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The in situ measurement of the effect of plastic deformation on the magnetic properties of steel
Journal of Magnetism and Magnetic Materials 187 (1998) 353—365
The in situ measurement of the effect of plastic deformation on the magnetic properties of steel Part II — Permeability curves J.M. Makar*, B.K. Tanner Department of Physics, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, UK Received 20 January 1998; received in revised form 19 March 1998
Abstract We report measurements of the bulk magnetic properties of pearlitic steels recorded in situ during plastic deformation. The low-field total differential permeability was found to initially increase with increasing elastic tensile stress applied in the direction of the magnetic field. However, it was found to reach a maximum value well before the yield point and decreased rapidly after reaching that maximum. No qualitative change of behaviour was observed at the yield point. The initial increases in total differential permeability are attributed to the stress-induced removal of domains with magnetisation vectors that are approximately perpendicular to the field and stress direction. The subsequent decreases in total differential permeability are attributed both to stress-induced changes in the magnetic anisotropy and to new pinning sites generated during the plastic deformation process. The low-field reversible differential permeability appears to show similar behaviour with increasing stress, although the maximum value for this type of measurement is not as well defined. Measurements of the high-field total differential permeability decreased monotonically with stress in both the pre-yield and plastically deformed regions. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Steel; Stress; Permeability — differential
1. Introduction Curves of the permeability as a function of applied field are frequently used as tools in analysing magnetic behaviour. While the curve produced by the permeability (k"k~1B/H) is not particularly 0 useful, the differential permeability (k"k~1dB/dH) 0 * Corresponding author. Present address: Institute for Research in Construction, National Research Council of Canada, 1500 Montreal Road, Ottawa, Ontario, Canada K1A 0R6.
and the equivalent susceptibility (s"dM/dH" k!1) are important both for magnetic modelling [1,2] and interpreting the magnetisation process [3—5]. The differential nature of the curve highlights behaviour that is not readily seen in the hysteresis loop and assists in the interpretation as to which magnetisation mechanism is dominant at a particular field or magnetisation. In the remainder of this paper, the term permeability always refers to the differential permeability unless otherwise stated.
0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 1 4 1 - 3
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Permeability curves have an advantage over hysteresis loops in that they can be readily divided into reversible and irreversible components [3,6—8]. Magnetic materials show both types of behaviour, with the irreversible magnetisation changes coming from irreversible magnetisation vector rotations and domain wall motion past pinning sites, while the reversible changes are due to reversible magnetisation vector rotations and domain wall bowing around pinning sites. It is impossible to separate these two components on the hysteresis loop, but the different types of permeability can be measured by the process of producing a small field reversal at the point of the magnetisation curve under investigation [6]. The slope of that curve gives the reversible permeability, with the irreversible permeability being calculated by subtracting the reversible value from the total permeability measured at the same point on the hysteresis loop [7]. Examination of the irreversible and reversible components not only allows the determination of the relative importance of the two classes of magnetisation methods, but can also give information about the nature of pinning sites and how stress affects the magnetic properties of the material in question. Previous work on 2% Mn pipeline steel [7,8] has examined reversible and irreversible permeability variations under elastic stresses for initial magnetisation curves, major hysteresis loops and minor hysteresis loops. These results showed that both types of permeability increase under uniaxial tension applied in the direction of the applied magnetic field at low magnetisations and decrease at higher magnetisations above the knee of the hysteresis loops. The opposite effect was seen for compression. It was also noted that differences between the upper and lower branch permeabilities behaved in a fashion opposite to the overall values. Studies of the effects of stress on the total permeability of magnetic materials have been reported [9,10] that show similar results for steel to those described above. Total differential permeability results have also been reported for steel samples under conditions of residual stress [11,12]. However, to the best of the authors’ knowledge, no comprehensive studies of the effect of stresses that approach and exceed the yield point on magnetic materials appear to have been undertaken.
The work presented here is the second of a pair of papers that discuss changes in the magnetic properties of similar specially prepared iron—carbon alloys during the plastic deformation process. While the first paper [13] examined hysteresis loop behaviour, including the coercive field and remnant magnetisation as well as the corresponding magnetostriction, this paper examines the behaviour of the corresponding reversible and total relative differential permeability curves. Initial measurements were performed under conditions of elastic tension up to the yield point. These were followed by in situ measurements made on the same sample in the plastic deformation region of each sample’s stress— strain curve.
2. Experiment An extensive description of the experimental samples and apparatus has been presented in the first paper [13]. They will only be summarised here, while the procedures that are particularly relevant to the data presented below will be given in detail. A schematic diagram of the apparatus and sample dimensions have also been published elsewhere [9]. 2.1. Samples Five different iron—carbon alloys were used for the experiments reported here. These samples were cast by the University of Sheffield Materials Advisory Centre in the form of 5 kg ingots with carbon contents of 0.003, 0.15, 0.43, 0.68 and 0.86 wt%. There was less than 0.003 wt% sulphur in the each ingot and no other significant constituents. The ingots were cooled, reheated and then extruded as rods before being air cooled and sawn into 43 cm long sections. Many of the rods were bent as a result of the extrusion process but the straightest sections were retained for measurements in the as-delivered condition. The remaining sections were straightened mechanically and heat treated to remove residual stresses. All rods were machined before mounting so that instrumentation could be attached to the sample in the plastic deformation zone. While
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initial measurements were made on each of the sample materials before heat treatment, complete sets of magnetic measurements were made on 0.43 wt% C samples without heat treatment and samples with all five carbon contents after heat treatment. 2.2. Apparatus The instrumentation on each sample consisted of a strain gauge, which was used to measure the strain on the sample (e) as well as the magnetostriction (j), a coil that was used to measure the flux density (B) experienced by the sample, and a Hall probe that was used to measure the magnetic field (H) applied to the sample. The strain gauge, coil and Hall probe were each attached to the sample in the region where plastic deformation occurred. One leg of a Wheatstone bridge incorporated the strain gauge, with the other three legs of the bridge being made from ‘dummy’ strain gauges mounted on a piece of steel located well away from the electromagnet. The outputs from the Wheatstone bridge and the Hall probe were each amplified and filtered before being acquired by a PC mounted data acquisition board. The coil was connected to an integrating fluxmeter and the outputs from the fluxmeter also connected to the data acquisition board. A ‘C’ shaped electromagnet with slots cut in each end to accept the samples was used to magnetise them so that the field was applied parallel to the sample length. A computer controlled bipolar power supply was used to power the electromagnet, which was designed to be mounted from the Instrom testing machine so that measurements could be made while the sample was under applied elastic or plastic stresses. The data acquisition system was also used to measure the signals from the Instrom load cell, with the data being converted into stress (p) values based on the crosssectional area of the sample at any given point on its stress/strain curve. 2.3. Experimental procedure The experimental procedure depended on whether or not the measurements were to be made
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under stress. Measurements made under conditions of zero stress were performed outside the Instrom to ensure that no biaxial stress was present. For the measurements made under stress, the instrumented sample was mounted in the Instrom and a sufficient load applied to the sample to raise it to the desired stress level, at which point the Instrom was turned off. If the sample was below the yield point, no further changes in the strain or stress experienced by the sample took place. However, if the samples were above the yield point, a relaxation effect was observed in which the stress dropped slightly and the sample continued to yield, increasing its strain. Typically a period of between 3 and 6 hours passed before the deformation process halted. This condition approximates that seen in practice in engineering structures, where a constant load has caused localised plastic deformation. Note that this is a very different situation to the relaxed state of compressive strain that exists once a load that has produced plastic deformation has been removed. Once the relaxation process was completed, the electromagnet was positioned against the sample in such a manner that biaxial stresses were minimised. Whether or not the sample was mounted in the Instrom machine, the Hall probe was calibrated for each individual sample and stress level before taking magnetic measurements by driving the sample well into saturation and measuring the slope of the B—H curve in that region. Boundary conditions require that the component of the field immediately inside the surface of the sample and parallel to it be the same as that immediately outside it, so a comparison of the slope to the expected value (the permeability of free space) gives a correction factor for subsequent field measurements [14]. Initial magnetisation curves and near-saturation major hysteresis loops were recorded for each sample material, including both heat treated and un-heat treated materials. Small field reversals were made at each point measured in order that reversible permeability values could later be calculated [3]. These field reversals ranged in size from 10 A/m for the 0.003 wt% C sample to 50 A/m for the 0.86 wt% C sample. Once measurements were taken under a particular level of stress beyond the yield point, the sample
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was either subjected to increased stress or the applied stress on the sample was removed so that a measurement under conditions of residual stress could be made before higher levels of tension were applied. The choice of procedure did not affect the measurements made under applied stress. After the magnetic measurements were complete, the resulting hysteresis loops were analysed to produce values for the total (k ) and reversible (k ) 505 3%7 relative differential permeability. The former value was calculated directly from the slope of the hysteresis loop measurements, while the latter was calculated from the slopes along the small field reversals. Although the irreversible component of the differential permeability (k ) was also deter*33 mined by subtracting k from k , the results are 3%7 505 not shown. The value of k in these samples is 3%7 considerably smaller than that of k , so that a plot 505 of k appears to be nearly identical to that of k . *33 505 The resulting data were then analysed to determine the maximum values of k and k . These 505 3%7 data points are plotted against stress in the figures that follow.
3. Results Figs. 1 and 3, respectively show plots of the response of the total and reversible relative differential permeabilities to stress. Fig. 1a and Fig. 3a shows the behaviour of the 0.003 wt% C sample, Fig. 1b and Fig. 3b that of the 0.15 wt% C sample, while Fig. 1c and Fig. 3c shows that of the 0.86 wt% C sample. The results from the remaining samples (0.43 wt% C both before and after heat treatment and 0.68 wt% C) are similar to those shown in Fig. 1c and Fig. 3c. Note that the quoted uncertainty values are for the central area of the figures where the hysteresis loop changes very rapidly. The results at the extremes of the figures are much less uncertain. Each of the plots in Fig. 1 shows similar results, with k initially increasing with increasing stress, 505 followed by a decrease in value that starts well before the yield point of the material. There are, however, significant differences in the fractional changes in the permeability values. In Fig. 1a the 0.003 wt% C sample has the highest k values of 505
all the samples. The application of elastic stress increases this permeability by about one third before it starts to decrease again. The results from the 0.17 wt% C sample in Fig. 1b show significantly lower k values, but the total permeability more 505 than doubles in value with increasing stress before starting to decrease as the stress approaches the yield point. Finally, the results of Fig. 1c show both lower overall permeability values and a much smaller increase in the permeability with increasing elastic stress. Fig. 2 shows the behaviour of the maximum total differential permeability measured for each sample with increasing stress. All samples show a characteristic pattern of an initial increase in the magnitude of k at low stress levels, followed 505 by a decrease in value with further increases in stress. The yield points of the materials are marked by the solid diamond shapes and have been published in the previous paper [13]. Note that in each case this decrease began well before the yield point of the material and that there does not appear to be a qualitative change in behaviour of the samples at the yield point. The reversible differential permeability behaviour in Fig. 3 is not as consistent as that shown in Fig. 1 for the total differential permeability. The results for the 0.003 wt% C and 0.86 wt% C samples both show an initial increase in k 3%7 with increasing stress, followed by a decrease at higher stress levels. However, the 0.15 wt% C sample only shows decreasing values of k . 3%7 A similar pattern is illustrated in Fig. 4, where the maximum k values initially increase with stress 3%7 (p) before decreasing for the 0.003 wt% C, 0.68 wt% C and the after heat treatment 0.43 wt% C samples, decreases continuously for the 0.17 wt% C sample, and initially does not appear to vary within the uncertainty for the 0.68 wt% C and the before heat treatment 0.43 wt% C values. A second significant difference between the two types of permeability measurements is that, with the exception of the 0.15 wt% C sample, the first decreases in k appear to be taking place at or 3%7 near the yield point. Before yield, k is either 3%7 increasing or maintaining a constant value, while after yield further increases in the stress applied to the sample correspond to increasing reductions in the parameter.
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Fig. 1. The total relative differential permeability (k ) as a function of magnetisation for different values of applied stress. The curves 505 with maximum on the left- and right-hand side of the graph are produced from the upper and lower branches of the hysteresis loop, respectively. (a) 0.003 wt% C sample. Magnetisation $5000 A/m, k $60 maximum; (b) 0.15 wt% C sample. Magnetisation 505 $5000 A/m, k $20 maximum; (c) 0.86 wt% C sample. Magnetisation $5000 A/m, k $20 maximum. 505 505
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Fig. 1. Continued.
4. Discussion In Part I of this pair of papers [13] the behaviour of the coercive field (H ) and remnant # magnetisation (M ) was investigated. The re3 sults presented there are also important in interpreting what happens to k as the samples ap505 proach and exceed the yield point. H was found to # be constant or initially to decrease with increasing stress, but to increase with further increases in stress above the yield point. This increase was found to start at or just before yield and was attributed to increasing numbers of pinning sites produced by the plastic deformation process, with the number of pinning sites being found to increase such that N"kp4, where N is the number of pinning sites, p is the applied stress and k is a constant. M showed 3 a very different behaviour, with initial increases in value at low levels of applied tension, followed by decreasing values with further increases in stress. The decreases in M were found to begin well 3
before yield had taken place and no qualitative changes in behaviour were seen at yield. Magnetostrictive evidence also showed that all of those domains that had magnetisation vectors pointing more closely towards the perpendicular to the applied field (henceforth referred to as ‘near-perpendicular domains’, with those that point more closely to the direction of the field being referred to as ‘near-parallel domains’) were eliminated from the sample as the applied stress had been increased. The initial increase in M was attributed to the 3 removal of near perpendicular domains and the resulting loss in total area of domain walls. A subsequent decrease in M was attributed to the effects 3 of the stress energy causing the sample to move from behaving as a cubic polycrystalline material to an uniaxial one. This change in anisotropy then causes the corresponding decrease in M [15]. 3 4.1. Total differential permeability Figs. 1 and 2 show total differential permeability behaviour similar to that seen previously for M . 3
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Fig. 2. Maximum total differential permeability for each sample as a function of applied stress. Stress measurements are $5%, while k is $60 for the 0.003 wt% sample and $20 for the remaining samples. 505
Initially k increased with applied stress but after 505 reaching a maximum value it decreased with further increase in applied stress. No qualitative difference can be seen in the behaviour immediately before and after the yield point, with the change in slope of the curve instead occurring well below yield. The largest observed increase in maximum total differential permeability is approximately one third of the initial, unstressed value for each type of sample except for the 0.15 wt% C alloy, where k almost doubles. 505 4.1.1. Effect of low stress values The initial changes in k can be attributed to 505 stress-induced changes in the domain structure of the samples. If the field applied to a sample and the applied stress lie along the same axis, low levels of elastic stress will cause the decrease (for tension) or
increase (for compression) of the total volume of near-perpendicular domains [8,16]. As a result, low levels of applied tension tend to increase the magnetisation of the sample through changes in the domain structure, while the total differential permeability of the sample is increased by reducing the total area of domain walls (and thus the amount of domain wall pinning). The differences between the amount of increase in k of each of the samples suggests that the 505 0.15 wt% C samples may have a greater volume of near-perpendicular domains in the unstressed state than those of the other samples. The presence of increased numbers of these domains in the 0.15 wt% C sample can be attributed to the way in which the domain walls are pinned in the different samples, under the assumption that closure domains tend to form at the edges of grains, while
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Fig. 3. The reversible relative differential permeability (k ) as a function of magnetisation for various values of applied stress. The two 3%7 curves for each stress level represent data from the upper and lower branches of the hysteresis loop. (a) 0.003 wt% C sample. Magnetisation $5000 A/m, k $35; (b) 0.15 wt% C sample. Magnetisation $5000 A/m, k $10; (c) 0.86 wt% C sample. Magnet3%7 3%7 isation $5000 A/m, k $5. 3%7
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Fig. 3. Continued.
domains separated by 180° walls are found in their centre. Examination of the domain wall structures by Lorentz electron microscopy in unstressed thinned samples with the two lowest carbon contents supports this model. Observing the domain behaviour under the influence of an applied magnetic field also suggests that domain walls in the two lowest carbon content samples are pinned primarily at the grain boundaries, rather than in the centre of the grains. The pearlite in the higher carbon content samples produces strong domain wall pinning and low values of k , but the pearlite 505 is distributed throughout the grains so that much of that pinning takes place along the 180° domain walls. In contrast, metallographic and electron microscopy examination of the 0.15 wt% C sample shows that it has small pearlite grains located primarily around the boundaries of the larger ferrite grains. The closure domains should therefore be more strongly pinned in this material than the 180° domain walls. Finally, in the 0.003 wt% C sample no pearlite is present, so that while the domain walls are again primarily pinned at the grain boundaries, this pinning is weaker than in the
0.15 wt% C sample. Applying tension to these samples will eliminate the walls of the closure domains either by eliminating those domains themselves where they form the near- perpendicular domains in a particular grain or by causing them to grow in size until they are the only domains left in grains where the 180° domain walls separate the nearperpendicular domains. The application of low levels of stress to the 0.15 wt% C sample would therefore eliminate much of the pinning in that material, shifting its magnetic behaviour towards that of the lowest carbon sample. A proportionally smaller change is seen for the 0.003 wt% C sample simply because the walls are not as strongly pinned in the first place. Examination of the results in Fig. 1 shows that while the k curves for the 0.003 and 0.86 wt% 505 C sample appear to have a very similar shape throughout the range of the stresses shown, the shape of the 0.15 wt% C curve varies depending on the stress level. In particular, the curve at 36 MPa shows a different behaviour to all the others. A very sharp increase in k occurs with field, followed 505 immediately by a decrease that is almost as rapid.
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Fig. 4. Maximum reversible differential permeability for each sample as a function of applied stress. Stress measurements are $5%, while k is $35 for the 0.003 wt% sample, $10 for the 0.15 wt% C sample and $5 for the remaining samples. 3%7
Unlike any of the other curves, there is no reduction in the slope of k near the maximum value. 505 The slope of the permeability curve can be interpreted to give information as to the nature of the pinning within the sample. A flat total or irreversible permeability, such as reported previously for 2% Mn pipeline steel [8], suggests that the rate of change in the sample’s domain structure is equally constant, so that the domains in the sample undergo a period of restructuring that changes the magnetisation without dramatically altering the distribution of domains. One possible way to produce such an effect would be for domain wall movements to take place without the actual elimination of domains. By contrast, the continuous increase or decrease in the 36 MPa, 0.15 wt% C sample suggests that domains may continuously be formed or removed in this material.
4.1.2. Behaviour after the elimination of the near perpendicular domains The behaviour of the samples once the near perpendicular domains have been eliminated and the maximum values of k have been reached can be 505 understood as being due to both stress-induced changes in anisotropy and the increase in pinning sites within the samples, once the yield point has been reached. The overall shape of the curves in Fig. 2 appear to be produced by the samples undergoing a process of changing from cubic anisotropy to uniaxial anisotropy. Theoretical considerations suggest that uniaxial samples should have much finer domains than polycrystalline ones [17]. As the anisotropy changes in these samples, the number of domains grows. k then rapidly decreases 505 due to the increased magnetostatic energy associated with this type of anisotropy. As there is
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higher stored energy in the domain structure, a higher applied field is necessary to produce the same degree of change in magnetisation compared to a cubic material, lowering the differential permeability. In examining the maximum k values in Fig. 2, 505 it is apparent that there is a significant difference between the behaviour of the 0.15 wt% C sample and that of the other samples, once the maximum k values begin to decrease. The plots for the other 505 samples do not show qualitative changes in behaviour at the yield point. Instead, there is a smooth decrease in k as the stress is increased. However, 505 in the case of the 0.15 wt% C sample, there is an abrupt change in the slope of the plot as the material passes through the yield point. The initial decreases in k shown by this sample appear to be 505 due to behaviour similar to that presented by the 0.003 wt% C sample. Once the yield point is reached, the behaviour appears to be closer to that of the higher carbon samples. In addition, examination of the results in Fig. 2 from the higher carbon content samples shows that as the carbon in the sample increases, the rate of change of the maximum k curves decreases, with the 0.86 wt% C sample 505 having the lowest value. Our results for the remnant magnetisation [13] do not show an abrupt change in slope along the equivalent curve. Instead, both the 0.003 and 0.15 wt% C samples show similar, approximately constant slopes. This suggests that the change in the 0.15 wt% C k slope is not due to an effect of 505 the changing anisotropy in the sample. The previous results also showed that the effect of increasing stress levels beyond the yield point on the coercive field of the 0.003 wt% C sample is minimal, with the value of H for the 160 MPa measure# ment being less than that from the 0 MPa results. In contrast, the 0.15 wt% C sample shows a noticeable increase in H at 182 MPa as compared to # 0 MPa. This coercive field result suggests that the effect of dislocation generation on domain wall pinning in the lowest carbon sample is minimal, but that significant pinning due to dislocation generation does appear to be taking place above the yield point for the 0.15 wt% C sample. These results, together with the change in the slope of the maximum k curve mentioned in the previous para505
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graph, suggest that while the decreases in k , seen 505 for all the samples after the maximum k value has 505 been reached, are due to changes in the anisotropy of the samples, they are also significantly affected by the number of pinning sites produced in the sample by dislocation generation. The initial, very rapid decreases in k for the 0.15 wt% C sample 505 would then be due to the relative absence of pinning sites, with the abrupt change in the slope of the curve at yield occurring due to the formation of new pinning sites during the plastic deformation process. These results suggest that increasing numbers of pinning sites within the sample will tend to inhibit the transition to uniaxial anisotropy within the sample. This transition would require not only that the stress energy applied to the sample alter the anisotropy itself, but also that it supply the energy to change the domain structure within each grain. The presence of pinning sites will impede domain wall motion, producing a requirement for more energy to be supplied to the samples to allow the domain reorganisation to occur. As a result, the rate of decrease in k with increasing stress 505 is greatly reduced when strong pinning sites are present. 4.2. Reversible differential permeability The behaviour of the maximum value of k is 3%7 much less clear below the yield point. As shown in Fig. 4, only the 0.68 wt% C results show the pattern of initial increase in the value followed by a decrease before yield takes place that was seen for M and k . The 0.003, 0.43 and 0.86 wt% C re3 505 sults only show an increase in k before yield. The 3%7 results for the 0.43 wt% C sample before heat treatment appear to indicate an initial decrease in the maximum value with strain, followed by a return to the initial value, but the variations are within the uncertainty of the measurements. Beyond the yield point the behaviour changes, and k clearly de3%7 creases with further increases in strain for each sample. Further, it is apparent the high carbon content results appear to become very similar to each other as the stress applied to the sample is increased beyond yield. The results of Fig. 4 show that k is not affected 3%7 by the increasing stress in the same manner as
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k or M . The lack of a consistent pattern of an 505 3 initial increase in k followed by a decrease before 3%7 the yield point indicates that this parameter is not being strongly affected by the stress-induced change in anisotropy that is believed to be responsible for most of the changes in the total differential permeability and remnant magnetisation. Instead, it appears that changes in the pinning within the samples are responsible for the changes in k . The 3%7 0 MPa values of k can be correlated with the 3%7 number of pinning sites in the samples, with increasing values of H corresponding to decreasing # values of k . The similarity between the 0 MPa 3%7 values given by the two lowest carbon samples can be attributed to k being produced by domain 3%7 wall bowing, which will primarily be affected by pinning sites in the central area of the grains, rather than at the grain boundaries. The low number of pinning sites in the ferrite of both types of sample produces similar values of k . 3%7 Increasing levels of stress primarily affects k through the elimination of near-perpendicular 3%7 domains and their associated domain walls. These walls act as a source of pinning for the other, near-parallel domain walls, so their removal would be expected to increase k . This effect is most 3%7 visible for the lowest carbon sample where the near-perpendicular walls form an important source of pinning sites. (It should be noted that two data points are missing from the 0.15 wt% measurements as compared to those for H , M # 3 or k , so conclusions cannot be drawn directly 505 about the behaviour of this particular material. However, based on other similarities in magnetic behaviour between the two lowest carbon sample types, it appears likely that the reversible permeability of the two materials will also behave in a similar manner.) For the higher carbon samples, the pinning produced by the carbide lamella in the pearlite grains is predominant and little increase in k with increasing stress below the yield point 3%7 is seen. The decrease in k after the yield point can be 3%7 attributed to the increasing number of pinning sites that are produced in the samples after that point. Higher pinning site energies or increasing numbers of pinning sites tend to reduce k since they re3%7 strict the amount of bending that the domain walls
can undergo [18]. It is interesting to note that just as the H values for the 0.003 wt% C sample # at 160 MPa remain below the 0 MPa value despite having passed through the yield point, so does the k value at 160 MPa remain above that 3%7 of 0 MPa. However, the behaviour of the maximum values of k at higher stress in Fig. 4 suggests that there is 3%7 a limit to the effect of the generation of pinning sites on the value of k . The curves of the higher carbon 3%7 samples not only follow along the same path, but they also show a decreasing slope as the stress increases. At the highest stress levels the plot for the 0.86 wt% results reaches a slope of zero within the uncertainty of the measurements, despite the very rapid increase in the number of pinning sites described previously [13]. This suggests that there may be a minimum k curve for this type of material and 3%7 that the maximum value of k along that minimum 3%7 curve is approximately 35. Further research is necessary to determine if this is indeed the case. Comparison of the magnitude of the reversible relative differential permeability plots (Fig. 3) to the equivalent total relative differential permeability plots (Fig. 1) clearly shows that irreversible processes dominate the behaviour of these samples. k is typically 30 times greater in value than k . 505 3%7 This is in direct contrast to previous work on 2% Mn pipeline steel [8], where k forms 20—25% of 3%7 the total differential permeability. The range of reversible relative permeability values for the latter material is of the same order of magnitude as those shown here for the iron—carbon alloys, indicating that the differences in between the magnetic properties of the two classes of iron alloys are almost entirely due to changes in the behaviour of the irreversible component of the permeability.
5. Conclusions The work reported here is part of the first complete set of in situ measurements made on the effects of plastic deformation on the magnetic properties of iron—carbon alloys. k was found to in505 crease initially with increasing levels of stress for all of the samples. In each case a maximum permeability value was reached and then k began to 505
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decrease with further increase in stress. This change in behaviour was found to occur well before the yield point and, with the exception of the 0.15 wt% C sample, no qualitative changes in the behaviour of k were seen at yield. 505 The initial increases in k with increasing stress 505 were attributed to the elimination of near-perpendicular domains and domain walls from the sample by the applied tension. Subsequent decreases in k were attributed to a stress in505 duced change in the sample anisotropy from that typical of a cubic material to that of an uniaxial material. This stress-induced change caused the formation of new domains within the material to lower the magnetostatic energy of the sample, with the increase in number of domains producing the observed decrease in k . Increasing 505 the number of pinning sites in the samples appears to inhibit this stress-induced change in the anisotropy, resulting in a lower rate of change in k for 505 the three highest carbon quotient samples than that seen for the lowest carbon content sample. The increase in pinning sites in the 0.15 wt% C sample after yield was seen to alter its k behaviour 505 from acting in a manner similar to that of the lowest carbon sample to that of the higher carbon samples. k did not show a consistent pattern of behav3%7 iour below the yield point, although in general an increase in the value with increasing stress was seen. This increase was attributed to the elimination of pinning sites as the applied stress removed nearperpendicular domains from the samples. Above the yield point increasing stress levels caused a decrease in k due to the increase in the number of 3%7 pinning sites produced by the plastic deformation process. At higher stress levels the samples appear to have very similar k values despite having dif3%7 ferent numbers of pinning sites. This result suggests that there may be a limiting value to k for this 3%7 class of magnetic materials.
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Acknowledgements Financial support from the Engineering and Physical Science Research Council is gratefully acknowledged. Thanks are expressed to Dr. J.P. Jakubovics and Dr. A. Petford-Long for assistance in the Lorentz electron microscopy studies. J.M. Makar gratefully acknowledges the support of the National Research Council of Canada during the writing of this paper.
References [1] D.C. Jiles, D.L. Atherton, J. Phys D: Appl. Phys. 17 (1984) 1265. [2] M.J. Sablik, D.C. Jiles, IEEE Trans. Magn. 29 (1993) 2113. [3] S. Chikazumi, Physics of Magnetism, Wiley, New York, 1964, p. 16. [4] S. Thompson, P.J. Allen, B.K. Tanner, IEEE Trans. Magn. 26 (1990) 1984. [5] D.C. Jiles, J. Phys. D 21 (1988) 1186. [6] R.M. Bozorth, Ferromagnetism, Van Nostrand, New York, 1951, pp. 538—546. [7] D.L. Atherton, T. Sudersena Rao, M. Scho¨nba¨chler, IEEE Trans. Magn. 24 (1988) 2033. [8] J.M. Makar, D.L. Atherton, IEEE Trans. Magn. 30 (1994) 1380. [9] J.M. Makar, B.K. Tanner, NDT&E Inter. 31 (1998) 117. [10] C.S. Schneider, M. Charlesworth, J. Appl. Phys. 57 (1985) 4198. [11] S.M. Thompson, B.K. Tanner, J. Magn. Magn. Mater. 132 (1994) 71. [12] D.C. Jiles, J. Phys. D: Appl. Phys. 21 (1988) 1196. [13] J.M. Makar, B.K. Tanner, J. Magn. Magn. Mater. 184 (1998) 193. [14] M. Scho¨nbachler, M.Sc. (Eng.) Thesis, Queen’s University, Kingston, Ontario, 1987. [15] S. Chikazumi, Physics of Magnetism, John Wiley and Sons, New York, 1964, pp. 291—295. [16] B.D. Cullity, Introduction to Magnetic Materials, Addison-Wesley, Reading, MA, 1972, pp. 271—272. [17] S. Chikazumi, Physics of Magnetism, Wiley, New York, 1964, pp. 226—232. [18] S. Chikazumi, Physics of Magnetism, Wiley, New York, 1964, pp. 217—274.
The in situ measurement of the effect of plastic deformation on the magnetic properties of steel Part II — Permeability curves J.M. Makar*, B.K. Tanner Department of Physics, University of Durham, Science Laboratories, South Road, Durham DH1 3LE, UK Received 20 January 1998; received in revised form 19 March 1998
Abstract We report measurements of the bulk magnetic properties of pearlitic steels recorded in situ during plastic deformation. The low-field total differential permeability was found to initially increase with increasing elastic tensile stress applied in the direction of the magnetic field. However, it was found to reach a maximum value well before the yield point and decreased rapidly after reaching that maximum. No qualitative change of behaviour was observed at the yield point. The initial increases in total differential permeability are attributed to the stress-induced removal of domains with magnetisation vectors that are approximately perpendicular to the field and stress direction. The subsequent decreases in total differential permeability are attributed both to stress-induced changes in the magnetic anisotropy and to new pinning sites generated during the plastic deformation process. The low-field reversible differential permeability appears to show similar behaviour with increasing stress, although the maximum value for this type of measurement is not as well defined. Measurements of the high-field total differential permeability decreased monotonically with stress in both the pre-yield and plastically deformed regions. ( 1998 Elsevier Science B.V. All rights reserved. Keywords: Steel; Stress; Permeability — differential
1. Introduction Curves of the permeability as a function of applied field are frequently used as tools in analysing magnetic behaviour. While the curve produced by the permeability (k"k~1B/H) is not particularly 0 useful, the differential permeability (k"k~1dB/dH) 0 * Corresponding author. Present address: Institute for Research in Construction, National Research Council of Canada, 1500 Montreal Road, Ottawa, Ontario, Canada K1A 0R6.
and the equivalent susceptibility (s"dM/dH" k!1) are important both for magnetic modelling [1,2] and interpreting the magnetisation process [3—5]. The differential nature of the curve highlights behaviour that is not readily seen in the hysteresis loop and assists in the interpretation as to which magnetisation mechanism is dominant at a particular field or magnetisation. In the remainder of this paper, the term permeability always refers to the differential permeability unless otherwise stated.
0304-8853/98/$19.00 ( 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 9 8 ) 0 0 1 4 1 - 3
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Permeability curves have an advantage over hysteresis loops in that they can be readily divided into reversible and irreversible components [3,6—8]. Magnetic materials show both types of behaviour, with the irreversible magnetisation changes coming from irreversible magnetisation vector rotations and domain wall motion past pinning sites, while the reversible changes are due to reversible magnetisation vector rotations and domain wall bowing around pinning sites. It is impossible to separate these two components on the hysteresis loop, but the different types of permeability can be measured by the process of producing a small field reversal at the point of the magnetisation curve under investigation [6]. The slope of that curve gives the reversible permeability, with the irreversible permeability being calculated by subtracting the reversible value from the total permeability measured at the same point on the hysteresis loop [7]. Examination of the irreversible and reversible components not only allows the determination of the relative importance of the two classes of magnetisation methods, but can also give information about the nature of pinning sites and how stress affects the magnetic properties of the material in question. Previous work on 2% Mn pipeline steel [7,8] has examined reversible and irreversible permeability variations under elastic stresses for initial magnetisation curves, major hysteresis loops and minor hysteresis loops. These results showed that both types of permeability increase under uniaxial tension applied in the direction of the applied magnetic field at low magnetisations and decrease at higher magnetisations above the knee of the hysteresis loops. The opposite effect was seen for compression. It was also noted that differences between the upper and lower branch permeabilities behaved in a fashion opposite to the overall values. Studies of the effects of stress on the total permeability of magnetic materials have been reported [9,10] that show similar results for steel to those described above. Total differential permeability results have also been reported for steel samples under conditions of residual stress [11,12]. However, to the best of the authors’ knowledge, no comprehensive studies of the effect of stresses that approach and exceed the yield point on magnetic materials appear to have been undertaken.
The work presented here is the second of a pair of papers that discuss changes in the magnetic properties of similar specially prepared iron—carbon alloys during the plastic deformation process. While the first paper [13] examined hysteresis loop behaviour, including the coercive field and remnant magnetisation as well as the corresponding magnetostriction, this paper examines the behaviour of the corresponding reversible and total relative differential permeability curves. Initial measurements were performed under conditions of elastic tension up to the yield point. These were followed by in situ measurements made on the same sample in the plastic deformation region of each sample’s stress— strain curve.
2. Experiment An extensive description of the experimental samples and apparatus has been presented in the first paper [13]. They will only be summarised here, while the procedures that are particularly relevant to the data presented below will be given in detail. A schematic diagram of the apparatus and sample dimensions have also been published elsewhere [9]. 2.1. Samples Five different iron—carbon alloys were used for the experiments reported here. These samples were cast by the University of Sheffield Materials Advisory Centre in the form of 5 kg ingots with carbon contents of 0.003, 0.15, 0.43, 0.68 and 0.86 wt%. There was less than 0.003 wt% sulphur in the each ingot and no other significant constituents. The ingots were cooled, reheated and then extruded as rods before being air cooled and sawn into 43 cm long sections. Many of the rods were bent as a result of the extrusion process but the straightest sections were retained for measurements in the as-delivered condition. The remaining sections were straightened mechanically and heat treated to remove residual stresses. All rods were machined before mounting so that instrumentation could be attached to the sample in the plastic deformation zone. While
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initial measurements were made on each of the sample materials before heat treatment, complete sets of magnetic measurements were made on 0.43 wt% C samples without heat treatment and samples with all five carbon contents after heat treatment. 2.2. Apparatus The instrumentation on each sample consisted of a strain gauge, which was used to measure the strain on the sample (e) as well as the magnetostriction (j), a coil that was used to measure the flux density (B) experienced by the sample, and a Hall probe that was used to measure the magnetic field (H) applied to the sample. The strain gauge, coil and Hall probe were each attached to the sample in the region where plastic deformation occurred. One leg of a Wheatstone bridge incorporated the strain gauge, with the other three legs of the bridge being made from ‘dummy’ strain gauges mounted on a piece of steel located well away from the electromagnet. The outputs from the Wheatstone bridge and the Hall probe were each amplified and filtered before being acquired by a PC mounted data acquisition board. The coil was connected to an integrating fluxmeter and the outputs from the fluxmeter also connected to the data acquisition board. A ‘C’ shaped electromagnet with slots cut in each end to accept the samples was used to magnetise them so that the field was applied parallel to the sample length. A computer controlled bipolar power supply was used to power the electromagnet, which was designed to be mounted from the Instrom testing machine so that measurements could be made while the sample was under applied elastic or plastic stresses. The data acquisition system was also used to measure the signals from the Instrom load cell, with the data being converted into stress (p) values based on the crosssectional area of the sample at any given point on its stress/strain curve. 2.3. Experimental procedure The experimental procedure depended on whether or not the measurements were to be made
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under stress. Measurements made under conditions of zero stress were performed outside the Instrom to ensure that no biaxial stress was present. For the measurements made under stress, the instrumented sample was mounted in the Instrom and a sufficient load applied to the sample to raise it to the desired stress level, at which point the Instrom was turned off. If the sample was below the yield point, no further changes in the strain or stress experienced by the sample took place. However, if the samples were above the yield point, a relaxation effect was observed in which the stress dropped slightly and the sample continued to yield, increasing its strain. Typically a period of between 3 and 6 hours passed before the deformation process halted. This condition approximates that seen in practice in engineering structures, where a constant load has caused localised plastic deformation. Note that this is a very different situation to the relaxed state of compressive strain that exists once a load that has produced plastic deformation has been removed. Once the relaxation process was completed, the electromagnet was positioned against the sample in such a manner that biaxial stresses were minimised. Whether or not the sample was mounted in the Instrom machine, the Hall probe was calibrated for each individual sample and stress level before taking magnetic measurements by driving the sample well into saturation and measuring the slope of the B—H curve in that region. Boundary conditions require that the component of the field immediately inside the surface of the sample and parallel to it be the same as that immediately outside it, so a comparison of the slope to the expected value (the permeability of free space) gives a correction factor for subsequent field measurements [14]. Initial magnetisation curves and near-saturation major hysteresis loops were recorded for each sample material, including both heat treated and un-heat treated materials. Small field reversals were made at each point measured in order that reversible permeability values could later be calculated [3]. These field reversals ranged in size from 10 A/m for the 0.003 wt% C sample to 50 A/m for the 0.86 wt% C sample. Once measurements were taken under a particular level of stress beyond the yield point, the sample
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was either subjected to increased stress or the applied stress on the sample was removed so that a measurement under conditions of residual stress could be made before higher levels of tension were applied. The choice of procedure did not affect the measurements made under applied stress. After the magnetic measurements were complete, the resulting hysteresis loops were analysed to produce values for the total (k ) and reversible (k ) 505 3%7 relative differential permeability. The former value was calculated directly from the slope of the hysteresis loop measurements, while the latter was calculated from the slopes along the small field reversals. Although the irreversible component of the differential permeability (k ) was also deter*33 mined by subtracting k from k , the results are 3%7 505 not shown. The value of k in these samples is 3%7 considerably smaller than that of k , so that a plot 505 of k appears to be nearly identical to that of k . *33 505 The resulting data were then analysed to determine the maximum values of k and k . These 505 3%7 data points are plotted against stress in the figures that follow.
3. Results Figs. 1 and 3, respectively show plots of the response of the total and reversible relative differential permeabilities to stress. Fig. 1a and Fig. 3a shows the behaviour of the 0.003 wt% C sample, Fig. 1b and Fig. 3b that of the 0.15 wt% C sample, while Fig. 1c and Fig. 3c shows that of the 0.86 wt% C sample. The results from the remaining samples (0.43 wt% C both before and after heat treatment and 0.68 wt% C) are similar to those shown in Fig. 1c and Fig. 3c. Note that the quoted uncertainty values are for the central area of the figures where the hysteresis loop changes very rapidly. The results at the extremes of the figures are much less uncertain. Each of the plots in Fig. 1 shows similar results, with k initially increasing with increasing stress, 505 followed by a decrease in value that starts well before the yield point of the material. There are, however, significant differences in the fractional changes in the permeability values. In Fig. 1a the 0.003 wt% C sample has the highest k values of 505
all the samples. The application of elastic stress increases this permeability by about one third before it starts to decrease again. The results from the 0.17 wt% C sample in Fig. 1b show significantly lower k values, but the total permeability more 505 than doubles in value with increasing stress before starting to decrease as the stress approaches the yield point. Finally, the results of Fig. 1c show both lower overall permeability values and a much smaller increase in the permeability with increasing elastic stress. Fig. 2 shows the behaviour of the maximum total differential permeability measured for each sample with increasing stress. All samples show a characteristic pattern of an initial increase in the magnitude of k at low stress levels, followed 505 by a decrease in value with further increases in stress. The yield points of the materials are marked by the solid diamond shapes and have been published in the previous paper [13]. Note that in each case this decrease began well before the yield point of the material and that there does not appear to be a qualitative change in behaviour of the samples at the yield point. The reversible differential permeability behaviour in Fig. 3 is not as consistent as that shown in Fig. 1 for the total differential permeability. The results for the 0.003 wt% C and 0.86 wt% C samples both show an initial increase in k 3%7 with increasing stress, followed by a decrease at higher stress levels. However, the 0.15 wt% C sample only shows decreasing values of k . 3%7 A similar pattern is illustrated in Fig. 4, where the maximum k values initially increase with stress 3%7 (p) before decreasing for the 0.003 wt% C, 0.68 wt% C and the after heat treatment 0.43 wt% C samples, decreases continuously for the 0.17 wt% C sample, and initially does not appear to vary within the uncertainty for the 0.68 wt% C and the before heat treatment 0.43 wt% C values. A second significant difference between the two types of permeability measurements is that, with the exception of the 0.15 wt% C sample, the first decreases in k appear to be taking place at or 3%7 near the yield point. Before yield, k is either 3%7 increasing or maintaining a constant value, while after yield further increases in the stress applied to the sample correspond to increasing reductions in the parameter.
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Fig. 1. The total relative differential permeability (k ) as a function of magnetisation for different values of applied stress. The curves 505 with maximum on the left- and right-hand side of the graph are produced from the upper and lower branches of the hysteresis loop, respectively. (a) 0.003 wt% C sample. Magnetisation $5000 A/m, k $60 maximum; (b) 0.15 wt% C sample. Magnetisation 505 $5000 A/m, k $20 maximum; (c) 0.86 wt% C sample. Magnetisation $5000 A/m, k $20 maximum. 505 505
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Fig. 1. Continued.
4. Discussion In Part I of this pair of papers [13] the behaviour of the coercive field (H ) and remnant # magnetisation (M ) was investigated. The re3 sults presented there are also important in interpreting what happens to k as the samples ap505 proach and exceed the yield point. H was found to # be constant or initially to decrease with increasing stress, but to increase with further increases in stress above the yield point. This increase was found to start at or just before yield and was attributed to increasing numbers of pinning sites produced by the plastic deformation process, with the number of pinning sites being found to increase such that N"kp4, where N is the number of pinning sites, p is the applied stress and k is a constant. M showed 3 a very different behaviour, with initial increases in value at low levels of applied tension, followed by decreasing values with further increases in stress. The decreases in M were found to begin well 3
before yield had taken place and no qualitative changes in behaviour were seen at yield. Magnetostrictive evidence also showed that all of those domains that had magnetisation vectors pointing more closely towards the perpendicular to the applied field (henceforth referred to as ‘near-perpendicular domains’, with those that point more closely to the direction of the field being referred to as ‘near-parallel domains’) were eliminated from the sample as the applied stress had been increased. The initial increase in M was attributed to the 3 removal of near perpendicular domains and the resulting loss in total area of domain walls. A subsequent decrease in M was attributed to the effects 3 of the stress energy causing the sample to move from behaving as a cubic polycrystalline material to an uniaxial one. This change in anisotropy then causes the corresponding decrease in M [15]. 3 4.1. Total differential permeability Figs. 1 and 2 show total differential permeability behaviour similar to that seen previously for M . 3
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Fig. 2. Maximum total differential permeability for each sample as a function of applied stress. Stress measurements are $5%, while k is $60 for the 0.003 wt% sample and $20 for the remaining samples. 505
Initially k increased with applied stress but after 505 reaching a maximum value it decreased with further increase in applied stress. No qualitative difference can be seen in the behaviour immediately before and after the yield point, with the change in slope of the curve instead occurring well below yield. The largest observed increase in maximum total differential permeability is approximately one third of the initial, unstressed value for each type of sample except for the 0.15 wt% C alloy, where k almost doubles. 505 4.1.1. Effect of low stress values The initial changes in k can be attributed to 505 stress-induced changes in the domain structure of the samples. If the field applied to a sample and the applied stress lie along the same axis, low levels of elastic stress will cause the decrease (for tension) or
increase (for compression) of the total volume of near-perpendicular domains [8,16]. As a result, low levels of applied tension tend to increase the magnetisation of the sample through changes in the domain structure, while the total differential permeability of the sample is increased by reducing the total area of domain walls (and thus the amount of domain wall pinning). The differences between the amount of increase in k of each of the samples suggests that the 505 0.15 wt% C samples may have a greater volume of near-perpendicular domains in the unstressed state than those of the other samples. The presence of increased numbers of these domains in the 0.15 wt% C sample can be attributed to the way in which the domain walls are pinned in the different samples, under the assumption that closure domains tend to form at the edges of grains, while
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Fig. 3. The reversible relative differential permeability (k ) as a function of magnetisation for various values of applied stress. The two 3%7 curves for each stress level represent data from the upper and lower branches of the hysteresis loop. (a) 0.003 wt% C sample. Magnetisation $5000 A/m, k $35; (b) 0.15 wt% C sample. Magnetisation $5000 A/m, k $10; (c) 0.86 wt% C sample. Magnet3%7 3%7 isation $5000 A/m, k $5. 3%7
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Fig. 3. Continued.
domains separated by 180° walls are found in their centre. Examination of the domain wall structures by Lorentz electron microscopy in unstressed thinned samples with the two lowest carbon contents supports this model. Observing the domain behaviour under the influence of an applied magnetic field also suggests that domain walls in the two lowest carbon content samples are pinned primarily at the grain boundaries, rather than in the centre of the grains. The pearlite in the higher carbon content samples produces strong domain wall pinning and low values of k , but the pearlite 505 is distributed throughout the grains so that much of that pinning takes place along the 180° domain walls. In contrast, metallographic and electron microscopy examination of the 0.15 wt% C sample shows that it has small pearlite grains located primarily around the boundaries of the larger ferrite grains. The closure domains should therefore be more strongly pinned in this material than the 180° domain walls. Finally, in the 0.003 wt% C sample no pearlite is present, so that while the domain walls are again primarily pinned at the grain boundaries, this pinning is weaker than in the
0.15 wt% C sample. Applying tension to these samples will eliminate the walls of the closure domains either by eliminating those domains themselves where they form the near- perpendicular domains in a particular grain or by causing them to grow in size until they are the only domains left in grains where the 180° domain walls separate the nearperpendicular domains. The application of low levels of stress to the 0.15 wt% C sample would therefore eliminate much of the pinning in that material, shifting its magnetic behaviour towards that of the lowest carbon sample. A proportionally smaller change is seen for the 0.003 wt% C sample simply because the walls are not as strongly pinned in the first place. Examination of the results in Fig. 1 shows that while the k curves for the 0.003 and 0.86 wt% 505 C sample appear to have a very similar shape throughout the range of the stresses shown, the shape of the 0.15 wt% C curve varies depending on the stress level. In particular, the curve at 36 MPa shows a different behaviour to all the others. A very sharp increase in k occurs with field, followed 505 immediately by a decrease that is almost as rapid.
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Fig. 4. Maximum reversible differential permeability for each sample as a function of applied stress. Stress measurements are $5%, while k is $35 for the 0.003 wt% sample, $10 for the 0.15 wt% C sample and $5 for the remaining samples. 3%7
Unlike any of the other curves, there is no reduction in the slope of k near the maximum value. 505 The slope of the permeability curve can be interpreted to give information as to the nature of the pinning within the sample. A flat total or irreversible permeability, such as reported previously for 2% Mn pipeline steel [8], suggests that the rate of change in the sample’s domain structure is equally constant, so that the domains in the sample undergo a period of restructuring that changes the magnetisation without dramatically altering the distribution of domains. One possible way to produce such an effect would be for domain wall movements to take place without the actual elimination of domains. By contrast, the continuous increase or decrease in the 36 MPa, 0.15 wt% C sample suggests that domains may continuously be formed or removed in this material.
4.1.2. Behaviour after the elimination of the near perpendicular domains The behaviour of the samples once the near perpendicular domains have been eliminated and the maximum values of k have been reached can be 505 understood as being due to both stress-induced changes in anisotropy and the increase in pinning sites within the samples, once the yield point has been reached. The overall shape of the curves in Fig. 2 appear to be produced by the samples undergoing a process of changing from cubic anisotropy to uniaxial anisotropy. Theoretical considerations suggest that uniaxial samples should have much finer domains than polycrystalline ones [17]. As the anisotropy changes in these samples, the number of domains grows. k then rapidly decreases 505 due to the increased magnetostatic energy associated with this type of anisotropy. As there is
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higher stored energy in the domain structure, a higher applied field is necessary to produce the same degree of change in magnetisation compared to a cubic material, lowering the differential permeability. In examining the maximum k values in Fig. 2, 505 it is apparent that there is a significant difference between the behaviour of the 0.15 wt% C sample and that of the other samples, once the maximum k values begin to decrease. The plots for the other 505 samples do not show qualitative changes in behaviour at the yield point. Instead, there is a smooth decrease in k as the stress is increased. However, 505 in the case of the 0.15 wt% C sample, there is an abrupt change in the slope of the plot as the material passes through the yield point. The initial decreases in k shown by this sample appear to be 505 due to behaviour similar to that presented by the 0.003 wt% C sample. Once the yield point is reached, the behaviour appears to be closer to that of the higher carbon samples. In addition, examination of the results in Fig. 2 from the higher carbon content samples shows that as the carbon in the sample increases, the rate of change of the maximum k curves decreases, with the 0.86 wt% C sample 505 having the lowest value. Our results for the remnant magnetisation [13] do not show an abrupt change in slope along the equivalent curve. Instead, both the 0.003 and 0.15 wt% C samples show similar, approximately constant slopes. This suggests that the change in the 0.15 wt% C k slope is not due to an effect of 505 the changing anisotropy in the sample. The previous results also showed that the effect of increasing stress levels beyond the yield point on the coercive field of the 0.003 wt% C sample is minimal, with the value of H for the 160 MPa measure# ment being less than that from the 0 MPa results. In contrast, the 0.15 wt% C sample shows a noticeable increase in H at 182 MPa as compared to # 0 MPa. This coercive field result suggests that the effect of dislocation generation on domain wall pinning in the lowest carbon sample is minimal, but that significant pinning due to dislocation generation does appear to be taking place above the yield point for the 0.15 wt% C sample. These results, together with the change in the slope of the maximum k curve mentioned in the previous para505
363
graph, suggest that while the decreases in k , seen 505 for all the samples after the maximum k value has 505 been reached, are due to changes in the anisotropy of the samples, they are also significantly affected by the number of pinning sites produced in the sample by dislocation generation. The initial, very rapid decreases in k for the 0.15 wt% C sample 505 would then be due to the relative absence of pinning sites, with the abrupt change in the slope of the curve at yield occurring due to the formation of new pinning sites during the plastic deformation process. These results suggest that increasing numbers of pinning sites within the sample will tend to inhibit the transition to uniaxial anisotropy within the sample. This transition would require not only that the stress energy applied to the sample alter the anisotropy itself, but also that it supply the energy to change the domain structure within each grain. The presence of pinning sites will impede domain wall motion, producing a requirement for more energy to be supplied to the samples to allow the domain reorganisation to occur. As a result, the rate of decrease in k with increasing stress 505 is greatly reduced when strong pinning sites are present. 4.2. Reversible differential permeability The behaviour of the maximum value of k is 3%7 much less clear below the yield point. As shown in Fig. 4, only the 0.68 wt% C results show the pattern of initial increase in the value followed by a decrease before yield takes place that was seen for M and k . The 0.003, 0.43 and 0.86 wt% C re3 505 sults only show an increase in k before yield. The 3%7 results for the 0.43 wt% C sample before heat treatment appear to indicate an initial decrease in the maximum value with strain, followed by a return to the initial value, but the variations are within the uncertainty of the measurements. Beyond the yield point the behaviour changes, and k clearly de3%7 creases with further increases in strain for each sample. Further, it is apparent the high carbon content results appear to become very similar to each other as the stress applied to the sample is increased beyond yield. The results of Fig. 4 show that k is not affected 3%7 by the increasing stress in the same manner as
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k or M . The lack of a consistent pattern of an 505 3 initial increase in k followed by a decrease before 3%7 the yield point indicates that this parameter is not being strongly affected by the stress-induced change in anisotropy that is believed to be responsible for most of the changes in the total differential permeability and remnant magnetisation. Instead, it appears that changes in the pinning within the samples are responsible for the changes in k . The 3%7 0 MPa values of k can be correlated with the 3%7 number of pinning sites in the samples, with increasing values of H corresponding to decreasing # values of k . The similarity between the 0 MPa 3%7 values given by the two lowest carbon samples can be attributed to k being produced by domain 3%7 wall bowing, which will primarily be affected by pinning sites in the central area of the grains, rather than at the grain boundaries. The low number of pinning sites in the ferrite of both types of sample produces similar values of k . 3%7 Increasing levels of stress primarily affects k through the elimination of near-perpendicular 3%7 domains and their associated domain walls. These walls act as a source of pinning for the other, near-parallel domain walls, so their removal would be expected to increase k . This effect is most 3%7 visible for the lowest carbon sample where the near-perpendicular walls form an important source of pinning sites. (It should be noted that two data points are missing from the 0.15 wt% measurements as compared to those for H , M # 3 or k , so conclusions cannot be drawn directly 505 about the behaviour of this particular material. However, based on other similarities in magnetic behaviour between the two lowest carbon sample types, it appears likely that the reversible permeability of the two materials will also behave in a similar manner.) For the higher carbon samples, the pinning produced by the carbide lamella in the pearlite grains is predominant and little increase in k with increasing stress below the yield point 3%7 is seen. The decrease in k after the yield point can be 3%7 attributed to the increasing number of pinning sites that are produced in the samples after that point. Higher pinning site energies or increasing numbers of pinning sites tend to reduce k since they re3%7 strict the amount of bending that the domain walls
can undergo [18]. It is interesting to note that just as the H values for the 0.003 wt% C sample # at 160 MPa remain below the 0 MPa value despite having passed through the yield point, so does the k value at 160 MPa remain above that 3%7 of 0 MPa. However, the behaviour of the maximum values of k at higher stress in Fig. 4 suggests that there is 3%7 a limit to the effect of the generation of pinning sites on the value of k . The curves of the higher carbon 3%7 samples not only follow along the same path, but they also show a decreasing slope as the stress increases. At the highest stress levels the plot for the 0.86 wt% results reaches a slope of zero within the uncertainty of the measurements, despite the very rapid increase in the number of pinning sites described previously [13]. This suggests that there may be a minimum k curve for this type of material and 3%7 that the maximum value of k along that minimum 3%7 curve is approximately 35. Further research is necessary to determine if this is indeed the case. Comparison of the magnitude of the reversible relative differential permeability plots (Fig. 3) to the equivalent total relative differential permeability plots (Fig. 1) clearly shows that irreversible processes dominate the behaviour of these samples. k is typically 30 times greater in value than k . 505 3%7 This is in direct contrast to previous work on 2% Mn pipeline steel [8], where k forms 20—25% of 3%7 the total differential permeability. The range of reversible relative permeability values for the latter material is of the same order of magnitude as those shown here for the iron—carbon alloys, indicating that the differences in between the magnetic properties of the two classes of iron alloys are almost entirely due to changes in the behaviour of the irreversible component of the permeability.
5. Conclusions The work reported here is part of the first complete set of in situ measurements made on the effects of plastic deformation on the magnetic properties of iron—carbon alloys. k was found to in505 crease initially with increasing levels of stress for all of the samples. In each case a maximum permeability value was reached and then k began to 505
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decrease with further increase in stress. This change in behaviour was found to occur well before the yield point and, with the exception of the 0.15 wt% C sample, no qualitative changes in the behaviour of k were seen at yield. 505 The initial increases in k with increasing stress 505 were attributed to the elimination of near-perpendicular domains and domain walls from the sample by the applied tension. Subsequent decreases in k were attributed to a stress in505 duced change in the sample anisotropy from that typical of a cubic material to that of an uniaxial material. This stress-induced change caused the formation of new domains within the material to lower the magnetostatic energy of the sample, with the increase in number of domains producing the observed decrease in k . Increasing 505 the number of pinning sites in the samples appears to inhibit this stress-induced change in the anisotropy, resulting in a lower rate of change in k for 505 the three highest carbon quotient samples than that seen for the lowest carbon content sample. The increase in pinning sites in the 0.15 wt% C sample after yield was seen to alter its k behaviour 505 from acting in a manner similar to that of the lowest carbon sample to that of the higher carbon samples. k did not show a consistent pattern of behav3%7 iour below the yield point, although in general an increase in the value with increasing stress was seen. This increase was attributed to the elimination of pinning sites as the applied stress removed nearperpendicular domains from the samples. Above the yield point increasing stress levels caused a decrease in k due to the increase in the number of 3%7 pinning sites produced by the plastic deformation process. At higher stress levels the samples appear to have very similar k values despite having dif3%7 ferent numbers of pinning sites. This result suggests that there may be a limiting value to k for this 3%7 class of magnetic materials.
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Acknowledgements Financial support from the Engineering and Physical Science Research Council is gratefully acknowledged. Thanks are expressed to Dr. J.P. Jakubovics and Dr. A. Petford-Long for assistance in the Lorentz electron microscopy studies. J.M. Makar gratefully acknowledges the support of the National Research Council of Canada during the writing of this paper.
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