Astm stp 833 1984

Baseline fatigue crack growth tests [15] conducted with through-thickness edge cracks loaded in four-point bending at a frequency of 2 Hz gave the crack growth equation - $ - = 1.0702 X

PRKTURC mCCHMICS

pmccnTH ivffipoiium

R J n n P O R D editor

(jJJtMTPWS

FRACTURE MECHANICS:

FIFTEENTH SYMPOSIUM

Fifteenth National Symposium

on Fracture Mechanics sponsored by ASTM Committee E-24 on Fracture Testing College Park, Md., 7-9 July 1982

ASTM SPECIAL TECHNICAL PUBLICATION 833

R J Sanford, University of Maryland, editor

ASTM Publication Code Number (PCN) 04-833000-30

1916 Race Street, Piiiladelphia, Pa 19103

National Symposium on Fracture Mechanics (15th : 1982 :

College Park, Md.)

Fracture mechanics

(ASTM special technical publication ; 833)

"ASTM publication code number (PCN 04-833000-30)."

Includes bibliographies and index

1 Fracture mechanics—Congresses I Sanford, R J

II ASTM Committee E-24 on Fracture Testing

III Title IV Series

TA409.N38 1982 620.1'126 83-72816

ISBN 0-8031-0208-9

Copyright © by AMERICAN SOCIETY FOR TESTING AND MATERIALS 1984

Library of Congress Catalog Card Number: 83-72816

NOTE The Society is not responsible, as a body, for the statements and opinions advanced in this publication

Printed in Baltimore, Md, (b) September 1984

Dedication

The dedication of this publication in honor of Dr

George R Irwin on his 75th birthday recognizes his opment of the basic theory of linear elastic fracture me- chanics and its application in solving critical problems of national importance In particular, we honor Dr Irwin's continued counsel and guidance to ASTM Committee E-24 on Fracture Testing

devel-We wish Dr Irwin many years of good health and piness

hap-The 15th National Symposium on Fracture Mechanics was held at the

Uni-versity of Maryland, College Park, on 7-9 July 1982 ASTM Committee E-24

on Fracture Testing was sponsor R J Sanford, University of Maryland,

served as symposium chairman and has edited this publication

Fractography of Ceramic and Metal Failures, STP 827 (1984), 04-827000-30

Elastic-Plastic Fracture: Second Symposium, Volume I—Inelastic Crack

Analysis, STP 803 (1983), 04-803001-30

Elastic-Plastic Fracture: Second Symposium, Volume II: Fracture Resistance

Curves and Engineering Applications, STP 803 (1983), 04-803002-30

Probabilistic Fracture Mechanics and Fatigue Methods: Applications for

Structural Design and Maintenance, STP 798 (1983), 04-798000-30

Fracture Mechanics (Thirteenth Conference), STP 743 (1981), 04-743000-30

Fractography and Materials Science, STP 733 (1981), 04-733000-30

Elastic-Plastic Fracture, STP 688 (1979), 04-688000-30

to Reviewers

The quality of the papers that appear in this publication reflects not only

the obvious efforts of the authors but also the unheralded, though essential,

work of the reviewers On behalf of ASTM we acknowledge with appreciation

their dedication to high professional standards and their sacrifice of time and

effort

ASTM Committee on Publications

Janet R Schroeder Kathleen A Greene Rosemary Horstman Helen M Hoersch Helen P Mahy Allan S Kleinberg Susan L Gebremedhin

Introduction to the Geoi^e R Irwin Anniversary Volume 1

LINEAR ELASTIC FRACTURE MECHANICS

Transition of Part-Through Craclis at Holes into Through-the-Thickness

F l a w s — A F GRANDT, JR., J A BARTER, AND B I HEATH 7

Part-Through Flaw Stress Intensity Factors Developed by a Slice

Analysis and Growth of Cracks in Skins with Variable Thickness—

M M R A T W A N I A N D H P KAN 44

Mode I Stress Intensity Factors for Point-Loaded Cylindrical Test

Specimens with One or Two Radial Cracks—

A p PARKER AND C p ANDRASIC 5 7

Stress and Fracture Analysis of Tapered Attachment Lugs—

K KATHIRESAN, T M HSU, AND J L RUDD 72

An Elastic-Plastic Finite Element Analysis of Crack Initiation, Stable

Stress Intensity Distributions and Width Correction Factors for Natural

Cracks Approaching "Benchmark" Crack Depths—c w SMITH

AND G C KIRBY 118

Dynamic Crack Branching—A Photoelastic Evaluation—M RAMULU,

A S KOBAYASHI, AND B S.-J KANG 130

Recent Advances in Crack-Arrest Technology—A R ROSENFIELD,

p N MINCER, C Vf MARSCHALL, AND A J MARKWORTH 149

A Failure Assessment Approach for Handling Combined

Thermomechanical Loading—j M BLOOM AND S N MALIK 165

FATIGUE CRACK GROWTH

Fatigue Life of Welded Stiffeners with Known Initial Cracks—A SAHLI

AND P ALBRECHT 193

Fatigue Crack Growth Behavior of 7XXX Aluminum Alloys under

Simple Variable Amplitude Loading—p E BRETZ,

A K VASUDEVAN, R J BUCCI, AND R C MALCOLM 2 4 2

Effects of Specimen Configuration and Frequency on Fatigue Crack

Propagation in Nylon 66—R W LANG, M T HAHN,

R W HERTZBERG, AND J A MANSON 266

Fatigue Life Estimation of Notched Members—D F SOCIE,

N E D O W L I N G , AND P KURATH 2 8 4

Effects of Constraint Variation on the Fatigue Growth of Surface

Flaws—M JOLLES AND V TORTORIELLO 3 0 0

MATERIAL INFLUENCES ON FRACTURE

Temperature Dependence of Fracture Toughness of Large Steam

Turbine Forgings Produced by Advanced Steel Melting

Processes—V p SWAMINATHAN AND J D LANDES 315

Fracture Toughness of Stainless Steel Weldments at Elevated

Application of High-Temperature Fracture Mechanics to the

Prediction of Creep Crack Growth for a 7 - 7 ' Nickel-Base

Effect of Section Size on Transition Temperature Behavior of

Microstructural Aspects of the Fracture Toughness

Cleavage-Fibrous Transition for Reactor-Grade Steel—K OGAWA,

X J ZHANG, T KOBAYASHI, R W ARMSTRONG, AND

G R I R W I N 393

Influence of Inclusions on the Fracture Properties of A588A Steel—

A D WriLSON 4 1 2

Load History Effects on the Fracture Toughness of a Modified 4340

S t e e l — I D LANDES AND T R LEAX 4 3 6

Discussion 474

Fracture Tougliness of High Strength Steel Predicted from Charpy

Energy or Reduction in Area—i H UNDERWOOD AND

G S LEGER 481

Effect of Fast-Neutron Irradiation on Fracture Toughness of Alloy

A-286—w J MILLS 499

Wide Range Creep Crack Growth Rate Behavior of A470 Class 8

ELASTO-PLASTIC FRACTURE MECHANICS

/-Integral R-Cune Testuig of High Strength Steels Utilizing the

Direct-Current Potential Drop Method—M G VASSILAROS

AND E M HACKETT 535

Single-Specimen /-Resistance Curve Evaluations Using the

Direct-Current Electric Potential Method and a Computerized

Data Acquisition System—G M WILKOWSKI, J O WAMBAUGH,

AND K PRABHAT 553

(jeometiy and Size Effects on J-R and b-R Curves under Plane Stress

Conditions—D HELLMANN AND K.-H SCHWALBE 577

Effect of Specimen Dimensions on Critical/-Value at the Onset

of Crack Extension—p DE ROD, B MARANDET,

G PHELIPPEAU, AND G ROUSSELIER 606

Influence of Loadmg Rate on the Fracture Toughness of Some Structural

Steels in the Transition Regime—B MARANDET, G PHELIPPEAU,

AND G SANZ 622

Crack Growth Resistance Measurement by Crack Openuig

Displacement Methods—D E MCCABE AND H A ERNST 648

Post-Yield Crack Openmg Displacement of Surface Cracks in Steel

Weldments—Y w CHENG, R B KING, D T READ,

AND H I MCHENRY 666

T WEERASOORIYA 682

A Tearing Instability Analysis for Strain-Hardening Materials—

C H POPELAR, J PAN, AND M F KANNINEN 699

Application of a Tearing Instability Analysis for Strain-Hardening

Materials to a Cu^umferentially Cracked Pipe in Bending—

J PAN, J AHMAD, M F KANNINEN, AND C H POPELAR 721

Summary 749

Index 755

Introduction

The George R Irwin Anniversary Volume

The year 1982 marked a number of milestones in the history of fracture

me-chanics In this year the National Symposium on Fracture Mechanics held its

15th annual forum to discuss a wide range of topics related to the fracture of

materials It also marked the 25th anniversary of the rocket motor fractures

which led to the formation (December 1958) of the Special Committee on

Frac-ture Testing of High-Strength Metallic Sheet Materials (in later years this

committee was formally organized as ASTM Committee E-24 on Fracture

Testing) Finally, in 1982, George R Irwin, the major driving force in the early

development of the theory of linear elastic fracture mechanics (LEFM),

cele-brated his 75th birthday In commemoration of this latter event, the

sym-posium subcommittee of E-24 assigned the University of Maryland the task

of hosting this anniversary symposium and has dedicated this publication in

Dr Irwin's honor

George Rankin Irwin was bom in El Paso, Texas, in February 1907 His

school years were spent in Springfield, Illinois, where he attended Springfield

High School (1921-1925) and Knox College (1926-1931) Initially an English

and journalism major, he earned his bachelor's degree in English but

devel-oped a keen interest in physics and took additional courses in this area

Con-tinuing his studies, he attended the University of Illinois and obtained a

master's degree in physics and then a doctorate in physics in 1937 During the

latter stages of his doctorate study (1935-1936) he was an associate professor

at Knox College

In July 1937, George Irwin, with degree and wife, Georgia, moved to

Wash-ington, D.C., and joined the staff of the Ballistics Branch at the Naval

Re-search Laboratory (NRL) He was assigned the task of investigating the cause

of brittle failures of armor materials Early in these studies he observed

corre-lations between the energy absorbed in penetration and the appearance of the

fractured area These results would later be generalized to the strain energy

re-lease rate concept that was to become the cornerstone of the theory of LEFM

prior to 1957

During these early years, numerous conceptual advances in the theory of

fracture were made by Irwin and his co-workers at NRL Crack growth by

ad-vance nucleation was observed both in thin foils and brittle solids Compliance

calibration for characterizing fracture test specimens was developed The role

of fracture markings in postmortem analysis of fracture failures was

demon-strated and catalogs of features and their origins compiled Bifurcation as a

mechanism for energy consumption in dynamic fracture was proposed

In the mid 1950's Dr Irwin turned his attention to the analytical aspects of

fracture mechanics with particular emphasis on the characteristics of the

stress field in the neighborhood of the crack tip In 1957 he published a

land-mark paper in which the near-field stress equations were presented and the

concept of the strength of the stress singularity (now referred to as K) was

pro-posed The use of the Westergaard method to determine the stress intensity

factor for various geometric configurations followed Later, the plastic zone

correction concept was proposed as well as other conceptual ideas such as

vir-tual crack extension He was one of the founding members of the

aforemen-tioned ASTM Special Committee on Fracture Testing, and he continues to

participate in ASTM Committee E-24

After 30 years of federal service, George Irwin retired from the Naval

Re-search Laboratory and assumed the position of University Professor at Lehigh

University During this period in his career, he placed his emphasis on the

de-velopment of undergraduate and graduate courses in fracture mechanics

Dr Irwin retired from Lehigh in 1972 and accepted his current position as

Visiting Professor at the University of Maryland, where his primary interests

lie in guiding research in dynamic fracture He continues to serve as adviser to

government, university, and private industry, drawing on his vast experience

to propose solutions to problems in fracture mechanics

In recognition of his pioneering work he has received numerous awards and

honors including:

Navy Distinguished Civilian Service Award-1946

ASTM Dudley Medal-1960

ASME Thurston Lecturer-1966

U.S Navy Conrad Award-1969

SESA Murray Lecturer-1973

ASTM Honorary Member-1974

ASM Sauveur Award-1974

Societe Fran?aise de Metallurgie Grande Medaille-1976

National Academy of Engineering Membership-1977

ASME Nadai Award-1977

Honorary Doctor of Engineering, Lehigh University-1977

SESA Lazan Award-1977

ASTM (E-24) Irwin Medal-1978

Franklin Institute Clauier Medal-1979

At the symposium banquet Dr John S Toll, President of the University of

Maryland, presented to Dr Irwin on behalf of the Governor of Maryland, the

Honorable Harry Hughes, the Governor's Citation for distinguished service to

the State of Maryland In turn, George Irwin presented the 1982 medal named

in his honor jointly to Drs J R Rice and J Hutchinson of Harvard University

The Symposium Organizing Committee consisting of Prof D B Barker,

Prof W L Foumey, Mr John Gudas, Dr John Merkle, Prof R J Sanford,

and Dr H H Vanderveldt are pleased to have been involved in this effort to

honor this truly remarkable scientist and educator We would like to express

our thanks to the staff and students of the Mechanical Engineering

Depart-ment at the University of Maryland for their many efforts before and during

the symposium Finally, the committee is especially grateful to Mr R Chona,

symposium secretary, for his invaluable assistance during the planning of the

symposium and the preparation of this publication

R J Sanford

Department of Mechanical Engineering, versity of Maryland, College Park, Mary- land; symposium chairman and editor

Uni-Transition of Part-Through Cracks at

Holes into Through-the-Thicl<ness

Flaws

REFERENCE: Grandt, A F., Jr., Harter, J A., and Heath, B J., "Transition of

Part-Through Cracks at Holes into Part-Through-the-Thickness Flaws," Fracture Mechanics:

Fif-teenth Symposium, ASTM STP 833, R J Sanford, Ed., American Society for Testing

and Materials, Philadelphia, 1984, pp 7-23

ABSTRACT: This paper describes results of a numerical and experimental study of the

behavior of part-through cracks located at holes as they transition into uniform

through-the-thickness flaws Fatigue crack growth tests are conducted with transparent polymer

specimens which allow the crack plane to be photographed during the fatigue test Stress

intensity factors are computed by the three-dimensional finite-element-alternating

method for the measured crack shapes Both analysis and experiment indicate that the

crack growth rate varies along the flaw perimeter in a manner which encourages the

trail-ing edge of crack advance to "catch up" with the leadtrail-ing edge Once a uniform

through-thickness configuration is achieved, the trailing point then slows down to the growth rate

occurring at the point of maximum crack advance

KEY WORDS: fatigue cracks, surface cracks, stress intensity factors, fracture mechanics

Nomenclature

a Crack dimension measured along bore of hole as defined in Fig 1

c Crack dimension measured perpendicular to bore of hole as defined

in Fig 1 CL»CR.Ct Free face crack dimensions as defined in Fig 1

D Hole diameter

da/dN Fatigue crack growth rate

'Professor, School of Aeronautics and Astronautics, Purdue University, W Lafayette, Ind

47907

^Former Graduate Assistant, Purdue University, W Lafayette, Ind 47907; currently with

Northrop Corporation, Hawthorne, Calif 92644

•'Former Graduate Assistant, Purdue University, W Lafayette, Ind 47907; currently with

Garrett Turbine Engine Company, Phoenix, Ariz 85010

K Stress intensity factor

AA^ Cyclic stress intensity factor

I<l N u m b e r of elapsed cycles

x,y Coordinate axes defined in Figs 4 and 7

This paper describes results of an experimental and numerical study of the

behavior of part-through cracks as they grow into through-the-thickness

flaws As indicated in Fig 1, the paper is concerned with both corner and

embedded surface cracks located along the bore of fastener holes The

transi-tion period when the nonuniform through-the-thickness flaw grows to a stable

through-thickness shape is of main concern here

FIG 1—Schematic of flawed hole configurations

Several procedures have been described in the literature to analyze the

transition from surface to uniform through-the-thickness cracks Some

au-thors, for example, have conservatively assumed that when the thickness

di-mension penetrates the back surface, the crack can be instantly treated as a

uniform through-the-thickness crack [1,2] Others have developed more

elab-orate criteria based on back surface yielding [3], imaginary equivalent surface

cracks [4], or engineering weighting factors [5]

The goal of this paper is to examine the transition behavior in more detail

The cyclic growth and transition of initial part-through cracks into

through-the-thickness flaws is documented through experiments with specimens made

from polymethyl methacrylate (PMMA), a transparent polymer Since the

test specimens are transparent, it is possible to record crack growth,

includ-ing changes in internal dimensions, by time-lapse photography Stress

inten-sity factors are computed for the observed crack shapes by the

finite-element-alternating method Fatigue crack growth rate changes at various points

along the flaw border are correlated with the stress intensity factor solutions

Results are described for initial quarter-elliptical corner and embedded

semi-elUptical surface cracks located at open holes in large plates loaded in remote

tension, and for embedded surface cracks located along the bore of a

simu-lated pin-loaded attachment lug

Approach

This section reviews the experimental and numerical procedures used to

characterize the transition of surface and corner cracked holes into

through-the-thickness flaws Since these methods have been employed before, only

brief descriptions are given here

Numerical Technique

The finite-element-alternating method (FEAM) developed by F W Smith

and associates [6-9] was used to determine stress intensity factor variations

along the perimeter of embedded surface and corner cracked holes The

FEAM approach involves iterative superposition of the three-dimensional

finite-element solution for an uncracked body subjected to a specified surface

loading [10] with the solution for a flat elliptical crack in an infinite body

loaded with a nonuniform surface pressure [//] Iteration between the

cracked and uncracked solutions approximates the part-through crack

boundary conditions and provides Mode I stress intensity factors and crack

opening displacements for the three-dimensional flaw geometry

The specific computer codes used here were developed by Smith and

Kull-gren for the analyses described in Ref 9 Other applications of this numerical

method to cracks at holes are given in Refs 6 to 8 and 12 to 14 Prior

compari-sons of the FEAM results with other numerical and experimental techniques

indicate that the finite-element alternating-method provides excellent stress

intensity factor solutions for the part-through cracked fastener hole problem

Experimental Procedure

The fatigue crack growth tests were conducted on specimens prepared from

a single sheet of polymethyl methacrylate (PMMA), a transparent polymer

Since the specimens were transparent, it was possible to view internal crack

dimensions directly with the aid of a mirror placed at an angle over the

pol-ished end of the specimen The cracks were photographed with a 35-mm

cam-era during the fatigue test Subsequent measurement of the photographs gave

crack dimensions as a function of applied load cycles Figure 2 gives a

sche-matic view of the test apparatus, showing the grips which were bolted and

bonded to the tension specimen, the placement of the viewing mirror, and the

camera

Baseline fatigue crack growth tests [15] conducted with through-thickness

edge cracks loaded in four-point bending at a frequency of 2 Hz gave the

crack growth equation

- $ - = 1.0702 X 10-31 A/ir9-2i4 (1)

dN

Here the units of the fatigue crack growth rate da/dN are inch/cycle and the

cyclic stress intensity factor A A" is expressed in psi-in.*''^ Equation 1 is

re-stricted to crack growth rates which fall between 10"^ in./cycle and 2 X lO^^

in./cycle (2.54 X 10""^ and 5.08 X lO"-' cm/cycle) Equation 1 agrees well

with other data collected from the same sheet of PMMA [16,17] Additional

details of the loading apparatus, specimen preparation procedures, and

spec-imen material properties are given in Refs 15 to / 7

Discussion

This section discusses the behavior of the part-through cracks as they

tran-sition into through-the-thickness flaws Experimental and numerical results

are given for corner and embedded surface cracks located at open holes in

plates loaded in remote tension Experimental data are also described for

em-bedded surface cracks located along the bore of a pin-loaded hole

Comer Crack at Hole

This subsection describes the transition of corner cracks located at an open

hole loaded in remote tension The localized variation in stress intensity

fac-P fac-P

FIG 2—Schematic of experimental apparatus for flawed holes

tor during the transition period is demonstrated first with numerical results

for a hypothetical problem, and then by measured fatigue crack growth rates

and corresponding stress intensity factors for an actual transitioning fatigue

crack

Figure 3 presents results of a parameteric study of the transition of a

hypo-thetical corner crack into a uniform through-the-thickness flaw Here the

front face dimension c of the corner crack was fixed at a value c = D = hole

diameter = plate thickness T A series of quarter elliptical corner crack

shapes was then analyzed by the FEAM method The crack dimension a was

allowed to increase along the hole bore from an initial value a — 0.5 c,

pene-trate the back face, and then transition to a uniform through-the-thickness

configuration {a/c = 10) After back face penetration, the dimension a

repre-sents the major axis of the portion of the quarter ellipse used to define the

transitioning crack front, and is not an actual crack length dimension The

specific crack shapes analyzed are shown in Fig 3

Dimensionless stress intensity factors K/a-Jo are indicated directly on Fig

3 at the fixed front face point and at the changing comer/back face location

Since the remote stress a and hole diameter D are fixed for the present study,

these dimensionless results provide the relative variation in the actual stress

back face

K / o V D = 2-97 1-98 = K / o V D

a/c

0-5 0-7 0-9 1.1 1-5

FIG 3—Hypothetical corner crack transition showing crack shape changes and dimensionless

stress intensity factors at fixed front face point and changing hole bore/back face location

intensity factor as the crack shape changes Although stress intensity factors

at other points along the crack perimeter were also obtained from the FEAM

analysis, the corner points bounded the K variation and are the only results

given in Fig 3

Now compare the stress intensity factors at the fixed front face point and at

the changing hole bore/back face location Note that at the front face,

increases relatively uniformly from a value of 1.23 for the initial crack

shape a/c = 0.5 to a value K/a\fD = 1.87 for an aspect ratio a/c = 10,

although a slight oscillation is apparent around a/c = 1 As discussed in Ref

9, the present FEAM computer codes have a numerical programming

insta-bility at a/c = 1 which precludes analysis of flaw shapes 0.9 < a/c < 1.1

Note that the front face stress intensity factors are relatively unaffected by the

position of the back face penetration point (1.73 < K/a\[D < 1.87 for 0.9 <

a/c < 10) For comparison, the Bowie [18] analysis (as reported in Ref 19) for

a through-cracked hole gives i(r/(T\(D = 1.88 when c/D = 1 Thus, assuming

a uniform through-the-thickness flaw of maximum length c during the

transi-tion period, the Bowie through-crack analysis gives a reasonable estimate for

the stress intensity factor at the leading point of crack advance on the front

face

The stress intensity factor at the trailing point, where the crack penetrates

the back face, varies significantly during the transition period From Fig 3,

note that K/a\fD at the hole bore/back face point increases from 2.12 at the

initial crack shape ale = 0.5, reaches a value 2.97 at a/c = 1.1, and then

decreases to K/ayfD = 1.98 for the through-thickness shape a/c — 10 The

fact that the back face value only decreases to 1.98 at a/c = 10, rather than to

K/aylD = 1.87 as seen at front face, indicates that there must still be a crack

curvature effect even for this nearly uniform through-the-thickness flaw

From this change in K with increasing crack length, one would expect the

back face crack dimension to initially extend quite rapidly after back face

penetration, and then decrease to a slower growth rate as a uniform

through-the-thickness geometry is achieved As shown by the following discussion of

experimental results, this increasing/decreasing growth rate is also observed

in the fatigue tests

The first experiment described here was originally conducted by Snow [/ 7]

and has subsequently been examined in more detail by the present authors

Snow subjected a 7.95 in (20.2 cm) wide by 0.698 in (1.77 cm) thick PMMA

plate to a remote cyclic stress which varied between 18 and 630 psi (0.124 to

4.34 MPa) at a frequency of 2 Hz A corner crack was initiated at a small

notch machined at the edge of the 0.739 in (1.88 cm) diameter hole The

loading apparatus shown schematically in Fig 2 allowed the crack plane to be

photographed during the fatigue test (recall the specimen material is

trans-parent) Although Snow [17\ measured only the major and minor axes of the

corner crack, his original filmstrips have been re-examined with a photo

in-terpreter/digitizer, and a much more detailed record of crack growth has

been obtained [20] Figure 4 shows the growth of the original corner crack (as

measured 2000 cycles after the test began) into a fairly uniform

through-the-thickness flaw at a cyclic life of 24 900 cycles In Fig 4, the origin of the x-y

coordinate system is located at the intersection of the hole bore with the front

face of the plate, the individual data points represent points measured along

the crack front at various cyclic lives, and the solid lines are portions of

ellipses fit through the measured points (The actual coordinates of the

digitized crack shape measurements are given in Ref 20.) For purposes of

the present analysis, the ellipses were required to go through the point of

front face penetration (crack position where x = c,y — Q) and through the

hole bore/back faoe point (either the jc = 0,y = aotx = c^,y ~ Jlocation)

The ellipse origin coincides with the x-y origin in all cases Note that the

part-elliptical model represents the actual crack shapes quite well prior to

back face penetration During the transition/through-crack period, however,

the actual crack fronts lead the elliptical shapes somewhat in the specimen

interior

Figures 5 and 6 present the change in crack dimensions and corresponding

stress intensity factors as a function of applied load cycles during the

transi-tion period Examining the Fig 5 data first (open symbols), note the growth

of the hole bore (a) or back face (cj) crack dimension Initially, the crack

grows at a uniformly increasing rate along the bore of the hole until

eiEOO

£3000

nioo

MOO naoo

tmu turn

mm

' front face

0.4 0 6 X-AXIS (IN)

0 8 Ul, 1.0

FIG 4—Comparison of part-elliptical and natural fatigue crack shapes for corner and

through-cracked holes

ing the back face of the plate The back face dimension Ct then increases very

rapidly initially, but eventually slows down as a uniform through-thickness

configuration is achieved Finally, the back face length Ct approaches the

front face dimension c, and both crack lengths grow at similar rates until final

fracture (which occurred shortly after 24 900 cycles) Meanwhile, note that

the front face crack length c grows at a uniformly increasing rate during the

entire transition period, and is apparently little affected by the localized

in-crease/decrease/increase in rate at the back face

Now examine the dimensionless stress intensity factor curves given in Fig 6

(solid symbols) Here K/a\fD is plotted for the crack sizes and shapes

corre-sponding to the current cyclic life N (Note that the K/a\fD curves in Fig 6

are superimposed on the fatigue crack growth data given earlier in Fig 5.)

The finite-element-alternating method was used to compute K at the leading

front face crack position and at the trailing crack bore/back face location

For the FEAM analysis, it was assumed that the hole diameter equaled the

plate thickness, and that the fatigue cracks had the part-elliptical shapes

shown in Fig 4 For comparison purposes, the Bowie [18] analysis was also

used to compute stress intensity factors for uniform through-thickness cracks

FIG 6—Comparison of stress intensity factors and crack length changes as function of

elapsed cycles during transition of comer crack

of length c and is shown as the dashed line in Fig 6 The following

least-squares representation [21] of the Bowie solution was used here and is

ex-pected to be accurate within ± 3 % for a//? < 10

Kt — a\ira 0.8733 + 0.6762

As seen in the hypothetical transition case considered in Fig 3, K again

increases at the hole bore position as the flaw grows, jumps to a large value

along the back face immediately after transition, and then decreases along

the back face as the crack transitions to the uniform through-thickness

con-figuration Meanwhile, the stress intensity factor at the front face increases

smoothly as the front face crack length c slowly grows by fatigue Again, the

front face K is apparently little affected by the large changes occurring at the

hole bore/back face location Following back face penetration, the front face

stress intensity factor can be closely approximated by the through-thickness

solution for a crack of length c given by Eq 2

The numerical and experimental results can also be compared by

combin-ing the predicted stress intensity factors with the fatigue crack growth law

given by Eq 1 and integrating for cyclic life as a function of crack length The

predicted crack growth curves obtained in this manner are shown by the solid

lines in Fig 5 Here the stress intensity factor curves in Fig 6 at the hole bore

position (corresponding to crack dimension a), the back face location

(corre-sponding to crack length c,), and the front face position (crack length c) were

treated independently to make the crack length predictions shown in Fig 5

Thus the observed crack shape changes are incorporated in the stress

inten-sity solutions and are implicit in the life calculations The predictions begin

with the actual flaw dimensions at 22 000 cycles of life The Cj curve begins

with the first measured flaw shape after back face penetration {N = 24 100),

rather than at the predicted point where a = T{22 900 cycles), since the stress

intensity factors were obtained for the observed crack shapes

Although the crack shapes were constrained to the observed behavior and

not a free parameter in the life prediction scheme, the predictions of Fig 5 do

verify the FEAM calculations Note in Fig 5 that the front and back face

crack length predictions agree quite well with the experimental behavior The

predicted growth along the hole bore does, however, exceed the observed rate

prior to back face penetration

Embedded Surface Crack at Hole

Consider the results for the embedded surface crack hole configuration

re-ported in Figs 7 and 8 The PMMA specimen was 7.97 in (20.3 cm) wide,

0.720 in (1.83 cm) thick, and contained a hole with a 0.750 in (1.91 cm)

CYCLES

36600

tzooo

16300 WHSO

«1S0 WSOO

FIG 1 ^-Comparison of digitized measurement of surface crack profiles and semielliptical

model as function of elapsed cycles

diameter D The specimen was machined from a piece of the original PMMA

sheet tested by Snow [17] several years earlier Care was taken to maintain the

same crack orientation and to employ the same annealing cycle used by Snow

to minimize potential residual stress effects The specimen was subjected to a

558 psi (3.85 MPa) cyclic stress [R = 0.01) at a frequency of 2 Hz As before,

crack growth was recorded on film, and specific crack shapes were analyzed

by the finite-element-alternating method

Some of the digitized crack growth profiles are shown in Fig 7 (Additional

results for this specimen and for other similar tests are given in Ref 15.) In

Fig 7, thej;-axis is oriented with the right edge of the hole bore and thejc-axis

corresponds to the front face of the plate The back face of the plate is given

by the line>' = 0.72 in The symbols in Fig 7 represent digitized

measure-ments of the crack profiles at various cyclic lives, while the solid lines are

portions of ellipses used to model the crack shapes for the FEAM stress

FIG 8—Comparison of stress intensity factors and crack length changes as function of

elapsed cycles during transition of surface crack

sity factor analysis For this case, the ellipse origin was allowed to shift along

the>'-axis in order to better approximate the actual crack shapes

The fatigue crack growth curves and stress intensity factor results for the

transition period are summarized in Fig 8 in the same format employed

ear-lier in Fig 6 for the corner crack As before, the crack lengths are represented

by unconnected open symbols, while the dimensionless stress intensity factor

curves are given by solid symbols connected with lines The FEAM stress

in-tensity factors were computed for the specific crack size and shape

corre-sponding to the particular life in Fig 8

Examine the fatigue crack length measurements in Fig 8 first Here

mea-surements of the back face dimension CL, the midlength c, and front face

crack length CR are given as a function of elapsed cycles N Note that, in this

particular test, the initial embedded surface crack was not centered exactly

along the hole bore, so that back face penetration (CL versus N) occurred prior

to front face break through (CR versus N) Nevertheless, both CL and CR

ini-tially increase quite rapidly, but then grow at a slower rate as both dimensions

approach the mid crack length c This increase/decrease in crack growth

rate, as a uniform through-thickness configuration is achieved, agrees with

the corner crack results described earlier

Now consider the dimensionless stress intensity f&ctor K/a\fD results given

in Fig 8 Note that again the stress intensity factors at the front and back face

points are initially largest after penetration through the thickness, decrease as

the crack dimension CL or CR increase, and eventually approach the stress

in-tensity factor at the midpoint location along the crack perimeter as a uniform

through-thickness shape is achieved Note, however, that K/a\fD at the front

and back faces does not show the large changes seen earlier for the corner

crack configuration Perhaps this smaller K variation is due to the fact that

transition from an embedded surface crack to a through-thickness flaw does

not involve as dramatic a crack shape change as required for the corner crack

configuration

Two other points regarding the stress intensity factor results are of interest

in Fig 8 Firstly, note that the uniform through-thickness stress intensity

fac-tor solution labeled "Bowie K" (computed by Eq 2 for the maximum crack

length c) gives a fairly good "average" value of the stress intensity factor

across the perimeter of the transitioning flaw Also note in Fig 8 that at a

particular cyclic life, the stress intensity factor is always smallest at the

lead-ing position of crack advance (midpoint location) and largest at the traillead-ing

point (crack dimension CR) Thus the stress intensity factors vary along the

crack perimeter in a manner which encourages the crack to grow to a uniform

through-the-thickness shape

Pin-Loaded Holes

Reference 15 describes several experiments conducted to simulate growth

of surface cracks located along the bore of an attachment lug As shown

sche-matically in Fig 9, the specimens considered here had a width W = 6.75 in

(17.1 cm), a hole diameter!) - 2.25 in (5.72 cm), and a thickness T = 0.71

in (1.8 cm) The specimens were made from the same sheet of PMMA as

before One end of the specimen was loaded with the grip/mirror

arrange-ment described earlier, while the other end was loaded through a close fit steel

pin placed in the bore of the hole The cyclic load P varied between 10 and

1100 lb (44 and 4900 N) for Specimen PT4 at a frequency of 2 Hz, while the

load limits for specimen PT5 were 50 and 1200 lb (220 and 5300 N)

The transition portion of the surface crack growth for Specimens PT4 and

PT5 are given in Figs 10 and 11 These crack lengths were measured from an

image of the 35-mm negatives projected onto a screen

Note in Fig 10 that the surface crack transition is similar to the open hole

case discussed in the last section Initially, the dimensions CL and CR grow

quite rapidly after free surface penetration, slow down as the midlength crack

dimension c is reached, and then eventually all three crack dimensions

accel-®

CRACK

FIG 9—Schematic view of pin-loaded attachment lug specimen and loading apparatus

(0 liJ

t 4

THOUSANDS OF CYCLES

FIG 10—Fatigue crack growth curves showing changes in crack dimensions during transition

of surface crack at pin-loaded attachment lug

FIG 11—Fatigue crack growth curves showing changes in crack dimensions during transition

of eccentric surface crack at pin-loaded attachment lug

erate at a similar rate until final fracture Figure 11 presents much the same

behavior, except in this case the initial embedded crack was located

signifi-cantly away from the center of the hole bore, so that the crack penetrated

through one free face much earlier than the other Note that even for this

nonsymmetric crack shape, the transitioning crack front grows locally at

dif-ferent rates in an apparent attempt to reach a stable through-the-thickness

configuration No stress intensity factor results are available for the

pin-loaded configuration

Summary and Conclusions

This paper has described experimental and numerical results from a study

of the transition of initial corner and embedded surface cracks into

through-the-thickness flaws Fatigue crack growth tests conducted with a transparent

polymer provided a detailed record of the growth rates and crack shape

changes during the transition period A corresponding three-dimensional

stress intensity factor analysis was performed by the

finite-element-alternating method

Both experimental and numerical results indicate that the initial

part-through cracks try to grow into a stable uniform part-through-the-thickness

con-figuration The stress intensity factor varies along the crack perimeter so that

the maximum K occurs at the trailing, rather than the leading, point of crack

advance Moreover, at the trailing point, where the crack penetrates a free

face, K varies significantly with crack advance

The stress intensity factor reaches a large value initially after penetration,

but then decreases locally at that point as the free face crack length grows to a

uniform through-the-thickness shape The stress intensity factor at the

lead-ing point of maximum crack advance usually has a smaller magnitude than at

the trailing point, and is apparently unaffected by the large changes

occur-ring at the trailing position duoccur-ring transition The leading edge stress

inten-sity factor may often be approximated by the two-dimensional analysis for a

uniform through-the-thickness crack whose length equals the distance of

maximum crack advance

The measured fatigue crack growth rates agree with the computed stress

intensity factors as the crack perimeter advances locally at rates

correspond-ing to the K variation along the flaw perimeter In particular, the free face

crack dimension immediately grows quite rapidly after penetration, but then

decreases to the rate seen by the point of maximum crack advance Successful

predictions for the corner crack transition crack growth using the computed

stress intensity factors further verify the finite-element-alternating method

stress intensity factor calculations

Acknowledgments

Portions of this research were supported by AFWAL Flight Dynamics

Lab-oratory Contract F33615-81-K-3206, with J Rudd as technical monitor

As-sistance was also provided by T Nicholas of the AFWAL Materials

Labora-tory The authors deeply appreciate the many discussions with T, E KuUgren

regarding the finite-element-alternating method, and gratefully acknowledge

the computer support provided by G Griffin and C Malmsten, and the

assis-tance of T Myers in measuring the crack photographs

References

[/] Rudd, J L in Part-Through Crack Fatigue Life Prediction ASTM STP 687, J B Chang,

Ed., American Society for Testing and Materials, 1979, pp 96-112

[2] Chang, J B in Part-Through Crack Fatigue Life Prediction ASTMSTP687, i B Chang,

Ed., American Society for Testing and Materials, 1979, pp 156-167

[3] Peterson, D E and Vroman, G A in Part-Through Crack Fatigue Life Prediction, ASTM STP 687,1 B Chang, Ed., American Society for Testing and Materials, 1979, pp 129-

142

[4] Johnson, W S in Part-Through Crack Fatigue Life Prediction, ASTM STP 687, J B

Chang, Ed., American Society for Testing and Materials, 1979, pp 143-155

[5] Brussat, T R., Chin, S T., and Creager, M., "Flaw Growth in Complex Structure,

Vol-ume I—Technical Discussion," Technical Report AFFDL-TR-77-79, Vol I, Air Force

Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, Dec 1977

[6] Kullgren, T E., Smith, F W., and Ganong, G V.,Journal of Engineering Materials and

Technology, Vol 100, April 1978, pp 144-149

[7] KuUgren, T E and Smith, F Vf., International Journal of Fracture, Vol 14, 1968, pp

R319-R322

[8] KuUgren, T E and Smith, F "f^., Journal of Engineering Materials and Technology, Vol

101, Jan 1979, pp 12-17

[9] Smith, F W and KuUgren, T E., "Theoretical and Experimental Analysis of Surface

Cracks Emanating from Fastener Holes," Technical Report AFFDL-TR-76-104, Air Force

Flight Dynamics Laboratory, Wright-Patterson Air Force Base, Ohio, 1977

[10] Wilson, E L., "Finite Element Analysis of Mine Structure," Technical Report Bureau of

Mines OF 27-73, Denver Mining Research Center, Sept 1972

[//] Shaw, R C and Kobayashi, A ^., Engineering Fracture Mechanics, Vol 3, No 1, July

1971

[12] Grandt, A F., Jr., and KuUgren, T E., Journal of Engineering Materials and Technology,

Vol 103, No 2, April 1981, pp 171-176

[13] Grandt, A F., Jr., Engineering Fracture Mechanics, Vol 14, No 4, 1981, pp 843-852

[14] Grandt, A F., Jr., and KuUgren, T E., "A Compilation of Stress Intensity Factor

Solu-tions for Flawed Fastener Holes," Technical Report AFWAL-TR-81-4112, Air Force

Wright Aeronautical Laboratory, Wright-Patterson Air Force Base, Ohio, Nov 1981

(con-densed version appears m Engineering Fracture Mechanics, Vol 18, No 2, 1983, pp

435-451)

[15] Harter, J A., "Fatigue Crack Growth of Embedded Flaws in Plate and Lug Type Fastener

Holes," M.S thesis School of Aeronautics and Astronautics, Purdue University, W

Lafay-ette, Ind., May 1982

[16] Grandt, A F., Jr., and Hinnerichs, T D., "Stress Intensity Factor Measurements for

Flawed Fastener Holes," AMMRC MS 74-8, Army Materials and Mechanics Research

Center, Watertown, Mass., Sept 1974, pp 161-176

[17] Snow, J R., "A Stress Intensity Factor Calibration for Corner Flaws at an Open Hole,"

Technical Report AFML-TR-74-282, Air Force Materials Laboratory, Wright-Patterson

Air Force Base, Ohio, 1975

[/*] Bowie, O L., Journal of Mathematics andPhysics, Vol 35, 1956, pp 60-71

[19] Paris, P C and Sih, G C mFracture Toughness Testing and its Applications, ASTMSTP

381, American Society for Testing and Materials, 1964, pp 30-81

[20] Grandt, A F., Jr., and Macha, D E., Engineering Fracture Mechanics, Vol 17, No 1,

1983, pp 63-73

[21] Grandt, A F., Jr., International Journal of Fracture, Vol 11, No 2, April 1975, pp

283-294

Part-Through Flaw Stress Intensity

Factors Developed by a Slice

Synthesis Technique

REFERENCE: Saff, C R and Sanger, K B.; "Part-Through Flaw Stress Intensity

Fac-tors Developed by a Slice Synthesis Technique," Fracture Mechanics: Fifteenth

Sympo-sium, ASTM STP 833, R J Sanford, Ed., American Society for Testing and Materials,

Philadelphia, 1984, pp 24-43

ABSTRACT: Part-through-the-thickness flaws are the most common type of flaw

occur-ring in metal structure Accurate prediction of their growth is vital to ensure adequate

life The majority of stress intensity factor solutions for these flaws are developed through

finite-element analyses, iterative techniques, or less accurate superposition of simple

solu-tions Recently the authors have developed and extended a slice synthesis technique for

computation of stress intensity factors for part-through flaws This technique is tar less

expensive than finite-element methods, yet provides excellent accuracy This paper

presents the derivation of the method and recently obtained solutions for surface flaws,

corner cracks at holes, and corner cracks at the edge of plates in tension and bending

Simple analytical expressions have been fit to these solutions which can be incorporated

into computer routines for crack growth prediction

KEY WORDS: stress intensity factors, surface flaws, corner flaws, mathematical models,

crack propagation

Part-through-the-thickness flaws are the most common type of flaw

occur-ring in metal structure Accurate prediction of their growth is vital to ensure

adequate life The majority of stress intensity factor solutions for these flaws

are developed through finite-element analyses, iterative techniques, or

super-position of simple solutions [1] Recently the authors have extended

develop-ment of a slice synthesis technique for computation of stress intensity factors

for part-through flaws This technique is far less expensive than

finite-element methods, yet provides excellent accuracy

The slice synthesis technique was originally formulated by Fujimoto [2] and

'Technical Specialist and Engineer, respectively Structural Research, McDonnell Aircraft

Company, McDonnell Douglas Corporation, St Louis, Mo 63166

is an extension of the line-spring model proposed by Rice and Levy [3]

Fuji-moto's method was developed for analysis of flaws at fastener holes [4] Later

the method was extended by Dill and Saff to analysis of surface flaws in

ten-sion [4,5] Recent improvements in the weight functions and solution

tech-nique have allowed the authors to use the same formulation to compute stress

intensity factors for part-through flaws either centrally located or at holes in

plates

Derivation

As shown in Fig 1, the part-through flaw is idealized as a system of

hori-zontal slices in thex-y plane, each containing a central through-crack (with or

without a hole depending upon the solution desired) whose length is

deter-mined by the locations through the plate thickness at which the slice was

taken Each slice is considered to react independently to the applied stress

(a), but is coupled through the introduction of pressure distribution (p*)

act-ing on the faces of the cracks The pressure p* is determined by a second

system of vertical slices in the z-y plane Each of the vertical slices contains an

edge crack over which the pressure/?* acts in opposition to that applied to the

center crack slices

Thus there are two slice systems: center cracks and edge cracks These

sys-Vertical Slice Idealization

FIG 1—Slicing procedure for synthesizing three-dimensional crack solutions

tems are coupled by the pressure distribution/?* acting on the crack surfaces

of each system and causing the displacements of the two systems to be equal

Using the crack face pressure distribution, stress intensity factors at the

maximum depth (A) and at the surface (B) can be determined from the

The coefficients Wy are determined by the requirements for continuity of

displacements over the crack face:

Vh.s.Ụ z) = Vv.s.Ụ z) (4)

These displacements can be determined from the pressure distributions and

weight functions for each flawed slice system For the horizontal slice:

vh.s = ^ m)H.s.g(x, Or ,d^ (5)

where ? is radial crack length from 0 to c(z), and Ệs is the stiffness of the

horizontal slice and is discussed later

Substitution of the weight function representation for stress intensity factor

(Eq 2) into the displacement expression produces more consistent

computa-tion of displacements for each crack face displacement The resulting

The solution scheme used to find/j* is the same as that used by Fujimoto;

p* is expressed as a power series (from Eq 3):

To assure that the coefficients a,y represent the displacements over the

en-tire crack face, the continuity expression is evaluated at the 13 points shown

in Fig 2 Then a multiple linear regression scheme is used to determine ay

Once (X/j are found, the stress intensity factors at A and B become (from Eq 1

4

^ \ ^ 2

Numbers identity points at whicti

slice displacements are matctied

FIG 2—Surface flaw model

To determine the stiffness of the component slices, £'h.s and£'v s,, we

con-sidered the displacements of an embedded flaw (Fig 3) under uniform

re-mote tension (po)- Returning to a slice idealization we find that, because

cen-ter cracks take on elliptical displacements, only a constant pressure (|8po) will

be required between slice systems to bring their displacements into agreement

with the exact solution [6] The crack surface displacement field of the

hori-zontal slices is

and of the vertical slices is

v(;»:, + 0 , z) = 2i3 (1 - M^)Po«o >R5^ (14)

where M is Poisson's ratio

Equating these displacements we find

|8 = ao/(ao + bo) Sneddon and Lowengrub [6] give the exact displacement field as

, „ _ 2(1 - iJ?)poao v(x, + 0 , z ) - ~ ^ 1

bo/ \ao

(15)

(16)

a) Embedded Eliptlcal

Flaw Uniform Pressure pp Acts on Faces in Direction Normal to Plane of Page

(b) Typical Thru-Cracl(

Obviously, the accuracy of the slice model depends a great deal on the

ac-curacy of the weight functions used The weight functions used in this analysis

are not exact but are accurate approximations Their derivation is included in

the Appendix Three weight functions were developed: for single or double

flaws emanating from a hole, for edge crack slices where out-of-plane

defor-mation was allowed to occur, and for edge crack slices where out-of-plane

deformation was restrained

The weight function derived for the crack from hole slice was developed so

that the hole radius could be set to zero for simulation of surface flaws or to

infinity for simulation of plates having corner cracks at an edge The two

dif-ferent solutions for edge cracks were required to account for the effect of plate

continuity to restrain out-of-plane deformations occurring for deep flaws

Analysis of Plate Restraint

Because shear stresses acting on the faces of the "free" edge crack slice are

ignored in the slice synthesis model, these slices displace (Fig 4) In reality

-Slice A

FIG 4—Deflection of edge crack slices without moment restraint

these types of displacements do not occur because of plate stiffness Thus the model requires an estimate of the effect of plate stiffness in restraining out of plate deflection of "free" edge crack slices The following paragraphs de-scribe development of a restrained edge crack element based on analyses us-ing NASTRAN three-dimensional finite elements

Analyses were performed to determine the effect of rotational restraint fered by the plate to the edge crack slices As shown in Fig 5, the stress inten-sity, and hence displacements, for a "free" edge crack can be represented as

K, restrained — •''^fixed + ^n (20)

where K^ is related to the amount of rotation (0) due to the crack in the edge

cracked slice, determined by a simple coupled beam analogy:

Ki,= ^free ^fixed

1 + X( 1 + 6 - a

(21)

where a = $E/4a From Eq 21: