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The relationship between fracture surface roughness and fatigue load parameters
International Journal of Fatigue 23 (2001) S135–S142 www.elsevier.com/locate/ijfatigue
The relationship between fracture surface roughness and fatigue load parameters Takao Kobayashi *, Donald A. Shockey Center for Fracture Physics, SRI International, Menlo Park, CA 94025, USA
Abstract A procedure was sought for estimating fatigue loading information from roughness characteristics of fracture surfaces. Topographs of fracture surfaces produced in compact tension specimens of a titanium alloy in load-shedding and monotonically increasing ⌬K tests were analyzed with a fast Fourier transform. The resulting elevation power spectrum density (EPSD) curve was correlated with the stress intensity range, ⌬K, and the stress ratio, R. A plot of (EPSD)1/2 vs ⌬K showed three distinct regions, possibly reflecting the competition between surface roughening and surface flattening from partial crack closure as ⌬K increased. The plot further indicated the effect of the stress ratio. Although the influence of crack closure complicated the relationship among (EPSD)1/2 , ⌬K, and R, the EPSD curves and (EPSD)1/2 values systematically changed as a function of ⌬K and R. These systematic changes may serve as a base for extracting fatigue loading information from roughness characteristics of fracture surfaces. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Fracture surface topography analysis; Fatigue loading parameters; Fast Fourier transform analysis; Elevation power spectrum density (EPSD) distribution; Crack closure effect; Ti-6Al-4V
1. Introduction Fractography, the art of interpreting markings on fracture surfaces to deduce how and why a failure occurred, is an important aspect of many failure investigations [1]. However, fractographic results are generally qualitative and consist of the failure mode, the load conditions, the fracture initiation site, and the propagation direction. Although this qualitative information is useful, the failure analyst would often like to know additional and, if possible, quantitative information. For example, what were the loads (their magnitude, duration, and frequency) that led to the fracture? How long did it take the crack to form? And once formed, what was its growth history? Did it propagate at a uniform speed, or did it experience periods of acceleration, deceleration, or perhaps hesitation? Quantitative answers require a quantitative analysis of the fracture surfaces. We sought this information in the fracture surface topography. The topography of a fracture surface shows not only
the interaction of a crack with the microstructure, but also the details of the deformation occurring at the crack tip. Previous work has demonstrated that, by matching the conjugate fracture surface topographs and manipulating the spacing between them, the amount and distribution of inelastic deformation can be assessed and the crack history can be approximately reconstructed. This technique is now known as the FRASTA (Fracture Surface Topography Analysis) method [2,3]. Although the FRASTA technique has been applied to many fracture problems successfully, many other failure cases exist where the FRASTA technique cannot be used because one surface was damaged or lost. In seeking a method to quantitatively deduce load parameters and crack history through analysis of one surface, we began to explore mathematical analysis of fracture surface topography with fast Fourier transform methods. The results showed that the elevation power spectrum density (EPSD) curve could be a useful indicator of the loading conditions [4]. The work described here was performed to evaluate further the potential of EPSD curves as an index of fatigue load parameters.
* Corresponding author. 0142-1123/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 1 ) 0 0 1 4 9 - 9
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2. Laboratory fracture specimens and surface topography characterization We selected for analysis two compact–tension specimens of Ti–6Al–4V that were fatigue tested by Pratt & Whitney to examine crack growth behavior in the nearthreshold regime. Specimen dimensions were 6.45 mm by 50.8 mm by 60.96 mm. A crack had been driven through the specimens under a range of well-controlled and documented loading conditions (at constant stress ratio, R=Kmin/Kmax, and varying stress range, ⌬K=Kmax ⫺Kmin) according to the ASTM E647 requirements for load shedding [5]. Two values of R were examined: 0.1 and 0.8. The variation of load conditions for each specimen is shown in Fig. 1. Small areas on the fracture surface in each fatigue test section (in total, 13 load conditions and 119 fracture surface areas) were selected along the center line of the specimen at a spacing of approximately 0.55 mm in the crack growth direction, and the topographies of these areas were characterized with a FRASTAscope [2,3], a system developed at SRI and consisting of a confocaloptics-based scanning laser microscope; a precision, computer controlled x–y–q stage; and computer programs that control data acquisition, manipulation, and display. The size of each area was 140.3 µm in the crack growth direction and 466.5 µm in the orthogonal direction. The number of elevation data points was 600 in the crack propagation direction and 2000 in the orthogonal direction. Data spacing, thus, was 0.233 µm in both
Fig. 1. Stress intensity range, ⌬K, as a function of crack length during multiple fatigue tests in specimens 8353 (R=0.1) and 8354 (R=0.8).
directions. Fig. 2 shows the contrast image and grayscale elevation image of a typical area. These images appear slightly fuzzy because of data compression in creating the graphic image. All areas examined were nominally flat and perpendicular to the loading direction. Shear lips occurred in the latter part of the last region where monotonically increasing ⌬K loading was applied, but even in this section the fracture surface areas examined were flat and perpendicular to the loading direction.
3. Results of fast Fourier transform (FFT) analysis A fast Fourier transform (FFT) analysis was applied to rows of data points parallel to the crack growth direction, and the elevation power spectrum density distribution was calculated. Although each row contained 600 data points, only the central 512 (=29) points were analyzed. The EPSD curves for individual rows exhibit significant differences, as shown in Fig. 3, due to local differences in microstructure. Such microstructural effects, although important from a materials science viewpoint, obscure the effect of loading. Noting that the stress intensity range, ⌬K, used to represent the loading condition is an average value over the length of the crack front and not a local ⌬K acting on a single grain or a few grains, we depressed local microstructural effects
Fig. 2. Contrast and gray-scale elevation images of an area on the fracture surface of specimen 8353.
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Fig. 3. EPSD curves for the roughness profiles along several lines shown in Fig. 2 (specimen 8353). (a) EPSD curve of data along line 400. (b) EPSD curve of data along line 800. (c) EPSD curve of data along line 1200. (d) EPSD curve of data along line 1600.
and emphasized the effects of loading condition by averaging the EPSD results over 2000 lines, wavelengthby-wavelength. The averaged EPSD curves from the fracture surface areas at different crack lengths (different ⌬K values) are closely parallel between wavelengths of 0.5 µm and 20 µm (see Fig. 4, for example). Choosing the square root
of the EPSD [(EPSD)1/2] value at a wavelength of 5 µm (arbitrarily selected as a mid-point between 0.5 and 20 µm wavelength) as a representative index of the roughness of each area and plotting these values as a function of position on the fracture surface, we obtained the results shown in Fig. 5(a) and (b) for Specimens 8353 (R=0.1) and 8354 (R=0.8), respectively. Also plotted are
Fig. 4. Averaged EPSD curves from five different locations on the surface in Test Run 1 in specimen 8353.
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test, the (EPSD)1/2 values initially decreased before increasing. In seeking to unify these results, we plotted all the (EPSD)1/2 values in Fig. 5 as a function of ⌬K. The results in Fig. 6 show three distinct regions for Specimen 8353 (R=0.1): (1) a linear relationship in the low ⌬K region (below 7 MPa√m); (2) a transition region (between 7 and 16 MPa√m); and (3) another linear relationship region at higher ⌬K (above 16 MPa√m). For Specimen 8354 (R=0.8), the trend in the first two regions (lower ⌬K) is not as apparent. Although the individual load-shedding test results in Figs. 5 and 7 indicated decreasing EPSD values where ⌬K decreased below a certain ⌬K value, and the opposite trend above this value, when all the test data were plotted together, the scatter masked the trend in both the first and second regions. However, the third region where (EPSD)1/2 increases linearly with increasing ⌬K is clearly seen (beyond 9 MPa√m). Interestingly, the boundaries of different regions in Specimen 8353 (R=0.1) did not coincide with the boundaries of decreasing and increasing ⌬K testing conditions. At higher values of ⌬K in decreasing ⌬K tests, the (EPSD)1/2 values increase as ⌬K decreases, and during the lower values of ⌬K in increasing ⌬K tests, the (EPSD)1/2 values decrease. Furthermore, the trend in the (EPSD)1/2 values in the transition region of decreasing and increasing ⌬K tests merge smoothly without discontinuity. This trend is less clear for Specimen 8354 ( R=0.8) although the results of decreasing and increasing ⌬K tests overlap in a similar region in the transition zone. The trend of increasing fracture surface roughness
Fig. 5. (EPSD)1/2 values at 5 µm of wavelength and stress intensity range, ⌬K, as a function of crack length for two specimens 8353 and 8354. (a) specimen 8353 (R=0.1) and (b) specimen 8354 (R=0.8).
the ⌬K values corresponding to the fracture surface locations or crack lengths. At first inspection, the behaviors of (EPSD)1/2 and ⌬K as a function of crack length indicate no clear correlation between load and roughness. For example, in the load shedding tests where the ⌬K continuously decreased, the (EPSD)1/2 values first increased before decreasing. In the monotonically increasing ⌬K fatigue
Fig. 6. Summary of the relationship between (EPSD)1/2 and ⌬K for two specimens 8353 and 8354.
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4. Crack closure effect on the fracture surface topography
Fig. 7. Possible correlation of (EPSD)1/2 behavior with ⌬K behavior. (a) Results of 8353 (R=0.1) and (b) results of 8354 (R=0.8).
with decreasing ⌬K near the threshold stress intensity range, ⌬Kth, for various titanium alloys was also noted by Ogawa et al. [6]. They quantified the fracture surface topography using stereo pairs of SEM micrographs and calculated the ratio of the real area of the fracture surface to its projected area at different magnifications. Using this ratio as a measure of fracture surface roughness, they showed that roughness increased with decreasing fatigue crack growth rate from da/dN=1×10−4 mm/cycle to da/dN=1×10−6 mm/cycle. However, they offered no explanation for this trend. The result shown in Fig. 6 clearly illustrates that something is happening to the topography of the fracture surfaces as ⌬K increases or decreases in a certain range. These results were obtained by averaging an EPSD curve of 2000 lines of topographic profiles in each area (for each ⌬K value); thus, we believe that definite physical phenomena were captured by this analysis. We sought the physical explanation.
One plausible explanation for the existence of three distinct regions in Fig. 6 for the R=0.1 specimen is that two competing processes are responsible for the formation and deformation of the fracture surface topography: (1) the roughening of the surface due to increasing ⌬K or Kmax and (2) the rubbing and flattening of the surface due to crack closure. It is generally believed that crack closure is not significant at high stress ratios. However, Boyce and Ritchie [7], referring to recent in-situ SEM studies on the mechanisms for fatigue crack propagation in Ti–6Al–4V by Davidson [8], suggest that even at very high load ratios (R苲0.8) significant crack wake interference (closure) up to 苲100 µm behind the crack tip may occur; and while this near-tip closure would not be detected by standard closure measurements (i.e. compliance-based measurements), it could significantly shield the crack-tip from fully unloading. The results presented here provide supportive evidence of crack closure at high stress ratio (R=0.8), and indicate how the closure effects influence or alter the fracture surface roughness. Fig. 6 indicates that when the ⌬K values are low, the (EPSD)1/2 value increases with increasing ⌬K (clearly shown in the case of Specimen 8353 (R=0.1). In this region, the effect of surface roughening with increasing ⌬K is dominant, and alteration of the surface roughness due to crack closure may not be significant. Possible reasons are that the degree of crack opening would be small in this range and, as a result, deformation or rubbing of surfaces due to crack-opening-related mismatch may be minimal. Interestingly, the roughening behavior of the surfaces for R=0.1 and 0.8 in the low ⌬K range are similar. However, when ⌬K reaches a certain value, the (EPSD)1/2 starts to decrease as ⌬K continues to increase (transition region in Fig. 6). In this region, we believe that crack closure begins to flatten asperities on the fracture surfaces and gradually overcomes the surface roughening with increasing ⌬K. The surface flattening effect outpaces surface roughening as ⌬K increases. Thus, the (EPSD)1/2 values decrease despite the increase in ⌬K. The degree of reduction of the surface roughness (reduction in (EPSD)1/2) is more significant for the surface with R=0.1 than with R=0.8. Furthermore, the range of reduction of surface roughness is wider for the case with R=0.1 than with R=0.8, as would be expected from crack closure. These results are consistent with the results reported by Ogawa et al. [6]. As ⌬K continues to increase, surface roughening starts to overcome surface flattening, and the (EPSD)1/2 values start to increase again as ⌬K increases. This is the third region in Fig. 6. In this region the surface roughness increase for the R=0.8 specimen is very rapid and the
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Fig. 8.
Fig. 9.
Comparison of EPSD curves of tests under similar stress intensity range but different stress ratios.
Changes observed in the shape of EPSD curves as ⌬K increases in the transition zone of R=0.1 specimen.
slope of the roughness increase is similar to that for the slope of the curve of the R=0.1 specimen in the first region. This observation indicates that crack closure is not playing a major role in a high stress ratio situation. The level of surface roughness for the R=0.1 specimen in the third region is considerably lower than that for the R=0.8 specimen, and the rate of increase of surface roughness with increasing of ⌬K for R=0.1 is much less than that for R=0.8. All these observations suggest that the three regions of (EPSD)1/2 behavior result from the modification of surface roughness due to crack closure, and the phenomena observed in the two specimens tested under R=0.1 and R=0.8 can be explained equally well with the closure effect. We now examine the individual fatigue test results more closely. Fig. 7 is a magnified view of the results
shown in Fig. 5. In these load-shedding tests, the ⌬K was decreased as the crack extended. When ⌬K is above a certain boundary value as shown in Fig. 7, the (EPSD)1/2 exhibits low values, and the higher the value of ⌬K, the lower the (EPSD)1/2 value. If the ⌬K values are below the boundary value, the trend in (EPSD)1/2 change is similar to that of ⌬K. This boundary value trend is also observed for tests in which ⌬K was increased as the crack grew, as shown in Fig. 7(b). Here the changes in (EPSD)1/2 above the boundary line are the same as for the load-shedding test; i.e., the higher ⌬K is above the boundary line, the lower the (EPSD)1/2 value. This trend is consistent with crack closure modifying the fracture surfaces at ⌬K’s above the boundary value.
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Fig. 10. EPSD curves showing changes in their shape with ⌬K for two specimens. (a) EPSD curves of run 1 in specimen 8353 (R=0.1) and (b) EPSD curves of run 1 in specimen 8354 (R=0.8).
5. Additional effects of crack closure To delineate the effect of crack closure on the surface roughness, we compared the individual EPSD curves from the fracture surface areas in the third region in Fig. 6 at similar ⌬K’s of about 17 MPa√m in the two samples. The four curves in Fig. 8 (two for R=0.1 and the other two for R=0.8) show that, within the same Rvalue, the EPSD curves have the same shape and are parallel to each other; however, at different R-values, the curves have different shapes in addition to a significant level difference. The curves for R=0.8 are nearly straight between wavelengths of 0.5 µm and 20 µm; however, the curves for R=0.1 deviate from a linear line below 1 µm and above 4 µm. This suggests that crack closure more significantly affects the features below the scale of 1 µm and above 4 µm. The behavior of EPSD curves in the transition region
where the EPSD values decrease and the curves bow more as ⌬K increases, as shown in Fig. 9, suggests that, at R=0.1 and as ⌬K increases, crack closure effects become more important. The EPSD results from the load-shedding tests also suggest that higher ⌬K loading induces crack closure effects that cause bowing of the EPSD curves. We examined the first fatigue test in each sample where crack growth was initiated at ⌬K well above the boundary line shown in Fig. 7. EPSD values were initially low, but gradually increased. The shape of EPSD curves during this change is shown in Fig. 10. The EPSD curves corresponding to initial high ⌬K exhibit bowing; however, as ⌬K decreases below the boundary ⌬K, the EPSD curve becomes straighter. Again these curves suggest that the higher ⌬K loading induces crack closure effects that cause bowing of the EPSD curves.
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topography, less stress is necessary to open the crack. In other words, the lower the opening stress, the higher the effect of surface deformation and flattening of the surface. For the case of higher R-value, the Newman results suggest that the ratio of the opening stress to the maximum stress is independent of ⌬K. The EPSD results indicate that a small but fixed percentage of the crack closure effect operates over a wide range of ⌬K based on the slope of the curve shown in Fig. 6. If this is the case, we anticipate that the ratio of the opening stress to the maximum stress could be constant.
Acknowledgements
Fig. 11. Crack-opening stress ratios for 2024-T3 aluminum alloy at low and high stress ratios [9].
6. Discussion The results presented here show that application of FFT analysis to fatigue fracture surface topography can illuminate the interactive influence of loading parameters ⌬K and R on the extent of crack closure. At low R values, crack closure significantly modifies the fracture surface topography over a wide range of ⌬K; at high R-values, the closure and surface modification effect is small and requires a ⌬K above a certain level. Data for only two R values were analyzed. Analysis of additional specimens fatigued under other loading conditions may allow quantitative relationships among R, ⌬K, and surface roughness to be established. The determination of fatigue loading conditions from fracture surface topography analysis is not straightforward, because the roughness–load parameter relationship is not single-valued. However, the shape of the EPSD vs ⌬K curve is sensitive to both R and ⌬K, suggesting that these loading parameters affect different parts of the curve. If so, other analysis procedures such as wavelet techniques that can examine specific features at specific scales, may allow a unique determination of ⌬K and R from the fracture surfaces. The results presented here are in accord with observations by Newman of crack opening for 2024-T3 aluminum alloy at low and high stress ratios shown in Fig. 11 [9,10]. A striking similarity exists in the shape of the curve for R=0.1 with the (EPSD)1/2 curve for R=0.1 shown in Fig. 6. The curve in Fig. 11 suggests that, due to the crack closure and re-deformation of the fracture
The authors thank Dr. Dennis Healy of the Defense Advanced Research Projects Agency and Dr. Anna Tsau of the Center for Computing Sciences at the Institute of Defense Analyses for their interest and encouragement. Drs. Michael Gehron, Joe Bateman, and Jay Littles and their colleagues at Pratt & Whitney consulted with us and provided crack growth specimens for analysis. Dr. Theodore Nicholas of the Air Force Research Laboratory provided valuable technical advice throughout this effort.
References [1] ASM handbook, vol. 12, Fractography. American Society of Materials International; 1987. [2] Kobayashi T, Shockey DA. FRASTA: A new way to analyze fracture surfaces, Part 1: Reconstructing crack histories. Advanced Materials & Processes 1991;140(5):28–34. [3] Kobayashi T, Shockey DA. Fracture analysis via FRASTA, Part 2: Determining fracture mechanisms and parameters. Advanced Materials & Processes 1991;140(6):24–32. [4] Kobayashi T, Shockey DA, Schmidt CG, Klopp RW. Assessment of fatigue load spectrum from fracture surface topography. Int J Fatigue 1997;19(Suppl. 1):S237–44. [5] ASTM Std. E647 — 95a, Standard test method for measurement of fatigue crack growth rates. December 1995. [6] Ogawa T, Tokaji K, Ohya K. The effect of microstructure and fracture surface roughness on fatigue crack propagation in a Ti– 6Al–4V alloy. Fatigue Fract Engng Mater Struct 1993;16(9):973–82. [7] Boyce BL, Ritchie RO. On the definition of lower-bound fatiguecrack propagation thresholds in Ti–6Al–4V under high cycle fatigue conditions. http://www/lbl.gov/Richie/Programs/URI/ Boyce2/boyce2.html [8] Davidson D. AFOSR Report. Southwest Research Institute, 1998. [9] Newman JC Jr. Analyses of fatigue crack growth and closure near threshold conditions for large-crack behavior. NASA/TM1999-209133. VA (USA): Langley Research Center Hampton, 1999. [10] Donald JK, Paris PC. An evaluation of ⌬Keff estimation procedures on 6061-T6 and 2024-T3 aluminum alloys. In: Proceedings of Fatigue Damage of Structural Materials II, 1998 Sept 7– 11; Cape Cod (MA).
The relationship between fracture surface roughness and fatigue load parameters Takao Kobayashi *, Donald A. Shockey Center for Fracture Physics, SRI International, Menlo Park, CA 94025, USA
Abstract A procedure was sought for estimating fatigue loading information from roughness characteristics of fracture surfaces. Topographs of fracture surfaces produced in compact tension specimens of a titanium alloy in load-shedding and monotonically increasing ⌬K tests were analyzed with a fast Fourier transform. The resulting elevation power spectrum density (EPSD) curve was correlated with the stress intensity range, ⌬K, and the stress ratio, R. A plot of (EPSD)1/2 vs ⌬K showed three distinct regions, possibly reflecting the competition between surface roughening and surface flattening from partial crack closure as ⌬K increased. The plot further indicated the effect of the stress ratio. Although the influence of crack closure complicated the relationship among (EPSD)1/2 , ⌬K, and R, the EPSD curves and (EPSD)1/2 values systematically changed as a function of ⌬K and R. These systematic changes may serve as a base for extracting fatigue loading information from roughness characteristics of fracture surfaces. 2001 Elsevier Science Ltd. All rights reserved. Keywords: Fracture surface topography analysis; Fatigue loading parameters; Fast Fourier transform analysis; Elevation power spectrum density (EPSD) distribution; Crack closure effect; Ti-6Al-4V
1. Introduction Fractography, the art of interpreting markings on fracture surfaces to deduce how and why a failure occurred, is an important aspect of many failure investigations [1]. However, fractographic results are generally qualitative and consist of the failure mode, the load conditions, the fracture initiation site, and the propagation direction. Although this qualitative information is useful, the failure analyst would often like to know additional and, if possible, quantitative information. For example, what were the loads (their magnitude, duration, and frequency) that led to the fracture? How long did it take the crack to form? And once formed, what was its growth history? Did it propagate at a uniform speed, or did it experience periods of acceleration, deceleration, or perhaps hesitation? Quantitative answers require a quantitative analysis of the fracture surfaces. We sought this information in the fracture surface topography. The topography of a fracture surface shows not only
the interaction of a crack with the microstructure, but also the details of the deformation occurring at the crack tip. Previous work has demonstrated that, by matching the conjugate fracture surface topographs and manipulating the spacing between them, the amount and distribution of inelastic deformation can be assessed and the crack history can be approximately reconstructed. This technique is now known as the FRASTA (Fracture Surface Topography Analysis) method [2,3]. Although the FRASTA technique has been applied to many fracture problems successfully, many other failure cases exist where the FRASTA technique cannot be used because one surface was damaged or lost. In seeking a method to quantitatively deduce load parameters and crack history through analysis of one surface, we began to explore mathematical analysis of fracture surface topography with fast Fourier transform methods. The results showed that the elevation power spectrum density (EPSD) curve could be a useful indicator of the loading conditions [4]. The work described here was performed to evaluate further the potential of EPSD curves as an index of fatigue load parameters.
* Corresponding author. 0142-1123/01/$ - see front matter 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 1 4 2 - 1 1 2 3 ( 0 1 ) 0 0 1 4 9 - 9
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2. Laboratory fracture specimens and surface topography characterization We selected for analysis two compact–tension specimens of Ti–6Al–4V that were fatigue tested by Pratt & Whitney to examine crack growth behavior in the nearthreshold regime. Specimen dimensions were 6.45 mm by 50.8 mm by 60.96 mm. A crack had been driven through the specimens under a range of well-controlled and documented loading conditions (at constant stress ratio, R=Kmin/Kmax, and varying stress range, ⌬K=Kmax ⫺Kmin) according to the ASTM E647 requirements for load shedding [5]. Two values of R were examined: 0.1 and 0.8. The variation of load conditions for each specimen is shown in Fig. 1. Small areas on the fracture surface in each fatigue test section (in total, 13 load conditions and 119 fracture surface areas) were selected along the center line of the specimen at a spacing of approximately 0.55 mm in the crack growth direction, and the topographies of these areas were characterized with a FRASTAscope [2,3], a system developed at SRI and consisting of a confocaloptics-based scanning laser microscope; a precision, computer controlled x–y–q stage; and computer programs that control data acquisition, manipulation, and display. The size of each area was 140.3 µm in the crack growth direction and 466.5 µm in the orthogonal direction. The number of elevation data points was 600 in the crack propagation direction and 2000 in the orthogonal direction. Data spacing, thus, was 0.233 µm in both
Fig. 1. Stress intensity range, ⌬K, as a function of crack length during multiple fatigue tests in specimens 8353 (R=0.1) and 8354 (R=0.8).
directions. Fig. 2 shows the contrast image and grayscale elevation image of a typical area. These images appear slightly fuzzy because of data compression in creating the graphic image. All areas examined were nominally flat and perpendicular to the loading direction. Shear lips occurred in the latter part of the last region where monotonically increasing ⌬K loading was applied, but even in this section the fracture surface areas examined were flat and perpendicular to the loading direction.
3. Results of fast Fourier transform (FFT) analysis A fast Fourier transform (FFT) analysis was applied to rows of data points parallel to the crack growth direction, and the elevation power spectrum density distribution was calculated. Although each row contained 600 data points, only the central 512 (=29) points were analyzed. The EPSD curves for individual rows exhibit significant differences, as shown in Fig. 3, due to local differences in microstructure. Such microstructural effects, although important from a materials science viewpoint, obscure the effect of loading. Noting that the stress intensity range, ⌬K, used to represent the loading condition is an average value over the length of the crack front and not a local ⌬K acting on a single grain or a few grains, we depressed local microstructural effects
Fig. 2. Contrast and gray-scale elevation images of an area on the fracture surface of specimen 8353.
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Fig. 3. EPSD curves for the roughness profiles along several lines shown in Fig. 2 (specimen 8353). (a) EPSD curve of data along line 400. (b) EPSD curve of data along line 800. (c) EPSD curve of data along line 1200. (d) EPSD curve of data along line 1600.
and emphasized the effects of loading condition by averaging the EPSD results over 2000 lines, wavelengthby-wavelength. The averaged EPSD curves from the fracture surface areas at different crack lengths (different ⌬K values) are closely parallel between wavelengths of 0.5 µm and 20 µm (see Fig. 4, for example). Choosing the square root
of the EPSD [(EPSD)1/2] value at a wavelength of 5 µm (arbitrarily selected as a mid-point between 0.5 and 20 µm wavelength) as a representative index of the roughness of each area and plotting these values as a function of position on the fracture surface, we obtained the results shown in Fig. 5(a) and (b) for Specimens 8353 (R=0.1) and 8354 (R=0.8), respectively. Also plotted are
Fig. 4. Averaged EPSD curves from five different locations on the surface in Test Run 1 in specimen 8353.
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test, the (EPSD)1/2 values initially decreased before increasing. In seeking to unify these results, we plotted all the (EPSD)1/2 values in Fig. 5 as a function of ⌬K. The results in Fig. 6 show three distinct regions for Specimen 8353 (R=0.1): (1) a linear relationship in the low ⌬K region (below 7 MPa√m); (2) a transition region (between 7 and 16 MPa√m); and (3) another linear relationship region at higher ⌬K (above 16 MPa√m). For Specimen 8354 (R=0.8), the trend in the first two regions (lower ⌬K) is not as apparent. Although the individual load-shedding test results in Figs. 5 and 7 indicated decreasing EPSD values where ⌬K decreased below a certain ⌬K value, and the opposite trend above this value, when all the test data were plotted together, the scatter masked the trend in both the first and second regions. However, the third region where (EPSD)1/2 increases linearly with increasing ⌬K is clearly seen (beyond 9 MPa√m). Interestingly, the boundaries of different regions in Specimen 8353 (R=0.1) did not coincide with the boundaries of decreasing and increasing ⌬K testing conditions. At higher values of ⌬K in decreasing ⌬K tests, the (EPSD)1/2 values increase as ⌬K decreases, and during the lower values of ⌬K in increasing ⌬K tests, the (EPSD)1/2 values decrease. Furthermore, the trend in the (EPSD)1/2 values in the transition region of decreasing and increasing ⌬K tests merge smoothly without discontinuity. This trend is less clear for Specimen 8354 ( R=0.8) although the results of decreasing and increasing ⌬K tests overlap in a similar region in the transition zone. The trend of increasing fracture surface roughness
Fig. 5. (EPSD)1/2 values at 5 µm of wavelength and stress intensity range, ⌬K, as a function of crack length for two specimens 8353 and 8354. (a) specimen 8353 (R=0.1) and (b) specimen 8354 (R=0.8).
the ⌬K values corresponding to the fracture surface locations or crack lengths. At first inspection, the behaviors of (EPSD)1/2 and ⌬K as a function of crack length indicate no clear correlation between load and roughness. For example, in the load shedding tests where the ⌬K continuously decreased, the (EPSD)1/2 values first increased before decreasing. In the monotonically increasing ⌬K fatigue
Fig. 6. Summary of the relationship between (EPSD)1/2 and ⌬K for two specimens 8353 and 8354.
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4. Crack closure effect on the fracture surface topography
Fig. 7. Possible correlation of (EPSD)1/2 behavior with ⌬K behavior. (a) Results of 8353 (R=0.1) and (b) results of 8354 (R=0.8).
with decreasing ⌬K near the threshold stress intensity range, ⌬Kth, for various titanium alloys was also noted by Ogawa et al. [6]. They quantified the fracture surface topography using stereo pairs of SEM micrographs and calculated the ratio of the real area of the fracture surface to its projected area at different magnifications. Using this ratio as a measure of fracture surface roughness, they showed that roughness increased with decreasing fatigue crack growth rate from da/dN=1×10−4 mm/cycle to da/dN=1×10−6 mm/cycle. However, they offered no explanation for this trend. The result shown in Fig. 6 clearly illustrates that something is happening to the topography of the fracture surfaces as ⌬K increases or decreases in a certain range. These results were obtained by averaging an EPSD curve of 2000 lines of topographic profiles in each area (for each ⌬K value); thus, we believe that definite physical phenomena were captured by this analysis. We sought the physical explanation.
One plausible explanation for the existence of three distinct regions in Fig. 6 for the R=0.1 specimen is that two competing processes are responsible for the formation and deformation of the fracture surface topography: (1) the roughening of the surface due to increasing ⌬K or Kmax and (2) the rubbing and flattening of the surface due to crack closure. It is generally believed that crack closure is not significant at high stress ratios. However, Boyce and Ritchie [7], referring to recent in-situ SEM studies on the mechanisms for fatigue crack propagation in Ti–6Al–4V by Davidson [8], suggest that even at very high load ratios (R苲0.8) significant crack wake interference (closure) up to 苲100 µm behind the crack tip may occur; and while this near-tip closure would not be detected by standard closure measurements (i.e. compliance-based measurements), it could significantly shield the crack-tip from fully unloading. The results presented here provide supportive evidence of crack closure at high stress ratio (R=0.8), and indicate how the closure effects influence or alter the fracture surface roughness. Fig. 6 indicates that when the ⌬K values are low, the (EPSD)1/2 value increases with increasing ⌬K (clearly shown in the case of Specimen 8353 (R=0.1). In this region, the effect of surface roughening with increasing ⌬K is dominant, and alteration of the surface roughness due to crack closure may not be significant. Possible reasons are that the degree of crack opening would be small in this range and, as a result, deformation or rubbing of surfaces due to crack-opening-related mismatch may be minimal. Interestingly, the roughening behavior of the surfaces for R=0.1 and 0.8 in the low ⌬K range are similar. However, when ⌬K reaches a certain value, the (EPSD)1/2 starts to decrease as ⌬K continues to increase (transition region in Fig. 6). In this region, we believe that crack closure begins to flatten asperities on the fracture surfaces and gradually overcomes the surface roughening with increasing ⌬K. The surface flattening effect outpaces surface roughening as ⌬K increases. Thus, the (EPSD)1/2 values decrease despite the increase in ⌬K. The degree of reduction of the surface roughness (reduction in (EPSD)1/2) is more significant for the surface with R=0.1 than with R=0.8. Furthermore, the range of reduction of surface roughness is wider for the case with R=0.1 than with R=0.8, as would be expected from crack closure. These results are consistent with the results reported by Ogawa et al. [6]. As ⌬K continues to increase, surface roughening starts to overcome surface flattening, and the (EPSD)1/2 values start to increase again as ⌬K increases. This is the third region in Fig. 6. In this region the surface roughness increase for the R=0.8 specimen is very rapid and the
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Fig. 8.
Fig. 9.
Comparison of EPSD curves of tests under similar stress intensity range but different stress ratios.
Changes observed in the shape of EPSD curves as ⌬K increases in the transition zone of R=0.1 specimen.
slope of the roughness increase is similar to that for the slope of the curve of the R=0.1 specimen in the first region. This observation indicates that crack closure is not playing a major role in a high stress ratio situation. The level of surface roughness for the R=0.1 specimen in the third region is considerably lower than that for the R=0.8 specimen, and the rate of increase of surface roughness with increasing of ⌬K for R=0.1 is much less than that for R=0.8. All these observations suggest that the three regions of (EPSD)1/2 behavior result from the modification of surface roughness due to crack closure, and the phenomena observed in the two specimens tested under R=0.1 and R=0.8 can be explained equally well with the closure effect. We now examine the individual fatigue test results more closely. Fig. 7 is a magnified view of the results
shown in Fig. 5. In these load-shedding tests, the ⌬K was decreased as the crack extended. When ⌬K is above a certain boundary value as shown in Fig. 7, the (EPSD)1/2 exhibits low values, and the higher the value of ⌬K, the lower the (EPSD)1/2 value. If the ⌬K values are below the boundary value, the trend in (EPSD)1/2 change is similar to that of ⌬K. This boundary value trend is also observed for tests in which ⌬K was increased as the crack grew, as shown in Fig. 7(b). Here the changes in (EPSD)1/2 above the boundary line are the same as for the load-shedding test; i.e., the higher ⌬K is above the boundary line, the lower the (EPSD)1/2 value. This trend is consistent with crack closure modifying the fracture surfaces at ⌬K’s above the boundary value.
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Fig. 10. EPSD curves showing changes in their shape with ⌬K for two specimens. (a) EPSD curves of run 1 in specimen 8353 (R=0.1) and (b) EPSD curves of run 1 in specimen 8354 (R=0.8).
5. Additional effects of crack closure To delineate the effect of crack closure on the surface roughness, we compared the individual EPSD curves from the fracture surface areas in the third region in Fig. 6 at similar ⌬K’s of about 17 MPa√m in the two samples. The four curves in Fig. 8 (two for R=0.1 and the other two for R=0.8) show that, within the same Rvalue, the EPSD curves have the same shape and are parallel to each other; however, at different R-values, the curves have different shapes in addition to a significant level difference. The curves for R=0.8 are nearly straight between wavelengths of 0.5 µm and 20 µm; however, the curves for R=0.1 deviate from a linear line below 1 µm and above 4 µm. This suggests that crack closure more significantly affects the features below the scale of 1 µm and above 4 µm. The behavior of EPSD curves in the transition region
where the EPSD values decrease and the curves bow more as ⌬K increases, as shown in Fig. 9, suggests that, at R=0.1 and as ⌬K increases, crack closure effects become more important. The EPSD results from the load-shedding tests also suggest that higher ⌬K loading induces crack closure effects that cause bowing of the EPSD curves. We examined the first fatigue test in each sample where crack growth was initiated at ⌬K well above the boundary line shown in Fig. 7. EPSD values were initially low, but gradually increased. The shape of EPSD curves during this change is shown in Fig. 10. The EPSD curves corresponding to initial high ⌬K exhibit bowing; however, as ⌬K decreases below the boundary ⌬K, the EPSD curve becomes straighter. Again these curves suggest that the higher ⌬K loading induces crack closure effects that cause bowing of the EPSD curves.
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topography, less stress is necessary to open the crack. In other words, the lower the opening stress, the higher the effect of surface deformation and flattening of the surface. For the case of higher R-value, the Newman results suggest that the ratio of the opening stress to the maximum stress is independent of ⌬K. The EPSD results indicate that a small but fixed percentage of the crack closure effect operates over a wide range of ⌬K based on the slope of the curve shown in Fig. 6. If this is the case, we anticipate that the ratio of the opening stress to the maximum stress could be constant.
Acknowledgements
Fig. 11. Crack-opening stress ratios for 2024-T3 aluminum alloy at low and high stress ratios [9].
6. Discussion The results presented here show that application of FFT analysis to fatigue fracture surface topography can illuminate the interactive influence of loading parameters ⌬K and R on the extent of crack closure. At low R values, crack closure significantly modifies the fracture surface topography over a wide range of ⌬K; at high R-values, the closure and surface modification effect is small and requires a ⌬K above a certain level. Data for only two R values were analyzed. Analysis of additional specimens fatigued under other loading conditions may allow quantitative relationships among R, ⌬K, and surface roughness to be established. The determination of fatigue loading conditions from fracture surface topography analysis is not straightforward, because the roughness–load parameter relationship is not single-valued. However, the shape of the EPSD vs ⌬K curve is sensitive to both R and ⌬K, suggesting that these loading parameters affect different parts of the curve. If so, other analysis procedures such as wavelet techniques that can examine specific features at specific scales, may allow a unique determination of ⌬K and R from the fracture surfaces. The results presented here are in accord with observations by Newman of crack opening for 2024-T3 aluminum alloy at low and high stress ratios shown in Fig. 11 [9,10]. A striking similarity exists in the shape of the curve for R=0.1 with the (EPSD)1/2 curve for R=0.1 shown in Fig. 6. The curve in Fig. 11 suggests that, due to the crack closure and re-deformation of the fracture
The authors thank Dr. Dennis Healy of the Defense Advanced Research Projects Agency and Dr. Anna Tsau of the Center for Computing Sciences at the Institute of Defense Analyses for their interest and encouragement. Drs. Michael Gehron, Joe Bateman, and Jay Littles and their colleagues at Pratt & Whitney consulted with us and provided crack growth specimens for analysis. Dr. Theodore Nicholas of the Air Force Research Laboratory provided valuable technical advice throughout this effort.
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