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Growth of very long “short cracks” initiated at holes
International Journal of Fatigue xxx (2014) xxx–xxx
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Growth of very long ‘‘short cracks’’ initiated at holes P. Lorenzino, A. Navarro ⇑ Departamento de Ingeniería Mecánica y de los Materiales, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain
a r t i c l e
i n f o
Article history: Received 30 September 2013 Received in revised form 26 March 2014 Accepted 26 March 2014 Available online xxxx Keywords: Short crack Microstructural barrier Grain size Notch Fatigue crack initiation
a b s t r a c t The initiation and growth behavior of very long microstructurally short fatigue cracks formed at circular holes is described. Very long here means cracks which are several millimeters or even centimeters long. Microstructurally short refer to the fact that these cracks, in spite of their physical length, are still smaller than the grain size of the material and thus exhibit the characteristic features of such cracks. Growth retardation or even halt at grain boundaries and fluctuating crack growth rates can readily be observed with the naked eye by employing a experimental technique which allows one to increase the grain size of Al1050 Aluminum alloy until the centimeter scale by applying a series of mechanical and heat treatments. Once the thermo-mechanical treatment is completed and the desired grain size obtained, a circular notch is machined on each specimen, and the samples are subjected to fatigue loading. With this method, interactions between cracks and microstructural barriers can be studied with an unprecedented level of ease and detail. An interesting observation is that the location of the crack initiation point along the hole contour varies greatly with the ratio between the hole diameter and the grain size: for large ratios, the initiation point is located close to the point corresponding to the maximum circumferential stress (the horizontal symmetry axis in our case), but for smaller ratios, however, the point of crack initiation moves markedly away from the symmetry axis. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The problem of the characterization of the crack tip stress and strain fields for fatigue problem has a long history. The inauspicious beginnings of the characterization of fatigue crack growth rate of long cracks by the stress intensity factor range as described in the Paris law is a well-known episode in fracture mechanics lore. There, a single parameter embodies the physics of the problem and crack growth rate can be expressed as a unique function of DK irrespective of the geometry of the component or the type of load applied. Dimensional analysis arguments (Navarro and de los Rios [1]) show that fatigue crack growth rate can indeed be expected to depend on DK alone when the size of the plastic zone relative to the crack length does not change. Consider a fatigue crack growing in an infinite medium (see Fig. 1, taken from [1]). As it is well known, the growth of a crack by fatigue depends on the plasticity generated at the tip of the crack. The dimensional analysis should then consider the length a of the crack and a new characteristic length c representing the extent of the plastic zone. Then, the rate of crack growth can be shown to be given by
⇑ Corresponding author. Tel.: +34 954487311; fax: +34 954487295. E-mail address: [email protected] (A. Navarro).
an da ¼ C DK m dN c
ð1Þ
But, of course, for long cracks and for the values of applied stress in which LEFM is considered to hold, s=sy < 0:3, the Dugdale relation
a p ¼ cos c 2
s sy
ð2Þ
gives values of between 0.9 and 1 for the ratio a=c, and therefore, neglecting this variation above, the Paris law is obtained
da ¼ C DK m dN
ð3Þ
When plasticity is more extensive, the term a=c should be taken into account and the parameter DK on its own does not describe physical similarity either. Attention is also drawn to the idea that the Dugdale-type relationship describes the self-similar growth of the crack across an ideal homogeneous body, where there is no distinguishable microstructural feature that may be used to define any intrinsic unit of length. It is only in these conditions that single parameter characterizations of fatigue crack growth would be applicable. Microstructurally short fatigue crack growth is thus a prime example of a situation where a single parameter characterization of the crack stress and strain fields does not seem to be entirely appropriate. It is a very important problem from the practical point
http://dx.doi.org/10.1016/j.ijfatigue.2014.03.023 0142-1123/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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Fig. 1. A crack in an infinite body.
of view, for there are many situations where the life of the component is ‘‘decided’’ when the crack length is of the order of the grain diameter. This is particularly the case in long-cycle fatigue problems when the load is close to the fatigue limit. It is now well established that the fatigue limit in metallic materials is really a threshold condition for the propagation of very small cracks. Fatigue cracks have been reported to initiate at persistent slip bands (PSBs), grain boundaries, pores and non-metallic inclusions, see the two recent reviews by Chan [2] and by Sangid [3], where initiation mechanisms and the role of microstructure thereof are discussed at length. When the length of a advancing crack is comparable to major microstructural features, its growth rate is very sensitive to the distribution of phase and grain boundaries ahead of it. Evidence of grain or phase boundaries retarding or even permanently arresting the growth of short cracks has been reported in many materials including aluminum alloys [4], titanium alloys [5] a nickel base superalloy [6] and carbon steels [7]. At stresses below the fatigue limit, cracks start growing fast but then they stop and become non-propagating [8,9]. At stresses just above the fatigue limit, cracks decelerate and may temporarily halt a number of times, but they do not stop growing altogether. Later on they accelerate and finally reach a regime of apparent continuous propagation [4,10–14]. Microscopic observations have identified the locations of minimum crack growth as microstructural barriers to slip propagation such as grain or phase boundaries. The crack decelerates on approaching the grain boundary until a new slip band is initiated in the neighboring grain, along which the crack will propagate next. This nucleation process of slip bands has to be repeated afresh in each grain [15]. The study of the fatigue limit of notched components is another case of interest. Here the problem is one of short crack growth, with the additional difficulty that the stress field through which the crack is growing has usually a very steep gradient. It has been found that in sharp notches it may be possible for a crack to grow through a few grains and then to become non-propagating [16– 19]. This suggests that the critical event in notch fatigue may not be the initiation of the crack itself, but rather the relative capacity of this crack to overcome successive microstructural barriers when the stresses that are driving it diminish rapidly. Typical nonpropagating cracks found in sharp notched specimens of carbon steel have lengths of the order of a few tens of microns [17]. In the growth of microstructurally short cracks in metallic materials self–similarity is not preserved. The actual physical size of the crack is not as important as the relative size of the crack with
respect to the characteristic microstructural dimension [20–22]. The size of the crack at any point in time must be measured in terms of how much of the grain it has traversed or how many grains it already spans. We have devised [23,24] a simple experimental technique whereby all this can be studied with an unprecedented level of ease and detail. The innovative aspect of this technique is the use of specially developed test coupons with grain sizes of a few millimeters – or even centimeters – and the use of low magnification USB cameras by means of which the crack growth process and the interactions with the microstructure can easily be registered and examined. Digital image correlation techniques have also been employed to enhance the technique. In this paper we have used this new technique to analyze the behavior of very long ‘‘short’’ fatigue cracks formed at circular holes. Very long here means cracks which are several millimeters or even centimeters long. Holes can be made as small as to fit inside a single grain of the metal or as big as to cover a very large number of grains. An intriguing result is the observation that the location of the point of crack initiation along the contour of hole seems to vary with the ratio between the hole radius and the grain size. 2. Experimental procedure 2.1. Thermo-mechanical treatments Our studies show that it is possible to produce a substantial increase in the grain size of commercially pure aluminum sheets by means of a combination of two thermal treatments and an intermediate moderate cold working. The process is very simple and easy to control. It is quite feasible, through the correlations derived between the final grain size and the control parameters, to set the values of these parameters in order to obtain any desired grain size fixed in advance (within a certain range). The technique has been shown to produce highly consistent and repeatable results [23,24]. Aluminum 1050 99.5-H24 is used in sheets 4.0 mm thick. Chemical composition (% wt): 99.56; Cu: 0.08; Fe: 0.2; Si: 0.1. The aluminum sheets are cut into pieces of 45 300 mm, in parallel to the lamination direction. A tubular furnace is used for the thermal treatments (Carbolite model 215GHA12). The aim of the first thermal treatment is to obtain a deformation-free equiaxial structure; different treatments were tried, changing the recrystallization temperature, the heating rate and the period at constant temperature. After observing the resulting microstructures, a heating rate of 2.6 °C/min was chosen, from room temperature to 550 °C. Then the temperature was kept constant during 5 h, followed by air cooling. It has been found that if this temperature is maintained for more than 5 h, the surface grains become slightly larger than the inner grains, and this induces a non-uniform deformation across the specimen thickness during the following stage (mechanical treatment). Next, cold working is performed in a MTS 810 servo-hydraulic machine. The strain level applied at this stage will determine the size of the grains after the second recrystallization. The following deformation percentages were chosen: 0, 8%, 11%, 14% and 18%. Zero corresponds to the material undergoing only the first recrystallization. The third stage is again a temperature ramp from room temperature to T = 550 °C at a rate of 2.6 °C/ min; then, the temperature is to be kept constant during 15 h and then increased again up to 575 °C and maintained there during 1 h. At this stage new crystals growing at the expense of the old ones are formed. Finally, the specimens are air-cooled. As it has been said above, the final grain size obtained depends critically on the level of plastic strain applied during the cold working. Fig. 2 shows an example of the possible microstructures obtained and the corresponding level of total deformation applied,
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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Fig. 2. Different grain sizes obtained depending on the degree of applied strain.
the upper image corresponds to a specimen which has undergone 8% of total strain during the mechanical treatment and the bottom image corresponds to a specimen subjected to 14%. 2.2. Grain size The grain size measurement is made according to ASTM E112, using the software Simagis Live and following the procedure described in [24]. The obtained grain sizes are summarized in Table 1. 2.3. Fatigue testing Once the thermo-mechanical treatment is completed and the desired microstructure obtained, a circular notch is machined on each specimen, and the samples are subjected to axial fatigue loading. Several combinations of notch and microstructural sizes have been tested. A RUMUL resonant testing machine (Testronic 100 kN) was used for fatigue testing. All tests were load-controlled, with R = 0.1 (pull-pull) and maximum stress between 45 and 95 MPa. Resonance frequencies for the loading conditions and geometry of specimens are between 75 and 90 Hz. Crack propagation results in a decrease on the resonance frequency. After several trials, it was established that a frequency decrease of 0.7 Hz was a good indicator of impending failure. A larger decrease results in complete failure, where the crack faces rapidly separate and the remaining section is plastically deformed, making post mortem analysis very difficult. 2.4. Crack monitoring An experimental procedure for measuring crack length and growth was developed in order to observe crack propagation and interactions with the microstructure. Two low magnification optical microscopes showing real time video images were connected to a computer via USB. They are easily mounted onto a platform and, as shown in Fig. 3, they scan both sides of the
Table 1 Grain size distributions for the different thermomechanical treatments. Total strain (%)
18
14
11
8
–
Analyzed area (mm2) Number of grains Av. grain area (mm2) Av. grain size (mm) Std. Dev. ([mm) Mode (mm) Median (mm)
338 1758 0.19 0.394 0.20 0.25 0.32
5317 2149 2.47 1.41 0.69 1.38 1.24
26,272 1598 16.44 3.46 2.14 2 2.99
416,116 2966 140.30 9.74 6.50 6.25 7.47
0.85 173 0.0049 0.0659 0.0437 0.0375 0.0475
3
Fig. 3. Set-up for monitoring and recording crack growth.
specimen simultaneously. The magnification is adjusted by regulating the distance between the magnifier and the specimen. Since the materials used have low mechanical properties and strains are always kept in elastic region, the movements produced during specimen load and unload are very small; as a result, the recorded video is quasi-static and there is no need to stop the test, measure the crack length and resume. This facilitates the crack growth analysis, and largely reduces the amount of time needed for the test in contrast with other crack monitoring experimental techniques, such as replicas with acetate. This technique makes it possible to keep a complete record of every test and of any possible potential crack on both sides of the specimen. This is important because studies can be made either of the interaction between the crack and the microstructure and of the interaction between different cracks and/or different crack branches. A frame of the video is taken every 30 s in order to measure crack length. Next, the crack length is measured using the free image analysis software ImageJ. We measure the whole length of the crack trace in the surface, from notch edge to crack tip, accounting for all the zig-zag segments that are normally formed. The combination of this information with the test frequency is elaborated in plots of crack lengths vs. number of cycles from which graphs of growth rate vs. number of cycles or crack length can be derived. The representation in terms of number of cycles allows one to compare the growth rates of the different cracks at the same point in time. 3. Results and discussion Fig. 4 shows photographs extracted from the fatigue tests videos. Cracks grow at both sides of a circular notch with a diameter of 2 mm. The grain size is 9.74 mm. The test is load controlled (R = 0.1) with a maximum stress of 45 MPa; under these loading conditions and geometry of specimen, the resonance frequency is 82.13 Hz. The fatigue life is 4.09 106 cycles. The left image corresponds to the front surface of the specimen and the right image corresponds to the back surface. But in the latter, the image has been reversed so that the traces at the left of the notch in both images correspond to the same crack growing at one of the sides of the notch, and similarly at the right. The complicated, non-planar, nature of the cracks surfaces is apparent. In the video, it is possible to observe crack tip arrest at grain boundaries, as well as changes in growth direction. Fig. 5 shows a sequence of the crack propagation process. It corresponds to a test of a 2 mm notch machined in a 3.46 mm grain size microstructure. The maximum stress is 50 MPa (R = 0.1) and the total life is 2.25 106 cycles. Different stages of the crack growth process can be observed. (a) N = 1.27106. The first crack appears on the left side of the notch. (b) N = 1.41 106. A second crack appears on the right side of the notch. Both cracks are seen to grow in (c) N = 1.58 106 and (d) N = 1.78 106. (e) N = 1.88 106. The left
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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Fig. 4. Cracks growing from a circular notch with a diameter of 2 mm within a microstructure of 9.74 mm.
Fig. 5. Different stages of propagation of cracks growing from a 2 mm circular notch within a microstructure of 3.46 mm.
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crack reaches a grain boundary. (f) N = 1.97 106, the crack on the left, after been arrested for almost 9 104 cycles, is seen growing into the neighboring grain with a marked change of direction. (g) N = 2.04 106. The left crack has branched at the grain boundary. The two branches progress along two different paths, which might be associated with different slip systems, although this cannot not be stated with certainty until an analysis of the crystallographic orientations of the crystal is carried out. h) N = 2.21 106. In the competition between the two branches of the crack at the left side, the upper one seems to shield the lower one which remains arrested while the other one continues it propagation. The right crack has also propagated with a slight reorientation. Fig. 6 shows crack length in relation to number of cycles for three cracks growing from a 2 mm notch in a 3.46 mm grain size microstructure. The three of them show acceleration and deceleration patterns along their trajectory. Fig. 7 shows growth rate in relation to crack length for three cracks growing in different microstructures with grain sizes of 1.41, 3.46 and 9.74 mm. It is apparent that the spacing between consecutive halts of the crack increases in relation to the rise in the grain size of the material. There is a good correlation between average distance between halts and grain size. A very interesting observation is that in the majority of the cases studied the cracks did not initiate at the expected point of maximum stress concentration. Let us recall that the specimens are subjected to axial loads alone. It is surprising, for classical
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methods of notched fatigue limit analysis for this type of loading would indicate that the most likely initiation points should always be the points of the notch edge lying in the diameter perpendicular to the loading axis, the horizontal diameter in our case, since the loading axis is vertical. Of course, anisotropy considerations associated with the large size of the grains would alter that. But it is still nevertheless quite interesting to analyze the pattern that emerge from the observation of the angular location, along the notch edge, of the points where initiation takes place. Thus, we have recorded carefully the point of initiation for many cracks. We have measured the angle subtended at the hole center by the arc along the notch circumference delimited by the point where initiation takes place and the horizontal diameter. Fig. 8 illustrates how these measurements were performed. Several notch sizes and grain diameters were analyzed, totaling more than 500 cracks. Table 2 show the aggregate results. A graphical summary is provided in Fig. 9, where average values of the initiation points location angles have been represented with respect to the ratio between notch diameter and grain size. A general decrease of the angle with the notch diameter to grain size ratio is clearly displayed. Here we can see too that for each grain size the bigger the notch the smaller the location angle. Fig. 10 shows frequency distributions of the angle for two different grain sizes, the biggest and the smallest ones, namely, 9.74 and 0.066 mm. Notice that the values of notch diameter to grain size ratios in both graphs are very different: in the lower graph they are 0.1, 0.2 and 0.4 whereas in the upper one they are 30, 60 and 90. It is interesting to observe that for the larger grain size,
Fig. 6. Crack length vs. number of cycles for three cracks growing from a 2 mm notch in a 3.46 mm grain size microstructure.
Fig. 7. Crack growth rate vs. crack length.
Fig. 8. Measurement of the angle subtended at the hole center by the crack initiation point. Top: notch size 4 mm; grain size 0.066 mm. Bottom: notch size 4 mm; grain size 3.46 mm.
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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Table 2 Location of crack initiation point. Dg , grain size. DNotch , notch diameter. nc , number of cracks measured. Dg (mm)
DNotch (mm)
DNotch =Dg
nc
Average (degrees)
StDev. (degrees)
Min (degrees)
Median (degrees)
Max (degrees)
0.066 0.066 0.066 0.390 0.390 0.390 1.410 1.410 1.410 3.460 3.460 3.460 9.740 9.740 9.740
6 4 2 4 2 1 4 2 1 4 2 1 4 2 1
0.066 0.066 0.066 0.390 0.390 0.390 1.410 1.410 1.410 3.460 3.460 3.460 9.740 9.740 9.740
32 40 36 36 40 26 36 38 28 40 40 24 39 42 22
7.33 8.82 8.12 8.76 10.08 15.01 10.59 13.53 17.33 12.42 15.62 25.14 12.51 17.29 26.32
4.78 7.06 6.56 5.43 7.54 9.28 6.69 7.98 13.46 9.02 12.59 14.53 7.70 10.28 14.01
1.18 0 0 0.74 0 1.74 1.6 0 1.91 0 0 4.76 1.35 0 0
6.14 6.06 7.69 8.66 7.95 13.06 8.60 12.51 16.77 11.48 10.12 24.82 12.07 16.80 27.55
18.05 25.64 22.44 21.30 28.41 36.47 25.15 29.98 46.79 31.46 71.57 58.47 29.77 37.52 47.82
Fig. 9. Location of crack initiation point as a function of notch to grain size ratio.
with the smaller notch size to grain size ratios, there is a wider region where the likelihood of initiation is almost equally high. Fig. 11 shows the results for different holes radii and for several specimens with the same microstructural size of 9.74 mm. On the left, a diagram shows a polar coordinate system corresponding to a quarter of the notch and it presents the crack initiation points for three different radii (0.5, 1 and 2 mm). The graph on the right shows the corresponding frequency distributions. About 36 measurements were made for each notch radius in this figure. Again, it can be appreciated that when the size of the notch diminishes (in relation to the microstructure) the width of the frequency distribution becomes larger and the most frequent location of the initiation point value moves away from the horizontal axis. Compare the values of 12.51, 17.29 and 26.32 degrees obtained for notches of radii 2, 1 and 0.5 mm respectively. Conversely, bigger notch radii in relation to grain size result in narrower zones of likely initiation and bring them closer to the horizontal axis. Of course, for materials with ‘‘normal’’ microstructural sizes of less than a few tens of microns, and ‘‘normal’’ notch sizes of a few millimetres, where the notch is several times larger than the microstructure, this indicates that cracks would normally initiate in the expected location in the horizontal diameter, which is just as well. This observation clearly reveals that the application of the usual techniques of notch analysis (viz critical distance [25,26] or notch sensitivity index concepts [27,28]) may not be directly applicable in situations where the notch size is of the same order or even smaller that the microstructure and this points out that this region of notch sizes may be a good testing ground to discriminate among different notch fatigue calculation methods.
Fig. 10. Distributions of crack initiation location as a function of notch radius for (a) 0.066 mm grain size microstructure and (b) 9.84 mm grain size microstructure.
We must honestly say that at present we only have a rather loose reasoning to account for these results. As mentioned above, none of the currently accepted methods for assessing notched fatigue limits is able to provide an explanation for these findings. Take for example the theory of critical distances: one must locate the ‘‘hot spot’’ and then and look for failure over a certain distance below the surface in the direction of maximum principal stress. Therefore, both the location of the initiation point and the angle of the crack plane are fixed in advance. And there is no room for variations depending on the relative size of the notch and the microstructure.
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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Fig. 11. Location of crack initiation point in relation to notch size for a grain size of 9.74 mm.
The only thing we can do at present is to speculate that for very small holes compared to the grain size the influence of the notch disappears long before the crack reaches the grain boundary and so incipient cracks growing at slip bands initiated at any point on the contour of the hole, as far as the circumferential stress is not compressive, have an almost equal chance of becoming dominant. The hole only provides the marginally higher stresses need to trigger the nucleation of the slip bands, but does not contribute to the overcoming of the barrier to growth represented by the grain boundary. For holes which are bigger compared with the grain size, the stress concentration acts almost directly upon the grain boundary, and since the stress concentration decreases markedly when one moves away from the point of maximum circumferential stress (in the horizontal axis passing through the center of the hole, assuming tensile loading in the vertical direction), the cracks initiated there have a much greater incentive to become dominant. There is an obvious need to make these arguments quantitative. Because of this, we are working now in the formulation of a microstructural fracture mechanics model, based on dislocation theory and representing an extension of the Navarro–Rios model [15,29–33], of a crack growing from a circular hole in a plate subjected to in-plane and in-phase biaxial loading. But this is still in a preliminary stage. We hope to have some results worth reporting in the very near future. Finally, we would just like to present some preliminary results obtained by using Digital Image Correlation (DIC) techniques in our experiments with the big-grained material. Digital image correlation analysis is a relatively new and very useful tool for understanding material behavior, specially at the grain scale. Recent investigation shows its potencial for evaluating the accumulated plastic strain fields associated with fatigue crack growth. When the microstructural size is much smaller than the notch size the strain fields appear as more homogeneous, but when the crack size is about the grain size the strain fields vary a lot from grain to grain, and also within individual grains [34]. This phenomenon can be easily illustrated when working with large microstructures. In this case a high resolution imaging of the samples is not needed for obtaining grain-scale strain fields. In this section we show the strain inhomogeneities that arise when subjecting our samples to a remote strain (monotonic load) and the plastic strain near a growing fatigue crack for both a small size and for a large-size microstructure, in order to compare the effect of the crack to grain size ratio effect upon the strain fields. In order to perform the static loading analysis, once the microstructure is obtained, a speckle pattern is generated on the surface, and then the specimen is tested in tension. By taking
photographs of the surface on a regular basis, image correlation techniques can be applied. A high definition camera plus a longdistance microscope Questar QM 1 are used to capture the images. Software VIC-2DTM is used for acquisition and image processing. Tensile tests were conducted on an MTS 810 servohydraulic machine, displacement controlled at a rate of 0.2 mm/s. In some tests a contact extensometer was used to corroborate the correct measurement of the deformations obtained by digital image correlation (DIC). Pictures of the sample surface were obtained every 5 s. These sets of photos are used for obtaining the deformation fields. The analysis with VIC-2D™ software was perform with a subset of 21 and a step of 1. The resolution of the captured images is 20 pixel/mm. Fig. 12 shows some statistical parameters (maximum, minimum, average, and standard deviation values) of the strain measurements performed on specimens of different grain sizes. It can be observed that, for the same amount of applied total deformation (12%), increasing the grain size produces an increase in the standard deviation and in the maximum strain reached. This is due to the effect of measuring with more detail what happens inside each grains, as the speckle pattern begins to be smaller than the grain size, combined with effect of the reduction of the number of neighboring grains, which makes the constraint forces preventing the grains to rotate or deform freely smaller, and thus increasing the amount of the deformation of the grains which are better oriented with respect to the load axis [35].
Fig. 12. Applied strain, standard deviation and maximum and minimum strain values obtained as a function of grain size of the specimen.
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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Fig. 13. Specimen subjected to a total axial strain of 12% and results for DIC analysis,
Fig. 14. Evolution of local strain eyy along lines A and B in Fig. 13 for different values of total strain applied to the specimen (top, line A, bottom, line B).
eyy ; exx ; exy .
Fig. 15. Final strain state along lines A and B in Fig. 13 at the end of the test (top, line A, bottom, line B).
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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The strain inhomogeneities can thus be easily analyzed in the case of the large microstructures. Fig. 13 shows a specimen 2 mm thick, whose measured grain size on the surface is 12.6 mm. The grains are thus larger than the specimen thickness and it behaves as a sort of bidimensional array of single crystals. The sample has been subjected to a total strain of 12%. The picture on the left shows the specimen after straining. It is possible to appreciate with some detail how the grain boundaries have become marked. The inhomogeneity in the deformation can be observed too. The three graphs on the left show the deformation fields obtained by digital image correlation, eyy ; exx , exy . The load acts in the y-direction. It is interesting to observe how for a total strain level of 12% applied in the axial direction, some grains exceed 25% of strain while others reached deformation levels below 7%. Furthermore, Fig. 14 shows the evolution of eyy along lines A and B in the previous figure for different levels of total applied stress. It is possible to observe how the inhomogeneities in the strain behavior increase with the increasing applied strain. Fig. 15 shows the final strain state along lines A and B at the end of the test; it also includes the position of the grain boundaries. It is easy to see how the highest deformation gradients occur at the grain boundaries. These results help us to understand how important strain inhomogeneities can be when working at the microstructural level. The effect of the loading history can be very important at a later stage in the life of the specimen. If the specimen presented here were subjected to cyclic loading, even for an applied strain uniform over the cross section, the accumulated local strains will be very far from those expected without taking into account these microstructural effects. Obviously these effects will be more important when working with specimens having fewer grains across the section. The next step in the analysis is to see how the strain fields evolve when the cracks grow by fatigue. Fig. 16 shows an example. The grain size is 0.06 mm, while the hole diameter is 2 mm. The specimen is subjected to a 50 MPa tension–tension (R = 0.1) fatigue
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Fig. 17. A specimen as viewed by DIC from one side and normal video from the other side. The grains, of average size 8.22 mm, are big enough to pass across the thickness of the specimen (2 mm). The diameter of the hole is 2 mm. The grain boundaries have been manually drawn in both pictures.
Fig. 16. Evolution of the eyy strain field during crack propagation. Notch diameter 2 mm, grain size 0.07 mm. Number of cycles: (a) N = 100,000, (b) N = 130,000, (c) N = 160,000, (d) N = 180,000, (e) N = 210,000, and (f) N = 230,000.
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cycle. The total fatigue life at this level was N = 7.9105 cycles. Figures a) to f) show different stages of crack propagation. It is possible to observe the evolution of the strain field. Despite a little deviation from the horizontal axis, the deformation zones generated by the growing crack are fairly symmetrical for this case where the notch is relatively big as compared to the grain size. On the other hand, Fig. 17 corresponds to the case of a specimen with a grain size of 8.22 mm. The specimen thickness is only 2 mm, so there is only one grain across the thickness and we have again a kind of bidimensional arrangement of single crystals. This arrangement allows one to monitor crack propagation by means of DIC analysis on one side of the specimen and by conventional image analysis on the other side of the specimen, and the information relative to the same grains can be compared. The specimen was subjected to 45 MPa tension–tension (R = 0.1) fatigue loading. The total fatigue life was N = 2.38 106 cycles. The strain analysis show the longitudinal strain on the sample at the moment of the initiation of plastic collapse. In this case, the effect of the microstructure upon deformation is dramatically captured: when we observe the specimen at a scale comparable to the microstructural size, the apparent symmetry in the crack strain field observed before disappears completely. This is also an striking example of cracks that start growing in Stage II and then switch to Stage I, reversing the usual course of events. Furthermore, it is possible to observe how the left crack is arrested at a grain boundary and a new plastic zone is generated not in the contiguous grain but in another neighboring grain that it is probably more favorably oriented. In this case microstructural inhomogeneities make it possible to find grains accumulating high levels of plastic strain and neighboring grains remaining virtually unaffected. These type of results may help us to understand the mechanism of crack propagation when working with micro-components. The ongoing tendence to miniaturization of mechanical components leads to the fact that the size of the microstructure is no longer negligible with respect to the component size itself [36]. The intrinsic role of the microstructure on the size effect has been studied by different authors [35,37–39]. In these papers, monotonic mechanical properties are studied mainly. In this paper we have tried to present some aspects of the role of the microstructure on tensile behavior and fatigue properties of notched components where the size of the microstructure is not negligible as compared with the specimen size. In specimens whose dimensions are in the same scale of the grain size, microstructural inhomogeneities will play a prominent role and thus special care has to be taken when analyzing mechanical properties with models that do not take into account this phenomenon. 4. Conclusions The initiation and growth behavior of very long microstructurally short fatigue cracks formed at circular holes has been described. Very long here means cracks which are several millimeters or even centimeters long. Microstructurally short refer to the fact that these cracks, in spite of their physical length, are still smaller than the grain size of the material and thus exhibit the characteristic features, such as growth retardation or even halt at grain boundaries and fluctuating crack growth rates, of these type of cracks as observed when ‘‘normal’’, much smaller microstructures are considered [40,41]. This has been possible by employing an experimental technique which allows one to increase the grain size of Al1050 Aluminum alloy until the centimeter scale by applying a series of mechanical and heat treatments. With this method, interactions between cracks and microstructural barriers can be studied with an unprecedented level of ease and detail.
An interesting observation is that the width of the arc around the notch root where the initiation of the cracks take place (and the location of this arc) varies greatly with the ratio between the notch diameter and the grain size: for large ratios, the width is small and the arc centers in the symmetry axis of the notch, which corresponds to the point of maximum stress concentration. For smaller ratios, however, the width of the arc increases markedly and its position moves away from the symmetry axis. This study highlights the fact that the actual physical size of the crack is not as important as the relative size of the crack with respect to the characteristic microstructural dimension. The size of the crack at any point in time must be measured in terms of how much of the grain it has traversed or how many grains it already spans.
Acknowledgements The authors would like to thank the Spanish Government for its financial support through grants DPI2008-01100 and DPI201127019 and the Junta de Andalucía through grants TEP1752 and P07-TEP-03045. References [1] Navarro A, de los Rios ER. On dimensional analysis of fatigue crack growth rate and geometrical similitude of cracks. Fatigue Fract Eng Mater Struct 1987;9(5):373–8. [2] Chan KS. Roles of microstructure in fatigue crack initiation. Int J Fatigue 2010;32(9):1428–47. [3] Sangid MD. The physics of fatigue crack initiation. Int J Fatigue 2013;57(0):58–72. [4] Blom AF, Hudlund A, Zhao W, Fathulla A, Weiss B, Stickler R. Short fatigue crack behaviour in al 2024 and al 7475. In: Miller KJ, de los Rios ER, editors. The behaviour of short fatigue cracks. London: Mechanical Engineering Publications; 1986. p. 37–66. [5] Yoder G, Cooley L, Crooker T. Quantitative analysis of microstructural effects on fatigue crack growth in widmanstätten Ti–6Al–4V and Ti–8Al–1Mo–1V. Eng Fract Mech 1979;11(4):805–16. [6] Wang SZ, Miller KJ, Brown MW, De Los Rios ER. A statistical analysis of short fatigue crack growth. Fatigue Fract Eng Mater Struct 1991;14(2–3):351–68. [7] Rios ERdl, Tang Z, Miller KJ. Short crack fatigue behaviour in a medium carbon steel. Fatigue Fract Eng Mater Struct 1984;7(2):97–108. [8] de los Rios ER, Navarro A. Considerations of grain orientation and work hardening on short-fatigue-crack modelling. Philos Mag A 1990;61(3):435–49. [9] Miller K. Initiation and growth rates of short fatigue cracks. In: Fundamentals of deformation and fracture. Eshelby memorial symposium. Sheffield: Cambridge University Press; 1985. p. 477–500. [10] Taylor D. Euromech colloquium on short fatigue cracks. Fatigue Fract Eng Mater Struct 1982;5(4):305–9. [11] Yoder G, Cooley L, Crooker T. On microstructural control of near-threshold fatigue crack growth in 7000-series aluminum alloys. Scripta Metall 1982;16(9):1021–5. [12] Zurek A, James M, Morris W. The effect of grain size on fatigue growth of short cracks. Metall Trans A 1983;14(8):1697–705. [13] Bolingbroke R, King J. A comparison of long and short fatigue crack growth in a high strength aluminium alloy. In: Miller KJ, de los Rios ER, editors. The behaviour of short fatigue cracks. London: Mechanical Engineering Publications; 1986. p. 101–14. [14] Tokaji K, Ogawa T, Harada Y, Ando Z. Limitations of linear elastic fracture mechanics in respect of small fatigue cracks and microstructure. Fatigue Fract Eng Mater Struct 1986;9(1):1–14. [15] Navarro A, de los Rios ER. Fatigue crack growth modelling by successive blocking of dislocations. Proc R Soc London Ser A: Math Phys Sci 1992;437(1900):375–90. [16] Smith RA, Miller KJ. Prediction of fatigue regimes in notched components. Int J Mech Sci 1978;20:201–6. [17] Frost NE. Non-propagating cracks in Vee-notched specimens subjected to fatigue loading. Aeronaut Quart 1957;8:1–20. [18] Frost NE. A relation between the critical alternating propagation stress and crack length for mild steel. Proc Instn Mech Engrs 1959;173:811–27. [19] Frost NE, Pook LP, Denton K. A fracture mechanics analysis of fatigue crack growth for various materials. Eng Fract Mech 1971;3:109–26. [20] Suresh S. Fatigue of materials. Cambridge University Press; 1998. [21] Ravichandran K, Murakami Y, Ritchie R. Small fatigue cracks: mechanics, mechanisms and applications: mechanics, mechanisms and applications. Elsevier Science; 1999.
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P. Lorenzino, A. Navarro / International Journal of Fatigue xxx (2014) xxx–xxx [22] Miller K, de los Rios E. The behaviour of short fatigue cracks. Mechanical Engineering Publications; 1986. [23] Lorenzino P. Fatiga en componentes con concentradores de tensión bajo carga en modo I. Ph.D. thesis. Seville University; 2012. [24] Lorenzino P, Krupp U, Navarro A. Naked eye observations of microstructurally short fatigue cracks. Int J Fatigue 2013;56:8–16. [25] Taylor D. Geometrical effects in fatigue: a unifying theoretical model. Int J Fatigue 1999;21:413–20. [26] Taylor D. The theory of critical distances: a new perspective in fracture mechanics. Elsevier Ltd.; 2007. [27] Neuber H. Kerbspannungslehre. Springer Verlag; 1937. Translated into English, Theory of Notches, In: Edwards JW. editor. Ann Arbor, MI; 1946. [28] Peterson RE. Notch sensitivity. In: Sines G, Waisman JL, editors. Metal fatigue. McGraw-Hill; 1959. p. 293–307. [29] Navarro A, de los Rios ER. An alternative model of the blocking of dislocations at grain boundaries. Philos Mag A 1988;57:37–42. [30] Navarro A, de los Rios ER. Short and long fatigue crack growth: a unified model. Philos Mag A 1988;57:37–42. [31] Chaves V, Navarro A. Application of a microstructural model for predicting notch fatigue limits under mode I loading. Int J Fatigue 2009;31:943–51. [32] Chaves V, Navarro A. Fatigue limit of notches of arbitrary profile. Int J Fatigue 2013;48:68–79. [33] Navarro A, Vallellano C, Chaves V, Madrigal C. A microstructural model for biaxial fatigue conditions. Int J Fatigue 2011;33(8):1048–54.
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[34] Carroll JD, Abuzaid W, Lambros J, Sehitoglu H. High resolution digital image correlation measurements of strain accumulation in fatigue crack growth. Int J Fatigue 2013;57:140–50. [35] Miyazaki S, Shibata K, Fujita H. Effect of specimen thickness on mechanical properties of polycrystalline aggregates with various grain sizes. Acta Metall 1978;27:855–62. [36] Greer JR, Oliver WC, Nix WD. Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Mater 2005;53:1821–30. [37] Klein M, Hadrboletz A, Weiss B, Khatibi G. The size effect on the stress–strain, fatigue and fracture properties of thin metallic foils. Mater Sci Eng A 2001;321:924–8. [38] Keller C, Hug E. Hall–Petch behaviour of Ni polycrystals with a few grains per thickness. Mater Lett 2008;62:1718–20. [39] Janssen PJM, Keijser THD, Geers MGD. An experimental assessment of grain size effects in the uniaxial straining of thin Al sheet with a few grains across the thickness. Mater Sci Eng A 2006;419:238–48. [40] Herbig M, King A, Reischig P, Proudhon H, Lauridsen EM, Marrow J, et al. 3-D growth of a short fatigue crack within a polycrystalline microstructure studied using combined diffraction and phase-contrast X-ray tomography. Acta Mater 2011;59:590–601. [41] Krupp U, Düber O, Christ HJ, Künkler B, Koster P, Fritzen CP. Propagation mechanisms of microestructurally short cracks- factors governing the transition from short- to long-crack behaviour. Mater Sci Eng A 2007;462:174–7.
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Growth of very long ‘‘short cracks’’ initiated at holes P. Lorenzino, A. Navarro ⇑ Departamento de Ingeniería Mecánica y de los Materiales, Escuela Superior de Ingenieros, Universidad de Sevilla, Camino de los Descubrimientos s/n, 41092 Sevilla, Spain
a r t i c l e
i n f o
Article history: Received 30 September 2013 Received in revised form 26 March 2014 Accepted 26 March 2014 Available online xxxx Keywords: Short crack Microstructural barrier Grain size Notch Fatigue crack initiation
a b s t r a c t The initiation and growth behavior of very long microstructurally short fatigue cracks formed at circular holes is described. Very long here means cracks which are several millimeters or even centimeters long. Microstructurally short refer to the fact that these cracks, in spite of their physical length, are still smaller than the grain size of the material and thus exhibit the characteristic features of such cracks. Growth retardation or even halt at grain boundaries and fluctuating crack growth rates can readily be observed with the naked eye by employing a experimental technique which allows one to increase the grain size of Al1050 Aluminum alloy until the centimeter scale by applying a series of mechanical and heat treatments. Once the thermo-mechanical treatment is completed and the desired grain size obtained, a circular notch is machined on each specimen, and the samples are subjected to fatigue loading. With this method, interactions between cracks and microstructural barriers can be studied with an unprecedented level of ease and detail. An interesting observation is that the location of the crack initiation point along the hole contour varies greatly with the ratio between the hole diameter and the grain size: for large ratios, the initiation point is located close to the point corresponding to the maximum circumferential stress (the horizontal symmetry axis in our case), but for smaller ratios, however, the point of crack initiation moves markedly away from the symmetry axis. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The problem of the characterization of the crack tip stress and strain fields for fatigue problem has a long history. The inauspicious beginnings of the characterization of fatigue crack growth rate of long cracks by the stress intensity factor range as described in the Paris law is a well-known episode in fracture mechanics lore. There, a single parameter embodies the physics of the problem and crack growth rate can be expressed as a unique function of DK irrespective of the geometry of the component or the type of load applied. Dimensional analysis arguments (Navarro and de los Rios [1]) show that fatigue crack growth rate can indeed be expected to depend on DK alone when the size of the plastic zone relative to the crack length does not change. Consider a fatigue crack growing in an infinite medium (see Fig. 1, taken from [1]). As it is well known, the growth of a crack by fatigue depends on the plasticity generated at the tip of the crack. The dimensional analysis should then consider the length a of the crack and a new characteristic length c representing the extent of the plastic zone. Then, the rate of crack growth can be shown to be given by
⇑ Corresponding author. Tel.: +34 954487311; fax: +34 954487295. E-mail address: [email protected] (A. Navarro).
an da ¼ C DK m dN c
ð1Þ
But, of course, for long cracks and for the values of applied stress in which LEFM is considered to hold, s=sy < 0:3, the Dugdale relation
a p ¼ cos c 2
s sy
ð2Þ
gives values of between 0.9 and 1 for the ratio a=c, and therefore, neglecting this variation above, the Paris law is obtained
da ¼ C DK m dN
ð3Þ
When plasticity is more extensive, the term a=c should be taken into account and the parameter DK on its own does not describe physical similarity either. Attention is also drawn to the idea that the Dugdale-type relationship describes the self-similar growth of the crack across an ideal homogeneous body, where there is no distinguishable microstructural feature that may be used to define any intrinsic unit of length. It is only in these conditions that single parameter characterizations of fatigue crack growth would be applicable. Microstructurally short fatigue crack growth is thus a prime example of a situation where a single parameter characterization of the crack stress and strain fields does not seem to be entirely appropriate. It is a very important problem from the practical point
http://dx.doi.org/10.1016/j.ijfatigue.2014.03.023 0142-1123/Ó 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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Fig. 1. A crack in an infinite body.
of view, for there are many situations where the life of the component is ‘‘decided’’ when the crack length is of the order of the grain diameter. This is particularly the case in long-cycle fatigue problems when the load is close to the fatigue limit. It is now well established that the fatigue limit in metallic materials is really a threshold condition for the propagation of very small cracks. Fatigue cracks have been reported to initiate at persistent slip bands (PSBs), grain boundaries, pores and non-metallic inclusions, see the two recent reviews by Chan [2] and by Sangid [3], where initiation mechanisms and the role of microstructure thereof are discussed at length. When the length of a advancing crack is comparable to major microstructural features, its growth rate is very sensitive to the distribution of phase and grain boundaries ahead of it. Evidence of grain or phase boundaries retarding or even permanently arresting the growth of short cracks has been reported in many materials including aluminum alloys [4], titanium alloys [5] a nickel base superalloy [6] and carbon steels [7]. At stresses below the fatigue limit, cracks start growing fast but then they stop and become non-propagating [8,9]. At stresses just above the fatigue limit, cracks decelerate and may temporarily halt a number of times, but they do not stop growing altogether. Later on they accelerate and finally reach a regime of apparent continuous propagation [4,10–14]. Microscopic observations have identified the locations of minimum crack growth as microstructural barriers to slip propagation such as grain or phase boundaries. The crack decelerates on approaching the grain boundary until a new slip band is initiated in the neighboring grain, along which the crack will propagate next. This nucleation process of slip bands has to be repeated afresh in each grain [15]. The study of the fatigue limit of notched components is another case of interest. Here the problem is one of short crack growth, with the additional difficulty that the stress field through which the crack is growing has usually a very steep gradient. It has been found that in sharp notches it may be possible for a crack to grow through a few grains and then to become non-propagating [16– 19]. This suggests that the critical event in notch fatigue may not be the initiation of the crack itself, but rather the relative capacity of this crack to overcome successive microstructural barriers when the stresses that are driving it diminish rapidly. Typical nonpropagating cracks found in sharp notched specimens of carbon steel have lengths of the order of a few tens of microns [17]. In the growth of microstructurally short cracks in metallic materials self–similarity is not preserved. The actual physical size of the crack is not as important as the relative size of the crack with
respect to the characteristic microstructural dimension [20–22]. The size of the crack at any point in time must be measured in terms of how much of the grain it has traversed or how many grains it already spans. We have devised [23,24] a simple experimental technique whereby all this can be studied with an unprecedented level of ease and detail. The innovative aspect of this technique is the use of specially developed test coupons with grain sizes of a few millimeters – or even centimeters – and the use of low magnification USB cameras by means of which the crack growth process and the interactions with the microstructure can easily be registered and examined. Digital image correlation techniques have also been employed to enhance the technique. In this paper we have used this new technique to analyze the behavior of very long ‘‘short’’ fatigue cracks formed at circular holes. Very long here means cracks which are several millimeters or even centimeters long. Holes can be made as small as to fit inside a single grain of the metal or as big as to cover a very large number of grains. An intriguing result is the observation that the location of the point of crack initiation along the contour of hole seems to vary with the ratio between the hole radius and the grain size. 2. Experimental procedure 2.1. Thermo-mechanical treatments Our studies show that it is possible to produce a substantial increase in the grain size of commercially pure aluminum sheets by means of a combination of two thermal treatments and an intermediate moderate cold working. The process is very simple and easy to control. It is quite feasible, through the correlations derived between the final grain size and the control parameters, to set the values of these parameters in order to obtain any desired grain size fixed in advance (within a certain range). The technique has been shown to produce highly consistent and repeatable results [23,24]. Aluminum 1050 99.5-H24 is used in sheets 4.0 mm thick. Chemical composition (% wt): 99.56; Cu: 0.08; Fe: 0.2; Si: 0.1. The aluminum sheets are cut into pieces of 45 300 mm, in parallel to the lamination direction. A tubular furnace is used for the thermal treatments (Carbolite model 215GHA12). The aim of the first thermal treatment is to obtain a deformation-free equiaxial structure; different treatments were tried, changing the recrystallization temperature, the heating rate and the period at constant temperature. After observing the resulting microstructures, a heating rate of 2.6 °C/min was chosen, from room temperature to 550 °C. Then the temperature was kept constant during 5 h, followed by air cooling. It has been found that if this temperature is maintained for more than 5 h, the surface grains become slightly larger than the inner grains, and this induces a non-uniform deformation across the specimen thickness during the following stage (mechanical treatment). Next, cold working is performed in a MTS 810 servo-hydraulic machine. The strain level applied at this stage will determine the size of the grains after the second recrystallization. The following deformation percentages were chosen: 0, 8%, 11%, 14% and 18%. Zero corresponds to the material undergoing only the first recrystallization. The third stage is again a temperature ramp from room temperature to T = 550 °C at a rate of 2.6 °C/ min; then, the temperature is to be kept constant during 15 h and then increased again up to 575 °C and maintained there during 1 h. At this stage new crystals growing at the expense of the old ones are formed. Finally, the specimens are air-cooled. As it has been said above, the final grain size obtained depends critically on the level of plastic strain applied during the cold working. Fig. 2 shows an example of the possible microstructures obtained and the corresponding level of total deformation applied,
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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Fig. 2. Different grain sizes obtained depending on the degree of applied strain.
the upper image corresponds to a specimen which has undergone 8% of total strain during the mechanical treatment and the bottom image corresponds to a specimen subjected to 14%. 2.2. Grain size The grain size measurement is made according to ASTM E112, using the software Simagis Live and following the procedure described in [24]. The obtained grain sizes are summarized in Table 1. 2.3. Fatigue testing Once the thermo-mechanical treatment is completed and the desired microstructure obtained, a circular notch is machined on each specimen, and the samples are subjected to axial fatigue loading. Several combinations of notch and microstructural sizes have been tested. A RUMUL resonant testing machine (Testronic 100 kN) was used for fatigue testing. All tests were load-controlled, with R = 0.1 (pull-pull) and maximum stress between 45 and 95 MPa. Resonance frequencies for the loading conditions and geometry of specimens are between 75 and 90 Hz. Crack propagation results in a decrease on the resonance frequency. After several trials, it was established that a frequency decrease of 0.7 Hz was a good indicator of impending failure. A larger decrease results in complete failure, where the crack faces rapidly separate and the remaining section is plastically deformed, making post mortem analysis very difficult. 2.4. Crack monitoring An experimental procedure for measuring crack length and growth was developed in order to observe crack propagation and interactions with the microstructure. Two low magnification optical microscopes showing real time video images were connected to a computer via USB. They are easily mounted onto a platform and, as shown in Fig. 3, they scan both sides of the
Table 1 Grain size distributions for the different thermomechanical treatments. Total strain (%)
18
14
11
8
–
Analyzed area (mm2) Number of grains Av. grain area (mm2) Av. grain size (mm) Std. Dev. ([mm) Mode (mm) Median (mm)
338 1758 0.19 0.394 0.20 0.25 0.32
5317 2149 2.47 1.41 0.69 1.38 1.24
26,272 1598 16.44 3.46 2.14 2 2.99
416,116 2966 140.30 9.74 6.50 6.25 7.47
0.85 173 0.0049 0.0659 0.0437 0.0375 0.0475
3
Fig. 3. Set-up for monitoring and recording crack growth.
specimen simultaneously. The magnification is adjusted by regulating the distance between the magnifier and the specimen. Since the materials used have low mechanical properties and strains are always kept in elastic region, the movements produced during specimen load and unload are very small; as a result, the recorded video is quasi-static and there is no need to stop the test, measure the crack length and resume. This facilitates the crack growth analysis, and largely reduces the amount of time needed for the test in contrast with other crack monitoring experimental techniques, such as replicas with acetate. This technique makes it possible to keep a complete record of every test and of any possible potential crack on both sides of the specimen. This is important because studies can be made either of the interaction between the crack and the microstructure and of the interaction between different cracks and/or different crack branches. A frame of the video is taken every 30 s in order to measure crack length. Next, the crack length is measured using the free image analysis software ImageJ. We measure the whole length of the crack trace in the surface, from notch edge to crack tip, accounting for all the zig-zag segments that are normally formed. The combination of this information with the test frequency is elaborated in plots of crack lengths vs. number of cycles from which graphs of growth rate vs. number of cycles or crack length can be derived. The representation in terms of number of cycles allows one to compare the growth rates of the different cracks at the same point in time. 3. Results and discussion Fig. 4 shows photographs extracted from the fatigue tests videos. Cracks grow at both sides of a circular notch with a diameter of 2 mm. The grain size is 9.74 mm. The test is load controlled (R = 0.1) with a maximum stress of 45 MPa; under these loading conditions and geometry of specimen, the resonance frequency is 82.13 Hz. The fatigue life is 4.09 106 cycles. The left image corresponds to the front surface of the specimen and the right image corresponds to the back surface. But in the latter, the image has been reversed so that the traces at the left of the notch in both images correspond to the same crack growing at one of the sides of the notch, and similarly at the right. The complicated, non-planar, nature of the cracks surfaces is apparent. In the video, it is possible to observe crack tip arrest at grain boundaries, as well as changes in growth direction. Fig. 5 shows a sequence of the crack propagation process. It corresponds to a test of a 2 mm notch machined in a 3.46 mm grain size microstructure. The maximum stress is 50 MPa (R = 0.1) and the total life is 2.25 106 cycles. Different stages of the crack growth process can be observed. (a) N = 1.27106. The first crack appears on the left side of the notch. (b) N = 1.41 106. A second crack appears on the right side of the notch. Both cracks are seen to grow in (c) N = 1.58 106 and (d) N = 1.78 106. (e) N = 1.88 106. The left
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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Fig. 4. Cracks growing from a circular notch with a diameter of 2 mm within a microstructure of 9.74 mm.
Fig. 5. Different stages of propagation of cracks growing from a 2 mm circular notch within a microstructure of 3.46 mm.
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crack reaches a grain boundary. (f) N = 1.97 106, the crack on the left, after been arrested for almost 9 104 cycles, is seen growing into the neighboring grain with a marked change of direction. (g) N = 2.04 106. The left crack has branched at the grain boundary. The two branches progress along two different paths, which might be associated with different slip systems, although this cannot not be stated with certainty until an analysis of the crystallographic orientations of the crystal is carried out. h) N = 2.21 106. In the competition between the two branches of the crack at the left side, the upper one seems to shield the lower one which remains arrested while the other one continues it propagation. The right crack has also propagated with a slight reorientation. Fig. 6 shows crack length in relation to number of cycles for three cracks growing from a 2 mm notch in a 3.46 mm grain size microstructure. The three of them show acceleration and deceleration patterns along their trajectory. Fig. 7 shows growth rate in relation to crack length for three cracks growing in different microstructures with grain sizes of 1.41, 3.46 and 9.74 mm. It is apparent that the spacing between consecutive halts of the crack increases in relation to the rise in the grain size of the material. There is a good correlation between average distance between halts and grain size. A very interesting observation is that in the majority of the cases studied the cracks did not initiate at the expected point of maximum stress concentration. Let us recall that the specimens are subjected to axial loads alone. It is surprising, for classical
5
methods of notched fatigue limit analysis for this type of loading would indicate that the most likely initiation points should always be the points of the notch edge lying in the diameter perpendicular to the loading axis, the horizontal diameter in our case, since the loading axis is vertical. Of course, anisotropy considerations associated with the large size of the grains would alter that. But it is still nevertheless quite interesting to analyze the pattern that emerge from the observation of the angular location, along the notch edge, of the points where initiation takes place. Thus, we have recorded carefully the point of initiation for many cracks. We have measured the angle subtended at the hole center by the arc along the notch circumference delimited by the point where initiation takes place and the horizontal diameter. Fig. 8 illustrates how these measurements were performed. Several notch sizes and grain diameters were analyzed, totaling more than 500 cracks. Table 2 show the aggregate results. A graphical summary is provided in Fig. 9, where average values of the initiation points location angles have been represented with respect to the ratio between notch diameter and grain size. A general decrease of the angle with the notch diameter to grain size ratio is clearly displayed. Here we can see too that for each grain size the bigger the notch the smaller the location angle. Fig. 10 shows frequency distributions of the angle for two different grain sizes, the biggest and the smallest ones, namely, 9.74 and 0.066 mm. Notice that the values of notch diameter to grain size ratios in both graphs are very different: in the lower graph they are 0.1, 0.2 and 0.4 whereas in the upper one they are 30, 60 and 90. It is interesting to observe that for the larger grain size,
Fig. 6. Crack length vs. number of cycles for three cracks growing from a 2 mm notch in a 3.46 mm grain size microstructure.
Fig. 7. Crack growth rate vs. crack length.
Fig. 8. Measurement of the angle subtended at the hole center by the crack initiation point. Top: notch size 4 mm; grain size 0.066 mm. Bottom: notch size 4 mm; grain size 3.46 mm.
Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023
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Table 2 Location of crack initiation point. Dg , grain size. DNotch , notch diameter. nc , number of cracks measured. Dg (mm)
DNotch (mm)
DNotch =Dg
nc
Average (degrees)
StDev. (degrees)
Min (degrees)
Median (degrees)
Max (degrees)
0.066 0.066 0.066 0.390 0.390 0.390 1.410 1.410 1.410 3.460 3.460 3.460 9.740 9.740 9.740
6 4 2 4 2 1 4 2 1 4 2 1 4 2 1
0.066 0.066 0.066 0.390 0.390 0.390 1.410 1.410 1.410 3.460 3.460 3.460 9.740 9.740 9.740
32 40 36 36 40 26 36 38 28 40 40 24 39 42 22
7.33 8.82 8.12 8.76 10.08 15.01 10.59 13.53 17.33 12.42 15.62 25.14 12.51 17.29 26.32
4.78 7.06 6.56 5.43 7.54 9.28 6.69 7.98 13.46 9.02 12.59 14.53 7.70 10.28 14.01
1.18 0 0 0.74 0 1.74 1.6 0 1.91 0 0 4.76 1.35 0 0
6.14 6.06 7.69 8.66 7.95 13.06 8.60 12.51 16.77 11.48 10.12 24.82 12.07 16.80 27.55
18.05 25.64 22.44 21.30 28.41 36.47 25.15 29.98 46.79 31.46 71.57 58.47 29.77 37.52 47.82
Fig. 9. Location of crack initiation point as a function of notch to grain size ratio.
with the smaller notch size to grain size ratios, there is a wider region where the likelihood of initiation is almost equally high. Fig. 11 shows the results for different holes radii and for several specimens with the same microstructural size of 9.74 mm. On the left, a diagram shows a polar coordinate system corresponding to a quarter of the notch and it presents the crack initiation points for three different radii (0.5, 1 and 2 mm). The graph on the right shows the corresponding frequency distributions. About 36 measurements were made for each notch radius in this figure. Again, it can be appreciated that when the size of the notch diminishes (in relation to the microstructure) the width of the frequency distribution becomes larger and the most frequent location of the initiation point value moves away from the horizontal axis. Compare the values of 12.51, 17.29 and 26.32 degrees obtained for notches of radii 2, 1 and 0.5 mm respectively. Conversely, bigger notch radii in relation to grain size result in narrower zones of likely initiation and bring them closer to the horizontal axis. Of course, for materials with ‘‘normal’’ microstructural sizes of less than a few tens of microns, and ‘‘normal’’ notch sizes of a few millimetres, where the notch is several times larger than the microstructure, this indicates that cracks would normally initiate in the expected location in the horizontal diameter, which is just as well. This observation clearly reveals that the application of the usual techniques of notch analysis (viz critical distance [25,26] or notch sensitivity index concepts [27,28]) may not be directly applicable in situations where the notch size is of the same order or even smaller that the microstructure and this points out that this region of notch sizes may be a good testing ground to discriminate among different notch fatigue calculation methods.
Fig. 10. Distributions of crack initiation location as a function of notch radius for (a) 0.066 mm grain size microstructure and (b) 9.84 mm grain size microstructure.
We must honestly say that at present we only have a rather loose reasoning to account for these results. As mentioned above, none of the currently accepted methods for assessing notched fatigue limits is able to provide an explanation for these findings. Take for example the theory of critical distances: one must locate the ‘‘hot spot’’ and then and look for failure over a certain distance below the surface in the direction of maximum principal stress. Therefore, both the location of the initiation point and the angle of the crack plane are fixed in advance. And there is no room for variations depending on the relative size of the notch and the microstructure.
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Fig. 11. Location of crack initiation point in relation to notch size for a grain size of 9.74 mm.
The only thing we can do at present is to speculate that for very small holes compared to the grain size the influence of the notch disappears long before the crack reaches the grain boundary and so incipient cracks growing at slip bands initiated at any point on the contour of the hole, as far as the circumferential stress is not compressive, have an almost equal chance of becoming dominant. The hole only provides the marginally higher stresses need to trigger the nucleation of the slip bands, but does not contribute to the overcoming of the barrier to growth represented by the grain boundary. For holes which are bigger compared with the grain size, the stress concentration acts almost directly upon the grain boundary, and since the stress concentration decreases markedly when one moves away from the point of maximum circumferential stress (in the horizontal axis passing through the center of the hole, assuming tensile loading in the vertical direction), the cracks initiated there have a much greater incentive to become dominant. There is an obvious need to make these arguments quantitative. Because of this, we are working now in the formulation of a microstructural fracture mechanics model, based on dislocation theory and representing an extension of the Navarro–Rios model [15,29–33], of a crack growing from a circular hole in a plate subjected to in-plane and in-phase biaxial loading. But this is still in a preliminary stage. We hope to have some results worth reporting in the very near future. Finally, we would just like to present some preliminary results obtained by using Digital Image Correlation (DIC) techniques in our experiments with the big-grained material. Digital image correlation analysis is a relatively new and very useful tool for understanding material behavior, specially at the grain scale. Recent investigation shows its potencial for evaluating the accumulated plastic strain fields associated with fatigue crack growth. When the microstructural size is much smaller than the notch size the strain fields appear as more homogeneous, but when the crack size is about the grain size the strain fields vary a lot from grain to grain, and also within individual grains [34]. This phenomenon can be easily illustrated when working with large microstructures. In this case a high resolution imaging of the samples is not needed for obtaining grain-scale strain fields. In this section we show the strain inhomogeneities that arise when subjecting our samples to a remote strain (monotonic load) and the plastic strain near a growing fatigue crack for both a small size and for a large-size microstructure, in order to compare the effect of the crack to grain size ratio effect upon the strain fields. In order to perform the static loading analysis, once the microstructure is obtained, a speckle pattern is generated on the surface, and then the specimen is tested in tension. By taking
photographs of the surface on a regular basis, image correlation techniques can be applied. A high definition camera plus a longdistance microscope Questar QM 1 are used to capture the images. Software VIC-2DTM is used for acquisition and image processing. Tensile tests were conducted on an MTS 810 servohydraulic machine, displacement controlled at a rate of 0.2 mm/s. In some tests a contact extensometer was used to corroborate the correct measurement of the deformations obtained by digital image correlation (DIC). Pictures of the sample surface were obtained every 5 s. These sets of photos are used for obtaining the deformation fields. The analysis with VIC-2D™ software was perform with a subset of 21 and a step of 1. The resolution of the captured images is 20 pixel/mm. Fig. 12 shows some statistical parameters (maximum, minimum, average, and standard deviation values) of the strain measurements performed on specimens of different grain sizes. It can be observed that, for the same amount of applied total deformation (12%), increasing the grain size produces an increase in the standard deviation and in the maximum strain reached. This is due to the effect of measuring with more detail what happens inside each grains, as the speckle pattern begins to be smaller than the grain size, combined with effect of the reduction of the number of neighboring grains, which makes the constraint forces preventing the grains to rotate or deform freely smaller, and thus increasing the amount of the deformation of the grains which are better oriented with respect to the load axis [35].
Fig. 12. Applied strain, standard deviation and maximum and minimum strain values obtained as a function of grain size of the specimen.
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Fig. 13. Specimen subjected to a total axial strain of 12% and results for DIC analysis,
Fig. 14. Evolution of local strain eyy along lines A and B in Fig. 13 for different values of total strain applied to the specimen (top, line A, bottom, line B).
eyy ; exx ; exy .
Fig. 15. Final strain state along lines A and B in Fig. 13 at the end of the test (top, line A, bottom, line B).
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The strain inhomogeneities can thus be easily analyzed in the case of the large microstructures. Fig. 13 shows a specimen 2 mm thick, whose measured grain size on the surface is 12.6 mm. The grains are thus larger than the specimen thickness and it behaves as a sort of bidimensional array of single crystals. The sample has been subjected to a total strain of 12%. The picture on the left shows the specimen after straining. It is possible to appreciate with some detail how the grain boundaries have become marked. The inhomogeneity in the deformation can be observed too. The three graphs on the left show the deformation fields obtained by digital image correlation, eyy ; exx , exy . The load acts in the y-direction. It is interesting to observe how for a total strain level of 12% applied in the axial direction, some grains exceed 25% of strain while others reached deformation levels below 7%. Furthermore, Fig. 14 shows the evolution of eyy along lines A and B in the previous figure for different levels of total applied stress. It is possible to observe how the inhomogeneities in the strain behavior increase with the increasing applied strain. Fig. 15 shows the final strain state along lines A and B at the end of the test; it also includes the position of the grain boundaries. It is easy to see how the highest deformation gradients occur at the grain boundaries. These results help us to understand how important strain inhomogeneities can be when working at the microstructural level. The effect of the loading history can be very important at a later stage in the life of the specimen. If the specimen presented here were subjected to cyclic loading, even for an applied strain uniform over the cross section, the accumulated local strains will be very far from those expected without taking into account these microstructural effects. Obviously these effects will be more important when working with specimens having fewer grains across the section. The next step in the analysis is to see how the strain fields evolve when the cracks grow by fatigue. Fig. 16 shows an example. The grain size is 0.06 mm, while the hole diameter is 2 mm. The specimen is subjected to a 50 MPa tension–tension (R = 0.1) fatigue
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Fig. 17. A specimen as viewed by DIC from one side and normal video from the other side. The grains, of average size 8.22 mm, are big enough to pass across the thickness of the specimen (2 mm). The diameter of the hole is 2 mm. The grain boundaries have been manually drawn in both pictures.
Fig. 16. Evolution of the eyy strain field during crack propagation. Notch diameter 2 mm, grain size 0.07 mm. Number of cycles: (a) N = 100,000, (b) N = 130,000, (c) N = 160,000, (d) N = 180,000, (e) N = 210,000, and (f) N = 230,000.
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cycle. The total fatigue life at this level was N = 7.9105 cycles. Figures a) to f) show different stages of crack propagation. It is possible to observe the evolution of the strain field. Despite a little deviation from the horizontal axis, the deformation zones generated by the growing crack are fairly symmetrical for this case where the notch is relatively big as compared to the grain size. On the other hand, Fig. 17 corresponds to the case of a specimen with a grain size of 8.22 mm. The specimen thickness is only 2 mm, so there is only one grain across the thickness and we have again a kind of bidimensional arrangement of single crystals. This arrangement allows one to monitor crack propagation by means of DIC analysis on one side of the specimen and by conventional image analysis on the other side of the specimen, and the information relative to the same grains can be compared. The specimen was subjected to 45 MPa tension–tension (R = 0.1) fatigue loading. The total fatigue life was N = 2.38 106 cycles. The strain analysis show the longitudinal strain on the sample at the moment of the initiation of plastic collapse. In this case, the effect of the microstructure upon deformation is dramatically captured: when we observe the specimen at a scale comparable to the microstructural size, the apparent symmetry in the crack strain field observed before disappears completely. This is also an striking example of cracks that start growing in Stage II and then switch to Stage I, reversing the usual course of events. Furthermore, it is possible to observe how the left crack is arrested at a grain boundary and a new plastic zone is generated not in the contiguous grain but in another neighboring grain that it is probably more favorably oriented. In this case microstructural inhomogeneities make it possible to find grains accumulating high levels of plastic strain and neighboring grains remaining virtually unaffected. These type of results may help us to understand the mechanism of crack propagation when working with micro-components. The ongoing tendence to miniaturization of mechanical components leads to the fact that the size of the microstructure is no longer negligible with respect to the component size itself [36]. The intrinsic role of the microstructure on the size effect has been studied by different authors [35,37–39]. In these papers, monotonic mechanical properties are studied mainly. In this paper we have tried to present some aspects of the role of the microstructure on tensile behavior and fatigue properties of notched components where the size of the microstructure is not negligible as compared with the specimen size. In specimens whose dimensions are in the same scale of the grain size, microstructural inhomogeneities will play a prominent role and thus special care has to be taken when analyzing mechanical properties with models that do not take into account this phenomenon. 4. Conclusions The initiation and growth behavior of very long microstructurally short fatigue cracks formed at circular holes has been described. Very long here means cracks which are several millimeters or even centimeters long. Microstructurally short refer to the fact that these cracks, in spite of their physical length, are still smaller than the grain size of the material and thus exhibit the characteristic features, such as growth retardation or even halt at grain boundaries and fluctuating crack growth rates, of these type of cracks as observed when ‘‘normal’’, much smaller microstructures are considered [40,41]. This has been possible by employing an experimental technique which allows one to increase the grain size of Al1050 Aluminum alloy until the centimeter scale by applying a series of mechanical and heat treatments. With this method, interactions between cracks and microstructural barriers can be studied with an unprecedented level of ease and detail.
An interesting observation is that the width of the arc around the notch root where the initiation of the cracks take place (and the location of this arc) varies greatly with the ratio between the notch diameter and the grain size: for large ratios, the width is small and the arc centers in the symmetry axis of the notch, which corresponds to the point of maximum stress concentration. For smaller ratios, however, the width of the arc increases markedly and its position moves away from the symmetry axis. This study highlights the fact that the actual physical size of the crack is not as important as the relative size of the crack with respect to the characteristic microstructural dimension. The size of the crack at any point in time must be measured in terms of how much of the grain it has traversed or how many grains it already spans.
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Please cite this article in press as: Lorenzino P, Navarro A. Growth of very long ‘‘short cracks’’ initiated at holes. Int J Fatigue (2014), http://dx.doi.org/ 10.1016/j.ijfatigue.2014.03.023