47. Afm Characterization Of The Evolution Of Surface Deformation During Fatigue Evolution In Polycristaline Copper.pdf

Acta mater. 49 (2001) 3755–3765 www.elsevier.com/locate/actamat

AFM CHARACTERIZATION OF THE EVOLUTION OF SURFACE DEFORMATION DURING FATIGUE IN POLYCRYSTALLINE COPPER L. CRETEGNY1‡ and A. SAXENA2† 1

GE Corporate R&D Center, Schenectady, NY, USA and 2School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245, USA ( Received 27 December 2000; received in revised form 6 July 2001; accepted 6 July 2001 )

Abstract—Atomic force microscopy (AFM) is a relatively new tool that readily provides high resolution digitized images of surface features. AFM is used here to study the development of slip bands and protrusions in strain controlled fatigue tests on polycrystalline copper at 0.161 and 0.255% strain amplitudes. The average slip band heights at failure for both strain amplitudes conditions are comparable, implying that the growth of slip bands saturates at a specific height. A parameter, γirrev, is defined that is a measure of the local slip irreversibility at the surface and is applicable to any type of surface deformation feature, independently of the size of the fields of view. Thus, estimates of surface deformation developed in regions where fatigue crack nucleation is likely to occur can be obtained, from which a fatigue crack nucleation criterion is defined.  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Re´sume´—La microscopie a` force atomique (AFM) est un outil qui permet l’obtention d’images digitales a` haute re´solution d’e´le´ments de de´formation de la surface et leur analyse quantitative. Cette technique est utilise´e ici pour e´tudier le de´velopement de bandes de glissements et de protrusions lors d’essais de fatigue en de´formation controˆle´e sur du cuivre polycristallin a` des amplitudes de de´formation relative de 0.161 et 0.255%. Pour les deux conditions d’essais, la hauteur moyenne des bandes de glissement apre`s rupture e´tait comparable, ce qui signifie que la croissance des bandes de glissements sature a` une certaine hauteur. Un parame`tre, γirrev, est de´fini comme mesure de glissements locaux irre´versibles en surface. Ce parame`tre est inde´pendant de la taille du champ de vision et fournit des informations sur la distribution de la de´formation de surface le long de la section calibre´e, y compris des re´gions susceptibles de de´velopper des fissures.  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Atomic force microscopy (AFM); Copper; Fatigue; Slip bands; Crack nucleation

1. INTRODUCTION

Fatigue crack initiation and crack growth are the two main stages in the life of cyclically loaded structures. Following the formation of a macro-crack, crack growth and the occurrence of failure can be predicted by fracture mechanics. Even though engineering models are available for predicting crack initiation, damage evolution that leads to the formation of a macro-crack is difficult to predict, because no easily measurable parameter uniquely describes the state of damage during this stage. Most significant advances in the understanding of

† To whom all correspondence should be addressed. Tel.: +1-404-894-2888; fax: +1-404-894-9140. E-mail address: [email protected] (A. Saxena) ‡ Formerly Graduate Research Assistant in the School of Materials Science and Engineering at Georgia Institute of Technology

fatigue mechanisms were obtained from the study of dislocation arrangements in copper single crystals. Since fatigue crack nucleation is a surface phenomenon, fatigue damage is most likely better characterized by changes on the surface rather than by alterations in the interior of the material. Various concepts have been introduced to explain the formation of extrusions and intrusions based on either the glide of dislocations out of the crystal or the accumulation of point defects in the bulk [1–6]. When large clusters of extrusions and intrusions form, they are referred to as macro-PSBs and form either positive or negative protrusions, which have also been called encroachments. In polycrystalline copper, dislocation arrangements are similar to those that develop in single crystals. However, due to the constraint between grains, slip readily occurs on multiple slip systems and results in three-dimensional networks of dislocations even at low strain amplitudes. The detailed analysis of the

1359-6454/01/$20.00  2001 Acta Materialia Inc. Published by Elsevier Science Ltd. All rights reserved. PII: S 1 3 5 9 - 6 4 5 4 ( 0 1 ) 0 0 2 7 1 - 3

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mechanisms involved in fatigue deformation and crack nucleation is more difficult for polycrystalline materials than for single crystals. Theories by Essmann and colleagues [4, 5] and Tanaka and Mura [8] provide some insight on the interaction between PSB and grain boundaries leading to crack nucleation. These models predict that high applied strain amplitudes and large grain sizes favor the formation of intergranular cracks, and low strain amplitudes and small grain sizes promote the development of transgranular cracks in copper. However, when an environment is factored, intergranular cracking can also be rationalized at smaller grain sizes and low strain amplitudes because of the quasi-brittleness of the process associated with oxidation of copper. In fact, intergranular cracks have been observed in copper for a wide range of applied strain amplitude and grain sizes [5, 9–12]. The potential of high resolution devices capable of quantitatively describing the surface topography, such as scanning tunneling microscopy (STM) and atomic force microscopy (AFM), was discovered in the mid 1990s and used for accurately measuring the height of surface features in various materials [13–19]. Quantitative parameters were defined from these measurements to describe the state of surface deformation, such as the average slip distance [13] or the ratio of the average height to the average spacing between the surface features [14] and root mean square (RMS) height of the surface [16]. A direct relationship exists between these parameters and the average accumulation of surface plastic strain, which was exploited in these studies to determine the irreversibility of slip at the surface. Although, the use of average values of surface parameters is correct to describe the general state of surface damage, the extension of their use to the determination of the onset of the nucleation of fatigue cracks, which is by nature a heterogeneous process, is beyond the capabilities of such parameters. The objective of this study is to exploit the capabilities of AFM to accurately perform an analysis which takes into account length, height, count and orientation of the surface features over a significant portion of cyclically loaded copper specimens. This will provide the basis necessary for the development of a model describing the evolution of the topography that can be used for determining the onset of fatigue crack nucleation based on physical observations of the surface. The subsequent application of this model to cyclically loaded components has the potential to evaluate, from actual surface measurements, how much of their fatigue life is spent. 2. EXPERIMENTAL PROCEDURE

Strain controlled fatigue tests were performed on high purity C101 grade polycrystalline copper (OFHC) with an average grain size of about 40 µm (estimated by the mean intercept length method). Standard axial fatigue specimens were machined with

a gage diameter of 8.9 mm (0.35 in) and the surface was prepared by a combination of mechanical and chemical polish. The fatigue tests were performed at a constant strain rate of 0.005 s⫺1 and a stress ratio R = ⫺1, with strain amplitudes of 0.161 and 0.255% that yielded fatigue lives of 75 900 cycles and 6900 cycles with total cumulative plastic strains of 184 and 79, respectively. Several tests were performed at both strain amplitudes with interruptions at fractions of life of 0.5 and 0.9 for the 0.161% amplitude and at 0.25 for the 0.255% strain amplitude. Mechanical tests were followed by sectioning of the specimens for post-test observations by SEM and AFM. The AFM used in this study was an Aris-3500 with a long range scanning module METRIS-3070. Due to the curvature of the fatigue specimens and the limited vertical range of the AFM (about 7 µm), the fields of view (FOV) were limited to 30×30 µm2. About twenty scans were analyzed along the length and at several positions around the circumference of each specimen. The collection of AFM images was submitted to a verification procedure described elsewhere [20] to ensure that the area covered by the AFM scans was representative of the whole surface. If the statistical requirements were not met, that is, the area covered was deemed too small, additional AFM scans were obtained. 3. RESULTS AND DISCUSSION

In the following, the various surface features of fatigued specimens of copper are first identified by SEM and broad set of AFM measurements are also described. This allows us to place our results in context of previously published results on copper. Detailed AFM results follow this preliminary discussion and lead to the development of a formalism to quantitatively characterize the surface deformation. 3.1. Observation of surface damage The most common occurrence of surface deformation in cyclically loaded polycrystalline copper is in the form of slip bands regularly distributed within grains. An example of a typical arrangement of slip bands covering the entire width of a large surface grain in copper is shown in Fig. 1, taken from a specimen tested at 0.161% strain amplitude. In general, it was observed that the majority of the slip bands are extrusions and only a few intrusions are visible. From AFM measurements, the height of slip bands was found to vary between 30 and 900 nm, with an average height at about 200 nm. The highest slip bands were found exclusively at the latest stages of life, while small bands were present at all life fractions. This suggests that both the creation of new slip bands and their growth occur throughout the fatigue life. The measured heights of the extrusions in polycrystalline copper do not compare with those observed in single crystals, which were on average between 3 and 4 µm high [5, 7, 21, 22]. The smaller slip band sizes

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Fig. 1. Typical arrangement of slip bands at the surface of a large grain in cyclically loaded polycrystalline copper at 0.161% strain amplitude. (b) The magnified region within the square shown in (a).

in polycrystalline material is, however, not contradictory to the findings on single crystals, because the amount of material available below the surface to create the upset is far less than the entire specimen thickness and width that is available for single crystals. As previously mentioned, the development of slip bands in a given grain depends on the orientation of the active slip systems within that grain. However, if no active slip system is oriented towards the surface of the material or if the interaction of the several slip systems creates a three-dimensional arrangement of dislocations (e.g. cellular of labyrinth structure) which prevents the development of large scale slip along a single band, surface upsets cannot form and a smooth grain surface is observed. Fig. 2 clearly illustrates this phenomenon, where some grains have fully developed sets of slip bands while the neighboring grains have little or no surface deformation. In this study, slip bands were observed at both applied strain amplitudes, although towards the end of the fatigue life in 0.255% strain amplitude tests, protrusions became the dominant form of surface

damage with considerable bulging of the surface (about 1 µm) (Fig. 3). The width of the measured protrusions was between 10 and 20 µm, creating a height to width ratio smaller than measured in single crystals [7, 9]. This is consistent with findings on copper single crystals by Hunsche and Neumann [7] that showed a decrease in the height of the protrusions with a reduction of the thickness of the single crystals, explained by availability of less material to produce the upset. In the present study, the term protrusion is used when PSBs occupy the greater part of a grain and the density of the slip bands forming the PSB has reached a high enough level that no band of matrix is visible between the slip bands.

Fig. 2. Distribution of slip bands of various orientations and densities among the different grains, which shows regions where no surface deformation has occurred (⌬⑀/2 = 0.255%).

Fig. 3. AFM image of a protrusion in polycrystalline copper (⌬⑀/2 = 0.255%). The vertical scale is indicated on the right of the image in nanometers.

3.2. AFM analysis of surface features As mentioned earlier, slip bands are the primary form of surface damage and towards the end of life at the higher strain amplitude (0.255%) protrusions are also observed. In this section, these features are quantitatively analyzed and, since they all evolve from the same surface deformation mechanism, no distinction is made between them. For instance, a protrusion consisting of ten slip bands is treated as a group and individual dimensions of slip bands are computed as the average value for the protrusion. Three measures of surface damage evolution were considered that include: (a) the length of slip bands;

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(b) the number of slip bands; and (c) the height of slip bands. Figure 4(a) provides the average length of slip bands at a strain amplitude of 0.161% at various fractions of life and shown as a function of the angle that the slip bands form with the loading axis. This representation was chosen to highlight the possible influence that the orientation of the trace of the slip systems with the loading axis might have on the surface features. One notices from Fig. 4(a) that the average length of slip bands does not vary substantially with orientation or with the number of cycles. This correlates with the earlier observation that slip bands cover the entire width of a grain and hence cannot grow further. This, along with the fact that slip bands may be truncated by the edge of the FOV and appear shorter than they actually are, restricts the capacity of the average length of slip bands to be used as a representative parameter of surface deformation in polycrystals. On the other hand, the number of slip

Fig. 4. Distribution of (a) the average slip band length, (b) the number of slip bands and (c) the average height of slip bands in polycrystalline copper tested at 0.161%.

bands per unit area is not subject to the same drawbacks and is therefore a suitable surface damage parameter, as shown in Fig. 4(b). For additional information on the evolution of surface deformation, the average height of the slip bands as a function of their orientation with the loading axis is also computed, Fig. 4(c). The behavior for the strain amplitude of 0.255% was similar and was therefore not included in the figures. From Fig. 4(b), one notices a strong influence of the angle of orientation on the number of slip bands. At both strain amplitudes, the number of slip bands is much greater at angles between 30° and 80° with the loading axis. Although these angles do not correspond to the actual orientations of the active slip systems, they are consistent with favorable slip orientations for the fulfillment of the Schmid law. Indeed, the trace of a plane oriented at 45° with the longitudinal axis of a cylinder assumes a range of orientations comprised between 45° and 90° around the circumference. This explains the orientations of slip bands ⬎45°. On the other hand, slip bands oriented closer than 45° to the loading axis do not belong to a favorably oriented slip plane and must therefore be the result of constraint from neighboring grains that locally disturbs the local strain and stress fields. At 0.161% strain amplitude, the number of slip bands is not measurably affected by the number of cycles between N/Nf = 0.5 and 0.9, but increases between N/Nf = 0.9 and 1. The increase in the number of slip bands between half-life and failure is only 30%, which may imply that the multiplication of slip bands is not the leading mechanism of fatigue damage evolution. On the other hand, the height of the slip bands is affected more significantly by the increase in number of cycles, Fig. 4(c). At half-life, an average height of about 180 nm is measured for all orientations. It is noted from Fig. 4(c) that slip bands with orientations between 45° and 90° with the loading axis reach heights of about 250 nm at N/Nf = 0.9 and up to 400 nm at failure. This tends to show that the height of the slip bands is a good indicator of fatigue damage evolution. This is justified by the theory that slip bands create sharp discontinuities at the surface that cause the stress concentration responsible for crack nucleation and the higher the upset, the larger the stress concentration. However, the transgranular cracks that one would expect from these stress concentrations, and are observed in single crystals at the edges of PSBs, were not present here. On the other hand, based on surface observations, it is believed that critical conditions that lead to crack nucleation arise from the development of a large number of slip bands with significant height in a grain adjacent to a grain with little surface strain. It was shown above how important the orientation of the active slip system relative to the load axis is on the development of slip bands and adjacent grains may have significantly different lattice orientations, leading to a severe mismatch in the surface strains between the neighboring

CRETEGNY and SAXENA: SURFACE DEFORMATION DURING FATIGUE

grains, for example, Fig. 2, that is accompanied by residual stresses. These, in turn, favor crack nucleation along the grain boundary according to the mechanisms proposed by Essmann et al. [4, 5] or Tanaka and Mura [8]. This issue is further discussed below. At 0.255% strain amplitude, the height of the slip bands was not significantly affected by the orientation of the slip bands relative to the loading axis for angles of 30° and beyond. The average height at N/Nf = 0.25 is 108 nm, which shows that a significant growth occurs early in the fatigue life, and the height then increases to 247 nm at failure. The latter compares well to the average height of 271 nm obtained at failure for the 0.161% strain amplitude. That value is significantly smaller than the several µm measured on single crystals, a fact that actually supports the theory by Essmann et al. [4] that relates the surface roughness to point defect formation in the bulk, which is limited by the size of the grains in polycrystals. As a result, the growth of slip bands in polycrystalline copper with grains on average 40 µm in diameter seems to saturate at a height of about 250 nm. The number of slip bands per unit area at the 0.255% strain amplitude on the other hand is affected by the slip band orientation and is about three times larger than at 0.161% strain amplitude. The significantly larger number of slip bands at the higher strain amplitude is due to two factors that are inter-related. First, because of the higher applied stress and the disruption of the stress field by adjacent grains that are undergoing deformation, the resolved shear stress of slip systems not ideally oriented may reach levels high enough to cause plastic deformation, hence increasing the number of active slip planes. In addition, the formation of protrusions as shown in Fig. 3 occurs at this strain amplitude and their high slip band density increases the number of slip bands. The characteristics of protrusions are discussed next. 3.3. Protrusions versus slip bands Figure 5 compares total count and length of standalone slip bands versus those that are part of protrusions at a strain amplitude of 0.255%. A total of 638 and 808 slip bands at N/Nf = 0.25 and 1, respectively, were measured. Early in the fatigue life, surface upset occurs principally in the shape of slip bands, with protrusions covering only 10% of the total surface upset in count and length, Fig. 5(a). The presence of protrusions at this stage shows that their formation is uniquely related to the magnitude of the strain amplitude and not, for instance, to the cumulative plastic strain or the height of the slip bands themselves that are actually larger at the lower strain amplitude which did not produce any protrusions. At failure, the proportion of protrusions to slip bands is 1:1 in terms of both number and length of slip bands, Fig. 5(b). However, protrusions within them contain a much higher density of slip bands than regular arrangements of slip bands and the area

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Fig. 5. Comparison of the characteristics of protrusions and stand-alone slip bands at (a) quarter-life and (b) failure (⌬⑀/2 = 0.255%). The charts on the left indicate the relative distribution of protrusions and slip bands as a function of the angle with loading axis and, on the right, the fraction of the total count and length of each feature is shown.

covered by protrusions is hence overall smaller than the surface covered by an equivalent number of slip bands. By combining the results of the proportion of slip bands that are not part of protrusions and the total number of all slip bands per unit area, one derives a count of 0.9×0.035 = 0.032 slip bands per µm2 at N/Nf = 0.25 and only 0.5×0.050 = 0.025 slip bands per µm2 at failure for the slip bands that are not part of protrusions, while the total number of all slip bands per unit area actually increases from 0.035 to 0.050 µm⫺2. This decrease in the number of slip bands that are not part of protrusions shows that protrusion formation is an out-growth of a continuous process of multiplication of active slip planes within existing slip bands and not the growth of closely packed slip bands due to cyclic deformation in a grain. In addition, the fact that the distribution of the slip band length in and out of protrusions follows closely the count distribution, Fig. 5, indicates that protrusions are formed in grains of any size. Indeed, if protrusion formation was favored in larger grains and the number of slip bands in and out of protrusions was the same, the fraction of the total length of slip bands within protrusions would be proportionally larger, which is not the case. The orientations assumed by the protrusions are within the standard range of 30–80° that has been observed for surface features for fatigued copper in this study. Very few protrusions have orientations ⬍45° with the loading axis, whereas an appreciable quantity of slip bands assumed orientations as low as 30°. This implies that the formation of slip bands may be enhanced by the effect of constraint from neighboring grains and occur at various orientations with the loading axis, but protrusions develop only on slip planes that have a favorable orientation relative to the loading axis to satisfy the Schmid law.

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3.4. Slip band spacing The analysis of the spacing between slip bands reveals an influence of the applied strain amplitude on the distribution of slip bands at the surface, Fig. 6(a). Slip bands that are not part of protrusions are separated on average by about 1.6 µm at the higher strain amplitude and 2.3 µm at the lower strain amplitude. This difference of about 25% does not change throughout the fatigue life, as the average slip band spacing remains constant at both strain amplitudes. One has to note that these values correspond to averages and that spacings vary significantly from one region to another. At the 0.255% strain amplitude, the surface features consist of either stand-alone slip bands or slip bands that form protrusions. Obviously, the spacing between them is strongly affected by this factor as it drops from over 1.6 µm outside to less than 1 µm inside protrusions, Fig. 6(b). Here again, the number of cycles does not influence the spacing between slip bands, as only a slight decline is measured throughout the fatigue life. The smaller spacing between slip bands illustrates the strong contribution of protrusions towards the accumulation of dense surface deformation. 3.5. Crack nucleation Copper specimens cyclically loaded at 0.161 and 0.255% applied strain amplitudes show systematic

crack nucleation at the grain boundaries as shown in Fig. 7. In almost every case, cracks initiate at the boundary between one grain with considerable surface upset and another grain that does not show much trace of surface deformation. Intergranular cracks at various strain amplitudes and grain sizes were already reported by other authors mentioned earlier [5, 9–11]. Although not explicitly mentioned in those articles, most of the micrographs of interest showed that intergranular crack nucleation occurred at the boundary between two grains with significantly different amounts of surface deformation, as is the case in the present study. Incidentally, a study by Lin et al. [23] showed that the propagation of slip bands from one grain to another offers an alternative to crack nucleation along grain boundaries by relieving excessive stress at that location. They demonstrated that this type of stress relief occurs only when the adjacent crystals have favorable relative orientations. These findings support the results from the current study in which cracks form along grain boundaries between grains that have a strong mismatch in their respective amount of surface deformation. In conclusion, the mismatch in the deformation of two neighboring grains seems to be the driving force for the nucleation of an intergranular crack. 3.6. Derivation of the irreversible surface deformation parameter The above characterization by AFM of the features that developed at the surface of copper fatigued specimens consisted of count per unit area and average length and height. The emphasis of this section is on developing a single surface parameter that encompasses the diversity of the above measurements and can be used to determine a criterion for crack nucleation. For general applicability, a parameter that characterizes surface deformation must be independent of the type of surface features (e.g. slip bands, extrusions, protrusions, streaks, etc.) and the size of FOV. This is essential because the dimensions of the features of fatigue damage can range from a few nanometers up to several micrometers and different sizes of FOVs are then needed to obtain statistically significant results. Let us first consider the simple case of a single crystal oriented for single slip, Fig. 8. If pulled monotonically in tension, the crystal experiences an irreversible deformation, δ, that can be measured by the surface step height and is directly related to the applied plastic strain amplitude, γ, by the following relationship g⫽

Fig. 6. Comparison of (a) the spacing between slip bands that are not part of protrusions for 0.161% and 0.255% strain amplitude and (b) the spacing between slip bands at the 0.255% strain amplitude that stand alone (not part of protrusions) and that are part of protrusions.

d L

(1)

where L is the gauge length or length of the crystal in this case. The magnitude of the strain may vary if a different set of coordinates is chosen, or if the

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Fig. 7. Typical intergranular cracks that are formed in polycrystalline copper cyclically loaded at various strain amplitudes. Cracks are usually formed at the boundary between one with grain with considerable surface upset and another with much less visible deformation.

冘 n

girrev ⫽

Fig. 8. Final crystal shape of a single crystal oriented for single slip after (a) one half cycle and (b) one cycle if slip occurs on an adjacent crystallographic plane in the reverse loading direction.

emergence angle of the slip band changes. The above definition of strain based on d, the normal displacement of the surface deformation, was chosen because it is readily measurable by AFM. The use of an adequate geometric factor may improve the accuracy of g, but the factor will most likely remain constant throughout the fatigue life. Therefore, its omission, as implied by equation (1), will not affect monitoring of the relative changes of irreversible surface deformation with fatigue cycles. In the same crystal, if slip occurs in the reverse loading direction on a different but parallel slip plane, the final crystal shape may resemble that of Fig. 8(b) and the residual plastic strain is then twice the amount given in equation (1), even though the crystal seems to have returned to its original shape at the macroscopic scale. According to Mura [24], slip occurs on adjacent layers of materials in the forward and reversed loading directions due to accumulation of back stress, which contributes to the formation of surface upset (e.g. slip bands) separated by regions with no apparent slip. Subsequently, if n slip bands are created by cyclic loading, the irreversible plastic strain is given by

2|di|

i⫽1

L

.

(2)

It is important to note that γirrev is an average of the measurable irreversible deformation experienced by the material over the gauge length L and not the amount of strain within each slip band. Here, slip is assumed to occur in the shape of slip bands, but the same concept is applicable to other types of features that create a surface step. The theory applies to all forms of surface deformation features, because its concept is simply based on the permanent deformation of the surface that is the result of irreversible plastic strain. The above definition of normalized surface deformation is different from the one used by Harvey et al. [14], which was given by the ratio of the average height to the average spacing between slip bands. The latter is actually a measure of the average strain contained within the slip bands, while equation (2) is a measure of normalized surface deformation. When regular arrays of slip bands develop at the surface both definitions yield similar results. However, the definition by Harvey et al. cannot be applied to a case where isolated slip bands form, because it requires a measure of spacing between surface features, while a measure of surface deformation is still possible with the definition given by equation (2). Consequently, the advantage of the latter definition is its potential to analyze regions with little or no surface deformation, which permits the analysis of the distribution of deformation over the entire sample surface and not only of selected FOVs that contain surface deformation. One significant difference between single crystals and polycrystalline materials is that a slip band does not cross the full width of a specimen and, therefore, its contribution to the total surface deformation not only depends on its height but also on its length, which must be included in the measure of surface

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deformation. For example, Fig. 9 shows a FOV of width W and length L where a single slip band of height δ developed across the width a of a grain at an angle α with the loading axis. To account for the fact that the slip band does not always cover the complete width of the crystal, it is proposed to normalize the slip band length by the width of the FOV in the direction parallel to the slip band, or anorm Min

a W L , sin a cos a

冉

冊

(3)

where α is the angle between the slip band and the loading axis and Min takes the smallest value listed as arguments. Now, the amount of the normalized surface upset in one FOV containing n slip bands is obtained by the summation of the contribution of each slip band, that is:

冘 n

dnorm FOV ⫽

2|di|anorm i

(4)

i⫽1

If a constant normalizing factor had been chosen in equation (3) instead of the width of the FOV, such as the grain size, all FOVs would have to be the same size to compare results from equation (4) but, by normalizing relative to the width of the FOV, it is possible to perform the analysis with different size FOVs. This is useful if different surface features develop at various length scale (e.g. [20]) and different magnifications are required to fully analyze the surface deformation. To convert the surface upset in equation (4) to a normalized surface deformation (strain), one needs to divide the amount of normalized surface upset by the entire gauge length. In other words, when considering m FOVs, it becomes

Fig. 9. Example of a FOV containing several grains and a single slip band of height d and length a. The normalization of the slip band length is done relative to the width of the FOV in the direction parallel to the slip band, which can be determined from the angle a with the loading axis.

冘 冘 m

gnorm j

girrev ⫽

j⫽1

(5)

m

Lj

j⫽1

where the denominator represents the total length of the FOVs, which is effectively the gauge length. The above definition of γirrev is not affected by the size of the FOVs, because the parameter is normalized relative to both the width W and the length L of the FOV. However, an appropriate choice of size for the FOVs is critical to reach the best combination of resolution and range necessary for accurate measurements of the surface features over a surface area that is as large as possible. Indeed, if a small size is selected, more FOVs will be necessary to meet the aforementioned validation criterion that ensures that the measurements are representative of the entire surface. Conversely, the resolution is lowered on large FOVs and small surface features may not be included in the measurements. 3.7. Distribution of irreversible surface deformation Due to the high level of heterogeneity of the surface roughening process, average values of the irreversible surface deformation are not representative of the regions with high levels of deformation. Therefore, it is virtually impossible to predict experimentally the location of these regions and monitor changes in topography at the precise location where the fatal crack will nucleate. However, even though the local maximum of γirrev cannot be directly measured, the statistical method of describing the heterogeneity of surface slip irreversibility described below is capable of estimating its magnitude and thus has the potential for detecting the onset of fatigue crack nucleation. This concept, which is substantiated by experimental data, assumes that surface damage resulting from cyclic deformation occurs in a random fashion, which implies that only a few regions have extreme amounts of surface upset (high or low) and the majority of the surface has a local amount of damage that is close to the sample average. As a result, by statistically characterizing the distribution of surface deformation in a large number of FOVs, it is possible to estimate the amount of damage in the region with maximum local surface damage, even though that specific region is not directly included in the FOVs. Figure 10 is an example of the distribution of γirrev at a quarter of the fatigue life and failure for the specimens tested at 0.255% strain amplitude. The horizontal axis represents ranges of γirrev and the respective fractions of the surface that have reached these ranges are indicated on the vertical axis (i.e. the sum of all the bars for a given condition amounts to 100% of the surface). The distribution of the surface deformation of all specimens resembles those shown in Fig. 10, which are analogous to a Gaussian (normal)

CRETEGNY and SAXENA: SURFACE DEFORMATION DURING FATIGUE

Fig. 10. Distribution of the irreversible surface deformation on the surface of fatigue specimens tested at a strain amplitude of 0.255%. These plots are based on the irreversible surface deformation measurements collected from the multiple fields of view analyzed on each specimen.

distribution about the average value of the irreversible surface deformation. Consequently, the data can be equivalently represented with a continuous normal distribution function, Fig. 11(a), which has the advantage that it can be easily integrated and has an area

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under the curve equal to one, corresponding to 100% of the specimen’s surface. Other statistical functions may also be used to characterize the distribution of surface deformation. For example, applying a Weibull distribution to the current set of data did not noticeably affect the results from this analysis. Additional measurements would be necessary to determine with certainty the type of distribution that surface strains assume during uniaxial cyclic loading of a polycrystalline material. From the plots of Fig. 11(a), two main observations are made on the evolution of the distribution of the irreversible surface deformation. First, the average value of the normal distribution curves increases with the number of cycles, which is consistent with the results from the previous section on the average values of irreversible surface deformation and, secondly, the spread of the curves becomes larger as the number of cycles increases. The latter is due to the development of substantial surface upset in some regions, while others do not experience much perturbation of their surface topography due to unfavorable local crystallographic orientations of the slip planes and/or constraint effects between grains. The regions with significant increase of the surface upset eventually become zones of instability where crack nucleation is possible. Since the normal distribution curves of Fig. 11(a) reveal the extreme values of irreversible surface deformation reached in some regions, these plots directly provide information on the advancement of surface upset in the highly deformed regions. Thus, this data can be used to predict when the material reaches a critical state that will trigger the nucleation of a fatal crack. This analysis is however better performed on the cumulative version of these plots shown in Fig. 11(b), which is simply the integral of the normal distribution curves. The vertical axis on the left indicates the portion of the surface that contains amounts of irreversible surface deformation between zero and the amounts indicated on the horizontal axis of the graph. Conversely, the vertical axis on the right provides the portion of the surface that has developed amounts of irreversible surface deformation larger than the amount indicated on the horizontal axis. With the information provided in Fig. 11(b), one can assess the evolution of the deformation distribution over the specimen surface, including in the regions where extreme values of surface deformation have developed. 3.8. Crack nucleation criterion

Fig. 11. Representation of the distribution of the irreversible surface deformation at the surface of fatigue specimens using (a) a normal (Gaussian) distribution function and (b) a normal cumulative distribution of the irreversible surface deformation. In (b), the right axis corresponds to one minus the left axis, and the vertical dotted lines represent the critical values of the irreversible surface deformation.

The previous AFM and SEM observation showed that fatigue cracks nucleate in locations where the surface deformation is high. Since γirrev is capable of determining the maximum local values of surface deformation, it is ideally suited as an indicator of the probability of fatigue crack nucleation. One may argue that the magnitude of surface upset is not always the driving force for crack nucleation. For

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example, Pola´ k et al. [21] argue that, in copper single crystals, crack nucleation is driven by the creation of point defects in the bulk and occurs through the linking of intrusions that are formed at the edges of extrusions. In this mechanism, γirrev is not a direct measure of damage that leads to crack nucleation but, because the forms of damage during fatigue (e.g. point defect creation and development of surface upset) occur concurrently, monitoring one provides a good indicator of the other, and thus, the overall state of fatigue damage. Therefore, γirrev is a relevant parameter for monitoring all types of fatigue damage and a criterion for crack nucleation may be defined in terms of a critical local value of γirrev necessary for crack nucleation. This critical value of γirrev can be determined from AFM measurements by measuring the value of γirrev of FOVs that contain a crack nucleus. Once known, this value can be traced on the normal distribution curves of the irreversible surface deformation, as shown by the vertical lines shown on Fig. 11(b). It was found that the levels of local surface deformation necessary to trigger the nucleation of a fatigue crack are different between the high and the low strain amplitude tests. The criterion described above does not act like a “failure or no failure” switch that would imply that a crack is always nucleated once the critical value of surface deformation is achieved in a specific location. On the contrary, Fig. 11(b) clearly shows that a certain portion of the surface may develop levels of surface deformation well beyond the critical value without nucleating a fatal crack, as observed in all failed specimens and in some specimens tested to 90% of the life. The actual fraction of the surfaces that has reached or exceeded the crack nucleation criterion is indicated by the ordinate of the intersection of the distribution curves with the vertical line that specifies the criterion. Now, if one postulates that the likelihood of nucleating a crack increases proportionally with the fraction of the surface over which the critical level of surface deformation is exceeded, the distribution curve of the irreversible surface deformation provides an estimate of the probability for crack nucleation. For example, when several identical fatigue experiments are performed, failure does not necessarily occur after the same number of cycles because of the individuality of each specimen and the ability to predict failure at that point becomes an exercise of probability. According to the criterion defined here, the probability of failure is null as long as γirrev has locally not exceeded its critical value at least in some regions. On the other hand, as regions with favorable lattice orientations for formation of surface upset develop levels of surface deformation beyond that critical amount, the nucleation of a fatal crack becomes possible, and a measure of the probability that the specimen has nucleated a fatal crack may be directly read from the plots of Fig. 11(b). According to the above theory, the probability of failure at N/Nf values of 1 in this study was approxi-

mately 30%. Ideally, these experiments should be repeated with in situ AFM measurements where the progression of surface damage can be followed on the same specimen at several N/Nf values. 4. CONCLUSIONS

From the SEM and AFM analyses of the surface deformation and crack nucleation behavior of polycrystalline copper tested at 0.161% and 0.255% strain amplitude, the following conclusions were drawn. A strong influence of the orientation of the slip bands was observed at both strain amplitudes, as the number of slip bands that have an angle with the load axis between 30° and 80° was much larger than for other orientations. Both strain amplitudes yield similar average slip band height at failure of about 250 nm, which indicates that the growth of slip bands saturates at this height in polycrystalline copper with a grain size of 40 µm. Protrusions were present at the surface of copper specimens tested at 0.255% strain amplitude and developed in grains of various sizes, from a few to several hundred µm. They develop throughout the fatigue life, with the fraction of slip bands that are part of protrusion increasing from 10% at N/Nf = 0.25–50% at failure. Protrusions appear to form by multiplication of active slip planes within existing slip bands. Intergranular crack nucleation occurs at both strain amplitudes. Crack nuclei systematically develop at grain boundaries between a highly deformed grain and one with little evidence of surface upset. The strain mismatch between the two grains seems to be the driving force for intergranular crack nucleation. The definition of the surface deformation parameter, γirrev, was formulated to examine the distribution of surface deformation over the surface of any material, independently of the type of surface upset. This approach provides a means to estimate the maximum levels of surface deformation developed by critical regions where fatigue crack nucleation is likely to occur. A criterion for crack nucleation was defined as follows: fatigue crack nucleation is possible once the material develops, on a local scale, surface deformation of a magnitude that exceeds a critical value. This critical value can be determined from AFM images that contain crack nuclei. Acknowledgements—The authors are grateful for the financial support of the Office of Naval Research under the M-URI Program “Integrated Diagnostics” (Grant No. N00014-95).

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