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Measurement of crack closure after the application of an overload cycle, using moiré interferometry
International Journal of Fatigue 27 (2005) 1453–1462 www.elsevier.com/locate/ijfatigue
Measurement of crack closure after the application of an overload cycle, using moire´ interferometry L.J. Fellowsa,*, D. Nowellb a School of Technology, Oxford Brookes University, Gipsy Lane, Oxford, UK Department of Engineering Science, University of Oxford, Parks Road, Oxford, UK
b
Available online 31 August 2005
Abstract The crack closure behaviour on the application of a single overload cycle was studied in a Ti-6Al-4V specimen. Moire´ interferometry with photoresist gratings was used to measure crack displacements. During the overload cycle a large crack opening displacement was observed at the maximum load. This was similar to predictions from a Dugdale-type crack closure model. When the load was taken back to zero, the crack was open at the crack tip due to the high levels of plastic deformation during the overload cycle. As the crack was grown there was some evidence of the deformed material on the crack faces. Moire´ interferometry provided displacement data close to the crack faces, even when the crack had grown to over two-and-a-half times the overload crack length. When the overload was applied the crack bifurcated, and the Dugdale-type model under-predicted the crack opening. q 2005 Published by Elsevier Ltd. Keywords: Overload; Crack closure; Dugdale-type model; Moire´ interferometry
1. Introduction In 1970 Elber [1,2] discovered that cracks peel open during a load cycle and he put forward the idea of plasticityinduced crack closure. This discovery is clearly very important in the prediction of fatigue crack growth and has been used to explain R-ratio phenomena, the stress intensity factor threshold and the delaying effect of overloads [3]. However, in the past 30 years there has been much controversy about which mechanisms account for crack closure and whether closure can account for these phenomena [4]. Part of the problem is that the closure is very hard to measure as small displacements are involved. Many methods have been used that have subsequently been found to be inaccurate or difficult to interpret. The authors have chosen to use moire´ interferometry, which provides full-field displacement maps of the area around the crack to sub-micron accuracy. This method has the benefit that the crack can be seen to peel open, and the point of crack opening and closure at the surface can be found [5–7]. The displacement maps can be compared with numerical and analytical models. In order * Corresponding author.
0142-1123/$ - see front matter q 2005 Published by Elsevier Ltd. doi:10.1016/j.ijfatigue.2005.07.004
to be able to measure closure at positive R-ratios the authors used a moire´ interferometer built into a fatigue machine and photoresist gratings. An analytical model has been developed by Nowell [8] based on a Dugdale strip yield model, with the plastic wake ‘pasted onto’ the crack faces, giving plasticity-induced crack closure. This model has given good results for crack shape and crack closure under constant amplitude loading [7]. This paper describes the next step, which was to introduce an overload cycle and determine whether the model can predict the shape of the crack several hundred cycles after the application of the overload.
2. Specimen and grating details The specimens used were made of Ti-6Al-4V and were loaded in four point bending; the geometry is shown in Fig. 1. The material yield stress (0.1% proof stress) was 963 MPa, and the ultimate tensile strength was 1013 MPa, showing that the material exhibited very limited strain hardening (see [7]). This is appropriate for later application of the strip yield model, where ideal elastic/plastic material behaviour is assumed. The specimens were produced with photoresist gratings of 1200 lines per mm using a dyed photoresist (Hunts 514). To create the grating the specimens were dipped
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Raised area for grating application
110mm
7mm
8mm
60mm 150mm Fig. 1. Geometry of specimen and loading arrangement, reproduced from Ref. [7]. Table 1 Loads and crack lengths for the different data sets Specimen identifier
Data set identifier
Max load (kN)
Min load (kN)
Overload (kN)
Comment
Crack length (mm)
51
a b c d e f g 53
3.0
0.0
5.0
Cycle before overload Overload cycle Cycle after overload
1.5
0.0
3.0
Overload cycle
0.438 0.438 0.451 0.654 0.700 0.916 1.158 1.881
53
in photoresist, exposed to an Argon laser, developed and gold-sputtered. The details of the grating production process are given in [9,10]. The gratings have a thickness of 0.75 mm and are well adhered to the specimen, giving faithful readings around the crack tip. After creation of the grating, specimens were mounted in the four point bend fixture and cracks were grown from a small EDM starter notch. Once a crack had started a 2 mm layer was machined from the top of the specimen in order to remove the starter notch and a bead of photoresist at the edge of the specimen. The crack was then grown under cyclic sinusoidal loading to the required crack length. At this point the moire´ measurements were taken at a number of points in the load cycle, referred to here as load steps. The authors have used this method for extensive numbers of specimens at constant amplitude [7]. However, variable amplitude testing was more difficult, and of the ten tests carried out only two were successful. The main problems were that the experiments were time consuming and the cracks tended to bifurcate. The two specimens presented in this paper each underwent an overload cycle. For specimen 51 the displacement data was recorded at the overload and then the crack was grown and displacement data recorded at several crack lengths after the overload (see Table 1). For specimen 53, only the overload cycle was recorded.
3. Analysis of the data A typical moire´ fringe pattern (interferogram) is shown in Fig. 2a. The crack can clearly be seen
Fig. 2. Moire´ displacement data from a typical specimen. (a) interferogram, (b) Phase Map, reproduced from Ref. [7].
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Fig. 3. Processed data—3D plot of v-displacement for a region around the crack tip, showing crack opening as a discontinuity in displacement, reproduced from Ref. [7].
positioned centrally and running vertically from the top of the image to the centre. Five interferograms are taken at each load step, using phase stepping to interpolate between the fringes and then they are converted into phase maps (see Fig. 2b). The phase map is unwrapped and converted into full field displacement maps (see Fig. 3). This process and the subsequent data analysis are described elsewhere [7]. It is worth emphasising an additional step in data processing. The grating that was put on the specimen has been deformed by the crack growth and associated plastic flow. Hence, even at zero load the crack appears open and this is because the grating has been stretched. Hence, at each load step, the displacements at the ‘null field’ (at the lowest load) are subtracted from the displacements at the current load step. This then gives only the displacement change between the current load and the lowest load. If the crack is closed then the displacement change will be zero. Hence, all the displacement values obtained were measured relative to this ’null field’.
4. Modelling To compare the results with predictions from a crack closure model, a plane stress strip yield model was employed. The model is fully described in [8] and is similar to that proposed by Newman [11]. The inequalities for yield and crack face contact were solved by a process of constrained minimisation using a quadratic programming algorithm. This method is numerically efficient and requires little user intervention. An elastic/ideal plastic material model is employed, and only two input parameters are required: the R-ratio of the load cycle and the maximum stress (normalised by the material yield stress). A selfsimilarity argument is used for cases of constant applied stress so the crack does not need to be grown from a small initial defect. Whilst this condition does not strictly hold for a bending stress field the errors involved were not found to be significant. The far-field stress value was the stress in the specimen remote from the crack at the same y-position. Output from the model was processed in a similar manner to the experimental data (i.e. by subtracting the crack profile at
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Position from crack tip (mm) Fig. 4. Data set 53 (RZ0, aZ1.881 mm)—Comparison of experimental and model results during overload cycle (a) 0.4 kN, (b) 0.6 kN, (c) 1.0 kN, (d) 3.0 kN, and (e) 0.4 kN (unloading), (f) 0.0 kN (unloaded) (g) Comparison of experimental results during overload cycle when applied force is 0.4 kN, 0.6 kN, 1.0 kN and 3.0 kN.
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Fig. 5. Data set 51 (RZ0, aZ0.438 mm) - Comparison of experimental and model results before overload (a) 0.5 kN, (b) 1.5 kN, (c) 3.0 kN, (d) 0.5 kN.
minimum load) in order to enable comparison. Since the model is numerical it does not yield simple formulae predicting closure.
5. Results The moire´ interferometry method described above generates a significant amount of data and space precludes a full presentation of the results. We will therefore choose to concentrate on the most significant data, commencing with data set 53, where a single cycle was analysed, then moving onto specimen 51, where seven cycles were analysed. 5.1. Specimen 53 Specimen 53 was cycled between 0 and 1.5 kN until the crack was 1.881 mm long. An overload of 3 kN was then applied to the specimen. Fig. 4(a)-(f) show the experimental and analytical results for loads of 0.4, 0.6 and 1.0 kN on loading, the maximum load of 3.0 kN, and 0.4 and 0.0 kN on unloading. Fig. 4(g) shows the crack opening displacements for the experimental measurements on loading. Closure is evident at the crack tip in Fig. 4(a), since there is very little change from the zero
load condition. It should also be noted that the model appears to slightly over predict the crack opening displacement. By 1.0 kN, Fig. 4(c), on loading the crack is open to the tip and the model predicts the crack opening displacement well close to the crack tip. At the maximum load, Fig. 4(d), the crack opening displacement is under predicted by the model along the whole crack. In Fig. 4(e), at 0.4 kN on unloading, the crack can be seen to be open along the whole length. This is due to a large amount of plastic deformation ahead of the crack tip when the overload was applied. In the model the plastic zone is confined to a line and a bulge can be seen ahead of the crack tip, corresponding to the boundary between forward and reverse plastic zones. In the experimental data this effect can also be seen but is much less prominent as the plastic deformation occurs over a much larger area. In Fig. 4(f) the specimen has been unloaded. It can be seen that the crack is propped open at the tip due to the plastic deformation experienced during the overload. The model is in reasonable agreement with the experiment about the percentage of the crack that is open during the cycle. However, the model predicts that the crack will be fully open from 0.8 kN and remain so for the rest of the cycle. In contrast, the experiment shows a lower crack opening load (0.6 kN).
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Specimen 51 5.0kN Maximum Load Overload Cycle
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–0.1
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Experimental
crack length = 0.451mm
Position of crack tip at overload 1000
crack length = 0.438mm 0.1
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crack length = 0.451mm 2000
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Specimen 51 0.0kN Unloading Overload Cycle
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Specimen 51 0.5kN Loading Cycle after overload
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Position from crack tip (mm) Fig. 6. Data set 51 (RZ0, aZ0.438 mm, Overload cycle)—Comparison of experimental and model results during overload (a) 5.0 kN, (b) 0.0 kN.
(c)
C.O.D. [nm]
Specimen 51 was also an overload test. The crack was grown to 0.438 mm at an R-ratio of zero under loads cycling between 0 and 3 kN. Moire´ readings were taken for three successive cycles: a cycle at 0.438 mm, then the following cycle when an overload was applied, then the cycle immediately after the overload. The crack was then grown and moire´ readings were taken when the crack was at lengths of 0.654, 0.700, 0.916 and 1.158 mm. For this test 89 load steps were analysed and hence only a small proportion of the data produced is shown and can be found in Figs. 5–12. The results are affected by a large bifurcation—the arrested part of the crack grew to 0.25 mm from the tip of the crack when the overload was applied. The plastic zone sizes and maximum K values for three of the cycles where moire´ readings were taken are given in the Table 2 Fig. 5(a)-(d) are for the cycle immediately before the overload was applied. At 0.5 kN on loading, Fig. 5(a), the model and experiment have approximately the same level of closure but the model over estimates the crack opening displacement. In Fig. 5(b), at 1.5 kN, the crack is on the point of opening for both the model and the experimental
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crack length = 0.451mm
Position of crack tip at overload
Experimental Model
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–0.1
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0.1
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0.4
0.5
Position from crack tip (mm) Fig. 7. Data set 51 (RZ0, aZ0.451 mm)—Comparison of experimental and model results immediately after overload (a) 0.5 kN, (b) 3.0 kN, (c) 0.5 kN.
data. The maximum load is shown in Fig. 5(c) and it can be seen that although the crack tip opening displacement is described well by the model, the crack opening displacement is over predicted. Fig. 5(d) shows the crack shape at 0.5 kN on unloading. The crack shapes on the overload cycle should be similar to the previous cycle until the load exceeds the previous maximum load. Fig. 6(a) shows the maximum load during the overload cycle. Again, the model over predicts the crack opening displacement along the entire length of the crack. When the load is reduced to zero, Fig. 6(b), the crack is open
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Fig. 8. Data set 51 (RZ0, aZ0.654 mm)—Comparison of experimental and model results 0.214 mm after overload (a) 0.5 kN, (b) 3.0 kN, (c) 0.5 kN.
Fig. 9. Data set 51 (RZ0, aZ0.700 mm)—Comparison of experimental and model results 0.262 mm after overload (a) 0.5 kN, (b) 3.0 kN, (c) 0.5 kN.
to the tip. Fig. 7(a)-(c) are taken from the cycle immediately after the overload. These figures show that there is no closure during this cycle and the model over predicts the crack opening displacement. The crack was then grown to 0.654 mm. Fig. 8(a)-(c) show crack shapes for this load cycle. Due to grating damage, data are not available for the first 0.1 mm of the crack. For the rest of the crack it can be seen that at low loads the model over predicts the crack opening displacement and for high loads the COD is under predicted. The crack was again grown, and this time readings were taken at a crack length of 0.700 mm. The model and experimental results for three loads are shown
in Fig. 9(a)-(c). The model under predicts the crack shapes at all times and the experimental results show no closure. Figs. 10 (a)-(c) and 11 (a)-(d) show further results when the crack was 0.916 and 1.158 mm respectively. Again the model tends to under predict the displacement of the crack faces. A bulge in the plastic wake is evident in these figures by the shape of the crack faces as predicted by the analytical analysis. In the model the bulge is not in the same place as the position of the crack tip when the overload was applied. In Fig. 11(a)(d) the crack has grown to over two-and-a-half times the length of the crack when the overload was applied. Generally, the amount of crack closure experimentally is
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L.J. Fellows, D. Nowell / International Journal of Fatigue 27 (2005) 1453–1462
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less after the overload than before. The overload is still affecting the amount of closure predicted by the model. This can be seen in Fig. 11(b), at 1.0 kN on loading, where the faces of the crack are in contact at the tip, then are not in contact, then come back into contact 0.45 mm away from the crack tip due to a bulge in the plastic wake. Fig. 12 shows the percentage of the crack that was open for the cycle immediately prior to the overload. The experimental and analytical data correlate very well, although some of the experimental data points are missing because the grating became poor close to the crack tip. The model predicts that the crack will first open to the tip at 2.0 kN and begin to close at 1.5 kN on
0.2
Position of crack tip (mm)
Position of crack tip (mm) Fig. 10. Data set 51 (RZ0, aZ0.916 mm)—Comparison of experimental and model results 0.478 mm after overload (a) 0.5 kN, (b) 3.0 kN, (c) 0.5 kN.
Experimental Model
Crack length = 1.158mm
Position of crack tip at overload Crack length = 1.158mm
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Position of crack tip (mm) Fig. 11. Data set 51 (RZ0, aZ1.158 mm)—Comparison of experimental and model results 0.720 mm after overload (a) 0.5 kN, (b) 1.0 kN, (c) 3.0 kN, (d) 0.5 kN.
L.J. Fellows, D. Nowell / International Journal of Fatigue 27 (2005) 1453–1462 100 Specimen 51
Percentage open
95 90 85 80 75
opening - experimental closing - experimental opening - model closing - model
70 65 60 0
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1
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Load (kN) Fig. 12. Data set 51 (RZ0, aZ0.438 mm)—Variation of opening point with applied load.
Table 2 Maximum Stress intensity value and plastic zone sizes for the first cycle, the overload cycle and the last cycle measured in specimen 51, and for specimen 53 Specimen
Maximum Load (kN)
Maximum Stress Intensity Factor K (MPaOm)
Crack Length, a (mm)
Plastic zone size, rp (mm)
Crack plus plastic zone, aCrp (mm)
53 51
3 3 5 3
43.20 20.87 34.78 33.77
1.881 0.438 0.438 1.158
0.29 0.07 0.188 0.177
2.17 0.506 0.626 1.335
unloading. Experimentally the crack both opens and closes at 2.0 kN.
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crack bifurcation was observed at the crack length where the overload was applied. This seems to give a significant increase in the crack opening displacement; hence the correlation between the model and the experiment is poor. A number of other overload tests were attempted, but in each case the crack bifurcated on the application of the overload. It can be seen from the model that as the crack grows the material that was stretched during the overload forms part of the plastic wake. It creates a bulge in the faces, causing then to contact prematurely and hence increases closure. In both the overload experiments the crack no longer closes immediately after the overload, as expected. In Figs. 10(b) and 11(c) the position of the crack tip when the overload was applied is marked. It can be seen that this is not in the same place as the bulge causing premature closure in the model. This may be because as the crack grows, the crack faces compress the bulge and as the bulge is wedge shaped, the bulge moves along the crack faces towards the current crack tip. By comparison of different crack lengths there is evidence that the position of the bulge moves along the crack faces. This does not happen in the experiment as the crack faces have little contact due to the bifurcation. In a previous paper [7] the model and experimental results correlate well. However, the correlation is poor for the overload tests. In the model, the simplified plastic zone is inappropriate for a crack that has bifurcated, the plastic zone being better represented by two lines of plasticity radiating from the crack tip. Further work is required with the current model and a less severe overload to assess whether the model can predict the crack closure after an overload if the crack has not bifurcated.
6. Discussion and conclusions Using the moire´ interferometry, typically the crack is displayed by over one hundred data points, each giving the separation of the crack faces with a notional sensitivity of less than 100 nm. However, where the grating is damaged, the accuracy can be closer to 1000 nm.The photoresist gratings showed remarkable adhesion to the specimen, even close to the crack faces after thousands of cycles. The crack can be seen to peel open and closed. The closure zones are relatively small but are large enough to support the closure forces as the model shows that the stresses in this region are high—up to KsY. The crack in specimen 51 did not behave as expected. Rather than there being increased closure after the overload, there appeared to be reduced closure. In addition to this, the crack in specimen 51 grew more quickly than the cracks in other specimens that had not experienced an overload, contrary to what was expected. When the crack was examined under a microscope, a
Acknowledgements The authors would like to thank Dr Salih Gu¨ngo¨r for his help with the experimental measurements and Dr Dougan McKellar for assistance with interpreting the results.
References [1] Elber W. Fatigue crack closure under cyclic tension. Eng Fract Mech 1970;2:37–45. [2] Elber W, The significance of fatigue crack closure. In: Damage tolerance in aircraft structures, ASTM STP 486. Philadelphia: ASTM; 1971. p. 230–42. [3] Suresh S. Fatigue of materials. Cambridge: Cambridge University; 1991. [4] Kemp R.M.J. Fatigue crack closure—a review. Royal Aerospace Establishment, Technical Report 90046 ICAF Document 1776; 1990. [5] Gu¨ngo¨r S, Fellows LJ. Combined fatigue rig and moire´ interferometer to measure crack closure. In: Allison IM, editor. Experimental mechanics. Rotterdam: Balkema; 1998. p. 1071–6.
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[6] Fellows LJ, Nowell D, Hills DA, Comparison of crack closure measurements using moire´ interferometry and results from a Dugdaletype crack closure model. In: Karihaloo BL, Y-W, editors. Advances in fracture research, Proceedings of the 9th International Conference on Fracture, Sydney, April 1997. p. 2551–58. [7] Fellows LJ, Nowell D. Crack closure measurements using moire interferometry with photoresist gratings. Int J Fatigue 2004;26/10: 1075–82. [8] Nowell D. A boundary element model of plasticity-induced fatigue crack closure. Fatigue Fract Eng Mater Struct 1998;21:857–71.
[9] Fellows LJ, Fatigue crack growth under variable stress ratios and complex load history. DPhil Thesis, University of Oxford, 1999. [10] Fellows LJ, Gungor S. Fabrication of photoresist diffraction gratings on Ti–6Al–4V beam specimens for use in moire´ interferometry. Meas Sci Technol 1998;9:1963–8. [11] Newman JC Jr, A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading. In: Chang JB, Hudson CM, editors. Methods and models for predicting fatigue crack growth under random loading, ASTM STP 748. Philadelphia; ASTM; 1981. p. 53–84.
Measurement of crack closure after the application of an overload cycle, using moire´ interferometry L.J. Fellowsa,*, D. Nowellb a School of Technology, Oxford Brookes University, Gipsy Lane, Oxford, UK Department of Engineering Science, University of Oxford, Parks Road, Oxford, UK
b
Available online 31 August 2005
Abstract The crack closure behaviour on the application of a single overload cycle was studied in a Ti-6Al-4V specimen. Moire´ interferometry with photoresist gratings was used to measure crack displacements. During the overload cycle a large crack opening displacement was observed at the maximum load. This was similar to predictions from a Dugdale-type crack closure model. When the load was taken back to zero, the crack was open at the crack tip due to the high levels of plastic deformation during the overload cycle. As the crack was grown there was some evidence of the deformed material on the crack faces. Moire´ interferometry provided displacement data close to the crack faces, even when the crack had grown to over two-and-a-half times the overload crack length. When the overload was applied the crack bifurcated, and the Dugdale-type model under-predicted the crack opening. q 2005 Published by Elsevier Ltd. Keywords: Overload; Crack closure; Dugdale-type model; Moire´ interferometry
1. Introduction In 1970 Elber [1,2] discovered that cracks peel open during a load cycle and he put forward the idea of plasticityinduced crack closure. This discovery is clearly very important in the prediction of fatigue crack growth and has been used to explain R-ratio phenomena, the stress intensity factor threshold and the delaying effect of overloads [3]. However, in the past 30 years there has been much controversy about which mechanisms account for crack closure and whether closure can account for these phenomena [4]. Part of the problem is that the closure is very hard to measure as small displacements are involved. Many methods have been used that have subsequently been found to be inaccurate or difficult to interpret. The authors have chosen to use moire´ interferometry, which provides full-field displacement maps of the area around the crack to sub-micron accuracy. This method has the benefit that the crack can be seen to peel open, and the point of crack opening and closure at the surface can be found [5–7]. The displacement maps can be compared with numerical and analytical models. In order * Corresponding author.
0142-1123/$ - see front matter q 2005 Published by Elsevier Ltd. doi:10.1016/j.ijfatigue.2005.07.004
to be able to measure closure at positive R-ratios the authors used a moire´ interferometer built into a fatigue machine and photoresist gratings. An analytical model has been developed by Nowell [8] based on a Dugdale strip yield model, with the plastic wake ‘pasted onto’ the crack faces, giving plasticity-induced crack closure. This model has given good results for crack shape and crack closure under constant amplitude loading [7]. This paper describes the next step, which was to introduce an overload cycle and determine whether the model can predict the shape of the crack several hundred cycles after the application of the overload.
2. Specimen and grating details The specimens used were made of Ti-6Al-4V and were loaded in four point bending; the geometry is shown in Fig. 1. The material yield stress (0.1% proof stress) was 963 MPa, and the ultimate tensile strength was 1013 MPa, showing that the material exhibited very limited strain hardening (see [7]). This is appropriate for later application of the strip yield model, where ideal elastic/plastic material behaviour is assumed. The specimens were produced with photoresist gratings of 1200 lines per mm using a dyed photoresist (Hunts 514). To create the grating the specimens were dipped
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in photoresist, exposed to an Argon laser, developed and gold-sputtered. The details of the grating production process are given in [9,10]. The gratings have a thickness of 0.75 mm and are well adhered to the specimen, giving faithful readings around the crack tip. After creation of the grating, specimens were mounted in the four point bend fixture and cracks were grown from a small EDM starter notch. Once a crack had started a 2 mm layer was machined from the top of the specimen in order to remove the starter notch and a bead of photoresist at the edge of the specimen. The crack was then grown under cyclic sinusoidal loading to the required crack length. At this point the moire´ measurements were taken at a number of points in the load cycle, referred to here as load steps. The authors have used this method for extensive numbers of specimens at constant amplitude [7]. However, variable amplitude testing was more difficult, and of the ten tests carried out only two were successful. The main problems were that the experiments were time consuming and the cracks tended to bifurcate. The two specimens presented in this paper each underwent an overload cycle. For specimen 51 the displacement data was recorded at the overload and then the crack was grown and displacement data recorded at several crack lengths after the overload (see Table 1). For specimen 53, only the overload cycle was recorded.
3. Analysis of the data A typical moire´ fringe pattern (interferogram) is shown in Fig. 2a. The crack can clearly be seen
Fig. 2. Moire´ displacement data from a typical specimen. (a) interferogram, (b) Phase Map, reproduced from Ref. [7].
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Fig. 3. Processed data—3D plot of v-displacement for a region around the crack tip, showing crack opening as a discontinuity in displacement, reproduced from Ref. [7].
positioned centrally and running vertically from the top of the image to the centre. Five interferograms are taken at each load step, using phase stepping to interpolate between the fringes and then they are converted into phase maps (see Fig. 2b). The phase map is unwrapped and converted into full field displacement maps (see Fig. 3). This process and the subsequent data analysis are described elsewhere [7]. It is worth emphasising an additional step in data processing. The grating that was put on the specimen has been deformed by the crack growth and associated plastic flow. Hence, even at zero load the crack appears open and this is because the grating has been stretched. Hence, at each load step, the displacements at the ‘null field’ (at the lowest load) are subtracted from the displacements at the current load step. This then gives only the displacement change between the current load and the lowest load. If the crack is closed then the displacement change will be zero. Hence, all the displacement values obtained were measured relative to this ’null field’.
4. Modelling To compare the results with predictions from a crack closure model, a plane stress strip yield model was employed. The model is fully described in [8] and is similar to that proposed by Newman [11]. The inequalities for yield and crack face contact were solved by a process of constrained minimisation using a quadratic programming algorithm. This method is numerically efficient and requires little user intervention. An elastic/ideal plastic material model is employed, and only two input parameters are required: the R-ratio of the load cycle and the maximum stress (normalised by the material yield stress). A selfsimilarity argument is used for cases of constant applied stress so the crack does not need to be grown from a small initial defect. Whilst this condition does not strictly hold for a bending stress field the errors involved were not found to be significant. The far-field stress value was the stress in the specimen remote from the crack at the same y-position. Output from the model was processed in a similar manner to the experimental data (i.e. by subtracting the crack profile at
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minimum load) in order to enable comparison. Since the model is numerical it does not yield simple formulae predicting closure.
5. Results The moire´ interferometry method described above generates a significant amount of data and space precludes a full presentation of the results. We will therefore choose to concentrate on the most significant data, commencing with data set 53, where a single cycle was analysed, then moving onto specimen 51, where seven cycles were analysed. 5.1. Specimen 53 Specimen 53 was cycled between 0 and 1.5 kN until the crack was 1.881 mm long. An overload of 3 kN was then applied to the specimen. Fig. 4(a)-(f) show the experimental and analytical results for loads of 0.4, 0.6 and 1.0 kN on loading, the maximum load of 3.0 kN, and 0.4 and 0.0 kN on unloading. Fig. 4(g) shows the crack opening displacements for the experimental measurements on loading. Closure is evident at the crack tip in Fig. 4(a), since there is very little change from the zero
load condition. It should also be noted that the model appears to slightly over predict the crack opening displacement. By 1.0 kN, Fig. 4(c), on loading the crack is open to the tip and the model predicts the crack opening displacement well close to the crack tip. At the maximum load, Fig. 4(d), the crack opening displacement is under predicted by the model along the whole crack. In Fig. 4(e), at 0.4 kN on unloading, the crack can be seen to be open along the whole length. This is due to a large amount of plastic deformation ahead of the crack tip when the overload was applied. In the model the plastic zone is confined to a line and a bulge can be seen ahead of the crack tip, corresponding to the boundary between forward and reverse plastic zones. In the experimental data this effect can also be seen but is much less prominent as the plastic deformation occurs over a much larger area. In Fig. 4(f) the specimen has been unloaded. It can be seen that the crack is propped open at the tip due to the plastic deformation experienced during the overload. The model is in reasonable agreement with the experiment about the percentage of the crack that is open during the cycle. However, the model predicts that the crack will be fully open from 0.8 kN and remain so for the rest of the cycle. In contrast, the experiment shows a lower crack opening load (0.6 kN).
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Specimen 51 was also an overload test. The crack was grown to 0.438 mm at an R-ratio of zero under loads cycling between 0 and 3 kN. Moire´ readings were taken for three successive cycles: a cycle at 0.438 mm, then the following cycle when an overload was applied, then the cycle immediately after the overload. The crack was then grown and moire´ readings were taken when the crack was at lengths of 0.654, 0.700, 0.916 and 1.158 mm. For this test 89 load steps were analysed and hence only a small proportion of the data produced is shown and can be found in Figs. 5–12. The results are affected by a large bifurcation—the arrested part of the crack grew to 0.25 mm from the tip of the crack when the overload was applied. The plastic zone sizes and maximum K values for three of the cycles where moire´ readings were taken are given in the Table 2 Fig. 5(a)-(d) are for the cycle immediately before the overload was applied. At 0.5 kN on loading, Fig. 5(a), the model and experiment have approximately the same level of closure but the model over estimates the crack opening displacement. In Fig. 5(b), at 1.5 kN, the crack is on the point of opening for both the model and the experimental
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data. The maximum load is shown in Fig. 5(c) and it can be seen that although the crack tip opening displacement is described well by the model, the crack opening displacement is over predicted. Fig. 5(d) shows the crack shape at 0.5 kN on unloading. The crack shapes on the overload cycle should be similar to the previous cycle until the load exceeds the previous maximum load. Fig. 6(a) shows the maximum load during the overload cycle. Again, the model over predicts the crack opening displacement along the entire length of the crack. When the load is reduced to zero, Fig. 6(b), the crack is open
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Fig. 9. Data set 51 (RZ0, aZ0.700 mm)—Comparison of experimental and model results 0.262 mm after overload (a) 0.5 kN, (b) 3.0 kN, (c) 0.5 kN.
to the tip. Fig. 7(a)-(c) are taken from the cycle immediately after the overload. These figures show that there is no closure during this cycle and the model over predicts the crack opening displacement. The crack was then grown to 0.654 mm. Fig. 8(a)-(c) show crack shapes for this load cycle. Due to grating damage, data are not available for the first 0.1 mm of the crack. For the rest of the crack it can be seen that at low loads the model over predicts the crack opening displacement and for high loads the COD is under predicted. The crack was again grown, and this time readings were taken at a crack length of 0.700 mm. The model and experimental results for three loads are shown
in Fig. 9(a)-(c). The model under predicts the crack shapes at all times and the experimental results show no closure. Figs. 10 (a)-(c) and 11 (a)-(d) show further results when the crack was 0.916 and 1.158 mm respectively. Again the model tends to under predict the displacement of the crack faces. A bulge in the plastic wake is evident in these figures by the shape of the crack faces as predicted by the analytical analysis. In the model the bulge is not in the same place as the position of the crack tip when the overload was applied. In Fig. 11(a)(d) the crack has grown to over two-and-a-half times the length of the crack when the overload was applied. Generally, the amount of crack closure experimentally is
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less after the overload than before. The overload is still affecting the amount of closure predicted by the model. This can be seen in Fig. 11(b), at 1.0 kN on loading, where the faces of the crack are in contact at the tip, then are not in contact, then come back into contact 0.45 mm away from the crack tip due to a bulge in the plastic wake. Fig. 12 shows the percentage of the crack that was open for the cycle immediately prior to the overload. The experimental and analytical data correlate very well, although some of the experimental data points are missing because the grating became poor close to the crack tip. The model predicts that the crack will first open to the tip at 2.0 kN and begin to close at 1.5 kN on
0.2
Position of crack tip (mm)
Position of crack tip (mm) Fig. 10. Data set 51 (RZ0, aZ0.916 mm)—Comparison of experimental and model results 0.478 mm after overload (a) 0.5 kN, (b) 3.0 kN, (c) 0.5 kN.
Experimental Model
Crack length = 1.158mm
Position of crack tip at overload Crack length = 1.158mm
6000 4000 2000 0 –0.2 –0.1 0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Position of crack tip (mm) Fig. 11. Data set 51 (RZ0, aZ1.158 mm)—Comparison of experimental and model results 0.720 mm after overload (a) 0.5 kN, (b) 1.0 kN, (c) 3.0 kN, (d) 0.5 kN.
L.J. Fellows, D. Nowell / International Journal of Fatigue 27 (2005) 1453–1462 100 Specimen 51
Percentage open
95 90 85 80 75
opening - experimental closing - experimental opening - model closing - model
70 65 60 0
0.5
1
1.5
2
2.5
3
Load (kN) Fig. 12. Data set 51 (RZ0, aZ0.438 mm)—Variation of opening point with applied load.
Table 2 Maximum Stress intensity value and plastic zone sizes for the first cycle, the overload cycle and the last cycle measured in specimen 51, and for specimen 53 Specimen
Maximum Load (kN)
Maximum Stress Intensity Factor K (MPaOm)
Crack Length, a (mm)
Plastic zone size, rp (mm)
Crack plus plastic zone, aCrp (mm)
53 51
3 3 5 3
43.20 20.87 34.78 33.77
1.881 0.438 0.438 1.158
0.29 0.07 0.188 0.177
2.17 0.506 0.626 1.335
unloading. Experimentally the crack both opens and closes at 2.0 kN.
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crack bifurcation was observed at the crack length where the overload was applied. This seems to give a significant increase in the crack opening displacement; hence the correlation between the model and the experiment is poor. A number of other overload tests were attempted, but in each case the crack bifurcated on the application of the overload. It can be seen from the model that as the crack grows the material that was stretched during the overload forms part of the plastic wake. It creates a bulge in the faces, causing then to contact prematurely and hence increases closure. In both the overload experiments the crack no longer closes immediately after the overload, as expected. In Figs. 10(b) and 11(c) the position of the crack tip when the overload was applied is marked. It can be seen that this is not in the same place as the bulge causing premature closure in the model. This may be because as the crack grows, the crack faces compress the bulge and as the bulge is wedge shaped, the bulge moves along the crack faces towards the current crack tip. By comparison of different crack lengths there is evidence that the position of the bulge moves along the crack faces. This does not happen in the experiment as the crack faces have little contact due to the bifurcation. In a previous paper [7] the model and experimental results correlate well. However, the correlation is poor for the overload tests. In the model, the simplified plastic zone is inappropriate for a crack that has bifurcated, the plastic zone being better represented by two lines of plasticity radiating from the crack tip. Further work is required with the current model and a less severe overload to assess whether the model can predict the crack closure after an overload if the crack has not bifurcated.
6. Discussion and conclusions Using the moire´ interferometry, typically the crack is displayed by over one hundred data points, each giving the separation of the crack faces with a notional sensitivity of less than 100 nm. However, where the grating is damaged, the accuracy can be closer to 1000 nm.The photoresist gratings showed remarkable adhesion to the specimen, even close to the crack faces after thousands of cycles. The crack can be seen to peel open and closed. The closure zones are relatively small but are large enough to support the closure forces as the model shows that the stresses in this region are high—up to KsY. The crack in specimen 51 did not behave as expected. Rather than there being increased closure after the overload, there appeared to be reduced closure. In addition to this, the crack in specimen 51 grew more quickly than the cracks in other specimens that had not experienced an overload, contrary to what was expected. When the crack was examined under a microscope, a
Acknowledgements The authors would like to thank Dr Salih Gu¨ngo¨r for his help with the experimental measurements and Dr Dougan McKellar for assistance with interpreting the results.
References [1] Elber W. Fatigue crack closure under cyclic tension. Eng Fract Mech 1970;2:37–45. [2] Elber W, The significance of fatigue crack closure. In: Damage tolerance in aircraft structures, ASTM STP 486. Philadelphia: ASTM; 1971. p. 230–42. [3] Suresh S. Fatigue of materials. Cambridge: Cambridge University; 1991. [4] Kemp R.M.J. Fatigue crack closure—a review. Royal Aerospace Establishment, Technical Report 90046 ICAF Document 1776; 1990. [5] Gu¨ngo¨r S, Fellows LJ. Combined fatigue rig and moire´ interferometer to measure crack closure. In: Allison IM, editor. Experimental mechanics. Rotterdam: Balkema; 1998. p. 1071–6.
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[6] Fellows LJ, Nowell D, Hills DA, Comparison of crack closure measurements using moire´ interferometry and results from a Dugdaletype crack closure model. In: Karihaloo BL, Y-W, editors. Advances in fracture research, Proceedings of the 9th International Conference on Fracture, Sydney, April 1997. p. 2551–58. [7] Fellows LJ, Nowell D. Crack closure measurements using moire interferometry with photoresist gratings. Int J Fatigue 2004;26/10: 1075–82. [8] Nowell D. A boundary element model of plasticity-induced fatigue crack closure. Fatigue Fract Eng Mater Struct 1998;21:857–71.
[9] Fellows LJ, Fatigue crack growth under variable stress ratios and complex load history. DPhil Thesis, University of Oxford, 1999. [10] Fellows LJ, Gungor S. Fabrication of photoresist diffraction gratings on Ti–6Al–4V beam specimens for use in moire´ interferometry. Meas Sci Technol 1998;9:1963–8. [11] Newman JC Jr, A crack-closure model for predicting fatigue crack growth under aircraft spectrum loading. In: Chang JB, Hudson CM, editors. Methods and models for predicting fatigue crack growth under random loading, ASTM STP 748. Philadelphia; ASTM; 1981. p. 53–84.