The use of eta factors to describe the plastic contribution to the J-integral

Inr. 1. Engng

Sci Vol. 29, No. 6, pp. 709-716, Printed in Great Britain. All rights resewed

1991

0020-7225191

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Copyright @I1991Pergamon Press plc

THE USE OF ETA FACTORS TO DESCRIBE THE PLASTIC CONTRIBUTION TO THE J-INTEGRAL E. SMITH Manchester University-UMIST, Materials Science Centre, Grosvenor Street, Manchester Ml 7HS, U.K. (Communicated

by B. A. BILBY)

AL&r&-The paper defines the condition for which the plastic component J, of the deformation J integral for a non-growing crack can be expressed in terms of energy and implemental energy integrals using a two eta factor description; the single eta factor description involving only the energy integral is treated as a special case. A two eta factor description, valid for all levels of deformation, is shown to be strictly accurate only for situations where the solid has a single characteristic dimension, i.e. for the bending of a small remaining ligament where q = 2 and qc = 0, and for the tension of a small ligament where ?J= 1 and ?J~= -1. With more general situations, e.g. the compact tension specimen geometry, assumptions must be introduced in order to effect a two factor description; these assumptions are highlighted with regard to attempts that have been made to use the two eta factor representation. 1.

INTRODUCTION

mere are many engineering situations, particularly in the nuclear field, where it is desirable, if not essential, to have a reliable method for obtaining a material’s J-crack growth resistance curve, which can be used as a basis for the assessment of a cracked engineering structure. One potentially very attractive way is to relate the J integral to load, load-point displacement and crack extension measurements obtained from laboratory experiments. In obtaining such a relation it is necessary, as a basis, to have a reliable estimate of the magnitude of the deformation J-integral (JD) for a non-growing crack. Since Jo can be separated into an elastic component JE, which is directly related to the stress intensity factor K,, and a plastic component Jp, the determination of the value of Jo essentially depends upon the determination of the value of Jp. JE provides the dominant contribution to JD at low deformation levels, whereas Jp provides the dominant contribution at high deformation levels. It is particularly for these high levels that a reliable estimate of Jp is required, although to cover all eventualities, it is desirable to have a reliable estimate of Jp throughout the complete spectrum of deformation levels, from small-scale yielding to post-general-yield situations. Against this background, there has been extensive discussion [l-lo] of the description of Jp for a non-growing crack in terms of the load P and the plastic component Ap of the load-point displacement, using the energy and complementary energy integrals and a relation of the form P 4 PdA,+$ A, dP Jp=-$/ (1) I0 0 B and b being respectively the thickness and width of the untracked ligament (Mode I plane strain deformation is assumed), while 9 and qc are dimensionless factors that depend solely on geometrical parameters but not on the level of deformation; a special case is where Q is zero, when the expression for Jp involves only the energy integral. If n and n, are known, it is a relatively straightfo~ard matter to obtain Jp from load (P) and load-point displacement (A,) measurements via relation (1). The objective of this paper is to critically review the description of Jp in terms of energy and complementary energy integrals, and to introduce some fresh perspectives on the topic. The structure of the paper is as follows. Section 2 focusses on the condition for which Jp can be expressed by means of relation (1) for all deformation levels. Section 3 then shows that the condition is satisfied for only a strictly limited range of geometrical configurations, i.e. for cases where there is a small remaining ligament in an otherwise very large solid. Section 4 reviews previous attempts to express Jp via relation (1) while, finally, Section 5 reviews the current status of the problem in the light of the fresh perspectives introduced in the paper, and suggests avenues for future research. 709

E. SMITH

710 2. THE

CONDITION

FOR

WHICH J, CAN BE EXPRESSED ETA FACTORS

IN TERMS

OF TWO

There has been detailed discussion [5-71 of the condition for which Jp can be expressed sotely in terms of the energy integral, i.e. relation (1) with ?I, = 0. The present section defines the condition for which Jp can be expressed more generally in terms of two n factors, the single eta factor condition being treated as a special case. Now A, can always be expressed as a power series in P in the form: A $=P:{fo+fiP*+fiP:+..*}

(2)

an expression that satisfies the requirement [8] that Ap ap3 for small scale yielding conditions; P* = PIBWY, W being the specimen width and Y the flow stress of the material. The functions fs are functions of the ligament width b, specimen width W, and any other dimensions (L1, Lz, etc.) of the solid, i.e.

with the dimensions L,, L2, etc., along with W, remaining constant during crack growth. Now Jp can be expressed [l] in the equivalent forms: dP

(4)

d4

(5)

and dA,=; AP

where a = W - b is the crack length, and it follows from relations (2), (3) and (4) that: (6) where fl is equal to dA/db. Since relation (2) gives the energy integral: AP P dA,

and the complementary

T&+Tf,+$&...}

(7)

$,+$fi+?A++**

(8)

energy integral:

it follows, from relations (l), (6), (7) and (8), by equating terms involving powers of P*, that in order for Jp to be described by the form (1) for all values of the load P, the following functional relation must be satisfied: -bff

= ((3 + 3)~ + n,lf,

(9)

for s = 0, 1, 2, etc. This functional relation integrates to give:

A=Amp( -1 {(s + 3)77+ rlc> db b

}

(10)

where the A, are functions which are independent of the load and ligament width. It then follows, from relations (2) and (lo), that in order for Jp to be described by the form (2) for all levels of deformation, A,, must be expressible in the functional form:

(11)

The plastic cont~bution to the J-integral or equivalently

711

in the form: $ = @(b)H(W(b))

(12)

where @ and +J are functions of the ligament width b, but not of the load P. If Ap is expressible in the form (12), Jp can be expressed in terms of the energy

complementary expressions:

and energy integrals via relation (l), with the 7 and Q factors being given by the .=_b!Y! @ db

(13)

bdQ,

“c=-Pz

Relation (12) can be inverted to give the equivalent functional form:

with @(b) = l/$,(b) and I@) = l/W,(b). For Jp to be expressible in terms of a single q factor, i.e. expression (1) with qC= 0, it follows from relations (12) and (14) that Ap must be expressible in the functional form:

with q being given by relation (13). Alternatively,

P must be expressible

in the functional form

(a special case of relation (15)): p =

W*W*( $f}

07)

i.e. there should be separability of the b and A, terms; this point has with V(b) = U&(b), been recognised in earlier considerations [6,7]. As indicated by Paris et al. [6], if the solid’s configuration degenerates so as to give pure bending of a small remaining ligament, this being the only characteristic dimension of the solid apart from the thickness B, dimensional considerations give a relation of the form (16) with q(b) a l/b’; in this case there is a single eta factor which is given by relation (13) as q = 2. Returning to the more general two eta factor description, where Q # 0, with the case of tensiie loading of the small remaining ligament associated with a symmetrical deep doublenotched specimen geometry, where the ligament width is the only characteristic dimension apart from the thickness, dimensional considerations give a relation of the form (12) with 4(b) 0: b and q(b) 0: l/b; in this case there are two eta factors which are given by relations (13) and (14) as 17= 1 and qC= - 1[7]. The next section shows that these two small remaining ligament situations are the only situations for which an eta factor description (with q and vC being independent of the level of applied load) is strictly valid.

3. THE CONDITION

FOR Ap TO BE EXPRESSIBLE IN THE FORM TO GIVE A TWO ETA FACTOR DESCRIPTION

(12)

SO AS

The preceding section has shown that for J, to expressible in terms of two eta factors for all levels of deformation from small-scale yielding to post general-yield deformation, A, must be expressible in the functional form (12). It has furthermore been indicated that two situations for which this is possible are: (a) bending of a small remaining ligament when q = 2 and qC= 0, and (b) tension of a small remaining ligament when q = 1 and qC= -1. It will now be shown that these are special cases and that, in general, Ap cannot be expressed in the form (12); such a description is possible only when the solid’s geometry involves a single iength parameter, i.e. the ligament width, apart from the solid thickness.

E. SMITH

712

By considering the displacement distribution associated with a dislocation or a line force applied to a crack, it will be shown that in general, the expression for AP is of the form: P WBY’

A

d=F

W

b b W’L,’

b -... L2

(18)

I

where L1, L2, etc. are other characteristic dimensions of the solid. This being the case, it immediately follows that, in general, the functional form as given by relation (12) is not satisfied. To make the demonstration, consider the configuration in Fig. 1 where a semi-infinite solid of thickness B contains a very deep crack such that the remaining ligament width is b, the solid being subjected to tensile loads that are applied at a distance W from the right-hand free surface. The Dugdale-Bilby-Cottrell-Swinden [ll, 121 strip yield representation is used to describe the plastic deformation at the crack tip, Y being the tensile stress within the simulated yield zone which extends to a distance s from the free surface. The plastic contribution AP to the load-point displacement is of the form:

while, in order for there to be finiteness of stress at the tip of the strip yield zone, the yield zone size must satisfy a relation of the form:

(20) where F,, F2, F3 and F4 are functions that need not be determined. By elimination follows from relations (19) and (20) that A, is given by a relation of the form:

of s, it

$‘=F{&,$}

(21)

This relation is clearly more general than relation (12) and consequently, in general, the expression for AP is not of the required form to give a two eta factor representation for Jp. This example clearly shows that a two eta factor description for Jp, valid for all loading levels, is possible only when the solid’s geometry involves a single length parameter, i.e. the ligament width, apart from the solid thickness. This means that if a two eta factor description is )

P

-

s

.

M

b

. w

Fig. 1. The model of a semi-infinite solid containing a very deep crack, the remaining ligament width being b. The solid is subjected to tensile loads that are applied at a distance W from the right-hand free surface.

The plastic contribution to the J-integral

713

used for more general situations, the description must be viewed as giving only an approximate value for J,. It is against this background that the next section reviews the important attempts that have been made to express JP via the two eta factor description, with the compact tension (CT) specimen geometry particularly in mind.

4. REVIEW

OF RELEVANT

PREVIOUS WORK WHERE TERNS OF ETA FACTORS

J, IS EXPRESSED

IN

The pioneering work of Rice et al. [l] considered a deeply cracked specimen subjected to bending deformation; in this case the ligament width is the only characteristic dimension, and in accord with the previous section’s considerations a two eta factor description for JP is valid for all levels of deformation, with n = 2 and qc = 0 (see Section 2). Subsequent work, particularly with the compact tension (CT) specimen geomet~ in mind, has focussed on the use of eta factors for situations where the ligament width is not necessarily small. Thus Merkle and Corten [2] used a simulation mode1 very similar to that shown in Fig. 1, where the presence of all other surfaces, other than the back surface, are ignored. They assumed limit load conditions and made certain assumptions concerning the deformation of the ligament region; in this way they were able to effect a two eta factor description, and obtained expressions for 11and Q in terms of b/w. Ernst et al, [3] used d~ensional considerations to obtain a relationship between load, load-point displacement and ligament width, and assumed the displacement and ligament width terms to be separable, thus allowing for an eta factor description. Their resulting J solution appears in the ASTM Standards 1131, and is widely used for defining a material’s crack growth resistance. Ernst [4] refined the Merkle-Corten analysis [2] by introducing different assumptions concerning the ligament deformation, while still assuming limit load conditions. He thus arrived at a two eta factor description, but with slightly different q and nc values to those obtained by Merkle and Corten. Zahoor [lo] (who also has considered the three-pointbend specimen geometry [9]) inferred a solution between the load, load-point displacement, and crack length from the contained yielding behaviour of the CT specimen, and by making certain simplifying assumptions in his analysis, was able to obtain a functional relation of the form (12), which accordingly allows for a two eta factor des~~ption for Jp. A detailed critique of Zahoor’s procedure is presented elsewhere [14]. The conclusion emerging from this brief survey is that, not surprisingly in view of the proceding section’s considerations, assumptions must be introduced in order for Jp to be expressible in terms of two eta factors via a relation of the form (1) that is valid for all deformation levels. It is only where a solid has a single characteristic dimension that such a representation is strictly accurate.

5. DISCUSSION

Against the background of the necessity to have a reliable method for quantifying a material’s J-crack growth resistance curve, which can be used as a basis for assessing the integrity of a cracked engineering structure, this paper has been concerned with the estimation of the magnitude of the plastic component (J,) of the deformation J-integral (JD) for a non-growing crack, recognising that it is desirable to have a reliable estimate of Jp throughout the complete spectrum of deformation levels, from small-scale yielding to post-general-yield situations. The paper has focused on the representation of Jp 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 two eta factor description. Section 2 defined the condition for which such a description is possible, with the single eta factor description involving only the energy integral being treated as a special case. A two eta factor description, valid for all levels of deformation, is possible provided that AP can be expressed in terms of ligament width and applied load via a functional relation of the form (12). Section 3 then demonstrated that such a functional relation is satisfied for only those geometrical configurations where the solid has a ES 296-E

714

E. SMITH

single characteristic dimension, i.e. for the case of bending of a small remaining ligament when 7 = 2 and Q = 0 and tension of a small remaining ligament when rl= 1 and rlC= -1. Section 4 has highlighted the various attempts that have been made to effect a two eta factor description for the compact tension specimen geometry; against the background of the present paper’s considerations, assumptions must be introduced in order for a two eta factor description to be possible, and these assumptions have been indicated. The Merkle-Corten [2] and Ernst [4] approaches should give reasonably reliable estimates for Jp at high applied load levels, since their assumptions have been related to the deformation patterns pertaining to the limit load state. This is reasonable for very ductile materials where Jp will provide the dominant contribution to J D. But to encompass the behaviour of less ductile materials, it is desirable to have a reliable estimate of Jp throughout the complete spectrum of load levels. Future work will use the eta factor values obtained from these limit-load approaches, and will examine how the Jp values that they give for the small-scale yielding situation compare with the Jp value obtained directly for this special situation; it is possible to obtain an expression for Jp for the small-scale yielding case in terms of the crack tip stress intensity KI [15]. it should therefore be possible to see whether the eta factor description, based on limit load solutions, is an adequate description for the complete spectrum of deformation levels. This paper has concentrated on the determination of Jp for a non-growing crack but, as indicated in the Introduction, such an assessment is a pre-requisite for quantifying a material’s crack growth resistance curve. If a two eta factor description is reasonably accurate, it has been shown that Jp for a crack which grows from a length a0 to a length a is given by the expression ]lO, 161:

Jp = J,a + a(J,,lb) 9{ 1+ vc - q-i- (1 - ar)r,Qda + j” (rclW36) * PA,, * (qz - r.Qda (22) i all

a0

where Jpo is given by relation (l), but with b, q and Q moved inside the integrals. rll and rlz are defined by the relations: (23) the primed quantities being total derivatives with respect to a/W = cu; Jpo is calculated using current crack length and associated quantities. With the aid of relation (22) it is possible to obtain the material’s crack growth resistance curve in terms of P-A-& data from a single specimen. It should also be noted that serious consideration is currently being given to the modified J integral, i.e. JIM, instead of JD, as an alternative and possibly more accurate description of crack growth. This is defined by the relation [17]: da

JM=JD-

(24)

With this alternative description of the J integral, the solution for the plastic component has been given as [lo]: JIM=

I

(rf/Bb)P

. dh, +

i

(vcJBb)Ap

+dP

of JM

(25)

where b is the current remaining ligament width and rl and rlC are defined using the current crack length. (There is currently some dispute over the accuracy of relation (29, since Ernst [16] believes that the right-hand-side of relation (25) should contain an additional term, a view with which the present author concurs.) Finally, it is worthy of mention that there is a relation between this paper’s considerations with regards to the two eta factor description of Jp and the R-6 description [18] of crack extension. With the R6 method, where the failure curve is expressed in terms of a unique relation between Kf/K,, and P/PL (as with the Option 1 and Option 2 fo~ulations), where PL is the limit load, implicit is the assumption that the J integral, and therefore the plastic

The plastic contribution to the J-integral

715

component Jp of the J integral, can be expressed in the functional form:

where pt, = P/B WY. Following arguments similar to those used in Section 3 with regard to AP, a formulation of Jp of the type given by expression (26), is possible only when the sohd’s geometrical configuration involves a single characteristic dimension, apart from the solid thickness. Thus the only cases where a unique relation between K,/K,, and P/P, is strictly accurate in predicting crack extension are the bending and tensile deformation of a small remaining ligament. As emphasized in this paper, these two cases are the only ones for which the two eta factor description of J,, is strictly accurate.

6. CONCLUSIONS The paper has defined the condition for which the plastic com~nent of the defo~ation J integral for a non-growing crack can be expressed in terms of energy and complementary energy integrals using a two eta factor description. The single eta factor description involving only the energy integral is treated as a special case. A two eta factor description, valid throughout the complete spectrum of deformation levels, is strictly accurate only for cases where the solid has a single characteristic dimension, i.e. the bending and also the tension of a small remaining ligament. With more general geometrical configurations, e.g. the compact tension specimen geometry, assumptions must be introduced in order to effect a two eta factor description. A simiiar state of affairs exists with regard to the R6 formulation of crack extension; a failure curve expressed in terms of a unique relation between KlfKfc and P/P,, as with the Option 1 and Option 2 formulations, is strictly accurate only where a solid has a single characteristic dimension. Acknowledgements-This work has been undertaken as part of the EPRI Programme on Component Reliability and the author thanks Mr T. J. Griesbach for his encouragement and for valuable discussions during the course of the investigation.

REFERENCES [l] J. R. RICE, P. C. PARIS and J. G. MERKLE, ASTM STP 536, p. 231 (1973). [2] J. G. MERKLE and H. CORTEN, ASME J. Press. Vess. Tec~nol. 286 (1974). [3] H. A. ERNST, P. C. PARIS and J. LANDES, ASTM STP 743, p. 476 (1981). [4] H. A. ERNST, ASTM STP 791, P. 499 (1983). [S] C. E. TURNER, ASTM STP 700 P. 314 (1980). [6] P. C. PARIS, H. A. ERNST and C. E. TURNER, ASTM STP 7#, P. 338 (1980). [7] E. SMITH, Int. J. Fruct. 26, R.55 (1984). [8] H. A. ERNST and P. C. PARIS, Report prepared for U.S. Nuclear Regulatory Commission. NUREG/CR-1222 (January 1980). [9] A. ZAHOOR, J. Engng Mater. Technol. 111,132(1989). [lo] A. ZAHOOR, J. Engng Mater. Technol. 111, 138 (1989). (111 D. S. DUGDALE, J. Mech. Phys. Sol& 8, 100 (1960). [12] B. A. BILBY, A. H. COTTRELL and K. H. SWINDEN, Proc. R. Sot. A 272, 304 (1963). [13] Standard test method for determining J-R curves. ASTM Methods E813 and E1152, Section 3, Annual Book of ASTM Standards (1988). 1141 E. SMITH, An appraisal of Zahoor’s analysis for estimating the plastic component of the deformation J integral for a stationary crack. f. Engng Muter. Technol. Accepted for publication.

716

E. SMITH

[15] E. SMITH, unpublished work. [16] R. M. GAMBLE, A. ZAHOOR, A. HISER and H. A. ERNST, Evaluation of upper shelf toughness requirements for reactor pressure vessels. Report prepared by Novetech Corporation, Materials Engineering Associates (MEA) and Georgia Institute of Technology for EPRI (March 1989). [17] H. A. ERNST, Elastic-Plastic Fracture: Second Symposium, Vol. 1. ASTM STP 803, p. 191 (1983). [18] LFJiNE, R. A. AINSWORTH, A. R. DOWLING and A. T. STEWART, ht. J. Press. Vess. Piping 32, 3

(Received

13 September

1990; accepted 3 October 1990)