Astm stp 868 1985

B., "Stress Intensity Factors for a System of Cracks in an Infinite Strip," Fracture Mechanics: Sixteenth Symposium, ASTM STP 868, M.. KEY WORDS: crack propagation, fracture materials,

FRACTURE MECHANICS:

SIXTEENTH SYMPOSIUM

Sixteenth National Symposium

on Fracture Mechanics sponsored by

ASTM Committee E-24

on Fracture Testing Columbus, Ohio, 15-17 August 1983

ASTM SPECIAL TECHNICAL PUBLICATION 868

M F Kanninen, Southwest Research Institute, and A T Hopper, Battelle's Columbus

National Symposium on Fracture Mechanics (16th: 1983; Columbus, Ohio)

Fracture mechanics

(ASTM special technical publication; 868)

Includes bibliographies and index

"ASTM publication code number (PCN) 04-868000-30."

1 Fracture mechanics—Congresses I Kanninen, Melvin F II Hopper, A

III ASTM Committee E-24 on Fracture Testing IV Title V Series

Printed in Baltimore, MD August 1985

1915-1982

Dedication

George E Pellissier contributed significantly to the cess of ASTM Committee E-24 on Fracture Testing He was a member of the committee from 1966 until his death

suc-on 25 June 1982, and was the first chairman of mittee 1 on Fracture Testing (now E24.01 on Fracture Me- chanics Test Methods)

Subcom-George received bachelor's (1936) and master's (1938) degrees in chemistry from Cornell and a bachelor's degree (1941) in metallurgical engineering from Carnegie-Mellon

The completion of his thesis for a doctor's degree was cluded by World War 11 Early in his career he worked for Inco, Columbia University, Union Carbide, and Carnegie Illinois Steel Corporation in such diverse areas as powder metallurgy, nondestructive testing, corrosion, and mechan- ical metallurgy He was considered a pioneer in the fields

pre-of electron microscopy and spectrographic analysis pre-of molten steel

George then went to the U.S Steel Research tory, where he held the posts of Research Associate, Di-

Labora-Senior Research Consultant He was involved in the areas

of chemical, crystal, and microstructural analyses; defect detection; oxidation and chemisorption; and toughness and failure mechanisms of high-strength steels He originated the concept of dual-mechanism strengthening of alloy steels; developed a noncontact thickness gage for thin sheet and coatings; and helped develop a new class of low-car- bon, weldable, high-strength/high-toughness alloy plate steels

From 1968 to 1982 George worked for E F Fullam, RRC International, and Mechanical Technology, where he used his extensive experience to provide internal and exter- nal consulting services on a broad range of metallurgical problems George was a charter member of the Electron Microscopy Society of America, a Fellow of the American Society for Metals and the American Institute of Chemists,

a member of various ASTM committees (including E-2 on Emission Spectroscopy, E-4 on Metallography, and E-24

on Fracture Testing), a member of The Electrochemical Society and Sigma Xi, and a licensed professional engineer

in Pennsylvania He published 26 technical papers

The Sixteenth National Symposium on Fracture Mechanics was held at

Battelle's Columbus Laboratories, Columbus, Ohio, on 15-17 August 1983

ASTM Committee E-24 on Fracture Testing was the sponsor M F

Kan-ninen Southwest Research Institute, and A T Hopper, Battelle's Columbus

Laboratories, served as symposium chairmen and have edited this

pub-lication

Related ASTM Publications

Methods for Assessing the Structural Reliability of Brittle Materials, STP

844 (1984), 04-844000-30

Damage Tolerance of Metallic Structures: Analysis Methods and

Applica-tions, STP 842 (1984), 04-842000-30

Fracture Mechanics: Fifteenth Symposium, STP 833 (1984), 04-833000-30

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

Environment-Sensitive Fracture: Evaluation and Comparison of Test

Elastic-Plastic Fracture: Second Symposium—Volume II: Fracture

Resis-tance Curves and Engineering Applications, STP 803 (1983),

04-803002-30

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

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

ASTM Editorial Staff

Allan S Kleinberg Janet R Schroeder Kathleen A Greene Bill Benzing

Introduction 1

LINEAR ELASTIC ANALYSES

Stress Intensity Factors for a System of Cracks in an Infinite Strip—

M B, CIVELEK 7

Wide-Range Displacement Expressions for Standard Fracture

Mechanics Specimens—j. A KAPP, G S LEGER,

AND B, GROSS 2 7

Evaluation of Analytical Solutions for Corner Cracks at Holes—

J B MECKEL AND J L RUDD 4 5

Stress Distribution at the Tip of Cracks Originating from a Circular

Wide-Range Weight Functions for the Strip with a Single Edge

Crack—T.w ORANGE 95

Analysis of an Externally Radially Cracked Ring Segment Subject to

Three-Point Radial Loading—B. GROSS, J E SRAWLEY, AND

J L SHANNON, JR 106

TTie Dugdale Model for Compact Specimen—s MALL AND

J C NEWMAN, JR 1 1 3

TEMPERATURE AND ENVIRONMENTAL EFFECTS

Internal Hydrogen Degradation of Fatigue Hiresholds in HSLA

Effect of Hydrogen on Crack Initiation and Growth in 18Mn-4Cr

Steel—Y.-j KIM, B. MUKHERJEE, AND D W CARPENTER 149

A Model for Creep/Fatigue Interactions in Alloy 718—

T NICHOLAS, T WEERASOORIYA, AND N E ASHBAUGH 167

Critical Load Assessment Method for Stable Crack Growth Analysis—

H C RHEE 183

Analysis of Fracture Parameters for Bending-Type Specimens—

T AIZAWA AND G YAGAWA 197

Hiree-Dimensional Elastic-Plastic Finite Element Analysis of

AND R H DODDS 214

Fracture Toughness Improvement in a Carbon Steel Due to

Normalization—B D MACDONALD 238

Studies on Size Effects and Crack Growth of Side-Grooved CT

Specimens—M KiKUCHi, s N ATLURI, AND H MIYAMOTO 251

ELASTOPLASTIC EXPERIMENTS

J-R Curve Determination Using Precracked Charpy Specimens and

the Load-Drop Method for Crack Growth Measurements—

J A KAPP 281

Strain-Hardening Effects on Fracture Toughness and Ductile Crack

Growth in Austenitic Stainless Steels—P BALLADON,

J HERITIER, AND C JARBOUI 293

Axial Fracture Toughness Testing of Zr-2.5Nb Pressure Tube

Material—p H DA VIES AND C P STEARNS 308

J A VAN DEN AVYLE, T, J LUTZ, AND W L BRADLEY 328

FATIGUE CRACK G R O W T H

A Similitude Criterion for Fatigue Crack Growth Modeling—

D BROEK 347

Effects of Load Gradient on Applicability of a Fatigue Crack Growth

Rate-Cyclic / Relation—M JOLLES 381

Initiation in li-6Al-4V—G R YODER, L A COOLEY,

AND T W CROOKER 392

Discussion 403

DYNAMIC FRACTURE MECHANICS

Strain-Rate Dependence of the Deformation at the Tip of a Stationary

Cracli—R HOFF, c. A RUBIN, AND G T HAHN 409

Cracli Tip Plasticity of a Tearing Crack—o s LEE AND

A, S KOBAYASHI 431

Dynamic Crack Propagation Ilirough Welded HY-80 Plates under

Blast Loading—c R. BARNES, J AHMAD, AND M F KANNINEN 451

Crack Arrest Behavior of a High-Strength Aluminum Alloy—c LIN

AND R G HOAGLAND 467

BASIC CONSIDERATIONS AND APPLICATIONS

In Situ SEM Observation of Fracture Processes in Short Glass Fiber

Reinforced Thermoplastic Composite—N. SATO, T KURAUCHI,

S SATO, AND O KAMIGAITO 493

Some Hiree-Dimensional Aspects of Subcritical Flaw Growth as

Measured in a Transparent Polymeric Material—

C W SMITH AND J S EPSTEIN 504

Stress Intensity Factors for Surface Cracks with Arbitrary Shapes in

Plates and Shells—T MiYOSHi, M SHIRATORI, AND O TANABE 521

On the Three-Dimensional Implications of LEFM: Finite Element

Analysis of Straight and Curved Through-Cracks in a Plate—

J S SOLECKI AND J L SWEDLOW 535

Fracture Behavior of a Uranium or Tungsten Alloy Notched

Component with Inertia Loading—J H UNDERWOOD

AND M A SCAVULLO 554

Strain-Rate Effects on the Ductile/Brittle Transition in Steels—

G A, KNOROVSKY 569

C A SCIAMMARELLA 597

Fracture Mechanics Analysis of a Pressure Vessel with a

Semi-Elliptical Surface Crack Using Elastic-Plastic Finite Element

Calculations—D AURiCH, w. BROCKS, H.-D, NOACK,

AND H VEITH 617

Application of Maximum Load Toughness to Defect Assessment in a

Ductile Pipeline Steel—A A, wiLLOUGHBY AND s J GARWOOD 632

SUMMARY

Summary 659

Author Index 665

Subject Index 667

Introduction

The sixteenth ASTM National Symposium on Fracture Mechanics, held in

Columbus, Ohio, in August 1983, was conceived two years previously at the

fourteenth symposium in Los Angeles The latter was attended by several

Battelle staff members, all of whom had been and continued to be greatly

impressed by the number and enthusiasm of the attendees at this series of

annual symposia

Upon learning that the sites subsequent to the fifteenth symposium were

not committed (the fifteenth symposium having been scheduled for the

Uni-versity of Maryland), the Battelle group decided to propose that the sixteenth

symposium be held at their home institution An organizing committee was

promptly formed, institute support obtained, and a formal request submitted

to the series sponsor, ASTM Committee E-24 on Fracture Testing Earnest

planning began when acceptance from Committee E-24 was formally received

in November 1981

The organizing committee consisted of nine individuals In addition to the

undersigned, they were Brian Leis, Samuel Smith, Charles Marschall, and

Gery Wilkowski of Battelle's Columbus Laboratories; Richard Hoagland

and Carl Popelar of The Ohio State University; and David Broek,

Fractu-Research, of Columbus The very able clerical assistance given to the

com-mittee by Ms Louisa Ronan of Battelle should also be recognized along with

management support as represented by David Snediker and Fred Milford

Each of the aforenamed individuals played an important role in the success

of the symposium

The first major decision of the organizing committee was to set the

sympo-sium date While constrained to the middle months of 1983, complete

flexi-bility within that period was possible As appropriate for a technical meeting,

the search was made on a scientific basis It included (1) determining the

schedules of competing conferences, (2) assessing the availability of hotel

and other lodging space in Columbus, (3) correlating the attendances at the

preceding symposia with the dates on which they were held, and (4)

investi-gating the Columbus weather records to find the most climatically favorable

time Items (1), (2), and (4) clearly pointed to mid-May But, as the previous

symposia held at that time had attracted significantly less than average

at-tendances, the next best choice was adopted Thus the sixteenth symposium

came to be scheduled, and eventually held, on 15-17 August 1983

The organizing committee was eager to adopt innovative ideas for the

con-duct of the symposium Its two main goals were to maximize the technical

level and to ensure ample opportunities for informal interactions among the

attendees The committee recognized, however, that this would have to be

done while still preserving the traditions of this series of symposia For the

latter reason, an open call for papers was timed to coincide with the fifteenth

symposium In line with the main goals, special invitations were given to a

number of outstanding technical people to have them participate in one

ca-pacity or another While too many to list individually here, we would be

re-miss indeed to not specifically mention two key contributors These were,

firstly, the stimulating invited keynote lecture on the evolution of fracture

mechanics by Professor George T Hahn of Vanderbilt University, and,

sec-ondly, the informative banquet talk by Jules Duga of Battelle on the

eco-nomic impact of fracture

Some 56 papers were selected for presentation The selection was based

upon technical content and relevance to the theme of the symposium as

evi-denced in three to five page summaries that included key figures and

referen-ces (Bound copies of the summaries were made available to the symposium

participants through the courtesy of Battelle's Columbus Laboratories.) The

selected papers fitted nicely into eight sessions:

1 Elastoplastic analyses

2 Linear elastic analyses

3 Temperature and environment effects

In addition to the formal sessions, two workshop sessions were also held

These were the Workshop on High Temperature Crack Growth and Fracture,

organized by A Saxena and V Kumar, and the Workshop on Nonlinear and

Dynamic Fracture Mechanics, organized by M F Kanninen and C H

Popelar

To conclude, we are pleased to have been associated with this distinguished

series of symposia for many reasons Firstly, we are happy to have been able

to organize and conduct a technical meeting where about 130 individuals

en-riched themselves both technically, through hearing so many fine papers,

and socially, through improved associations with their colleagues in fracture

mechanics Secondly, having Professor Jerry Swedlow of Carnegie-Mellon

University be specially honored for his contributions to ASTM at the

sym-posium banquet was a distinct pleasure to us as well as to his many friends

Thirdly, we are glad to have helped in producing this Special Technical

Pub-lication, which will take its place with its predecessors in this illustrious series

of volumes Finally, we are proud to have helped continue the tradition of

these annual symposia which, at the time of this writing, are being scheduled

through its twentieth year May it continue to grow and prosper!

Linear Elastic Analyses

Stress Intensity Factors for a System

of Cracks in an Infinite Strip

REFERENCE: Civelek, M B., "Stress Intensity Factors for a System of Cracks in an

Infinite Strip," Fracture Mechanics: Sixteenth Symposium, ASTM STP 868, M F

Kan-ninen and A T Hopper, Eds., American Society for Testing and Materials,

Philadel-phia, 1985, pp 7-26

ABSTRACT: The plane elasticity problem is considered for an infinite strip containing

arbitrarily arranged cracks The method of solution involves representing cracks by

continuous distributions of dislocations, and results in a system of singular integral

equations The geometry and other parameters may be arbitrarily specified, including

the number, locations, angles, and lengths of the cracks as well as the loading

param-eters The problem of a semi-infinite strip is solved by letting one of the cracks cut the

strip through Stress intensity factors are presented for various geometries and

loadings

KEY WORDS: crack propagation, fracture (materials), stress analysis, cracks, stress

intensity factors, singular integral equations, numerical quadrature

The plane elasticity solution for an infinite strip containing a line crack

has been considered by Isida [ i ] , Sneddon and Srivastav [2], Krenk [5],

and Erdogan and Arin [4'] In Refs 7 and 2 the crack is perpendicular to the

boundary; in Refs 3 and 4 the crack is arbitrarily oriented The problem of

two symmetrically located collinear cracks perpendicular to the boundary

has been studied by Gupta and Erdogan [5], and recently the problem of

multiple cracks perpendicular to the boundary was treated by Civelek and

Erdogan [(?] The formulation of the basic problem of multiple cracks in an

infinite plane can be found in papers by Datsyshin and Savruk [7] and

Isida [81

The purpose of the present paper is to treat arbitrarily arranged cracks in

an infinite strip under arbitrary loading First, the stress solutions for edge

dislocations arbitrarily placed in an infinite strip are obtained Then, with

the cracks represented by continuous distributions of dislocations, singular

integral equations are systematically derived The numerical solution follows

the Gauss-Chebyschev quadrature method [9,70] Numerical examples are

given for several typical cases of practical importance involving a single

crack or two cracks The crack problem for a semi-infinite strip is also

obtained by considering two cracks, one of them perpendicular to the

boundaries of the strip, and moving the ends of the perpendicular crack to

the strip boundaries

The Integral Equations

Consider an infinite plane containing a pair of point dislocations with

Burgers vectors bx (parallel to x-axis) and by (parallel to >'-axis) located at the

point X — a,y = Q The stress state in the infinite plane may be expressed as

aUx.y) = — cUx.y.a) + — FUx.y,a) ( l a )

Oyyix.y) = -^ Gyy{x.y,a) H Fiy{x.y,a} (lb)

oiyix.y) = — Giy{x,y,a) + — fUx.y.a) (Ic)

In Eq 2 /u is the shear modulus and K = 3 — 4v for plane strain and

K = (3 — v)/{l + f) for generalized plane stress, r being the Poisson's ratio

To determine the stress state due to point dislocations in an infinite strip

0 < X < h parallel to the >'-axis (Fig 1), consider the Airy stress function

equation and solving for </>, one obtains

- / •

^ Jo

U{x,y) =— [ ( ^ , + xA2)e-'"' + (A^ + xA,)e'"'-\{Z7y] da (6)

The stress components will be expressed by using the relations

d^U d^U d^U

(7)

dy dx dxdy

The strip boundary conditions for the residual loading problem are

o,.iO,y) = -aU0,y) (8a) o.y{0,y) = -aiy{0,y) (8fc) oxxih.y) = -aix(h,y) (8c) Oxyih.y) — -aiyih.y) (Sd)

The terms Ai, Ai, A3 and An are determined from these conditions after

taking the transforms of ^^(O.j), aiy{0,y), aix(h,y), and aiy(h,y) Use is made

of the Fourier transforms listed in the Appendix

Substituting Ai {i = \, ,4) in the stress expressions and adding the

infinite plane solution, one obtains

1 Dx i

Oxx{x,y,a) = {—Gxx(x,y,a) + Qi(x,y,a) + Qx(h — x,y,h — a)

7r /•» J

+ / ^, ^ [S\(x,a,a) + S2{x,a,a.) + S\{h — x,h — a,a)

Jo D{a)

+ S2{h — x,h — a,a)]sinaj' da} (9a)

1 Dx i

Oyy{x,y,d) = {-Gyy{x,y,a) + Q2ix,y,a) + Q2{h - x,y,h - a) TT

Jr°° 1 ' ^, , [T[(x,a,a) + T2(x,a,a) + T\(h — x,h — a,a)

0 D{a) + T2(h — x,h — a,a)]sina_>' da] (9b)

~ Uiih — x,h — a,a)]cosaj da\ (9c)

for the case Z);^ 5^ 0, Z)y = 0 (denoted by superscript 1) and

a\x{x,y,a) = {—Fix{x,y,d) + R\(x,y,d) — R\{h — x,y,h — a)

+ / -r-— [Vx{x,a,a) + Viix.a.a) - V,(h - x,h - a.a)

Jo ^ ( a )

~ Viih — x,h — a,a)'\cosay da} (lOo)

Oyy{x,y,a) = {—Fyy(x,y,a) + Riix.y.a) — Ri(h — x,y,h — a)

TT

+ \ ——- {Y^ix.a.a) + Yi{x.a,a) - Yi{h - x,h - a,a)

Jo D(a)

— Yiih — x,h — a,a)]cosay da} (lOft)

aly(x.y,a) = {-Fiy{x,y,a) + Ri(x,y,a) + Ri(h - x.y.h - a)

n

—;— [Zi{x,a,a) + Z2(x,a,a) + ZI(/J — x,h — a.a)

D(a) + Zz(h — x.h — a,a)~\%inay da} (10c)

for the case Dx = Q, DyT^ 0 (denoted by superscript 2) The expressions of

Qu Ri, Si, Ti, Ui, Vi, Yi, Zi 0 = 1 , ,3) and Z>(a) are given in the Appendix

To simulate a line crack in the s direction the stress distributions due to the

dislocations with Burgers vectors bs and br will be used as Green's functions

(Fig 1) The stress distributions can be obtained by resolving the vectors bs

and br into components in the x and y directions and summing the stresses

set up by these components

Introducing

a = cos d , ^ — sm 6

I + K I + K

the stress state due to bs and b'r could be expressed as

oUx.y) = — [(/8A - aDs)G;,x{x,y ~ 0t,d + at)

TT

i^Ds + aDr) F.,(x,y - 0t,d + at)] (1 la)

oyyix.y) = — [(/8A — aDs)Gyy(x,y ~ ^f,d + at)

IT

(PDs + aDr) Fyy(x,y - 0lj + at)] (116)

osyix.y) = — [(PDr - aDs)Gxy(x,y - pt.d + at)

and so on The coordinates of a point on the i-axis are x = d + as, y = 0s;

hence the normal and shear stress components orr and Osr at this point in the

[s, r} coordinate system may be written as

arr{s,0) = 0^axx{d + as,0s) + a^Oyyid + as,0s) - la0(Jxy{d + as.0s) (l2a)

CTsr(j,0) = (a^ - 0^)axy(d + tts,0s) + a0[ayy{d + as,0s) - a«(c/ + ois,0s)]

02b)

A crack along the 5-axis may be represented by continuous arrays of

dislo-cations in the interval (—ai,fli) with the density functions

/ W = T - f - — [v(0\^) - v(0-,5)] {\3a)

I + K ds

g(s) = ~ - -^ [u(0\s} - u(0\s)-\ (13b)

I + K ds

where u and v are displacement components in the {s, r] coordinate system

and for \s\ > ai:

f(s) = g(s) = 0

To obtain the stresses due to continuously distributed edge dislocations, Dr

and Ds are replaced by —f(t)dt and —g{t)dt, respectively, and integrated

along the line crack The unknown density functions will be determined from

the conditions that stresses are prescribed for on the crack surface, which

yield

OrAs.0) = — I k(s,t)fit)dl-\ / h{s.t)git)dt = p{s)

where (—oi < i < ai) (14a)

1 C" 1 C"

<Jsr{s,Q) = — I l{s,t)f(t)dt + — I m(s,t)g(t)dt = q{s)

where (-oi < s < a\) (146)

in which/>(j) and q{s) are crack surface tractions and the kernels are defined by

k{s,t) = p\oiFtx - PG*.) + a\aF*y - ^0%) - 2aP{aF% - PG%) (15a)

h{s.t) = P\PF*, + aG*.) + a\PF*y + aG*y) - lapi/^F^ + aG^) {I5b)

t(s.t) = ia^-p'){aF*y-pGty)

+ aP[a{F^y - F*,) - l3(G*y - G*.)] (15c) m{s,t) = (a' - P'XPFty + aGty)

+ aPllSiFyy - Ft.) + a{G*y - G.*)] (15^)

For the internal crack, the integral equations (I4a and 146) have ordinary

Cauchy kernels and their solution may be obtained by expressing

^(,)=.^m^ , git)=-M= iMa,b)

and using Gauss-Chebyshev integration formula after normalization of the

interval [9,10] Once the density functions/and g are determined, the stress

intensity factors may be evaluated from

Ki(-aO = - ^ lim s/27r(s + «,) /(s) (1 Sa)

2n Kiiai) = - - - p - lim ^/iMaT^) f{s) (18fc)

1 + K '-<•<

2/u

Kui-Oi) = — f - Hm y/2iT{s + a,) g{s) (18c)

2/x

Kuiai) = - —~- lim y/2n(a, - s) gis) (I8d)

where Ki and Kn are Modes I and II stress intensity factors

In the case of an edge crack, the integral equations (14^ and I4b) and the

kernels (Eqs 15a to I5d) remain the same However, the single-valuedness

conditions (Eqs 16a and 16Z») are no longer valid For this case again, the

forms given by Eqs 17a and 176 will be assumed and at the strip boundary

the 4) and i/> functions will be taken equal to zero Thus the functions/and g

will be bounded at the free end

The formulation given above is for a single crack and can easily be

ex-tended for the case of arbitrarily oriented cracks In this case the coordinate

systems [si, r,} (/ = 1,2, ,«) are used (Fig 2) and the expressions for

crack surface tractions yield the following system of singular integral

equations:

— E / kij(s„tj)fj(tj)dtj + I h,j{si.tj)gj(tj)dtj\ = pi{si)

n i=\ L J~a, J-a, J

where —a, < Si < ai (( = 1, ,n) (19a)

E [ J tiAsi.tj)fj(tj)dtj + I my(Si,tj)gj{tj)dfj\ = ^,(.$0

where —a, < st < a (/ = 1, ,n) {\9b)

where {—aj.aj) denotes the tips of t h e y t h crack, and Pi{si) and qi{si) are the

n o r m a l a n d shear stresses acting o n the crack surfaces, respectively The

ker-nels of the integral equations are given by

kij{s,tj) = p,\ajFl - PjGl) + ot^{ajF*y - HjG^)

- 2aiPi((XiF% - ^jG%) (20a) hij{Si,tj) = l3i\l3jF*, + ajG*,) + a^{lijF*y + ajGyy)

- 2aifiifijF*y + ajGty) (206) l,j{Si,tj) = ( « , ' - l3d(ajF*y - pjG*y) + a,l3i[aj(F^y - F*,)

- Pj{G*y - G*.}] (20c) mij{si,tj) = (a," - l3i'mF*y + ajG%) + a,A[A(^*^ - ^ * )

Again, to provide continuity of the material outside the cracks, the valuedness conditions

The technique for solving the system of singular integral equations (19a

and I9b) is the same as for the single crack case The stress intensity factors

will be defined for each individual crack in a similar fashion

Results and Discussion

The first problem considered is the problem of a single crack in an infinite strip In Fig 3 a uniform pressure is applied on the crack surfaces and in Fig

4 the crack is loaded by uniform shear Note in Fig 3 that Kn changes sign at

FIG 4—Normal and shear components of the stress intensity factors (at s = a\)for a crack in

an infinite strip, under uniform shear TO

6 — 60° for the uniform pressure loading These results are in good

agree-ment with the results given in Ref 3

In Fig 5 the strip is under uniform tension and contains an oblique edge

crack Under plane strain conditions this problem corresponds to the long

part-through crack problem It may be seen from the figure that K'l decreases

with increasing 6 while ^"'11 reaches a peak value then it decreases As may be

seen, both stress intensity factors tend to zero when the crack tip approaches

the free surface as 6 tends to 90°

Figures 6 and 7 show some results for the interaction of two internal

cracks in a strip under uniform tension For this geometry take n = 2,

d\ = di = 0.5h and define 01 = ^2 = c, d\ = ~di = 6, ei = —ei = e From

the results for c/h = 0.1 it may be concluded that ^i values always decrease

with increasing 6 and decreasing e, while Ku values show different variations

FIG 5—Normalized stress intensity factors for an oblique edge crack in an infinite strip (K'l :

Ki/oo\/ST K'li = Kn/aoyfrri)

FIG (>—Normal components of the stress intensity factors/or two internal cracks in an infinite

strip under uniform tension (K./' = Ki(—c)/ao\/7rcT Ki"^ = Ki(c)/ao\/7rc)

Kn ,Kn

e

FIG 7—Shear components of the stress intensity factors for two internal cracks in an infinite

strip under uniform tension (Kn = Kn(—c)/oo \Arc, Kn = Kii (c)/(7o V ^ )

-As expected, ^ii as a function of 6 is maximum for d in the neighborhood of

45° When e gets very large the single crack solution is obtained for both

cracks More results on crack interaction problems (especially for edge

cracks) can be found in Ref 6 These results will not be reproduced here In

Ref 6 the two edge cracks considered are symmetrically located in the strip

With the present formulation, the edge cracks could be arbitrarily oriented;

however, only the case of two edge cracks located on opposite sides will be

considered and the crack lengths will be taken as equal (Fig 8) As expected,

^11 becomes zero when the cracks are collinear or far enough

In the last example two cracks, one of them perpendicular to the strip

edge, are considered; the perpendicular crack cuts the strip through This

way half the strip can be removed and the other semi-infinite strip contains a

crack Figure 9 displays the variation of the stress intensity factors of a crack

OA 0.6

h

FIG 8—Stress intensity factors Ki and Ku for the crack on the left in the case of two edge

cracks on opposite sides

K'l.K',

FIG 9—Normalized stress intensity factors for a crack in a semi-infinite strip under uniform

tension (Ki = Ki/ooVirai, Kii = Kii/CTo\/7rai)

in a semi-infinite strip that is under uniform tension when the center of the

crack is fixed and its orientation is varied In Fig 10 the crack is parallel to

the short boundary and it is loaded by uniform pressure If a small crack

length {lax = 0.1) is chosen, the stress intensity factors are almost identical,

with the values belonging to the pressurized crack located parallel to the

sur-face of an elastic half-space It may also be seen that if the crack length, 2a\,

FIG 10—Normalized stress intensity factors for a crack in a semi-infinite strip under uniform

pressure CTO (Ki = Ki(ai)/oo\/7rai, K'n = Kii(ai)/ao\/'rai)

is equal to 0.8A the stress intensity factors increase rapidly as the crack

ap-proaches the short boundary These calculated stress intensity factors will be

useful when an appropriate mixed-mode criterion is used to predict failure;

they can also be used in the maximum tensile stress theory [72] to predict the

crack growth direction

Conclusions

1 Using dislocation solutions and employing superposition principles

have proven to be helpful in fracture analysis

2 It has been demonstrated that by letting one of the cracks cut the strip

through the semi-infinite strip geometry is obtained

3 The formulation presented offers a certain flexibility that becomes very

useful in dealing with the cracks in nonhomogeneous materials

APPENDIX

Below are the Fourier transforms used for the evaluation of the transforms of

aix(0,y), aiy(0,y), aix(h,y), and aiy(h,y) which appear in Eq 8:

' sin(ay)dy = — e~°"0 + aa) (a > 0)

, , ^ , 2 ^m{ay)dy = ^ e-^(\ - aa) (a > 0)

The functions D(a) and g„ i?„ Si, Ti, Ui Vi, K,, Z, (; = 1, ,3), which appear in

Eqs 9 and 10, are defined as follows:

D{a) = Wh^ - e'"* - c"'"'' + 2

/ + (fl + xf [>•' + (a + xff [y' + (fl + xff

y 2X3fl-x)(fl + x) , 4ax>'[3(a + X ) ' - / ] QiiXA'.a) = :; , :; r-r H ; r-;

y' + {a + x)' [y' + ia + xfr [y'+ (a + xff (X - a)[(a + xf - y'-j , 4ax(a + x)[(,a + xf - 3y']

Qi(x,y,a) — ; T— —,

[y' + (a + xff [y' + ia + xfV {x - a)[(a + xf - / ] 4ax(a + x)[{a + xf ~p^]

R,(x,y.a) =

R2(x,y,a) =

[y^ + (a + xff h^ + (a + xff 2(a + x) (3a + x)[(a + xf - / ] y' + (a + xf [}' + (a + xff

+ la'ax + la'ah ~ laV + la^hxy^'*'^

Si(x.a.a) = [(1 - aa + ax)(e''"^ - 1) + lah(aa - ah - \)

- 2a^hx(\ + 2ah - 2aa)]e°"'""

Ti(x,a.a) = [(4aV - ^"^°'')(1 + 3aa - a x - la^ax) + 1 + 3aa - a x

+ lah - la'ax - 2a'ah + 2a'h' - 2o'/ix]e"""'*"'

Ttix.a.a) = [(1 - ax + aa){e'^'^ - 1) - 2ah(aa - ah - \)

+ 2ah{ax - 2)(1 + 2ah - 2aa)]e°'°""

U\{x,a,a) — [(e'^"'' — 4a^h^)(ax — aa + 2a^ax) — ax + aa — 2a^ax

- 2a'ah + 2a'h' - 2a'hx]e-°"*'^

U2{x,a,a) = [(aa - ax)(e''^°'' - I) - 2ah(aa - ah - I) - 2ah(l - ax)

(l+2ah- 2aa)]e''"'""

Vi(x,a,a) = [(e"^"* - 4a^h^)(ax - aa - 2 a W ) + aa - ax + 2a^ax

+ 2a^ah - 2aW + 2a^hxY''^''*'^

V2(x,a,a) = [(aa - ax)(e'^°'' - I) + 2ah(ah - aa) - 2a^hx(\ - 2ah

+ 2afl)]e'"''""'

Yi(x,a.a) = [(e"^"* - 4a^;j^)(2 - 3aa - ax + 2a^ax) - 2 + 3aa + a;«: + 4ah

- 2a'ax - 2a'ah + 2a'h' - 2a^hx]e-''""^

Y2(x,a,a) = [(ax - aa - 2)(e"^"'' - 1) + 2aA(aa - ah) + 2ah(ax - 2)

(\-2ah + 2aa)]e""'""

Zi(x,a,a) = [(e"^"* - 4a^ft^)(aa + a x - 1 - 2a^ax) + \ - aa - ax - 2ah

+ 2a^ax + 2a^ah + 2a'hx - 2a^/j']e"''"'*''

Z2(x,a,a) = [(aa - a x + l)(e'^°* - 1) + 2a;j(a;j - aa) + 2ah(\ - ax)

[3] VivenV.,S., InternationalJournal of Solids and Structures, Vol 11, 1975, p 693

[4] Erdogan, F and Arin, K., International Journal of Fracture, Vol 11, No 2, 1975, p 191

[5] Gupta, G D and Erdogan, F., Journal of Applied Mechanics, Transactions of ASME, Vol

41, No 4, 1974, p 1001

[6] Civelek, M B and Erdogan, F., InternationalJournal of Fracture, Vol 19, 1982, p 139

[7] Datsyshin, A P and Savruk, M P., Journal of Applied Mathematics and Mechanics, Vol

37, 1973, p 326

[5] Isuia, M., Bulletin of the Japan Society of Mechanical Engineers, Vol 13, 1979, p 635

[P] Erdogan, F and Gupta, G D., Quarterly of Applied Mathematics, Vol 29, p 525

[70] Krenk, S., Journal of the Institute of Mathematics and Its Application, Vol 22, 1978, p 99

[/;] Head, A K., Proceedings of the Physical Society (London), Vol, 66B, 1953, p 793

[ i 2 ] Erdogan, F and Sih, G C , Journal of Basic Engineering, Transactions ofASME, Vol 85,

1963, p 519

Wide-Range Displacement

Expressions for Standard Fracture

Mechanics Specimens

REFERENCE: Kapp, J A., Leger, G S., and Gross, B., "Wide-Range Displacement

Expressions for Standard Fracture Mechanics Specimens," Fracture Mechanics: Sixteenth

Symposium, ASTM STP 868, M F Kanninen and A T Hopper, Eds., American

So-ciety for Testing and Materials, Philadelphia, 1985, pp 27-44

ABSTRACT: Wide-range algebraic expressions for the displacement of cracked fracture

mechanics specimens are developed For each specimen two equations are given: one

for the displacement as a function of crack length, the other for crack length as a

func-tion of displacement All the specimens that appear in ASTM Test for Plane-Strain

Fracture Toughness of Metallic Materials (E 399) are represented in addition to the

crack mouth displacement for a pure bending specimen For the compact tension

sample and the disk-shaped compact tension sample, the displacement at the crack

mouth and at the load line are both considered Only the crack mouth displacements

for the arc-shaped tension samples are presented The agreement between the

displace-ments or crack lengths predicted by the various equations and the corresponding

nu-merical data from which they were developed are nominally about 3% or better These

expressions should be useful in all types of fracture testing including J\c, KR, and fatigue

crack growth

KEY WORDS: fracture testing, displacement solutions, displacement expressions

In order to determine the many fracture properties available to characterize

materials many parameters of the specimens used must be known Foremost

among these parameters is the crack length Measuring properties such as

J-resistance curves, K^-J-resistance curves, and fatigue crack growth rates, the

crack length changes must be measured during the actual test Several

tech-niques have been devised to measure crack growth by instrumenting the

sample Perhaps the simplest method is the so-called "compliance"

tech-nique where the elastic compliance of the specimen is measured by

simul-' Materials Engineer, U S Army Armament, Munitions, and Chemical Command, Armament

Research and Development Center, Large Caliber Weapon Systems Laboratory, Benet Weapons

Laboratory, Watervliet, NY 12189-5000

' Graduate Student, Mechanical Department, University of New Mexico, Albuquerque, NM

87106

taneously measuring the load and displacement of the sample Since the

elas-tic load-displacement characteriselas-tics of any cracked body are a function of

the elastic properties of the material tested and specimen parameters

(includ-ing crack length), the elastic properties or the crack length for any sample

can be determined if the other is known

Presented in this paper are algebraic equations that allow for the easy

cal-culation of either crack length or elastic properties for many standardized

specimens

In order to determine the elastic properties of the material from which the

specimen was made, an expression which gives the load-displacement

charac-teristics as a function of crack length must be developed Similarly, to

calcu-late the crack length, an expression must be developed where the crack length

is a function of the load and displacement measured Expressions of both

types were developed for all the standardized specimens in ASTM Test for

Plane-Strain Fracture Toughness of Metallic Materials (E 399) and for a

rec-tangular pure bending sample

The expressions were developed by first establishing the appropriate

limit-ing displacements as the crack approaches zero length and as the remainlimit-ing

ligament approaches zero This was accomplished using the Paris equation

[7] based on Castigilliano's theorem These limiting solutions serve as guides

when choosing a nondimensional form of the specimen displacements which

has finite limits at short and long crack lengths Numerically determined

displacements were then normalized to the nondimensional form derived

from the limiting solutions, and multivariable regression was used to fit a

polynomial to these data The resulting algebraic equations represent the

displacement solutions nominally within 1% over a wide range of specimen

parameters

Procedure for Developing Nondimensional Displacements

To determine an appropriate nondimensional form of displacements for

the various specimens considered, Paris's application of Castigilliano's

theo-rem to crack problems [2] was used For the general two-dimensional cracked

body shown in Fig 1, the displacement due at the location at the applied

force F in the direction of F is

2 f dKf

— da (1)

where Kp is the stress intensity factor due to the force P, KF is the stress

inten-sity factor due to the force F, and £" is Young's modulus {E) for plane stress

or E/(\ — V ) {v = Poisson's ratio) for plane strain

For the short crack limit we use the K solution for a finite crack in a

semi-FIG 1—A general two-dimensional cracked body

infinite medium By placing a dummy load F a t the crack mouth and replacing

the load P with a uniform stress a, the crack mouth opening displacement

can be derived The various K solutions necessary for Eq 1 are [2]

A similar approach was taken for the long crack limit In this case, we

as-sume that for tension samples, the normal component is negligible Thus only

one A'solution is necessary, namely that of a semi-infinite crack in a

semi-in-finite medium subjected to a moment M For Eq I, A'F = ^^ = 3.915M/b^'^, where b is the uncracked ligament {W — a) With this approach, integrating

Eq 1 gives us the angle of rotation of the two crack surfaces {6m)'

EXW-afB