Post yield fracture mechanics

Materials Science and Engineering American Society for Metals, Metals Park, Ohio, and Elsevier Sequoia S.A.. Lausanne - Printed in the Netherlands

129

Post Yield Fracture Mechanics P. T. HEALD, G. M. S P I N K and P. J. W O R T H I N G T O N Central Electricity Research Laboratories, Kelvin A venue, Leatherhead ( Gt. Britain)

(Received February 17, 1972)

Summary* A simple crack model, which &cludes a representation of plastic relaxation, & used to discuss the j?acture of high and low strength materials in the post yieht regime. A correspondence is set up between theJractur e criteria given by the present model and the familiar results of linear elastic ,fracture mechanics. The model is used to discuss the fracture of reinforced composite materials, polymers and turbine and pressure vessel steels. An apparent fi"acture toughness is defined and it is shown that this quantity reduces to the plane strain .lracture toughness Jor cases where there is limited plastic deJbrmation. It is found that the plane strain fracture toughness controls the mechanics of fracture even in the post yield regime. The effect of notch root radius on measured fi'acture toughness values is also discussed.

INTRODUCTION

It is possible, by the use of linear elastic fracture mechanics, to estimate the defect tolerance of structures under plane strain conditions and at stresses of up to approximately one-half of the general yield stress. Recently much effort has been made to develop a criterion which describes fracture behaviour in the post yield regime. This is particularly important for discussing the fracture characteristics of tough materials since, for these materials, failure occurs after appreciable plastic deformation has taken place. In the linear elastic fracture mechanics theory of Griffith I and Irwin 2 it is implicitly assumed that the material ahead of the crack can support infinite stresses. Any theory which describes post yield conditions must contain some representation of plastic deformation at the For French and German translations of the Summary, p. 137. *

Mater. Sci. Eng., tO (1972)

see

crack tip. In this paper we apply the Bilby Cotrell Swinden 3 model of plastic relaxation around a crack to the problem of fracture in the post yield region. This model was originally presented by Vitvitskii and Leonov 4 in 1960 and, from a different point of view, by Bilby, Cottrell and Swinden in 1963. Dugdale 5 has used the model to derive an equation which gives the extent of plastic relaxation ahead of a crack but did not use the model to discuss the fracture process. A closely related model was presented by Barenblatt 6 in 1959. Although Barenblatt's model gives a clear picture of the nature of the cohesive forces at the crack tip it is only applicable to situations involving small amounts of plastic relaxation. The present paper is complementary to those of Cottrell 7'8 in so far as we relate the theory to macroscopic behaviour as Cottrell did to the various fracture mechanisms. It is shown that, even in the post yield regime, the plane strain fracture toughness, Klc, controls the fracture behaviour. We define an apparent fracture toughness, as obtained from a "non-valid" A.S.T.M. test, and derive an expression which relates this to the plane strain fracture toughness. The associated problem of size effect is discussed and it is shown that although small laboratory samples cannot always reproduce the constraints existing in large structures it is possible to determine the fracture behaviour of such structures from small scale tests. In addition the effect of crack root radius on the measured fracture toughness value is examined. In particular, an equation is derived which relates the apparent toughness measured on a specimen with finite root radius to the plane strain fracture toughness.

POST YIELD FRACTURE

In the Vitvitskii-Leonov, Bilby Cottrell-Swinden theory it is assumed that the material ahead of

P. T. HEALD, G. M. SPINK, P. J. W O R T H I N G T O N

130

'1

the crack can only support a finite stress, a~, which is characteristic of the material; hence, some consideration must be given to the non-linear mechanical behaviour there. In regions far from the crack tip the stresses are such that linear elasticity is applicable. However, in the vicinity of the crack tip deviations from Hooke's law come from the rupture or non-linear distortion of the atomic bonds between the atomic planes. Bilby, Cottrell and Swinden 3 represent this non-linear region by a distribution of dislocations coplanar with the crack; this enabled them to use linear elasticity everywhere since the Burger's vectors of the dislocations can account for the "extra" non-elastic displacements. This model is equivalent to assuming that yielding is confined to a plane surface ahead of the crack and is, of course, only a rough representation of the physical situation. The characteristic stress o7 is of particular importance since, if we denote the relative displacement between the atomic planes in the crack tip region by ~b, ai(qS) is the law of cohesive force between atomic layers in the non-linear region. That is, ai(~b) is equivalent to the stressstrain curve of the material. For simplicity Bilby, Cottrell and Swinden consider a law of force of the form a~(qS) = al, 0~< qS< ~bc~.. (1)

0, ¢~>~o

m

-.<.

a

ZONES OF PLASTICRELAXATION

Fig. la. Schematic representation of the crack model.

-a

-

~

+ c

+ a

Fig. lb. The distribution of dislocations which simulates a relaxed crack.

,j----a

i +c

-c

i +a

Fig. lc. The displacement of the upper crack surface.

STRESS

-a

(2

c

In addition we have, from the analyses mentioned above 3'4, the profile of the crack surface. In particular, the relative displacement at the crack tip is ~b(c) - 4 ( 1 - v ) a l c In sec re#

,

(3)

-c

+c

+a

Fig. ld. Schematic representation of the distribution of stress on the plane y = O.

ai

a

I

iI

- 1.

Mater. Sci. Eng., 10 (1972)

/ c

J

~bc is a critical displacement beyond which the atomic layers do not interact. For more general distributions of cohesive force it is necessary to use numerical methods to solve the mixed boundary value problem (see, for example, Bilby and Swinden9). To avoid unnecessary complications at this stage we shall refer to a crack in an infinite plate under tensile loading; the effect of different geometrical situations will be referred to later. The situation is shown schematically in Fig. 1. If the stresses at the extremity of the relaxed zone are finite (this is equivalent to requiring that the atomic planes meet in a cusp there) then the length, s, of the relaxed zone is related to the crack length, 2c, and the applied stress, a, by - = sec

Ty

\

CRACK

iI

"~\ \,.

Fig. le. Laws of force between atomic planes; the solid line corresponds to the fictitious distribution of cohesive force given by eqn. (2) and the broken line is a schematic representation of the actual forces of cohesion.

13 1

POST YIELD FRACTURE MECHANICS

where/~ and v are the shear modulus and Poisson's ratio respectively. This is especially important since this crack is, energetically, in neutral equilibrium 1° and hence some crack extension condition must be postulated. Cottrell la has suggested that crack growth will occur whenever the relative displacement at the crack tip exceeds a critical value qSc. That is a = af, the fracture stress when ~b(c) >~ q5c .

(4)

Since we may define the fracture surface energy, 7, quite generally as the area under the cohesive force-displacement curve we have 27 =

a~(4)d4,

(5)

,, 0

which gives, using the force law defined by eqn. (1) 27 = ol ~bc.

(6)

Combining eqns. (3), (4) and (5) gives the appropriate fracture condition for this model:

singularity which would otherwise occur at the crack tip. As the stress is further increased intense shear bands form and microcracking occurs in the shear bands. Since our miniature tensile specimens are elements of the relaxed zone at the crack tip it is not unreasonable to say that they have yielded when the stress on them is equal to the yield stress of the material and that they fail (and crack growth takes place) when the stress on them is equal to the ultimate tensile strength of the material. Therefore, we suggest that in eqn. (2) a 1 should be set equal to O'y giving - = sec c

exp-

2(1 -v)a~c

.

(7)

If we expand the right-hand side of eqn. (7) for small values of 7r#7 2 ( 1 - v)a21c we obtain

(8) the well known Griffith ~ fracture condition. As c--*0 eqn. (7) gives af--,a~ in contrast with eqn. (8) which gives crr~ oo. We have already mentioned that the law of force between atomic planes a~(q~) should contain all the information about the stress-strain curve of the material but we have not discussed the specific value of a~. Bilby et al. 3 set al equal to the lower yield stress ay; however, this will only be the case if there is no work hardening. The following argument is consistent with Cottrell's view of the crack tip~l : imagine the atomic bonds or cohesive forces in the vicinity of a crack tip to be represented by a series of miniature tensile specimens. As the applied stress is increased from zero, yielding occurs across these miniature specimens in order to relieve the stress Mater. Sei. Eng., 10 (1972)

(9

while in the fracture condition, eqn. (7), al is equal to the ultimate tensile stress, au. The fact that eqn. (7) reduces to Griffith's criterion when the crack length is large (i.e. large compared with ~7/a 2) suggests that this model may be used as a natural extension of linear elastic fracture mechanics. We have, from the normal definitions of fracture mechanics, the plane strain fracture toughness K,o = t ( 1 - Q ~

ar = - - a l C O S - 1 rc

- 1,

'

hence we may write eqn. (7) in the form af=~auCOS-'

exp-

(10)

While this equation was derived assuming a plane strain stress system the only factors in the equation which depend on the state of stress are the fracture toughness and possibly the ultimate tensile stress (due to the different constraints). If the fracture stress, % is plotted as a function of crack length using both eqn. (8) and eqn. (10) then the two curves "merge" for crack lengths greater than

c* ~ ( K ~ I 2 \ a./

For crack lengths less than c* linear elastic fracture mechanics is inadequate but eqn. (10) suggests that Kac may be determined from tests in the post yield regime. Indeed, we may define an apparent fracture toughness in the post yield region and relate it to the plane strain fracture toughness through eqn. (10) thus:

(11)

132

P. T. HEALD, G. M. SPINK, P. J. W O R T H I N G T O N

As c becomes large we enter the fracture mechanics regime and KA--*Klc. Equations (10) and (11) show that the plane strain fracture toughness controls the fracture behaviour even in the post yield region• The model may be extended to take into account the effect of notch root radius on fracture toughness measurements. Smith's 12 result for the propagation of fracture from a semi-elliptical notch of semimajor axis c and semi-minor axis b (root radius p = b2/c) may be written or -- [ 1 + au (plc)q {2n c°s-1

• [cxp - \ 8 a 2 c J j + (-Pc)Y}. 1 (12) (Strictly Smith's result is for anti-plane deformation only but it should be approximate for plane strain deformation•) The apparent fracture toughness is (nc)½a,

I 2

KA(P) - [1 + (p/c)~] •~ -n c o s -

fracture mechanics result in the limit of large crack lengths•

DISCUSSION

In this section we shall compare the results obtained in the previous section with experimental data and discuss the consequences of the results. In Fig. 2 eqn. (9), which gives the plastic zone size as a function of the applied stress, is compared with the experimental results of Dugdale 5 on mild steel and Tetelman 13 and Hahn and Rosenfield 1. on silicon iron. In all cases we have increased the value of the yield stress quoted by the authors by four per cent. This is rather arbitrary but it allows for some work hardening; no significance is attached to this increase• The agreement between eqn. (9) and the experimental data is satisfactory•

1 A MILD STEEL: OUGDALE (1960), INTERNAL SLITS

• [exp - ~ , ~ ] j

+ (P)~}

(13,

•

MiLD STEEL: DUGDALE (1960), EDGE NOTCHED

4- SILICON [RON: HAHN AND ROSENFIELO (196S), EDGE NOTCHED x SILICON IRON: TETELMAN ([964), PRE-MICROCRACKED

and as p ~ O. KA(p) reduces to the result previously given in eqn. (11). One further effect of geometry is worth noting; in order to apply eqn. (10) to more complicated geometries it is necessary to introduce a correction factor into these equations. Recalling that in linear elastic fracture mechanics the stress intensity factor may be written as K = Q . a"

where Q is a function of the specimen geometry, we may write the fracture condition in either of the equivalent forms (nc)~ aequiv >/K,c

(14a)

or

K~o,

a ( n Cequiv) ½ •

(14b)

where a~qui~= Qa and Cequiv = cQ 2. Thus if we require eqn. (10) to reduce to the corresponding fracture mechanics result in the limit of large crack length we may write 2

Q "af

= --/.¢a

u

~ (nK~,'~[ COS-1 (exp - \8a2cJ j

Ix O x

(15a)

_0..~0~

0x

or

2

1{

af = ~- a n cos- _exp

_ { nKZ~)~ ~8a~cQ2/IJ.

(lSb)

Both eqns. (15a) and (15b) reduce to the correct Mater. Sci. Eng., 10 (1972)

o.i

0.2

0.3

0.4

o.s

0.6

0.7

o.0

I

0.9

aa I

Fig. 2. The plastic zone size as a function of the applied stress from eqn. (9).

133

POST YIELD FRACTURE MECHANICS

TUNGSTEN FIBRE REINFORCED COPPER:

COOPER AND K E L L Y ( 1 9 6 7 )

160 ~u = 145 k . s . i .

K c = 125 k . s . i . , , ~ m

140

z

o Iu u.J

120

z

I00

w ~: m

80

•~

60 O

~-

41/

2(1

I 2

I

I

I

I

I

I

I

4

6

8

I0

12

14

16

CRACK LENGTH,

I

I

18

20

I 22

I 24

mm

Fig. 3a. T h e fracture stress, o-f, as a function of crack length for tungsten fibre reinforced copper. T h e solid line is a plot of eqn. (10) with cru = 145 k.s.i, and K c = 125 k.s.i. ~/in. The solid points represent s l o w initial crack growth while the open ones c o r r e s p o n d to fast fracture.

In Figs. 3a-3c the fracture condition, eqn. (10), is compared with the experimental results of Cooper and Kelly 15 on tungsten reinforced copper, of Berry 16 on polystyrene and of Lubahn and Yukawa 17 on turbine steel. The experimental conditions

for the results of Cooper and Kelly 15 were probably plane stress rather than plane strain since the specimens were thin. For such a wide range of material properties and fracture mechanisms (the reinforced copper fails in a discontinuous manner, polystyrene

70

LO

POLYSTYRENE BERRY (1961) t 1 ~ It

•

ou = 6.8 k.s.i.. KIC = 2.7 k.s.i.

60

o-~,~ ~ o ~

%o ig'

50

w

ua

u-

o

40

o ~oo~a"

o8

~l.

3,0 &

•

20

•

qa

I0

I 4

I 8

I 12

I I I 16 20 24 CRACK LENGTH x 102 INCHES

I 28

I 32

Fig. 3b. T h e fracture stress, crf, as a function of crack length for polystyrene. T h e solid line is a plot of eqn. (10) with cru = 6.8 k.s.i, and K k = 2.7 k.s.i, x/in. T h e o p e n points c o r r e s p o n d to e x t e n s i o n rates of 0.02 in./min while the solid points c o r r e s p o n d to e x t e n s i o n rates of 0.2 in./min. Circular points d e n o t e s p e c i m e n s of 0.42 × 0.2 inch cross section, square points 0.98 × 0.2 inch cross section a n d triangular points 1.42 × 0.2 inch cross section.

Mater. Sci. Eng., 10 (1972)

134

P.T. HEALD, G. M. SPINK, P. J. W O R T H I N G T O N

N i H o V STEEL : LUBAHN AND YUKAWA (1958) au=

I I 0 k.s.i.,

KLC= 60 k.s.i. JT~n

MAXIMUM OUTER FIBRE STRESS : 217 k.s.i. 240 220 200 180 [60 140 [20 100 80 60 40 20

I

I

0.2

0.4

I

I

I

1.0

2.0

4.0

I 10.0

SPECIMEN SIZE. (inches) = 5 x CRACg LENGTH

Fig 3c. The equivalent fracture stress, af, as a function of crack length in NiMoV steel for geometrically similar three point bend specimens. The solid line is a plot of eqn. (10) with Klo = 60 k.s.i. ~/in. and a tensile strength of 110 k.s.i.

40 - -

• i

30

--

20

--

/

<

•/

10

• . . . . . . - - - 2. - - " ' -

ALUHINIUM ALLOY 20[4 - T651 : CORN (1966)

/

ou = 9Sk.s.i.,

I 0.01

KIC= 4Sk.s.i.i.,/~-n

I

I

I

I

I

I

i

I

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

DETECT SIZE.

[ 0.10

INCHES

Fig. 4a. The apparent fracture toughness as a function of crack length for the aluminium alloy 2014-T651. The solid line is a plot of eqn. (11) with K l c = 4 5 k.s.i. ~/in. and au=95 k.s.i. The defect size is cQ 2.

Mater. Sci. Eng., 10 (1972)

POST YIELD FRACTURE

135

MECHANICS

0 6 - AC (LOW STRENGTH): RANDALL (1966) 140

a u = 242k.s.i.,

Kic = 186k.s.i. I-J~'n

130

00

0

120

o y/° ~ ~:

j

100

o

o~

o

O

80

70 001

I 0.02

I

i

I

0.04

0,03

0.05

l 0.06

I

I

0.07

0.08

L

I

0.09

0 I0

I

L

0.1[

0.12

L

DEFECT SIZE. inch

Fig. 4b. The a p p a r e n t fracture t o u g h n e s s as a function of crack length for low strength D 6 AC. The solid line is t a k e n from eqn. (11) w i t h K l c = 186 k.s.i, x/in. a n d o-,,=242 k.s.i. The defect size is cQ 2.

FERRITIC-PEARLITIC STEEL: WILSHAW, RAU AND TETELMAN (1968) % : 190 k.s.i., KIC = 21.3 k.s.i.-~-In 2.5

5

I0

20

30

40

I

I

i

I

i

[

50

I J

pX 103 inch

60

I S0

i

40

o

30

o

~

20

'°f

I

I

I

I

I

I

I

[

[

I

I

I

2

4

6

8

10

12

14

16

18

20

22

24

(ROOT RADIUS)

'/z

x

102 ~

Fig. 5. The a p p a r e n t fracture t o u g h n e s s as a function of n o t c h root radius. The solid line is a plot of eqn. (13) with K ~ c = 2 1 . 3 k.s.i. x/in., ~ru = 190 k.s.i, a n d c = 2 ram.

M a t e r . Sci. Eng. , 10 (1972)

136

P.T. HEALD, G. M. SPINK, P. J. W O R T H I N G ] ON 1.0

0.9 B A T T E L L E PIPE TESTS: DUFFY. EIBER AND MAXEY (1969) O A P I 5 L X - 5 2 GRADE PIPE, a u = 81 k.s.i.,

0.8

K c = 217 k.s.i.dTm'in

O A P I S L X - E 0 GRADE PIPE, o U : 84.4 k.s.i.. K c : AS.TM

0.7 • 0.6

226 k.s.i.,fq'nn

AI06B GRADE PIPE, =u = 77.5 k.s.i., K c = 165 k.s,i, la'¥

A.S.T~M. AI06B GRADE PIPE, a u = 74.5 k.s.i., K c = 136 k.s.i,JTff

O A . S , T . M AI06B GRADE PIPE, a u = 74 k.s,i., K c = 107 k.s.i.

O.S

0.4

0.3

0.2

°

J

8au2C C2 )2 ~Kc2 (1+ 0.015 ~ -

0.1 26 I

I

30 I

I

2

4

6

8

34 I 10

I 12

3R r

I

I

I

14

16

18

20

I

I

I

22

24

26

Fig. 6. The fracture stress as a function of crack length for a number of low strength materials. The solid line represents eqn. (15b).

crazes and the turbine steel fails by a brittle mechanism) the agreement between eqn. (10) and the experimental data is excellent. A constant value of fracture toughness should be obtained from tests using specimens with varying crack lengths according to linear elastic fracture mechanics. However, it has been known for some time that the apparent fracture toughness falls below the plane strain fracture toughness for small crack lengths; this is a consequence of plastic deformation at the crack tip. The experimental results on the aluminium alloy 2014-T651 (Corn 18) and on low strength D6-AC Steel (Randall 19) are compared with the apparent fracture toughness given by eqn. (11)~ with Q=~/1.2/~, in Figs. 4a and 4b. (~b is a complete elliptic integral of the second kind which depends on the shape of the crack.) The effect is shown most strikingly for the steel in Fig. 4b. None of the results is valid according to A.S.T.M. criteria; however, from eqn. (11) we calculate the fracture toughness to be 45 k.s.i. ~/in. for the aluminium alloy and 186 k.s.i. ~/in. for the steel. In some fracture toughness tests on high strength materials the apparent toughness increases as the crack length is decreased; this would seem to be at variance with eqn. (11). However, from the slip line Mater. Sci. Eng., 10 (1972)

field calculations of Ewing 2° if the crack length to specimen width ratio is less than a critical value then yielding to the back surface occurs. This results in stress relaxation at the crack tip (Knott21), thus the stress required to cause fracture, and hence the apparent fracture toughness, is correspondingly higher than would be predicted by eqns. (10) and (11). The effect of notch root radius on fracture toughness measurements has previously been studied by Wilshaw, Rau and Tetelman 22. Their theoretical result is clearly incorrect since it predicts that KA(p)-+0 as p---~0. Their experimental data are compared with our result (eqn. 13) in Fig. 5. For small root radii the experiment points are somewhat lower than the theoretical curve. This is not unreasonable since Cottrel123 has pointed out that there will be a limiting sharpness which governs the scale of microscopic size effects at sharp notches. This limiting sharpness will be determined by some microstructural feature. One final remark on geometrical effects concerns eqns. (15a) and (15b). In a recent paper Folias 24 has used eqn. (15a)to discuss the fracture of cylindrical pipes and spherical shells; there is good agreement between eqn. (15a) and the experimental results. In

137

POST Y1ELD FRACTURE MECHANICS

addition eqn. (15b) can be used to discuss similar situations; for example, Fig. 6 shows a comparison between the experimental results of Duffy, Eiber and Maxey 25 and eqn. (15b) with Q = ( l + 0 . 0 1 5 , where R is the pipe radius and t its thickness. Although this is somewhat different from the original factor given by Folias 26 it provides a good correlation over a wide range of material properties and pipe geometries. For the comparison we have replaced K ~ by K~ since pipe tests are usually under conditions of plane stress. The method used by the C.O.D.A. panel 27 for incorporating the Folias bulging factor into the equation for the crack opening displacement is clearly incorrect since in the limit of small scale stresses it does not reduce to the equivalent fracture mechanics result. In conclusion, we have shown that the results given here can extend the range of fracture mechanics into the regime of post yield failures. In particular the fracture condition given in eqn. (10) shows that even in the post yield region the plane strain fracture toughness controls the mechanics of fracture. The use of eqn. (11) makes it possible to relax the present A.S.T.M. recommended practice for plane strain fracture toughness testing since K i~ can be calculated from the apparent toughness, K A, obtained from an A.S.T.M. "non-valid" test. Equation (13), which takes into account a finite root radius, leads to the possibility of using machined test pieces rather than fatigue cracked specimens in fracture toughness testing.

ACKNOWLEDGEMENTS The work was carried out at the Central Electricity Research Laboratories and the paper is

Mkcanique des ruptures se produisant aprks le dObut de la d@)rmation plastique #knkraliske Les auteurs utilisent un mod61e simple de fissure, dans lequel ils ont inclus une repr6sentation de la relaxation plastique, pour analyser la rupture des mat6riaux ~t forte et fi faible r6sistance, lorsque celle-ci se produit apr6s le d6but de la d6formation plastique g6n6ralis6e. Ils 6tablissent une correspondance entre les crit6res de rupture d6duits du pr6sent mod61e et les r6sultats bien connus de la m6canique lin6aire 61astique de la rupture. Ils Mater. Sei. En9., 10 (1972)

published by permission of the Central Electricity Generating Board. REFERENCES 1 A.A. Griffith, Phil. Trans. Roy. Soc. London, A221 (1920) 163. 2 G. R. Irwin, J. Appl. Mech., 24(1957) 361. 3 B.A. Bilby, A. H. Conrell and K. H. Swinden, Proc. Roy. Soc. (London), A272 (1963) 304. 4 P. M. Vitvitskii and M. Ya. Leonov, Vses. lnst. Nauchn.Tekhn. Inform. Akad. Nauk. SSSR Pt. 1 (1960) 14. 5 D.S. Dugdale, J. Mech. Phys. Solids, 8 (1960) 100. 6 G. 1. Barenblatt, Prikl. Mat. Mek., 23 (1959)434. 7 A. H. Cottrell, in C. J. Osborn (ed.), Fracture. Buttcrworths, London, 1965, p. 1. 8 A. H. Cottrell, Proc. Roy. Soc. (London), A285 (1965) 10. 9 B. A. Bilby and K. H. Swinden, Proc. Roy. Soc. (London), A285 (1965)22. 10 K. H. Swinden, Thesis, University of Sheffield. 1964. 11 A. H. Cottrell, lron Steel Inst. Spec. Rept. No. 69, p. 281. 12 E. Smith, Proc. Roy. Soc. (London), A299 (1967)455. 13 A. S. Tetelman, Acta Met., 12 (1964) 993. 14 G. T. Hahn and A. R. Rosenfield, Acta Met., 13 (1965) 293. 15 G. A. Cooper and A. Kelly, J. Mech. Phys. Solids, 15 (1967) 279. 16 J. D. Berry, J. Polymer Sci., 50 (1961)313. 17 J. D. Lubahn and S. Yukawa, Am. Soc. Testin# Mater., Proc., 58 (1958) 661. 18 D. L. Corn, Rept. SM-49149, Douglas Aircraft Co,, 1966. 19 P. N. Randall, Am. Soc. Testing Mater., Spec. Tech. Publ. No. 410, 1966, p. 88. 20 D. J. F. Ewing, J. Mech. Phys. Solids, 16 (1968) 205. 21 J. E. Knott, J. Mech. Phys. Solids, 15 (1967)97. 22 T. R. Wilshaw. C. A. Rau and A. S. Tetelman, En~t. Fracture Meeh., 1 (1968) 191. 23 A. H. Cottrell, The Mechanical Properties o! Matter, Wiley, New York, 1963. 24 E. S. Folias, Eng. Fracture Mech., 2 (1970) 151. 25 A. R. Duffy, R. J. Eiber and W. A. Maxey, Practical Fracture Mech. Jor Structural Steels, Risley, U.K.A.E.A.. Chapman and Hall, London, 1969. 26 E. S. Folias, lntern. J. Fracture Mech., 1 (1965) 104. 27 C.O.D.A. panel report, Practical Fracture Mech..[br Structural Steels, Risley, U.K.A.E.A., Chapman and Hall, London, 1969.

Bruchmechanik jenseits der Streckorenze Der Bruch von Materialien mit hoher und niedriger Festigkoit im Dehnbereich jenseits der Streckgrenze wird anhand eines einfachen Rigbildungsmodells diskutiert, das die plastische Relaxation berficksichtigt. Es wird ein Zusammenhang zwischen den Bruchkriterien des vorliegenden Modells und ~ihnlichen Ergebnissen der linearen elastischen Bruchmechanik hergestellt. Der Bruch von verst~irkten Kompositmaterialien, Polymeren, TurN° nensfiihlen und von Druckkesselsfiihlen wird mit

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utilisent le mod61e pour 6tudier la rupture de mat6riaux composites renforc6s, de polym6res, ainsi que d'aciers pour turbines et r6servoirs sous pression. Ils d6finissent une t6nacit6 de rupture apparente et montrent que cette quantit6 devient 6gale au facteur d'intensit6 de contrainte critique en d6formation plane, dans les cas off il ne se produit qu'une d6formation plastique limit6e. Ils trouvent que le facteur d'intensit6 de contrainte critique, en d6formation plane, continue ~ jouer un r61e pr6pond6rant dans la m6canique des ruptures se produisant apr6s le d6but de la d6formation g6n6ralis6e. Ils examinent 6galement l'influence du rayon h fond d'entaille sur les valeurs mesur6es de la t6nacit6.

Mater. Sci. Eng., 10 (1972)

P.T. HEALD, G. M. SPINK, P. J. WORTHINGTON

Hilfe dieses Modells diskutiert. Eine scheinbare Bruchfestigkeit vcird definiert und es wird gezeigt, dab sich diese Gr6Be in Fallen mit begrenzter plastischer Verformung auf die Bruchfestigkeit bei ebener Dehnung reduzieren l~il3t. Es ergibt sich, dab die Bruchfestigkeit bei ebener Dehnung selbst im Dehnbereich jenseits der Streckgrenze die Bruchmechanik bestimmt. Der EinfluB des Kerbradius auf die gemessenen Werte der Bruchfestigkeit wird ebenfalls diskutiert.