a combined numeri~al/experimental study of ductile crack growth

&n@wering Fmctum Mechctnks Vol 23, NO. 3. pp. 537-550. 1986 0013-7944186 $3.00 + .@J

Printed in Great Britain. 0 1YRh Pergamon Press Ltd

A COMBINED NUMERI~AL/EXPERIMENTAL STUDY OF DUCTILE CRACK GROWTH AFTER A LARGE UNLOADING, USING T”, J AND CTOA CRITERIA

F. W. BRLJSTt Battelle Columbus Laboratories, 505 King Avenue, Columbus, OH 43210, U.S.A.

J. J. MCGOWAN* ORNL, Oak Ridge, TN 37830, U.S.A.

and

S. N. ATLURIB Center for the Advancement of Computational Mechanics, Georgia Institute of Technology,

Atlanta, GA 30332. U.S.A.

Abstract-The J-integral is commonly used to determine the resistance of plastically deformed structures to continued crack growth. This approach is only valid for monotonic loading and small crack growth; yet it is believed conservative for all general loadings. Another parameter, T*, has been formulated, which is theoretically valid under a wide range of conditions. These two plastic fracture criteria, along with CTOA (crack tip opening angle), were examined after a cycle of loading/unloading to zero load followed by reloading. The results of the numerical/ experimental study showed that the T* parameter accurately predicted the behavior (crack growth after 50% reloading), while the other parameters (J and CTOA) were seriously anti- conservative.

INTRODUCTION

ELASTIC-PLASTIC fracture mechanics has seen a rather rapid development in recent years. Much of this development can be attributed to the discovery of the J-integral. In fact, through the use of J, a very practical engineering approach has been developed for assessing the integrity of flawed structural components which undergo plastic deformation[ 11. The J-approach, however, can give crack growth predictions that are so conservative that they can greatly under- estimate the load-carrying capacity of a cracked component, as discussed by Kanninen et ~1.121. Nonetheless, this approach is very useful since the engineer can very quickly make a decision about the integrity of a flawed component.

More accurate assessments can be made by using a combined JICTOA (crack tip opening angle) approach. CTOA levels out to a constant value once steady-state crack growth has been achieved. With this approach, the J-resistance curve is used to characterize crack initiation and smalf amounts of growth, and then CTOA is used to characterize further growth. The CTOA method requires a finite element analysis of the flawed component, which is a serious disadvantage compared to the engineering approach. However, more accurate assessments clearly result from JICTOA.

Recently, a path-independent integral, labeled T*, has been identified as a practically useful parameter for characterizing stable elasto-plastic crack growth[3]. T* is equivalent to J under conditions where J is theoretically valid (i.e. deformation theory plasticity and a stationary crack) and is also very nearly equaf to J for small amounts of crack growth. Like CTOA. T* levels out to a constant value once steady-state conditions have been achieved. The T*-resistance curve approach for characterizing fracture has several advantages over the JICTOA technique. First, with JICTOA a decision must be made as to when the switch from J to CTOA- controlled growth is necessary. Secondly, CTOA is obtained from elements that are in the wake of the extended crack. It is unreasonable to expect an accurate solution from these highly deformed elements. Hence, the constant CTOA that results is probably only a convenient

tPh.D. Candidate, Georgia Institute of Technology. $Research Engineer. IRegents’ Professor of Mechanics.

537

538 F. W. BRUST ef nl.

artifact from the analysis and not necessarily the true value. On the other hand, T* is calculated close to the crack tip but away from these wake elements. Hence, T* should be accurate and not merely a useful numerical tool. Thirdly, there is reason to believe that an engineering approach in the spirit of Ref. [l] is possible with T*[4].

The above plastic fracture parameters were developed by assuming that the cracked body was loaded monotonically with the crack growing in a stable manner until failure. Indeed, the J-resistance curves are obtained by monotonically loading the specimen to the point of steady- state crack growth or failure. The effect of unloading the specimen and then reloading after a period of stable crack growth has not been considered. This effect has important practical implications. Engineering structures are generally loaded many times throughout their design life.

The purpose of this paper is to show the effect of low-cycle fatigue on elastic-plastic stable crack growth. This paper will present results showing that

The behavior of a-fracture specimen in which stable crack growth has occurred followed by unloading and reloading is different from a monotonically loaded specimen. Continued crack growth upon reloading commences at loads which are a fraction of the previous maximum load. f and CTOA approaches cannot predict growth during reloading. Moreover, they give nonconservative predictions. T* accurately predicts the fracture behavior of such an unload/reload situation.

ANALYTICAL PRELIMINARIES

The Tsth-independent J-integral of Rice[S] (actually the component of a vector-integral along the Xi axis, denoted herein by Ji). valid for a deformation theory of plasticity, is denoted as (Fig. 1)

J, = [Wnt - t;ui,t] ds = I

LWn, - tiui,l] ds. (1) l I‘IZW

In (1) W is a single-valued function of the strains such that

Uij = tJW/hiij, (2)

where ni is the component of a unit normal to the path in the Xi direction, t; is the traction vector such that ti = njo/j, and Ui is the component of displacement. The path independence of Ji for deformation theory plasticity is easily seen from eqn (1):

I [Wn, - tiui,,] dS = I

LW,I - oij Ei,i,ll da, (3) - 1‘123450 v- VE

Fig. 1. Path integral definitions in a cracked soiid.

Ductile crack growth using T*, J and CTOA criteria 539

CRACK GROWTH

Fig. 2. Typical T*- and J-resistance curves for an elastic-plastic stable crack growth specimen,

and since eqn (2) holds,

(4)

Substituting (4) into (3) proves path independence. Now, if we consider an incremental theory of plasticity and redefine W as the total ac-

cumulated increments of stress working density, a path-independent integral, from (1) utilizing (3), may be defined as

T: = - tju;,,] ds

:: I I.‘12345 [Wnt - tiu;,,] ds + I v- v, LW,, - (~i.jui.l)~~l dt)*

The path independence embodied in eqn (5) is valid for arbitrary Ioading/unloading and crack growth of the flawed structure as long as W is appropriately defined.

The typical far-field Jr and T: resistance curves for a fracture specimen are illustrated in Fig. 2. It is seen that J and T* are equivalent during crack initiation and for small amounts of crack growth. After some amount of growth, J and T* begin to deviate from one another. When the specimen reaches steady-state crack growth conditions, T* levels out to a constant value; and J1 continues to rise. From eqns (1) and (5), it is clear that the only difference between the two is the presence of the volume term in (5). The contribution to TT from this volume term becomes larger and larger as crack growth proceeds. Indeed, the rise of J during steady-state growth is due to the fact that it is a far-field parameter. J becomes more and more divorced from crack tip events as growth proceeds.

One can take a formal increment in Tf and show[3 and the references therein]

ATT = f, [Awn! - (ti + Atif Aui,r - Atiui,,] ds I

=: [A Wnl - (ti + Atj) Aui., - Atiui.,] ds

+ /

[Auij(E;,j.l + 4 Aei,j,i) VI.-v,

(6)

540 F. W. BRUST et al.

Here AW is the increment in stress working density. It is clear that

T: =

and either form [eqn (5) or eqn (6)] may be used to calculate TT. As discussed in Ref. [3], the incremental form of ATT can be more accurate during a numerical simulation of crack growth than the total form: and the incremental form is useful as a rate parameter for time-dependent inelastic deformation. Also, it is a simple matter to generalize eqns (5) and (6) to conditions of finite deformation and strain[4]. TT is a valid path-independent fracture parameter for an arbitrary constitutive relation.

In the laboratory, J, can be calculated accurately when crack growth occurs as[7]i

(7)

where h; is the instantaneous length of the remaining ligament, W is the specimen width, q and (Y are known functions of bl W, and a. ,+ I and a; are the crack lengths evaluated at steps i + 1 and i. A,,;+ I is the area under the load displacement record between lines of constant displacement at steps i and i + 1. .J, , calculated from eqn (1) along a path at the boundary of the specimen, is nearly equivalent to eqn (7), as discussed in Ref. [3]. The strong dependence of Ji on the area under the load displacement curve in eqn (7) is emphasized.

Elastic-plastic finite element analyses were carried out on 1-T compact tension specimens of A533B steel, as discussed in Ref. [3]. The displacement resistance curve of the experiments reported in Ref. [6] was used as input to the model, and the output consisted of the load vs crack growth along with the Ji-, CTOA- and TT-resistance curves. Plane stress conditions were asssumed, and the node release technique was used to model crack growth. A detailed de- scription of the finite element model, the typical degree of mesh refinement, and the material properties utilized are presented in Ref. [3] and [4]. The Ji- and Ty-resistance curves resulting from these analyses are shown in Fig. 3, and the CTOA-resistance curve is shown in Fig. 4 (for uo/W = .5). These curves are reproduced from Ref. [3].

In the second phase of the study, the analysis described above for aO/ W = .5 was restarted, wherein after 1 .O mm of crack growth, the specimen was unloaded (displacement control). After a certain amount of unloading, the nodes along the crack faces near the tip began to come into contact. Here the analysis was stopped since no attempt was made to model the crack face contact. The specimen was then reloaded. Upon reloading, three different criteria were used to permit continued crack growth:

1. J-control (Fig. 3) 2. CTOA-control (Fig. 4) 3. TT-control (Fig. 3)

For example, for the J-controlled analysis, upon reloading crack growth was permitted to occur once again when J = 260 N/mm.

The computed load vs displacement record for the J-controlled analysis is shown in Fig. 5. Here it is seen that there is a nearly vanishing hysteresis loop; and since Ji is related to the area under this curve, the behavior of the specimen was nearly identical to that of the monotonically loaded specimen. The load vs displacement record for the CTOA-controlled analysis was very similar to this.

The computed load vs displacement record for the TT-controlled analysis is shown in Fig. 6. Here, the crack was predicted to grow at roughly half the original load before unloading. Also, it is seen that by the time the load reached slightly below the original load level, the area under this record is again hardly changed. Hence, if continued crack growth in such a circum- stance is truly TF-controlled, it is seen that J is unaffected by the unloading/reloading effect.

Kompare this general form given here to eqn (4.3) of Ref. [4], which is the particular form valid for the compact specimen.


a-a,(mm)

Fig. 3. J- and T*-resistance curves for A533B steel.

cToA (RADIANS)

0.2 0.4 06 0.8 1.0 1.2 14 1.6 I. 8 2.0 22

a-a, Cm@

Fig. 4. CTOA-resistance curve for A533B steel.

542 F. W. BRUST et (I/.

P NJ)

- Actual Specimen

W Unloading at a-&= l.Omm

x-_mC Reloading ot a-o.9 I.Omm continued crock growth governed by J-Realstance

curve

%I n

c-z- qo=2.2

-

Crack Growth

4 - 4.0 KN

Fig. 5. Load vs load-point displacement record of specimen. After unloading, continued crack growth governed by J-resistance curve (A533B steel).

P(KN) 6

+e-a Actual Speclmrn

I m-e-a Unloading at a- a. * LOmm n-16-# Reloading at a- go = 1.0 mm

continurd crack growth governad by l$ Roslrtanco curve (e= 0.3mm)

/ 4

0.1 0.2 0.3 0.9 1.0 1.1 I.2

b (mm1

Fig. 6. Load vs load-point displacement record of specimen. After unloading, continued crack growth governed by P-resistance curve (A533B steel).


P CKNI

J (or) C T 0 A -Control

1

‘4

~

,P 8

Y h

L ,

-Actual Spoclmrn a-o-o ltnloodlng or o-o, I I.Omm +w6 Roloodlng 0t a- 00 = i.Omm

continued crock growth governed by T Resirtonco

curve (E= 0.3mm)

! : : ! : : ! : : : : ,

0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2.0 2.2 -4.0 KN

Q-l& h-d

Fig. 7. Load vs crack growth record of specimen (AS33B steel).

A more graphic demonstration of the difference between the analyses is shown in Fig. 7. Here it is seen that f- or CTOA-control predicts essentially no change from the monotonically loaded analysis. However, TT-control predicts roughly a 10% increase in crack growth by the time the original load level is once again reached.

These results were expected. Upon unloading, reverse plastic deformation clearly occurs in front of the crack tip. ‘I’?, being a near-field parameter, can capture this effect; but J, being a far-field parameter, cannot. Also, CTOA appears to lose physical significance when reverse plastic deformation occurs near the crack tip.

The results presented in Figs 4-6 were obtained by hypothesizing J, CTOA or TF to be valid fracture parameters. The specimen was unloaded after an arbitrary amount of stable crack growth and reloaded with continued crack growth governed by the hypothesized fracture parameters. Up to this point, these were completely numerical simulations-no experiments studying this effect had been carried out. Indeed, at this point, it was unknown which parameter could correctly capture the unload/reload effect because the behavior of the laboratory specimen was unknown.

MODELING OF AN ACTUAL UNLOADING EXPERIMENT

To establish the crack growth behavior of an actual test specimen, a set of experiments was carried out at Oak Ridge National Laboratories. Displacement control was used for these experiments, as was the case for the numerical experiments above. The material used for these experiments was A508 steel at 220°F. As measured for this material, the yield stress is 510 MPa, E = 206,430 MPa, and n and B, the material parameters for the Ramberg-Osgood relation, are 6.9 and 996.5 MPa, respectively. The load-point displacement vs crack growth record obtained from these laboratory experiments was used as input to the finite element analysis. The potential drop technique is used to measure the crack growth. (Further details on the experimental technique are given in the Appendix.) The TF- and J-resistance curves which resulted from this numerical simulation are compared to the measured behavior in Fig. 8, The experimental J-resistance curve was calculated using the modified Ernst method. A plane strain analysis would fall above this experimental curve; and, as usual, the plane stress J-resistance curve falls below the experimental curve. Again, T* is equivalent to J for small amounts of growth, after which TT levels out to its constant steady-state value, while J continues to rise.

F. W. BRUST et (il.

J = YZinTp(P~a~~ STRESS)

Fig. 8. /- and P-resistance curves for A508 steel.

CTOA (RADIANS]

1

0 I. I I1 I I a I. c g :

0.2 0.4 0.6 0.8 I.0 1.2 1.4 1.6 1.8 2.0 2.2

a-cI,(mm>

Fig. 9. CTOA- resistance curve for A508 steel.

LOAD

(KN)

Ductile crack growth using T*, J and CTOA criteria

O-O-0 Soble crack growth

w Unloading

W Reloading and Con?inwd growth controlled by

545

15.- A5088 Steel

EXPERIMENT T; Reststance curve

0.5

mm. . _

1.0 1.5 2.0 2.5 2.7

a-ao(mm)

Fig. 10. Experimental and numerical load vs crack growth record for A508 steel including the effect of unloading/reloading.

160”

T+(& IW-

crack growth again proceeds after unloading/reloading

2.3 2.4 25 2.6 2.7 2.8 2.9

8tmm)

Fig. I I. Behavior of T* during unloading.

546 F. W. BRUST et ui.

tp

O-0-0 Actual Specimen

e-B-8 Unloadmg (a-q,= 1.0) & Reloading a( a-q= l.Omrn

Conlmued Crack Growth

Governed by J-R curve.

f ( : i : : ; t ; :

.I ,2 .3 .5 1.0 1.2

8 (mm)

Fig. 12. Behavior of J during unloading.

The CTOA-resistance curve is shown in Fig. 9. CTOA starts out as a large value and then reaches its constant steady-state value after roughly 1 mm of crack growth.

The experimental specimen was unloaded after about 2.2 mm of crack growth and then reloaded again, After unloading the specimen at a - a 0 = 2.2 mm in the numerical simulation, the TT-resistance curve of Fig. 8 was again used to control crack growth during reloading. That is, crack growth upon reloading was begun when T? reached 1.50 N/mm, which is the constant value attained during steady-state crack growth. As reloading continued, each time TT reached this level, an increment of crack growth was modeled, whereupon TT dropped. Displacements were again appplied, until TF = 1.50 again; and another increment of crack growth was permitted.

The experimental and numerical load vs crack growth records for this are revealed in Fig. 10. Remarkable agreement between the analytical and numerical results is seen here although the overall plane stress results are lower than the experimental results in the usual fashion for a plane stress analysis. If the J-resistance curve of Fig. 8 or the CTOA-resistance curve of Fig. 9 are used to control continued crack extension upon reloading, very little change in the monotonically loaded specimen is observed. This was also the case for the A533B specimen, as illustrated in Fig. 6.

The behavior of T” and Jr during unloading is shown in Figs, t I and 12, respectively. T? appears to nearly Ievel out after unloading, while J is nearly unchanged. The maximum percentage difference of T? between paths (neglecting the path closest to the crack tip) was about 2% just before unloading, about 12% after unloading, and about 5% after reloading and during continued crack growth. The maximum percentage difference in J is about 100% just before unloading, 300% after unloading, and about 100% again after reloading to its original far-field level. Again this extreme path dependence in J further illustrates that it has no relation to the near-field fracture events, since it is the fur-$eld Jpath that is calculated in experiments and estimated via the engineering approach.

DISCUSSION

The problem of low-cycle fatigue has important practical implications. Many structural components are such that they are only unloaded between very long intervals of time. As shown


here, the integrity of a flawed structure experiencing low-cycle fatigue could be seriously over- estimated when the integrity assessment is made via the engineering approach utilizing J as developed in Ref. [l]. Moreover, it is also seen that CTOA produces nonconservative predictions. CTOA also appears to lose physical significance once reverse plastic deformation occurs at the crack tip during global unloading of the flawed structure.

TT appears to accurately account for this effect, although a more thorough study is clearly in order. Other experiments have been made in which about 10 cycles of unloading/reloading were applied to a specimen during stable crack growth. During each cycle of reloading, crack growth reinitiated at roughly half the original load before unloading; and significant extra crack growth had occurred by the time the original load level (before unloading) was reached. It should also be clear that T? could be used to characterize high-cycle fatigue in similar situations where a large amount of plastic deformation occurs.

A unique value of J to characterize fracture after unloading is not possible. In fact, the value of J which will predict fracture after unloading/reloading is entirely dependent on the amount of unloading. Moreover, once the original load level is reached, and corresponding original J-value is obtained, we obtain two identical values of J with different amounts of Au. Hence, J is not unique. A nonunique fracture parameter under these circumstances is non- workable. Note that T* predicts fracture upon reloading at precisely the same value of T* before unloading; hence, it is unique.

Isotropic hardening was used for these analyses. Since there was reverse loading at the crack tip as the specimen was unloaded, using isotropic hardening is not sufficient. However, since these were very low hardening materials and since there was not an excessive amount of reverse loading which occurred (unloading was terminated when the crack faces came into contact), this effect should not influence the results significantly. More importantly, the effect of neglecting large deformations and strains must be investigated. This will not be a simple chore. McMeeking[S] had to model an initially blunted crack tip in his large deformation analyses of a stationary crack, since the crack tip singularity mitigated this effort. Modelling a moving crack tip while including large deformations and strains should be much more difficult. Nonetheless, large deformation analyses of moving cracks using constitutive models which can account for cyclic plasticity more effectively[%lO] should be undertaken.

REFERENCES

(11

VI

[31

[41

[51

[61

[71

PI

[91

[lOI

V. Kumar, M. D. German and C. F. Shih, An engineering approach for elastic-plastic fracture analysis. EPRI Research Project NP-1931. 1237-l. prepared by G.E., July (1981). M. F. Kanninen, C. H. Popelar and D. Broek, A critical survey on the application of plastic fracture mechanics to nuclear pressure vessels and piping. NRC report NLJREGICR-2110, BMI-2080, May (1981). F. W. Brust, T. Nishioka, S. N. Atluri and M. Nakagaki, Further studies on elastic plastic stable fracture utilizing the T* integral. En~ng Fructrtre Mech. 22, 1079-I 103 (1985). F. W. Brust, The use of new path independent integrals in elastic-plastic and creep fracture. Ph.D Thesis, Georgia Institute of Technology (1984). J. R. Rice, A path independent integral and the approximate analysis of strain concentration by notches and cracks. J. Appl. Mech. 35, 379-386 (1968). J. P. Gudas et (I/., A summary of recent investigations of compact specimen geometry effects on the J,-R curve of high strength steels. NRC report NUREGICR-1813. R5. November (19801. H. A. Ernst,P. C. Paris and J. A. Landes, Estimations on J-integral and tearing modulus Tfrom a single specimen test record. ASTM STP 743, 476-502 (1981). R. M. McMeeking, Finite deformation analysis of crack tip opening in elastic-plastic material and implications for fracture initiation. Brown University Report COO-3084144, May (1976). 0. Watanabe and S. N. Atluri. Constitutive modeling of cyclic plasticity and creep, using an internal time concept. Georgia Institute of Technology, Center for the Advancement of Computational Mechanics, May (1984). 0. Watanabe and S. N. Atluri, Internal time, general internal variable and multi-yield-surface theories of plasticity and creep: A unification of concepts. Georgia Institute of Technology, Center for the Advancement of Compu- tational Mechanics, August (1984). H. H. Johnson, Calibrating the elastic potential method for studying slow crack growth. Muf. Res. Stundurds 5, 422 (1965). R. 0. Ritchie and K. J. Bathe, On the calibration of the electrical potential technique for monitoring crack growth using finite element methods. Inr. J. Fructrrre 15, 47-55 (1979). B. L. Freeman and G. J. Neate, The measurement of crack growth during fracture at elevated temperatures using the dc potential drop technique. In The Meusurement qf Crack Length und Shape During Fracture and Futigue. ISBN o-9506820-04, Materials Advisory Services, Ltd., Warley, England (1980). “Standard Test for JI, , A Measure of Fracture Toughness”, ASTM E813-81. In Annuul Book ofA.STM Stundurds. Pt. 10. pp. 810-828. American Society for Testing and Materials, Philadelphia (1981). H. A. Ernst, Material resistance and instability behond J-controlled crack growth. Scientific Paper 81-107.JINTF- P6, Westinghouse Research and Development Center, Pittsburgh (1981).

548

[I61

[I71

F. W. BRUST et 01.

G. M. Wilkowski and W. A. Maxey, Review and applications of the electric potential method for measuring crack growth in specimens, flawed pipes, and pressure vessels. In Fracfure Mechanics: Fourteenth Symposium, pp. II- 266-294. ASTM STP 791, American Society for Testing and Materials, Philadelphia (1983). F. J. Loss, ed., Structural Integrity of Water Reactor Pressure Boundary Components, Annual Report, Fiscal Year 1979, NUREGICR-I 128. NRL Memorandum Report 4122, Naval Research Laboratory, Washington, D.C. (19791.

(Receiwd I5 Janrrnr:v 1985)

APPENDIX USE OF THE POTENTIAL DROP TECHNIQUE FOR CRACK GROWTH MEASUREMENT

For accurate determination of crack length by the potential drop (PD) method, an accurate calibration is essential. Several calibrations exist in the literature11 I-131 for various fracture mechanics specimens. However, there are no calibrations of sufficient accuracy for the complex Ji, -specimen modified for load-line displacement. Three separate calibrations were performed: two experimental and one numerical. The current input and voltage probe locations are shown in Fig. Al. Note that both active and reference probes were used. The probe locations were selected for maximum output and minimum sensitivity to errors in probe location. The current input location was selected for maximum output. The ADINAT finite element program, with eight-node rectangular isoparametric elements, was used to perform the numerical calibration. The finite element grid. shown in Fig. Al, utilized 85 elements and 299 nodes. Two experimental calibrations were performed: (I) 0.7-mm-thick type 304L stainless steel sheet model, and (2) 0.03-mm-thick aluminum sheet model. Each model was precisely machined to the planar dimensions of a 50.8-mm-thick Ji,-specimen. Current was applied to the models, and the dc potential was measured at the desired locations for each crack length selected.

REFERENCE PROBE dc CURRENT INPLiT

/ I I I

I 1 I /

I I I RCTIVE PROBE5

Fig. Al. JI, -specimen geometry and probe location.

JIC SPEC PD CAL 1 0 . 03 mm AL SHEET

0 .7 mm 55 SHEET l FE RNRLYSIS

.6 -

.4 -

.2 -

0 J1l’lll’lll’lll’lll 0 .2 .4 .6 .a 1

a/W

Fig. A2. Results of calibration models.

Ductile crack growth using T*, J and CTOA criteria

HEWLETT-PACKARD 9836 DESK TOP COMPUTER

549

DIGITRL SYSTEM

t

RNRLOG SYSTEM

Fig. A3. Schematic of desktop computer control system.

The results of the three models are shown in Fig. A2. Note that the results are presented as a ratio of dc potentials. This ratio is a property only of the geometry and does not depend upon input current or resistivity. The agreement among the three models is quite good, with a difference in a/~ of 1% for 0.5 < N/W < 0.9.

The system for performing J-integral resistance curve tests by the PD technique utilized a Hewlett-Packard (HP) 9836 desktop computer, a custom-designed interface, a standard MTS 800 testing system, a 90-A dc power supply, two high-gain microvolt amplifiers, two HP-3478A S-digital voltmeters, and a 12-bit digital-to-analog converter. The desktop computer controls the test by clip gage displacement through the custom interface (consisting of two l6-bit analog-to-digital converters and a digitally programmed analog ramp generator). The computer also controls the power supply to optimize the dc potential output. The dc power supply is cycled off and on to permit correction for any thermal voltage from the dc potential probes. A schematic of the computer testing system is shown in Fig. A3. The computer testing system controls the test, provides printed and graphical output during the test, and records data on disk for future analysis. Fig. A4 shows the load vs load-point displacement trace for the J,, -test.

A comparison of J-resistance curves by the potential drop and unloading compliance techniques is shown in Fig. A5. The J-integral was calculated by the modified Ernst method[l5]. The PD crack length used in this figure was the value measured immediately before the unloading. The figure shows a positive offset in the PD crack length at the

2TC870 23-RPR-84 104 DEGREES C

50

40

Z A 30

g 20 0 -I

10

0 d 4 6 B I0

DISPLACEMENT (mm)

Fig. A4. Load vs displacement trace for _I,,. test.

550 F. W. BRUST et (11.

2TC870 23-RPR-84 104 DEGREES C

. POTENTIRL DROP DELTR-A

-.2 1.8 3.8 5.E 7.8 3.3

CRACK EXTENSION (mm)

Fig. AS. Comparison of J-resistance curves for the unloading compliance and potential drop techniques.

2TC870 23-APR-84 104 DEGREES C I0 r

0

0

+-BLUNTING

--TUNNELING

INITIRTION

0 2 4 6 8 I0


Fig. A6. Trace of load-line displacement vs crack extension for potential drop technique (note that the traces during unloading have been omitted for clarity).

2TC870 23-RPR-84 104 DEGREES C 2000 MODIFIED-mNST J-INTttHRL

cu _- POWER LRW FIT . JIC(E-8131

< - - OFFSET LINES . JIC(PWR LRWI

E 1600 - ‘--,r BLUI;ITING LINE . RCTURL OELTR-R

\ c, 2 - 1200

2 % a00

W

5 400

H

c, 0

E 2 4 6 8 I0


Fig. A7. J-resistance curve for potential drop technique showing power law curve tit

beginning of the test. The reason for this offset is the sensitivity of the PD technique to local crack tip changes in resistivity and geometry prior to initiation of crack extension, a direct result of plastic blunting. This blunting behavior is illustrated in Fig. A6, which shows the three stages of crack growth. This behavior has been reported previously by others, e.g. Wilkowski and Maxeyll61. This offset, however, can be calculated easily through the power law Jt,- procedure[l7]. In this procedure, a curve tit Atr = C + AJB is determined by nonlinear regression techniques for all J-Au pairs up to the IS-mm-offset line (as shown in Fig. A7). The value of C is then used to offset the PD crack length values. The resulting Ji,. (power law method) for the PD technique (158.4 kJ/m’l is quite close toJi,.-(power law method) value for the UC technique (124. I kJ/m’); this represents a difference of 167c, within the characteristic variance for this parameter. The good agreement between Ji,-values obtained by ASTM ES13 and the power law method is not affected by the use of the PD technique.

Documents

a combined numeri~al/experimental study of ductile crack growth