Cullity B.D. Introduction to Magnetic Materials. Second Edition (2008)
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9.9 RESIDUAL STRESS |
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Stresses may be divided into two categories: applied stress (the stress in a body due to external forces) and residual stress (the stress existing in a body after all external forces have been removed). Residual stress is often mistakenly called “internal stress,” which is both uninformative and incorrect. All stress, whether applied or residual, is internal.
Residual stress may in turn be divided into macro and micro, depending on scale. Residual macrostress is reasonably constant in magnitude over distances large compared to the normal grain diameter, while residual microstress varies rapidly in magnitude and sometimes in sign over distances about equal to, or smaller than, the normal grain diameter. Residual macrostress, which is the kind of most concern to the engineer because of its effect on such phenomena as fatigue and fracture, is often due to nonuniform plastic flow that has occurred at some time in the previous history of the material; it can be removed by an appropriate annealing treatment. Residual microstress is caused by crystal imperfections of various kinds, particularly dislocations. Although it may be reduced to quite low levels by annealing, it is never entirely absent.
The distinction between residual macrostress and microstress is not sharp. A more practical distinction can be made on the basis of the effect of the stress on the X-ray diffraction pattern of the material. The depth to which X rays penetrate a metal surface is typically of the order of 20–30 mm, while the grain size of most metals lies in the range 10–100 mm. Thus a macrostress will be essentially constant over the depth examined by X rays, while a microstress will vary within this depth. As a result, residual macrostress causes an X-ray line shift, a change in direction of the diffracted beam, because the lattice-plane spacing d in Equation 5.31 is essentially constant, but different from the stress-free value, in the region examined. On the other hand, residual microstress causes line broadening, because of the variation of plane spacing within this region.
Residual macroand microstress are quite often found together. For example, the residual macrostress caused by grinding the opposite faces of a metal strip or plate with an abrasive wheel is shown at the left of Fig. 9.30. Each ground surface is in residual compression in a direction parallel to the surface. This stress rapidly decreases to zero with depth and changes to tension in most of the interior. Residual stresses must form a balanced force system. Regions in residual compression must be balanced by regions in tension. X-ray examination of the material at the surface, or just below (accomplished by etching part of the material away), shows the X-ray lines to be both shifted and broadened. We can then imagine the microstress distribution in one small region to resemble that sketched at the right of Fig. 9.30.
Fig. 9.30 Residual stress distribution due to grinding.
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There is, however, an ambiguity in the interpretation of line broadening, because two effects can cause line broadening: microstress and small crystal size, less than about
m˚
0.1m (1000 A). Thus line broadening cannot be interpreted solely in terms of either cause unless the other can be shown to be absent.
As noted above, residual microstress is never entirely absent, even in a well-annealed specimen. Consider these effects:
1.Dislocations. All specimens, except a few of the most carefully grown single crystals, contain a substantial number of dislocations and each has a stress field associated with it, because the dislocation distorts the surrounding material. In a real material, dislocation lines run in many different directions, forming a complex network and a very irregular distribution of microstress. The diameter of the stress field around a dislocation is generally less than the thickness of a domain wall, and dislocation lines will in general not lie parallel to the plane of a domain wall. The result is that the interaction of an isolated or widely spaced dislocations with moving domain walls is expected to be small.
2.Magnetostriction. When a ferroor ferrimagnetic is cooled below the Curie point, spontaneous magnetostriction acts to distort different domains in different directions. Because the domains are not free to deform independently, microstresses are set up. The same argument shows that the stress inside a domain wall differs from the stress in the adjoining domains. Figure 9.31 shows some examples for a material, like iron, in which the magnetostriction is positive in the direction of the spontaneous magnetization. Three domains are shown in Fig. 9.31a, separated by one 908 and one 1808 wall. Dashed lines indicate the dimensions of the various portions if they were free to deform in the y direction; strains in the x direction are ignored here. The corresponding variation of the y component of stress, sy, with x is shown in Fig. 9.31b, where the zero-stress level refers to the paramagnetic material above the Curie point. Stresses of
magnetostrictive origin are rather small, being of the order of lsiE, where E is Young’s modulus (the stresses are generally less than 1000 lb/in2 or 7 MPa). They
Fig. 9.31 Microstresses due to magnetostriction.
9.9 RESIDUAL STRESS |
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are large enough, however, to cause interactions between domains, or domain walls, and crystal imperfections. (In fact, if one takes the point of view that anything which sets up microstress is a crystal imperfection, then domains and domain walls are crystal imperfections.) The localized distortion within domain walls can be revealed by special methods of X-ray diffraction.
Plastic deformation (cold work) has marked effects on many magnetic properties, and it has often been studied. In this context, two kinds of plastic deformation should be distinguished:
1.Nonuniform, such as the rolling of sheet, the drawing or swaging of wire and rod through a die, bending, or twisting. These processes invariably produce both macro and micro residual stress. The resulting macrostress distribution depends on the kind and amount of working, and its form is not always predictable in advance. Figure 9.32, for example, shows two stress distributions that can be produced by cold drawing, depending on the shape of the die; in Fig. 9.32a the rod is composed of an outer “case” in compression and an inner “core” in tension; in Fig. 9.32b this distribution is reversed. Nonuniform modes of deformation are quite unsuitable for fundamental studies of the effect of cold work, because the inside and outside of the same specimen can have very different magnetic properties.
2.Uniform, such as uniaxial tension and compression. These processes produce only micro residual stress, as indicated in Fig. 9.33 for a stretched rod. However, the microstress distribution is irregular, as shown in the sketch at the right: most of the lattice, estimated at some 90%, is in residual compression, balanced by small regions under high tensile stress. Although the details of this stress distribution are hypothetical, as is true of any microstress distribution in the present state of our knowledge, its general form is supported by both X-ray and magnetic evidence. While tensile deformation has the advantage over swaging, for example, of not introducing macrostress, the amount of uniform tensile deformation that can be achieved is limited by the onset of “necking” (local contraction); this limits the amount of uniform deformation, as measured by the reduction in cross-sectional area, to 25% or less in most materials. Reductions of more than double this amount can usually be achieved by swaging or wire drawing before cracking begins.
Fig. 9.32 Possible distributions of longitudinal residual stresses across the diameter of a cold-drawn metal rod.
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Fig. 9.33 Distribution of longitudinal residual stress across the diameter of a rod after plastic elongation in the y direction.
The reader should note one simplification running through all the above discussion of residual stress: only the stress acting in one direction has been considered. The real situation may be much more complex. At the surface of a body the residual stress can be, and usually is, biaxial (two stresses at right angles) and, in the interior, triaxial (three stresses at right angles).
9.10HINDRANCES TO WALL MOTION (MICROSTRESS)
Residual microstress hinders domain wall motion because of magnetostriction. The behavior of 908 walls (non-1808 walls) and 1808 walls is quite different. When a 908 wall moves, the direction of magnetization is altered in the volume swept out by the wall motion, and because of magnetostriction there is an elastic distortion of this volume. This distortion interacts with the local stress distribution, generally in a way that tends to keep the domain wall in its original position. When a 1808 wall moves, on the other hand, only the sense of the magnetization direction is altered, and no magnetostrictive strain occurs. The only effect of local microstress is to change the domain wall energy, by adding a stress anisotropy Ks ¼ 32ls to the crystal anisotropy K (here s is the stress, not the wall energy).
It is possible to calculate the effect of an assumed stress distribution on the motion of an isolated 908 or 1808 domain wall, but this is not a very useful exercise for at least two reasons: First, the actual microscopic stress distribution in real materials is generally unknown, and second, isolated domain walls do not occur in real materials. Domains exist in an interconnected network, in which no single wall can move without influencing the position of its connected walls. Not much can be said beyond the obvious general conclusion that domain walls move most easily when the magnetostriction and the stresses are small.
9.11HINDRANCES TO WALL MOTION (GENERAL)
At zero applied field, a domain wall will be in a position which minimizes the energy of the system, where the “system” means the wall itself and the adjoining domains and domain
9.11 HINDRANCES TO WALL MOTION (GENERAL) |
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walls. Thus a 1808 wall will tend to bisect an inclusion and be located at the point where the microstress goes through a minimum. If a small field is then applied, the wall will move, but it moves against a force tending to restore it to its original position. This restoring force, if caused by an inclusion, is due mainly to the increase in wall energy and magnetostatic energy resulting from the wall motion. If caused by microstress, the restoring force is due to an increase in wall and magnetoelastic energy.
Hindered wall motion of either kind may be discussed in terms of the variation of a single energy E with wall position x, where E stands for any or all of the various energies mentioned above. We may suppose that E varies with x as in Fig. 9.34a. The gradient of the energy is shown in Fig. 9.34b, along with the line cH representing the pressure of the field on the wall. The value of the constant c depends on the kind of wall and the orientation of the field. For the case of a 1808 wall, c ¼ 2Ms. At H ¼ 0, the wall is at position 1, in an energy minimum. As H is increased from zero, the wall moves reversibly to 2; if the field were removed in this range, the wall would return to 1. But point 2 is a point of maximum energy gradient (maximum restoring force); if the field is sufficient to move the wall to 2, it is sufficient to make the wall take an irreversible jump to 3, which is the only point ahead of the wall with an equally strong restoring force. This is a Barkhausen jump. If the field is then reduced to zero, the wall will go back, not to point 1, but to 4, which is the nearest energy minimum. The wall thus exhibits the phenomena of hysteresis and remanence. A reverse field will then drive the wall reversibly from 4 to 5 and by another
Fig. 9.34 Reversible and irreversible domain wall motion.
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Barkhausen jump from 5 to 6. If the diagram in 9.34b is rotated 908, it takes the form of the elemental hysteresis loop shown in Fig. 9.34c, because wall motion x is equivalent to change in magnetization M, and dE/dx is proportional to H.
This hysteresis loop refers to only one small region of a specimen and to a restricted range of H. The hysteresis loop of a real specimen is the sum of a great number of these elemental loops, of various shapes and sizes, summed over the whole volume of the specimen. Although we are ignorant of the exact form of the E, x curve, diagrams like Fig. 9.34 have proved to be valuable aids to thought in many magnetic problems. And, as will become clear later, Fig. 9.34 can be generalized to include magnetization changes by rotation as well as by wall motion.
One form of energy change not accurately described by Fig. 9.34a is the sudden motion of a wall as it jumps away from the tubular domains connecting it to an inclusion (Fig. 9.29). Further comment on this inherently irreversible process will be made in Section 13.3.
9.12MAGNETIZATION BY ROTATION
We have already made some calculations, in Chapter 7, of magnetization change by rotation in single crystals of particular orientations. We now need to obtain more general results. As it was convenient in the preceding sections on wall motion to consider only one wall, it will now be convenient to consider the rotation process in isolation. We can do this by treating only single-domain particles, in which there are no domain walls. This problem was examined in detail in a classic paper [E. C. Stoner and E. P. Wohlfarth, Phil. Trans. Roy. Soc., A240 (1948) p. 599]. Their calculations have an important bearing on the theory of perma- nent-magnet materials, because some of these materials consist of single domains.
When an applied field rotates the Ms vector of a single domain out of the easy direction, the rotation takes place against the restoring force of some anisotropy, usually the shape, stress, or crystal anisotropy, or some combination of these. We will treat the problem in terms of shape anisotropy, in which most of the other forms can be included. By letting the particle have the shape of an ellipsoid of revolution, we include all the particle shapes of physical interest: rod (prolate spheroid), sphere, and disk (oblate or planetary spheroid).
9.12.1Prolate Spheroid (Cigar)
Let c be the semi-major axis, the axis of revolution, and a the semi-minor axis. Then c is the easy axis of magnetization, and the anisotropy energy is given by
Ea ¼ Ku sin2u, |
(9:35) |
where u is the angle between Ms and c, and the uniaxial anisotropy constant Ku can be written in terms of the demagnetizing coefficients along a and c by means of Equation 7.62. For the present, though, we leave Equation 9.35 in its simple form, because it can also represent the anisotropy energy of a spherical crystal subjected to a stress or that of a spherical crystal with uniaxial crystal anisotropy. The values of Ku for these three forms of anisotropy have been given in Table 8.2.
Let the applied field H make an angle a with the easy axis, as in Fig. 9.35a. Then the potential energy is
Ep ¼ HMs cos (a u), |
(9:36) |
9.12 MAGNETIZATION BY ROTATION |
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Fig. 9.35 Rotation of magnetization by a field applied to a single-domain ellipsoid; h is a reduced field H/HK (see text).
and the total energy is |
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E ¼ Ea þ Ep ¼ Ku sin2 u HMs cos (a u): |
(9:37) |
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The equilibrium position of Ms is given by |
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dE |
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¼ 2Ku sin u cos u HMs sin (a u) ¼ 0, |
(9:38) |
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du |
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and the component of magnetization in the field direction is given by |
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M ¼ Ms cos (a u): |
(9:39) |
Suppose the field is normal to the easy axis, so that a is 908. Then |
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2Ku sin u cos u ¼ HMs cos u,
and
M ¼ Ms sin u:
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Therefore,
M
2Ku Ms ¼ HMs:
Let M/Ms ¼ m ¼ normalized magnetization. Then
m ¼ H |
Ms |
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(9:40) |
2Ku |
This shows that the magnetization is a linear function of H, with no hysteresis. Saturation is achieved when H ¼ HK ¼ (2Ku/Ms) ¼ anisotropy field, as we saw in Equation 7.34 for a similar problem. If we put h ¼ normalized field ¼ H/HK ¼ H(Ms/2Ku), then m ¼ h when a is 908.
For the general case, Equations 9.38 and 9.39 may now be written
sin u cos u h sin (a u) ¼ 0, |
(9:41) |
m ¼ cos (a u), |
(9:42) |
where a is the angle between the field and the easy axis.
Suppose now that the field is along the easy axis (a ¼ 08), and that H and Ms both point along the positive direction of this axis. Then let H be reduced to zero and then increased in the negative direction (a ¼ 1808). Although H and Ms are now antiparallel and the field exerts no torque on Ms, the magnetization will become unstable at u ¼ 08 and will flip over to u ¼ 1808 (parallel with H ) when H reaches a sufficiently high value in the negative direction. To find this critical value we note that a solution to Equation 9.41 does not necessarily correspond to a minimum in the total energy E, a point of stable equilibrium. It might also correspond to an energy maximum (unstable equilibrium), depending on the sign of the second derivative. If d2E/du2 is positive, the equilibrium is stable; if it is negative, the equilibrium is unstable; if it is zero, a condition of stability is just changing to one of instability. Thus the critical field is found by setting
d2E |
¼ cos2 u sin2 u þ h cos (a u) ¼ 0: |
(9:43) |
du2 |
Simultaneous solution of Equations 9.42 and 9.43 leads to the following equations, from which the critical field hc and the critical angle uc, at which the magnetization will flip, may be calculated:
tan3 uc |
¼ tan a, |
(9:44) |
hc2 |
¼ 1 43 sin2 2uc: |
(9:45) |
When a ¼ 1808, uc ¼ 0 and hc ¼ 1, or H ¼ HK. The hysteresis loop is then rectangular, as shown in Fig. 9.38.
The way in which the total energy E varies with the angular position u of the Ms vector for a ¼ 1808 is shown in Fig. 9.37b for various field strengths. It is apparent there how the
9.12 MAGNETIZATION BY ROTATION |
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original energy minimum at u ¼ 0 changes into a maximum when h ¼ hc. These curves are plotted from Equation 9.37. They are the analogs, for the rotational process, of the curve of Fig. 9.34a, which showed how energy varies with domain wall position.
The reduced magnetization m as a function of reduced field h for an intermediate angle, say a ¼ 208, is calculated as follows. For positive values of h, the angle u will vary between 0 and 208. For selected values of u in this range, corresponding values of h and m are found from Equations 9.41 and 9.42. When h is negative, a ¼ 1808 2 208 ¼ 1608, and Equation 9.44 gives the critical value uc at which the magnetization will flip. Values of h and m are again found from Equations 9.41 and 9.42, with a equal to 1608, for selected values of u in the range 0 to uc.
Figure 9.36 shows hysteresis loops calculated for various values of a. In general, these loops consist of reversible and irreversible portions, the latter constituting large single Barkhausen jumps. We see that reversible and irreversible changes in magnetization can occur by domain rotation as well as by domain wall motion. The portion of the total change in m due to irreversible jumps varies from a maximum at a ¼ 08 to zero at a ¼ 908. The critical value of reduced field hc, at which the Ms vector flips from one orientation to another, decreases from 1 at a ¼ 08 to a minimum of 0.5 at a ¼ 458 and then increases to 1 again as a approaches 908. These critical values are equal for any two values of a, such as 208 and 708, symmetrically located about a ¼ 458. On the other hand, the reduced intrinsic coercivity hci (the value of h which reduces m to zero) decreases from 1 at a ¼ 08 to zero at a ¼ 908. For values of a between 0 and 908, cyclic variation of H in a fixed direction has a curious result: Ms makes one complete revolution per cycle, although it does not rotate continuously in the same direction.
Fig. 9.36 Hysteresis loops for single domain particles with uniaxial anisotropy; a is the angle between the field and the easy axis.
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Fig. 9.37 Hysteresis loop of an assembly of noninteracting, randomly oriented, uniaxial single domain particles.
Stoner and Wohlfarth also calculated the hysteresis loop of an assembly of noninteracting particles, with their easy axes randomly oriented in space so that the assembly as a whole is magnetically isotropic (Fig. 9.37). This hysteresis loop is characterized by a retentivity mr of 0.5 and a coercivity hci of 0.48. The stipulation that the particles are not interacting means that the external field of each particle is assumed to have no effect on the behavior of its neighboring particles. This is a serious restriction, which is discussed in Section 11.3.
We will now translate the results of the above calculations from the rather abstract normalized field h into the more concrete actual field H for three kinds of uniaxial anisotropy (shape, stress, and crystal). By definition, H ¼ h(2Ku/Ms). Combining this with the values of Ku from Table 8.2, we have
(shape) |
H ¼ h(Na Nc)Ms, |
(9:46) |
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3lsi |
s |
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(stress) |
H ¼ h |
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(9:47) |
Ms |
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2K |
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(crystal) |
H ¼ h |
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(9:48) |
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Ms |
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where Na and Nc are the demagnetizing factors parallel to the a and c axes, lsi the saturation magnetostriction (assumed isotropic), s the stress, and K1 the crystal anisotropy constant. If we wish to calculate the coercivity, for example, under similar conditions (same inclination a of easy axes to field direction), then h is the same in each of these three equations. This means that the intrinsic coercivity Hci varies directly as Ms for shape anisotropy but inversely with Ms for stress and crystal anisotropy. Thus, to maximize Hci, both the anisotropy and the saturation magnetization of the material must be considered.
As an indication of the coercivities that result from shape anisotropy alone, Table 9.1 gives values of (Na 2 Nc) for various values of c/a, along with calculated Hci values for iron particles (Ms ¼ 1714 emu/cm3 ¼ 1.714 MA/m). Crystal anisotropy is taken to be small, so that Hci for spherical particles is assumed to be zero. We note that the particle shape has to depart only slightly from spherical in order to produce a coercivity of