Some aspects of corner fatigue crack growth from holes

Int J Fatigue 13 No 3 (1991) pp 233-240

Some aspects of corner fatigue crack growth from holes C.S. Shin

The early stages of fatigue failures often involved the initiation and growth of corner cracks. To assess the residual strengths and to predict the remnant lives of components containing corner cracks, the associated stress intensity factors must be known. These stress intensity factors are sensitive to crack shape as well as crack size and vary along the crack front. The complex nature of the problem often precluded analytical treatments so that numerical methods are generally employed to obtain the related stress intensity factors. Different numerical solutions exist. These solutions usually assumed the corner cracks are exactly elliptical in shape. Each solution uses its own interpolation method to arrive at the stress intensity for different crack sizes and crack shapes. There exist few experimental data to check the usefulness of these solutions. In the current work, direct observations of corner fatigue crack growth were made. It was attempted to calibrate the stress intensity factor of some corner cracks using a backtracking method. Details about the experiment are reported. The empirical stress intensity factors are compared with two published numerical solutions. The agreement is found to be reasonable and the possible causes of discrepancy are discussed. Key words: corner crack; stress intensity factor; experimental method; PMMA; fatigue crack propagation

Laboratory fatigue experiments are mainly based on straightedged and through-thickness cracks, yet practical failures often involve part-through corner cracks (see, eg, Ref. 1). Accurate stress analyses of this type of cracked components are needed for reliable prediction of their residual service lives and fracture strengths. Unlike that of a straight-edged throughthickness crack, the stress intensity factor of a comer crack varies from point to point on the curved crack front. Few exact stress intensity solutions for these cracks exist and numerical techniques are often resorted to (see, eg, Refs 2, 3). Numerical solutions have mostly concentrated on large specimens and are computed for discrete crack geometries. Some forms of interpolation are needed in order to apply the solutions to practical situations where the crack geometries differ from those analysed (see, eg, Refs 4,5). Moreover, numerical approaches usually assume that the corner cracks take up an exactly elliptical shape while the stress intensity factor is sensitive to the actual crack shape. There have been only very limited experimental investigations to check the validity of the numerical solutions as well as to substantiate the various assumptions the numerical approaches have made. Most experimental work has only deduced the maximum stress intensity or the end point stress intensities 6-9 of a corner crack. The current work employs the backtracking method proposed by James and Anderson. 1° This method, when suitably applied, may provide the variation of stress intensity along the whole crack front. The development of the crack shape is also recorded. The assumption about the crack shape and the validity of the numerical solutions can then be checked.

The backtracking method Figure 1 shows a pair of corner cracks emanating from a hole. The stress intensities at the points A, B, C, D and E are different. The stress intensity factor Ki at any particular point i can be expressed as Ki = F i c r ( = a / q ) '~2

(1)

where ~ is the remotely applied stress, a is the crack length along the hole surface, Q is a shape factor and Fi is the geometry correction factor. The shape factor Q can be approximated as follows:S Q = 1 + 1.464(a/c) 1"6s

(2a)

for a / c <~1 and Q = 1 + 1.464(c/a) 1"6s

(2b)

for a / c > 1. Different points on the crack front are identified by their different values of the parametric angle ~. The angle ~ for any point i on the crack front is defined in Fig. 2. For a particular crack geometry and loading, the variation in stress intensity along the crack front is reflected in the geometry correction factor Fi. In other words, Fi varies from point to point on the same crack front. When a component is subjected to a cyclic loading, cr in Equation (1) is replaced by the stress range Act, and a stress intensity range AK is obtained. Owing to the variation in AK, the amount of fatigue crack growth increment at different points on the crack front

0142-1123/91/030233-08 © 1991 Butterworth-Heinemann Ltd Int J Fatigue May 1991

233

Plate width = 2b Fatigue test on through-thickness crack

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Fig. 2 Definition of the parametric angle P

varies. By comparing successive crack profiles obtained after loading for a number of cycles, the crack growth rates at different points on the corner crack front can be evaluated. Meanwhile, the fatigue crack growth rate (da/dN) against stress intensity range (At/Q data for the same material can be obtained by carrying out some standard fatigue testing. Knowing the comer crack growth rate at a particular point on the crack front, the associated stress intensity can then be deduced from the baseline da/dN against ArK data. This backtracking technique is summarized schematically in Fig. 3. Experimental

Corner

crack

Transparent

procedures

Corner-cracked specimens were machined from a 20 mm thick polymethylmethacrylate (PMMA) plate. Each specimen had a length of 280 mm and a width of 200 mm. A central hole of 20 mm diameter was drilled in each specimen. Both the symmetric and single corner-cracked cases were tested. In the former case, an end miller was used to produce a pair of small symmetric corner flaws on one surface of a specimen. In the case of a single corner crack, a small corner flaw was produced by forcing a razor blade into one corner of the central hole. All specimens were stress relieved at 80 °C for one hour before testing. Constant-amplitude sinusoidal tensile loading with a frequency of 1 Hz was applied to the cornercracked specimen through a fixture as shown in Fig. 4. An R-ratio (minimum load/maximum load) of 0.1 was used in all the tests, Corner cracks emanating from the central hole were observed through the bottom of the specimen with a mirror inclined at 45 ° to the horizontal. An ordinary mirror reflects the crack once from the front glass surface and again from the rear silvered surface. The two images overlap but are slightly offset from each other, making precise measurement difficult. A front-surface-reflecting mirror made

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Fig. 4 The experimental set-up to evaluate the fatigue crack growth behaviour of corner cracks

by vacuum depositing aluminium vapour on a glass slide was used to obtain a single image. To help define the crack front more clearly, red ink was introduced into the crack. Successive crack profiles were recorded with a camera. Figure 5 shows a typical photographic record of a pair of corner cracks. Baseline fatigue crack propagation data for this material were obtained by carrying out tatigue testing with constant loading amplitudes under the same conditions on centrecracked specimens made from the same PMMA plate. Red ink was also used to keep conditions as similar to the corner crack case as possible. Crack length was measured by a

Int J Fatigue M a y 1991

direction of this vector is continuously changing. At any particular point il on profile Pt, the vector is normal to P, at il and intersects profile P2 at point i2. im is the mid-point between il and i2. The locus of the mid-point im of thls traversing normal vector defines the mean profile of P1 and P2. The average crack growth increment associated with point im on the mean profile is given by the distance between the two intersection points il and i2. The average growth rate at im is then obtained by dividing the above crack growth increment by the number of cycles elapsed between P1 and P2. Crack growth rates at different points on each mean profile were computed and compared against the baseline da/dN against ArKdata. The ArKassociated with any particular point on each mean profile can then be deduced. Fig. 5 A photographicrecord of a pair of corner cracks

Results a n d discussion Crack shape The initial crack starter notches for the symmetric corner cracks produced by an end miller were shallow circular arcs. The starter notches for single comer cracks produced by razor blades were arbitrary smooth arcs. On applying cyclic loading, crack initiation may only start at some point of the notch and spread over the notch front gradually. Hence, at the initial stage, the comer cracks are far from being quarter elliptical. From the digitized crack fronts, it was found that the corner crack eventually conformed gradually to quarterelliptical shape. Figure 7 shows some digitized crack fronts from one typical test together with fitted quarter-elliptical curves. Each crack front was digitized at fifteen points. The fifth and the thirteenth points (counting from the front surface of the specimen) were assumed to lie on a quarter ellipse. The coordinates of these two points were used to calculate the major and minor axes of this ellipse, which forms the fitted quarter-elliptical curve. Figure 7 shows that the larger part of each crack front lies on the corresponding fitted curve. The assumption of a quarter-elliptical crack shape is therefore justified except near the two free surfaces. Slight lagging of

travelling microscope capable of resolving 0.01 mm crack growth increments. A seven-point quadratic fit method was employed to compute crack growth rate in this throughthickness crack case. The growth rate against fiaKdata fall in a scatter band the centre line of which can be described by the Paris law ~/,:b'V

=

C(~rC)m

(3)

with C = 0.1843, m = 10.456; da/dN is in mm/cycle and M¢ is in MPa Vmm. The corner crack front profiles were recorded photographically once for every growth increment of about 0.5 mm. These profiles were digitized using a projector system that has a resolution of 0.02 mm. Figure 6 shows a series of crack front profiles of a pair of symmetric corner cracks regenerated from digitized photographic records. The crack growth rate at any point on the corner crack front was calculated as follows: for two successive profiles (say P~ and P2), imagine that an outward-pointing normal vector is traversing P~. The

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Int J Fatigue May 1991

235

,-^

If the elliptical curves were computed using the two end points on the free surfaces, the rest of the crack front would not lie on the resultant curve. This is the reason that prompts the use of interior points rather than the points on the two end surfaces for elliptical curve fitting. Crack aspect ratios (a/c) which are required in later comparisons between numerical and experimental results will be computed based on elliptical curves fitted to the interior points.

A

E E

Reproducibility of the backtracking method for stress intensity

¢=

Empirical stress intensities along a comer crack front are presented in the form of the normalized stress intensity factor (Fi in Equation (1)) against a normalized parametric angle (26/~') for comparison. With regard to the reproducibility of this backtracking method, Figs 8(a) and (b) compare the empirical stress intensities for crack profiles having similar shapes (same values of a/c ratio) and crack depths (same values of a/t). Figure 8(a) compares two crack profiles taken from the left- and right-hand sides of a specimen with symmetric comer cracks. The profiles in Fig. 8(b) were taken from two different specimens with the single comer crack configuration. It can be seen that the stress intensity factors from two different crack profiles are well within 10%. It is considered that a certain amount of experimental error is inevitable because of scatter in the baseline fatigue crack growth data and the difficulty in locating and measuring the crack front precisely. However, based on the above comparison, the reproducibility is considered to be satisfactory.

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(full curves)

Comparison between empirical and numerical stress intensity solutions

the crack front was observed in most profiles near the front surface of the specimen and surface of the hole. This phenomenon may be attributed to two effects: 1)

Two published numerical stress intensity solutions are compared with empirical results from the current work. One of the numerical solutions was obtained by Raju and Newman using a three-dimensional finite-element analysis. 3 Newman and Raju had consolidated their discrete finite-element results into closed-form equationss which are employed in the current comparison. The other numerical solutions were given by Rudd et aL 4

crack closure being more significant at the surface than in the interior of a comer crack, 11'12 a consequence of the three-dimensional nature of the crack tip stress field where a crack intersects a free surface.

2)

These effects will be discussed in greater detail later. 2.0

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236

Int J Fatigue May 1991

Figure 9 shows the comparison for the two symmetric corner crack cases. Open circles denote empirical results. Triangles denote the solution of Ruddet al 4 while the dotted line represents the results from the Newman and Raju stress intensity equations, s The two numerical solutions are roughly equal for small a/t except near the hole surface region. As crack depth increases, both solutions have a similar trend of variation along the crack front but Newman and Raju's solution becomes larger than the solution of Rudd et aL The two numerical results differ by about 20% at a/t = 0.84. At small crack lengths, empirical results are larger than both numerical solutions. With increasing crack depth, the empirical results fall below the solution of Newman and Raju and lie somewhere between the two numerical solutions. In the worst case, the deviation of the numerical data from the experimental results is about 20% while the majority numerical data are within 10% of the experimental results. Two systematic errors are apparent on comparing the empirical results with the numerical solutions. One systematic error occurs near the hole surface (ie d~ near ~r/2). In this region, both numerical solutions increase gradually to above the experimental solutions. The other systematic error occurs at large crack depth ratios (a/t > 0.56). In these cases, the solutions of Newman and Raju are higher than the empirical results near the specimen surface as well as near the hole surface. Figure 10 shows the comparison between empirical and numerical solutions for the single corner crack case. To apply to single corner cracks, numerical data were modified by a correction factor proposed by Schijve:" g o n e crack

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1) 2) 3) 4) 5)

deviation from symmetry in the symmetric corner crack case, crack shape deviates from being truly elliptical, inherent scatter and uncertainty in the experimental data, different degrees of crack closure along the crack front, three-dimensional nature of the crack tip singular stress field at points where a crack front intersects the free surfaces.

These will be discussed in further detail in the ensuing paragraphs. For the symmetric corner crack case, comparison of the corresponding crack profiles (see Fig. 6) shows that the cracks on both sides were initially of the same size. Then one side had grown faster and become larger than the other. The difference in size became more pronounced as the cracks grew, reaching a maximum of about 10% when the larger crack transformed into a through-thickness crack. This

Int J Fatigue May 1991

phenomenon arose out of the difference between the crack initiation times on the two sides. When the cracks were still small, the growth rate was slow so that the difference in crack sizes was not prominent. As the cracks grew, the growth rates increased and so did the size difference. In the case of a single corner crack, the problem of lack of symmetry does not occur. Incidentally, comparing Figs 9 and 10 shows the agreement between the numerical and empirical results is better in the single corner crack case. It is evident from Fig. 7 that although the corner crack fronts were very close to the elliptical shape, they were not exactly elliptical. These two geometric deviations will probably affect the stress intensity factors of the actual corner cracks, causing some discrepancy between the empirical and the numerical solutions. Premature crack closure may well account for part of the discrepancy between the empirical and numerical solutions. It has been observed that in a one inch thick structural steel specimen, fatigue crack closure occurred for a larger fraction of loading cycle near the free surface than in the mid-thickness region.14 The same trend exists along a semielliptical surface crack front in a structural steel specimen, is Optical interferometric measurement showed that a similar phenomenon occurred in corner-cracked PMMA specimens: n at an R-ratio of 0.1, the crack front near the specimen surface and the hole surface regions was closed before the minimum load was reached. The crack opening load decreases progressively as we go to the interior. The part of the crack front with qb from about 10° to 85° remained open at minimum load. As a result, the AK deduced at regions near the free surfaces will be about 5% smaller than it should actually be. In Figs 9 and 10 if the empirical results are corrected for the crack closure effect, they will be in better agreement with the numerical solutions and in general will be closer to the Newman-Raju solution. Analytical and numerical investigations into the nature of the crack tip stress singularity in three-dimensional cracks showed that the familiar r -~/2 singularity no longer holds when the crack front intersects a free surface at right angles. .6'17 The elliptical corner crack in Newman-Raju's finite-element model intersects the specimen and hole surfaces orthogonally. Thus the concept of stress intensity factor breaks down and the validity of the numerical solution is in doubt there. Hence comparison between numerically and experimentally derived K-values is not meaningful at these points. On the other hand, numerical analysis 17 suggested that the crack front of a propagating mode-I crack must intersect a free surface obliquely (ie the surface point lags behind the interior crack front points). This is exactly what is observed in the corner crack front profiles (see Fig. 7). In the light of the above discussion, the use of a numerical solution based on the K-concept from the planar elasticity problem to predict corner fatigue crack growth must be treated with caution. This is especially so if only the end points on the free surfaces are used. However, it is worth pointing out that a number of predictions about corner and surface crack propagation in different materials have followed this line and given acceptable results. 1s,19 The fact that the approach is not rigorous yet produced a reasonable estimate is both interesting and has important practical implications. More effort to clarify this point seems to be worthwhile. The above discussions show that the empirical backtracking method is feasible for calibrating the stress intensities of corner cracks. It may not be a useful alternative to replace the numerical method because of the difficulty in obtaining the desired crack geometry and also of the inevitable experimental scatter that limits its accuracy. However, it can

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Int J Fatigue

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239

provide a useful comparison to substantiate the numerical solutions. This is important because in numerical analysis the crack shape has to be given and simple curves such as an ellipse are often assumed. If an incorrect shape is used, one may end up with a solution accurate in itself but not for the real problem. Moreover, the backtracking technique can easily be applied to finite-sized and single corner crack specimens. Empirical stress intensity solutions derived from these tests may be correlated with existing numerical solutions for two symmetric corner cracks in a large plate. Based on this correlation, empirical correction factors relating the large plate symmetric crack solutions to finite-plate single corner cracks may then be formulated or validated. In this way the application of existing numerical solutions can be extended to a much wider scope.

(American Society for Testing and Materials, 1983) Vol I pp 1238-1265 6.

Hall, L.R. and Finger, R.W. 'Fracture and fatigue growth of partially embedded flaws' Proc Air Force Conf on Fatigue and Fracture of Aircraft Structures and Materials, AFFDL-TR-70-144 (US Air Force Systems Command, Wright-Patterson Air Force Base, Ohio, 1970) pp 235-262

7.

McGowan, J.J. and Smith, C.W. 'Stress intensity factors for deep cracks emanating from the corner formed by a hole intersecting a plate surface' Mechanics of Crack Growth, ASTM STP 590 (American Society for Testing and Materials, 1976) pp 460-476

8.

Smith, C.W., Jolles, M. and Peters, W. H. 'Stress intensities for crack emanating from pin-loaded holes' Flaw Growth and Fracture, ASTM STP 631 (American Society for Testing and Materials, 1977) pp 190-201

9.

Snow, J.R. 'A stress intensity factor calibration for corner flaws at an open hole' AFML-TR-74-282 (Air Force Materials Laboratory, Wright-Patterson Air Force Base, Ohio, 1975)

10.

James, L.A. and Anderson, W.E. 'A simple experimental procedure for stress intensity calibration' Eng Fract Mech 1 3 (1969) p 565 Ray, S.K., Perez, R. and Grsndt, A.F., Jr 'Fatigue crack closure of corner cracks located at holes loaded in tension or bending' Fatigue Fract Eng Mater Struct 10 3 (1987) pp 239-250

Conclusions

A backtracking method for calibrating the stress intensity of corner cracks from fatigue crack growth tests was attempted. The stress intensity solution obtained through this method is reproducible to within 10%. The empirically obtained stress intensity factor of corner cracks agrees reasonably well with the existing numerical solutions in regions that are not near the intersection of a crack front with the free surfaces. At the points where the crack front intersects the free surface, such a comparison is not meaningful. However, the oblique intersection suggested in three-dimensional singular stress field analysis is indeed observed. Acknowledgement

The author is obliged to an anonymous reviewer for his stimulating discussion on the three-dimensional nature of the stress field on the free surface. References 1. Gran, R.J., Orazio, F.D., Paris, P.C., Irwin, G.R. and Hertzberg, R. 'Investigation and analysis development of early life aircraft structural failures' AFFDL-TR-70-1439 (Air Force Flight Dynamics Laboratory, 1971) 2.

3.

Kullgren, T.E. and Smith, F.W. 'The finite elementalternating method applied to benchmark no 2' Int J Fract 14 (1978) pp R319-R322 Raju, I.S. and Newman, J.C., Jr 'Stress-intensity factors for two symmetric corner cracks' Fracture Mechanics, ASTM STP 677 (American Society for Testing and Materials, 1979) pp 411-430

4.

Rudd, J.L., Hsu, T.M. and Wood, H.A. 'Part-through crack problems in aircraft structures' Part-Through Crack Fatigue Life Prediction, ASTM STP 687 (American Society for Testing and Materials, 1979) pp 168-194

5.

Newman, J.C., Jr and Raju, I.S. 'Stress-intensity factor equations for cracks in three-dimensional finite bodies' Fracture Mechanics, Fourteenth Symp, ASTM STP 791

240

11.

12.

Schijve, J. 'Comparison between empirical and calculated stress intensity factors of hole edge cracks' Eng Fract Mech 22 1 (1985) pp 49-58

13.

Schijve, J. 'Stress intensity factors of hole edge cracks. Comparison between one crack and two symmetric cracks' /nt J Fract 23 (1983) pp R I l l - R 1 1 6

14.

Fleck, N.A. 'An investigation of fatigue crack closure' Technical Report, CUED/C-MATS/TR. 104 (Cambridge University Engineering Department, UK, 1984)

15.

Fleck, N.A., Smith, I.F.C. and Smith, R.A. 'Closure behaviour of surface cracks' Fatigue Eng Mater Struct 6 3 (!983) pp 225-239

16.

Benthem, J.P. 'State of stress at the vertex of a quarterinfinite crack in a half-space' Int J Solids Struct 13 (1977) pp 479- 492 Bazant, Z.P. and Estenssoro, L.F. 'Surface singularity and crack propagation' Int J Solids Struct 16 (1979) pp 405- 426 Heckel, J.B. and Rudd, J.L. 'Evaluation of analytical solutions for corner cracks at holes' Fracture Mechanics, Sixteenth Symp, ASTM STP 869 (American Society for Testing and Materials, 1985) pp 45-64

17.

18.

19.

Newman, J.C., Jr and Raju, I.S. 'Prediction of fatigue crack growth patterns and lives in three dimensional cracked bodies' Proc of 6th Int Conf Fracture (ICF6), New Delhi, India, 1984 (Adv Fract Res 3 pp 1597-1608)

Author

The author is with the Department of Mechanical Engineering, National Taiwan University, Republic of China.

Int J Fatigue M a y 1991