The definition and measurement of crack closure

0013-7944/92 $5.00 + 0.00 0 1992 Pergamon Press Ltd.

Engineering Fracfure Mechanics Vol. 43, No. 1, pp. 109-I 15, 1992 Printed in Great Britain.

TECHNICAL NOTE THE DEFINITION

AND MEASUREMENT CLOSURE

OF CRACK

DAVID TAYLOR Mechanical Engineering Department, Trinity College, Dublin, Ireland Abstract-The conventional method of measurement of closure, using compliance measurements to define K,, and AK&. is reassessed from a theoretical standpoint. The method is clearly simplistic in that it assumes that closure occurs entirely at K,,, when it is now well established that cracks close gradually over a range of K during unloading. However, results obtained from this method are often very useful in explaining fatigue behaviour. This apparent anomaly is explained if we recall that AK may be defined in terms of the release. of elastic energy. It can be shown that, by measuring a change in compliance, the observer is actually making a direct assessment of the energy change associated with the closing crack and that the conventional measure of A& is, fortuitously, a simple approximation of that energy change. However, the measurement is not precise because it omits the energy term associated with the curved portion of the unloading line, below 4,. When this term is included, a more accurate assessment of AX;, can be made, which we define as AK,,. The use of AK,, is able to resolve some anomalies which are present in experimental data, such as the variation of AK,,,c, with R ratio.

THE MEASUREMENT- OF CLOSURE USING COMPLIANCE VARIATION TORE AREa number of methods of measuring crack closure, but by far the most common is the measurement of specimen compliance using, for example, a crack mouth gauge or back face strain gauge. Figure 1 shows typical experimental results [l]; a plot of displacement, CJ,as a function of load, P, is used to define PC,,, the value of P at which the compliance begins to decrease during unloading. Ignoring hysteresis, it is usual to define the effective stress intensity range, A&, as:

This method is relatively simple to implement and has been, on the whole, very successful in explaining the role that crack closure plays in crack propagation. A-variety of effects has been successfully ixplained by using closure, such as the effect of R ratio .12.31. -. microstructure 14.51 and environment 16.71. To take one example, the effect of R ratio on the threshold value AK,,, can oAen be explain& d terms of the increased crack closure which &curs at low R [2,3]. The success of this method seems somewhat surprising when one considers how simplistic it actually is. One objection which can be raised is that the method involves a very simple picture of the way that a crack opens and closes. The definition of AK& only really makes sense if we assume that all the closure occurs instantaneously at K = Kc,,, so that the crack is completely closed, and therefore ineffective, for all K less than K,, In practice, cracks are observed to close very gradually,

Crack fully open

l

Crack

opening point

X Crack closing point

Crack partially open )

o Crack fully closed

Creek fUftV CiOsed

P, K

l-

Displecement, 6

Fig. I. Typical load-displacement EFM 13,1--H

results [l] showing the definition of closure parameters. 109

110

Technical

Note

C

(Crack

Length

=

a)

C” C’

(Crack

length

=

a+da)

U Fig. 2. Schematic

P/U lines for crack

lengths

a and a + ha in perfectly

linear elastic material.

with closure usually

spreading backwards from the crack tip as the load is reduced. Thus the crack is partially open at K < Kc,,, and presumably exerts some influence on crack propagation. These objections may be resolved if we consider more carefully the measurement which is being made. Since we are essentially measuring a change in compliance, it is useful to remember that stress intensity can be defined fundamentally in terms of an energy release rate, which can be deduced from changes in compliance. This approach constitutes the standard energy balance definition of K, and appears in the compliance calibration method for standard specimen geometries. This paper reviews briefly the appropriate theory and goes on to show that the conventional definition of AK,, is in fact equivalent to a simplified use of the energy balance method.

THE DEFINITION OF AK The standard

energy

balance

derivation

of K using linear elastic fracture E6W K= ! ssa-

mechanics

‘2

may be written:

(1)

!

Thus the stress intensity, K, is proportional to the square root of the elastic energy change, 6W, for constant Young’s modulus (E), specimen thickness (8) and crack extension increment (da). Since K is proportional to P, the value of AK at R =0 will be proportional to the square root of the area between the two loading lines OC and OC’ in Fig, 2.

AK AT POSITIVE R RATIOS When R > 0, AK is conventionally defined as K,,, - K,,, , i.e. the monotonic value of K is defined at K,,, and K,,,,, and the difference taken. Referring to Fig. 2, this amounts to evaluating the difference between the square root of area OCC’ and the square root of area OAA’. It can be shown by simple geometry that this is equivalent to taking the square root of area ACC”, where AC” is parallel to OC’: that is. - -~~ ~~~ j&~~6?” = ./area OCC’ - Jarea OAA’. (2) This can be appreciated if we note that it is equivalent to shifting the origin to (Cm,“, P,,,,,)and raising the line for (a + 6~) so that it passes through this new origin. This construction will be used later to obtain the conventional evaluation for AK. It is noted in passing that a more appropriate definition of AK at positive R ratios might be made by taking the square root of the difference between the two areas OCC’ and OAA’, which is the same as the square root of area ACC’A’. Discussion of this point is, however, beyond the scope of this paper. whose purpose is to consider the effect of closure on the conventional definition of AK.

THE EFFECT OF CLOSURE The existence of closure modifies the P/U plot as shown in Fig. 1, introducing a change in the slope of the lines at low loads. We consider first a simplified form of the plot, as shown in Fig. 3. Here closure is assumed to occur at a single value of P, termed P,,, or P,, so that, at point B on the plot, the slope changes from that of the cracked specimen to that of the untracked specimen. The R ratio is assumed to be zero. The lines for crack lengths a and a + da will be coincident along OB. Lines OD and OD’ represent the same specimen if no closure occurs; thus OD is parallel to BC and OD’ is parallel to BC’. The conventional method of measurement of AK,, amounts to defining APee by: AP,, = P,,,

- P,,,,

with AK proportional is exactly equivalent AK, is proportional

to AP. It is a simple matter to show that, for the case described by Fig. 3, this definition of AK,, to calculating AK using the energy approach outlined above. The energy method would deduce that to the square root of the triangular area BCC’, whereas the conventional value of AK (uncorrected for closure) is found from area ODD’. If we define points E and E’ such that the areas of triangles BCC’ and OEE’ are the same, then: P, = P,,, - PC,, = APeR.

Technical Note

111

0 “uE Fig. 3. Schematic P/U lines for specimens showing closure (OBC/OBC’) and no closure (OED/OE’D’), assuming closure occurs piecemeal at point B. Thus the (closing) specimen’s behaviour is the same as if it were not closing but loaded from zero to Pa. Therefore we will deduce the same value for AK,, whether we use the conventional closure method or the energy approach.

A COMPLICATION: THE CURVED PORTION OF THE P/U PLOT A complication arises when we note that experimental P/Ucurves (Fig. I) differ from Fig. 3 in that they are significantly curved below PGlO.Typically the curvature is continuous between P,,, and zero, so that the slope of the line only becomes equal to the compliance of the untracked specimen at P = 0, as shown in Fig. 4. This figure provides the basis for much of the following discussion, so it is important to outline the assumptions used in its construction. (a) For a crack of length a, the loading line follows OBC; compliance varies from Co (compliance of the untracked specimen) to C, (compliance of cracked specimen) as P varies from P,, to P, Compliance is assumed to vary linearly with P, thus: C=C,+(C,-C&P-P&P,-P,). (3) The line BC is a straight line of slope C,. (b) For a crack of length a + 6a, the loading line OB’C’ is constructed in the same way as OBC. The reduced compliance is denoted C,.; the untracked compliance is of course unchanged. (c) The pomt B’ occurs at the same value of U as the point B; the significance of this assumption will be discussed below.

CONSEQUENCES:

THE UNDERESTIMATE OF AK&

Referring to Fig. 4, &,, will conventionally be defined using PclO,which is equal to P, . As shown above, this will lead to an estimate of AKJAK which is exactly the same as that predicted by the energy approach so long as all the lines are straight (as in Fig. 3). However, in Fig. 4, we must allow for the extra energy available, expressed as the area between the two loading lines at P < Pe: the area OBB’. Since the total area between the lines is now larger, AK, will be larger, i.e. the compliance method will lead to an underestimate of AK,,. To avoid confusion we define the result of the energy method as AK,,, , to distinguish it from the conventional result for AK,,, defined as K,,,,, - Kc,,.

EXPERIMENTAL EVIDENCE Can this underestimate be demonstrated experimentally? Consider the threshold value AKth. It is often proposed that, for example, the effect of R ratio on threshold can be explained in terms of closure; if so, then the effective stress intensity

0

0.2

0.4

0.6

0.6

R Fig. 4. Schematic P/U lines showing closure occurring gradually between B (or B’) and 0.

lb) Fig. 5. Measurements of AK,, and AK,h.eRat various R ratios (from Blom [2]).

Technical Note

112

1.0

0.5

0

R ratio

Fig. 6. Data on the effect of R ratio on AK,,, taken from sources on a wide range of materials from Taylor (81. The solid data points are from datasets which tend to show a plateauing effect at high R.

at the threshold, AK,,,c,, should be a constant for all R. Experimental values for this parameter are somewhat scattered due to measurement errors, but in many cases there seems to be a trend in which AK,,,e, increases with R. Figure 5 shows an example of this effect, from results due to Blom [2]. Here the measured value of AK,,,, varies by a factor of 2.2 across the range of R ratios used. This increase is to be expected from the above argument. At high R, when there is no closure, AKth = AK,,,c, = AK,h,rrue. As R decreases and Km,” drops below Kc,, , the measured value of AKeRwill be less than AK,,,,, as outlined above, and this discrepancy will increase as R decreases. The same effect can be inferred when we look at a large amount of data on AKth as a function of R, taken from a wide range of materials (Fig. 6, using data from ref. [8]). A simple closure argument, in which K,, is taken to be constant, will conclude that all this data should lie on a line given by K,,,,, = constant, provided R is small enough for some closure to occur. In fact this line tends to form an upper bound to the data for R < 0.6 (for higher R a plateauing effect occurs since most materials show no crack closure). This implies that AK,,,, is constant in some cases, but decreases with R in the majority of cases, giving lower thresholds than expected at low R. From Fig. 6, the ratio between AK,,,c, at R = 0.5 and at R = 0 varies from unity to 2.5-a result which is very similar to the factor of 2.2 shown by the results of Blom in Fig. 5. It is concluded that there is some experimental evidence to support the suggestion that Ai& is being underestimated when measured using compliance methods.

QUANTITATIVE PREDICTION Using Fig. 4 it is possible to make a quantitative estimate of AKcRand AK,,,, at various R ratios, for comparison with experimental results. The conventional definition of AK,, is equivalent to the assumption that the two loading lines follow O’BC and O’BC”, where the slope of O’B corresponds to the compliance of the untracked specimen; thus: AK:, = a(area BCC”) where a is a constant found from eq. (I). The true stress intensity factor is given by the real area between the loading lines, i.e.: AK&, = a(area BCC” + area OBB’ + area BCX’B’).

(4)

(5)

It follows from eq. (3) that the line OB is described by: (i=c”P+(c~-c0)P2/2Pa.

(6)

Rearranging this in terms of P: p, ____ iic; + zic, - c,)u/P*)“* p = (C, - C,)

- C,).

(7)

113

Technical Note

0

75

25

0

100

87.5

37.5

U

100

U

Fig. 7. P/U curves for closing and non-closing cases, used in the numerical example (units of P and U are arbitrary).

Fig. 8. P/U curves as in Fig. 7 for the case where closure occurs gradually (as in Fig. 4).

Integrating gives the area under OB as:

P’B 3(Ca - Q2

((c: + 2(Cs - c,)cJs/Ps)“*

- c;, -s,

B

0

The equations for the line OB’ are the same as those above, with C,. replacing Ca. The value of P, is the experimentally measured value of PC,,; Ps can be found by noting that Us = UW.

AN EXAMPLE CALCULATION It is convenient to demonstrate the approach using some concrete numbers. If we take the compliance values to be C, = 1.0, C,. = 1.01, and C, = 0.5, and load from P = 0 to 100 units, with closure occurring at PC,, = 50,then the simple linear case (Fig. 3) will be as shown in Fig. 7. Note that the gaps between the loading lines have been exaggerated somewhat for clarity. Calculating the areas of the triangles and taking the constant a to be unity, gives AK = 7.07 if no closure is assumed and AK,, = 3.54 if closure occurs. This is exactly the same as would conventionally be predicted for AK&, since P,,,,,/P,,, = 2,thus AK/AKcff = 2.This illustrates the equivalence of results in the simple case where no curvature occurs on the P/U plot. Using the same figures but introducing curvature gives the results in Fig. 8. Here the lines for the no-closure case have been omitted for clarity. The value for AK,,,, now increases to 5.91: larger than AKdf by a factor of 1.67.

THE EFFECT OF R RATIO As R increases, the difference betwen AKeR and AK,,,, will tend to decrease, with the ratio AK,,/AK, approaching unity as P,,,* approaches Pc,o. In order to make predictions at R values greater than zero it is necessary to redraw the curves as shown in Fig. 9. Load and displacement at the minimum point are defined as P, and U*,; the two curves are required to meet at this point. Figure 10 shows predicted values of AK,,,/AK, as a function of R for various values of the closure ratio, PJP_ (= K,,/K&).The closure ratio is the only variable parameter in the model. The experimental data from Fig. 5 are reproduced here; in this case the experimental closure ratio was between 0.6 and 0.7, so it can be seen that the predictions are quite accurate. The general range of results, from unity to 2.56, is similar to the range noted in the data of Fig. 6; thus the discrepancy between simple closure theory and the true R ratio effect can be explained by using AK,,. Materials in which the closure ratio is relatively large (i.e. Kclo is relatively high) will show the largest discrepancy.

U Fig. 9. Schematic P/U curves for the case of cycling at a positive R ratio.

114

Technical

Note

O-Experimental (l&&,=0.6-0.7)

u

Fig. 10. Comparison

0:l

0.'25 015 R RATIO

Data

0:7

of experimental data (from ref. [2]) with predictions lines), at various values of P,,,/Pm_.

from the present

analysis

(solid

This approach also predicts that there will be differences between different types of specimen, since specimen geometry will affect the relationship between P and K. However, discussion of this problem is beyond the scope of the present paper.

RELATIONSHIP

TO AJ

The above energy-based approach is essentially the same as the use of the J integral, since the material is being treated as a non-linear elastic solid. The definition of AJ, as well as of the areas defined above, is open to interpretation, especially at positive R ratios, but the present work amounts to a justification of the use of some form of AJ approach. The concept of a closure corrected AJctr then amounts to making the correction twice over, and is thus inappropriate.

THE B-B’ ASSUMPTION In Fig. 4 it was assumed that the points B and B’ occur at the same value of U, i.e. that the curved portion ends at the same U value for both crack lengths. Strictly speaking we should place B’ so that K,,, is constant, but then the exact position of B’ will depend on specimen type because it depends on the relationship between C and a. Possible extreme positions of B’ are: (a) the present position and (b) at a load P = P,. In any case the error involved is small, and is of the same order as the compliance change between a and a + ha, so it disappears as the compliance difference is reduced.

FINAL

REMARKS:

THE USE OF AK,,,,

In conclusion, this analysis has been able to explain the relative success of the compliance method for measuring crack closure by arguing that the method is actually measuring a fundamental parameter, namely the change in elastic energy from which K is directly derived. This does not diminish the importance of crack closure as a phenomenon, but rather strengthens it by showing the strong effect which it has on the energy release rate. The conventional measurement of closure is simple to carry out, and in many cases will continue to be a useful indication of material behaviour. However, it is clearly simplistic since we know that the crack does not close entirely at a single value of K, but gradually over a wide range of K below some upper value, Kc,,. Anomalies occur, such as the variation of AKlhCRwith R, which limit the usefulness of AKeR. These anomalies may be resolved by using the new parameter, as defined above. This method is relatively easy to implement, since it uses the same experimental data which are AK,,, presently being collected by anyone using the compliance method, namely P/U curves at various crack lengths. It should not be difficult, therefore, for experimenters to check the author’s findings for themselves.

CONCLUSIONS (1) The conventional method of measurement of AKeR is equivalent to a calculation of AK via the change in compliance or enerttv release rate of the specimen. This explains the relative success of AKeR in explaining fatigue phenomena, despite obvious objections to it on mechanistic grounhs. (2) However. the use of AR involves certain simnhfying assumptions which become unnecessary if AK is calculated from the energy change directly. Tiis leads to the definition of a new parameter, AK,,,,. (3) The use of AK,,,, is able to resolve some anomalies in the experimental data; the variation with R of measured values of AKWR can be explained in terms of AK,,,,, which remains constant with R and appears to be the ‘intrinsic’ value of the threshold. I_

REFERENCES [l] N. A. Fleck, Compliance methods, in Crack Length Measurement (Edited by K. J. Marsh, R. A. Smith and R. 0. Ritchie), Chapter 3. EMAS, U.K. (1990). [2] A. F. Blom, Near-threshold fatigue crack growth and crack closure in 17-4 PH steel and 2024-T3 aluminium alloy, in Fatigue Crack Growth Threshold Concepts, pp. 263-280. TMS-AIME, U.S.A. (1984).

Technical Note

115

[31K. A. Esaklul, A. G. Wright and W. W. Gerberich, An assessment of internal hydrogen versus closure effects on near-threshold fatigue crack propagation, in Fatigue Crack Growth Threshold Concepts, pp. 299-326. TMS-AIME, U.S.A. (1984). [4] P. Renaud, P: Violan, J. Petit and D. Ferton, Microstructural influence on fatigue crack growth near threshold in 7075 Al alloy. Scripta Metall. 16, 1311-1316 (1982). [5] K. Minakawa, G. Levan and A. J. McEvily, Metall. Trans. 17A, 1787 (1986). [6] R. 0. Ritchie, S. Suresh and C. M. Moss, Near threshold fatigue crack growth in ZiCr-1Mo pressure vessel steel in air and hydrogen. J. Engng Mater. Technol. 102, 293-299 (1980). [7] G. C. Salivar and D. W. Hoeppner, A Weibull analysis of fatigue crack propagation data from a nuclear pressure vessel steel. Engng Fracture Mech. 12, 181-184 (1979). [8] D. Taylor, Fatigue Thresholds, p. 113. Butterworth, U.K. (1989). (Received 5 August 1991)