Experimentally determined stress intensity factors for several contoured double cantilever beam specimens

Engineeriny

Fracrure

Mrchonics,

197 1, Vol. 3, pp. 27-43.

Pergamon

Press.

Printed

in Great

Britain

EXPERIMENTALLY DETERMINED STRESS INTENSITY FACTORS FOR SEVERAL CONTOURED DOUBLE CANTILEVER BEAM SPECIMENS? J. P. GALLAGHER University of Illinois. Urbana, 111.61803. U.S.A.

Abstract-Experimentalstress intensity coefficients KBWVP

are found for several contoured double cantilever beam specimens using the Irwin-Kies method. Least squares procedures and the movable strip technique were employed to estimate the derivative of compliance data. The experimental stress intensity coefficients were determined for both plane stress and plane strain states of stress and were compared to the results of Gross and Srawley’s boundary value collocation procedure. The results are also discussed with respect to the strength of materials approach used by Ripling, Mostovoy and Crosley.

INTRODUCTION level (K) is chosen as the experimental control, use of a contoured double cantilever beam (contoured DCB) specimen[l] should be considered (see Fig. 1). A contoured DCB Specimen, if properly designed, produces a stress intensity level that can be considered independent of crack length for several inches of crack length. Several investigators have already had success using this type of specimen for stress corrosion tests [2], corrosion fatigue tests [3], and fracture initiation and arrest studies[4]. Probably because enough data is not presently available to develop guidelines for specimen design, this specimen has not been used to any large extent. The stress intensity relations which have demonstrated the independence of stress intensity for ranges of crack length have been found using either a boundary value collocation procedure [ 11, a strength of materials approach [4] or the experimental method suggested by Irwin and Kies[2-51. The approach presented in this paper employs the Irwin-Kies method which is based on the derivative of a specimen’s compliance with respect to its crack length. This method presents a serious problem because the compliance derivative must be obtained from a set of discrete and unevenly spaced compliance vs crack length data points. Since errors in experimental measurements affecting the compliance data can be magnified by a differentiation process, the technique which is used to calculate the derivative of the data has to be carefully chosen. Two techniques frequently used to obtain an estimate of this derivative either rely on finding the slope of tangents that are drawn through portions of a compliance curve hand fitted to the data[6,7] or on the differentiation of a compliance function that is fit to all the data[&lO]. While these techniques are useful, they give no direct indication of the accuracy of their respective derivative estimates. In this investigation, estimates of compliance derivatives for several contoured DCB specimens were obtained using least squares procedures [ 1 l] in conjunction with the movable strip technique[l2]. Applying this technique to compliWHEN stress intensity

lPresented at the Second National Symposium on Fracture Mechanics, Lehigh University, Pa., 17-19 June, 1968. 27

Bethlehem.

28

J. P. GALLACiHER

Fig.

I. Geometric

parameters specimens.

for contoured

DCB

ante data results in a smooth function that approximates the compliance data and that when it is differentiated estimates the compliance derivative. The accuracy of the approach employed in this paper is indicated by comparing the integrated compliance derivative fitting function to the original compliance vs. crack length data. The experimental results in terms of a stress intensity coefficient [ I, 131 are compared to the results of Gross and Srawley’s collocation procedure. The usefulness of Ripling, Mostovoy and Crosley’s strength of materials approach is discussed. EXPERIMENTAL

PROCEDURE

Geometry and material properties

The contoured DCB specimens analyzed in this investigation are designated as Types 1,2A, 2% and 3. The mechanical properties of the specimens are listed in Table 1 and their dimensions are given in Figs. 2-4. The side grooves shown in Fig. 3 were only used for the Type 2S specimen. The crack length vs. compliance data used in this

Holes for Lccoting Extemunater

T”

I ::

:

I

Fig. 2. Specimen geometry of the Type toured DCB specimen.

I 1 con-

Stress intensity Table

factors

for double

1 [2, 14. 151. Mechanical

beam specimens

Properties

of Specimens

Modulus of elasticity (psi)

Specimen

Material

Type 1

AISI 4340 7075T6 I020 Mild Steel 18 Ni (200) Marginal Steel

Type 2A Type 2s Type 3

cantilever

Poisson’s ratio

28.900.000 10.500.000 30,000,000

0.290 0.333 0.290

27.000,OOO

0.300

Fig. 3. Specimen geometry of the Type 2A and 2s contoured DCB specimens. Type 2A specimen did not have the side groove.

Cutter

Fig. 4. Specimen

geometry

SA-705

of the Type 3 contoured

DCB specimen.

29

J. P. GALLAGHER

30

investigation were made available by Van Der Sluys[2, 141 and U.S. Steel Corp. [15] and are shown in Figs. 5-8. The data points were collected for crack lengths measured. with a microscope to within an accuracy of 0.005 in. The compliance values were determined by averaging four or more deflection vs. applied load curves. The cracks were extended using a band saw with 0.060 in. thick blades. The deflection measurements were made using the extensometer described by Van Der Sluys[2] or the Srawley and Brown type COD gage [ 131. Experimental

analysis of specimen

The stress intensity coefficient KBWVP was determined experimentally for each specimen by using the Irwin-Kies approach and the 9 to K transformation equations. For completeness, a short outline of this method is given in the paragraphs following. Irwin and Kies [5] found that when a cracked plate experiences an infinitesimal increment of crack extension (da) upon application of a force (P), the strain energy 4-0-

44-

40-

36-

32-

24.

16-

Crock

Fig. 5. Compliance

Length

-1”.

data for the Type 1 specimen function.

and the resultant

integrated

Stress

0

I

intensity factors for

2

3

31

double canitlever beam specimens

4 Crack

5 Length

6

7

8

9

IO

-1”

Fig. 6. Compliance data for the Type 2A specimen and the resultant integrated function.

released per unit thickness (9) was given by

where B, represents the thickness through the specimen at the crack front and C represents the compliance (de~ection/app~ied load) of the specimen. When the specimen has modulus of elasticity, E, length W, and a thickness B which does not vary with crack length (a), (1) can be put in the following form

(2) The stress intensity factor (K) is related to the strain energy released by one of two transformation equations. These equations are [ 161 K = [5FE]“2

(3)

32

J. P. GALLAGHER

44L

40-

36-

l2-

a-

4-

0

2

I

3

5

4 Crock

Fig. 7. Compliance

Length

6

7

8

9

I

-I”

data for the Type 2S specimen and the resultant integrated function.

and

K=

YE [

(l--v”)

1

II2

which apply to the plane stress and plane strain states of stress, respectively. tion of (3) and (4) to (2) results in two equations which can be written as

(4) Applica-

(5)

for plane stress and

KB;? for plane strain,

[p__)

(!!w)]“’

where B* E [B,B]l/Z v = Poisson’s

ratio.

(6)

33

Stress intensity factors for double cantilever beam specimens

36-

32-

24-

16 -

12 -

8-

4-

O_ 0

I

2

4

6

8

IO

I

12

I

I4

16

I8

Crock Lenglh - m

Fig. 8. Compliance data for the Type 3 specimen and the resultant integrated function.

The collected data was suitable only for plane-stress analysis. However, (6) was used to provide an alternative estimate of K in central ‘plane-strain’ regions of the crack. By considering the parameter KB” WVP after the plate geometry, method of loading and material are chosen, one finds that for any specimen the parameter variables, stress intensity and force, are related to the compliance derivative function. More explicitly, if the load were held constant during an experiment in which the crack extended, the stress intensity would vary as the square root of the compliance derivative function. If a plate specimen could be found which provided a relatively constant compliance derivative for some range in crack length, the use of the stress intensity as an experimental control would be simplified. The difficulty resulting from application of this experimental stress intensity approach centers on the determination of the compliance derivative function. This derivative must be obtained from a set of discrete and unevenly spaced points. The method of approach employed in this investigation is detailed in the paragraphs which follow.

EFM Vol. 3 No. 1-C

34

J. P. GALLAGHER

Determination of the compliance derivative function The movable strip technique was employed to smooth and to analyze the data by grouping all the discrete compliance data points into a collection of data point sets, each containing n data points. A smoothing operation was performed first on each data point set by using least squares procedures. The resulting set of data obtained with the smoothing operation was processed in the same manner so that differentiation of the least square functions fitted to each set in the new collection yielded estimates of the compliance derivative. An approximation to the compliance derivative was obtained by least squares fitting an eighth order polynomial to the estimated compliance derivative vs. crack length data points. Integration of this function yielded a ninth order polynomial which closely approximated the original compliance vs. crack length data. The approximating compliance derivative function and compliance fitting function were obtained by the procedure listed in the paragraph below. The parameters used in this investigation are listed after the method of approach is presented. First, form a collection of data point sets from the compliance data. The data point sets each contain n discrete points and the first set contains the data points corresponding to the n shortest crack lengths. The elements of the other sets are chosen so that each set contains the data point corresponding to the next larger crack length, which is not contained in the preceding set, along with all the points of the preceding set except the one which corresponds to the shortest crack length. The collection of data point sets becomes closed when the largest crack length data point becomes an element of a data point set. Second, choose a function that can be least squares fitted to each data point set in the collection. Call this function the data point set function. For example, one could choose a polynomial of low order or a power function as the data point set function. Third, after coefficients are found for a particular data point set function that fits the n data points, evaluate the function at the mean of the n crack length values. This operation results in a set of smoothed compliance vs. crack length points. Fourth, repeat the first two operations on the smoothed compliance vs. crack length points. Again choose a data point set function and find the coefficients which correspond to each data point set. Differentiate the data point set functions and evaluate them at the mean crack length value of their corresponding data point set. Fifth, the discrete estimated compliance derivative vs. crack length data points are least squares fitted with a polynomial function of crack length to yield an approximate compliance derivative function. Sixth, integrate this compliance derivative function and add to it a constant of integration to find a function that approximates the compliance data. Call this function the resultant integrated function. The constant of integration is chosen to be the mean of the difference between the integrated compliance function and the compliance values at each of the crack length data points. The resultant integrated function then fits the compliance data and its derivative fits the estimated compliance derivative data. The general method contains several parameters which must be chosen for a particular set of data. These are the number n contained in a data point set, the data point set function and the order of the compliance derivative function. The number n is chosen large enough to smooth out possible random experimental compliance errors. The data point set function can be selected on the basis of assumed behavior of the data. The order of the compliance derivative polynomial function is chosen large enough to fit the estimated compliance derivative data points. For this investigation, the following choices were found to give satisfactory results: a number n equal to 4, a data

35

Stress intensity factors for double cantilever beam specimens

point set function that had the form Y = b,(X)**, and the compliance derivative polynomial function was an eight order least squares polynomial. RESULTS AND DISCUSSION The method used to generate the compliance derivative and the resultant integrated functions for the contoured DCB specimens was evaluated by, comparing the fit of the compliance data with the resultant integrated function and by comparing the results obtained from (5) and (6) with analytically developed KBW’YP parameters.

Because the compliance of the contoured DCB specimens changes by an order of magnitude between the specimen’s smallest crack length value and its largest crack length value, the accuracy of fitting the compliance data with the resultant integrated function was determined by a relative error. The relative error for a given crack length (ai) was found by dividing the absolute error by the compliance value at ai. The absolute error was the difference at ai between the resultant integrated function and the compliance value. Relative errors for the contoured DCB specimens are summarized in Table 2. Several data points corresponding to crack lengths greater than those noted did not fall Table 2. Error in using the resultant integrated function to fit the compliance data points Points which do not fit within overall relative error Overall relative error per cent

Point (a, c x lo+@)

Relative error (5%)

Absolute error ( 10eB in/lb)

Type ?A$

t3 51.5

Type 2%

t2

Type 3

+1

(3.764. I 1.30) t 1.500. 12.85) (1.780. 5.58) (3.515.13g4) (1.510. 1.27) t 15.978.43*00)

t9. I 12.0 -2.6 -3.5 +2.7 +6.2

+I.08 10.26 -0.15 -0.48 +0.03 +2.68

Specimen Type

1”;

Figure No. for compliance curve

5 6 7 8

tFor crack lengths greater than I .3 in. f For crack lengths greater than 1.O in.

within the overall relative error. For convenience, these points are also listed in Table 2, along with their relative and absolute errors. Results of the compliance fitting are shown in Figs. 5-8. The relative errors observed here are about as small as those that would be obtained by fitting a fifth or sixth order polynomial to the compliance data directly. But since the derivative fitting polynomial was found to fit the estimated compliance derivative within 1-l per cent for almost all of the generated derivative values for each specimen, this numerical technique provided an estimate of the accuracy of the compliance derivative, one definite improvement over previously reported methods. correlation

of experimental

resdts

with analytic

resl~lts

Srawley and Gross found that two analytical models describe the dominant stress effects in the contoured DCB specimen. For smaller relative crack lengths (a/W) the dominant stress effect comes from the wedging action of the applied forces, and the

36

J. P. GALLAGHER

wedge dominant (or ( W-a) indifferent) KB WVP

parameter was found to be [ l] (7)

where the parameters of the specimen are defined in Fig. 1 and A varies with H,le as given in Table 3. The crack length (a) throughout this paper refers to the crack length Table 3 [I]. Relationship between HpIe and 11

-HP e 0.0 0.1 0.2 0.3 0.4 0.45 0.53

h 346 3.26 3.10 2.98 2.88 2.84 2.78

as measured from the load points. For the larger values of relative crack length, bending stresses over the end portion of the specimen dominate over wedge opening stresses and the end dominant (or (W-u) dependent) KB W*/2/P parameter is given by [ 1, 171 KBW1’2 ---= P

0.537+2.17(1 +a/W)/(l (1 -u/W)“2

-u/W) .

(8)

The results obtained from applying (5) and (6) to the compliance data is shown in Figs. 9-12. Except for the Type 2S specimen, the thickness parameter B* reduces to the uniform thickness B of the specimen. With these results, (7) and (8) are plotted for purposes of comparison. It should be noted that in Figs. 9, 10, 12, the plane strain experimental results do not correlate with the analytical results. This strengthens several statements made by Gross, Srawley, and Brown[l8] concerning the correlation between analytical and experimental results. The main difference between the curves in each case is that the plane strain results (6) are approximately 5 per cent higher than the plane stress results (5). A detailed discussion of the experimental plane stress results and their correlation with the analytical models is examined separately because of the differences encountered. Type 1 specimen

The correlation between the analytical and experimental KB W112/P parameters was represented graphically in Fig. 9. Examination of Fig. 9 indicated that the correlation between the analytical and experimental parameter values was poor except in the end dominant region of the specimen. In this region, the experimental results indicated that the upward trend started approximately at a relative crack length of 0.6 and closely approximated the analytical end dominant KB W1’2/P. In the wedge dominant region of

37

Stress intensity factors for double cantilever beam specimens

I

0.1

0.2

t

8

0.3

0.4

Relative

0.5 06 Crock Length -b/WI

0.7

0.8

0.9

10

Fig. 9. Correlation of the experimental KBW*‘2/P parameter with the analytical parameters for the Type I Specimen.

the specimen, the experimental KBW*‘2/P parameter was substantially lower than the analytical wedge dominant K13W1~YP parameter. Since the specimen was actually a straight DCB specimen for relative crack lengths less than O-32, the specimen was expected to exhibit a lower compliance in this region than a specimen that had been contoured throughout its length. At a relative crack length of approximately O-37, the experimental curve crosses the analytical wedge dominant curve and continues to in-

J. P. GALLAGHER

Experimental

Plane Strain 4’

1

0.1

0.2

0.3

Relative

Fig.

10. Correlation

3

I

05

0.4 Crack

06 Length

8

I

07

0.0

09

-(a/W)

of the experimental KBW’VP parameter meters for the Type 2A Specimen.

with the analytical

para-

crease, but at a decreasing rate, until the end dominant region of the specimen is reached. The experimental curve only levels out between relative crack lengths of 0.475 and 0.575. The constant portion occurs at a level 6 per cent higher than predicted, which might be due to effects caused by the straight DCB or end dominant regions of the specimen. If one accepts the experimental evidence, one must conclude from this study that if the straight DCB portion of the specimen constitutes approximately one-third of

39

Stress intensity factors for double cantilever beam specimens

01

, 0.2

0.3

I

0.4

0.5

I

06

I

I

0.7

0.8

I

0.9

Relal~ve Crock Lerqih - (a/W)

Fig. 11. Correlation

of the experimental KBW*‘2/P parameter with the analytical parameters for the Type 2s Specimen.

the specimen length, the specimen behaves more like the straight DCB than like the contoured DCB, and has little or no constant compliance derivative region. Type 2A specimen

The analytical and experimental KB WVP parameters were represented graphically in Fig. 10. It is seen that the plane stress experimental values closely agree with the

40

J. P. GALLAGHER

Experimental

0.1

02

03

Plal~

Strom

04 Relatlve

,_

05 Crack

06 Length

07

08

0.9

-(a/W)

Fig. 12. Correlation of the experimental KBW’VP parameter with the analytical parameters for the type 3 specimens.

I.

Stress intensity factors for double cantilever beam specimens

41

analytical wedge effect parameter. The compliance data of this specimen was only sufficient to obtain experimental results for relative crack lengths between 0.20 and 0.55. In the relative crack range between 0.25 and O-50, the parameter value and hence the stress intensity does not change more than 2 per cent. Since the specimen’s length was 10 in., this specimen was considered appropriate for controlling a constant stress intensity level for a 2.5 in. crack length range. Type 2S specimen

The analytical KBW1’21P and the experimental KB* WVP parameters for this specimen were represented graphically in Fig. 11. In the wedge dominant region of this specimen, the difference between the plane stress experimental and wedge effect parameters was less than 4.5 per cent. Since the specimen was made from a low yield strength material, this difference between the experimental and analytical parameters might be a material effect. Examination of the experimental curve indicated that this specimen had a stress intensity that changed less than 2 per cent between 0.25 and 0.45 relative crack length. In the end dominant portion of the specimen, the experimental results showed an upward trend at a relative crack length of approximately O-55, indicating this overriding effect. Type 3 specimen

The experimental and analytical parameter results are shown in Fig. 12. The experimental plane stress parameter was seen to be 3 per cent greater than the analytical wedge effect parameter. The probable reason for this difference was attributed to additional bending which resulted from the specimen’s shape in its initial portion. While this additional bending may have affected the magnitude of the experimental values, it did not seem to affect its shape. The experimental results indicate that the specimen has a constant (within 2 per cent) stress intensity range between 0.3 and 0.6 relative crack length. The experimental curve also exhibits an upward trend in the end dominant region of the specimen. The experimental results of this specimen indicate that it provides the longest constant compliance region of the three geometries investigated. One reason for this is that the magnitude of the wedge effect is large enough to delay the upward trend caused by the end dominant effect. Discussion

of strength of materials approach

An attempt was made to correlate the compliance derivative results with the Ripling, Mostovoy and Crosley strength of material approach [4]. For purposes of comparison, (9) was used. (9) = a+a, a = real crack length a, = empirical correction H = H(a), the height of the specimen at crack length a.

wherea*

42

J. P. GALLAGHER

It was not possible to find a value for the empirical crack length correction using the data presented here. The form of (9) does not differ appreciably from an equation derivable from (7) if one assumes that a,=cH

(10)

Since the shape of the experimental KBW112/P curve is predicted by the SrawleyGross analytical wedge effect equation, (7), in the region of constant compliance, the importance of finding a, is more academic than critical.

CONCLUSIONS From the results of this investigation, the following conclusions appear reasonable. (1). The use of a programmed numerical method for evaluation of the compliance derivative provides experimentally determined stress intensity with limits on its accuracy. (2). The use of this numerical method indicated that the compliance derivative and thus the stress intensity level of the contoured DCB specimens remains relatively constant for a range of crack length. (3). The results of the Srawley-Gross investigation[l] can be used for designing contoured DCB specimens if the shape of the specimen closely approximates the shape shown in Fig. 1. (4). The plane stress experimental parameter, (5), should be used for purposes of comparison with the Srawley-Gross analytical models, (7) and (8).

REFERENCES [ll

J. E. Srawley and B. Gross, Stress intensity factors for crackline-loaded edge-crack specimens. NASA Tech. Note. NO. NASA-TD D-3820 (1967). 121W. A. Van Der Sluys, Mechanisms of environment induced subcritical flaw growth in AlSl 4340 steel. Theoretical and Applied Mech. Dept. Rep. No. 292. Univ. of IN. (I 966). assisted fatigue crack growth rates in SAE 4340 steel. Ph.D. dissertar31 J. P. Gallagher, Environmentally tion. Univ. of Illinois (1968). [41 E, J: Ripling, S. Mostovoy, and P. B. Crosley, Use of crackline loaded specimens for measuring plane fracture toughness. Mater. Res. Lab., Inc. Rep., Richton Park, Illinois (1966). t51 G. R. Irwin, and J. E. Kies. Critical energy rate analysis of fracture strength. Weld. J. 33, 193-198 (I 954). fracture toughness tests. Mater. Res. Stand. 4, WI A. M. Sullivan, New specimen design for plane-strain 20-24 (1964). determination of energy release rates for notch bending and notch tension. r71 J. D. Lubahn, Experimental ASTM. 59,885-Y 13 (lY5Y). [8] R. G. Hoagland, On the use of the double cantilever beam specimen for determining the plane strain fracture toughness of metals. ASME. Paper No. 67-Met-A (1967). [9] W. K. Wilson, Review of analysis and development of WOL specimen. Westinghouse Research Lab. Rep. No. 67-7D7-BTLPV-RI (I 967). [lo] J. E. Srawley. M. H. Jones, and B. Gross, Experimental determination of the dependence of crack extension force on crack length for a single-edge-notch-tension-specimen. NASA Tech. Note. No. NASA TN D-2396, (I 964). [l I] E. T. Whittaker and G. Robinson, The Calculus of Observations, 2nd Ed. Blackie and Son (1926). [12] C. Lanczos, AppliedAnalysis. Prentice Hall, Englewood Cliffs, N.J. (1956). [13] W. F. Brown, Jr., and J. F. Srawley, Plane strain crack toughness testing of high strength metallic materials. A.S.T.M. Spec. Tech. Pub!. No. 4 10 (1967). [14] W. A. Van Der Sluys, Unpublished compliance data. [IS] A. K. Shoemaker, U.S. Steel Corp., Private Communication (1967). [16] G. R. Irwin, Fracture Mechanics, Structural Mechanics, pp. 557-594. Pergamon Press. Oxford (1960).

Stress intensity factors for double cantilever beam specimens

43

[17] P. C. Paris and G. C. M. Sih, Fracture toughness testing and its applications. A.S.T.M. Spec. Tech. Publ. No. 381, pp. 30-83 (I 965). [18] B. Gross, J. E. Srawley, W. F. Brown, Jr., Stress-intensity factors for a single-edge-notch tension specimen by boundary collocation of a stress function, NASA Tech. Note, No. NASA TN D-2395 (1964). (Received 1April 1968)