Nodes

51.3

91.

291.

281.

281.

28-0

.49

-0.5

0-0

.86

54.4

01.

291.

281.

281.

28-1

.09

-1.0

4-1

.02

57.6

61.

301.

271.

281.

27-2

.20

-2.1

2-2

.41

60.4

61.

311.

281.

281.

28-2

.69

-2.6

6-2

.68

63.5

31.

321.

271.

281.

27-3

.40

-3.3

6-3

.66

66.1

61.

331.

271.

271.

27-4

.04

-4.0

2-4

.03

69.1

11.

331.

271.

271.

27-4

.64

-4.6

1-4

.89

71.6

11.

341.

271.

271.

27-5

.20

-5.1

8-5

.20

74.4

91.

351.

271.

271.

27-5

.67

-5.6

7-5

.92

76.8

51.

351.

271.

271.

27-6

.12

-6.1

1-6

.12

79.7

31.

351.

271.

271.

26-6

.45

-6.4

5-6

.69

81.8

91.

361.

261.

261.

26-6

.75

-6.7

5-6

.75

84.8

91.

361.

261.

261.

26-6

.93

-6.9

3-7

.17

86.3

81.

361.

261.

261.

26-7

.07

-7.0

7-7

.07

90.0

01.

361.

281.

281.

28-5

.74

-5.9

1-5

.97

151

Tab

leE

.18

Model

9(a

/t=

0.8,

a/c=

1.0)

:K

-Fac

tor

dat

afr

omA

BA

QU

S

Angl

eN

ewm

an-R

aju

FE

AK

FE

AK

%diff

%diff

(deg

rees

)D

irec

tly

from

Jel

FE

AD

irec

tfr

omJ

el

0.00

1.54

1.59

1.59

3.54

3.51

5.60

1.47

1.50

1.50

2.12

2.03

11.2

51.

411.

441.

442.

352.

7016

.79

1.35

1.38

1.38

2.15

2.25

22.5

01.

311.

341.

342.

572.

8227

.98

1.27

1.30

1.30

2.65

2.76

33.7

51.

241.

271.

272.

803.

0339

.15

1.21

1.24

1.24

2.46

2.56

45.0

01.

191.

221.

222.

182.

3850

.29

1.18

1.20

1.20

1.45

1.55

56.2

51.

171.

181.

180.

841.

0361

.36

1.17

1.17

1.17

-0.1

1-0

.01

67.5

01.

161.

151.

16-0

.89

-0.7

272

.24

1.16

1.14

1.14

-1.7

6-1

.68

78.7

51.

161.

131.

14-2

.36

-2.2

082

.05

1.16

1.13

1.13

-2.7

5-2

.72

90.0

01.

161.

181.

171.

330.

66

APPENDIX F: FULLY PLASTIC RESULTS FOR COARSE MESHES

152

153

Table F.19 Model 1 (a/t=0.2, a/c=0.2): h1 data from ABAQUS

Angle h1, h1, h1,(degrees) n = 5 n = 10 n = 15

0.00 0.42 0.48 0.543.04 0.17 0.19 0.216.10 0.23 0.26 0.309.00 0.27 0.31 0.3511.93 0.31 0.35 0.4114.60 0.34 0.40 0.4617.30 0.38 0.45 0.5119.73 0.42 0.49 0.5622.19 0.46 0.53 0.6124.25 0.48 0.56 0.6426.35 0.52 0.61 0.7028.23 0.54 0.63 0.7230.15 0.58 0.68 0.7931.88 0.60 0.70 0.8033.64 0.64 0.76 0.8735.24 0.64 0.75 0.8636.87 0.69 0.81 0.9341.60 0.76 0.91 1.0446.71 0.83 0.98 1.1350.74 0.88 1.05 1.2155.15 0.93 1.11 1.2858.76 0.97 1.16 1.3462.80 1.01 1.20 1.3966.12 1.04 1.24 1.4369.95 1.06 1.27 1.4773.03 1.08 1.29 1.4976.79 1.10 1.31 1.5279.57 1.11 1.33 1.5483.44 1.12 1.34 1.5585.36 1.13 1.35 1.5690.00 1.14 1.37 1.58

154



0.00 0.63 0.71 0.805.60 0.32 0.33 0.3611.25 0.42 0.47 0.5316.79 0.43 0.48 0.5422.50 0.47 0.54 0.6127.98 0.49 0.57 0.6533.75 0.52 0.61 0.6939.15 0.54 0.63 0.7345.00 0.57 0.67 0.7650.29 0.59 0.69 0.7956.25 0.61 0.72 0.8361.36 0.63 0.74 0.8567.50 0.65 0.77 0.8872.24 0.66 0.78 0.8978.75 0.67 0.80 0.9282.05 0.68 0.80 0.9290.00 0.77 0.92 1.05

155



0.00 0.56 0.66 0.755.60 0.36 0.40 0.4511.25 0.46 0.56 0.6416.79 0.43 0.51 0.5822.50 0.45 0.56 0.6427.98 0.45 0.56 0.6533.75 0.46 0.57 0.6639.15 0.45 0.57 0.6745.00 0.46 0.57 0.6750.29 0.45 0.57 0.6756.25 0.45 0.58 0.6861.36 0.45 0.57 0.6767.50 0.45 0.58 0.6872.24 0.45 0.58 0.6878.75 0.45 0.58 0.6882.05 0.45 0.58 0.6890.00 0.64 0.82 0.96

156

Table F.22 Model 4 (a/t=0.5, a/c=0.2): h1 data from ABAQUS (Part 1)


0.00 2.97 5.92 10.033.04 1.44 2.95 5.126.10 1.99 4.22 7.369.00 2.25 4.85 8.5611.93 2.52 5.35 9.3314.60 2.86 6.06 10.6017.30 3.12 6.52 11.2619.73 3.43 7.17 12.3622.19 3.72 7.67 13.1424.25 3.96 8.21 14.0826.35 4.25 8.72 14.8528.23 4.46 9.17 15.6430.15 4.74 9.67 16.4131.88 4.92 10.03 17.0033.64 5.20 10.53 17.7135.24 5.33 10.78 18.1936.87 5.57 11.19 18.7339.40 5.89 11.79 19.6742.02 6.15 12.20 20.2444.32 6.42 12.65 20.9246.71 6.66 13.00 21.3448.83 6.86 13.33 21.8251.06 7.07 13.62 22.1753.05 7.24 13.86 22.5055.15 7.41 14.10 22.7957.05 7.54 14.27 23.0159.05 7.70 14.47 23.2760.87 7.80 14.58 23.3962.80 7.93 14.76 23.6564.55 8.01 14.82 23.6666.42 8.11 14.98 23.9068.12 8.17 15.01 23.8969.95 8.26 15.13 24.08

157



71.60 8.30 15.14 24.0373.40 8.38 15.25 24.2275.00 8.41 15.23 24.1476.79 8.47 15.34 24.3278.34 8.48 15.31 24.2280.13 8.53 15.40 24.3881.62 8.54 15.36 24.2783.44 8.58 15.44 24.4284.81 8.57 15.39 24.3086.72 8.60 15.47 24.4587.68 8.59 15.41 24.3290.00 8.67 15.60 24.65



0.00 2.74 4.43 6.735.60 1.76 2.78 4.1611.25 2.38 4.02 6.2116.79 2.37 4.11 6.4022.50 2.57 4.54 7.1527.98 2.66 4.73 7.4933.75 2.79 4.97 7.8739.15 2.85 5.08 8.0445.00 2.93 5.21 8.2050.29 2.96 5.26 8.2756.25 3.02 5.36 8.4161.36 3.02 5.32 8.3467.50 3.05 5.38 8.4372.24 3.03 5.31 8.3178.75 3.05 5.34 8.3582.05 3.03 5.28 8.2590.00 3.33 5.82 9.07

158



0.00 2.10 3.20 4.685.60 1.60 2.40 3.4611.25 2.06 3.39 5.0516.79 1.94 3.15 4.7522.50 2.01 3.41 5.2327.98 1.99 3.41 5.2833.75 1.98 3.44 5.3539.15 1.95 3.42 5.3545.00 1.93 3.40 5.3150.29 1.89 3.33 5.2056.25 1.86 3.29 5.1561.36 1.82 3.19 4.9767.50 1.80 3.15 4.8972.24 1.76 3.06 4.7378.75 1.75 3.03 4.6882.05 1.73 2.98 4.5890.00 2.23 3.88 5.98

159



0.00 8.12 20.06 43.862.44 6.91 18.09 39.684.88 9.73 26.15 58.387.20 10.76 29.41 65.689.53 11.84 32.11 71.7211.64 13.02 34.88 77.3013.77 13.88 36.47 80.2315.68 14.98 39.05 85.2617.61 15.88 40.65 88.0619.22 16.74 43.00 92.7720.84 17.65 44.84 96.4722.31 18.36 46.63 99.8523.79 19.18 48.31 103.0725.12 19.79 49.63 105.6726.48 20.59 50.96 107.4327.71 20.99 51.83 109.2528.96 21.61 52.65 110.3030.69 22.40 54.46 114.0332.46 22.88 55.00 114.2734.05 23.37 55.65 115.5635.66 23.75 56.25 116.4037.13 24.04 56.43 116.4638.62 24.29 56.73 117.2340.00 24.44 56.84 117.1041.41 24.61 56.83 117.5542.72 24.65 56.90 117.4044.05 24.74 56.80 117.9045.29 24.70 56.75 117.3146.57 24.74 56.83 118.4747.76 24.62 56.35 116.9848.99 24.61 56.56 118.5450.14 24.47 56.16 117.4351.32 24.41 55.79 116.2252.43 24.21 55.41 115.6553.58 24.13 55.18 114.8954.66 23.90 54.52 113.50

160



55.77 23.80 54.35 112.9956.83 23.55 53.55 111.1257.91 23.43 53.35 110.8458.94 23.18 52.65 109.2360.00 23.05 52.42 108.8861.01 22.80 51.82 107.8562.05 22.67 51.72 107.5463.03 22.43 51.18 106.8664.06 22.33 51.16 106.7865.02 22.11 50.81 106.3266.03 22.02 50.81 106.2266.98 21.83 50.45 105.5667.98 21.76 50.52 105.7868.91 21.59 50.15 104.8969.89 21.54 50.25 105.2570.82 21.39 49.85 104.2171.79 21.36 49.97 104.6472.70 21.22 49.54 103.5073.67 21.21 49.66 104.0074.57 21.07 49.20 102.8575.52 21.07 49.33 103.2876.41 20.94 48.86 102.3777.36 20.94 48.99 102.5378.24 20.83 48.57 101.9479.19 20.83 48.73 102.4480.06 20.72 48.28 101.4581.01 20.73 48.48 102.1181.86 20.63 48.07 101.2682.82 20.65 48.26 101.6983.66 20.55 47.92 101.1784.62 20.58 48.12 101.4185.43 20.49 47.80 101.0586.42 20.53 48.04 101.2887.17 20.45 47.73 100.9788.21 20.50 47.99 101.2188.73 20.43 47.70 100.9390.00 20.56 48.15 101.57

161



0.00 5.34 10.81 20.904.03 5.76 12.35 24.448.09 7.63 17.11 34.6411.97 7.51 17.14 35.1215.92 7.94 18.52 38.3319.55 8.07 18.96 39.3223.26 8.13 18.94 39.2726.62 8.17 19.24 39.8530.09 8.18 18.99 39.1033.00 8.12 18.97 39.0736.02 8.17 19.03 38.9938.72 8.03 18.67 38.1741.52 8.05 18.63 37.9644.02 7.86 18.09 36.7746.62 7.89 18.07 36.6548.95 7.60 17.33 35.1651.39 7.55 17.24 35.0554.40 7.41 16.97 34.7057.66 7.18 16.39 33.5160.46 7.04 16.14 33.1863.53 6.94 16.11 33.4466.16 6.83 15.92 33.1169.11 6.79 16.07 33.6771.61 6.72 15.91 33.3074.49 6.72 16.15 33.9576.85 6.66 15.95 33.4779.73 6.68 16.14 33.9081.89 6.63 15.95 33.5384.89 6.66 16.05 33.6486.38 6.61 15.95 33.6090.00 6.87 16.55 34.66

162



0.00 4.13 7.75 14.125.60 4.66 9.48 18.0011.25 5.83 12.63 24.6916.79 5.31 11.71 23.3622.50 5.48 12.58 25.4927.98 5.23 12.09 24.5833.75 5.06 11.65 23.7539.15 4.78 11.01 22.3245.00 4.55 10.31 20.7050.29 4.31 9.68 19.3556.25 4.12 9.26 18.5461.36 3.88 8.76 17.6167.50 3.74 8.44 16.9172.24 3.61 8.29 16.7778.75 3.57 8.29 16.8782.05 3.49 8.11 16.5490.00 3.82 8.87 18.12

APPENDIX G: INCREMENTAL PLASTICITY TABLES

163

164

Table G.30 Stress vs. strain data at n = 15, based on Equation 3.13

Stress Elastic Strain Plastic Strain Elastic to Plastic Ratio34000 0.0017 5.82× 10−5 29.1935200 0.00176 9.8E-05 17.9636400 0.00182 0.000162 11.2337600 0.00188 0.000264 7.1338800 0.00194 0.000422 4.6040000 0.002 0.000667 3.0041200 0.00206 0.001039 1.9842400 0.00212 0.001598 1.3343600 0.00218 0.002428 0.9044800 0.00224 0.003649 0.6146000 0.0023 0.005425 0.4247200 0.00236 0.007982 0.3048400 0.00242 0.011633 0.2149600 0.00248 0.016797 0.1550800 0.00254 0.024042 0.1152000 0.0026 0.034124 0.0853200 0.00266 0.048049 0.0654400 0.00272 0.067142 0.0455600 0.00278 0.093139 0.0356800 0.00284 0.128302 0.0258000 0.0029 0.175561 0.0259200 0.00296 0.23869 0.0160400 0.00302 0.322525 0.0161600 0.00308 0.433231 0.0162800 0.00314 0.578633 0.0164000 0.0032 0.768614 0.0065200 0.00326 1.0156 0.00

165


Stress Elastic Strain Plastic Strain Elastic to Plastic Ratio30000 0.0015 3.75× 10−5 39.9548849335200 0.00176 0.000185667 9.4793204639600 0.00198 0.000602921 3.28401024544000 0.0022 0.001729162 1.27229285548400 0.00242 0.004485 0.5395763752800 0.00264 0.010706513 0.24657887957200 0.00286 0.023837962 0.11997670161600 0.00308 0.050016805 0.06157930366000 0.0033 0.099712174 0.03309525770400 0.00352 0.190123333 0.01851429874800 0.00374 0.348597932 0.01072869279200 0.00396 0.617391497 0.00641408383600 0.00418 1.060160459 0.003942799

166


Stress Elastic Strain Plastic Strain Elastic to Plastic Ratio22000 0.0011 3.36× 10−5 32.7826400 0.00132 8.35× 10−5 15.8130000 0.0015 0.000158 9.4835200 0.00176 0.000352 5.0039600 0.00198 0.000634 3.1244000 0.0022 0.001074 2.0548400 0.00242 0.001729 1.4052800 0.00264 0.002672 0.9957200 0.00286 0.003986 0.7261600 0.00308 0.005774 0.5366000 0.0033 0.008153 0.4070400 0.00352 0.011258 0.3174800 0.00374 0.015245 0.2579200 0.00396 0.020288 0.2083600 0.00418 0.026585 0.1688000 0.0044 0.034358 0.1392400 0.00462 0.04385 0.1196800 0.00484 0.055333 0.09101200 0.00506 0.069105 0.07105600 0.00528 0.085493 0.06110000 0.0055 0.104851 0.05114400 0.00572 0.127567 0.04118800 0.00594 0.15406 0.04123200 0.00616 0.184783 0.03127600 0.00638 0.220223 0.03132000 0.0066 0.260903 0.03136400 0.00682 0.307384 0.02140800 0.00704 0.360265 0.02145200 0.00726 0.420186 0.02149600 0.00748 0.487828 0.02154000 0.0077 0.563913 0.01158400 0.00792 0.649209 0.01162800 0.00814 0.744528 0.01167200 0.00836 0.850727 0.01171600 0.00858 0.968713 0.01176000 0.0088 1.099441 0.01

167

Table G.33 Stress vs. plastic strain data at n = 15, used for ABAQUS models

Stress Plastic Strain34000 0.0035200 9.8× 10−5

36400 0.00016237600 0.00026438800 0.00042240000 0.00066741200 0.00103942400 0.00159843600 0.00242844800 0.00364946000 0.00542547200 0.00798248400 0.01163349600 0.01679750800 0.02404252000 0.03412453200 0.04804954400 0.06714255600 0.09313956800 0.12830258000 0.17556159200 0.2386960400 0.32252561600 0.43323162800 0.57863364000 0.76861465200 1.0156

168


Stress Plastic Strain30000 0.0035200 0.00018566739600 0.00060292144000 0.00172916248400 0.00448552800 0.01070651357200 0.02383796261600 0.05001680566000 0.09971217470400 0.19012333374800 0.34859793279200 0.61739149783600 1.060160459

169


Stress Plastic Strain22000 0.0026400 8.35× 10−5

30000 0.00015835200 0.00035239600 0.00063444000 0.00107448400 0.00172952800 0.00267257200 0.00398661600 0.00577466000 0.00815370400 0.01125874800 0.01524579200 0.02028883600 0.02658588000 0.03435892400 0.0438596800 0.055333101200 0.069105105600 0.085493110000 0.104851114400 0.127567118800 0.15406123200 0.184783127600 0.220223132000 0.260903136400 0.307384140800 0.360265145200 0.420186149600 0.487828154000 0.563913158400 0.649209162800 0.744528167200 0.850727171600 0.968713176000 1.099441

VITA

Eric Quillen was born 1968 in Charlottesville, Virginia. He has an Associates

in General Engineering from Virginia Highlands Community College, a Bachelors of

Science in Manufacturing Engineering Technology from East Tennesse State Univer-

sity. He has been pursuing a masters degree while working full time at Fleetguard,

Inc. He is now a candidate for a Master of Science in Mechanical Engineering.

170

an abstract of a thesis - ttu cae networkchriswilson/theses/quillen_ms.pdf · an abstract of a...

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