Astm stp 803 v1 1983

These theioretiCal and analyti-cal developments, combined with progress in test method development and ductile fracture toughness characterization led to substantial growth in engi-neeri

ASTM SPECIAL TECHNICAL PUBLICATION 803

C F Shih, Brown University, and

J P Gudas, David Taylor Naval Ship R&D Center, editors

ASTM Publication Code Number (PCN) 04-803001-30

#

1916 Race Street, Ptiiladelphia, Pa 19103

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

Printed in Baltimore, Md (b) November 1983

Foreword

The Second International Symposium on Elastic-Plastic Fracture

Mechan-ics was held in Philadelphia, Pennsylvania, 6-9 Oct 1981 This symposium

was sponsored by ASTM Committee E-24 on Fracture Testing C F Shih,

Brown University, and J P Gudas, David Taylor Naval Ship Research and

Development Center, presided as symposium chairmen They are also editors

of this publication

ASTM Publications

Fracture Mechanics (13th Conference), STP 743 (1981), 04-743000-30

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

Crack Arrest Methodology and Applications, STP 711 (1980), 04-711000-30

Fracture Mechanics (12th Conference), STP 700 (1980), 04-700000-30

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

A Note of Appreciation

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 Virginia M Barishek

Acknowledgments

The editors would like to acknowledge the assistance of Professor G R

Irwin, Dr J D Landes, Professor P C Paris, and Mr E T Wessel in

plan-ning and organizing the symposium We are grateful for the support provided

by the ASTM staff, particularly Ms Kathy Greene and Ms Helen M

Hoersch The timely submission of papers by the authors is greatly

appreci-ated Finally, this publication would not have been possible without the

tre-mendous effort and dedication that was put forth by the many reviewers

Their high degree of professionalism ensured the quality of this publication

The editors also wish to acknowledge the diligent assistance of Ms Susan

Beigquist, Ms Ann Degnan, Mr Steven Kopf, and Mr Mark Kirk in

preparing the index

J P Gudas

C F Shih

Introduction I-l

ELASTIC-PLASTIC CRACK ANALYSIS Dynamic Growth of an Antipiane Shear Crack in a Rate-Sensitive

Elastic-Plastic Material—L B FREUND AND A S DOUGLAS 1-5

Elastic Field Surrounding a Rapidly Tearing Crack—A s KOBAYASHI

AND O S LEE 1-21

Elastic-Plastic Steady Crack Growth in Plane Stress—A H DEAN 1-39

A Finite-Element Study of the Asymptotic Near-Tip Fields for Mode I

Plane-Strain Cracks Growing Stably in Elastic-Ideally Plastic

S o l i d s — T - L SHAM 1-52

Crack-Tip Stress and Deformation Fields in a Solid with a Vertex on

Its Yield Surface—A NEEDLEMAN AND V TVERGAARD 1-80

The Je^-Integral Based on the Concept of Effective Energy Release

R a t e — H MIYAMOTO, K KAGEYAMA, M KIKUCHI, AND

K MACHIDA 1-116

A Criterion Based on Crack-Tip Energy Dissipation in Plane-Strain

Crack Growth Under Large-Scale Yielding—M SAKA,

T SHOJI, H TAKAHASHI, AND H ABE 1-130

Discontinuous Extension of Fracture in Elastic-Plastic Deformation

Field—M P WNUK 1-159

Influence of Compressibility on the Elastic-Plastic Field of a

Growing Crack—Y.-C GAO 1-176

Material Resistance and Instability Beyond /-Controlled Crack

G r o w t h — H A ERNST 1-191

An Elastoplastic Finite-Element Investigation of Crack Initiation

Under Mixed-Mode Static and Dynamic Loading—T AHMAD,

C R BARNES, AND M F KANNINEN 1-214

Elastic-Plastic Analysis of a Nozzle Comer Crack by the

Finite-Element Method—w BROCKS, H H ERBE, H D NOACK,

AND H VEITH 1-240

Elastic-Plastic Finite-Element Analysis for Two-Dimensional Crack

FULLY ELASTIC CRACK AND SURFACE FLAW ANALYSIS

Bounds for Fully Plastic Crack Problems for Infinite Bodies—

M Y HE AND J W HUTCHINSON 1-277

Penny-Shaped Crack in a Round Bar of Power-Law Hardening

Material—M Y HE AND J W HUTCHINSON 1-291

Elastic-Plastic and Fully Plastic Analysis of Crack Initiation, Stable

Growth, and Instability in Flawed Cylinders—v KUMAR,

M D GERMAN, AND C F SHIH 1-306

A Superposition Method for NonUnear Crack Problems—

G YAGAWA AND T AIZAWA 1-354

Consistency Checks for Power-Law Calibration Functions—

D M PARKS, V KUMAR, AND C F SHIH 1-370

Ductile Growth of Part-Through Surface Cracks: Experiment and

Analysis—c s WHITE, R O RITCHIE, AND D M PARKS 1-384

Evaluation of J-Integral for Surface Cracks—M SHIRATORI AND

T M I Y O S H I 1-410

Effects of Thickness on J-Integral in Structures—M SAKATA,

s AOKI, K KISHIMOTO, M KANZAWA, AND N OGURE 1-425

J-Integral Analysis of Surface Cracks in Pipeline Steel Plates—

R B KING, Y.-W CHENG, D T READ, AND H I McHENRY 1-444

Use of J-Integral Estimation Techniques to Determine Critical

Fracture Toughness in Ductile Steels—G GREEN AND

L MILES 1-458

Evaluation of Plate Specimens Containing Surface Flaws Using

J-Integral Methods—w G REUTER, D T CHUNG, AND

C R EIHOLZER 1-480

Conditions—H RIEDEL 1-505

Stable Crack Extension Rates in Ductile Materials: Characterization

by a Local Stress-Intensity Factor—E W HART 1-521

Cracks in Materials with Hyperbolic-Sine-Law Creep Behavior—

J L BASSANI 1-532

Stress Concentrations Due to Sliding Grain Boundaries in Creeping

A l l o y s — G W LAU, A S ARGON, AND F A McCLINTOCK 1-551

Steady-State Crack Growth in Elastic Power-Law Creeping

Materials—c Y HUI 1-573

Effects of Creep Recovery and Hardening on the Stress and

Strain-Rate Fields Near a Crack Tip in Creeping

Materials—s KUBO 1-594

On the Time and Loading Rate Dependence of Crack-Tip Fields

at Room Temperature—A Viscoplastic Analysis of

Tensile Small-Scale Yielding—M M LITTLE, E KREMPL,

AND C F SHIH 1-615

Boundary-Element Analysis of Stresses in a Creeping Plate with a

Estimates of the C"' Parameter for Crack Growth in Creeping

Materials—D J SMITH AND G A WEBSTER 1-654

Crack Growth in Creeping Solids—v M RADHAKRISHNAN AND

A I McEVILY 1-675

Parametric Analysis of Creep Crack Growth in Austenitic Stainless

Microstructural and Environmental Effects During Creep Crack

Influence of Time-Dependent Plasticity on Elastic-Plastic Fracture

Toughness—T INGHAM AND E MORLAND 1-721

Index 1-747

STP10803-EB/NOV 1983

Introduction

In October 1981, ASTM Committee E24 sponsored the Second

Interna-tional Symposium on Elastic-Plastic Fracture Mechanics which was held in

Philadelphia, Pennsylvania The objective of this meeting was to provide a

forum for review of recent progress and introduction of new concepts in this

field The impetus for this symposium was the historical development of

elas-tic-plastic fracture technology Concepts such as the J-Integral, COD, and

HRR field generated tremendous interest which led to the First International

Symposium held in 1977 The presentation and publication of works on such

topics as J-controlled crack growth, tearing instability, and numerical

de-scription of crack tip fields, among others, led to major, broad-based

re-search activities and application-oriented developments This sustained

growth provided the motivation for another meeting devoted solely to

elastic-plastic fracture

The call for papers for the Second International Symposium generated an

overwhelming response This was reflected by the number of papers

pre-sented, and the attendance which exceeded 300 participants The papers

sub-mitted to this symposium underwent rigorous review The works contained in

these two volumes reflect the high degree of interest in this subject and the

quality of the efforts of the individual authors

In the first ASTM publication devoted to elastic-plastic fracture (ASTM

STP 668), there were three major groupings including elastic-plastic fracture

criterten and analysis, experimental test techniques and fracture toughness

data, amd applications of elastic-plastic methodology The present collection

of papers shows stittstantial growth in theoretical and analytical areas which

now include topics fanging from fundamental analysis of crack growth under

static and dynamic conditions, finite strain effe^s at the crack tip, elevated

temperature effects, visCO-plastic crack analysis, and tractable treatments of

fully plastic crack probfems and surface flaws These theioretiCal and

analyti-cal developments, combined with progress in test method development and

ductile fracture toughness characterization led to substantial growth in

engi-neering application of elastic-plastic fracture methodologies as evidenced by

the large selection of papers on this topic

The papers in these two volumes ha«e %&SM grouped into six topic areas

incloding elastic-plastic crack analysis, fii% plastic crack and surface flaw

analysts, visco-plastic crack anjalysis and ecw^fcition, engineering

applica-tions, test methods and gpomietry effects, and cysfic plasticity effects and

ma-1-1

terial characterization The first grouping contains papers on crack

propaga-tion under static and dynamic condipropaga-tions, crack growth and fracture criteria,

finite strain effects on crack tip fields, and plasticity solutions for important

crack geometries and structural configurations Fully plastic crack solutions,

elastic-plastic line-spring models, approximate treatment of surface flaws,

and surface flaw crack growth correlations are the focus of the second

group-ing of papers An area not addressed in the first symposium and which has

since attracted significant attention is crack growth at elevated temperature,

and time dependent effects Papers on this topic address theoretical aspects

of creeping cracks, computational procedures, microstructural modelling,

creep crack growth correlations, and materials characterization

The second volume of this publication begins with the major section on

engineering applications This includes several papers on tearing instability,

J-based design curves, evaluations of several fracture criteria, further

devel-opments of fracture analysis diagrams, and flawed pipe analyses This is

fol-lowed by a series of papers on test methods and geometry relationships A

majority of the papers focus on /jc and Jj-R-curve test procedures and

compu-tations, and several papers address the test specimen geometry dependence of

these parameters In the last section several papers are devoted to prior load

history effects, crack growth in the elastic-plastic regime under cyclic loading,

and micromechanism studies of the fracture process

The collections of papers from this and the previous symposium contain

many of the major works in the rapidly evolving subject of elastic-plastic

frac-ture It is hoped that these two volumes will serve to stimulate further

prog-ress in this field

Elastic-Plastic Crack Analysis

Dynamic Growth of an Antiplane

Shear Crack in a Rate-Sensitive

Elastic-Plastic Material

REFERENCE: Freund, L B and Douglas, A S., "Dynamic Growth of an Antiplane

Shear Crack in a Rate-Sensitive Elastic-Plastic Material," Elastic-Plastic Fracture:

Sec-ond Symposium, Volume I—Inelastic Crack Analysis, ASTM STP 803, C F Shih and

J P Gudas, Eds., American Society for Testing and Materials, 1983, pp I-5-I-20

ABSTRACT: A numerical study of the dynamic steady-state antiplane shear (Mode III)

crack growth process is described The study is based on continuum mechanics, and the

material is modeled as being elastic-viscoplastic Using the small-scale yielding concept of

fracture mechanics and allowing only for steady-state crack growth, but including the

effects of material inertia explicitly, a full-field solution for the deformations is obtained

by means of an iteration procedure involving the finite-element method Numerical

results for the strain distribution in the active plastic zone are presented The fracture

criterion for materials which fail in a locally ductile manner proposed by McClintock and

Irwin [/],•' which stipulates that crack growth will proceed such that a critical strain level

is maintained at a characteristic distance ahead of the crack tip, is adopted This criterion

is coupled with the results of numerical calculations to develop theoretical dynamic

frac-ture toughness versus crack tip speed relationships for two levels of critical strain For

rate-dependent materials a range of toughness-versus-speed relations is manifested,

rang-ing from that similar to rate-independent materials for low rate sensitivity to a

relation-ship which rises dramatically for even a small increase in crack speed at low crack speeds,

but which levels off for higher speeds for extremely rate-sensitive materials

KEY WORDS: fracture (materials), elastic-plastic crack propagation, dynamic crack

propagation, crack propagation, small scale yielding, finite element method,

elastic-plastic fracture

The in-plane modes of deformation (Modes I and II) are of main interest in

fracture mechanics, but mathematical difficulties encountered in their

analy-sis have prevented complete descriptions of their features, especially for

dy-'Professor, Division of Engineering, Brown University, Providence, R.I 02912

^Assistant professor, Department of Mechanical Engineering, Johns Hopkins University,

Baltimore, Md.; formerly, research assistant Brown University, Providence, R.I 02912

'The italic numbers in brackets refer to the list of references appended to this paper

1-6 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

namic or elastic-plastic systems The difficulties for the antiplane mode of

deformation (Mode III) are not as severe, and several features of in-plane

crack deformation have been anticipated from Mode III solutions Here,

dy-namic steady-state crack growth in the antiplane shear mode through a

strain-rate-sensitive elastic-plastic material is studied The inertial resistance

of the material to motion is included explicitly but for this steady-state

situa-tion (in which the crack-tip speed is constant, and has been for all time), the

deformation is time independent as viewed by an observer fixed at the moving

edge of the crack The full deformation field is determined by means of an

iteration procedure involving the numerical finite-element method, under

conditions of the small-scale yielding hypothesis of fracture mechanics [2],

According to this hypothesis, the elastic-plastic field in the crack-tip region is

controlled by the surrounding elastic field A useful measure of this

surround-ing elastic field, for any crack tip speed v, is the linear elastic stress-intensity

factor A:, which is assumed to be known from the solution of the relevant

elas-tic crack problem

The same problem as that being considered here, but for rate-independent

material response and with inertial effects neglected, was studied in some

de-tail by Rice [3] and Chitaley and McClintock [4] Each employed incremental

plastic stress-strain relations based on the associated flow law and a yield

con-dition suitable for antiplane shear in a nonhardening material They showed

that the principal shear lines in the active plastic zone are straight, and Rice

[3] determined the distribution of plastic strain on the line directly ahead of

the crack tip in terms of the unknown distance between the tip and the

elastic-plastic boundary Chitaley and McClintock [4] went on to integrate

numeri-cally the field equations In a more recent study of the same problem Dean

and Hutchinson [5] noted some discrepancies between their finite-element

calculations and the numerical results of Chitaley and McClintock Whereas

the latter authors assumed that the principal shear lines in the active plastic

zone were all members of a centered fan, the results of Dean and Hutchinson

suggest that, in the portion of the active plastic zone adjacent to the

unload-ing boundary, the shear lines do not form a centered fan

Progress on the corresponding steady-state dynamic problem, with

mate-rial inertia taken into account but for rate-independent matemate-rial response,

has been reported by Douglas, Freund, and Parks [6] and by Freund and

Douglas [7] In Ref 6, a numerical procedure similar to that developed by

Parks, Lam, and McMeeking [8] and by Dean and Hutchinson [5] was used

to find the complete deformation field for several values of crack propagation

speed It was found that the amount of plastic strain at a fixed distance ahead

of the moving crack tip decreased with increasing crack-tip speed, for a fixed

level of remote stress-intensity factor An analysis leading to an exact

expres-sion for the strain distribution on the crack line within the active plastic zone

for any crack propagation speed was presented in Ref 7, and the strain

distri-butions for selected speeds obtained previously by means of the numerical

method compared very well with the exact results Furthermore, the

analyti-cal and numerianalyti-cal results were combined with the "critianalyti-cal plastic strain at a

characteristic distance" crack growth criterion in Ref 7 to generate

theoreti-cal fracture toughness versus crack speed relationships The results showed a

strong dependence of fracture toughness on crack speed for even moderate

crack growth rates Because the material response was independent of strain

rate in the material model used, the results suggested that the influence of

inertia on fracture resistance is quite significant

During rapid crack propagation, material particles very near to the crack

tip experience very high strain rates Thus, for materials for which the flow

stress has a significant dependence on strain rate, it is expected that

strain-rate effects will also influence the fracture toughness versus crack speed

rela-tionship To quantify the level of this influence, the numerical method which

has been used in our earlier work on dynamic elastic-plastic crack growth has

been modified to accommodate a particular model of elastic-plastic material

response in which the flow stress depends on the instantaneous strain rate

The features of the material model and the results of some calculations are

described in the following sections

Governing Equations

All variables are referred to a translating set of Cartesian axes labelled

{xi, JC2> ^3) in the body The Xyaxis coincides with the crack edge, and this

edge moves in the jrj-direction with speed v The displacement vector is in the

:c3-direction and its magnitude U3 depends only on position {xi, X2)- The

equation of motion is

3ffi3 da23 _

+ -T P«3 (1)

dxi dx2 where a^j and ff23 are the nonzero stress components and p is the mass density

of the material The dot denotes the material time derivative Because only

steady-state solutions are sought, the material time derivative may be

re-placed by a spatial gradient as follows

"3 = ~v—— (2)

axi The shear strain components corresponding to the stress components 0^3 and

ff23 are €13 and 623, respectively It is convenient to introduce the following

nondimensional quantities:

1-8 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

To = flow stress of the material for very slow loading in shear,

fi = elastic shear modulus,

c = yln/p = elastic wave speed, and

a = Vl — vVc^

The strain-displacement relations in nondimensional form are

7 = ^ ^ T, = a - g ^ (4)

The relationship between the stress and elastic strain, and the additivity of

elastic and plastic strain components, may be expressed as

rp = 7 / = 7^ - 7 / &=x ov y (5)

Using Eqs 3 and 4, the equation of motion may be written in the form

bx^ by^ a^ dx a dy The asymptotic behavior of tM stress components as r = yJx^ + y^ -* «

is chosen to be consistent with the small-scale yielding hypothesis Thus,

the stress components are required to approach the elastic near-field limits,

that is

sin (e/2) cos (6/2)

7^=^ Tv -" n,— • (7)

where 6 = arctan (y/x) as r ^ 00 The crack faces are traction-free

The constitutive model used is of the overstress type which was introduced

by Malvern [9] in his study of rate effects in uniaxial stress wave propagation

Malvern's stress-strain rate relation was subsequently generalized to general

stress states by Perzyna [10], and these results were embedded in a general

framework for time-dependent plastic response by Rice [11] It is assumed

here that the material responds isotropically, that the strain rate vector is

co-linear with the stress vector, and that the magnitude of the strain rate is

pro-portional to the difference between the current stress magnitude and the

ini-tial yield stress magnitude raised to some power In terms of nondimensional

variables, the plastic strain rate is given by

where h is the actual strain rate sensitivity parameter The latter is defined as

that strain rate at which the instantaneous flow stress magnitude is twice the

slow-loading flow stress TQ The response is governed by Eq 8 as long as the

strain rate is high enough to result in a positive stress rate f If the strain rate

magnitude falls below the level required to meet this criterion, but the strain

magnitude continues to increase, then the deformation proceeds with f = 0

If the strain magnitude begins to decrease, then the material responds

elasti-cally This description of behavior seems to be consistent with the limited

ex-perimental data obtained for changes in strain rate at the very high strain

rates which are anticipated in the crack-tip region during rapid crack

propa-gation (see Lipkin, Campbell, and Swearengen [12] and Chiem and Duffy

[13]) The essential features of the material model are shown for uniaxial

re-sponse in Fig 1

Numerical Procedure

In order to develop the numerical procedure used, the governing field Eq 6

is recast into variational form Both sides of Eq 6 are multiplied by an

arbi-trary function, say w*, which has the same smoothness properties and

satis-fies the same kinematic constraints as w, and are then integrated over any

portion A of the x,y-plant not containing the crack itself The result of

apply-ing the divergence theorem to the equation obtained is

bA = closed boundary of A,

V = two-dimensional gradient operator, and

n = unit normal to bA pointing out of the region A

1-10 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

FIG 1 —Schematic diagram of uniaxial material response for representative strain rate

histo-ries: (1) uniform "low" strain rate, (2) uniform "high" strain rate, and (3) nonuniform strain

rate

The rectangular region — X<x<X,Q<y< Y (see Fig 2) is then

divided into finite elements, and the displacement field is approximated by

discrete interpolation functions <A,-, that is,

where summation is implied and 6, are the values of the displacement at the N

discrete node points Likewise,

x - r , 2^0 2 ,

^ — TIP ^ V x.T(f

CRACK

FIG 2—Plane of deformation, showing a coarse representation of the finite-element mesh

With the finite-element formulation and with 6,- arbitrary, the variational

statement of the governing Eq 10 may be reduced to the system of linear

(15)

1-12 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

where the path L^ is the perimeter of the rectangular region shown in Fig 2

except for the portion of the left-hand boundary which is in the wake region

and which is identified a s i As is evident in Eq 15, the boundary tractions

imposed on Lg are consistent with the far-field small-scale yielding stresses

(Eq 7) For this identification to be reasonable, points on Lg should be far

from the crack tip when compared with plastic zone size

In order to solve these equations for the unknown displacements 6,-, the

plastic strains must be known These are, however, unknown a j^non, and an

iterative procedure must be adopted There are two main steps to the iteration

procedure used here One step involves the determination of the total strain

distribution satisfying the momentum equation, subject to the specified

boundary conditions, for any distribution of plastic strain This is

accom-plished with the finite-element method The other step involves the

calcula-tion of the stress distribucalcula-tion for a given distribucalcula-tion of total strain through

integration of the incremental stress-strain relation along a j ; = constant

"pathline" from x = X to x = —X If the strain rate is large enough for Eq 8 to

hold, then the stress at an integration station is determined by

straightfor-ward application of Newton's rule to Eq 8 at the previous integration station

on the same pathline If the strain rate falls below the level required for Eq 8

to hold, but plastic flow continues, then the stress is determined by sequential

application of the method of Rice and Tracey [14] at integration stations along

a pathline The details of this numerical integration procedure are described

inRef/5

The stress and deformation fields for a range of values of the parameter H

are obtained by starting with the value H = 0, which reproduces the

well-known elastic results This parameter is then increased a small amount and

the iteration procedure, starting from the final results for the previous value

oiHas an initial guess, is followed until convergence is achieved The steps in

the numerical procedure may be summarized as follows:

1 Calculate the elastic displacements (that is, those for J^ = 0) at the

nodes Use these as the first estimate, say 5,(1)

2 Increase the value ofHa small amount

3 Calculate the total strains 7^(1) from the rth estimate of displacement

7^(0 by means of the strain-displacement relations

4 Determine whether or not the stress response at each integration station

is rate-sensitive or slow-loading; if rate-sensitive, go to Step 5; if slow-loading,

go to Step 6

5 Integrate Eq 8 to calculate the stresses Go to Step 7

6 Use the method of Rice and Tracey [14] to determine the stresses Go to

Step 7

7 Determine the plastic strains y/ii) according to Eq 5

8 Calculate the (i + l)th estimate of displacement 8j{i + 1) from Eq 13

9 Compare dj{i + 1) with 6,(z) If the difference is sufficiently small, go to

Step 8; otherwise, increment the index i by 1 and return to Step 3

10 liH is large enough, the process is complete; otherwise, reset / = 1 and

return to Step 2

The finite-element calculations were based on an array of 2800 rectangular

elements, each comprising four constant strain triangles formed by the

diago-nals Nodal points were concentrated near the crack tip and element size

in-creased with distance from the crack tip The size of the smallest element was

approximately 0.2 percent of the maximum extent of the plastic zone The

distance to the remotely applied elastic stress field, that is, the value of ^ and

Y, was chosen to be at least ten times the maximum plastic zone size A coarse

representation of a typical mesh is shown in Fig 2 This choice of element

array led to a few elements with very small aspect ratios but they gave rise to

no numerical difficulties, perhaps because they occurred only in regions

where all fields had a high degree of smoothness

Results

To illustrate the accuracy of the numerical method, the distribution of

strain on the crack line within the active plastic zone for a rate-independent

material has been computed for several values of crack-tip speed and the

results are compared with the exact analytical results in Fig 3 These data are

reproduced from Ref 7 The general features of the computed strain

distribu-tions, including the very large strains at the crack tip, are consistent with the

exact distributions and the computed values are approximately correct The

strain distributions shown in Fig 3 are also obtained from the calculation for

a rate-dependent material if a very large value of H is assumed, which would

correspond to a relatively rate-insensitive material (h -» oo) or to a very slow

crack speed (v ^ 0)

The computed strain distributions on the crack line within the active

plas-tic zone for a rate-dependent material are shown in Figs 4 and 5 for the

non-dimensional crack speeds m = 0.3 and m = 0.5, respectively The

calcula-tions were done with the value of the exponent « = 5 and with the various

values of the rate-sensitivity parameter H shown in the graphs For purposes

of comparison, the corresponding strain distributions for a rate-independent

material are also shown in the same figures No analytical results are available

with which these computed results may be compared for rate-dependent

ma-terial response From these figures, it appears that the strain distributions for

the rate-dependent materials are more strongly singular at the crack tip, and

that the size of the region over which the stronger singularity applies increases

with increasing speed The strength of the singularity may be estimated from

an examination of the equation of motion (Eq 6) and the stress-strain relation

as represented by Eqs 5 and 8 Suppose that the strain varies with distance

ahead of the crack tip x as an inverse power, say x"', and consider the

possi-bility that the elastic strain dominates the plastic strain Then the governing

equations are essentially the elastic equations and s must have the value V2

1-14 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

FIG 3—Normalized shear strain versus distance x, along the crack line in the active plastic

zone for a rate-independent elastic-ideally plastic material; Tg is the distance along the crack line from the crack tip to the elastic-plastic boundary

But if s = V2 then the elastic strain can dominate the plastic strain at the crack tip only if « < 3 Next, consider the possibility that the plastic strain

dominates the elastic strain at the crack tip The equation of motion suggests

that the strength of the singularities in stress and total strain are the same

Then, in the stress-strain relation, there is no singularity stronger than the

elastic singularity to balance the plastic strain singularity and, therefore, the

magnitude of the plastic strain singularity is zero For n > 3, this leaves as the

only possibility that the strengths of the elastic and plastic strain singularities

are the same, which implies that 5 = l/(« — 1) If this reasoning is correct, then the strain distributions in Figs 4 and 5 are both singular at the crack tip

as jc"'^'' for the case n = 5 It is interesting to note that the strain singularity

anticipated here for « > 3 is the same as that predicted by Hui and Riedel

[16] in their study of the asymptotic strain field near the tip of a growing crack

1

-ElasUc (ro=kV2»rTo*) Rate Independent

FIG 4—Normalized shear strain versus distance Xj along the crack line in the active plastic

zone for rate-sensitive material; tg is the distance along the crack line from the crack tip to the

elastic-plastic boundary H and n are defined through Eq 8

under creep conditions However, because of the presence of the inertia term

in the governing equations, it appears that the arguments concerning the

as-ymptotic field cannot be made in the same concise way in which they are

pre-sented in Ref 16 The range over which such an asymptotic field might be

valid cannot be established without a more complete solution of the problem

Such a complete solution was presented for the rate-independent material in

Ref 7, where it was shown that the range of validity of the asymptotic solution

did indeed increase with increasing crack tip speed

To determine the level of the remotely applied stress-intensity factor k

re-quired to sustain crack growth at the predetermined speed v, a ductile

frac-ture criterion must be introduced The fracfrac-ture criterion adopted is that

pro-posed in Ref /, namely, that a crack will grow if a critical level of plastic strain

occurs at a point on the line directly ahead of the crack at a characteristic

1-16 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

FIG 5—See Fig 4

distance from the tip If 7y and df represent the critical (nondimensional)

plastic strain and the characteristic length, respectively, then the crack will

grow with 7 / = 7y at Xi = df on X2 = 0 For levels of plastic strain below 7y

atxj = cfythe crack cannot grow, and strains above 7/atXi = d^are

inacces-sible For present purposes, it is useful to eliminate df in favor of the critical

stress-intensity factor kc (assuming small-scale yielding) The value of kc is

defined as that stress-intensity factor required to satisfy the fracture criterion

for a stationary crack in an elastic-ideally plastic material under slow-loading

conditions Thus, according to Rice [2], these material parameters are related

by

dfijf + 1) = (kc/Toy (16) which, on using Eqs 3, gives the nondimensional characteristic distance in

terms of the failure strain and the ratio of the stress-intensity factor k needed to

sustain crack advance at speed v to the static stress-intensity factor k^

For a given critical strain jy, the stress-intensity factor (for the speed v) can be

found from the strain versus nondimensional distance ahead of the crack tip

Xf through Eq 17 This allows construction of a fracture toughness versus

crack speed relationship for a material which fails in a locally ductile manner

The next objective is to generate theoretical fracture toughness versus crack

speed relationships based on the critical plastic strain criterion Whereas

such a relationship should be obtained for a fixed set of material parameters,

the calculated strain distributions were obtained for fixed values of H It is

clear from its definition (Eq 9) that H depends on k and v as well as on

mate-rial parameters However, in view of Eqs 16 and 17, H may alternatively be

expressed by

2uhdf 1 h*

H = -^—^ = (18)

CTQ mXf mxf

where h* depends only on the material parameters, including the rate

sensi-tivity parameter A In order to obtain toughness values for a range of values of

crack speed m, but for fixed values of A*, jy, and df, it is necessary to guess a

value for J¥ for each particular crack speed ratio m and critical strain jf The

value for h* is then given by Eq 18, where Xf is obtained from the strain versus

nondimensional distance ahead of the crack x for the particular critical strain

jf In order to obtain results for values of h* within 1 percent of a desired

value, typically three runs for each crack velocity and critical strain were

needed However, with information gained from previous calculations, good

estimates of the H required to give a particular h* can be made

Estimates of the driving force ratio k/kc required to sustain crack growth

were obtained for « = 5 and two values of the nondimensional critical plastic

strain, 7/ = 2 and 5 Calculations were performed for crack velocities of 1, 5,

10, 20, 30, 40, and 50 percent of the elastic shear wave speed c Values of H

were selected to result in graphs of k/k^ versus m for A* = 0.02, 0.1,1.0, and

10.0 The results are shown in Figs 6 and 7 The results suggest that the

influence of strain rate sensitivity is greatest at the lower crack speeds, where

the influence of inertia is least Although the calculated toughness continues

to increase with increasing speed for all levels of rate sensitivity, the rate of

increase is lower than for the corresponding rate-independent material for the

higher values of speed In fact, for the case of jf = 5, the curves seem to have

a common crossover point at about m = 0.45 For speeds beyond the

cross-over point, the results shown an inverse dependence of calculated toughness

on crack propagation speed This feature may be connected with the

increas-ing domain of validity of the strong algebraic sincreas-ingularity in strain at the crack

tip with increasing speed

1-18 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

The calculations have been performed with a critical plastic strain 7y which

is independent of crack tip speed or strain rate If it were assumed that the value of 7y increased with crack-tip speed, then the toughness would increase more rapidly with speed and the crossover point just mentioned would be moved to the right or even eliminated On the other hand, if it were assumed that the value of 7y decreased with crack-tip speed, then the toughness would increase less rapidly with speed In this situation, it would be possible to gen-

erate toughness versus speed relationships which have local maxima and

min-ima If the value of A: at a local minimum should fall near or below the value of

kc, it would have important implications in considering the fast fracture

ar-rest capabilities of the material A thorough discussion of this issue has been

presented recently by Dantam and Hahn [17\

5.0

o 3.0

-FIG 1—See Fig 6

For the sake of brevity, attention was restricted here to the case of M = 5

Corresponding results for the case of « = 1 have also been obtained and they

are reported in Ref 15

Finally, it is noted that a large amount of experimental data on fracture

toughness versus crack speed for metals which fracture in a locally ductile

manner are presented in Ref 17 In all cases for which data are available, the

toughness is a monotonically increasing function of crack speed

Further-more, materials with significant strain rate sensitivity show the reversed

cur-vature behavior exhibited in Figs 6 and 7 The range of values of A * for which

calculations were done seems to include realistic values For example, if the

rate sensitivity parameter is in the range 10^/s < h < 10^/s, the yield strain of

the material is TO/JU = 0.002, and the critical distance df = 10'"' m, then the

value of A* is in the range 0.02 < h* < 2.2

1-20 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

Acknowledgments

The project described here is supported by the National Science

Founda-tion, Solid Mechanics Program, Grant No CME 77-15564 All computations

were done in the Brown University, Division of Engineering VAX-11/780

Computer Facility The acquisition of this computer was made possible by

grants from the National Science Foundation (Grant No ENG78-19378), the

General Electric Foundation, and the Digital Equipment Corp

Helpful discussions with Professor D M Parks of Massachusetts Institute

of Technology concerning the numerical finite-element method and with Dr

K K Lo of Chevron Research Laboratories concerning the crack-tip strain

singularity for rate-sensitive materials are gratefully acknowledged

References

[/] McClintock, F A and Irwin, G R in Fracture Toughness Testing and Its Applications,

ASTMSTP381, American Society for Testing and Materials, 1965, pp 84-113

[2] Rice, J R in Fatigue Crack Propagation, ASTM STP 415, American Society for Testing

[5] Dean, R H and Hutchinson, J W in Fracture Mechanics: Twelfth Conference, ASTM

STP 700, American Society for Testing and Materials, 1980, pp 383-405

[6] Douglas, A S., Freund, L B., and Parks, D M m Proceedings, Fifth International

Con-ference on Fracture, Cannes, France, D Francois, Ed., Pergamon Press, New York, 1981,

pp 2233-2240

[7] Freund, L B and Douglas, A S., Journal of the Mechanics and Physics of Solids, Vol 30,

1982, pp 59-74

[5] Parks, D M., Lam, P S., and McMeeking, R M in Proceedings, Fifth International

Conference on Fracture, Cannes, France, D Francois, Ed., Pergamon Press, New York,

1981, pp 2607-2614

[9] Malvern, L ^., Journal of Applied Mechanics, Vol 18, 1951, pp 203-208

[10] Perzyna, P., Quarterly of Applied Mathematics, Vol 20, 1963, pp 321-331

[//] Rice, J R., Journal of Applied Mechanics, Vol 37, 1970, pp 728-737

{12\ Lipkin, J., Campbell, J D., and Swearengen, J C, Journal of the Mechanics and Physics

of Solids, Vol 26, 1978, pp 251-268

[13] Chiem, C Y and Duffy, J., "Strain Rate History Effects in Aluminum Single Crystals

During Dynamic Loading in Shear," Division of Engineering Technical Report, Brown

University, Providence, R.I., 1981

[14] Rice, J R and Tracey, D M in Numerical and Computer Methods in Structural

Mechan-ics, S J Fenves, Ed., Academic Press, New York, 1973, pp 585-623

[15] Douglas, A S., "Dynamic Fracture Toughness of Ductile Materials in Antiplane Shear,"

Ph.D Thesis, Division of Engineering, Brown University, Providence, R.I., Aug 1981

[16] Hui, C Y and Riedel, H., "The Asymptotic Stress and Strain Field Near the Tip of a

Growing Crack Under Creep Conditions," Division of Engineering Technical Report,

Brown University, Providence, R.I., 1979

[17] Dantam, V and Hahn, G., "Evaluation of the Crack Arrest Capabilities of Ductile Steels,"

presented at the American Society of Mechanical Engineers Pressure Vessel and Piping

Conference, Denver, Colo., June 1981

Elastic Field Surrounding a Rapidly

Tearing Crack

REFERENCE: Kobayashi, A S and Lee, O S., "Elastic Field Surrounding a Rapidly

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

Analysis, ASTM STP 803, C F Shih and J P Gudas, Eds., American Society for Testing

and Materials, 1983, pp I-21-I-38

ABSTRACT: The transient elastic stress field surrounding a rapidly tearing crack in a

single-edged notch (SEN) specimen is analyzed by dynamic photoelasticity using a

16-spark gap Cranz-Schardin camera system The 1.6-mm-thick annealed polycarbonate

SEN specimens produced rapidly tearing cracks with approximately U-mm-long necked

regions which were modeled as Dugdale strip yield zones In addition, the wake of the

residual compressive stress along the rapidly tearing crack was modeled statically by a

single force parallel to the crack and acting close to the physical crack tip Theoretical

isochromatics generated by this residual-stress-corrected, propagating Dugdale model of

a rapidly tearing crack agreed reasonably well with those obtained experimentally by

dy-namic photoelasticity

The experimental analysis showed that the length of Dugdale strip yield zone remained

essentially constant through much of the rapid tearing The crack-tip opening angle also

remained constant, while the crack-mouth opening displacement increased during the

terminal stage of crack propagation

KEY WORDS: dynamic fracture, rapid tearing, ductile fracture, Dugdale strip yield

zone, dynamic photoelasticity, elastic-plastic fracture

Unlike the efforts in elastodynamic fracture, little information is available

on the transient state involved in dynamic ductile fracture, particularly in the

presence of large-scale yielding Earlier investigations on dynamic ductile

fracture, including those of Kanninen et al [1],^ Hahn et al [2], and

Ogasa-vs^ara et al [3] as well as by Koshiga et al [4\ and Nora et al [5] involved

phe-nomenological modelings of specific experimental observations While these

results are useful in predicting dynamic ductile fracture responses in specific

' Professor and graduate student, respectively Department of Mechanical Engineering,

Uni-versity of Washington, Seattle, Wash 98195

^The italic numbers in brackets refer to the list of references appended to this paper

1-22 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

structures, the lack of data on the near-field states of stress and strain makes

it difficult to extract a dynamic ductile fracture criterion from these studies

A major obstacle in experimental research in dynamic ductile fracture

studies is the lack of a suitable experimental technique with which the

dy-namic states of stress, strain, plastic yield zone, etc., can be measured directly

for detailed scrutiny In contrast, dynamic photoelasticity and dynamic

caus-tics provide an optical image of the elastodynamic field surrounding the

run-ning crack from which the dynamic stress-intensity factor, K/y^, among

oth-ers, can be extracted for elastodynamic fracture analysis The plasticity

counterpart of these optical techniques, that is, dynamic photoplasticity and

dynamic caustics with plasticity, cannot be used because the theoretical

dy-namic near-field states of stress, strain, and displacement, which are needed

for data interpretation, are not explicitly known at this time [6,7],

Com-pounding this difficulty are the uncertainties in the theory of static

photoplas-ticity [8-10] and hence in the theory of dynamic photoplasphotoplas-ticity The science

of caustics in the presence of plastic yielding is in a similar state [11] although

efforts have been made to model the plasticity effect by a propagating

Dugdale-type strip yield zone [12]

Despite the aforementioned uncertainties, dynamic photoelasticity does

provide information on the surrounding elastic field from which an estimate

of the influence of a propagating crack-tip yield zone can be made For

exam-ple, a dynamic extension of Irwin's plasticity correction factor [13] can be

extracted from this surrounding elastic field and used to correct the otherwise

elastic dynamic stress-intensity factor, Ki^^" The experimentally determined

elastodynamic field can also be used as a prescribed boundary condition for

an elastoplastic dynamic finite-element analysis of the interior plastic field

surrounding the propagating crack tip Alternatively, the surrounding elastic

field can be used to develop a mathematical model of the propagating plastic

yield zone for the purpose of extracting fracture parameters which could

re-late to the elastoplastic dynamic fracture process

The purpose of this paper is to develop such an elastic-plastic model using

the dynamic isochromatics associated with a rapidly tearing crack The

Dugdale strip yield zone is used in this paper for its mathematical simplicity

as well as for its apparent simulation of the observed strip yield zone ahead of

the rapidly tearing crack in polycarbonate fracture specimens The

polycar-bonate sheet, 1.6 mm thick, used in this investigation exhibited negligible

strain-hardening prior to necking and thus the strain-hardening effect was

not incorporated in the Dugdale strip yield zone

Theoretical Background

Moving Dugdale Strip Yield Zone

The static and dynamic near-field states of stress in the vicinity of a

Dugdale strip yield zone shown in Fig 1 were derived in Ref 14 and Refs 12

FIG 1—Dugdale strip yield zone at the crack tip of a semi-infinite crack

and 15, respectively In terms of the local polar coordinate system shown in

Fig 1, the dynamic maximum shear stress, T„, can be represented in the

three stress components a^x, Oyy, and a^y which were derived from Ref 76 as

"JV -JSL] + 0 - ^ 2

1/2

(la) where

(Txy-Oy Ml log

l + i + 2 i^sin^

l + i - 2 E s i n ^

(16)

1-24 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

, , o 9^2 1 + - ^ + 2 - ^ Sin ^

1 + ^ - — 2 - ^ sin

—-ri y r2 2

(Ic)

ri = r(cos26l + /3i sin^e)!/^ ^2 = rCcos^e + /Sj ^m^ef^ (Id)

tan ^1 = |8i tan 6 tan 02 = &2 tan 0 (le)

1 + ^2^ „ 4gig2

^1 = - ^ ^ ^2 = - ^ T o 2, (1/)

D = 4^,^2 - (1 + /322)2 {\g)

and

Ofy^ = remote stress component,

Oy = yield stress in uniaxial tension,

Vy = length of the Dugdale strip yield zone, and

C, Ci, C2 = crack, dilitational, and distortional wave velocities, respectively

Residual Stress Field in the Wake of a Propagating Crack

The effect of residual stresses in the wake of the propagating ductile crack

cannot be ignored when large-scale yielding is involved The residual

plane-stress deformation field in the wake of a moving Dugdale strip yield zone has

been analyzed by Budiansky and Hutchinson [17] The a^ component in the

wake of this Dugdale model vanishes while experimental results, as shown

later, indicated the existence of a substantial residual a^^ component in the

unload region immediately behind the moving crack tip

As a simplified modeling of the residual stress effect in the wake, the state

of stress can be approximated as a varying uniaxial stress along the crack

surface An estimate of this normal stress distribution can be made by

observ-ing the termini of the isochromatics at the wake of the propagatobserv-ing crack in

photoelastic specimens Further simplification can be made by replacing the

residual normal stress distribution by the average stress in the wake of the

crack and subsequently with a concentrated force Q, acting close to the

physi-cal crack tip, that is, at a distance ry from the Dugdale crack tip, as shown in

Fig 2 The static near-field state of stresses at the Dugdale crack tip due to

this concentrated force can then be derived from a solution in Ref 18 The

FIG 2—Residual stress field in wake of crack propagation

sum of the two normal stresses for this static boundary-value problem for

ry <ii a bec(

<^xy —

jmes

„, _ 2Q(r^ + r cos 6)^

iriry^ + 2ry rcosd + r^)^

, _ IQr^ sin^e (r, + r cos 6)

1-26 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

Moving Dugdale Strip Yield Zone with a Trailing Wake

If the Dugdale strip yield zone is propagating slowly, that is, at a crack

velocity of C/Ci < 0.1, then for all practical purposes the stress wave effect

can be ignored and the static and dynamic crack-tip stress fields coincide

Equations 16, Ic, and 2 can then be superimposed and, with Eqs \d to \h, be

used through Eq la to construct near-field isochromatics at the tip of a

propa-gating Dugdale strip yield zone The validity of this model is then verified by

comparing the experimentally determined dynamic photoelastic fringe

pat-terns with those generated by Eqs 1 and 2 using experimentally determined

parameters Vy and Q Once the moving Dugdale strip yield zone is verified as a

viable model of a rapidly tearing crack, then the associated parameters such

a s / a n d crack-opening displacement 8, [19] can be studied in detail as

plausi-ble fracture parameters associated with rapid tearing

Experimental Analysis

Polycarbonate SEN Specimens

The experimental program consisted of dynamic photoelasticity using a

thin, that is, 1.6 mm thick, polycarbonate single-edged notch (SEN)

speci-men which is shown in Fig 3 The annealed polycarbonate, from which the

specimens were machined, has a well-defined static yield point of 44 MPa,

flows under high stresses, and exhibits tensile instability with accompanying

Lueder's bands Under high strain rate loading, that is, about 1 0 " ' 1/s, such

as in the vicinity of a propagating crack tip, the strain-rate-sensitive

polycar-bonate yields at about 60 ~ 65 MPa [79], and fractures in a ductile mode with

a 100 percent shear lip together with an apparent strip yield zone preceding

the propagating crack tip Details on this ductile brittle transition in dynamic

loading of polycarbonate is discussed in Ref 20

In the as-received condition, the polycarbonate sheet exhibits considerable

residual stress distribution and thus the sheets were annealed after rough

cut-ting This annealing, which caused some distortion and shrinkage, consisted

of overnight heating at 160°C, followed by gradual cooling at the rate of 5°C

per hour The starter crack consisted of a 0.4-mm-wide edge crack

approxi-mately 25 mm in length which was machined and then chiseled to simulate a

somewhat blunt crack tip Dynamic stress-fringe constant, dynamic elastic

modulus, dynamic yield stress, and Poisson's ratio at various strain rates were

determined by a standard split Hopkinson bar system described in previous

publications A static modulus of elasticity of £• = 2.00 GPa and yield

strength of 62 MPa at a strain rate of 10~' were determined separately for this

annealed polycarbonate sheet

The polycarbonate SEN specimens with blunt starter cracks of about 25

mm in length were loaded to failure and the dynamic photoelastic patterns

FIG 3—Polycarbonate single-edged notch specimen

surrounding the propagating crack tips were recorded The 16-spark gap

Cranz-Schardin camera and the associated dynamic photoelastic systems

used to record 16 discrete dynamic photoelastic patterns has been described

in many previous papers and thus will not be repeated here

Data Reduction Procedure

Previous experimental investigations [14,15] showed that annealed thin

polycarbonate fracture specimens can exhibit either cleavage or ductile

frac-ture with shear lips depending on the loading conditions For dynamic ductile

1-28 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

fracture with about 50 percent shear Hps, the propagating crack tip was

pre-ceded by a small strip yield zone of r^ = 1.3 mm in length In these previous

experimental investigations, a propagating Dugdale model was postulated

and Yy was computed from the experimentally determined isochromatics

These computed Vy coincided with the lengths of the darkened apparent strip

yield zones preceding the propagating crack Thus, the length of this

appar-ent strip yield zone represappar-ented by the darkened strip zone in the dynamic

photoelastic record was used as the measured Vy in this investigation

As mentioned previously, the elastic region surrounding the moving yield

zone provides the elastodynamic state which is used to extract the apparent

elastic fracture parameter for characterizing dynamic ductile fracture In the

static \14\ and dynamic \15\ procedures used in previous investigations,

theo-retical near-field isochromatics with disposal parameters, such as Ty and a^,

were least-squares fitted to the experimentally determined isochromatics and

numerical values for these disposal parameters were extracted For this

inves-FIG Aa—Dynamic isochromatics of rapidly tearing polycarbonate SEN specimen (Specin

No 70981P-0SL5 Frame No 1)

tigation, only Oox remained unknown since r^, was determined directly from

experimental results

The second disposal parameter in this investigation was the force Q which

models the effect of the compressive residual stress in the trailing wake of the

propagating crack shown in Fig 2 Figures 4 and 5 show typical dynamic

photoelastic patterns recorded in two rapidly tearing polycarbonate SEN

specimens The isochromatic loops which terminate at the trailing wake of

the crack are approximately one half of CT^ at this terminus since o^ = 0 on

the crack surface From these terminating isochromatics, an estimate of the

residual stress distribution in the trailing wake can be obtained Q can then

be approximated by the average of the total compressive a^x distribution in

the trailing wake and in part of the strip yield zone Q is also assumed to act in

a region adjacent to the physical crack tip for the reason described later in the

Results section In the actual data reduction procedure, Q was set as a disposal

parameter which was determined together with r^ from the recorded

isochro-• isochro-• isochro-• isochro-• " * isochro-• *

FIG Ab—Dynamic isochromatics of rapidly tearing polycarbonate SEN specimen (Specimen

No 70981P-0SL5 Frame No 6)

1-30 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

(cm

FIG 5a—Dynamic isochromatics of rapidly tearing polycarbonate SEN specimen (Specimen

No 70981P-0SL4, Frame No 1)

matic loops The computed Q was then checked against Q estimated from the

experimental results following the procedure described in the foregoing

Having determined ry directly from the dynamic photoelastic results, the

overdeterministic least-squares method [21] was used to determine two

un-knowns, (Jocc and Q, in Eqs 1 and 4 This data reduction procedure for static

and dynamic photoelasticity has been described by the authors and others in

many previous papers and thus will not be elaborated on here

Results

Crack Velocities

A total of four polycarbonate SEN specimens was tested and the crack

ve-locities were determined by measuring the recorded instantaneous crack

lengths at discrete time intervals Measured crack velocities as shown in Fig 6