An experimental and finite element investigation of fracture in aluminium thin plates

0022-5096/80/100~167

.I. Mech. Phys. Solids Vol. 28, pp. 167-189 0 Pergamon Press Ltd. 1980. Printed in Great Britain

$02.00/O

AN EXPERIMENTAL AND FINITE ELEMENT INVESTIGATION OF FRACTURE IN ALUMINIUM THIN PLATES W.

T.

EVANS

Department of Civil Engineering, The Polytechnic of Wales,

Pontypridd, Mid Glamorgan CF37 IDL, Wales,

M. F. LIGHT Lloyds Register of Shipping, 71 Fenchurch Street, London EC3M 4BS, England

and A. R. LUXMOORE Department of Civil Engineering, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, Wales (Received I5 October 1979)

ABSTRACT DETAILED strain and crack-opening measurements have been made on centre crack and double edge crack specimens of aluminium alloys L70 and L71 in plates 1.6 mm thick, using the Moire technique to measure strains in front of the instantaneous crack tip during stable crack growth. The specimen load-displacement curves were also monitored during stable crack growth, using a displacement-controlled loading machine, and fracture resistance (R-) curves calculated. A two-dimensional elastic-plastic finite element program was used in a numerical model of the experiments, with stable crack growth being simulated by relaxation of the nodal reactions ahead of the initial crack tip. The numerical models were used to calculate the energy changes occurring during crack growth and also the Rice J-integral. These parameters were then examined as possible fracture criteria, and a simple energy balance proved adequate in explaining the main features of stable and unstable crack growth. This result was confirmed by modelling fracture data on L70 obtained by other investigators.

NOTATION a a’

Err I n fJ CY

k

observed crack length plastically-corrected crack length strain component parallel to specimen axis distance measured from crack tip Ramberg-Osgood hardening index stress yield stress elastic stress intensity coefficient 167

168 Kit

G

GC J E

QC T

u, u U’ up u* U’ &

fracture resistance parameter strain energy release rate strain energy release rate at crack instability Rice path-independent line integral force difference between elastic and plastic energy release rates difference between elastic and plastic energy release rates for crack initiation traction on boundary displacements in (x, y)-directions, parallel and perpendicular to crack total elastic energy according to incremental theory of plasticity total plastic work according to incremental theory of plasticity strain energy per unit volume according to deformation theory of plasticity specimen width crack tip opening displacement at onset of crack growth

1.

INTRODUCTION

CRACK propagation in thin sheets of high strength alloys is usually accompanied by extensive yielding, which complicates the stress analysis, and provides corresponding difficulties in the establishment of a fracture criterion. In these circumstances, most practical fracture toughness procedures, such as R-curve analysis (KRAFFT,SULLIVAN and BOYLE, 1961) are based on pseudo-elastic principles, i.e. using equivalent elastic crack lengths to allow for small amounts of plastic flow at the crack tip, and they become increasingly inaccurate for the more ductile alloys, where general yield occurs prior to fast fracture. Recent developments in finite element methods of stress analysis have provided a method for analysing elastic-plastic stress fields around crack tips, so that quantities not easily obtained by experiment, such as stresses, elastic strain energy etc., can be related to the critical experimental parameters and tests carried out for different geometries in order to investigate the existence of a geometry-independent fracture parameter. Hence it is important that accurate experiments and stress analyses be synchronized for this purpose. Many types of criteria have been proposed for use in general yield fracture mechanics, but the most promising ones appear to be the following: a Griffith type energy balance; the Rice J-integral (which is usually considered a crack-tip characterizing parameter); the R-curve analysis (which is based on an energy balance approach); and a critical strain or displacement near the crack tip. Apart from the R-curve, these criteria each involve a single parameter, which could be equivalent to each other, i.e. unique relations could exist between the energy changes, J-integral, strains and displacements, so that any one of them may be used. If this is so, then certain conditions are imposed on the stress and strain fields ahead of the crack tip. The energy balance does not necessarily depend on any unique stress and strain distribution around the crack tip (although a unique distribution would imply a single parameter criterion), but the usefulness of the J-integral depends on it being path-independent, which is only possible for certain stress and strain distributions

Fracture in aluminium thin plates

169

(LIGHT, 1975). A critical strain/crack-opening displacement criterion also implies a unique strain distribution near the crack tip just prior to fracture, and if the R-curve is independent of geometry, then the area of the plastic zone at fracture must be reasonably constant for different constraints for given material and thickness. The present study investigates the applicability of these parameters for the case of certain aluminium alloys in thin plate form.

2.

EXPERIMENTAL DETAILS

Tests were conducted on centre notch (C.N.) and double edge notch (D.E.N.) specimens with either a rectangular outline (Fig. l(a)) or a contoured side (C.S.) outline (Fig. l(b)) (see EVANS(1975)). The latter was used in an attempt to overcome the buckling inherent in the fracture of thin plates, 1.6 mm thick, which were used exclusively in the present study. For the rectangular outline, low friction straps were used to prevent buckling. Cracks were produced by sawing with a specially ground hacksaw blade 0.15 mm thick, and the final radius was obtained with a razor blade, producing a notch-tip

+++++++

+++++++

Ia) RECTANGULAR

Ilb)

SPECIMEN

305mm

CONTOUREO-SIOE

SPECIMEN

FIG. 1. Geometry of test specimens.

170

T. EVANS. M. F. LIGHTand A. R. LUXMIXRE

W.

radius of between O-025 and O-037mm. (BRADSHAW and WHEELER(1974) have shown that little difference occurs in the fracture resistance between fatigue and sawcut cracks.) Initial total crack lengths were approximately 25 mm long except for the L70 D.E.N. specimen which had a 40 mm total crack length, i.e. 20 mm each side. Two aluminium alloys, L70 and L71, were used (these were basically the same alloy, but aged in different ways) with significantly different work-hardening rates (see Fig. 2). Strain measurements were made around the crack tip using the Moire technique, and crack-opening displacements (C.O.D.) were measured at magnification x 800 using transmitted illumination. During slow crack growth, strains were measured along the actual crack line, which deviated from the original crack line due to the oblique direction of fracture in the specimens. Crack resistance curves were obtained from load vs displacement curves (BRADSHAW and WHEELER,1974) using a plasticity correction in the form of an equivalent elastic crack length, giving K, = km Jd. The value of a’ was determined from identical specimens with a range of sawcut cracks, loaded in the elastic region

500-

LONGITUDINAL ___--__------

71 LONGITUDINAL

---__--L70

LOO 350 “E - 300. E t $j

u ;;

250. 200 150 100 50 L

0

FIG.2. Stress-strain

I

0 05

I

01 STRAIN

curves for aluminium

I

0 15

I

-

02

alloys L70 and L7 1.

171

Fracture in aluminium thin plates

only. These results were plotted so that any elastic-plastic specimen compliance could be related to an elastic crack length (which would be longer than the observed crack length, a). 3.

EXPERIMENTAL RESULIX

3.1 Strain distributions The strain distributions were similar to those observed by KOBAYSHI, ERGASTRON and SIMON(1969) and BRADSHAW, LACEYand WHEELER(1972). The plastic zones grew from the crack tip in the form of elongated ellipses comparable to the model proposed by DUGDALE(1960), and as yielding progressed, the zones expanded away from the crack line, with high strains concentrated near the tip. At the onset of slow crack growth, strip necking zones were very small compared with the overall plastic zones, and were characterized by the appearance of twin strain peaks on either side of the crack axis (LUXMOORE, LIGHTand EVANS,1977). Within a relatively small distance from the crack tip (x 3 mm) these peaks merged into a single low peak along the crack axis. Similar results were observed by GAVIGAN,KE and LIU (1973) and SCHAEFFER, Lru and KE (1971), who apparently confirmed the suggestion by LIU (1961) that the two strain-peaks are produced by the shearing action on two planes at 45” to the specimen surface. However, the writers have reproduced this double strain-peak with a plane stress finite element computation (LUXMOORE, LIGHT and EVANS,1977), and this indicates that the formation of the strip necking zone is related to purely two-dimensional behaviour, which is modified later by threedimensional effects. Once the crack propagated, the strip necking was left behind, and the maximum strain on the surface of the specimen was ahead of the new crack line (which was offset from the original crack line by the action of the slant fracture). Presumably, an original crack cut at 45” to the specimen surface would not produce these twin peaks. r lmml 3 I

2 I

L I

5 I

6 7 6 9101112 15 I111111 I

20 1

_ -0030

K

nW -0020 "\ '1. -0010 -0.009

.

-0.006 -0 007

\.

EYY

-0006 -0005 \

-0.OOL '\,

-0.003 . -0002

FIG.

3. Residual strain distribution at crack initiation for centre notch contoured side specimen of L70 alloy.

W. T. EVANS,M. F. LIGHT and A. R. LUXMOORE

172

rimm)

oSTWIN

AT 70 kN

+ RESIDUAL

LOAD

STRAIN

+\

t

+

0 05

%

OOL

t

Lo 03

FIG. 4. Comparison

between

residual

strains and strains under specimen of L70 alloy.

load for a centre

notch

contoured

side

Distribution of the strain E,,,, (parallel to the specimen axis) beneath the static crack at the onset of slow crack growth is illustrated in Fig. 3. Two regions can be distinguished, and in both of them the variation of cYY can be represented approximately by ++ = Cr-“, where r is-the distance beyond the crack tip, and C and n are constants for a particular load. (In the contoured-side specimens, only residual strains were measured after unloading the specimens, but the slope of the strain distribution was generally unaffected by unloading (see Fig. 4).) These results show good agreement with those of GAVIGAN, KE and LIU (1973) for 6 mm thick aluminium specimens, except that they identified a third change in slope farther out from the crack tip, outside the range of our measurements. The first region lay just within the limits of the strip necking zone, but the twin strain-peaks could only be measured by a specialized Moirt interpolation procedure, and for the determination of 11, and C, (the subscript 1 referring to the region closest to the crack tip and subscript 2 to the furthest) a coarser technique was used, which averaged the strains in this region over the width of the strip necking zone (z 1.5 mm). This averaging over the strip necking zone may be the cause of the change in the value of the strain gradient index II. rimml

10 0

IO 1 III1 7 k N

LOAD=67 a,_32

_..i_miI

Lrnrn

a2.3L

0:

2 mm \_

a, .CRACK ON LEFTOF SPECIMEN

HAND

o2=CRACK ON RIGHT-HAND OF SPECIMEN

FIG. 5. Strain

distribution

under

CRACK

0,

+ =CRACK

“2

SIDE

SIDE

+q, \o

+

load ahead of both cracks in a wide plate centre specimen of L71 alloy.

notch

rectangular

173

Fracture in aluminium thin plates L70

C N

L70DEN

0

CRACK

l

CRACK

a2

0

CRACK

a,

.

CRACK

a2

a,

70-

i 5 60% 3 50-

LO

1,’ 0 01

, 0 02

I

0 03

I

0 04

I

0 05

I

0 05

I

0 07

I

0 08

C:, imml

FIG. 6. Relation between load and C, for both C.N. and

D.E.N. specimens in L70 and L71 alloys.

HUTCHINSON (1968) showed that, for materials obeying the Ramberg-Osgood power law, the strain distribution beyond a static crack tip in plane stress conditions should vary as r-“‘(“+I). The stress-strain curve for L70 can be represented with n = 10, which produces strains varying as r-o.91 according to Hutchinson. The residual strain measurements on the contoured side C.N. specimen gave the strain variation as r-o’44 in region 1 (nearest the tip) and r -“‘* in region 2. The strain variation in region 1 is probably affected by three-dimensional deformation, in both behaviour and measuring accuracy (mentioned previously), but it is interesting to note that Hutchinson predicted a r-* -type of strain variation near the crack tip when using bilinear elastic-plastic material behaviour. Also, the same two strain-regions were observed in front of the moving crack (discussed below) where no strip necking zone could be detected. Active strain measurements were continued during slow crack growth for the rectangular specimens of alloys L70 and L71. The same general behaviour was found as in the case of the static crack, with two distinct regions, represented by E = Cr-” (Fig. 5) being identified. However, the value of the slope in region 2 (the region furthest from the tip) tended to reduce with increasing crack length (Table 1). When the values of C, were plotted against the load P (Fig. 6) a linear relation was obtained, apparently independent of both geometry and material. If the data for L70 and L71 are plotted separately, then a small difference can be assigned to the P-C, relation in each case. A similar relation was found when plotting P against C, although there were less data in this case. The slope of the P-C, line was twice that of the P-C, line, within experimental error. These results show that, for a fixed value of I, strain is proportional to load, even though this position is moving due to crack growth. This result appears independent of geometry, and, to a lesser extent, of material, and suggests that there is a characteristic stress and strain distribution within the plastic zone surrounding the crack tip. This characteristic zone could have the same importance as the established crack-tip stress field in elastic fracture mechanics.

Load (kNf

55.8 63.3 66.8

51.0 G-9

58.9 67.7 68.2

47.3 53.4 56.5

Material

L70 C.N.

L70 D.E.N.

L71 C.N.

L71 D.E.N.

24.1 26.33 268

27.8 32.4 354

25.3 1 26.3

26.02 2694 27.75

A

B

24-6 24.85 25.13

28.05 34.2 35.8

25.31 26.4

25.95 27.25 28.23

Crack length @ml

0022 0.03

0.04 0056

0.033 O-043

0.03 0.05 O-064

CI

0.51 0.73

038 0,613

0.45 0.58

0.63 O-63 0-s

nl

0.02 0.023

0.04 0.056

0,028 0.043

0,035 0048 O-065

c,

0.6 0.49

0.59 0.61

0.55 058

063 0.63 0‘6

fll

&I = C1rmn~ Crack A Crack B

0.017 0027

0.03 0.034

0.026 O-023

0.021 0.029 O-058

0.012 0.027

0.025 0.034

0036 O-023

0.024 0.033 0~0%

Strain limit for El A B

O-018 0032 0043

0% o-073 0082

0,037 0056

0,036 0.067 O-064

C,

1.03 0.87 0.97

1.07 0.87 084

0.85 077

0.87 0.9 0.75

n2

0.018 003 0.043

0.053 o-073 0.089

0,033 0.056

0042 0.054 0*064

1.03 @89 0.97

1.07 0.87 0.84

0.8 0.77

0.90 0.81 o-75

n2

Crack B C,

Ed = Gzr-“=

specimens.

Crack A

around both cracks in the rectangular

Region 1 is nearest to and region 2 is furthest from crack tip

TABLE 1. Summary of results from the strain distributions

Fracture in aluminium

175

thin plates

3.2 Crack opening displacement

The onset of stable crack propagation was not determined very accurately from the crack-opening profile, and it proved more satisfactory to plot the C.O.D. at the original crack tip (taking an average for both tips) for increasing crack length, and then to extrapolate back to the original crack length. The critical C.O.D. values for both C.N. and D.E.N. geometries were very close, being 0.08 mm for L71 and 0.16 mm for L70. 3.3 Fracture resistance (R-) curves The resistance (R-) curves are shown in Fig. 7(a, b) for L70 and L71 respectively. Good agreement is obtained with the experimental curves obtained from BRADSHAW and WHEELER’S (1974) C.N. specimens and there is no significant difference for the two geometries. A close similarity exists between the curves for the two alloys, which could be connected with the similarity of strains in the materials during slow crack growth. This similarity may be fortuitous, as Bradshaw and Wheeler have shown that substantially different R-curves can be obtained from other geometries. Examination of their results also shows that R-curves for different materials appear very similar for small C.N. specimens, where general yield conditions are most likely to be approached (or exceeded). In our L70 specimens, general yield occurred at an early stage of stable crack growth, while the L71 specimens were close to general yield at C.N

0 +

DEN

e

DEN

6

CN

-

CONTOURED

SIDE

RECTANGULAR

CN

BRADSHAW

GENERAL

YIELD

SPECIMEN

SPECIMEN

AND

WHEELER

119741

DEN

REC TANG”>>> SPECIMEN

@

/

ik~~~~;%&EN

GENERAL YIELD SPECIMEN

0

I 2

I

I 6

I

I

I

I

curves for C.N. and D.E.N.

I,

IL

10 a-a0

FIG. 7a. Resistance

C.N

CONTOURED

,

16

,

22

SIDE

,

,

26

lmml

contoured

side and rectangular

specimens

of L7O alloy.

176

W. ‘1’. EVANS.M. F. LIGHT and A. R. LUXMU~RE

0

CN

+

0.E N EXPERlMENT

EXPERIMENI

m

-1-r-ia

22

26

la’-aolmm

FIG. 7b. Resistance curves for C.N. and D.E.N. rectangular specimens of L71 alloy.

the maximum load (these estimates being based on net section stresses). It is probable that the extensive yielding masks the effect of stable crack growth, producing similar load-displacement curves for the different materials, hence leading to similar R-curves. In these cases, the pseudo-elastic analysis will probably be subject to considerable errors.

4. 4.1

NUMERICAL

~~ODELS

Enmg_v balance

For quasi-static crack propagation under fixed-grip conditions (corresponding to the screw-driven Instron testing machine used in the present experiments), the energy balance for a small crack extension can be stated as work of s~puratjo~~equafs r~~~as~d potential energy minus plastic dissipation. Expressing this result in the form of workrates, we have Q = dU”/da - dU’/da.

(1)

An important assumption in this energy balance regards the plastic energy dissipation rate dU*/da as a part of the constitutive description of a material, rather than as a part of a modified surface-energy term. Local plasticity may occur in the immediate vicinity of the crack surface during separation, and this localized dissipation should be included in Q (RICE and DRUCKER, 1967). The energy balance can be modelled numerically with a finite element program (details of which are given by LIGHT (1975)). Previous work (LUXMOORE, LIGHT and EVANS, 1977) has shown that a two-dimensional elastic--plastic program based on incremental (rate) theory of plasticity could predict both the overall and local deformation of static cracks, For crack propagation, however, the oblique fracture

Fracture in aluminium thin plates

111

associated with thin plates could not be modelled in a two-dimensional representation, and in the numerical models, the crack was advanced along the line of the original crack axis. For a symmetrical specimen, the cracks lie on a plane of symmetry, which is taken as one boundary in the numerical model (see Fig. 8). Under load, the solid part of the boundary is restrained from moving by nodal reactive forces. The crack is propagated along the solid boundary, whilst the specimen is under load, by applying external forces equal and opposite to the reactions at successive nodes. The work done by these external forces in cancelling the nodal reactions along any specified crack increment Au, i.e.

i=l

is external work, which, by conservation conditions, gives

of energy for a specimen under fixed-grip

”

iF, S Pidvi = AU’-AUP

= QAa

per unit thickness. This work-term, which is treated as external work by the finite element model, is equivalent to a surface energy term in a real material (i.e. the work done by internal forces being reduced to zero as the crack propagates). The program was modified to allow crack extension after either any specified load increment or any critical value of Q calculated according to (I). Tests were carried out using either one, two or four nodes for each increment of crack extension, the multinode extensions having the cancellation forces applied simultaneously rather than sequentialIy. (The program used parabolic isoparametric etements in all the numerical work, so that one node release corresponded to a crack extension across one half-element, two nodes across one element, and four nodes across two elements.) For identical geometries and loading patterns, the one-node crack extension produced a Q-value lo-20 per cent lower than for either the two-node or four-node extensions, which were virtually identical. The two-node extension was considered to give adequate flexibility, and was used exclusively in all subsequent work.

Ap,

ANO

FORCES

Af’,,i Pn AND

SIMULTANEOUSLY

FIG. 8.

ARE &+I

INCREMENTS WHICH

TO ZERO

OF REACTIVE

ARE

OVER

REDUCED

8 ,NcREMENT!j

Crack advancement, one element at a time.

178

W. T. EVANS,M. F. LIGHTand A. R. LUXMOORE

Applications of the opening forces in toto produced large increases in the nodal residuals after crack extension. This was avoided by applying these forces incrementally. A sequence of 8 increments was found to be satisfactory and this reduction in the residual forces was improved by using large increments initially. The energy balance described above is a simple extension of linear elastic fracture mechanics, and the main assumption is that it is possible to separate the macroscopic plastic flow during crack growth from the local plasticity adjacent to the crack surface, which is included in the crack separation rate Q. After this work was presented for publication, the writers’ attention was drawn to a paper by RICE (1966) in which he used continuum mechanics arguments to show that, for elastic-perfectly plastic material behaviour, the difference between the elastic and plastic energy rutes tended to zero as the crack increment 6a tended to zero. This result must also apply to any materiai where the stress ahead of the crack tip saturates at a finite strain. Hence, a simple parameter based on this difference, such as the parameter Q proposed by the writers, cannot be, by itself, a sufficient fracture criterion. Rice’s (1966) analysis was confirmed by DE KONING (1975) who showed that, for a finite element crack extension model using constant strain triangles, the energy difference must always be linearly related to the crack increment, because of the infinitely sharp crack. RICE (1978) produced a more general version of this result which depended only on the growing crack having an infinitely sharp tip, and he also showed, on the basis of continuum mechanics, that this must be the case even though the initial (static) crack would deform under load to produce some crack tip blunting, corresponding to a finite C.O.D. Because of the success of the finite element model in predicting stable crack growth in dissimilar specimens, described below in Section 5, the writers felt that their numerical results were adequately representing, at least empirically, the fracture behaviour of these particular specimens, despite the theoretical objections discussed above. This feeling was reinforced by a detailed study of the numerical model of the C.N. specimen, where the value of Q was found to be essentially independent of the crack increment &z, at the loads at which crack growth occurred (BLEACKLEYand LUXMOORE, 1978). This result was explained in terms of an effective C.O.D. provided at the crack increment tip by a combination of the large plastic flow under plane stress conditions and the crack increment flexibility. It must be emphasized that the writers do not find this result (which could be an artefact) with all their computations, and other work modelling plane strain elastic-plastic specimen behaviour, using the same finite element program, produces a marked dependence of Q on da, as predicted by RICE (1978). The writers present their results on the use of the Q-parameter for further consideration by other workers, and do not suggest that it is useful for any other fracture situation than the particular one described here.

4.2 Calculation

of J-integral

In a previous investigation (LIGHT and LUXMOORE, 1977) the finite element program was used to evaluate the J-integral (RICE, 1968; see also KNOTT (1973, pp.

Fracture in aluminium thin plates

179

164 et seq.)), where

The validity of the numerical evaluation was tested by modelling the experiments of BEGLEY and LANDES(1972), and good agreement was obtained. The same program was used to evaluate .I, at the initiation of crack growth in the present study (the J-integral becomes meaningless after cracking has initiated, due to unloading of the specimen behind the advancing crack tip).

5.

COMPARISONOF NUMERICAL ANDEXPERIMENTAL RESULTS

5.1 EvaEuation ofQ The numerical models were based on a mesh containing 52 isoparametric parabolic elements (see Fig. 9) which is equivalent to approximately 800 constant strain triangles. The crack was advanced along a line of equal elements 08 mm long (a finer mesh would have involved excessive computer time). Reduced integration of the element stiffness was employed (ZIENKIEWICZ, TAYLORand Too, 1971). The first numerical crack extension was allowed at the displacement point corresponding to the first experimentally observed crack extension. At this stage, the value of Q was calculated from (I) and this value was then used as a critical parameter at subsequent load increments to test for further crack extensions. The load-displacement and load-crack extension curves for L70 are compared in Figs 10 and 11 respectively. Agreement is fairly good, consideiing the difficulties in detecting visually the onset of crack growth. The critical values of Q (denoted Q,) were 18.0 kN mm- * for the C.N. and 19.0 kN mm-’ for the D.E.N. specimen giving an average value of 185 kN mm-‘.

FIG. 9. Mesh used in modelling slow crack growth for L70 C.N. and D.E.N. specimens.

180

W. T. EVANS,M. F. LIGHT and A. R. LUXMOORE EXPERIMENIAL CN

CURVE I

0 FINITE

ELEMENT5

CN

+ FINITE

ELEMENTS

DEN

i

DISPLACEMENT

imm!

FIG. 10. Comparison between experimental and finite element load displacement curves during fracture for both D.E.N. and C.N. specimens of L70 alloy.

The beginning of crack growth could not be determined with sufficient accuracy for the L71 specimens, and so several computations were made with different Q,-values which straddled the most probable starting-point. The load-displacement curves giving the best fit to the experimental data were selected for comparison. Similar agreement was obtained for L71 as for L70, and the critical parameters are summarized in Table 2. If the Qc values are divided by %/z, they are converted to the crack extension force for the true area of the 45” oblique crack rather than the projected area perpendicular to the plane of the plate. The values are 13.0 kN mm-’ for L70, and 11.2 kN mm-’ for L71. These are very close to the typical G,, values for medium to high strength aluminium alloys, i.e. 612 kN mm- 2. There is no direct thick plate specification equivalent to L70, but for L71, the equivalent thick plate specification is DTD 5020 which has a plane strain G,, value of 10.8 kN mm-’ (Mr. C. WHEELER, Materials Department, Royal Aircraft Establishment, Farnborough, private communication, 1974). 5.2 Numerical

modelling of Bradshaw and Wheeler’s (1974) data

BRADSHAW and WHEELER (1974)

have also investigated

the fracture

toughness

of

Fracture

in aluminium

L70

L70

I

I

20

LO TOTAL

FIG. 11. Comparison

of numerical

CN

DEN

0

NUMERICAL

0

EXPERIMENTAL

I 60 CRACK

181

thin plates

I

80 EXTENSION

I 10 0

imml

and experimental values, load vs crack extension specimens of L70 alloy.

for C.N. and D.E.N.

L70 and they kindly allowed the present writers access to detailed information from their unpublished investigations. They used the same thickness sheets (1.6 mm) in different sized specimens based on compact tension (C.T.) and C.N. geometries. For the numerical models, two sizes of the C.T. specimens were chosen, which, in their work, they refer to as CT2-120 and CT2-240. The same numerical crack propagation procedure was used as described previously and the mesh was very similar in distribution to that for the C.N. and D.E.N. geometries, the number of elements being increased to 64. The results are shown in Figs 12 and 13, and the agreement seems closer than that for the C.N. and D.E.N. specimens. The Q,-values were 18.4 and 17.0 kN mme2 for the CT2-120 and CT2-240 specimens respectively, and all the Q,-values are summarized in Table 2.

5.3 Calculation

of J-integral

The J-integral was evaluated for 8 different paths at different distances from the crack tip, as well as for 2 closed loops near the crack tip. No systematic trend could be IO

450

L71

C.N. D.E.N.

25 25

25 2oF, 42~

25 40

C.N. D.E.N.

D.E.N. CT2-120 CT2-240

(mm)

Specimen

2a

Crack length

70.6 86.3 45.9 35.3

61.4 51.6

145.2 128.9

163.2 147.9 89.0 103.3

AUP/Aa (first cracking) (kN mmm2)

AU’/Aa (first cracking) (kN mmA2)

of possible fracture

t Results obtained in present work. $ Results obtained by ALLEN (1975). 9 Crack length measured from load line to crack tip.

312

L70

Material(MN(r&-2)

stress

Yield

TABLE 2. Comparison

10.9 Il.5

13.0 12.0

_ 18.4 17.0 15.5 16.3

12.7 13.4

,5Q (k$mi-*)

18.0 19.0

QE

(kN mm-*)

-

-

46.4 48%

34.3 32.9

49.0 -

50.8 64.8

45.5 63.6

JC+’ (kN mmm2)

(kN mmm2)

JCt

parameters for L70 and L71 (for unit thickness)

0.08

0.16

6, (mm)

+c

36.0

49.9

(kN mmm2)

Fracture

in aluminium

thin plates

183

20

NUMERICAL

RESULTS

z 1 9 s

Za mm

I’

IO

1’

L1 5

2’

L2 51 16 0

3’ 4’

50 2 53

5’ 6’

I 565

/ FlNlTE

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FIG. 12a. A comparison

of numerical and experimental stable crack growth WHEELER, 1974, CT2-120 specimen).

10

SPECIMEN

s r

values

(BRADSHAWand

CT2-120

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BRADSHAW

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NUMERICAL

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WHEELER

119741

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9

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FIG. 12b. A comparison

of numerical and experimental stable crack growth WHEELER, 1974, CT2-240 specimen).

values (BRADSHAWand

184

W. T. EVANS,M. F. LIGHTand A. R. LUXMOORE

L. 0

NUMERICAL

0

EXPERIMENTAL

i

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5 CRACK

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EXTENSION

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FIG. 13. Comparison of numerical and experimental load vs crack extension (BRADSHAW and WHEELER. 1974, compact tension specimens, L70 alloy).

seen in the results for different paths, but the average deviation from the mean was +5 per cent, or f3 kN mm-‘, compared with a maximum J-value from the closed loop (which should be zero) of -03 kN mm- 2. If the values of the closed loop are used as an indication of the numerical errors, then the J-values vary by a significant amount. For L70, the J,-values (Table 2) are reasonably constant for the two C.T. specimens and one C.N. specimen, but there is a considerable increase for the D.E.N. specimen. Doubt was cast on the validity of the D.E.N. result, especially as it was difficult to determine the beginning of crack growth precisely, but it was also noted that this specimen had a different total crack length compared with the C.N. specimen (Table 2). Both experimental and numerical work was repeated for new specimens of L70 and L71 by ALLEN (1975). He used a total crack length of 25 mm in each specimen, but he also studied an L70 D.E.N. specimen with a 40 mm crack length. His results (Table 2) indicate that the odd result for the L70 D.E.N. specimens is connected with the variation in crack length, and not with uncertainties in the beginning of crack growth (errors in this point could not give such a large variation in J-values).

Fracture in aluminium thin plates

185

For the L71 specimens, Allen obtained consistent J,-values which were significantly lower than those for L70, This lower value correlated with the smaller amount of yielding which occurred in L71 prior to fracture. The Dugdale model predicts that J, = (T,,&, and these values are included in Table 2 for comparison. They agree quite well with the finite element values of J,, apart from the odd value obtained from the L70 D.E.N. specimen with the 40 mm crack length. This anomaly is puzzling, although it is possible that the same error has been made twice for this crack length. More probably, the longer crack length produces conditions for which the J-integral is no longer valid. The J,-values deviated quite significantly from the Qc-values, as found by BOYLE (1972), and they do not correspond with any of the other energy parameters in Table 2. The relation between J and Q is undoubtedly complex, but as the extent of yielding at fracture is decreased, they will tend to the same value (equivalent to G, for the purely elastic case). 5.4 Comparison of numerical and experimental R-curves

The values of dU’/da and dUP/da obtained from the program are a direct measure of the “fracture resistance”, and have been plotted against crack extension for comparison with the ex~rimentally determined R-curves for L70 (see Fig. 14). Values have only been taken up to the point of maximum load on each specimen, as it is difficult to assess the accuracy of the numerical values past this point (after maximum load, the crack kept advancing in the numerical model after only one or two further increments of load, and the process became very dependent on the size of the load increment, which was chosen arbitrarily). Though the curves agree closely for the first millimetre of crack growth, large discrepancies occur with subsequent growth. The two C.T. specimens produce the expected physical behaviour, but the C.N. specimen behaves differently. This was associated with general yield occurring in this specimen during slow crack growth, and BRADSHAW and WHEELER(1974) also report this phenomenon for their C.N. specimens. The values of G, (the R-value at maximum load) from the numerical and experimental calculations are compared in Table 3. Reasonable agreement is obtained for the C.T. specimen, but there is a wide discrepancy for the C.N. and D.E.N. specimens. As pointed out by Bradshaw and Wheeler, where failure occurs after the net section stress has exceeded the yield stress (which was the case for the C.N. and D.E.N. specimens), G,-values predicted from resistance curve analysis will be seriously underestimated. This is due to the failure of the elastic correction factor to account adequately for general yield conditions.

6.

DISCUSSION

Experimental investigations have shown that the strain distribution ahead of the crack tip can be divided into at least two regions, each with a distinctive straindistance relation of the type, E = Cr”. For both regions, specific relationships were

W. T. EVANS,M. F. LIGHTand A. R. LUXMWRE

186

--

-

R-CURVE ELASTIC

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120

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FIG. 14. Energy rates vs crack extension (L70 alloy).

TABLE

3. Co~~~risofz

of energy

L70 120 BRADSHAW and WHEELER (1974) C.T. 240 BRADSHAW and WHEELER (1974) C.N. BRADSHAW and WHEELER (1974) C.N.

CT.

D.E.N. t Present work

release rates from calcufations

expey~~nt

Experimental elastic energy release rate at instability, G, (kN mm-*)

and unite

Numerical elastic energy release rate at instability, G, (kN mm-‘)t

151

127

244

171

120 122

131

element

Fracture in aluminium thin plates

187

found between the constant C and load during slow crack growth and these relationships appeared independent of geometry and, possibly, material. This result was taken as indicating a unique strain distribution ahead of the crack tip in the elastic-plastic stress field, analogous to the purely elastic situation. This work supports KE and LIU’S (1973) contention that a critical strain value can be used as a fracture parameter. The experimental results also show that the C.O.D. values at initiation are independent of geometry, but not of material, which does not correlate with the strain measurements. However, strain measurements were only taken to within 0.55 mm of the crack tip, and no measurements could be made of the large gradients which exist close to the tip. The numerical crack growth model could not represent the 45” through thickness slope of the growing crack surface, and there are serious theoretical objections to a simple one-parameter energy balance model (Section 4.1). Despite these drawbacks, the quasi-static parameter Q could be used to predict the important features of stable crack growth in the four specimens studied. The success of the numerical fo~ulation can be attributed, at least in part, to two factors. First, the extensive plastic flow at crack propagation allows the greater part of the work done to be calculated from regions away from the crack tip, where the coarse mesh is known not to model the deformation accurately (an attempt to compare experimental and numerical strains near the crack tip was unsuccessful because of the coarseness of the finite element mesh used for the crack propagation study, though previous work (LUXMOORE,LIGHT and EVANS, 1977) had shown that a finer mesh will give realistic results). Secondly, the values dU’/da and dUP/da are calculated from the difference in elastic and plastic energies before and after crack extension. If the numerical model is too stiff (and stiffness increases with coarseness of mesh), then the same stiffness applies before and after crack extension, and though the values of dreads and dUp/da are underestimated, their difference will be much less affected. In addition, it is generally accepted that finite element techniques can produce energy values more accurately than stress and strain values, because of the averaging effect when computing the energy over an element. No comparison was made between numerical and experimental C.O.D.% for the reasons given above. The occurrence of general yield during slow crack growth invalidated the experimental R-curves, and this was confirmed by the numerical results, which predicted much higher plastic work terms than those obtained from the pseudoelastic analysis. In these circumstances, the pseudo-elastic theory will not predict crack instability correctly. The J-values were reasonably constant for each material (with the exception of the anomalous L70 D.E.N. specimen), but they did not correlate with the Q,-values, although they confirmed the Dugdale-model relation J, = oYSc. The deformation theory of plasticity, on which J is based, cannot model crack extension in elasticplastic fields correctly because of unloading behind the crack tip, although it may provide an adequate model for the crack-tip stress field just prior to fracture. The uniqueness of the stress and strain fields, indicated by the strain measurements, is in agreement with the contention that J can be used to predict the onset of crack growth because it characterizes the crack tip field. However, the strain measurements also

188

W. T. EVANS,M. F. LIGHTand A. R. LUXM~~RE

indicated a unique field ahead of the growing crack, which suggests that J might be a suitable parameter for predicting stable crack growth, and hence may be used to predict cracking instability. This value of J would still be load path-dependent, and necessitate calculation by incremental theory.

7.

CONCLUSIONS

For the materials considered, the energy balance provides the most satisfactory explanation of slow crack growth followed by unstable fracture, and gives a simple explanation in terms of a parameter similar in magnitude to the linear elastic fracture parameter. The general form of the R-curve can also be explained by this model, but the numerical work was not sufficiently precise to allow comparison of strains and C.O.D.‘s. An apparent anomaly occurred when using the J-integral in connection with large-scale yielding, and further work is needed in this area.

ACKNOWLEDGEMENTS The writers are indebted to the Science Research Council (U.K.) for financial support, and also to the late Dr. F. J. BRADSHAWand Mr. C. WHEELER (Materials Department, Royal Aircraft Establishment, Farnborough) for useful discussions and free access to their very comprehensive unpublished experimental data.

REFERENCES ALLEN, N. G.

1975

BEGLEY, R. and LANDES, J. D.

1972

BLEACKLEY,M. H. and LUXMOORE,A. R.

1978

BOYLE, E. F.

1972

BRADSHAW, F. J., LACEY, D. and WHEELER, C.

1972

BRADSHAW, F. J. and WHEELER, C.

1974

A Numerical and Experimental Investigation of Stresses and Strains around Crack Tips in High Strength Aluminium Alloys. M.Sc. Dissertation. University of Wales. Fracture Toughness (Proceedings of the 1971 National Symposium on Fracture Mechanics, Part II), ASTM STP 514, p. 1. Proceedings of First International Conference on Numerical Methods in Fracture Mechanics (Swansea. 9-13 January, 1978), (edited by A. R. Luxmoore and P. R. J. Owen), p. 508. University College of Swansea, Wales. The Calculation of Elastic and Plastic Crack Ph.D. Dissertation. E.utension Forces. Queen’s University, Belfast. The Effect of Section Thickness on the Crack Resistance of Aluminium Alloy L93. with Measurements of Plastic Strain. Tech. Report No. 72039, Materials Department, Royal Aircraft Establishment, Farnborough. The Crack Resistance of Some Aluminium Alloys and the Prediction of Thin Section Failure. Tech. Report No. 73 191, Materials Department, Royal Aircraft Establishment, Farnborough.

Fracture in aluminium thin plates DE KONING, A. U.

1975

DUGDALE,D. S. EVANS,W. T.

1960 197.5

GAVIGAN,W. J., KE, J. S. and LIU, H. W. HUTCHINSON,J. W. KE, J. S. and LIU, H. W. KNOTT, J. F.

1973

KOBAYASHI,A. S., ERGASTRON,W. L. and SIMON,B. R. KRAFFT, J. M., SULLIVAN,A. M. and BOYLE,R. W.

1969

1968 1973 1973

1961

LIGHT, M. F.

1975

LIGHT, M. F. and LUXMOORE, A. R. LIU, H. W.

1977

hJXMOORE,

1977

A. R., LIGHT, M. F. and EVANS,W. T. RICE, J. R.

1961

A Contribution to the Analysis of Slow Stable Crack Growth. Report NLR MP 75035, National Aerospace Laboratory, Amsterdam, The Netherlands. J. Mech. Phys. Solidr 8, 100. An Experimental Study of Fracture Criteria under Plane Strain and Plane Stress Conditions. Ph.D. Dissertation. Council for National Academic Awards, U.K. Int, J. Fract. 9, 255. J. Mech. Phys. Solids 16, 13. Eng. Fract. Mech. 5, 65. Fundamentals of Fracture Mechanics. worths, London. Exp. Mech. 9, 163.

Butter-

Proceedings of the Crack Propagation Symposium (Cranfield. September 1961), Vol. 1, p. 8. College of Aeronautics (now Cranfield Institute of Technology), Cranfield. A Numerical investigation of Post- Yield Fracture. Ph.D. Dissertation. University of Wales. J. Strain Anal. 12, 306. Proceedings of the Crack Propagation Symposium (Cranfield. September 1961), Vol. 2, p. 514. College of Aeronautics (now Cranfield Institute of Technology), Cranfield. J. Strain Anal. 12, 208.

1967

Proceedings of the First International Conference on Fracture (Sendai. 12-17 September 1965), (edited by T. Yokobori, T. Kawasaki and J. L. Swedlow), Vol. 1, p. 309. The Japanese Society for Strength and Fracture of Materials, Tokyo. Trans ASME 90, Series E, J. app. Mech. 35, 379. Proceedings of First International Conference on Numerical Methods in Fracture Mechanics (Swansea. 9-13 January 1978), (edited by A. R. Luxmoore and P. R. J. Owen), p. 434. University College of Swansea, Wales. int. J. Fract. Mech. 3, 19.

1971

Exp. Mech. 11, 1.

1966

1968 1978

RICE, J. R. and DRUCKER,D. C. SCHAEFFER,B. J., LIU, H. W and KE, J. S. ZIENKIEWICZ,0. C., TAYUIR, R. L. and Too, J. M.

189

Int. J. Num. Methods 3, 275.