Astm stp 1074 1990

In the linear-elastic regime characterized by small-scale plastic deformation the relationship of stress ahead of the crack tip to CTOD is identical for the short crack and the deep cr

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ISBN: 0-8031-1299-8

ISSN: 1040-3094

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Foreword

The A S T M Twenty-First National Symposium on Fracture Mechanics was held in

Annapolis, Maryland, on 2 8 - 3 0 June 1988 Its sponsor was Committee E-24 on Fracture

Testing

The co-chairmen for this symposium were John P Gudas, David Taylor Research Center;

James A Joyce, United States Naval Academy; and Edwin M Hackett, David Taylor

Research Center They have also served as editors of this volume

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University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized

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Contents

An Analytical Comparison of Short Crack and Deep Crack C r O D Fracture

Specimens of an A36 Steel w.A SOREM, R H DODDS, JR., AND S T ROLFE

Direct J-R Curve Analysis: A Guide to the Methodology R HERRERA AND

J D LANDES

Application of the Method of Caustics to J-Testing with Standard Specimen

Geometries R J SANFORD AND R W JUDY, JR

Extrapolation of C(T) Specimen J-R Curves G M WILKOWSKI, C W MARSCHALL,

AND M P LANDOW

Application of J-Integral and Modified J-Integral to Cases of Large Crack

Extension J A JOYCE, D A DAVIS, E M HACKETT, AND R A HAYS

Impact Fracture of a Tough Ductile Steel A s DOUGLAS AND M S SUH

Dynamic Fracture Behavior of a Structural Steel K CliO, J P SKLENAK, AND

J DUFFY

Discussion

Dynamic Key-Curves for Brittle Fracture Impact Tests and Establishment of a

Transition T i m e - - w BOHME

Explosive Testing of Full Thickness Precracked Weldments L N GI~ORD,

J R CARLBERG, A J WIGGS, AND J B SICKLES

Magnetic Emission Detection of Crack Iuitiation s R WINKLER

TRANSITION FRACTURE

Effect of Biaxial Loading on A533B in Ductile-Brittle Transition s J CARWOOD,

T G DAVEY, AND Y C WONG

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Fracture Toughness in the Transition Regime for A533B-I Steel: The Effect of

Specimen Sidegrooving E MORLAND

Analysis of Fracture Toughness Data for Modified SA508 C12 in the Ductile-to-

Brittle Transition Region M T MIGLIN, C S WADE, AND

W A VAN DER SLUYS

Discussion

Effects of Warm Pre-Stressing on the Transition Toughness Behavior of an A533

Grade B Class 1 Pressure Vessel Steel r) LIDBURY AND P BIRKETT

215

238

263

264

Unique Elastic-Plastic R-Curves: Fact or Fiction? M R ETEMAD AND

C E TURNER

Adhesive Fracture Testing M F MECKLENBURG, C O ARAH, D McNAMARA,

H HAND, AND J A JOYCE

Discussion

Evaluation of Elastic-Plastic Surface Flaw Behavior and Related Parameters

Using Surface Displacement Measurements w R LLOYD AND

Effect of Dynamic Strain Aging on Fracture Resistance of Carbon Steels

Operating at Light-Water Reactor Temperatures c w MARSCHALL,

M P LANDOW, AND G M WILKOWSKI

Prediction of Fracture Toughness by Local Fracture Criterion T MIYATA,

A OTSUKA~ M MITSUBAYASHI, T HAZE, AND S AIHARA

Microscopic Aspects of Ductile Tearing Resistance in AISI Type 303 Stainless

S t e e l - - A SAXENA, D C DALY, H A ERNST, AND K BANE1LII

Microstructure and Fracture Toughness of Cast and Forged Ultra-High-Strength,

Low-Alloy (UHSLA) Steels J ZEMAN, S ROLC, J BUCHAR, AND J POKLUDA

Simulation of Crack Growth and Crack Closure under Large Cyclic Plasticity

K S KIM, R H VAN STONE, J H LAFLEN, AND T W ORANGE

Comparison of Elastic-Plastic Fracture Mechanics Techniques F w BRUST,

M NAKAGAKI, AND P GILLLES

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Treatment of Singularities in a Middle-Crack Tension S p e c i m e n - -

K N SHIVAKUMAR AND I S RAJU

Assessment of Influence Function for Elliptical Cracks Subjected to Uniform

Tension and to Pure Bending M PORE

Finite Element Meshing Criteria for Crack P r o b l e m s - - w n GERSTLE AND

FRACTURE MECHANICS TESTING

Closure Measurements via a Generalized Threshold Concept -G MARCI,

D E CASTRO, AND V BACHMANN

Use of the Direct-Current Electric Potential Method to Monitor Large Amounts

of Crack Growth in Highly Ductile Metals c w MARSCHALL, P R HELD,

M P LANDOW, AND P N MINCER

Load-Point Compliance for the Arc-Bend/Arc-Support Fracture Toughness

Specimen F I BARATTA, J A KAPP, AND D S SAUNDERS

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Introduction

The success of the Twenty-First National Symposium on Fracture Mechanics, held on

28-30 June 1988 in Annapolis, Maryland, and sponsored by ASTM Committee E-24 on

Fracture Testing, demonstrated the continued rapid development occurring in this field

Papers were solicited from all areas of fracture mechanics and its applications Contributions

representing a wide range of topics came from the United States and six foreign countries

New work is presented in elastic-plastic fracture, dynamic fracture, transition fracture in

steels, micromechanical aspects of the fracture process, computational mechanics, fracture

mechanics testing, and applications of this technology Each area poses its own challenges,

and developments proceed somewhat independently This volume aids the researcher in

keeping abreast of these varied aspects of the discipline of fracture mechanics

The diligent work of the Symposium Organizing Committee, the authors, and the review-

ers is gratefully appreciated We would particularly like to recognize the efforts of the ASTM

staff including Mr Hans Greene, Ms Kathy Friend, Ms Wendy Dyer, Ms Kathy Greene,

Ms Monica Armata, Ms Rita Harhut, and Mr Allan Kleinberg Finally, the assistance of

Mrs Mary Cropley and Ms Amanda Ewen of the David Taylor Research Center is gratefully

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Elastic-Plastic Fracture Mechanics (I)

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IV A Sorem, ~ R H Dodds, Jr., 2 and S T Rolf8 3

An Analytical Comparison of Short Crack and Deep Crack CTOD Fracture Specimens of an A36 Steel

REFERENCE: Sorem, W A., Dodds, R H., Jr., and Rolfe, S T., "An Analytical Comparison

of Short Crack and Deep Crack CTOD Fracture Specimens of an A36 Steel," Fracture Mechanics: Twenty-First Symposium, ASTM STP 1074, J P Gudas, J A Joyce, and E M Hackett, Eds., American Society for Testing and Materials, Philadelphia, 1990, pp 3-23

ABSTRACT: The effect of crack-depth to specimen-width ratio on crack tip opening displacement (CTOD) fracture toughness is an important consideration in relating the results of laboratory tests to the behavior of actual structures Deeply cracked three-point bend specimens with crack-depth to specimen-width ratios (a/W) of 0.50 are most often used in laboratory tests However, to evaluate specific weld microstructures or the behavior of structures with shallow surface cracks, specimens with a~ W ratios much less than 0.50 often are required Lab- oratory tests reveal that three-point bend specimens with short cracks (a/W = 0.15) exhibit significantly larger critical CTOD values than specimens with deep cracks (a/W = 0.5) up to the point of ductile initiation

In this study, finite element analyses are employed to compare the elastic-plastic behavior of square (cross-section) three-point bend specimens with crack-depth to specimen-width ratios

(a/W) ranging between 0.50 and 0.05 The two-dimensional analysis of the specimen with an

a/Wratio of0.15 reveals a fundamental change in the deformation pattern from the deep crack deformation pattern The plastic zone extends to the free surface behind the crack concurrent with the development of a plastic hinge For shorter cracks (a/W = 0 l0 and 0.05), the plastic zone extends to the free surface behind the crack p n o r to the development of a plastic hinge For longer cracks (a/W > 0.20), a plastic hinge develops before the plastic zone extends to the free surface behind the crack

These results prompted further study of specimens with an a/W ratio of 0.15 using three- dimensional, elastic-plastic finite element analyses Results of the short crack (a/W = 0.15) analysis are compared to the results of the deep crack (a/W = 0.50) analysis reported previously by the authors In the linear-elastic regime (characterized by small-scale plastic deformation) the relationship of stress ahead of the crack tip to CTOD is identical for the short crack and the deep crack specimens At identical CTOD levels in the elastic-plastic regime (large- scale plasticity, hinge formation), the crack tip stress is significantly lower for specimens with

a~ W = 0.15 than for specimens with a~ W = 0.50 Correspondingly, at equivalent stress levels, the CTOD for the short crack is approximately 2.5 times the CTOD for the deep crack This observation has considerable significance in the application of CTOD results to failure analysis

or specification development where the fracture mechanism is cleavage preceded by significant crack tip plasticity

KEY WORDS: elastic-plastm fracture mechanics, CTOD, crack depth, short crack, toughness, finite element, constraint

University of Kansas, Lawrence, KS 66045; currently at Exxon Production Research Company, Houston, T X 77252

2 University of Illinois, Urbana, IL 61801

3 University of Kansas, Lawrence, KS 66045

Copyright 9 1990by ASTM International www.astm.org

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4 FRACTURE MECHANICS: TWENTY-FIRST SYMPOSIUM

The crack-depth to specimen-width ratio (a/W) has a significant effect on the fracture

toughness results in the elastic-plastic regime where brittle fracture is preceded by significant

crack tip blunting but no ductile tearing Typically, deep cracks (a/W = 0.5) are tested in

laboratory specimens to develop maximum constraint (stress triaxiality) at the crack tip and

therefore provide conservative estimates of material toughness While this approach may be

appropriate for the general characterization of material fracture toughness, in the evaluation

of existing flaws in structures it is more appropriate to model the actual degree of constraint

present in the structural component In weldments, for example, the testing of short cracks

becomes particularly important, since the region of lowest toughness does not necessarily

occur at a crack depth halfway through the specimen With the variation of microstructures

through the heat-affected zone (HAZ) and local brittle zones (LBZs), it is probable that a

deeply cracked laboratory specimen would not produce the conservative fracture toughness

values expected

Several methods of laboratory testing for fracture toughness within the elastic-plastic

regime are standardized The two most widely used are the J-integral and the crack tip open-

ing displacement (CTOD) test procedures Both procedures could be extended to include

short crack specimens Currently, the CTOD test procedure (BS 5762, "Methods for Crack

Opening Displacement Testing") has a major advantage in that it is the only standard which

allows testing throughout the entire realm of fracture toughness from linear-elastic to fully-

plastic behavior J-integral standards (e.g., ASTM E 813, "Jic, A Measure of Fracture Tough-

ness") are currently limited to determining the initiation of ductile tearing (J~c), but have

often been extended to quantify brittle fracture (Jc) prior to initiation of ductile tearing

Recent experimental investigations [I-7] have examined the effect of a/W ratio on the

fracture behavior of three-point bend specimens Both critical values and ductile initiation

values of CTOD and J-integral were reported A variety of materials were tested including

low-strength steels, high-strength steels, and weldments The experimental studies [1-7] have

demonstrated CTOD and J-integral values for short crack specimens (a/W < 0.20) to be

significantly larger than for deep crack specimens (a/W = 0.50)

Most CTOD-based studies have used the BS 5762 CTOD versus crack mouth opening

displacement (CMOD) relation to analyze the behavior of short crack specimens This rela-

tion is based on a small-scale yielding component and a plastic rotation component A major

difficulty for CTOD testing is assessing the plastic rotation factor for specimens with a~ W

ratios less than 0.2 This rotation factor has been experimentally determined with dual clip-

gage techniques [3, 5] and rubber crack replication techniques [3, 4, 6, 7] The rotation factor

for specimens with a~ W ratios of approximately 0.15 has been reported as low as 0.20 to as

high as 0.45

This investigation focuses on three objectives The first is to determine the a~ W ratio at

which the plastic zone extends from the crack tip to the free surface behind the crack for a

material with significant strain hardening This a~ W ratio is expected to define the boundary

between short crack behavior and deep crack behavior The second objective is to establish

a relationship to calculate CTOD from the measured load-CMOD record or alternative mea-

surement of specimen response Ideally, this relationship would merely extend the current

BS 5762 equation The third objective is to determine the effects of the crack depth on

stresses near the crack tip in relation to the CTOD levels

Finite element analyses are conducted to study the effect of crack depth on the nonlinear

behavior of CTOD fracture toughness test specimens Two-dimensional (plane-stress and

plane-strain) analyses are conducted using the uniaxial stress-strain properties of an A36

steel Square cross-section, three-point bend specimens with crack-depth to specimen-width

ratios (a/W) of 0.50, 0.20, 0.15, 0.10, and 0.05 are analyzed The stress distributions are

compared to determine the effect of crack depth on specimen behavior and crack tip con-

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SOREM ET AL ON SHORT AND DEEP CRACK FRACTURE SPECIMENS 5

straint A three-dimensional, elastic-plastic analysis is performed on full-size (31.8 by 31.8

by 127 m m (1.25 by 1.25 by 5.0 in.)) and sub-size (12.7 by 12.7 by 50.8 m m (0.50 by 0.50

by 2.0 in.)) three-point bend specimens with a~ W ratios of 0.15 Comparisons of the numerical results demonstrate the effect of crack depth relative to specimen size and also the effect

of absolute crack depth Results of the short crack ( a / W = 0.15) C T O D specimens are compared to numerical results of the deep crack ( a / W = 0.50) C T O D specimens previously analyzed by Sorem et al [8]

Material Properties

The material properties for the finite element analysis were taken from the corresponding experimental study on a 31.8 m m (1.25 in.) thick A36 steel plate in its as-rolled condition The engineering stress-strain curve obtained from a standard 12.8 m m (0.505 in.) diameter longitudinal tensile test conducted at a slow loading rate is shown in Fig 1 The A36 steel had an ultimate stress to yield stress ratio of 1.86 and a strain hardening exponent of 0.23 Tensile tests were conducted at room temperature and thus typify the stress-strain properties over a temperature range of 0*C (32~ to 2 I*C (70*F) These temperatures corresponded to the transition region between brittle and ductile behavior of the A36 steel

Finite Element Analysis Procedure

Elastic-plastic finite element analyses were conducted on square three-point bend specimens with a / W ratios ranging from 0.05 to 0.50 Finite element meshes for two of these specimens are shown in Fig 2 The analyses predicted the deformation for both small- and large-scale plasticity with no simulation of crack growth The finite element solutions employed the conventional, linear strain-displacement relations based on small geometry

STRAIN (mm/mrn) FIG l A36 steel tensile test showing modification for finite element analysis

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ROLLERS

I o / w = 0.s0 I o/w = o l s

specimens

change assumptions Numerical computations were performed with the POLO-FINITE

structural mechanics system [9, I0] The analytical procedure was identical to that adopted

for the study of deeply cracked square specimens considered by Sorem et al [8] Details of

the finite element procedure are provided in the Appendix

Finite Element Results

Plastic Zone Distributions

Two-dimensional finite-element analyses (FEA) were performed on the full-size, square

C T O D specimen geometry with crack-depth ratios (a/W) of 0.50, 0.20, 0.15, 0.10, and 0.05

Plastic zone sizes obtained from the 2-D FEA were compared at various linear-elastic and

elastic-plastic C T O D levels The plastic zones developed based on the von Mises equivalent

stress for the five plane-strain models at applied C T O D levels of 0.0254, 0.0533, and 0.109

m m (1.00, 2.10, and 4.30 mils) are shown in Fig 3 Plastic zones for the specimens with

crack-depth ratios of 0.05 and 0.10 extended from the crack tip to the free surface behind

the crack before a plastic hinge formed For the specimen with a crack-depth ratio of 0.15,

the formation of a plastic hinge coincided with the plastic zone extending back to the free

surface The specimen with an a~ W ratio of 0.20 developed a plastic hinge well before the

plastic zone extended back to the free surface The plastic zone of the deep crack specimen

(a/W = 0.50) was contained completely in the plastic hinge region and never reached the

free surface behind the crack The boundary between short crack and deep crack specimens

apparently occurs at an a~ W ratio of about 0.15 for this structural steel which undergoes

significant strain hardening

This definition of short and deep crack behavior agrees with the theoretical slip-line field

discussed by Matsoukas et al [6] and the results of several experimental studies [3-6] The

crack depth at which the slip-line field first extends to the free surface was found to be

0.177 W ( a / W = 0.177) for a rigid-plastic material This does not imply that specimens with

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specimens

a~ W ratios greater than 0.177 behave the same as deep crack ( a / W = 0.50) specimens The

plastic zone for the deep crack specimen was always confined to the hinge region and did

not extend to the back surface Even though the plastic hinge formed before the plastic zone

reached the back surface in the specimen with a / W = 0.20, continued strain hardening of

the material caused the plastic zone to eventually reach the back surface

Comparison of Numerical and Experimental Results

Load versus crack mouth opening displacement (CMOD) records from the plane-strain,

plane-stress, and three-dimensional finite element analyses for the square ( W = 31.8 m m

(1.25 in.)) C T O D model are compared to a measured load versus C M O D record in Fig 4

The measured load versus C M O D record is typical of specimens tested in a temperature

range between 0~ (32~ and 21 ~ (70~ At these temperatures, the stress-strain properties

of the A36 material are similar to the room temperature tensile properties input to the finite

element model

At equivalent C M O D levels, the plane-strain analysis provides an upper bound and the

plane-stress analysis provides a lower bound to the experimentally measured load The

C M O D at the center plane of the three-dimensional model is plotted versus load, since it

matches the location of the C M O D measurement in the test specimen The load-CMOD

record of the 3-D finite element model accurately predicts the experimental load-CMOD

record

Although very good agreement is achieved between the finite element and the experimen-

tal load-CMOD records, this "global" agreement does not necessarily verify the accuracy of

predicted response in the crack tip region, specifically the CTOD A procedure was developed

to provide a comparison between the crack opening profile predicted by the 3-D finite ele-

ment analysis and the actual crack profile

A short crack ( a / W = 0.15) test specimen was loaded well beyond plastic hinge develop-

ment to a C M O D of approximately 0.427 m m (16.8 mils) and then the load was removed

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= 0.15

The residual plastic component of CMOD was approximately 0.366 mm (14.4 mils) as illus-

trated in Fig 5 The specimen was cut in half longitudinally through the center plane, pol-

ished, and etched Micrographs were taken of the crack profile at the center plane of the

unloaded specimen (Fig 5a) These micrographs show the residual plastic displacement of

the crack profile in the unloaded specimen The specimen was then reloaded in a special

fixture to the original CMOD of 0.427 mm (16.8 mils) Micrographs of the crack profile of

the reloaded specimen were taken and compared to the predicted crack profile (superim-

posed) at the equivalent CMOD level (Fig 5b) The crack profile of the test specimen is

predicted very accurately by the finite element model This comparison provides the needed

validation of the 3-D finite element modeling procedures

C M O D - C T O D Relation

The CMOD-CTOD relation for the short crack specimen is developed from the finite ele-

ment results Variations of the CTOD and CMOD through the thickness clearly demonstrate

the three-dimensional character of the short crack specimen The CTOD values are taken

directly from the deformed finite element mesh using the 90* intercept method [11] at six

positions through the half-thickness of the specimen

The CMOD-CTOD results for the full-size square specimen are shown in Fig 6 Load step

20 and load step 24 designated on the figure show the relationship of CTOD levels and

CMOD levels through the thickness of the specimen The CTOD remains nearly constant

over the center 70% of the specimen and then decreases significantly near the outside free

surface The CMOD exhibits the opposite behavior Unlike the deep crack specimen, which

has no variation of CMOD over the thickness of the specimen, the CMOD for the short crack

specimen is smallest at the center plane and increases as the outside free surface is

approached Similar CMOD variations are observed in the laboratory specimens

The CMOD-CTOD results of the sub-size square specimen are shown in Fig 7 Again,

load steps 20 and 24 designated on the figure show the through-thickness relationships of

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SOREM ET AL ON SHORT AND DEEP CRACK FRACTURE SPECIMENS 9

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5.0

C T O D and CMOD levels At step 20 (CMOD of 0.32 m m (13 mils)), response of the sub- size specimen is similar to that of the full-size specimen The CTOD remains nearly constant over the center 70% of the specimen and decreases near the outside surface The CMOD is largest at the outside surface The relative deviation between outside surface measurements and center plane measurements is more consistent with step 24 of the full-size specimen

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SOREM ET AL ON SHORT AND DEEP CRACK FRACTURE SPECIMENS 1 1

( C M O D o f 0.72 m m (28 mils)) At step 24 o f the sub-size specimen analysis (CMOD of 0.70

m m (27 mils)), the C M O D relationships through the thickness are consistent with previous

steps, but the C T O D relationships are reversed The C T O D is smallest at the center plane,

increases slightly over the intermediate layers, and then increases to a m a x i m u m at the out-

side surface The cause of this reversal in trends is not yet established Experimental data are

insufficient to distinguish this response as the actual response or an anomaly of the finite

element modeling and solution (e.g., large displacement effects and mesh refinement near

the free surface)

Analysis of BS 5762 Equation

A C M O D - C T O D relation for the short crack specimen is required to determine the C T O D

from the measured l o a d - C M O D record Conducting a finite element analysis for each spec-

imen type and each material is impractical An empirical relation similar to the existing BS

5762 equation is desirable

The BS 5762 equation relating C T O D to applied load and C M O D consists of two parts:

(1) a small-scale yielding (SSY) contribution, which is often referred to as the elastic contri-

bution, and (2) a fully plastic contribution:

6 - K2(I v2) + R F ( W - - a)Vp

2%sE R F ( W - a) + a where

K = theoretical stress intensity factor = Y P / B ' x / W ,

Y = stress intensity coefficient for a three-point bend specimen having span (5) = 4 W:

6(a/W)t/2(l.99 - a / W [ l - a/W][2.15 3.93 a / W + 2.7(a/W)2])

O-y s = 0.2% offset yield strength at temperature of interest,

E = Young's modulus at temperature of interest,

R F = plastic rotation factor,

a = physical crack length (initial length + stable crack growth), and

Vp = plastic c o m p o n e n t of clip-gage displacement

The elastic (SSY) c o m p o n e n t of C T O D is a function of the stress intensity factor, K~ Three

methods are used to calculate the stress intensity factor from the linear-elastic finite element

analysis First, K~ is calculated from the displacement of the quarter-point crack tip elements

[12] Second, J-integral calculations [13] are correlated to K~ Third, K~ is calculated from

the stress at the Gauss points ahead of the crack tip [14] The KI values obtained from the

three methods are nearly identical The K~ value is constant over the center 80% of the spec-

imen and then gradually decreases near the outside surface The calculated Kj value at the

center plane agrees within 2% of the theoretical equation (ASTM E 399, "Plane-Strain Frac-

ture Toughness of Metallic Materials") The free surface value of K~ is approximately 15%

less than the center plane value

The contribution to C T O D from the SSY c o m p o n e n t is determined from the elastic-plas-

tic finite element analysis For the first few load steps, the plastic c o m p o n e n t is essentially

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12 F R A C T U R E M E C H A N I C S : T W E N T Y - F I R S T S Y M P O S I U M

zero; C T O D is simply a function of the SSY component K, The C T O D of the displaced finite element mesh is compared with the theoretical K, value to determine the value of the constraint factor (m) The constraint factor is 1.7, which is 15% less than the BS 5762 equa-

tion but within the bounds of the 1.0 to 2.0 value described by Dawes [15] The effect of

changing the constraint factor from 2.0 to 1.7 is negligible in the elastic-plastic regime because the plastic component is the dominant contribution to CTOD Since using 2.0 yields

a conservative estimate of elastic CTOD, it is unchanged in the BS 5762 equation

The fully-plastic component of C T O D is based on the rigid body rotation of the specimen about a point ahead of the crack tip (Fig 8) For deeply cracked specimens, the plastic rota-

tion factor has been shown to be approximately 0.45 [16] The plastic rotation factor for the

short crack specimens is determined from the FEA following two independent procedures First, the outer four nodes on the crack profile at the center plane (closest to the crack mouth) are used to locate the plastic center of rotation The linear-elastic displacements of these nodes are subtracted from the total displacements to determine the location of the nodes due only to the plastic deformation Performing a simple linear regression on these node locations provides the location of the plastic rotation point The second method is a calculation

of the rotation point using the FEA load-CMOD record and the measured C T O D value Each term in the equation is known (K, v, ays, E, W, a, Vp) with the exception of the rotation factor (RF) The rotation factor is simply back calculated from the 90* intercept value of CTOD

Both methods predict rotation factors between 0.2 and 0.3 for the short crack ( a / W =

0.15) specimen Therefore the apparent center of plastic rotation for the short crack speci-

men is closer to the crack tip than for the deep crack ( a / W = 0.50) specimen This value of

the rotation factor is for a low strength (A36) steel and has not been shown to be material independent

The C T O D is calculated from the 3-D finite element load-CMOD record and the BS 5762 equation with the adjusted rotation factor A comparison of the calculated C T O D and the measured C T O D for the full-size square specimen is shown in Fig 9 Use of a plastic rotation

!

II I illlll

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S O R E M ET AL ON S H O R T A N D DEEP C R A C K F R A C T U R E SPECIMENS 13

CMOD eenterplone (ram)

FIG 9 - - C T O D versus CMOD for square (31.8 by 31.8 mm) A36 steel spectmens with

CMOD centerplone (ram)

FIG I O CTOD versus CMOD for square (12.7 by 12 7 ram) A36 steel specimens with

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1 4 FRACTURE MECHANICS: TWENTY-FIRST SYMPOSIUM

in the back-calculated rotation factor The rotation factor calculated from the slope of the crack flanks does not indicate a similar apparent change of rotation factor

Stress Distributions

Stresses near the crack tip at locations through the thickness are compared for the two short crack specimens at the same absolute position from the crack tip (r = 0.36 m m (0.0142 in.) and 0 = 55~ This location corresponds to the first Gauss quadrature point to yield near the crack tip outside of the degenerated isoparametric elements and the zone of large strain effects Since this location corresponds to that considered by Sorem et al [8] for the deep crack specimens, a comparison of the short crack square specimens with the deep crack square specimens is possible

The through-thickness variation in crack opening stress (~y) is shown in Figs 11, 12, 14, and 16 for a number of C T O D levels throughout the linear-elastic and elastic-plastic regions The axes are nondimensionalized to accommodate the different specimen thicknesses The through-thickness dimension (Z) is divided by one half the specimen thickness (t); on the horizontal axis, 0.0 corresponds to the specimen center plane and 1.0 corresponds to the outside free surface The opening mode stress is divided by the room-temperature static yield stress (248 MPa (36 ksi))

The transverse stress (~x) and through-thickness stress (at) are not shown in these figures The through-thickness variation of crx is similar to that of ay but has approximately one half the magnitude The magnitude of az is approximately one half of av at the specimen center plane, but decreases to zero at the outside surface

The distribution of the opening mode stress for essentially linear-elastic behavior is shown

in Fig 11 The C T O D is 0.0002 m m (0.0125 mils), which corresponds to a stress intensity factor of 5.5 MPak/-m (5 ksi i ~ The stress distribution for the full-size short crack specimen is nearly identical to that of the full-size deep crack specimen The opening mode stress

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is constant over the center 70% of the specimen and then decreases slightly at the outside

(free) surface Very small plastic zones exist at the crack tip for this level of loading

The opening mode stress distributions for a C T O D of 0.028 m m (1.1 mils) are shown in

Fig 12 for full-size specimens with a~ W ratios of 0.15 and 0.50 The maximum stress occurs

at the center plane and is nearly constant over the center 60% of the specimen Similar

through-thickness variations are exhibited by the short crack and deep crack specimens, but

the m a x i m u m stress developed in the short crack specimen is approximately 20% less than

for the deep crack specimen At a C T O D of 0.067 m m (2.6 mils), the short crack specimen

develops an opening mode stress equivalent to the deep crack specimen at a CTOD of 0.028

m m (1.1 mils) Thus the C T O D of the short crack specimen is approximately 2.4 times the

C T O D o f the deep crack specimen at equivalent values of opening mode stress

The corresponding von Mises stresses are shown in Fig 13 At a C T O D of 0.028 m m (1.1

mils), the yielded regions are contained at the crack tip and at the point of load application

for both the short crack and the deep crack specimens and are similar in absolute size Sig-

nificant yielding occurs behind the crack tip of the short crack specimen reaching toward the

free surface At a C T O D of 0.067 m m (2.6 mils), a plastic hinge develops and the yielded

region extends from the crack tip to the surface behind the crack tip Development of a

plastic hinge does not correspond to the attainment of a limit load, since the low-strength

steel undergoes significant strain hardening Nevertheless, there is a significant increase in

specimen compliance

The opening mode stress distributions for a C T O D of 0.102 m m (4.0 mils) are shown in

Fig 14 for full-size specimens with a~ W ratios of 0.15 and 0.50 Similar stress distributions

are exhibited for the short and deep crack specimens The maximum stress occurs over the

center 50% of the specimen and then decreases at the outside surface The maximum stress

developed in the short crack specimen is approximately 85% of the value in the deep crack

specimen at the same CTOD At a C T O D of 0.251 m m (9.9 mils), the short crack specimen

develops an opening mode stress equivalent to the deep crack specimen at a C T O D of 0.102

Trang 23

i iiiiiiii ,oEo RE ,ON AT-

c oo = o

FIG 13 yon Mzses stress distributions for 31.8 by 31.8 m m A36 steel specimens

m m (4.0 mils) Thus, to obtain a maximum opening mode stress equivalent to that of the deep crack specimen at a C T O D of 0.102 m m (4.0 mils), the C T O D of the short crack specimen must be approximately 2.5 times larger than the C T O D of the deep crack specimen The corresponding yon Mises stress at the center plane of each specimen is shown in Fig

15 At a C T O D of 0.102 m m (4.0 mils), a plastic hinge develops in the deep crack specimen Further yielding occurs in the short crack specimen at C T O D levels of 0.102 m m (4.0 mils) and 0.251 m m (9.9 mils) and extensive yielding occurs on the free surface behind the crack tip

Trang 24

S O R E M E T A L O N S H O R T AND DEEP CRACK FRACTURE SPECIMENS 17

(4.0 mils) CTOD 0.251 mm

(99 re,s)

CENTER

FIG 15 von Mises stress dlstrlbutions Jbr 31.8 by 31.8 m m A36 steel specimens

A comparison of the opening mode stress distribution for the subsize specimen with a / W

0.15 to the stress distribution for the full-size specimen with a / W = 0.15 is shown in Fig

16 The stress distributions are compared at the same CTOD levels as previously discussed

For nearly linear-elastic behavior (K~ = 5.5 M P a k / m (5 k s i ' ~ ) ) the opening mode stress

of the sub-size specimen is identical to that of the full-size specimen At a CTOD level of

0.028 m m (1.1 mils) the behavior of the sub-size specimen is similar to that of the full-size

Trang 25

18 FRACTURE MECHANICS: TWENTY-FIRST S Y M P O S I U M

specimen, but the m a x i m u m opening mode stress is slightly less (2%) and the stress decreases more quickly near the outside free surface on the sub-size specimen than on the full-size specimen Similar behavior occurs at a C T O D of 0.102 m m (4.0 mils) The opening mode stress is consistently 4% less for the sub-size specimen than for the full-size specimen at the same relative location through the thickness of the specimen

At the (center plane) C T O D of 0.380 m m (14.9 mils), the C T O D is significantly higher at the outside surface than at the center plane of the sub-size specimen (Fig 7) The stress is constant over the center 50% of the specimen and then decreases significantly near the outside free surface Therefore the increased surface C T O D is not reflected in the opening mode stress distribution through the thickness of the specimen, The opening mode stress is approximately 4% less for the sub-size specimen than for the full-size specimen

A comparison of the plastic zones developed based on the von Mises stresses for the full- size and the sub-size short crack specimens is shown in Fig 17 The performance of the sub- size specimen at a C T O D o f 0.170 m m (6.7 mils) is very similar to the full-size specimen at

a C T O D o f 0.378 m m ( 14.9 mils) The extent of elastic-plastic behavior appears proportional

to the specimen dimensions The yielded region of each specimen is extensive and undergoes considerable strain hardening The region of fully-plastic behavior (on the plateau of the stress-strain curve) is very small and limited to the crack tip elements At a C T O D of 0.425

m m (16.7 mils), the sub-size specimen exhibits more extensive elastic-plastic behavior, but the fully plastic zone is still small

To summarize the effect of crack depth on the near crack tip stress field, the opening mode stress at the center plane of the short crack and deep crack specimens is shown in Fig 18 for the entire C T O D range Throughout the linear-elastic regime, the short ( a / W = 0.15) and the deep ( a / W = 0.50) crack specimens exhibit the same level of stress at the same CTOD Throughout the elastic-plastic regime, the stress in the specimen with a / W = 0.50 is approximately 20% higher than the specimen with a~ W = 0.15 at the same C T O D level Therefore,

at equivalent stress levels, the C T O D of the short crack specimen is approximately 2.5 times

as large as the C T O D of the deep crack specimen

Trang 26

Investigations of the short crack three-point bend specimen [3-6] show that the boundary

separating short crack and deep crack specimens occurs between the crack-depth to speci-

men-width ratios ( a / W ) of 0.15 and 0.20, depending on the degree of strain hardening

Between these a~ W ratios, the yielded region at the crack tip also reaches the free surface

behind the crack When this occurs, the stress triaxiality near the crack tip is relaxed and a

loss of constraint occurs Milne and Chell [17] and Pisarski [18] have used the R K R model

proposed by Ritchie, Knott, and Rice [19] to describe the effects of constraint on cleavage

fracture In the R K R model, cleavage fracture occurs when the maximum principal stress

ahead of a sharp crack exceeds a critical value (a0 over a characteristic distance Cleavage

fracture generally occurs if there is a large degree of triaxiality because yielding is restricted

and the maximum principal stress ahead of the crack tip is elevated

In the region of linear-elastic (SSY) fracture behavior, the plastic zone is contained at the

crack tip and negligible loss of constraint occurs Similar stress states are developed at the

crack tip irrespective of the specimen crack depth Therefore the short crack and deep crack

specimens produce equivalent C T O D or K~ results, reaffirming the applicability of a single

parameter characterization

In the region of elastic-plastic fracture behavior, the plastic zone is no longer limited to

small-scale yielding at the crack tip Plasticity in the short crack specimen extends to the free

surface behind the crack and a loss of constraint occurs at the crack tip This loss of con-

straint (or loss of stress triaxiality) requires that the short crack specimen undergo consider-

ably larger strains than the deep crack specimen to develop equivalent opening mode stresses

at the crack tip Larger strains imply increased crack tip blunting and higher CTOD values

Therefore, if fracture occurs when the maximum principal stress ahead of the crack exceeds

the critical value, then short crack specimens should exhibit significantly larger critical

C T O D values than deep crack specimens

Trang 27

These observations developed from numerical analyses are substantiated by the prelimi-

nary results of the corresponding experimental comparison of short crack and deep crack

C T O D specimens [20] The experimental test results of the short crack ( a / W = 0.15) spec-

imens are shown in Fig 19 A lower bound curve was constructed through the lowest exper-

imental values of the deep crack specimens The deep crack lower bound curve was magni-

fied by 2.5 in the lower transition region and agreed extremely well with the lowest

experimental values from the short crack specimens

C o n c l u s i o n s

Elastic-plastic finite element analyses were conducted on a low-strength structural steel

(A36) to determine the effect of crack depth on the behavior of square three-point bend

specimens Two specimen sizes were modeled ( W = 31.8 m m (1.25 in.) and W = 12.7 m m

(0.50 in.)) using the standard geometry of W b y W b y 4W The analyses were employed to

predict the deformation up to the point of a cleavage-type failure with no stable crack growth

The short crack results were compared to results for a deep crack ( a / W = 0.50) specimen

previously analyzed by Sorem et al [8] The results of this study on the short crack specimen

and the comparisons with the deep crack specimen may be summarized as follows:

1 A comparison of the 2-D finite element results of various crack-depth to specimen-

width ratios ( a / W ) showed a fundamental change in the nonlinear stress field at an a / W

value of approximately 0.15 Specimens with shorter cracks ( a / W = O 10 and 0.05) showed

yielding to the free surface behind the crack (back surface) well before the formation of a

plastic hinge Specimens with deeper cracks ( a / W = 0.20) developed a plastic hinge before

the plastic zone extended from the crack tip to the back surface

2 From the 3-D finite element analysis conducted on the specimen with a~ W = 0.15, an

analytical C T O D - C M O D relationship was developed With this relationship, the BS 5762

Trang 28

procedure for calculating CTOD from the load-CMOD record worked very well if the plastic

rotation factor was decreased from 0.40 to 0.20

3 A comparison of square specimens with a/Wratios of0.15 and 0.50 showed that at the

same linear-elastic KI level, short crack specimens and deep crack specimens developed

equivalent opening mode stresses near the crack tip consistent with a single parameter char-

acterization of fracture driving force At equivalent CTOD levels in the elastic-plastic regime,

the short crack specimens exhibited significantly lower opening mode stresses near the crack

tip compared to the deep crack specimens Correspondingly, at equivalent stress levels, the

short crack specimens exhibited CTOD values approximately 2.5 times larger than the deep

crack CTOD values

In summary, the results of this study suggest that the short crack CTOD specimen loses

crack tip constraint when the yielded region at the crack tip extends to the free surface behind

the crack This loss of constraint requires that the short crack specimens undergo consider-

ably more crack tip blunting and plastic zone development than the deep crack specimen to

develop the same opening mode stress near the crack tip These results have considerable

significance in the application of CTOD values to failure analysis or specification develop-

ment where the fracture mechanism is cleavage preceded by significant crack tip plasticity

A P P E N D I X

The finite element solutions employed conventional, linear strain-displacement relations based on small geometry change assumptions Crack growth was not simulated in the finite

element analyses Numerical computations were performed with the POLO-FINITE struc-

tural mechanics system [9, I0]

The 2-D model for both plane-strain and plane-stress analyses employed the 8-node iso-

parametric element Due to symmetry of the 3-point bend specimen, only one (right) half of the specimen was modeled A total of 71 elements and 262 nodes were incorporated in the

model The crack tip was represented with 8-node isoparametric elements collapsed into

triangles This is a common technique to develop a strain singularity of the order 1/'x/~ for linear analysis [21] and 1/r for elastic-plastic analysis [22] The size of the crack tip elements was adjusted for each geometry to be < 5% of the crack depth The collapsed 8-node elements

provided a convenient means to extract CTOD from the displacements of the initially coin-

cident nodes at the crack tip using the 90 ~ intercept method

The 3-D finite element model consisted of 20-node isoparametric elements The element

mesh was constructed by replicating the 2-D grid pattern in the through-thickness dimension

to define five element layers through the half thickness (Fig 2) Element layers varied in

thickness, with thinner elements on the free surface to accommodate the sharp transition to

a traction-free external surface from essentially plane-strain conditions on the interior near

the crack front Symmetry constraints were applied over the remaining ligament on the crack

plane and the longitudinal center plane; only one quarter of the specimen was modelled

given the two symmetry planes The nonlinear model contained 355 elements and 2052

nodes

The 3-D, 20-node isoparametric elements were collapsed into triangular prisms at the

crack tip Their behavior is analogous to that of the collapsed 2-D, 8-node isoparametric

element [23] The coincident corner nodes of the collapsed element develop a strain singu-

larity of the order 1/V~ for linear analysis and 1/r for elastic-plastic analysis The coincident

mid-side nodes of the collapsed element do not develop the same extent of singular region

The CTOD was obtained from the displacements of the initially coincident corner nodes of

the collapsed elements using the same procedure previously outlined for the 2-D analysis

Trang 29

Thus six C T O D values were available through the specimen half thickness for the 3-D analysis

The finite element model for the sub-size square specimen was generated by simply scaling the coordinates of the full-size square specimen The models were scaled to insure identical locations for comparison of stresses and strains (radius and angle from the crack tip) independently of the overall specimen size Identical element aspect ratios were also maintained Reduced integration (2 by 2 for 2-D and 2 by 2 by 2 for 3-D) was used in the analysis to eliminate artificial locking under the incompressibility conditions imposed by plastic defor-

mation [24] N o r m a l integration (3 X 3 X 3 for 3-D) was used for the degenerated crack tip

elements o f the 3-D model to prevent spurious zero-energy modes Plasticity was modeled using incremental theory with a yon Mises yield surface, associated flow rule, and isotropic

hardening [25]

The stress-strain properties of the material for the finite element analysis were derived from

a standard 12.8 m m (0.505 in.) diameter uniaxial tensile test (Fig 1) A piecewise linear

a p p r o x i m a t i o n provided a better match with the experimental curve than did a power law approximation However, due to the Newton-Raphson iteration technique employed to solve the equilibrium equations, a "stiffening" material caused the solution to diverge when

large load steps were taken [26] Consequently, the small yield plateau was replaced with a

straight line from the yield point to a point of tangency on the strain hardening curve to produce a stress-strain curve with a monotonically decreasing slope

The finite element models were loaded directly below the uncracked ligament By impos- ing load increments, rather than displacement increments, the solutions converged much faster Approximately 25 load steps were imposed on each finite element model A conver- gence tolerance of 0.5% on the ratio o f the norm of the residual load vector to the n o r m of the applied load increment was maintained throughout the analyses Initially, the load was imposed directly below the uncracked ligament at a single node for 2-D analysis and at a line

o f nodes through the specimen thickness for 3-D analysis As the size of the plastic zone increased, it was necessary to distribute the load on the adjacent nodes to be consistent with the local deformation observed at the loading roller in the laboratory tests

References

[1] de Castro, P M S T., Spurrier, J., and Hancock, P., "An Experimental Study of the Crack Length/

Specimen Width (a/W) Ratio Dependence on the Crack Opening Displacement (COD) Test Using Small-Scale Specimens," in Fracture Mechanics (Eleventh Conference), ASTM STP 677, C W

Smith, Ed., American Society for Testing and Materials, Philadelphia, 1979, pp 486-497

[2] Sumpter, J D G., "The Effect of Notch Depth and Orientation on the Fracture Toughness of

Multi-Pass Weldments," International Journal of Pressure Vessels and Piping, Vol 10, 1982, pp

169-180

[3] Cotterell, B., Li, Q.-F., Zhang, D.-Z., and Mai, Y.-W., "On the Effect of Plastic Constraint on

Ductile Tearing in a Structural Steel," Engineering Fracture Mechanics, Vol 21, No 2, 1985, pp

239-244

[4] Li, Q.-F., "A Study About aT, and 6, in Three-Point Bend Specimens With Deep and Shallow

Notches," Engineermg Fracture Mechanics, Vol 22, No 1, 1985, pp 9-15

[5] Li, Q.-F., Zhou, L., and Li, S., "The Effect of a~ W Ratio on Crack Initiation Values of COD and J-integral," Engineering Fracture Mechanics, Vol 23, No 5, 1986, pp 925-928

[6] Matsoukas, G., Cotterell, B., and Mai, Y.-W., "Hydrostatic Stress and Crack Opening Displace-

ment in Three-Point Bend Specimens with Shallow Cracks," Journal of the Mechanics and Phystcs

of Solids, Vol 34, No 5, 1986, pp 499-510

[7] Zhang, D Z and Wang, H., "On the Effect of the Ratio a / W on the Values of 6, and J, in a Structural Steel," Engineering Fracture Mechanics, Vol 26, No 2, 1987, pp 247-250

[8] Sorem, W A., Dodds, R H., Jr., and Rolfe, S T., "An Analytical and Experimental Comparison

of Rectangular and Square Crack-Tip Opening Displacement Fracture Specimens of an A36

Steel," Nonlinear Fracture Mechanics: Vol II Elastic-Plastic Fractures, ASTM STP 995, Amer-

ican Society for Testing and Materials, Philadelphia, 1989, pp 470-494

[9] Lopez, L A., "Finite: An Approach to Structural Mechanics Systems," Internattonal Journal for

Numerical Methods in Engineering, Vol 11, No 5, 1977, pp 851-866

Trang 30

[10] Dodds, R H and Lopez, L A., "A Generalized Software System for Nonlinear Analysis," Inter-

national Journal for Advances in Engineering Software, Vol 2, No 4, 1980

[11] Rice, J R., "A Path Independent Integral and the Approximate Analysis of Strain Concentration

by Notches and Cracks," Journal of Apphed Mechamcs, Transactions of the American Society of

Mechanical Engineers, Vol 35, June 1968, pp 379-386

[12] Shih, C F., de Lorenzi, H G., and German, M D., "Crack Extension Modeling with Singular

Quadratic Isoparametric Elements," International Journal of Fracture, Vol 12, No 4, 1976, pp

647-651

[13] Dodds, R H., Jr., Carpenter, W C., and Sorem, W A., "Numerical Evaluation ofa 3-D J-Integral

and Comparison with Experimental Results for a 3-Point Bend Specimen," Engineering Fracture

Mechantcs, Vol 29, No 3, 1988, pp 275-285

[14] Westergaard, H M., "Bearing Pressures and Cracks," Transactions of the American Society of

Mechanical Engineers, Journal of Apphed Mechamcs, 1939

[15] Dawes, M G., "Elastic-Plastic Fracture Toughness Based on the COD and J-Contour Integral

Concepts," in Elastw-Plastw Fracture, ASTM STP 668, J D Landes, J A Begley and G A

Clarke, Eds., American Society for Testing and Materials, Philadelphia, 1979, pp 307-333

[16] Matsoukas, G., Cotterell, B., and Mai, Y.-W., "On the Plastic Rotation Constant Used in Standard

COD Tests," International Journal of Fracture, Vol 26, No 2, 1984, pp R49-R53

[ 17] Milne, I and Chell, G G., "Effect of Size on the J Fracture Criterion," in Elastic-Plastic Fracture,

ASTM STP 668, American Society for Testing and Materials, Philadelphia, 1979, pp 358-377

[18] Pisarski, H G., "Influence of Thickness on Critical Crack Opening Displacement (COD) and J

Values," International Journal of Fracture, Vol 17, No 4, Aug 1981, pp 427-440

[19] Ritchie, R O., Knott, J F., and Rice, J R., "On the Relationship Between Critical Tensile Stress

and Fracture Toughness in Mild Steel," Journal of the Mechanws and Physics of Solids, Vol 21,

1973, pp 395-410

[20] Sorem, W A., Rolfe, S T., and Dodds, R H., Jr., "An Experimental Comparison of Short Crack

and Deep Crack CTOD Fracture Specimens of an A36 Steel," to be published

[21] Barsoum, R S., "Application of Quadratic Isoparametric Finite Elements in Linear Fracture

Mechanics," International Journal of Fracture, Vol 10, No 4, Dec 1974, pp 603-605

[22] Barsoum, R S., "Triangular Quarter-Point Elements as Elastic and Perfectly-Plastic Crack Tip

Elements," International Journal for Numerical Methods m Engineering, Vol 11, No 1, 1977, pp

85-98

[23] Bloom, J M and Van Fossen, D B., "An Evaluation of the 20-Node Quadratic Isoparametric

Singularity Brick Element," International Journal of Fracture, Vol 12, No 1, Feb 1976, pp 161-

163

[24] Dodds, R H., "Effect of Reduced Integration on the 2-D Quadratic Isoparametric Element in

Plane Strain Plasticity," International Journal of Fracture, Vol 19, No 3, 1982, pp R75-R78

[25] Nayak, G C and Zienkiewicz, O C., "Elasto-Plastic Stress Analysis: A Generalization for Various

Constitutive Relations Including Strain Softening," International Journal for Numerical Methods

in Engineering, Vol 5, No 1, 1972, pp 113-135

[26] Wellman, G W., Rolfe, S T and Dodds, R H., Jr., "Three-Dimensional Elastic-Plastic Finite

Element Analysis of Three-Point Bend Specimens," in Fracture Mechanics: Sixteenth Symposium,

A S T M STP 868, American Society for Testing and Materials, Philadelphia, 1985, pp 214-237

Trang 31

R Herrera 1 a n d J D L a n d e s 2

Methodology

REFERENCE: Herrera, R and Landes, J D., "Direct J-R Curve Analysis: A Guide to the

Methodology," Fracture Mechanics: Twent.v-First Symposium, ASTM STP 1074, J P Gudas,

J A Joyce, and E, M Hackett, Eds., Americian Society for Testing and Materials, Philadel-

phia, 1990, pp 24-43

ABSTRACT: A direct method for evaluating J-R curves from load displacement data is pro-

posed This method eliminates a need for automatic crack length monitoring equipment The

method uses the geometric normalization suggested by Ernst for deeply cracked bend speci-

mens to develop calibration curves which relate load, displacement, and crack length Given

two of these parameters, the third can be determined from the calibration curves These curves

are developed by assuming a functional form with unknown constants and evaluating the

unknowns at calibration points in the test This method was previously proposed using a power

law functional form In this paper a more general functional form is proposed which combines

a power law and straight line The method is studied for a variety of materials and specimen

sizes From the results a recommendation is made on how to generally apply the method

KEY WORDS: elastic-plastic fracture, J integral, fracture toughness, J-R curve, normalization

method, plastic deformation pattern

J-R curves represent a basic fracture toughness property for materials which fail in a ductile m a n n e r [I] Standard tests for developing J-R curves rely on automatic methods for measuring crack advance [2]; the elastic unloading compliance method is most frequently used [3,4] The automatic crack measuring methods require sophisticated equipment and techniques; sometimes a good deal of effort is required to achieve good results

Landes and Herrera have recently proposed a method for directly determining a J-R curve from a load versus displacement record This method does not require automatic crack length measuring equipment [5,6] The method is based on the principle of normalization

of deformation properties of a material for which load, displacement, and crack length can

be functionally related [7,8] If an appropriate functional form is assumed, the crack length can be determined directly from corresponding load and displacement values

The functional form previously assumed was a power law [6] The method was applied to

a single material, A508 steel, where compact specimens of different sizes were analyzed to

develop J-R curves by this direct method of normalization These R curves were compared with ones determined by elastic compliance measurements The direct method worked well for this material, possibly because a power law gave a good representation of the material stress-strain properties A power law is not a general representation for deformation and therefore may not generally apply for the direct analysis of the J-R curve

The method, however, only requires that a known functional form be used; this functional

National University of Mar Del Plata, Mar Del Plata, Argentina

2 Department of Engineering Science and Mechanics, The University of Tennessee, Knoxville, TN

37996

24 Copyright 9 1990by ASTM International www.astm.org

Trang 32

HERRERA AND LANDES ON DIRECT J-R CURVE ANALYSIS 25 form does not necessarily have to be a power law Therefore to make the method more appli-

cable, a general representation of deformation is needed This paper proposes a general func-

tional form which can be used in the direct method Using this general function the method

is evaluated by applying it to several steels and specimen sizes The results are used to suggest

general guidelines for analyzing the load versus displacement record of any metal undergoing

ductile fracture to develop the J-R curve without automatic crack length monitoring

equipment

Review of Normalization

The principle of normalization follows from the work of Ernst et al [7,8], who showed

that load, P, can be written as separable multiplicative functions of crack length, a, and

displacement, v, when the plastic c o m p o n e n t of displacement is used:

where a and vp~ are normalized by a width dimension, W, and displacement is separated into

elastic and plastic components:

where C ( a / W ) is a compliance function relating load and elastic displacement

For deeply cracked bend type specimens the function form for the crack length depen-

dence, G(a/W), is known [8] The load can be normalized by this function:

so that normalized load, IN, is only a function o f vpff IV, that is, the deformation character

of the material The functional form of H(vp~/W) should then reflect the form of the material

stress-strain relationship This stress-strain relationship can often be a power law [9] There-

fore it seems reasonable to assume that a power law functional form might work for

H( Upl / W)

A power law function given by

W

where A and m are unknown constants, was used to analyze an A508 steel where compact

specimens varied in size from W = 25 m m to W = 500 m m [6] The unknown constants

were evaluated by taking load and displacement values at the initial and final crack lengths,

where crack lengths were measured from the surface of the broken specimen The method

for choosing the appropriate values for m and A is described in Ref 6 The resulting J-R

curve from this analysis was compared with the J-R curve generated by the elastic unloading

compliance method W h e n the compliance method was successful in determining the phys-

Trang 33

ically measured crack length, the two J - R curves agreed very well W h e n compliance method predicted the crack length poorly, the method of normalization was judged to improve the quality o f the J-R curve A n example of a successful use of both is given in Fig 1, where Fig

1 a shows the normalized load PN as a function of Vp,/W and Fig 1 b shows the resulting J-R curve

The success with which the method was applied to the A508 steel could be directly related

to the stress and strain properties which can be well represented by a power law relationship

[10] There are other materials where a power law is not a good description of the stress- strain relationship A different functional form should be postulated for the H(vp]/W) values

[] COMPLIANCE I + POWER LAW

Trang 34

H E R R E R A A N D L A N D E S ON D I R E C T J-R C U R V E A N A L Y S I S 27

o f these materials A n i m p o r t a n t consideration in choosing a functional form is that

unknown constants can only be evaluated when load, crack length, and displacement are

known for a c o m m o n point Only initial and final crack lengths can be determined from a

specimen fracture surface, in principle leaving only two points for determining unknowns

The choice o f the functional form for H ( v J W ) must then be one which is compatible with

the stress and strain relationship and allows unknown parameters to be evaluated at the

known values o f crack length

Trang 35

Deformation Relationships

Austenitic stainless steel is a material whose stress-strain relationship does not follow a

power law A previous study had been conducted to develop J-R curves for stainless steel specimens using the elastic compliance procedure [11] An example test was re-analyzed using the normalization procedure with a power law fit for PN versus Vpl/W following the

method described in Ref 6 The result (Fig 2) was not particularly good This result rein- forces the idea that a power law stress-strain relationship is necessary for the method to work

with a power law H(vp# W) function

COIVPLIANCE POWER LAW

A a , m m

FIG 2 J versus Aa R-curve for Type 304 stainless steel, W = 50 mm, comparing power

law with compliance

Trang 36

HERRERA AND LANDES ON DIRECT J-R CURVE ANALYSIS 29

The study in Ref 1 1 had suggested that series of power laws could be used to model the stress-strain behavior of the stainless steel with a different exponent being used over different ranges of strain (Fig 3a) If the stress-strain relationship is to be used as a model of the Pu versus vp~/W relationship, a similar relationship could be used However, this does not fit the criteria suggested in the previous section of keeping the relationship simple enough to be described by two unknown constants

A careful examination of the stress-strain relationship for this material shows that a straight line gives a better fit of the data over a larger range of strain (Fig 3b) Using this as

True Strain, ram/ram

F I G 3 a - - T r u e stress versus true strata for Type 304 stamless steel showing power law fits

F I G 3 b - - T r u e stress versus true strain for Type 304 stainless steel adding straight hne l~t

Trang 37

30 F R A C T U R E M E C H A N I C S ' T W E N T Y - F I R S T S Y M P O S I U M

a model for the PN versus % j / W relationship would suggest that the H ( v j W) function in Eq

4 could be replaced by the straight line relationship

where D~ and Dz are unknown constants This has only two unknowns which can be determined at the initial and final crack lengths

Using the straight line relationship in Eq 5 and a procedure similar to one described previously (Eq 6), the normalization method was applied to the stainless steel specimen of Fig

line, and power law (W = 50mm)

Trang 38

HERRERA AND LANDES ON DIRECT J-R CURVE ANALYSIS 31

2 The resulting J-R curve is shown in Fig 4, where all three m e t h o d s - - c o m p l i a n c e , power

law H ( v J W), and straight line H(v~,/W) are compared The straight line function gives a

much better match to the one developed from the compliance procedure The three methods

are also c o m p a r e d in terms of PN versus vpt/W in Fig 5 This shows that the deformation

behavior is better described by a straight line

This result suggests that the normalization procedure can work well for determining J-R

curves if the functional form of H(up,/W) is known That functional form could be similar

to the one from stress and strain relationship However, given a record of load and displace-

L

ment for an arbitrary material that has no stress-strain data available, a proper choice of the

H ( v j W) functional relationship may be difficult It would be helpful to develop some gen-

eral guidelines for the arbitrary choice of an H(vp,/W) function

§

+ ~ + ~

+ a + a +

Trang 39

Comparing the normalized load and plastic displacement for the stainless steel specimen

in Fig 5 with that for the A508 steel specimen in Fig 1 a an obvious difference is in the range

o f vp,/W needed to develop the J-R curve For the A508, the m a x i m u m vp~/W value is about 0.05, but for the stainless steel this value is greater than 0.20 Evaluation of other materials showed that generally specimens which had a vp~/W range of 0.05 or less worked well with a power law fit to the normalized curve whereas a range o f vp~/Wgreater than 0.1 worked well with a straight line fit

The conclusion from this is that the material deformation pattern seems to describe a power law for the short Vpl/W range and approaches a straight line at larger Vp~/W A more accurate way to approximate the functional form o f H(vp,/W) would be to start with a power law for low vp,/W a n d ' t h e n change to a straight line at some value of vpdW around 0.05 Although this appears to be the ideal way to develop a functional form of H(vp,/W), it pre- sents a procedural difficulty in that an intermediate calibration point cannot be determined from the information available Only an initial and a final value of crack length can be used with the appropriate load and displacement pairs as calibration values for determining unknown constants However, for m a n y of the specimens crack growth begins after some

Trang 40

straight line (W = 50 ram)

a m o u n t o f v J W has been developed Therefore several pairs o f PN and vp~/W could be asso-

ciated with the initial crack length and used to determine the exponent in the power law An

example of this is given in Fig 6, where the two axes are plotted in a log-log format so that

the exponent can be identified from the slope o f the resulting line This figure shows that

initially a power law relationship is established However, when the point associated with the

final crack length is plotted, it is obviously not part of the same power law relationship

Therefore, using the combined two functional forms, power law and straight line, appears

appropriate but the point o f transition between the two must be determined from some addi-

tional information Looking at the data from Fig 6 replotted on linear axes (Fig 7), a simple

transition point is available A line drawn from the final calibration point tangent to the top

Tiêu đề	Fracture Mechanics: Twenty-First Symposium
Tác giả	J. P. Gudas, J. A. Joyce, E. M. Hackett
Trường học	University of Washington
Chuyên ngành	Fracture Mechanics
Thể loại	Symposium
Năm xuất bản	1990
Thành phố	Philadelphia

Định dạng
Số trang	615
Dung lượng	13,2 MB