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INTERIM PROGRESS REPORT:

l-INTEGRAL METHOD FOR FITNESS-FOR-SERVICE ASSESSMENT

D.T. Read

H.l. McHenry

Fracture and Deformation Division

National Measurement Laboratory

National Bureau of Standards

Boulder, Colorado 80303

Sponsored by:

David Taylor Naval Ship Research & Development Center

Annapolis, MD 21402

May 1981

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INTERIM PROGRESS REPORT:

MHTEGRAl METHOD FOR FITNESS-FOR-SERVICE ASSESSMENT

D.T. Read

H.l. McHenry

Fracture and Deformation Division

National Measurement Laboratory

National Bureau of Standards

Boulder, Colorado 80303

May 1981

Sponsored by:

David Taylor Naval Ship Research & Development Center

Annapolis, MD 21402

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Director

Interim Progress Report:

J-integral Method for Fitness-for-Service Assessment

D. T. Read* and H. I. McHenry*

National Bureau of Standards

Boulder, Colorado 80303

Progress to date on the J-integral fitness-for-service assessment

project is reported. The goal of this project is to produce a method

for evaluation of material toughness requirements, allowable defect

sizes, and sustainable stresses and strains. Yielding fracture mechanics

and, specifically, the J-integral are employed. The J-integral is

regarded as the crack driving force; applied J values above some critical

value are predictive of material tearing. The applied J value is a function of applied stress and strain, crack length, and material tensile

properties. The critical J-integral value is measured in material

fracture-toughness tests. The experimental focus of the present study

is on measurement of the applied J-integral as a function of stress and

strain in HY130 specimens with different crack lengths. Neither experimental data nor established predictive theories for the behavior of

applied J values are available in the literature for short cracks, the

type most likely to be found in a structural element. Results for

relatively large cracks indicate that displacements imposed at the

specimen ends are transmitted in full to the cracked cross section. The

displacements add to the strain field of the crack and produce a large

applied J value. This behavior is described as net section yielding.

For relatively short cracks, displacements imposed at the specimen ends are absorbed as plastic strain all along the specimen length, add only

slightly to the strain field of the crack, and produce small applied J values. This behavior is described as gross section yielding. It is

important to note that cracks as small as 5 % of the total cross section

have been found to exhibit net section yielding.

Key Words: Crack opening displacement; design curve; elastic-plastic

fracture mechanics; fitness-for-service; fracture mechanics; J-integral.

*Fracture and Deformation Division, National Measurement Laboratory

1
ADMINISTRATIVE INFORMATION

This report was prepared as part of the Fracture Control Technology

Program under the sponsorship of Dr. H. H. Vanderveldt, Naval Sea Systems

Command (SEA 05R15). The effort was monitored by Mr. John P. Gudas,

David Taylor Naval Ship Research and Development Center, Code 2814,

under Program Element 627IN, Task Area SF 61-544-504.

2
INTRODUCTION

In 1978, the Naval Sea Systems Command started a program entitled

"Fracture Control Technology for Ships and Submarines." The initial

tasks included the identification of Navy fracture control requirements [l] 1

and an assessment of fracture control practices used by United States [2]

and foreign [3] industries. The review of Navy requirements identified

several questions that advances in fracture control technology may help

to answer:

1. How much toughness is sufficient for a given application?

2. How large a defect can be tolerated in a given structural

element, particularly in welds and castings?

3. How can one take the results of a fracture toughness test and

apply them to a complex structural configuration?

4. How can elastic plastic fracture mechanics be reduced to a

form useful to the Navy?

The assessments of fracture control practices revealed a technical

approach that may serve as a useful pattern for the development of a

method to answer these questions: f itness-for-service evaluation based

on the crack-opening-displacement (COD) design curve [4]. This approach

has been widely used, particularly in Europe, to answer similar questions

for a variety of structural applications [5].

The key ingredient of a fitness-for-service assessment methodology

is an elastic-plastic-fracture-mechanics relationship between applied

strain, flaw size, and required toughness. Given such a relationship,

one can proceed to answer the questions cited earlier. Minimum toughness

^Numbers in brackets indicate the literature references at the end of

the paper.

3
requirements can be based on conservative estimates of the anticipated

service stress and strain levels and flaw sizes. Allowable defect sizes

and associated repair decisions can be based on specified toughness

levels and maximum credible stress levels. Conceptually speaking,

elastic-plastic-fracture mechanics could be readily applied to a wide

range of complex structures in a manner analogous to the way linearelastic-fracture mechanics (LEFM) is applied to aircraft structures.

The purpose of this program is to develop a fitness-for-service

approach that would be applicable to the high-strength, high-toughness

alloys used by the Navy. The specific objectives are:

1. To develop experimental methods to relate required toughness,

flaw size, and applied strain under elastic-plastic loading conditions;

2. To relate required toughness to applied strain experimentally

for a variety of flaw sizes and configurations in specimens that

simulate structural elements;

3. To develop analytical methods that are reliable predictors of

the experimental results and, implicitly, of structural behavior;

4. To demonstrate the validity of such methods by using wide

plate tests on HY130 weldments.

The present report is the first interim report on the continuing

program to accomplish these objectives; the experimental methods and

experimental results for single-edge-notched tensile panels are described.

This report consists of a discussion of the technical approach and a

4
brief summary of the experimental procedures and results. The two

appendices are technical publications in which the experimental procedures

and the results for a C-Mn ship steel and for HY130 steel are presented

in detail

.

TECHNICAL APPROACH

The technical approach is modeled after the COD design curve developed

by the British Welding Institute [4]. There are, however, two fundamental

differences in this approach: the J-contour integral, J, is used as the

fracture criterion instead of the COD, and the experimental methods of

developing the relationship of the required toughness to strain and flaw

size are completely different. In this section, the COD-design-curve

approach is reviewed and the J-integral approach is introduced and

described.

The COD Design Curvp

The COD design curve relates required fracture toughness, 6 c

,

applied strain, e, and allowable flaw size, a, as shown in figure 1.

The curve is an empirical lower bound derived from the results of wide

plate tests. A given data point (x,y) used to derive the curve consists

of (1) an x-coordinate of e/Sy, where e is the failure strain in the

wide-plate test and is the yield strain of the test material, and

(2) a y-coordinate of 6 /2tt£ a, where 6 is the COD fracture toughness

c y c measured with a ful 1 -thickness bend test on the same material used in

the wide-plate test and a is the effective defect size parameter for the

crack in the wide plate test. Thus, a given data point requires a

fracture- toughness test and a wide-plate test.

The COD design curve is used to make fitness-for-service assessments

in cases where two of the three variables -- 6 , e, and a -- can be

5
safely estimated. From the design curve, the third variable is assessed,

e.g., minimum toughness requirements can be established if e and a are

given, etc. The parameters e and aare defined to permit assessment of a

wide range of conditions. The effective defect size parameter, ais the

half-length of an equivalent through-thickness crack. The design is

used in accordance with rules for calculating afor a range of defect

types including surface defects, embedded defects, clustered defects,

and defects adjacent to fillets or holes. The applied strain, e, is

the sum of the average membrane strain, bending strain, resiudal strain,

and peak strain due to a local stress concentration.

The COD design curve is useful in its present form to evaluate

fi tness- for- service for a wide range of structures and materials [5].

However, it is not developed sufficiently for use in critical Naval

applications. The underlying empirical data base does not include

enough results on the high-strength, high-toughness alloys used in Naval

vessels and many of the underlying assumptions have not been critically

evaluated, e.g., the calculation of a for a surface flaw, the treatment

of stress concentrations and residual stresses, and the assumption of

uniform strain after yielding. To overcome these shortcomings, it was

decided to use the COD design curve as the model for the development of

a method for fitness- for-service assessment using materials of interest

to the Navy.

The J-integral Approach

Given the task of developing a fitness-for-service assessment

methodology for the Navy, the first consideration was selection of a

6
fracture criterion. The J-contour integral described by Rice [6] had

two critical attributes: first, it had been used as an elastic-plastic

fracture criterion [7,8,9]; second, it had been measured as a function

of strain in single specimens in configurations different from standards

fracture- toughness specimens [10,11,12]. It was not considered feasible

to measure COD as a function of strain for nonstandard specimen geometries.

No well-developed elastic-plastic-fracture-mechanics fracture criterion

other than J or COD was available. Therefore, the J-contour integral

was selected as the fracture criterion for use in a fitness-for-service

assessment method patterned on the COD-design-curve approach.

The needed relationship of the required J-integral fracture toughness

to applied strain and flaw size was derived by assuming that the required

material toughness at a given strain level and defect size was simply

equal to the J contour integral measured at that strain level and defect

size. This was seen as a natural extension of the assumption used in

J-integral fracture- toughness testing that the applied J contour integral

at crack initiation is taken as the material fracture toughness. In

defect size assessment, the largest acceptable defect size would be the

size at which the J contour integral was equal to the material fracture

toughness. Similarly, the largest acceptable strain level would be the

level for which the applied J for the defect size under consideration

was equal to the material fracture toughness. Thus a complete conceptual

scheme for fitness-for-service assessment using the J-integral was

formulated.

The next step was to develop relationships of the J contour integral

to applied strain and crack size. Curves displaying J as a function of

e for a given specimen configuration with a series of flaw sizes are

7
shown schematically in figure 2a. If the strain pattern remains uniform

as crack size increases, the curves of figure 2a can be collapsed into a

single curve for that specimen configuration (fig. 2b). Rules to develop

an equivalent flaw size, a, for a variety of configurations would lead

to a single design curve. If, on the other hand, the form of the strain

pattern were to change with flaw size, the individual J-e curves would

not collapse onto a unique design curve, and approximate methods for

describing this behavior would be required for fitness-for-service

assessments. To put the latter situation in perspective, recall that

the LEFM analysis procedures are configuration dependent. This dependence

is expressed mathematically by writing

K = a ika Y ( a ) ( 1

)

where K is the stress intensity factor, a is the stress, and Y(a) is a

geometry-dependent function. To provide experimental data for the Jvs.- strain curves described here schematically, the techniques referred

to earlier were applied.

Previous techniques for measurement of J as a function of strain

include the compliance technique and two techniques for direct measurement

of the J-contour integral. The compliance technique [7] was tested in

the work on the C-Mn ship steel described in Appendix A. This method

was rejected because it was not considered appropriate for cracks 25% or

less of the specimen width. A technique for direct measurement of the

J-contour integral was developed and used for the linear elastic case by

Kino and Herrmann [12], and a different technique applicable to rubber

8
was developed by Oh [11]. The method developed for the present study

extended the experimental evaluation of Rice's path independent Jcontour integral [6] to the elastic-plastic regime. According to Rice,

the J-integral is given by

J =j Wdy - f~|i ds (2)

where W is the strain-energy density, f is the traction vector normal to

the contour r surrounding the crack, x and y are Cartesian position

coordinates, u is the displacement vector, and ds is a length increment

along r.

The strain-energy density is evaluated from strain gages mounted

along a convenient path surrounding the crack tip using the relation

(3)

where the a-e curve of an uncracked specimen serves as the function

a 22

(

e 22 ^ ’ anc* ot *ier com Ponents of the stress tensor are zero

because the contour follows the free surface. The rotational contribution

f

ell •— is evaluated by using displacement gages mounted along the xdirection component of the contour r. The nature of the strain energy

and rotational contributions to J are shown in figure 3. Experimental

details for measurements on single-edge-cracked specimens are given in

Appendix B. The method of direct measurement of the J contour integral

was found to be suitable for evaluating the applied J-integral as a

function of strain for several different nonstandard specimen geometries

with short cracks.

9
EXPERIMENTAL PROGRAM

Material

s

Specimens of two different steels were tested. Preliminary experiments

were conducted using C-Mn ship steel specimens. This steel was selected

because of its extremely high toughness to prevent specimen tearing from

interfering with measurements of the applied J-integral. This material

had a yield strength of 340 MPa (49.3 ksi) and a tensile strength of 483

MPa (70.1 ksi). Its chemical composition is listed in table 1. HY130

steel is the principal specimen material of this program. The plate

used had a yield strength of 933 MPa (135 ksi) and an ultimate tensile

strength of 964 MPa (140 ksi). Its chemical composition and heat treatment

are 1 i sted i n tabl e 1

.

Table 1. Chemical compositions by weight percent and treatment for

steels used in the experimental program.

C-Mn Shi p Steel

Chemical composition, weight percent:

C M P S Si Cr Ni Mo Cu V Al_

0.11 1.44 0.10 0.007 0.21 0.08 0.30 0.06 0.20 0.001 0.038

Ce Fe

0.0 bal

Treatment: Silicon killed, vacuum degassed, aluminum added, normalized.

HY130 Steel

Chemical Composition, weight percent:

Ni Cr Mo V 5.00 0.42 0.53 0.043

S Cu A1 Co

0.004 0.022 0.021 0.02

Treatment: Austenitized, quenched.

C Mn P Si

0.11 0.76 0.005 0.03

Ti _ Fe

0.008 bal

tempered, quenched.

10
Techni que

The experimental technique developed for the present study, direct

measurement of the J-contour integral, is described in detail in Appendix B.

This method uses strain gages and LVDT's to measure the quantities

inside the J-integral directly; numerical integration is used to calculate

the J-integral from these data.

Specimens

In this study, a total of 9 measurement runs have been completed to

date. These include 4 single-edge-cracked specimens of C-Mn ship steel

and 5 single-edge-cracked HY130 specimens. All specimens were approximately

9-cm wide and were pin loaded. Saw cuts were used to simulate cracks

in most specimens. In others, fatigue cracks were used.

RESULTS

The results for the C-Mn steel specimens are discussed in Appendix A;

the results for the HY130 single-edge-cracked specimens are interpreted

briefly below.

Data accumulated on single-edge-cracked specimens (shown in fig. 4)

indicate some important features of the dependence of J on applied

strain and crack length. First, the functional form of the dependence

of J on strain is usually parabolic, then linear. All credible theories

and most of the experimental data support this behavior. The parabolic

region at low strains is the region of 1 inear-elastic-fracture-mechanics

.

Here the J-integral approach reduces to the well-known stress intensity

factor approach. Second, the slopes of the linear parts of the Jvs. -strain curves depend on crack length. For fairly large cracks in

11
tension (as opposed to bending), the slopes are weakly dependent on

crack length. As the crack length decreases to low values, the slope of

the J-vs. -strain plot decreases sharply. In figure 4, the crack length

at which this slope change occurs is less than 4 mm in a 90-mm wide

specimen.

The interpretation of these results is believed to be as follows:

for relatively large cracks, displacements imposed at the specimen ends

are transmitted in full to the cracked cross section; they add to the

strain field of the crack and produce a large applied J value. This

behavior is described as net section yielding [13]. For relatively

short cracks, like the 1-mm crack in figure 4, displacements imposed at

the specimen ends are absorbed as plastic strain all along the specimen

length; they add only slightly to the strain field of the crack and

produce small applied J values. This behavior is described as gross

section yielding [13]. It is important to note that cracks as small as

5 % of the total cross section can exhibit net section yielding.

CONCLUSIONS

From the effort expended so far in this program the following

conclusions have been reached:

1. Direct measurement of the J contour integral, supported by

finite element analysis and theoretical modeling, is capable

of providing accurate evaluation of the applied J-integral in

the presence of short cracks (down to about 1 mm) in specimens

simulating structural situations.

12
2 . For relatively large cracks, displacements imposed at the

specimen ends are transmitted in full to the cracked cross

section; they add to the strain field of the crack and produce

large applied J values (net section yielding). For relatively

short cracks, displacements imposed at the specimen ends are

absorbed as plastic strains all along the specimen length; they

add only slightly to the strain field of the crack and produce

small applied J values (gross section yielding). Cracks as

small as 5% of the total specimen cross section have been

found to exhibit net section yielding.

WORK IN PROGRESS

Work in progress includes testing and analysis for a series of

different specimen geometries, including center, part-through, and

double-edge-cracked base-metal specimens, center-cracked base-metal wide

plates and welded specimens in small and large sizes. Finite element

analytical studies are in progress at the University of Kansas to allow

analytical replication of the experimental ly observed behavior.

ADMINISTRATIVE INFORMATION

This report was prepared as part of the Fracture Control Technology

Program under the sponsorship of Dr. H. H. Vanderveldt, Naval Sea Systems

Command (SEA 05R15). The effort was monitored by Mr. John P. Gudas,

David Taylor Naval Ship Research and Development Center, Code 2814,

under Program Element 6276IN, Task Area SF 61-544-504.

13
References

[1] Judy, R. W., Jr.; Crooker, T. W. ; Corrads, J. A. ; Beach, J. E.;

Hauser, J. A. Review of fracture control technology in the U.S.

Navy. Presented at the Second Program Review, Fracture Control

Technology for Ships and Submarines; 1979 November 30; Annapolis, MD.

[2] McHenry, H. I.; Rolfe, S. T. Fracture cotnrol practices for metal

structures. National Bureau of Standards (U.S.) NBSIR 79-1623; 1980

January. 114 p.

[3] Brock, D. Fracture control approaches in Europe. Final report to

the Naval Ship Research and Development Center; Battelle Columbus

Laboratories, Columbus, OH; 1978 July 25. 45 p.

[4] Anon.; Guidance on some methods for the derivation of acceptance

levels for defects in fusion welded joints. British Standards

Institution, London, England; 1980. 64-93.

[5] Harrison, J. D. ; Dawes, M. G. ; Archer, G. L. ; Kamath, M. S. The

COD approach and its application to welded structures, in Elastic

plastic fracture, ASTM STP 668. J. D. Landes, J. A. Begley, and

G. A. Clarke, eds . Philadelphia, PA: American Society for Testing

and Materials; 1973. 1-20.

[6] Rice, J. R. A path independent integral and the approximate analysis

of strain concentration by notches and cracks. J. Appl . Mech. 35:

379-386; 1968 June.

[7] Begley, J. A.; Landes, J. D. The J integral as a fracture criterion,

in Fracture toughness, ASTM STP 514. Philadelphia, PA: American

Society for Testing and Materials; 1973. 1-20.

14
[8] Landes, J. D. ; Begley, J. A. The effect of specimen geometry on Jj ,

in Fracture toughness, ASTM STP 514. Philadelphia, PA: American

Society for Testing and Materials; 1972. 24-39.

[9] Longsdon, W. A. Elastic plastic (Jj

c

) fracture toughness values; their

experimental determination and comparison with conventional linear

eastic (Kj

c

) fracture toughness values for five materials, in

Mechanics of crack growth, ASTM STP 560. Philadelphia, PA: American

Society for Testing and Materials; 1976. 43-60.

[10] Bucci , R. J. ; Paris, P. C. ; Landes, J. D. ; Rice, J. R. J integral

estimation procedures, in Fracture toughness, ASTM STP 514.

Philadelphia, PA; American Society for Testing and Materials;

1972. 40-69.

[11] Oh, H. L. A simple method for measuring tearing energy of nickel

rubber strips, in Mechanics of crack growth, ASTM STP 590.

Philadelphia, PA: American Society for Testing and Materials; 1976

104-114.

[12] Herrmann, G.; Kino, G. S. Ultrasonic measurements of inhomogeneous

stress fields. Proceedings of ARPA/AFML review of progress in

quantitative NDE, AFML-TR-78-205 . 1979 January. 447-451.

[13] Turner, C. E., Methods for post-yield fracture safety assessment,

in Post-yield fracture mechanics. D. G. H. Latzko, ed. London:

Applied Science Publishers; 1979. 23-210.

15
Toughness

/

Flaw

size, 6

c

/27T€y

Allowable flaw size , a

fracture toughness, 8 C

applied strain, €

material properties, c

y = ^

Figure 1. Crack-opening-displacement (COD) design curve.

16
J=f(e,a, material)

riigure 2b. Single curve formed from the curves of Fig. 2a by

dividing ordinate, J, by crack length a, and a factor which includes material tensile properties,

17
DEFORMATION PATTERNS THAT CONTRIBUTE

TO THE J ~ INTEGRAL

Figure 3. Sources of the strain energy and rotational con- tributions to J.

18
GAGE LENGTH STRAIN

Figure 4 Measured J-integral plotted against strain for single-edgecracked speicmens. Crack lengths are noted on the graph.

19

J-INTEGRAL,

Ib/in
APPENDIX A

EXPERIMENTAL METHOD FOR DIRECT EVALUATION OF THE J CONTOUR INTEGRAL

D. T. Read

Fracture and Deformation Division

National Bureau of Standards

Boulder, Colorado 80303

Key Words: Contour integral; direct evaluation; displacement; evaluation;

experimental; J-integral, line integral; numerical integration

path independent; strain
ABSTRACT

A method for direct experimental evaluation of the J contour integral has been developed and used to measure J as a function of strain

in tensile panels of high strength steel (a^ = 900 MPa) under elasticplastic loading conditions. The principle of the present method is to

measure the integrand terms of J at suitable intervals along an appropriate contour and then to evaluate the integral. Because the resulting

J values are based directly on the definition of J, no assumptions about

the crack size or the stress-strain fields in the vicinity of the crack

tip are necessary. This advantage is crucial in cases of very short

cracks (less than 10% of the panel width), which are of special interest

for structural applications. The integrand terms are measured with

strain gages and linear variable displacement transducers (LVDTs) placed

on an appropriate contour surrounding the crack. Load, LVDT-displacement, strain-gage, and crack-mouth-opening-displacement measurements are acquired, stored, and printed at each deformation increment, and the Ji ntegral is evaluated numerically by a minicomputer. J-integral values

are plotted against the average specimen strain in real time. Directly

measured J-integral values and values calculated using linear elastic

fracture mechanics agreed within 12% for average strain values between

10 and 90% of yield. The experimental uncertainties become significantly less for strains of several times yield than for strains below

yield. This technique has been used to measure J as a function of

strain in edge-cracked and center-cracked tensile panels with short cracks

at strains beyond yield, cases for which closed- form solutions for J are not available.

21
INTRODUCTION

This paper describes a method for experimental evaluation of the Jintegral as a function of stress or strain for single-edge cracks. The

method is also applicable to through cracks and may be of interest for

studies of surface cracks. The principle of the present method is to

measure the integrand terms of J at suitable intervals along an appropriate contour and then evaluate the integral. Because the resulting

J value is based directly on the definition of J, no assumptions about

the crack size or the behavior of stress-strain fields produced by the

action of the crack tip are necessary. This advantage is crucial in

cases of very short cracks, which are of special interest for structural

applications. The present method extends previous direct experimental

evaluations of the J-integral in metals in the elastic range [1] and in

rubber [2] to the case of metals in the plastic-strain range.

The J contour integral has been applied widely in characterizing

the fracture toughness of metals [3-5] but its application to evaluation

of the durability of structures containing cracks or similar defects has

been delayed by the absence of information on the material Jj

c value

needed in a structural element containing a crack and subjected to a

certain stress or strain. The present study, therefore, is complementary to current work advancing the use of the J-integral for materials characterization ; it also extends current knowledge of applied Jintegral values beyond the linear-elastic case where well-known and

widely used handbook solutions [6] for the stress intensity K can be

used to calculate the applied J-integral.

22
METHOD

The definition of the J-integral as given by Rice [7] is:

J = / W dy - f*8u/8x ds , (1)

r where W is the strain energy density, f is the traction vector, u is

the displacement vector, x and y are Cartesian position coordinates, and

s is arc length along the contour. The contour chosen for direct measurement

of the J-integral is indicated in Fig. la. Because of symmetry about

the plane of the crack, only half of each specimen was instrumented.

Tensile panels were selected because they have more accessible cracks, a

larger possible range of crack-1 ength-to-specimen-width ratios, and a

greater relevance to structural situations than bend bars. The contour

follows specimen free surfaces along the y-direction and traverses the

tensile axis at a location away from the crack. Measurement of the Jintegral along this contour requires determination of the strain energy

density along the y-direction contour segments and measurement of traction

and displacement vectors along the x-direction segments. The strain-energy

density was obtained by using strain gages along the contour (Fig. lb) and

referring these strains to the stress-strain curve of an uncracked tensile

specimen. The energy density of the uncracked specimen was calculated

simply as the area under its stress-strain curve. Because the only nonzero

stress component at the strain gages along the y-direction contour segments

was the axial tensile stress, the strain-energy density value for a given

strain was the same as that for the tensile specimen. The y-components of

the traction and displacement vectors along the x-direction contour

segments were measured using strain gages and LVDT's, respectively. Because

the horizontal contour segment was chosen to be away from the crack, only the

y-components of the traction and displacement were significant.

Since the only significant stress at the contour was the y-direction

23
tensile stress, it could be obtained from the strain measured at the

contour using the stress-strain curve of the uncracked tensile bar. The

y-component of the traction was numerically equal to this stress. The

y-direction displacement was measured at three locations along the xdirection contour segment. The change in displacement with position,

du/dx, was calculated from the measured displacements. After determination

of the terms of the integrand, the J-integral was calculated either (a)

by numerical integration using predetermined values for the length of

contour covered by each strain-energy density or traction-bending term

or (b) by fitting the strain along each of the two y-direction contour

segments with polynomials or selected nonlinear functions and then

calculating strain-energy density values and performing the required

integrations using the strains as given by the fitting functions. The

method involving nonlinear fits to the measured strains was finally

judged to be superior.

The specimen material used in the present study was HY-130 steel;

the plate used had a yield strength of 933 MPa and a tensile strength of

964 MPa. Tensile panels with gage sections 30-cm long by 9-cm wide by

1-cm thick were used. Saw-cut notches were usually used instead of

fatigue cracks to postpone tearing. Specimens having single-edgenotches with lengths of 1 , 2, 4, 7, and 20 mm were tested. The specimens

were instrumented as in Fig. lb. Approximately eight strain gages were

placed along each y-direction contour segment. Very small gages (active

length of about 0.75 mm) were used near the mouth of short cracks.

Three LVDT's were mounted along the x-direction contour. Strain gages

were placed on the specimen between LVDT attachment locations. A clipon gage was mounted in the crack mouth to measure crack-mouth-opening

24
displacement (CMOD). One 9- x 30-cm surface of each specimen was coated

with brittle laquer to reveal strain patterns. The strain patterns

produced were suitable for recording by photography.

A servohydraul ic testing machine with a load capacity of 1 MN was

used. The tests were run in displacement control. A commercially

obtained analog-to-digi tal (A-D) conversion system with a 16-bit resolution

was connected to the strain gages, LVDT's, testing-machine load cell,

and CMOD gage. The A-D converter was connected to a commercially

supplied minicomputer. The minicomputer system consisted of a processor,

a dual floppy-disk data-storage unit, a digital plotter, and a printer.

The servohydraul ic testing machine was controlled manually, but the

data acquisition, on-line analysis, storage, printing, and plotting for

each point of data were controlled by the minicomputer. For each deformation

increment, each sensor was polled and its signal was converted from

electrical to physical units and stored on floppy disk. Then the Jintegral was calculated as described above. The J-integral, stress,

CMOD, average strain (defined below), measured displacement values, and

measured strain values were printed. Finally, each strain-gage output

was plotted on a bar graph, and the calculated J-integral value was

plotted against average strain. The average strain for this purpose was

taken as the average of the three LVDT displacements divided by the

specimen gage length. These displacements included contributions from

the strain field of the crack as well as contributions from the strain

that would have existed in the absence of the crack.

25
In the course of a typical test, first the instrumented specimen

was mounted in the testing machine, and all of the instrumentation was

connected and checked for proper function. When all the apparatus was

performing properly, a zero-load data point was taken, and the specimen

was extended by a preselected amount. Then the computer was allowed to

proceed through the data acquisition-storage-calculation-print-plot

procedure described above. The specimen was then extended further and

the procedures repeated. The strain patterns revealed by the brittle

laquer coating were photographed periodically throughout the test.

Tests were terminated for two reasons: specimen tearing or gage failure.

Tearing occurred before strain-gage failure in almost all cases.

About a hundred data points, each consisting of about twenty-five

measured strain, displacement, and load values, were obtained during a

typical test. The maximum average strain (defined above) attained

during a test ranged from around 1.5 times yield for relatively deep

notches to near 9 for shallow notches.

The stored data were used for posttest reanalysis, corrections, and

replotting. A large-scale computer was used for the nonlinear fits and

for detailed plotting of the strain and displacement data.

RESULTS AND DISCUSSION

The J-vs. -strain data for the tests described here are shown in

Fig. 2. These curves consist of an elastic part at low strains and a

plastic part at high strains. In the elastic region, the J-vs. -strain

26
curve is parabolic, and in the plastic part the curve is linear for most

specimens, both as expected from previous studies [8,9].

The present technique of direct evaluation of the J contour integral

requires subtraction of the energy density integral along the notched

side (AB in Fig. lb) from that along the unnotched side (CD in Fig. lb)

to obtain the energy density term, / Wdy, and subtraction of the displacement measured between points A and F from that between C and E to

obtain the traction-bending term, fl • (du^/dx) *ds . (The result for the

traction-bending term is insensitive to the displacement measured between points G and H because it is essentially cancelled out in the

integration procedure.) The shorter the crack, the more nearly equal

will be the two terms of each of these subtractions; it is the equality

of the terms in each subtraction that causes J to be zero when the crack

length is zero. The experimental uncertainty in the differences between

the two nearly equal energy density integrals and the nearly equal

displacements is much larger than the experimental uncertainty in

either individual energy-density integral or displacement. This is

shown in Table 1, which lists experimental uncertainties in energy

density integrals, displacements, and the J-integral for the specimen

with the 4-mm crack. The uncertainties in individual measured quantities were estimated from differences among measured values and differences between measured and expected values. Uncertainties in the energy

density and J-integral s were estimated from the uncertainties in the

individual measured quantities by using standard procedures from the

theory of propagation of errors. Table 1 also shows that the uncertainty in J in the plastic-strain range is noticeably smaller than that

in the elastic-strain range.

27
Table 1. Propagation of experimental uncertainties in direct measurement

of the J contour integral. Data for HY-130, single edge

notch, a = 4 mm.

Uncertainty Elastic part Plastic part

In larger integrated

energy density term

1% 1%

In J from energy

density terms alone

7.6% 2%

In LVDT displacement 1% 1%

In J from LVDT

displacement alone

9.5% 2%

In J (combined uncertainty) 12 % 3%

The energy density integrations were performed numerically directly

on the experimental data, on polynomials fitted to the data, and on nonlinear

functions fitted to the data. The method involving nonlinear fits was

judged to be the best. Figure 3 shows the measured strains as a function

of position along the contour segment DC for the 4-mm crack-length

specimen for a point of data in the plastic-strain range. Shown for

comparison are three functional forms for the strain used in the energy

density integral. Although the nonlinear function consisting of a

Gaussian peak plus a three-term polynomial appears to give the best fit,

the peak of the fitted function is well above the maximum measured

strain. Examination of the specimen strain pattern as revealed by the

brittle laquer coating showed that the highest strains occurred at the

strain gage at x = 5.4 cm, and therefore, that the fitting function was

in error. By adding a fictitious strain value linearly interpolated

halfway between the values at 5.4 and 7.9 cm from the plane of the

28
crack, the fitted curve was brought down to the locus shown as a dashed

line. From this exercise it was concluded that the uncertainty in the

J-integral from improper fitting could be about ± 8%, but that careful

analysis could bring this down.

Lower uncertainties could also be obtained by using more strain

gages along contours, such as DC, or by using the correct functions for

strain vs. position along contour segments AB and CD to fit the measured

strain values. Unfortunately, the correct functional form for the

specimens of this study is unknown.

The comparison between the measured results and the linear-elastic

handbook formula for K was made by converting the measured J-integral

values to stress intensity (K) values through the equation

K = (JE) 1/2

(2)

where E is Young's modulus, and plotting K against nominal applied

stress. An example is shown in Fig. 4. In this figure a linear relationship

between experimentally measured K values and stress, with some experimental

scatter superposed, is seen for the low- and medium-stress regions. At

high stresses, K increases rapidly with stress because of the formation

of a plastic zone of significant size at the crack tip. In linear

elastic fracture mechanics, K is related to stress by equations of the

form

K = a-f (3)

where a is the applied stress and f is a function of crack length and

specimen width. Experimental values of the coefficient f were obtained

by manually fitting the linear parts of the experimental K-a data sets

29
to straight lines. The slopes of the lines were taken as the experimental

f values. By comparing the experimental and theoretical f values, the

agreement between measured and theoretical J-integral values can be

assessed. In Table 2, reasonable agreement between measured and theoretical

values of f can be seen.

The results obtained with both the nonlinear fit procedure and the

no-fit procedure were close to the expected values. The polynomial

fitting procedure was rejected because the fitted strains were not

consistently close to the measured strains. The nonlinear fit procedure

was adopted for final data analysis because it was considered to be less

subjective than the no-fit procedure. For the numerical integration of

strain-energy values in the no-fit procedure, the contour segments AB

and CD were divided into intervals such that each interval contained one

strain gage. A consistent procedure for selecting the boundaries between intervals was followed, but this procedure was selected in an

essentially arbitrary manner and was considered subjective. An element

of subjectivity was also present in the selection of the fitting functions

used in the nonlinear fit method, but since these functions provided

consistently good fits to the data, the results were considered to be

approximate! y equivalent to the results that would have been obtained if

the correct functional form were known.

No plasticity corrections were used in the comparisons of Table 2.

The appropriateness of using only the linear-elastic terms is seen from

the linear dependence of K on a in the region of the comparison and is

confirmed by the linear relation between CMOD and stress in the region

30
Table 2. Calculated and measured values of f = dK/da for single-edgenotched HY-130 tensile panels.

Crack Length

(mm)

Calculated f Measured f

(no fit)

(,

1/2

)

Measured f

(polynomial fit)

(m1/2

)

Measured f

(nonlinear fit)

(.

1/2

)

2 0.0898 0.0999 0.0728 0.089

4 0.1287 0.1366 0.1242 0.140

7 0.1747 0.1668 0.1493 0.163

20 0.3587 0.3567 0.3444 0.355

of the comparison (Fig. 5). Plasticity effects on the effective crack

length show up in Fig. 5 only at stresses above 720 MPa for the specimen

with the 4-mm notch.

It is concluded that good agreement has been obtained between Jintegral values predicted from 1 inear-el astic-fracture mechanics and

values obtained experimentally by direct evaluation of the J contour

integral. The agreement obtained in the linear-elastic region indicates

that the uncertainty in the J-integral values obtained in the plastic

region, which are the real focus of the present study, may be expected

to be about 5 % or less.

Another estimate of the experimental uncertainty was made by comparing values in the plastic strain region calculated by the no-fit and

non-linear-fit methods. Differences of up to 10% were obtained. Accordingly, an uncertainty of ± 10% was ascribed to the data produced by the

present method for direct evaluation of the J contour integral. Larger

errors can occur if the measured displacements include erroneous contributions from, for example, bending of the specimen. The measured displacements for the 2-mm-crack case listed in Table 2 required a correction for bending. Such bending might occur if the specimen were not

31
machined to adequate straightness or if the pin holes were not centered.

These errors can be detected by comparison of measured displacements

with expected results obtained from experience or analysis.

The present results are now compared with the previously developed

extended-perfect-pl astici ty theory [10]. The J-integral -vs . -strain curves

for each specimen can be divided into elastic and plastic regions. For

the elastic region, linear elastic fracture mechanics (lefm) provides an

appropriate theoretical description of the experimental situation. The

experimental data were shown earlier to be in agreement with lefm predictions for the elastic region. The extended-perfect-plasticity

theory is now applied to the plastic-strain range. First, the total Jintegral is regarded as the sum of an elastic and a plastic part:

J = J + J e P (4)

The linear dependence of Jp on gage-length strain is expressed as

J = M*a_p*v^

P f P (5)

where is the material flow stress and v is the plastic part of the

f p displacement applied at the specimen ends.

Differentiating we obtain

(dj/dv) •( l/a

f

) = M. (6)

A straight line was determined by eye through each data set for J vs.

strain in the plastic region, and experimental M values were calculated

from the slopes of these lines and Eq. 6; these values are plotted in

Fig. 6. Also shown in Fig. 6 is a dashed line calculated by the extended32
perfect-plasticity theory:

M _ ! . f. +

r.2f + ( 1 -2g) ( 1 -f ' )

]

/l + 2f ( 1 -2a)-2a( 1 -a)

where a = crack length, a = a/W, W = specimen width, f' = 3f/3a, f

= -a/a

Q

a (1-e ), a = a/W, and a Q

= an adjustable parameter. The value of

the parameter a Q

used in the calculations was chosen as 11 mm to optimize

the fit of the theory to the data. The utility of this extended-perfectplasticity model is that it provides a method for interpolating between

large crack sizes, where theories of the type discussed by Rice, Paris,

and Merkle [11] are expected to describe experimental results and very

short crack sizes, where J must approach 0 as crack length approaches 0.

The author believes that this region of short crack sizes will prove

important in yielding fracture mechanics.

CONCLUSION

Direct experimental measurements of the J contour integral have

been accomplished in a high-strength steel in the elastic-and plasticstrain ranges.

ACKNOWLEDGMENT

Helpful discussions with H. I. McHenry, technical assistance by

J. D. McColskey, and the financial support of the United States David

Taylor Naval Ship Research and Development Center, Annapolis, MD through

J. P. Gudas, Project Monitor, are appreciated.

33
REFERENCES

1. H. L. Oh, "A Simple Method for Measuring Tearing Energy of Nickel

Rubber Strips," Mechanics of Crack Growth , ASTM STP 590, American

Society for Testing and Materials, Philadelphia, 1976, pp. 104-114.

2. G. Herrmann and G. S. Kino, "Ultrasonic Measurements of Inhomogeneous Stress Fields," Proceedings of ARPA/AFML Review of Progress

in Quantitative NDE , AFML-TR-78-205, Air Force Materials Laboratory,

1979, pp. 447-451.

3. W. A. Logsdon, "Elastic Plastic (Jj

c

) Fracture Toughness Values:

Their Experimental Determination and Comparison with Conventional

Linear Elastic ( K

j

c

) Fracture Toughness Values for Five Materials,"

Mechanics of Crack Growth , ASTM STP 560, American Society for

Testing and Materials, Philadelphia, 1976, pp. 43-60.

4. J. A. Begley and J. D. Landes, "The J Integral as a Fracture Criterion," Fracture Toughness , ASTM STP 514, American Society for

Testing and Materials, Philadelphia, 1972, pp. 1-20.

5. J. D. Landes and J. A. Begley, "The Effect of Specimen Geometry on

Jj

c

," Fracture Toughness , ASTM STP 514, American Society for Testing

and Materials, Philadelphia, 1972, pp. 24-39.

6. H. Tada, P. Paris, and G. Irwin, The Stress Analysis of Cracks

Handbook , Del Research Corporation, Hellertown, Pennsylvania, 1973.

7. J. R. Rice, "A Path Independent Integral and the Approximate Analysis of Strain Concentration by Notches and Cracks," Journal of

Applied Mechanics 35., 1968, pp. 379-386.

8. R. J. Bucci, P. C. Paris, J. D. Landes, and J. R. Rice, "J Integral

Estimation Procedures," Fracture Toughness , ASTM STP 514, American

Society for Testing and Materials, Philadelphia, 1972, pp. 40-69.

34
9. J. A. Begley, J. D. Landes, and W. K. Wilson, "An Estimation Model

for the J-Integral ," Fracture Analysis , ASTM STP 560, American

Society for Testing and Materials, Philadelphia, 1974, pp. 155-169.

10. D. T. Read and H. I. McHenry, "Strain Dependence of the J-Contour

Integral in Tensile Panels," Proceedings of the Fifth International

Conference on Fracture , 1981, to be published.

11. J. R. Rice, P. C. Pris, and J. G. Merkle, "Some Further Results of

J-Integral Analysis and Estimates," Progress in Flaw Growth and

Fracture Toughness Testing , ASTM STP 536, American Society for

Testing and Materials, Philadelphia, 1973, pp. 231-245.

35
Specimen Specimen

with

Contour

J^Wdy-T-ffds

x,y Position coordinates

W Strain energy density

T Traction vector

d Partial derivative with

dx respect to x

y Displacement vector

Figure 1 (a). Contour chosen for direct evaluation of the J contour

integral for single-edge-cracked specimens.

36
I

I

I

Specimen with

Instrumentation

Small strain gages

1 D Strain gages

LVDT’s

T

=0 CMOD gage

Figure 1 (b) Instrumentation placement for direct evaluation of the

J contour integral for single-edge-cracked specimens.

37
GAGE LENGTH STRAIN

Figure 2. Measured J-integral plotted against strain for single-edge

cracked specimens. Crack lengths are noted on the graph.

38

J-INTEGRAL,

Ib/in
STRAIN

STRAIN GAGE LOCATION, cm

• Meas. value Non-linear fit —Polynomial fit Non-linear insert

fit with fabricated

point (o)

Figure 3. Strain as a function of location along contour segment AB

(Fig. lb). The zero of location is the crack plane.

Included are measured values, showing position increment

used for direct integration, polynomial fit, nonlinear fit,

and nonlinear fit to real data plus one fictitious data

poi nt.

39
,

MPa

Vm

Figure 4. Stress intensity, K, plotted against nominal applied stress

for the specimen with the 4-mm crack.

40
C.M.O.D.,

mm

STRESS, ksi

0 40 80 120

Figure 5. Crack-mouth-opening-displacement (CMOD) plotted against

nominal applied stress for the specimen with the 4-mm

crack.

41

C.M.O.D.,

inches

x

10
NORMALIZED

SLOPE

OF

J

vs

v

RESULTS,

AJ/CT

y

Av

NORMALIZED CRACK LENGTH, a/W

Figure 6. Measured values of M = (1/a ) dJ/dV for four single-edgecracked specimens plotted against crack length. The curve through the data was determined by eye. The dashed line

is an approximate theoretical result.

42
APPENDIX B

Reprinted from

ADVANCES IN FRACTURE RESEARCH

Edited by D. FRANCOIS et al PERGAMON PRESS OXFORD and NEW YORK 1980

STRAIN DEPENDENCE OF THE J-CONTOUR INTEGRAL

IN TENSILE PANELS*

D. T. Read and H. I. McHenry

Fracture and Deformation Division

National Bureau of Standards

Boulder, Colorado 80303

ABSTRACT

The J contour integral has been experimentally measured as a function of applied

strain for single edge notch tensile panels under elastic-plastic loading conditions.

The results have been compared to analytical predictions based on finite element

analysis and to theoretical estimates based on models representing the two behavioral

extremes, uniform strain and perfect plasticity. The experimental, analytical and

theoretical results have the expected form: the J-integral initially increases as the square of the applied strain, and at strains above yield the J-integral is a linear function of strain. The experimental and analytical results are in reasonable

agreement, while the uniform strain and perfect plasticity models under- and over- estimate J respecti vely. An extension of the perfect plasticity model is proposed to treat behavior between these two limiting cases.

KEYWORDS

Crack driving force; elastic-plastic conditions; J-integral; fracture mechanics;

tensile panels; finite-element-analysis; theoretical models

INTRODUCTION

The J-integral has gained widespread acceptance as a measure of the driving force for

fracture under elastic-plastic conditions, that is, when notch tip plasticity is extensive. This report describes experimental measurements of applied J-integral

values as a function of strain in simple configurations relevant to structural com- ponents. The experimental results are compared with results obtained from finite

element analysis and theoretical models.

EXPERIMENTAL PROGRAM

Experimental measurements of the applied J value as a function of strain for several

notch lengths have been performed. Three different experimental approaches were applied to allow verification of results: direct measurement of the contour integral;

measurement of the pseudo-potential -energy change with crack extension (compliance

technique); and measurement of crack tip opening displacement (CTOD). Each approach

applies a known relationship between J and some set of experimentally measurable

quantities, as described in detail by Read and McHenry (1980).

43
Experimental Procedures

The material chosen for the present study was a normalized C-Mn steel with a yield

strength of 340 MPa. Single-edge-notched tensile panels with gage section 100 mm wide, 348 mm long, and 12 mm thick were tested with notch lengths of 0, 2, 10, 30 and 46 mm. Tension-tension fatigue precracking was used to sharpen the crack tips

(except for the 2 mm notch which was tested with the notch tip as cut with a jeweler's

saw)

.

The instrumentation required for direct measurement of the J contour integral con- sisted of twenty electrical resistance strain gages and three linear variable displacement transducers (LVDT's) mounted on the specimen (Fig. 1) and a minicomputer

for acquisition and storage of the strain and displacement values. The terms of the

integrand were derived from the measured strain and displacement data and the integration was performed numerically using the trapezoidal rule.

A qualitative measure of the shape of the strain fields around the crack tips was obtained by coating the specimen gage sections with brittle lacquer before straining

and then observing darkened regions in the lacquer which corresponded to regions of

high specimen strain. Photographs were made of the strain patterns at regular in- tervals during the test.

LOAD

I

l

u = LVDT

displacement

LOAD

Fig. 1. Experimental arrangement

for direct measurement

of the J-contour-i ntegral

.

Fig. 2. Load plotted against displacement for all speci- mens .

RESULTS

The load-displacement records are shown in Fig. 2. J-integral values were determined

as a function of strain for the four notched tests. Strain was defined as the

average displacement measured by the three LVDT's over a gage length of 348 mm. Data points were obtained at strain intervals of about 0.00015. J-values were determined

by the direct evaluation of the J-contour-i ntegral , Fig. 3, and by the compliance

method, Fig. 4. The CT0D at maximum strain was measured using the replication technique; the results are shown in Table 1. The general form of the J vs. e curves is

a parabolic dependence of J on strain at low strains and a linear dependence at strains above yield. The parabolic-then-linear form of the J-e curves consistent

_ with previous experimental studies employing the compliance techique by Bucci , Paris,

44
Landes and Rice (1972), with theoretical studies by Begley, Landes and Wilson (1974),

and with theoretical and experimental studies in support of British COD design curve (Burdekin and Stone, 1966). Significant deviations from the parabolic-then-linear

dependence of the J on strain were observed in certain of the tests of the present

study. In the 2 mm notch, the rapid rise in J at the yield strain occured as narrow

shear bands emanated from the crack tip, Fig. 5. At higher strains (>0.0025), the

shear bands spread due to strain hardening, and effectively masked the 2 mm notch so that J no longer increased with strain. In the 10 mm notch test, the shape of the

J-e curve was influenced by plastic deformation near the holes machined through the

specimen to attach the LVDT's. Note that these holes did not cause plastic deformation (detectable by cracking of the brittle lacquer) in the deeper-notch tests; holes were not used in the 2 mm notch specimen. The results for the 46 mm notch test were influenced by the loading history which included complete unloadings at strain values

of 0.0012 and 0.0019. In addition, shear strains, which are greater for the deeply

notched case, were not accounted for in the direct measurement of J.

Table 1 Crack tip opening displacements (CTOD) and resulting J values.

calculated using m

Notch length (mm)

=

_7T7;

Strai

n

CTOD (mm) J(kN/m)

10 0.011 1.57 1152

30 0.0077 1.90 1394

46 0.0034 0.56 409

Fig. 3. Experimental results for the

J-integral obtained by direct

measurement of the contour

integral

.

Fig. 4. Experimental results for the

J-integral determined by the

compliance technique.

Analytical Results

The two-dimensional elastic-plastic finite element analysis computer code of Gifford

(1975, 1978) and Hilton and Gifford (1979) was used to calculate J-integral vs. strain. This program incorporated special nonlinear crack tip elements and con- ventional 12-node quadril aterial isoparametric elements. Finite element analysis

calulations of J vs strain were carried out for 30 and 46 mm single-edge-notches to model the experimental situation. Limitations of the finite element analysis program

in the treatment of large strains prevented calculation of results for the 2 mm

45
notch and prematurely terminated the results for the 10 mm notch. The general form

of the finite element results, shown in Fig. 6, was parabolic-then-linear as expected.

C-Mn STEEL FINITE ELEMENT RESULTS /'

1 S00 3 NOTCH LENGTHS /

< 1 000 -

CL

CD

30

/ ^ 46 /S' 1 0 mm

I 0.000 0 002 0.004 0.006 0.008 0 010 STRAIN

Fig. 5. Photograph of strain pattern Fig.

revealed by brittle lacquer,

in C-Mn steel specimen with

2.25 mm notch.

6. Finite element results simulating

the behavior of two specimens of

the present study in the 46 mm notch

and the 30 mm notch, and partial

results for the 10 mm notch.

THEORETICAL MODELING

Three simple theoretical models giving J as a function of strain were considered in the present study. The case of a small notch in a large, uniformly strained panel was treated by Begley, Landes, and Wilson (1974); their result is referred to as the

uniform strain model. The case of perfect plasticity was treated by Rice, Paris, and

and Merkle (1973). The perfect plasticity model was extended for the present study

by adding a term to explicitly account for the stress-strain singularity at the

crack tip in a simple manner.

The uniform strain model as developed by Begley, Landes, and Wilson (1974) resulted

in the following expression for J:

for e/ey <_ 1 (la)

1 ) for e / Ey >_ 1 (lb)

These formulas were derived for the case of a notch that is small enough that plastic

strains are distributed throughout the panel rather than concentrated at the notch

section. This situation mininizes J, and therefore, the uniform strain model is a lower bound solution for J vs. e.

The elastic-perfectly plastic model for J as a function of strain follows the work of

Paris, Tada, Zahoor, and Ernst (1979). Limit load per crack tip, P, is governed by

yielding of the ligaments (W - a):

e \ 2

! = 1 tt 3 (—

)

J E y

J

= -^ Tra (2e/ey

46
P = 0 ( 2 ) ^

(W - a) T

and plastic displacements, Q^, are concentrated at the crack tips, that is,

Op = Qp»y

(3)

where J is the plastic component of J. Adding the elastic component, J , yields:

P c

J = J e + J

p " 0 7Ta/E + ^p°y (4)

The strain distribution assumed in this model maximizes J, and therefore the perfect

plasticity model usually provides an upper bound solution for J vs. e.

Equation 2 neglects the existence of a stress-strain singularity at the crack tip.

A simple way to approximately include the singularity is to postulate a line tensile

force which operates along the crack tip perpendicular to the crack plane. This

force was assumed to be proportional to a for a/W £ 0.1 and to approach a constant

value for longer cracks. A convenient functional form for such a force is the ex- ponential relationship:

F = F q (1-e

- a/a

o). (5)

Here, F is the force across the crack tip per unit thickness and Fq and a need to be chosen. A relationship between Fq and a Q was derived by requiringj to approach zero

as a approaches zero. After application of this relationship only one adjustable

parameter, a , is needed to fix F.

This force is regarded as a simplification of the Hutchinson (1968), Rice and

Rosengren (1968) (HRR) Stress-strain field. In the extended perfect plasticity

model, the spatial dependency of the HRR strainfield is ignored; the force, F, is considered to be concentrated at the crack tip. The strain independence of this

force is consistent with the low strain hardening of the C-Mn steel used in the

experimental program.

Using the crack tip force, the load required to extend a notched specimen

at displacements above the yield displacement is calculated by assuming that the

tensile and compressive forces across the plane of the notch consist only of tensile

(or compressive) stresses equal to the yield strength all along the ligament, as before, plus the crack tip force. The load is calculated by requiring the net load

and net moment applied to each half of the specimen to be nil. Once the load has

been calculated, Eq. 8 is used to calculate J. The result for Jp, is:

J

p

= Q

p (ay

“ 3F/3a)

(6)

The result for the si ngl e-edge- notched specimen is:

Jp = Q

p

where f = F/ct TW, a = a/w, and f

1 = df/da. The full expressions for the applied J value were formed by adding the linear elastic part to the plastic part.

Compari son of Resul ts

The experimental load-displacement data for the specimen with the 30 mm notch are compared to calculated values from finite element analysis and the extended perfect

plasticity theory in Fig. 7. The differences between experimental and calculated

f

. +

[2 f + ( 1 -2a) ( 1 - f

1 )]

/l + 2f (1 - 2a ) - 2a ( 1 -a

)

(7)

47
values are attributed to strain-hardening effects not accurately accounted for in the calculations. Figure 8, in which load at a strain of 2 times yield is plotted

against crack length, shows that the crack length dependences of both the experimental and the analytical results are well represented by the extended perfect

plasticity theory.

Fig. 7. Experimental, analytical, and Fig.

theoretical results for the single-edge-notched specimen with the

30 mm notch; included are experimental, finite element analysis

(FEA), and extended perfect plasticity (EPP) theory results.

8. Load at a displacement of 2 times

yield as a function of crack length; experimental, finite element

analysis, and extended perfect

plasticity (EPP) theory results

are shown.

Figure 9 displays the contour integral, compliance, CTOD, finite element analysis and

the extended perfect plasticity model results for the 30 mm notch plotted as functions of strain. The agreement among these different methods of determining the J- integral was not as good for the other notch lengths, because of the experimental

problems noted above.

Practically all the measured and calculated J integral results had the same type of

strain dependence. But the results of the theoretical models differ significantly in their dependence on crack length. This is shown in Fig. 10, which displays J as a function of crack length at a strain of 4 times the yield strain. Experimental

results by the compliance and contour integral techniques, finite element analysis

results, and the three theoretical models are plotted. This figure shows that the

experimental results disagree with both the perfect plasticity and the uniform strain

theories. The extended perfect plasticity theory predicts J values which lie between

those of the uniform strain and perfect plasticity models, and best represents the

experimental and analytical results. Similar conclusions were drawn from Fig. 11, in which the uniform strain, perfect plasticity, and extended perfect plasticity models

are compared with one another and with finite element results for a center-notchedpanel .

48
Fig. 9. Experimental, analytical, and Fi theoretical results for singleedge-notched specimen with 30 mm

notch; included are direct contour

integral (Cl), compliance technique (CT), finite element analysis

(FEA), and extended perfect plasticity theory (EPP) results.

.10. Experimental, analytical, and

theoretical results for the singleedge-notched specimens; included are direct contour integral (Cl), com- pliance technique (CT), finite element

analysis (FEA), uniform strain theory

(US), perfect plasticity theory (PP)

and extended perfect plasticity theory

(EPP) results.

DISCUSSION

When this study was begun, it was hypothesized that the uniform strain model could be

verified over a significant range of crack lengths. However, the experimental and

analytical studies consistently produced J values several times those predicted by

the uniform strain model; they were often in the neighborhood of the perfect plasticity result. But the perfect plasticity model was clearly unsatisfactory for short

notch lengths. A physical interpretation of this result has been developed over

the course of this study. This interpretation is that the uniform strain model holds

only when plasticity conditions are such that plastic strains are spread over the

whole length of the strained panel and are prevented from concentrating at the crack

tip. A dramatic case of strain concentration at a crack tip is illustrated in Fig. 5. Slip bands at ± 45° and 90° to the tensile axis emanated from the crack tip.

Each increment of applied displacement contributed to the strain in the slip bands. Because the slip bands terminated at the crack tip, all the strain contributed to the opening of the crack. The photograph in Fig. 12 shows that the strains were

not concentrated at the tip of the 10 mm notch as much as for the 2 mm notch, and

the J values shown in Fig. 3 for strains between 0.002 and 0.003 are lower for the

10 mm notch.

It is concluded that the limiting cases for the behavior of J as a function of strain

in tensile panels are provided by the uniform strain and perfect plasticity models.

In the uniform strain model only limited strain concentration at the crack tip is allowed; in the perfect plasticity model all the plastic strain is concentrated at

the crack tip. The extended perfect plasticity theory is intermediate between these,

two extremes.

49
k • Wk '

•

'

i$| i

? *1

"*! : -

- V

+ yAt

Fig. 11. Theoretical and analytical

results for a center-notchedpanel .

Fig. 12. Strain pattern, revealed by

brittle lacquer, in ship steel

specimen with 10 mm notch.

ACKNOWLEDGMENTS

The financial support of the U.S. Naval Sea Systems Command through J. P. Gudas,

Project Monitor, and the technical assistance of J. W. Elmer, V. Hauptman and J. D. McColskey are gratefully acknowledged.

REFERENCES

Begley, J. A., J. D. Landes, and W. K. Wilson (1974). An Estimation Model for the

Application of the J-Integral. Fracture Analysis , ASTM STP 560 , American Society

for Testing and Materials, 155-169.

Bucci , R. J., P. C. Paris, J. D. Landes, and J. R. Rice (1972). J Integral Estimation Procedures. Fracture Toughness , ASTM STP 514 , American Society for Testing

and Materials, 40-69.

Burdekin, F. M., and D. E. W. Stone (1966). The Crack Opening Displacement Approach

to Fracture Mechanics in Yielding Materials. J. Strain Analysis , 1_> 145-153.

Gifford, L. N. (1975). APES-Second Generation Two-Dimensional Fracture Mechanics and

Stress Analysis by Finite Elements. David Taylor Naval Ship Research and Development Center, Bethesda, MD 20084, Report 4799.

Gifford, L. N. (1978). J-Integral Analysis of a Compact J, Specimen. David Taylor

Naval Ship Research and Development Center, Bethesda, MD 20084, Report M-4.

Hilton, P. D. , and L. N. Gifford (1979). Evaluation of Some Crack Tip Finite

Elements for Elastoplastic Fracture Analysis. David Taylor Naval Ship Research and

Development Center, Bethesda, MD 20084, Structures Department Research and Develop- ment Report DTNSRDC-79/052.

Hutchinson, J. W. (1968). Singular Behavior at the End of a Tensile Crack in a Hardening Material. _J. Mech . Phys . Sol i ds , 16, 13-31.

Paris, P. C., H. Tada, A". Zahoor, and H. ErnstTl 979) . The Theory of the Tearing

Mode of Elastic-Plastic Crack Growth. Elastic Plastic Fracture , ASTM STP 668 , American Society for Testing and Materials, 5-36. , - .. Read, D. T. and H. I. McHenry (1980) Detailed Report on Strain Dependence of the

J-Contour Integral. To be published.

Rice, J. R., P. C. Paris, and J. G. Merkle (1973). Some Further Results of J-Integral

Analysis and Estimates. Progress in Flaw Growth and Fracture Toughness Testing , ASTM STP 536, American Society for Testing and Materials, 231-245.

Rice, J. R. , and G. F. Rosengren (1968). Plane Strain Deformation Near a Crack Tip

in a Power-Law Hardening Material. jJ. Mech . Phys . Sol i ds , 1

6

, 1-12.

50
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

4. TITLE AND SUBTITLE

PUBLICATION OR

REPORT NO.

2. Performing Organ. Report No. 3. Publication Date

NBSIR 81-1648 May 1981

Interim Progress Report: J-integral Method for Fitness-for-Service Assessment

5. AUTHOR(S)

D. T. Read and H. I. McHenry

6. PERFORMING ORGANIZATION (If joint or other than NBS. see in struction s) 7. Contract/Grant No.

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

8 . Type of Report & Period Covered

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Naval Sea Systems Command Annapolis Laboratory

John P. Gudas, Project Monitor Annapolis, Maryland 21402

Code DTNSRDC 2814

David Taylor Naval Ship R & D Center

10. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most sign ificant information. If document includes a significant

bi bl iography or literature survey, mention it here) Progress to date on the J-integral f itness-for-service assessment project is re- ported. The goal of this project is to produce a method for evaluation of material

toughness requirements, allowable defect sizes, and sustainable stresses and strains.

Yielding fracture mechanics and, specifically, the J-integral are employed. The Jintegral is regarded as the crack driving force; applied J values above some critical

value are predictive of material tearing. The applied J value is a function of applied

stress and strain, crack length, and material tensile properties. The critical Jintegral value is measured in material fracture-toughness tests. The experimental

focus of the present study is on measurement of the applied J-integral as a function of

stress and strain in HY130 specimens with different crack lengths. Neither experimental

data nor established predictive theories for the behavior of applied J values are available in the literature for short cracks, the type most likely to be found in a structural element. Results for relatively large cracks indicate that displacements imposed

at the specimen ends are transmitted in full to the cracked cross section. The displacements add to the strain field of the crack and produce a large applied J value.

This behavior is described as net section yielding. For relatively short cracks, displacements imposed at the specimen ends are absorbed as plastic strain all along the

specimen length, add only slightly to the strain field of the crack, and produce small

applied J values. This behavior is described as gross section yielding. It is important to note that cracks as small as 5% of the total cross section have been found to

exhibit net section yielding.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Crack opening displacement; design curve; elastic-plastic fracture mechanics; f itnessfor-service; fracture mechanics; J-integral.

13. AVAILABILITY

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| | For Official Distribution. Do Not Release to NTIS

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PRINTED PAGES

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