On the relationship between ductility and fracture toughness in an Al-6.0% Zn-2.5% Mg alloy

Acta .~etallurgica Vol. 29. p p 229 1o 239 Pergamon Press Lid 1981 Printed in Great Britain

O N THE R E L A T I O N S H I P B E T W E E N D U C T I L I T Y A N D F R A C T U R E T O U G H N E S S IN A N A1-6.00oZn-2.5°/0 Mg A L L O Y T A K E S H I K A W A B A T A and O S A M U I Z U M I The Research Institute for Iron. Steel and Other Metals. Tohoku Universit.~. qendzi

QRO

tartan

(Received 24 Jammry 1980) Abstract--The fracture toughness (K~c) of an AI-6.00oZn-2.5°o Mg allo3 heat-treated variously was examined in relation to tensile properties using the notched and unnotched specimens. As usual trends. the fracture strain (ef) showed minimum at the peaks of yield stress {ao.2) and ultimate tensile stress (trt:xs). The K~c and ~: when aged isothermally varied similarly, but when isochronall.', aged Ktc and Cro.2 (and arts) showed a similar trend. An intimate relationship was found between the dimple size at the portion fractured intergranutarly and the fracture strain (~f). An equation showing the relation between K~c and tensile properties such as ao.2. ~trs (the strain up to ultimate tensile stress) and n tthe work hardening parameter) could be introduced, based on a new model of the plastic zone. The model was verified experimentally. R6~am~-Nous avons 6tudi+ la resistance a la rupture tK~c) d'un alliage AI 6,0°0 Zn-2,50o Mg ayant subi divers traitements thermiques, en relation avec ses proprietes en traction en utilisant des 6prouvettes entaill6es et non entaill6es. D'une mani/~re gen6rale, la d6formation /l la rupture (~:) presentait un minimum pour les pics de la limite 61astique lao.2) et de la contrainte finale en traction (acv'r). Darts le cas de revenus isothermes. Kic et E: variaient de maniere analogue, alors que dans le cas de revenus isochrones ce sont K.c et ao.2 ~et acw) qui presentaient les m~mes caracteristiques principales. Nous avons trouv6 une relation etroite entre la faille des rides sur la r6gion de rupture intergranulaire et la d6formation/~ la rupture (e:). Nous avons pu introduire une ~quation montrant la relation entre K~c et des propri6t~s/a la traction telles que ao,,, acvr (la deformation ~t la contrainte de traction finale); cette 6quation repose sur un nouveau mod+le de la zone plastique. Nous avons verifie experimentalement ce module. Zusammenfassung--Die Bruchz~ihigkeit Kic der w~.rmebehandelten Legierung AI-6,0°,oZn-2,5°:O Mg wurde im Hinblick auf das Zugverhalten mittels gekerbter und ungekerbter Proben untersucht. Wie tiblich wies die Bruchdehnung ~: ein Minimum bei den Maxima yon FlieBspannung ao,2 und Zerreil3spannung trvrsauf. Nach isothermer Auslagerung ~inderten sich Ktc unde.: ~.hnlich, jedoch nach isochroner Auslagerung zeigten Ktc und ao.., {und tre~s) ~ihnliche Tendenz. Es wurde ein enger Zusammenhang zwischen Grtibchengr613e an der intergranularen Bruchfl/iche und Bruchdehnung aufgefunden. Aufbauend auf einem neuen Modell der plastischen Zone konnte eine Gleichung eingefiihrt werden, die den Zusammenhang zwischen K~c und den Zugeigenschaften -beschrieben etwa durch ao.2, ~t:rs (Zerrei[~dehnung) und n [Verfestigungsparametert-erfaBt. Dieses Modell wurde experimentell erh~rtet. 1, I N T R O D U C T I O N It is very useful industrially if the toughness of material can easily be deduced from the tensile test using u n n o t c h e d specimens. The relationship between fracture toughness (Klc) in notched tensile test and ductility in unnotched ones has been discussed by Krafft [1,2], Tetelman and McEvily, Jr [3]. H a h n and Rosenfield [4. 5], and Peel and Forsyth [6]. Their proposed theories are shown in Table 1. The first hypothesis proposed by Krafft (equation (2T) in Table 1) was corrected to equation (3T). Peel and Forsyth have also presented a model, similarly to the Krafft model, in which a cylindrical plastic zone was assumed to be formed at a crack tip. The model stands on the strain in the plastic zone to be uniform, the crack to develop when • = • : . where e: is the fracture strain and the stress-strain curve to follow the Ludwik's law. i.e., a = aoe". where ao is the value of tr at • = 1, n the work hardening parameter. They confirmed equation (7T) experimentally for three kinds of A I - Z n - M g alloy except failing to explain systematically an impurit? effect. 229

Both theories either by Krafft or by Peel and Forsyth assumed that the strain in the plastic zone is uniform and a large strain occurs in whole plastic zone when E1 = n (Krafft) or • = ~: (Peel and Forsyth). However, in the actual fracture of the notched specimen of A I - Z n - M g alloys the large strain occurs only in the interior of precipitate free zone fractured intergranularly and in the grain fractured transgranularly. The strain in the remainder of the plastic zone should be smaller than that predicted by Krafft and by Peel a n d Forsyth. This view is supported by the optical micrographs of Fig. 1 showing the structures near the fracture surface and by the transmission electron micrographs of Fig. 5 in the Ref. [7]. Since grains are deformed by coarse slip b a n d s spaced widely [7], the average strain in the deformed grain would not be so large. Therefore, for deducing the relationship between ductility and fracture toughness, the real strain distribution in the plastic zone must be considered. The purpose of the present work was to investigate the relationship between fracture toughness a n d ten-

230

KAWABATAandlZUMI:

DUCTILITY AND FRACTURE TOUGHNESS

Table I. Theories having been proposed on the relationship between K~c and ductilit3 Author Krafft

Year 1964 1968

Criterion between KK- and strain K l c = El "El2rtd, I1 2 where ~ = n

(IT) 12T) ~3T)

~ = n 2 + arE

Tetelman and McEvily. Jr.

1967

Ktc = [ E ' 2 a r p ¢ / ( c j l l

Hahn and Rosenfield

1968

K1¢ = [(2 3)EG.l*q] ~ "¢;Ic} -~ 11 3)E~,

I5TI 16T)

Peel and Fors3th

1973

K~c(l - v 2) E

(7Tj

-

- v2j] ~ 2

droo~)) -m

I + n

sile properties using the notched and unnotched specimens of AI-6.0°;, Zn-2.5°o Mg alloy heat-treated variously. 2.

EXPERIMENTAL PROCEDURE

The chemical composition of the specimen used in the present study was Z n : 6.00o, Mg: 2.5°~, impurity elements Fe, Si, Cu < 0.01°o (weight per cent), respectively. The size of the gage part of the unnotched tensile specimen was 4 m m in diameter and 40 m m in length. The notched specimen had a nominal sectional dimension of 40 m m in width and 15 m m in thick-

[4T)

Nomenclature E: the Young's modulus n: the parameter of strain hardening d,: the diameter of lhc plastic zone a~: the tensile )ield stress v: the Poisson's ratio Er : the true fracture strain Etlcl: the ductilit3 of the specimen at the crack tip p: the effective radius of the crack tip I*: the length of plastic zone at the onset of cracking

ness with a notch and fatigue crack of ~ 13 mm in length, and with the grooves of 2.5 m m in depth on both the surfaces of the specimen. The notched and unnotched specimens were solution-treated at 733 K for 10.8ks and interrupt-quenched at 6 1 3 K for 0 ~ 300 s then quenched into iced water, and further pre-aged at 373 K for 0 ~ 300 ks then subsequentt3 aged at 433 K for 3.6 ~ 36 ks. Tensile test of the unnotched specimens was done using an lnstron-type testing machine (TOM-5000) at the initial strain rate of 2 × 10-'*s -~ at room temperature. The notched specimens were pulled using the electric tube type universal testing machine at the loading rate of 33 M P a . s - a . 3. E X P E R I M E N T A L R E S U L T S 3.1 C h a n g e in m e c h a n i c a l p r o p e r t i e s with a~ling period

Figure 2 shows the change in mechanical properties with aging period where Kw is the fracture toughness. arTS the ultimate tensile stress, ao2 the 0.2°o proof stress and EI the fracture strain when single-step aged at 433 K after solution-treated at 733 K and quenched into iced-water. The Ktc value reaches the peak at 3.6ks then decreases, and after 180ks it increases again. O n the other hand. acts and a0.2 show the peaks between 14.4 ~ 36 ks. Ej- is lowest at the peak of strength.

¢,/~'

3.2 Effect o f pre-agin9 on m e c h a n i c a l p r o p e r t i e s ,

, ~ °%

.

:,

:

Fig. 1. Optical micrographs showing the feature of deformation observed near fracture surface of the notched specimen solution-treated at 733K for 3.6ks, iced-water quenched and aged at 433 K for 14.4ks in an AI-6.0°o Zn-1.5°o Mg alloy.

Figure 3 shows the mechanical properties of specimens which were pre-aged at 373 K for 3' s and subsequently aged at 433 K to give the highest hardness after solution-treated at 733K for 3.6 ks then quenched into iced water. Although the value of Kjc does not vary so much with the pre-aging period, aLrs and a0. 2 increase obviously at the region y > 3 ks. ¢/ shows the opposite tendency to the strength, i.e.. the brittleness appears at the highest strength. 3.3 Effect o f the i n t e r r u p t - q u e n c h i n 9 properties

on m e c h a n i c a l

Figure 4 shows the mechanical properties when the specimen was aged at 437 K for 14.4 ks after solution-

K A W A B A T A and I Z U M I :

DUCTILITY AND FRACTURE TOUGHNESS ISOTHERMAL AGING

80

l

1

I

I

S.T ; 733 K,3.6x I03 s

/////// KIc~

?0

'E 60 Z

3E v50 u

I

d" : 53.8 m~

- 400

o"

:l¢

40 3OO

30

D 'E

20

Z

:E

/ /

200

/ /

/

0.3

/

I00

0-2 Ef 0.1 I

0

I

0

I

,

lo6

103 104 105 AGING PERIOD AT 4 3 3 K ( s )

Fig. 2. Changes in mechanical properties (K=c. o'u'rs,ao.2 and e•) with aging period at 433 K.

70

ISOTHERMAL TWO-STEP I .... I I

i

S.T. : 733K, 3.6x103 s

AGING I 1

1

d'~ = 53.8 m'~

~60 'E

z=1.44 xlO~ S

z 50

7.2x103

v

1-26x10~

u_ 40

1.26x1~

7.2x103

- 500

30

~UTf5 / ~ -

z -

~/,,////

400 :Z

/ //

-

~o;

~

-

d - 300

0.1 Ef

0

2O0 ,|.

0

I

I

!

I

102 103 10~ 105 PRE-AGING PERIOD AT 373K ( s )

Fig. 3. Changes in mechanical properties (K:¢, (~CTS,(~o.2 and Et) with pre-aging period at 373 K.

231

232

KAWABATA and IZUMI: DUCTILITY AND FRACTURE TOUGHNESS 90

I

I

I

I

I

SIT:733K 108x10~or 1.44x10 ~$

,:o z= 1.44x104 s

.E z

/ \ K1c

/

60-

=T SO / 40

400 Z~

30-

"x

n,\ %\ \ -

N

E z 3E

~u,s "1~-"'--

300

-"-'a"-

0.2

- 200 Ef

E~ 0.1 0

~ I

"

~ I

~ I

-

I

I

0 3 30 300 3000 INTERRUPTED PERIOD AT 613K, x (s)

Fig. 4. Mechanical properties vs interrupted period at 613 K. Specimens were solution treated at 733 K for 10.8 ~ 14.4 ks, interrupt-quenched at 613 K for X s, subsequently iced-water quenched and aged at 433K for 14.4ks.

treated at 733 K for 10.8 ks or 14.4 ks and interruptquenched at 613 K for x s. K~c shows a maximum value at x = 3 s then decreases with x. The values aurs and ao.2 of the specimens interrupt-quenched up to x = 3 ks are lower than those of the specimens directly quenched (x = 0). 3.4 Mechanical properties when isochronally aged Figure 5 shows the mechanical properties of unnotched and notched specimens isochronally aged at 293 ~ 573K for 14.4ks after solution-treated at 733 K for 1.08 ks and quenched into iced water. After the K~c value reaches the peak at 373 K, then decreases with the increase of temperature. The values of OUTSand oo.z have the maxima at about 400 K, at which, correspondingly, ej- becomes lowest. 3.5 Fractography Figures 6 A, B and C are the examples of the fracture surfaces of specimens applied usual single step aging after the interrupt-quenching at 613 K for 3, 30 and 3000s, respectively. With increasing the interrupt-quenching period, the size of grain boundary precipitate becomes large and the large precipitates fractured are observed as shown in Fig. 6 C. Most of the precipitates are fractured and a part of them remains on the fracture surface. Therefore, the bonding of the interface between the precipitate and the

matrix seems to be rather good, and the crack may develop in consequence of the initial fracturing of the grain boundary precipitates followed by the fracturing of the interior of the precipitate free zone between those fractured precipitates. 3.6 Effect of pre-aging on fracto#raphy Figure 7 A, B and C are the scanning electron micrographs of the fracture surface of the largenotched specimen pre-aged at 373 K for y = 0, 3 and 300 ks, respectively. The dimple size decreases with increasing y. This trend is more clearly observable at low magnification as shown in Figs 7 D, E and F, corresponding to Figs 7 A, B and C, respectively. By two-step aging, the roughness of the surface fractured intergranularly becomes small and the area fraction of the region fractured transgranularly decreases. 3.7 Fracture surface when single step aged at 433 K Figures 8 A, B and C are the scanning electron micrographs showing the fracture surfaces of the specimens solution treated and aged at 433 K for 36 ks and for 360 ks, respectively. Figure 8 A is the transgranularly fractured surface consisting of highly ductile dimples. The observation with lower magnification shows the partial occurrence of the intergranular fracture.

KAWABATAandlZUMI: DUCTILITY AND FRACTURE TOUGHNESS I5OCHRONAL 70

1

1

AGING I

I

S.T.: 733K, 1-08xlO4s d ~ = 50.7rn'~

60 -

233

-

'E 50 :E 40 u

400

3O

300~ 200

o"

0.2

g ff

~

loo

0.1

o

L

,

,

AGING TEMPERATURE

' 600

,o;o °

( K )

Fig. 5. Changes in mechanical properties of notched and unnotched specimens with isochronal aging temperature. Specimens were solution-treated at 733 K for 10.8 ks, iced-water quenched then aged isochronally for 14.4 ks. 4. DISCUSSION 4.1 The relationship between the peak of Klc and fro.2, aurs and Ef As shown in Figs 2.3, 4 and 5, the peak of Klc does not coinside with the peak of ao,2 and (rUTS. This trend would be owing to EI being small at the peak of strength, K~c is a measure expressing the fracture energy. Therefore, Kjc relates not only to the strength level but also to the strain to fracture.

4.2 The relation of the size of plastic zone at a crack front and ductility to fracture toughness A schematic figure of plastic zone at a crack front has been shown in Fig, 9 of the Ref. [8].

The plastic strain in a plastic zone would be large at the crack front then decrease to zero at the end of the plastic zone. The level of stress at a given portion in the plastic zone would also be related to the plastic strain at the portion. In the preceding investigation [8], the plastic zone was divided into three portions corresponding to the strain, i.e., the strongly deformed layer of the portion fractured transgranularly, the interior of precipitate free zone at the portion fractured intergranularly and the remain deformed weakly in the plastic zone. The equation (16) in the Ref. [8], showing the relationship between the fracture toughness and the area fraction of transgranularly fractured part, represents that the

Fig. 6. Scanning electron micrographs. Changes in the feature of fracture surface with interrupt quenching period. Specimens were solution treated at 733 K for 10.8 ~ 14.4 ks, interrupt quenched at 613 K for (A) 3, (B) 30 and (C) 3000 s. subsequently iced water quenched and aged at 433 K for 14.4 ks.

234

KAWABATAandlZUMI:

DUCTILITY AND FRACTURE TOUGHNESS

m

Fig. 7. Scanning electron micrographs showing the effect of pre-aging period on the feature of fracture surface• Specimens were solution treated at 733 K for 3.6 ks. iced-water quenched, pre-aged at 373 K for (A and D: high and low magnification, respectivelyt 0. (B and El 3 and IC and FI 300 ks. subsequently aged at 433 K for 7.2--14.4 ks to the peak hardness. plastic strain energy in the interior of precipitate free zone at the portion fractured intergranularly is negligibly small compared with the plastic strain energy at the other portions. That is. the ratio of the plastic strain energy in the other portions to the whole fracturing energy is comparatively large because the plastic strain energy per unit volume is small but the volume fraction is large. From the above mentioned, it should be emphasized that the size of plastic zone and the distribution of the energy density of the plastic strain are in due consideration of discussing the fracture energy or the fracture toughness•

Then. we put some simple assumptions to the shape of plastic zone. the distribution of plastic strain and the relationship between the plastic strain and the stress. Based on those assumptions the relationship between the fracture toughness and the size of plastic zone can be discussed in relation to the plastic properties of the alloy. When a load is applied to a tensile specimen with a notch (containing a fatigue crack), the plastic zone at the notch front will be formed in accordance with the applied load. Then it is assumed that the crack will grow under the following conditions:

Fig. 8. Scanning electron micrographs showing the effect of aging period on the teature of fracture surface, Specimens were solution-treated at 733 K for 3.6 ks, iced-water quenched and aged at 433 K for [A) 0. (B) 36 and [CI 360 ks.

KAWABATA and IZUMI:

DUCTILITY AND FRACTURE TOUGHNESS

235

CRACK

\

v[.

-i (a)

(b)

Fig. 9. A schematic figure of plastic zone at a crack front used for calculation. (l) the shape of plastic zone is cylindrical with the radius r v centering around the notch front,t (2) the plastic strain in the plastic zone decreases linearly from the maximum value of ep at the notch front to zero at the end of radius rv,~ (3) when the plastic strain at the notch front reaches the critical value Ev = ec, the crack begins to grow, then the radius of the plastic zone reaches rv = re,§ ~"The shape of plastic zone is not cylindrical actually. However, the plastic work, when a crack propagates a unit length, depends on the distribution of strain to the y direction in Fig. 9. Therefore, the assumption (l) does not affect the result. The plastic work in the plastic zone has not been calculated precisely, because the conditions of stress and strain in the plastic zone are tri-axial and very complex. Furthermore, the histories of stress and strain are not simple increasing, i.e., both processes of increasing and decreasing are mixed. However, as the first approximation only the increasing process should be considered and the profile of strain distribution is important. In the present qudy, for the simplicity the strain distribution is assumed as a linear function of radius, though it is more suitable that the strain distribution should be a quadratic function. The strain in the plastic zone is tri-axial so that the equivalent strain should be considered. § When ductile fracture is the process composed of formation and coalesence of voids, Ec is affected by the level of triaxial stress (i.e., the hydrostatic tensile stress). However, the level of the hydrostatic tensile stress at the front of crack would he not so different from the hydrostatic tensile stress at fracturing portion in the uniaxial tension. HStrictly speaking, the profile of strain distribution in a specimen with finite width would vary with developing of crack. We consider here the profiles of strain distribution before and after the crack developed by a very short distance. ¶ Ludwik's law is usually applicable to uniaxial tensile test. However, in the deformation under the triaxial stress and strain, Ludwik's law can also be applied to the octahedral shearing stress and the octahedral unit shear [9]. t t We consider a case that the size of plastic zone is larger than the grain size (e.g., when 2r > 3d, where r is the radius of the plastic zone and d the grain size). The plastic constrain between grains in the specimen under uniaxial tension will be rather strong when there are a large number of grains in the section of the specimen. Therefore, the plastic constrain in the plastic zone surrounded by elastic region should be approximately the same order as that at the center of the section in uniaxial tension.

(4) before and after the initiation of crack growth, there is no difference in the profile that the plastic strain decreases linearly with the distance from the notch (or crack) front, ll (5) the stress and strain in whole part at the inside of the plastic zone follow the Ludwik's law, i.e. = aoE"

(l)

where n is the strain hardening parameter, txo the stress at the true strain of unity, tr the true flow stress under uniaxial tension,¶ (6) the plastic properties at the local part of the crack front obey the macroscopic plastic properties of the unnotched tensile test.tt Fracture toughness for plane strain, Kjc is related to the critical strain energy release rate for unstable crack extension, G~c as follows, =r K,c

G'cE l'/2 L1_--Z-~. ]

.

(2)

Then, we divide the plastic strain energy for propagating the crack into two parts: (1) the plastic strain energy W~ before the first fracturing begins at the front of the fatigue crack (the point A in Fig. 9 a), (2) the additive plastic strain energy W~ which is necessary for propagating the crack for a distance of the unit length. W~vis expressed as follows (see Fig. 9)

W~ =

2~rE(r)dr

(3)

where E(r) is a function of the plastic strain energy per unit volume at the point r, i.e., the energy density, which is given as E(r)

=

f~'~ a(r) de.

(4)

F r o m equations (1) and (3), E(r)

-

ao [~(r)]" + i

n+1

(4')

236

KAWABATA and IZUMI:

DUCTILITY AND FRACTURE TOUGHNESS

where E(r) is the plastic strain at the point r near the notch front. Substituting equation (4)' to (3), 2ttao f , Wv = - n - - ~ Jo" r[E(r)]"+l dr i

(5)

treatments, i.e., cc is proportional to r~,

(assumption (2)).

(6)

( , ).+, dr

(I 3)

m-

n+2 2

2Err ° ~ k (7)

(14)

=

"]1 ~2

(1 - v2)(n + 1)(n + 2)J

(8)

The plastic strain energy Wv which is necessary for propagating the crack by a unit length at the beginning is as follows,

Wv = W~ + W~.

(9)

The plastic strain energy W~ which is necessary for propagating by the further unit length is

w ; = w~.

(10)

The critical release rate of strain-energy for unstable crack extension, G~c is equal to W~,, i.e., W~, and putting the right sides of equations (2) and (8) being equal, the following relationship is obtained.

2Eao ] 1/2 ~'(a+1)/2 wl/2 (11) (1 - v2)(n + 1)(n + 2 ) J - ' "

where E, is the true strain at the maximum load. Then assuming that in the same material, the value of e,/r, is approximately constant under various heat I

I

I

I lli

i

4O

4.3. The comparison and verification of the calculated curve and the experimental data Figure 10 shows the relationship between K~c and the true strain at the ultimate tensile stress %TS obtained from the tensile tests using the large notched and the unnotched specimens, respectively. The curve A is of the two step aging. The plots marked by the arrow show the data on the specimens fractured without any local necking when the load is increasing. This brittle behavior might be owing to the large grain size, i.e., the sectional area containing only scores of grains. In the larger grained specimen, the constraint between grains is weaker than in the specimen containing the greater number of smaller grains in the same sectional area, i.e., the larger the grain is, the larger the grain boundary sliding becomes. (As described elsewhere [10], the sliding at the middle part of grain boundary will be proportional to the length of grain boundary.) Therefore, when the fracture occurred, after the local necking developed and the load decreased gently, the measured %TS would shift I

!

I

i

i I ill

I

I

~

I

I-1

A __.O.---~F

30 2O

10

I 'lO -3

!

l

| llllJ

(13')

That is, the Kmc vs e, relationship on the logarithmic graph becomes linear.

" " 100

'E 80 ~,, 60 x

(15)

or

lnK.c = Ink + mln~,.

2O'oE~+ I r, (n + 1)(n + 2)"

I

key

and

Then the plastic strain energy W~ is

Kjc =

=

r -- -+ 1 gr

2 naoE~ +~ r,2 = (n + 1)(n + 2)(n + 3)'

W~ =

Kic where

Equations (5) and (6) yield n-+l

(12)

we get

where E(r) is assumed as

E(r) = c~ - -rc + 1

1 E, = ~ r,,

l 10-2

I

J

i llllJ 10-1

EUTS

Fig. 10. Relationship between K~c and the true strain at the ultimate tensile stress. The curves A and B are of two step aged- and interrupt quenched-specimens, respectively.

KAWABATA and IZUMI: DUCTILITY AND FRACTURE TOUGHNESS to the larger side of E than the position shown in Fig. 10. The curve B is of the plots of specimens applied the interrupt-quenching treatment (x = 3 ~ 30 s, y = 0, z = 3.6 ~ 36 ks). The whole specimens in this group fractured after the local necking developed. Therefore when verifying the relationship between the K~c value and Ctrrs the curve B shows the more accurate relationship than that of the curve A. Then the reasonableness of equation (13) will be verified, using the parameters decided from the curve B and the properties obtained from the tensile test on the unnotched round specimens. We will study on the specimen after solution treated at 733K for 14.4ks interrupt-quenched at 613 K for 3 s, quenched into iced water, then aged at 433 K for 14.4ks. If the stress-strain curve obeys to equation (1) (the assumption (5)), a 0 and n are 457 MPa and 0 . 1 4 1 , respectively. Moreover E = 71.54GPa [11], v = 0.33 [11"] andE~ = 0.104 (the present work). From the feature of the fracture surface, r, is assumed as 1.6mm. The substitution of these values to equations (12, 14 and 15) yields k "- 680MN.m -3n and m = 1.071. The values of k and m of the curve B shown in Fig. 10 are 6 7 0 M N , m -3/2 and 1.07, respectively. These values coincide very well with the values calculated from equations (14 and 15) assuming r, as a reasonable value. This would show that the equation representing the relationship between K~c and ~c is reasonable. 4.4 The comparison of the results calculated in the

present study and those from the equations of Krafft and Peel and Forsyth From equations (IT and ll), taking the ratio of dr/2rc

dr -

-

2r~

~oE~+ " =

2=EE~(I -- v2)(1 + n)(2 + n)

(16)

is obtained. Now assuming that E ~ 100o0, a~ - E~ "- n (the condition of Krafft). The rough estimation by substituting those values to the above equation yields

dT

1

2re - 25

(17)

That is, the value calculated from the equation given by Krafft is about 1/25 of that in the present study. This value can not be verified by the experimental data because of a lack of the experiment measuring the size of plastic zone. However, if the reason that Krafft has corrected the value ~ to equation (3T) in the Ref. [2] is due to his finding out of the underestimation of the plastic zone size, his correction supports partly the result of the present calculation. By way of experiment, if the value ~1 obeys to equation (3T) the approximate value of equation (16) becomes d___L1 2r~ - 6"

(18)

237

Thus, the equation of Krafft would further underestimate the size of plastic zone to 1/6 of the present calculation. Subsequently we will compare the equation of Peel and Forsyth with the result of the present calculation. From the equations (7T) of Peel and Forsyth and (11), taking a ratio 2r/2rc, ~,l+n 2r---~= (2 + n)E} *"

2r

(19)

is obtained. In the present study we consider as ~ "- n so that we will assume ~s ~ 2n. The following approximate value is obtained by substituting these values to equation (19). 2r 1 1 2r c - (2 + n)-21+" - 5

(20)

This value coincides with the approximate value calculated from the equation corrected by Krafft suggesting a similar underestimation of the size of the plastic zone. Then the stress distribution, which is calculated based on the equation (1), corresponding to the strain distribution (equation (6)) is shown in Fig. 11. This curve would be more reliable than the curves of the stress distribution due to the perfect elastic solution and the elastic-plastic solution. Finally, the authors would like to mention the following thing. We have shown the relationship between K~c and the area fraction of the transgranular fracture (fr~) elsewhere [8]. In that case Ec was ignored and the present study also ignoredfr~. (In the equation of the Ref. [8] which is related to fre, k2 is the term on the size of plastic zone and is considered formally as a constant.) That is, since the strain distribution is assumed to be discontinuous in the equation offTr and continuous in the equation of ~c, respectively, two equations of different type are obtained. The nature as a parameter is different infr~ and ~c.frE is the characteristic of the fracture surface and ec is the parameter concerned with the elongation in the tensile test. The material with the large Ec has the large value offT~, i.e., there is a relationship between ~c and fr~. In other words, the difference between equation (13) in the present study and equation (16) in the ELASTIC SOLUTION 1-5 - ' ~ f i

o STRESS AT 02"1. STRAIN ELASTIC-PLASTIC

1.0

O" E~

SOLUTION

0.0525 0.1

~ 0.5 0

I.

05

I

1.0 r/E

I

1.5

I

2.0

Fig. 11. Stress distribution calculated by the strain distribution in Fig. 9.

238

KAWABATA and IZUMI:

DUCTILITY AND FRACTURE TOUGHNESS

reference (8) is caused from that models are simplified in order to clarify the relationship between those parameters. Originally, both of the equations should become the same equation, in which a relationship between fr~ and ec should be constructed. The most simple relationship is obtained from assuming k2 in the equation (16) of the Ref. [81 to be connecting with Ec. In the present model, it would be useful to discuss on the expression (13) of Ec only. However the result of another model will be shown in the appendix, in which the approximation of the strain distribution in the present model and that of the model in the Ref. [8] are combined. The result gives a similar expression as the equation (16) in Ref. I'8] substantially though it is more complex.

3. A. S. Tetelman and A. J. McEvily, Jr, Fracture of Structural Materials p. 60 Wiley, New York (1967). 4. G. T. Hahn and A. R. Rosenfield, Acta metall. 13, 293 (1965). 5. G. T. Hahn and A. R. Rosenfield, Applications Related Phenomena in Titanium Alloys ASTM STP 432, p. 5 0968). 6. C. J. Peel and P. J. E. Forsyth. Metal Sci. J. 7, 121 (1973). 7. T. Kawabata and O. Izumi, Acta metall. 24, 817 (1976). 8. T. Kawabata and O. Izumi, Acta metall. 25, 505 (1977). 9. A. Nadai, Theory of Flow and Fracture of Solids 2nd edn p. 413. McGraw-Hill, New York (1950). 10. T. Kawabata, Doctoral thesis, Tohoku University (1974). II. L. F. Mondolfo, The Aluminum--Magnesium-Zinc Alloys, A Review of the Literature p. 171, Research and Development Center Revere Copper and Brass Incorporated (1967).

SUMMARY Tensile tests and fractographic observation on unnotched and notched specimens of an A I - Z n - M g alloy heat treated variously were done. The relationship of fracture toughness, the size of plastic zone at a crack front and ductility were discussed, based on a model simplifying the distribution of plastic strain in a plastic zone, and the deduced expression is verified by the experiment and is compared with the equations of Krafft and Peel and Forsyth. Acknowledgements--The authors are grateful to Drs E. Hata and Y. Baba of the Sumitomo Light Metal Industry, Ltd., for the supply of materials, and to Emeritus Professor S. Shimodaira and Mr T. Sato given facilities for using the scanning electron microscope. This work was supported partly by the Keikinzoku Shogakukai (The Light Metal Educational Foundation, Incorporated).

REFERENCES 1. J. M. Krafft, J. appl. Mater. Res. 3, 88 (1964). 2. J. M. Krafft, Mechanical Behavior of Materials under Dynamic Loads (Edited by U. S. Lindholm) p. 134, Springer Verlag, Berlin (1968).

APPENDIX Relationship offTr and ec to Kic Figure I A (a) and (b) show the schematic figure for getting the plastic energies consumed in the three regions of a plastic zone, respectively. (a) shows the sizes of the portions fractured transgranularly and intergranularly after a crack developed, fTF is the area fraction of the portion fractured transgranularly. The width of the portion deformed heavily, in the intergranular fracture and transgranular fracture are 2ya and 2y2, respectively. (2ya equals the width of precipitate free zone) (b) shows the profile of strain with distance of y direction. The strain between AF of the transgranular fracture portion is assumed as e=

(')

- - - + 1 ~. Yl

(IA)

The strain between F ~ O is a constant Of E 2. And it is also assumed that the strain between A ~ G of the intergranular fracture portion follows equation (IA) and the strain between G ~ O (in the PFZ) is ca. The assumptions of I, 5 and 6 (in the text) are used additionally to the above. The plastic work, J,, expended at the part of A ~ G in

WIDTH OF PLASTIC ZONE

WIDTH OF REGION DEFORMED HEAVILY

WIDTH OF PFZ

,g

i

I

L

~r

1- fTF

REGION FRACTURED TRANSGRANULARLY

l'

\ \ REGION FRACTURED INTERGRANULARLY (a)

1

D

!

E

I

Y (b)

Fig. 1A. A schematic figure for getting the plastic energies consumed in the three regions of a plastic zone, respectively, fT~ is the area fraction of the portion fractured transgranularly. (I --fTF) shows the region fractured intergranularly.

KAWABATAandlZUMI:

DUCTILITY AND FRACTURE T O U G H N E S S

the region fractured intergranularly is Jl=

1

k2=

2£i~'' E l ( y l d y

.'l

_ _

L-.72-

per unit area of the crack fractured intergranularly, where E~0') is the distribution function of plastic work. From the assumption that the stress strain relationship follows the Ludwik's law.

t7o

El(y) = n + ~

(3A)

[{(3,)].-I

then. ao,7+1 El()') = -~ + 1

-

)~ + 1 3'1

+ I1))E

Yl/

+

ao(n'

e'J+l)'2

239

+

g 1

aoln' + 11 (14A)

Then using (I - x ) - I -~ 1 + x . when 1 > > x > 0 , and putting a o ~- %. n ~ n' and '2 ~- % = 'c- k'l and k2 are simplified as

(4A)

k'~

7+

-

1

Ya/t

and

2Oo"~*13'1 Jl

(

(n + l}(n + 2)

)'s "+" I - ~)

(5A'

x

[~

(1~1 ~-1 + y3i l -\,~/

The plastic work. J2, in the PFZ is

)'1

-

(,1),-~ 1}],

(16A)

(

The relationship between Km and fracture energy G~c is J2 =

.)f)'3 E s d y = 2Eay s

"7 0

(6A) Gw -

where E 3 is the density of plastic work in the PFZ and expressed as Es = f l ~o dE =

E~'+I. n'+~0 1

J2

n'+ 1

2°°E~' * 11'1 ( )'2~n+2 (~1+ l)(n + 2 ) 1 - ) ~ h /

(plane strain).

Klc = kl(.fTi-- + k2) 1 a

(7A)

(17A)

(18A)

where

(

(gA)

where o 0 and n' are the parameters in the PFZ having the same meaning of o0 and n. The plastic work Ja expended at the portion of A ~ F in the region fractured transgranularly becomes Ja

E

Putting Gic = W, so that we get,

Therefore,

2OoE.~'" I Y3

(l - v2)K~c

k, = \ l -- ;,2/ = [ ~ 2a°~+ Iy2E +l)(l_v2)(l

.

Ys')I1 - \~/(~')"+a}112 . (19A) - ~-27t

Furthermore. kl and k 2 might be simplified as

F

k, ~- L(n + 1)(1

(9A)

-

-

l

v2)_]

('1) '+1

and the plastic work J,~ at the portion (F ~ O) deformed heavily in the region fractured transgranularly is

(20A)

(21A)

2y2 \ ~ , / Similarly in equation (17), we assume as

J4 ~

2%,~" 13'2 -n+l

(10A)

Whole of the plastic work W when a crack propagates to unit area becomes as W = (J1 + J2)( 1 --.fTr) + (J3 + J4)f'rF

1 Ec = - Yl

(22A)

k 2 ~ --~-~c{El , / \|"+ I E 2)'2 \Ec J

(23A)

then

(IIA)

where .fT~ is the area fraction of the region fractured transgranularly. Substitution of equations (5A - 10A) to (1 IA) yields

E1/Ec might be a constant, k 2 is proportional to Ec, i.e., equation (18A) becomes

W = k'l(fTF + k2)

Kic = kl(fTv + k~Ec)1'2

(12A)

(24A)

where

and

+

E~+l)'2

(n' + 1)o" o

Equation (24A) has the same meaning as equation (16) in Ref. [8]. The difference between those is k2 being proportional to Ec in equation (24A).