The JM crack growth resistance curve expressed in terms of eta factors: Analysis of some idealized models

lnt. J. Engng Sci. Vol. 32. No. 6, pp. 93.5-944. 1994 Copyright 0 1994 Elsevier Science Ltd

Pergamon

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THE &r, CRACK GROWTH RESISTANCE CURVE EXPRESSED IN TERMS OF ETA FACTORS: ANALYSIS OF SOME IDEALIZED MODELS E. SMITH Manchester University-UMIST

Materials Science Centre, Grosvenor Street, Manchester Ml 7HS, U.K. (Communicated

by B. A. BILBY)

Abstract-Two recent theories have provided different expressions for the plastic component Jt,,P of the modified J integral JM for a growing crack, in situations where the plastic component Jop of the deformation J integral Jo for a non-growing crack is represented in terms of two eta factors and the energy and complementary energy integrals. In an earlier paper, the author has shown that the difference between the rival Jhlp expressions is significant for a particular configuration, namely that of a symmetrically double-notched solid with tensile loading of the small remaining ligament. It was therefore concluded that the difference between the rival JMp expressions cannot, in general, be disregarded when quantifying a material’s crack growth resistance. The present paper analyses other models, and the results provide underpinning of the conclusion drawn from the analysis of the double-notched configuration.

1.

INTRODUCTION

When assessing the integrity of a cracked engineering structure, it is important to have a quantitative description of the material’s crack growth resistance curve. This is usually expressed in terms of the deformation J integral Jb, though in some quarters the tendency is to use the modified J integral JM [l, 21 to characterize the resistance curve. A strong argument is that for a rigid perfectly plastic material, &, (unlike J,,), always satisfies the Rice condition [3] that the rate of increase of the characterizing parameter must not be a function of the rate of increase of crack length. Furthermore, it has been argued that, in some cases [4-71, much of the geometry dependence of a material’s crack growth resistance curve is removed when the curve is expressed in terms of & rather than I,,. A potentially very attractive way of obtaining a Jr, or &, description of a material’s crack growth resistance behaviour, is to use a relation between the J integral and load, load-point displacement and crack extension measurements, and then determine the J-crack growth resistance curve from laboratory experiments. As a basis, it is necessary to have a reliable estimate of the magnitude of Jn for a non-growing crack. Since JD can be separated into an elastic component &, which is directly related to the crack tip stress intensity factor K,, and a plastic component JDp, the determination of .Jb essentially depends upon the determination of the value of JDp. Against this background, there has been extensive discussion of the feasibility of representing JDp for a non-growing crack, in terms of the load P and the plastic component Ap of the load-point displacement via the energy and complementary energy integrals, using a relation of the form JDp=;

AP PdA +> ‘A dP I0 ’ B Io ’

B being the solid thickness with Mode 1 plane (strain or stress) deformation Such a description is assumed to be applicable for all levels of deformation, yielding to extensive deformation beyond general yield, with n and 77,.being have dimensions length-‘, but are independent of the thickness B and, most 935

(1) being assumed. from small-scale eta factors that importantly, the

936

E. SMITH

level of deformation. It has been shown ]s, 91 that for J bP to be given by relation (1), then A,, must be expressible in the functional form

where 4 and 4 are functions of the crack size a,, and any other geometrical parameters of the configuration, but not of the load P; the eta factors 7 and nC are then given by the expressions

PI

Most importantly, however, it has been shown [9] that the functional form (2) is possible only when the solid’s geometry involves a single length parameter, i.e. the ligament width or crack size, apart from the thickness B, the implication being that a two eta factor JDP description, applicable for all levels of deformation, is strictly accurate only when the solid has a single length parameter. A JDP description of the form (I), indeed a single eta factor description with 77,.= 0, is also applicable for the two separate situations of small-scale yielding and extensive deformation at limit load conditions irrespective of the configuration, though the two eta factors are in general different for the two cases. If a two eta factor JnP description is used for more general situations, i.e. for the complete spectrum of deformation levels and for a configuration with more than one length parameter other than B, the description must be viewed as giving only an approximate value for Jr)+ On the basis of expression (1) for J ,,,, for a non-growing crack, both Zahoor [8] and Ernst [lo] have obtained equivalent expressions for JbF for a growing crack in terms of P, Ap and crack size. They also obtained expressions for the plastic component .I,, of the modified J integral for a growing crack. However Ernst’s JMP expression differs from Zahoor’s expression, in that it contains an additional term. The present author has recently determined [ll] the magnitude of this difference between the rival J MP expressions by analysing a particular configuration, namely that of a symmetrically double-notched solid with tensile loading of the small remaining ligament. With this configuration, for which a two eta factor JDP description for a non-growing crack is appropriate with n = 1/2h and nC = - 1/2b (2b = ligament width), it was shown that the difference is significant. The implication is that the difference between the rival &, expressions cannot, in general, be disregarded when quantifying a material’s crack growth resistance. The present paper analyses other models, and the results provide underpinning of the conclusion drawn from the analysis of the double-notched configuration.

2. J,,

and JMp EXPRESSIONS

FOR A GROWING CRACK BY ZAHOOR AND ERNST

AS OBTAINED

For a crack (single or doubie ended) which grows from a length uow to n,, both Zahoor ]8] and Ernst [lo] have shown that JDp is given by the expression

+$Ia* r7c(vz- rl, PA, da,

(4)

alI*

with or = (q’ - ~,!)/(n - Q) and n2 = 77:./n,., where the primed quantities are total derivatives with respect to the crack size a,. Relation (4) allows JDP (and consequently Jr, = JE + JDP) for a growing crack to be obtained from P, A, (and consequently A = AE + AP, where AE is the

937

& crack growth resistance curve

elastic component of A) and crack extension measurements using a single specimen, provided that q and r],. are known. With JMP defined in accord with the relation [l]

JMP-JDP-[; (t$), da,

(5)

P

Ernst [lo] has presented for JMp:

the full details of an analysis which leads to the following expression

Zahoor [8] has also quoted a result for J MPthat is different to that given by relation (6), in that his JMp expression does not contain the last term in relation (6). However, he does not present the details of his analysis; the author has checked the Ernst J MP analysis very carefully, and believing it to be correct, is therefore supportive of the Ernst JMp formulation, i.e. relation (6) rather than the Zahoor JMp formulation. Relations (4), (5) and (6) give dJ MP_D_dJ dJDP -= --

da,

dla,

=(a-ri+

da,

%PDP

Thus with

the remainder of the paper analyses specific configurations, with the objective of assessing the magnitude of III (the additional term in the Ernst JMp formulation) in retation to the magnitude of I + II + III = (~J~~/~~*)~~, in order to see whether the additional term is significant. The configurations to be analysed, like the double-notch configuration analysed in the author’s eariier paper [ll], are all characterized by having 77,f 0; if Q = 0 then clearly the additional term in the Ernst JMp formulation is zero and the Ernst and Zahoor JMp formulations are then identical.

3. SEMI-INFINITE

CRACK

IN AN INFINITE ON THE CRACK

SOLID, FACES

SUBJECTED

TO

LOADS

Figure 1 shows the model of a semi-infinite crack in a solid of infinite dimensions in the plane of the figure but with thickness B in the direction of the figure normal. The crack faces are subjected to loads P at a distance behind the crack tip, and the plastic deformation is represented by a line plastic zone, [12,13] of length R, ahead of the crack tip, within which the P

P

Fig. 1. A semi-infinite crack in an infinite solid, with the crack faces subjected to loads P at a distance R behind the crack tip.

938

E. SMITH

tensile stress is Y. With this model, dimensional considerations show that the plastic component Ap of the load-point displacement is given by a functional relation of the form

Thus

with a, = a, by analogy

relations

with relation

(2), it follows

(3) show that a two eta factor J,,,, description

appropriate

that

C$la

[relation

with n = - 1 /a and 77,.= 1/a. It then follows

and $ x l/a,

whereupon

(l)] for a non-growing

crack is

that n, = n2 = - 1/a and then relations

(7) and (8) give --_=~--_

-

dJDdl,dJDpdJMp-( da da da

(10)

da I

with the II term being equal to zero. For this model, the crack tip stress

intensity

III

K for linear

elastic

behaviour

is given

2P K=---BV’Zii. The

stress

loadings

by the

[ 141

expression

is finite

at the tip of the plastic

and the restraining

stress

zone

Y are equal, 2P Bv2rc(a

+ R)

(11) if the stress

whereupon

intensities

standard

results

due to the applied [ 141 give

Xv= n

(“7)

or P

-=QGFJ

(‘3)

2BaY

and with x = RJa. JDp = (JD - JE) where JE = K’/Eo with EC,= E for plane stress deformation EC1= E/(1 - Y’) for plane strain deformation with E being Young’s modulus and v Poisson’s ratio; J,, = Y@ where @ is the crack tip opening displacement. Thus, since standard results [14] give V’ZEtvR @=-CL nE,,B In1 vla+R - * it follows,

after elimination

of P via relation

8YR 1

(13). that J,,lI is given by the expression

&,JDP

~=~(x)_8[2~sinh~~‘&-x Now Ap is given by an expression

(14)

nE<,

-x(1

+x)1.

(‘5)

of the form

&A, - Ya = g(x) where gives

the function

g(x) is as yet undetermined.

Differentiation

of relations

(13), (15) and (16)

(17) &,SJ,P Y2

= f(x)

6a + af’(x)

6x

E,,aAr = g(x) 6a + ag’(x) 6x. Y

(‘8) (‘9)

JM crack growth resistance curve

939

Table 1. Values of the terms I, III and (I + III) for the model of a semi-infinite crack in an infinite solid, subjected to loads on the crack faces x=- R a

E,,I

E,,III

Y2

Y*

&(I + III) Y*

0.5 1.0 1.5 2.0

-0.277 -1.291 -3.193 -6.072

0.374 1.752 4.371 8.384

0.097 0.461 1.178 2.312

Noting that JDP, AP, P and a are related by the expression (Z,.

(20)

= - f ($LP

if follows from relations (17)-(20) that &@)

t1+ 2x1

_

2x(1 +x)

g(x)=

(21)

-2$&)

which integrates to give after substitution for f(x) using relation (15) EoAr _ - g(x) = z [mln(l Ya

+ x) - m

It then follows from relation (15) and by reference the relation nE,,I -=2msinh_rX&x(2+x) 8Y2

+ sinh-‘G].

(22)

to relation (10) that the term I is given by

(23)

while relations (lo), (17), (19) and (22) show that the term III is given by the relation nEOIII -= 8Y2

2[-

- sinh-‘X&j ln(1 +x)

+ x) - m

[mln(l

’

+ sinh-%I.

(24)

The magnitudes of the terms I, III and (I + III) are shown in Table 1 for various values of n = R/a. Inspection of these results clearly shows that the III term, i.e. the additional term in the Ernst JMP expression [lo] is numerically greater than the I term, and therefore cannot be disregarded, especially as plasticity becomes more extensive, i.e. as x increases. Indeed, if it is disregarded, the predicted slope of the JM curve is greater than that of the Jr, curve, whereas the reverse is true if the III term is not disregarded.

4. FINITE

SIZE ON

CRACK THE

IN

CRACK

AN

INFINITE

FACES

AT

SOLID, THE

SUBJECTED

CRACK

TO

LOADS

CENTRE

Figure 2 shows the model of a crack of length a* = 2a in a solid of infinite dimensions in the plane of the figure but with thickness B in the direction of the figure normal. The crack faces are subjected to loads P at the crack centre, and the plastic deformation is again represented by line plastic zones of length R, ahead of each crack tip, the tensile stress within these zones being Y. With this model, dimensional considerations again show that A,, is given by a functional relation of the form

940

E. SMITH

V

V

*

A

il*=?H

R

e

R

Fig. 2. A crack of length O* = 20 in an infinite solid, with the crack faces subjected crack centrc.

to loads P at the

It then follows, using the same arguments as in the preceding section, that a two eta factor J,,,, description is appropriate with n = - l/a, = -1/2a and 7, = l/a, = 1/2a; relations (7) and (8) then give dD --_=

dl,

da,

da,

UDP ___

da,

3.J DP

d&W

da,=

i-1

da,

JDP

(26)

A,=<

or, with a, = 2a (27)

with the II term [see relations (7) and (S)] being equal to zero. For this model, the crack tip stress intensity K for linear elastic behaviour expression [ 141

is given by the

KYf-

(28)

BVGl

The stress is finite at the tips of the plastic zones if the stress intensities due to the applied loadings and the restraining stress Y are equal, whereaupon standard results [14] give P Bm

2Y

=-dx(a+R)sec-’ n

(29)

or

Gy =8set

0

(30)

with x = R/a and (1 +x) = set 8. With JD = Y@, where CDis the crack tip opening displacement, standard results [14] give @=xln

I&B

(a + R) + q(a + R)’ - a2

i (a + R) - q(a + R)2 - a*

]+~(aIn(a~)-\/(a+R)‘-a2secm’~~)} (31)

whereupon, noting that JDp = (JD - JE) and JE = K2/E,, with K being given by relation (28), it follows that JDp is given by the expression !%$+&[

OsecBln(secB+tan8)+lnsecB-etane-$sec’e

Now A, is given by an expression

1

(32)

of the form

GAP Ya

-km

(33)

941

&, crack growth resistance curve

where the function g(B) is as yet undetermined. gives

Differentiation

of relations (30), (32) and (33)

(34)

Noting that JDP,A,, P and a are related by the expression (371 it follows from relations 334)-(37) that

which integrates to give- after sub~ti~tjo~

for f(e)

using relation (32):

2 =g($) = sI. [ln(sec 8 -t-tan 0) + e set 8 In see e]. 32:

It then follows from relation (27) and relation (321, that the term I is given by the retation

EP

R&J

- z se&

-=esec8lrr(secB+tan8)Jrlnsec8-8tan8 8Y2

(40)

while reiations (28), (341, (36) and (391, show that the term III is given by the relation

The magnitudes of the terms I, IX1 and (I + III) are shown in Table 2 for various values of x = R/a. Inspection of these results clearly shows that the III term, i.e. the additional term in the Ernst JMP expression [lo) is numericaily greater than the I term, and thus cannot be disregarded, especialfy as plasticity becomes more extensive, i.e. as x increases. Indeed, if it is disregarded, the predicted slope of the & curve is greater than that of the JD curve, whereas the reverse is true if the III term is not disregarded.

Figure 3 shows the model of a crack of length Q, = 2a in an i~~~itely long panel, of width 2tu, and of thickness B in the direction of the figure normal. The panel is subjected to a tensile load Table 2. Values of the terms I, III and (I + III) for tke mode1 of a finite size crack in an infinite soiid, subjested to loads on the crack faces at the crack centre xc-

R a

E,,I Y2

E,,111 YZ

&“,,(If III) “.“__ P

942

E. SMITH

-

-7

R

Fig. 3. A crack of length N* = 2n in a panel of width 2~, and subjected to a tensile load E.

P, and the plastic deformation is again represented by line plastic zones of length R, ahead of each crack tip, the tensile stress within these zones being Y. Concentrating on the situation where the line plastic zones are at a great distance from the vertical panel surfaces. i.e. (a + R)/w is small, enables the infinite body results obtained by Bilby ef al. [13] to be used as a basis for the considerations. Thus AP is given by a functional relation of the form (42) and it thereby follows, by analogy with relation (2). that 4 ~a$ and (c, is independent of a,, whereupon relations (3) show that a two eta factor J,,I, description [relation (l)] for a non-growing crack is appropriate with q = 0 and 77,-=210, = I /cr. It then follows that 77l = q2 = - 1/a, = - 1/2a, and then relations (7) and (8f give dJD

d&l ~-------_ d&P dJNP

da,

G.

da,

da,

il.l~, p __

(43)

! da * i A,>

I)

or, with a, = 2a, dJ,> ~JM df,, --_-_~~-------_~ da da da

dJ,w

(44)

dn

T I

t III

with the II term [see relations (7) and (8)] being equal to zero. For this model, J,, = Y@, with Q1 being the crack tip opening displacement, and Jn is given by the expression [ 131 8Y2U J,, = -nF‘ In secx.

A0

(4Sj

943

JM crack growth resistance curve

With the tensile load P being equivalent x = n(~/2Y is given by the relation

to a tensile stress u = P/2Bw,

the parameter

this condition satisfying the requirement that the stress is finite at the outer extremities of the line plastic zones. Since the elastic contribution JE to JD is equal to K2/Eo with K = cr&, it follows from relation (45) that J,,=%lnsec*-F. 0

(47)

0

Thus by making use of the relation (48) it follows from relation (47) that Ap=g(tan*-X). 0 Relations relations

(44), (47) and (49) show that the I and III terms are given respectively

by the

2

%I

(50)

-=lnsecX-: 8Y2 and nE,,III _= 8Y2

- 2(tan x - x)’ tan2X ’

(51)

The magnitudes of the terms I, III and (I + III) are shown in Table 3 for various values of x = R/a. Inspection of these results again clearly shows that the III term, i.e. the additional term in the Ernst JMp expression [lo] is numerically greater than the I term, and thus cannot be disregarded, especially as plasticity becomes more extensive, i.e. as x increases. Indeed, if it is disregarded, the predicted slope of the JM curve is less than that of the J,, curve, whereas the reverse is true if the III term is not disregarded.

6. DISCUSSION

As indicated in Section 2, conflicting expressions have been presented [8, lo] for JMp, the plastic component of JM, the Ernst expression [lo], with which the present author concurs, containing an additional term when compared with the expression obtained by Zahoor [S]. The importance of this additional term has been demonstrated in an earlier paper [ll] for the model of a symmetrically double-notched solid with tensile loading of the small remaining ligament. Table 3. Values of the terms I, III and (I+ III) for the model of a centre-cracked panel subjected to a tensile load x=-

R

E,,III

&,(I + III)

a

YZ

Y’

-0.313 -0.797 - 1.243 -1.624

-0.181 -0.428 -0.620 -0.756

0.5 1.0 I.5 2.0

0.132 0.369 0.623 0.868

944

E. SMITH

It was therefore concluded that the difference between the rival JMr expressions cannot, in general, be disregarded when quantifying a material’s crack growth resistance. The present paper has analysed three models, all characterized by the feature that the deformation J integral for a non-growing crack can be represented in terms of two eta factors. 77 and n(., involving respectively the energy and complementary energy integrals. The results from the present paper’s analyses clearly underpin the conclusion drawn from the analysis of the double-notched configuration, in that they show that the additional term in Ernst’s .& formulation has a signi~cant effect on the di~ereI~ce between the J,, and JM crack growth behaviours, when plasticity becomes extensive. Further~~ore. the additional term can also provide a significant contribution to JW itself. Thus for example with the models (Figs I and ?j where the crack faces are loaded, if the additional term is III. then &III/Y IO for K/n -r 2, and this can be significant in comparison with the material tearing modulus (&‘Y’j d//d@, particularly when the material’s crack growth resistance is not especially cxcessivc. This conclusion accords with the results for the double-notched configuration, for which it was shown that the importance of the additional term is greatest when there is cxtensivc deformation after general yield. In assessing the importance of this paper’s results and those for the double-notched configuration, it is worth emphasizing that both sets of results have been with regard to situations where n, = 0, when the additional term in the Ernst JMp expression is non-zero [see relation (6) for example]. In situations where a single eta factor J,,, description for a non-growing crack is appropriate, i.e. qC.= 0, the arguments regarding the importance of the additional term in Ernst’s JMP expression become irrelevant. Bending of a small remaining ligament of width h, for which q = 2/h and 7,. = 0, falls into this category. Furthermore, it has been shown [IS] that a single eta factor J i,i, description is reasonably accurate for the compact tension configuration up to h/W <0.8, with h = ligament width and W L distance between loading points and back free surface. This configuration involves primarily bending deformation of the ligament, and it would therefore seem that it is for situations where this is not the case, i.e. with the models considered in this paper and with the double-notched con~guration~ where a single eta factor fr,P description with qC = 0 is inappropriate, that the implications of this paper’s analyses become important.

REFERENCES H. A. ERNST, ASTM STP 803, 191 (1983). H. A. ERNST, ASTM STP 995,306 (198Y). J. R. RICE, W. J. DRUGAN and T. L. SHAM, ASTM STP 700, 189 (IWO). D. E. MCCABE and J. D. LANDES, ASTM STP 803,353 (1983). D. E. MCCABE, J. D. LANDES and H. A. ERNST, ASTM STf 803,562 (19X3). G. P. GIBSON. S. G. DRUCE and C. E. TURNER, Inr .I. Fmcrrtre 32,219 (19%). H. A. ERNST, K. H. SCHWALBE, D. H. H~LLMANN and D. E. MCCABE, iac.J. Frtrinm 37, X3 (198X). A, ZAHOOR, J. Engng f~ufer. Technoi. 111, 13X (198Y). E. SMITH, Im. f. Engng Sci. 29,709 (1991). H. A. ERNST. in Evaluation of upper shelf tou8hness rcquir~~~nts for reactor prcssurc: vessels. Rcpart prepared by Novetech Corporation, Materials En8ineering Associates (MEA) and Georgia Institute of Technology for EPRI (March 1989). [ll] E. SMITH, Inr. J. Engng Sci. 30, 1621 (1992). [12] D. S. DUGDALE, J. Mech. Phys. Solids 8, 100 (1960). 1131 B. A. BILBY, A. H. COTTRELL and K. I-l. SWINDEN. Pmt. R. SW. A272,304 (1963). [ 14) H. TADA, P. C. PARIS and G. R. IRWIN. The Srress Ann[ysis of Cm& Handbook. Del Research Corporation. Hellertown, Pennsylvania, U.S.A. (1973). 115) E. SMITH and T. J. GRIESBACH, ASTM ST/’ 1171,418 (1993).

[l] [2] [3] [4] IS] f6] [7] [X] [9] ]lO]