A novel approach is introduced to map the mesoscale plastic strain distribution resulting from heterogeneous plastic deformation in complex loading and component geometries, by applying the discrete Fourier transform (DFT) to backscattered electron (BSE) images of polycrystalline patches. These DFTs are then calibrated against the full width at half the maximum of the central peak of the DFTs collected from the same material tested under in situ scanning electron microscopy uniaxial tensile conditions, which indicates a close relationship with the global tensile strain. In this work, the technique is demonstrated by measuring the residual strain distribution and plastic zone size around a fatigue crack tip in a commercially pure titanium compact tension specimen, by collecting BSE images in a 15×15 array of 115 μm square images around the fatigue crack tip. The measurement results show good agreement with the plastic zone size and shape measured using thermoelastic stress analysis.
The role of the plastic zone surrounding the tip of a fatigue crack has been recognized for a long time as important in determining crack growth rates, including the potential retardation of growth via mechanisms associated with crack closure (Elber 1970), which has been referred to recently, and perhaps more accurately or descriptively, as plastic shielding (Pacey et al. 2005). The mechanisms supporting plasticity-induced closure have been the subject of unresolved debate (James 1997), primarily because it has not been possible to make direct measurements of the size and shape of the crack tip plastic zone, much less evaluate the effect of variations in, for instance, loading or environmental conditions.
The common method of assessing crack closure effects is to use compliance measurements, which in the frequently used compact tension (CT) specimen are based on back-face strain gauge data (Pacey et al. 2005). These measurements are very remote relative to the crack tip and their interpretation is not straightforward.
Uguz & Martin (1996) reviewed a range of techniques that are available for measuring the plastic zone at the crack tip. Some of these techniques are forensic in nature, in that they are performed at the conclusion of a test or event, such as etching techniques, the recrystallization method and shear lip size measurements. Others, such as micro-hardness measurements and strain gauge measurements, provide only point measurements, so that evaluation of the shape of the plastic zone required many measurements. The optical shadow spot or caustics method uses a laser beam to make direct measurements of the plastic zone based on out-of-plane deformations of the specimen surface, which for a metal must be very highly polished. Caustics are not straightforward to use and so have not been adopted widely.
A number of other non-contact optical techniques, such as moiré interferometry and digital image correlation (DIC), can be used to obtain the displacement field around a propagating crack and the corresponding strain fields can be evaluated straightforwardly. In order to determine the extent of the plastic zone, however, an assumption has to be made about the value and component of strain at which plasticity begins to occur, and knowledge of the material properties is required. Significant uncertainty can be introduced to the process by both this assumption and the prerequisite knowledge.
Recently, monitoring the release of energy, in the form of heat associated with the movement of dislocations during plastic deformation, has been employed to identify the extent of plasticity (Patki & Patterson 2010). During cyclic loading, the release of heat causes a local phase difference between the time-varying temperature, owing to the thermoelastic effect, monitored in the far-field elastic region and the temperature measurements made in the plastic zone (Diaz et al. 2004). This approach has the significant advantage that the simultaneous measurement of amplitude of the temperature variation in the elastic field is routinely possible. This measurement is directly proportional to the first strain invariant so that the evaluation of the intensity factor, associated with the crack, is straightforward.
An alternative approach to measure the plasticity is to assess the accumulation of residual dislocations resulting from the plastic deformation. Groma & Ungar 1988 and Ungar & Groma (1989) have discussed how the peak broadening in X-ray diffraction is closely related to the dislocation density in materials. They (Groma & Ungar 1988) have built an effective model to relate statistically stored dislocations (SSDs) density with local plastic strain. More recently, Barabash et al. (2001, 2003) have used three-dimensional X-ray diffraction to characterize local orientations and stress states with micro-scale resolution in bulk materials. While diffuse peak broadening is associated with SSDs, peak streaking results from the geometrically necessary dislocations (GNDs), and the Burgers vectors associated with the GNDs can be identified by analysing the streaking direction of specific Laue reflections. However, because of the slow data acquisition and high intensity X-ray source needed for this technique, it is currently unavailable to a wide range of researchers.
A number of scanning electron microscopy (SEM) approaches have also been used to examine how the local changes in lattice curvature and dilation, associated with GNDs and SSDs, affect the lattice interaction with incident electron beams. Selected-area channel patterns, formed from the variation in intensity of backscattered electrons (BSEs) resulting from rocking the beam about a point on the surface of a single crystal or grain, are well known to deteriorate with increases in plastic strain (Joy 1972; Joy et al. 1972; Crompton & Martin 1980). Using SACPs, Davidson & Lankford (1980) found that fatigue crack plastic zone size is correlated with work performed in creating the new crack surface and the yield stress, rather than yield stress and stress intensity factor. Nevertheless, the application of the SACP approach is limited in that most SEMs are incapable of forming SACPs; and those with the capability typically can rock the beam only for an area in the tens to hundreds of micrometres in diameter, owing to spherical aberration and focusing errors. Consequently, the technique is effectively limited to the study of rather coarse-grained polycrystalline materials or single crystals. Electron backscattered diffraction patterns (EBSDs) are Kikuchi patterns that form as the BSEs from a stationary electron beam diffract when they exit the crystal. In a manner akin to that observed for SACPs, the patterns also break down with the accumulation of dislocations (Wilkinson & Dingley 1992; Wilkinson et al. 1993). Using EBSDs, Jia et al. (2008) quantitatively elucidated the change in grain orientation-dependent plastic behaviour in austenitic and ferrite steels during in situ tensile tests. EBSD has an advantage of being able to collect information from small areas under the electron beam, in the range of hundreds of nanometres. The quantification will, however, become difficult at large strain levels.
The electron channelling effect outlined above also plays a significant role in the contrast of BSE SEM images of single-phase polycrystalline specimens. Because the electron beam trajectory varies only slightly while scanning over an area to form an image; prior to deformation, most individual grains show very even contrast. Strong contrast variations are evident between the grains because the different orientations result in large differences in channelling behaviour. With deformation, as dislocations accumulate within grains, typically to the order of 102 μm−2 when plastic strain is about 5 per cent (Crimp et al. 2004; Ungar et al. 2008), significant lattice distortions are generated and stronger contrast variations develop within the grains. Consequently, the grain-to-grain contrast becomes less dominant with deformation, whereas the intra-grain contrast increases. While this phenomenon has been observed (Bieler et al. 2009; Yang et al. 2011), it has never been used to quantify the strain levels associated with localized plastic deformation. These contrast changes eventually lead to a change in the frequency distribution of features within the corresponding BSE image. Thus, if the frequency distribution of the BSE images can be quantified, then this may be related directly to the extent of plastic deformation. The discrete Fourier transformation (DFT) provides a measure of the frequency content of an image. In two-dimensions, the DFT of an m by n matrix X is given by another m by n matrix Y : 1.1where the complex roots are defined as ωm=e−2πi/m and ωn=e−2πi/n. A Fourier transformation of a two-dimensional image gives a complex-number image, i.e. two images are generated—identified here, for the sake of understanding—as the magnitude map and the phase map. The magnitude of a DFT provides the distribution of the frequency components in the original image, whereas the phase of the DFT gives the location of this frequency component in the original image. In the proposed technique, the magnitude of the DFT is considered and will be shown to reflect the contrast change within microstructure patches during plastic deformation.
In this study, the plastic zone size and plastic strain distribution are determined quantitatively by measuring the full width at half the maximum (FWHM) of the peak in the magnitude of the DFT of BSE images of the microstructure around a fatigue crack tip in a commercially pure titanium (Ti) specimen. According to the results of a calibration made from images collected during an in situ tensile test, the peak width has a close relationship with strain for images of multi-grain patches. On the basis of this calibration, a plastic strain distribution map around the crack tip has been produced. This map is compared with the plastic zone size determined using thermoelastic stress analysis (TSA) in order to elucidate the connection between these micro- and mesoscale measurements. The influences of an array of imaging parameters on the BSE/DFT strain measurements are also discussed.
2. Experimental details
The material used in this work was a commercially pure polycrystalline Ti sheet of thickness, B=0.7 mm, with an average grain size of about 10 μm as determined by the linear intercept method. The sheet had a crystallographic texture with the hexagonal  pole orientations being about five times greater in the normal direction of the sheet than in a material with a truly random grain orientation. A CT specimen, which conformed to the specification in ASTM E399, was manufactured from the sheet. This specimen had a height, H=24 mm, a width from the loading holes, W=20 mm and had an initial notch length of 3 mm. In addition, a small dog-bone tensile specimen was manufactured with dimensions suitable for loading within the SEM; i.e. a gauge length of 10 mm and width of 3 mm. A large dog-bone specimen, manufactured with a gauge length of 80 mm and width of 10 mm, was used to obtain the tensile stress–strain curve for the material. Prior to the fatigue and tensile tests, all specimens were mechanically polished, finishing with a 0.05 μm colloidal silica suspension. After mechanical polishing, the specimens were electro-polished using 10 vol% perchloric acid+90 vol% methanol at 20 V at −20°C for 30 s to get an even, flat and damage-free surface. Using a fine airbrush, the polished surface of the CT specimen was sprayed with a thin coat of matt black paint in order to provide uniform emissivity for the TSA measurement. The CT specimen was then loaded in air with a mean load of 600 N, and load amplitude of 150 N, at a frequency of 20 Hz. During the fatigue loading, TSA data were recorded at regular intervals using a Deltatherm instrument (Model 1550; Stress Photonics Inc., Madison, WI, USA), which has a staring array sensor (256×320 pixels) that was fitted with a two-position lens set in the magnification position so that one pixel was equal to 0.0625 mm on the specimen. The fatigue test was stopped after 27 252 cycles when the crack was of length 5.88 mm. The TSA instrumentation was adjusted so that the phase difference between the forcing and temperature signals was zero in the far-field relative to the crack, where the strain field was elastic. The movement of dislocations in the plastic zone generated a phase difference between the plastic zone and the far-field elastic region. Thus, the plastic zone was identified by applying a binary filter spatially to the phase difference between the load and temperature signals, following the methodology of Patki & Patterson (2010).
After the fatigue loading the matt black paint was removed from the specimen by ultrasonic cleaning in methanol. An array of 225 BSE images, as shown in figure 1, were collected at an instrument magnification of 500×(equivalent to3 pixels μm−1) using a CamScan 44FE scanning electron microscope (CamScan, Cambridge, UK) so that each image was 115×115 μm with a centre-to-centre distance of 187 μm. An in situ, uniaxial tensile test-frame (E. Fullam, Latham, NY, USA) was employed to conduct a calibration experiment using the small polished dog-bone tensile specimen. For each of six loading increments between 0 per cent and 22 per cent engineering strain, six BSE images were acquired in a rectangular array in the middle of the specimen under identical imaging conditions (brightness, contrast, magnification and working distance) as those used for the CT specimen. Additional images were collected under differing contrast and brightness levels, as well as at varying magnifications, in order to assess the influence of these variables on the analysis.
3. Analysis technique and results
(a) Thermoelastic stress analysis measurement of plastic zone
Data obtained using TSA are shown in figure 2 for a line through the crack tip and along the crack path. The uncalibrated TSA signal and its phase difference with respect to the applied loading are plotted. The phase signal is noisy ahead of the crack tip but in the far-field has a mean value that deviates by less than 5° from zero indicating essentially adiabatic conditions, as would be expected for elastically deforming material. Close to and ahead of the crack tip, the phase difference exhibits a large negative value with a maximum difference of about 22°. Immediately behind the crack tip, the phase difference is positive with a peak of about 10°. These two regions of phase difference with opposite sign have been attributed to heat generation occurring ahead of the crack tip in the loading part of the cycle as a result of dislocation movement and heat generation behind the crack tip in the unloading part of the cycle owing to contact of the recently formed flanks. Diaz et al. (2004) suggested that the transition between positive and negative phase provides an indication of the location of the crack tip along the direction of expected crack growth. Patki & Patterson (2010) applied a binary filter to the phase difference and identified the shape and size of the plastic zone from the region of significant phase difference ahead of the crack tip. This is the approach adopted here and the result is shown by the black area superimposed on figure 2, which is approximately circular with a radius about 0.7 mm.
(b) Backscattered electron assessment of plastic strain
A gradual change in the sharpness of the BSE images with distance from the crack tip can be observed in figure 3. At large distances from the crack tip, e.g. at position h14 in figure 3, the images display sharp, well-defined features, allowing easy identification of the individual grains, which tend to exhibit uniform brightness levels that are distinctly different to their neighbours. Images acquired close to the crack tip show significant variation in brightness within the grains, making the individual grains less apparent and the overall image less sharp. This is a result of significant crystal rotations and residual plastic strain within these grain patches. Grain boundary edges, slip lines and secondary grain boundary cracks were also observed in those grains close to the fatigue crack.
In order to quantify the changes in image characteristics, the DFT was performed on each BSE image using the fast Fourier transform algorithm applied via the ‘fft2’ function in Matlab (MathWorks, Natick, MA, USA). The ‘fftshift’ and ‘log2’ functions were employed to improve the output image quality by shifting the zero frequency to the DFT image centre, and enlarging the detailed frequency information, respectively. Figure 4 shows this analysis for four example images taken at 561 μm intervals directly ahead of the crack. Figure 4b shows the magnitude of the DFT and a distinct trend in the extent of the central peak with distance from the crack tip is apparent. This sharpening with increasing distance was quantified by plotting the profile of the magnitude component of the DFT across its horizontal centre line, as shown in figure 4c, and evaluating the FWHM in these profiles. In more detail, symmetry of the DFT plot was assumed and the left half of these profiles was smoothed using the adjacent-averaging method with a five-point window and then an exponential curve fitted prior to measuring the FWHM, as shown schematically in figure 5. Figure 6 shows the FWHM distribution in front of the crack tip. The FWHM decreases dramatically with increasing distance ahead of the crack tip. The FWHM remains constant beyond 2 mm, indicating a baseline resulting from the frequencies in the images associated with the undeformed microstructure.
The results from the in situ uniaxial tensile test were used to quantify the relationship between the residual plastic strain level and the FWHM of the DFT magnitude. In order to obtain a detailed stress–strain relationship, an additional ex situ tensile test was carried out on the large dog-bone specimen using a conventional tension–compression test machine (Denison Mayes SM100). Displacement data were determined using two-dimensional DIC employing a LaVision DaVis/StrainMaster v. 7.1 system. Strains were determined using a gauge length of 42.4 mm within the overall specimen gauge length of 80 mm so that, following the onset of plasticity, the images for the DIC were acquired in an equivalent region to the BSE images. The engineering stress–strain curve generated (figure 7) shows that the material yields at about 0.4 per cent strain. In figure 7, the arrows below the stress–strain curve indicate the five strain increments at which the in situ tensile test was interrupted to acquire the BSE images (115×115 μm) that were processed using the same procedure as described in the previous paragraph. The mean FWHM of the DFT at each strain increment was plotted, as shown in figure 8, and a polynomial fitted to create a calibration curve that could be used to determine the strain from the BSE images recorded from the CT specimen.
The calibration curve in figure 8 was used to estimate the level of plastic strain in each of the 225 BSE images shown in figure 3 after applying the DFT and calculating its FWHM. The results are shown in figure 9 as a map of residual plastic strain in which each coloured tile represents an average residual plastic strain for a 187×187 μm area based on 115×115 μm images spaced with a centre-to-centre distance of 187 μm.
The overall shape of the plastic zone around the crack tip, shown in figure 9, is oblate with a Dugdale-type area of high strain extending about 1.5 mm ahead of the crack and about 0.5 mm on both sides of the crack. The plastic strain exhibits high values in the crack wake and at the crack tip, with a maximum value of 18 per cent, which decreases dramatically with distance from the crack tip.
The effects of the instrument parameters associated with the electron imaging conditions need to be assessed before further discussion. It was expected that changes in brightness would not change the shape of the peak in the DFT, because the frequency distribution of the image remains the same. Increases in brightness may increase the counts within the DFT peak, which means that the peak will be extended along the y-axis, but the FWHM will not be changed. An increase, in contrast, will increase the difference in brightness level between bright and dark pixels, but will not change the image frequency distribution. Images of the same microstructure patch were collected using three different brightness levels and then using three widely varying contrast levels. The resulting FWHM analyses found variations of less than 2 per cent, confirming that neither the brightness nor contrast affects the underlying FWHM analysis.
Nevertheless, the magnification of a BSE image will influence the DFT peak because a different field of view will be present for the same image size. Thus, a low magnification BSE image will contain more grains and the channelling bands within the grains will be finer in scale than for a higher magnification image of the same sample, resulting in a wider DFT distribution for the lower magnification image. At the other extreme, when the magnification is too high, there will be insufficient grains sampled to provide statistically significant data, and observations show that the DFT become asymmetric and noisier. Therefore, it is important that images are acquired at an appropriate magnification for the grain size of the material being studied. In this study, the average grain size was approximately 10 μm. The appropriate magnification was found by plotting the FWHM for a data patch with a plastic strain of about 6 per cent containing a constant number of pixels as a function of magnification with the range defined by the limits described above. At 400–500× magnification, the FWHM was nearly constant and it diverged to high and lower values below and above this range, respectively. Thus, 500× was selected as an appropriate value for the magnification. It is also essential that the imaging conditions, such as working distance, objective aperture, beam convergence and spot size remain constant for both the calibration and fatigue samples in order to avoid introducing errors. The strain determined is an average for the field of view of the image so a smaller field of view would allow a higher spatial resolution for the evaluation of strain, but at a cost of more noise in the DFT. The spatial resolution of the technique can be varied by adjusting the centre-to-centre spacing of the images, i.e. overlapping the images would increase the spatial resolution compared with the example given here, which has a spatial resolution of ±93.5 μm.
The strain calibration procedure is another aspect that needs to be considered briefly. The assumption in the DFT calibration presented above is that an equi- valent breakdown in channelling conditions, associated with a given distribution of dislocations, occurs for the same equivalent strain, regardless of the strain path. Indeed, the two strain paths in this work are considerably different, i.e. a uniaxial strain distribution with measurements taken during monotonic loading compared with a tri-axial strain distribution at the crack tip induced by high-cycle tension–tension fatigue with measurements taken after loading. Previous studies (Davidson & Lankford 1980; Konig & Blum 1980) indicate that fatigue specimens can be expected to have a higher dislocation density for an equivalent strain level than uniaxial tension specimens. Thus, the strain deduced for the fatigue sample based on the calibration using the uniaxial sample might be an overestimate of the actual strain to some small extent. Nevertheless, the form of the relationship in figure 8 is consistent with the changes in angular resolution of channelling bands around crack tips in fatigued specimens and the subsequent conclusions of Crompton & Martin (1980).
An alternative to using a tensile test for calibration might involve carrying out an in situ uniaxial fatigue test with a prescribed stress amplitude and load ratio and collecting a series of BSE images as a function of cycles, but such a test would be somewhat difficult to define and execute, and would be limited by a lack of knowledge of the equivalent strain developed over a given number of cycles.
The error bars in figure 8 represent ±2 times the standard deviation, i.e. 95 per cent confidence limits, based on evaluations of the FWHM obtained from six different areas on the tensile specimen collected at each of the six strain levels. Based on these error bars, the precision of the technique would appear to be of the order of five pixels for the FWHM; thus, the strain sensitivity is of the order of 0.5 per cent strain. These values were confirmed by taking a series of six measurements at locations of zero strain 5 mm ahead of the crack and an additional six measurements on the virgin tensile specimen; the FWHM had a constant value of 65.5 and 68 pixels, respectively, with standard deviations of 3.209 and 3.899, respectively. These standard deviations represent the noise level in the measurements and correspond to the minimum measurement sensitivity, whereas the FWHM values provide the background or zero level in the absence of plasticity, which were subtracted from measured values before plotting them in figures 8 and 9.
An additional consideration is whether there is any change in the FWHM values for a region that has been plastically strained when the measurements are taken in the loaded or unloaded state. This was checked by considering a data patch in the calibration specimen that had been subjected to 22 per cent strain and it was found that the mean FHWM values for six measurements were 252.5 and 252.8 pixels for the loaded and unloaded cases, i.e. there is no significant difference.
Figure 9 clearly reveals the decay in the local plastic strain distribution with distance from the crack tip, which is to be expected from traditional fracture mechanics models. In addition, significant scatter is seen in the strain levels within this decaying plastic zone. This scatter is to be expected because of the material heterogeneity that arises from the inter-granular variations in material morphology, but some scatter will arise from measurement uncertainties. Nevertheless, three BSE images for each of positions h1–h15 were collected on two entirely separate occasions, providing a total of six different measurements at each position, and the difference in the processed FWHM was found to be within ±3 pixels. Thus, the variations in measured strain in the plastic zone, which are of the order of hundreds of pixels of FWHM, appear to be associated with real effects and not just experimental uncertainty.
Patki & Patterson (2010) compared the plastic zone size rp, estimated using TSA with those determined using the theory proposed by Irwin (Ewalds & Wanhill 1991): 4.1where 2ry is the Irwin plastic zone diameter, KI is the applied stress intensity factor and σy is the yield stress of the material. The agreement between experiment and theory was within about 0.2 mm when the stress intensity factor range, ΔKI, determined from TSA measurements was used in equation (4.1), rather than KI. Thus in figure 9, a comparison has been made of the crack tip plastic zones determined from the TSA and BSE data obtained in this study with those based on the expressions proposed by Dugdale and Irwin (Ewalds & Wanhill 1991). The value of ΔKI (23.8 MPa√m) was found using the multiple point over-deterministic method to fit a Muskhelishvili-type description of the crack tip stress field to the measured TSA data following the method described by Diaz et al. (2004), which was implemented in the software algorithm, FATCAT1 There is excellent agreement between the extent of the plastic zone ahead of the crack tip found using TSA measurements and the value estimated using Irwin's approach (figure 9); and good agreement between the size of the plastic zone found using the BSE measurements and the value estimated using Dugdale's expression. Although this level of agreement is encouraging, it should be noted these theoretical expressions are based on the continuum principles of linear elastic fracture mechanics, which take no account of dislocation distribution or movement, or how these will be affected by local grain structure and orientation variations.
The plastic zone found from the TSA data is smaller than that determined from the BSE images, but of the same order of magnitude with its boundary in the direction perpendicular to crack growth corresponding to 3–5% plastic strain measured from the BSE images. Ahead of the crack, the boundary of the plastic zone found from the TSA data corresponds to 5–7% plastic strain evaluated from the BSE images. The two experimental approaches are based on different physical principles and so some difference in results would be expected. The data from the BSE images are based on detecting the disruption of the grains by the residual dislocations resulting from plastic deformation. By definition, this disruption will not disappear when the load is removed, as was necessary for the fatigue crack observations in the SEM; however, the removal of the load could have caused some reverse plasticity as a consequence of compatibility requirements between the crack tip plastic zone and the surrounding elastic region. This reverse plasticity could have influenced the measurement of the plastic strain in the region closest to the crack tip but would not affect the maximum extent of plasticity detected. The evaluation of the plastic zone from the TSA data is based on identifying the region in which the generation of heat by dislocation movement is sufficiently large to cause adiabatic conditions to be lost, i.e. a heat flux or transfer occurs during fatigue loading. A higher density of dislocations will generate a larger heat flux, which will be easier to detect as a phase difference and will tend to occur closer to the crack tip. At lower loading frequencies, though still high enough to maintain adiabaticity in the elastic field, there is more time for heat transfer to occur and so the TSA measurements, based on phase difference, will show greater sensitivity to the heat generated by plastic deformation. Consequently, a lower density of dislocations is likely to generate sufficient heat to produce a measurable phase difference and so lower levels of plasticity will be detectable. This concurs with the observations by Tomlinson et al. (2011) that the plastic zone size measured using TSA was smaller at higher frequencies.
Both TSA and the new technique proposed here based on FWHM of BSE images have the significant advantage that they allow the identification of the elastic–plastic boundary without a prior knowledge of the elastic limit of the material. This is potentially important in materials for which the yield behaviour might be a function of the load history. The use of FWHM on BSE images, as described here, also allows the magnitude of the plastic strain to be evaluated within the plastic region. These novel attributes of the technique should be useful in enhancing our understanding of the development of fatigue damage in metals for which the development of plasticity ahead of a propagating crack is the first step and is followed by void formation and coalescence to form the crack.
BSE imaging is a common imaging mode in the SEM for studies on metals, so this approach could be easily applied to measure mesoscale heterogeneous plasticity in other kinds of deformation, such as bending, compression and in creep tests, or on components with complex geometries. However, there are some restrictions in terms of the types of materials because in order to acquire statistically significant information, there needs to be sufficient grains in the field of view and therefore, materials with a large grain size will decrease the spatial resolution of this method. Another consideration is that the materials need to be single-phase, or multi-phase with large primary-phase grains and very small secondary-phase particles, so that the DFT is effectively applied only to the primary-phase grains in the image. In materials with approximately equal amounts of two different phases, the contrast of BSE images will tend to be dominated by intensity differences between the phases, which result from the influence of atomic number on BSE yield, instead of the differences in the BSE yield resulting from variations in channelling behaviour arising from dislocations. Consequently, the DFT will be a function of the shape and quantity of these phases, which will mask the changes associated with changes in dislocation density.
Secondary electron images were considered as an alternative to BSE images and although on visual inspection they reveal a distinct variation in surface topology in the immediate vicinity of the crack tip, it was not possible to obtain quantitative information of the nature reported here. BSE images are much less sensitive to surface topography and contain information from greater crystal depths, allowing the crystallographic changes associated with changes in dislocation density to be assessed.
An additional major advantage of the proposed technique is, as previously mentioned, that it allows the extent of plasticity to identified without a prior knowledge of the elastic limit of the material, primarily because it is based on the underlying mechanism that drives plastic strain, i.e. the creation of residual dislocations. The technique is also non-destructive and has variable spatial and temporal resolutions that are functions of the pitch and size of the image patches or facets used to capture the data.
In its current state of development, this approach to mapping the strain is somewhat tedious. While it is beyond the scope of this study, the method is conducive to a high degree of automation. Using stage-controlled scanning, readily available on modern SEMs, the BSE image pattern collection can be easily automated. Likewise, it should be possible to integrate the post-processing and DFT calculations of the images with the collection automation, in a manner similar to that performed with the pattern collection and Hough transforms involved with modern EBSD systems (Adams et al. 1993). These steps should greatly enhance the speed and flexibility of the method.
The plastic strain around a fatigue crack has been successfully evaluated using a novel technique based on using the DFT to analyse the structure of BSE images from a CT specimen of commercial purity Ti. The FWHM of the magnitude of the DFT of the BSE images was used as a measure of the level of plasticity based on the distortion in the texture of grains caused by residual (GNDs and SSDs) dislocations. The technique was calibrated using images obtained from a uniaxial tension test performed in situ in the SEM so that the magnitude of the average plastic strain in the field of view of an image could be related to the FWHM for the image. After a fatigue crack had been propagated to a length of 5.88 mm, 225 BSE images 115×115 μm were collected in an array occupying a 2.8×2.8 mm square around the crack tip and a map of plastic strain evaluated for this area. The measured plastic zone size and shape were assessed during the fatigue loading using TSA and the results from the two techniques are in general agreement and correlate with estimates of plastic zone size based on the theoretical expressions owing to Irwin and Dugdale evaluated using the measured range of the stress intensity factor during the fatigue cycle. This new approach may prove to be useful for measuring local heterogeneous plastic deformation in other applications when there is a dominant primary phase present; however, it has great potential in assisting to identify the role of the crack tip plastic zone in crack closure and plasticity-induced shielding.
E.A.P. and R.A.T., respectively, acknowledge with gratitude the support of the Royal Society through a Royal Society Wolfson Research Merit Award and the support of the Engineering and Physical Science Research Council through an Overseas Travel grant. Y.Y. and M.A.C. gratefully acknowledge the support of the National Science Foundation, grant DMR 0710570.
- Received November 18, 2011.
- Accepted February 15, 2012.
- This journal is © 2012 The Royal Society