The Divergent Beam

(Kossel) X-Ray Method

and Its Uses in

Measuring Strain Contours

in an individual Grain of

Fe-3 Weight Percent

Si Transformer Sheet

ho. l^C
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m 3 0

Ot. 10 0 The Divergent Beam (Kosselj X-Ray

Method and Its Uses in Measuring Strain

^ /J Contours in an Individual Grain of Fe-3

Weight Percent Si Transformer Sheet

Harvey Yakowitz

Metallurgy Division

Institute for Materials Research

National Bureau of Standards

Washington, D.C. 20234

6 O

in 5ir fi

c

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS, Richard W. Roberts, Direc/or

Issued August 1973
Library of Congress Catalog Card Number: 72-600265

National Bureau of Standards Monograph 130

I Nat. Bur. Stand. (U.S.), Monogr. 130, 80 pages (Aug. 1973) CODEN: NBSMA6

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. C13.44:130). Price 95 cents. Stock Number 0303 -01120
Contents

Page

I. Introduction 1

II. Problems Relating to Fe-3 Si Transformer Sheet 2

III. The Kossel Method 4

A. The Observed Diffraction Pattern 4

B. Pattern Projection 8

C. Contrast in the Kossel Pattern 8

IV. Measurement of Lattice Strains 10

A. Error Evaluation 12

V. Stress-Strain Analysis 13

VI. Reliability Analysis 19

VII. Equipment for the Study of Materials by the Kossel Method 21

A. Loading Device for the KPG 23

B. Measuring Device for Kossel Patterns 24

VIII. Results of Electron Probe Microanalysis Investigation Seeking Local Chemical

Variations in Fe-3 Si 25

IX. Obtaining the Kossel Stress-Strain Analysis 25

X. Results of the Kossel Internal Stress-Strain Analysis 28

XI. Metallurgical Aspects Pertaining to the KISS Results 31

XII. References 36

Xm. Tables 39

XIV. Figures 60

iii

The Divergent Beam (Kossel) X-Ray Method and Its Uses in Measuring Strain

Contours in an Individual Grain of Fe-3 Weight Percent Si Transformer Sheet

Harvey Yakowitz

Residual and impressed stresses and strains in Fe-3 wt% Si alloy transformer sheet were studied. Results were obtained by the divergent beam (Kossel) x-ray microdiffraction techniques. The optically opaque Fe-3 wt% Si alloy was mapped for residual and impressed stresses and strains. These maps

are roughly analogous to polarized light stress analysis of transparent materials. The results showed

that a variable strain distribution existed in the alloy sheet at the time it was ready for insertion into a transformer core. Small applied compressive and tensile loads tended to rearrange this variable strain distribution but not to appreciably alter the total stored elastic energy.

Key words: Divergent beam x-ray diffraction; electron probe microanalysis; iron-silicon alloy; Kossel;

strain; stress; transformer sheet.

I. Introduction

Iron alloyed with about 3 percent by weight of sil- icon (Fe-3 Si) is one of the most important commercial materials used in the electrical industry. This

alloy is used as core material in virtually all transfor- mers presently manufactured. In fact, Fe-3 Si has

been used in transformers since the early part of the

century. As with most commercial materials, steady

improvements in the material were made with the

primary goal being to eliminate or, more realistically, to reduce to the lowest practical level hysteresis

losses in the transformer core. A prime source for the power losses is that fer- romagnetic crystals are very anisotropic in their

behavior. Body centered cubic (BCC) Fe-3 Si has the

(100) directions (the cube edges) as its easy magnetization directions. Thus, a relatively low field in any edge direction will magnetize Fe-3 Si to saturation. AU other directions are more difficult to magnetize; they cost more input energy for the same

result. For these reasons, Goss [i]' correctly postulated that producing Fe-3 Si having a special grain

orientation would reduce power losses. In Goss Fe-3 Si, the (110) plane is made parallel to the rolling direction which is the [001] direction. This preferred orientation is achieved by secondary

recrystallization. Today the power loss at 50 Hz and

15 kG is about 0.4 to 0.5 watts/pound for 12 mil (0.305

' Italicized figures in brackets indicate the literature references on pages 36 and 37.

mm) thick laminations when this material is used in transformers. This material has been in continuous use since 1937. In 1957, the so-called cube texture denoted by

(100) [001] orientation was developed by Assmus

[2]. This material has almost equal magnetic properties in two directions at right angles within the

sheet. Power losses can be reduced and the trans- former designer has a bit more latitude in working

with this material. But the manufacturing process for the cube texture is more expensive and difficult than

that for the Goss texture. Thus, an economic balancing act of improved performance savings versus in- creased material costs must be performed. At

present, economic considerations still favor the Goss

texture for most applications. Hence, the material

chosen for study here is the Goss oriented Fe-3 Si. For orientation purposes, the two textures are shown

schematically in figure 1.'^ Low level stresses in Goss Fe-3 Si can appreciably

alter both power losses and magnetostriction [3,4]-

Hysteresis losses pass through a minimum at tensile stresses of a few hundred psi [3,5]. This result sug- gests that inhomogeneous residual stresses may be

present in the original material which is coarse- grained (> 1 mm) polycrystalline sheet. The

presence of the grain boundaries is known to in- crease average power loss as well [5]. Another

source of loss is that temperature gradients of up to 200 °C can occur between the center and the edge of

^ AU figures are at the end of this publication beginning on page 60.

1
a transformer core. These in turn can generate inter- nal stresses in the core laminates of 400 to 500 psi.

Stresses above 500 psi (0.35 MN/m^) cause power

losses to increase rapidly [4].

To date, no one has correlated quantitative local

stress-strain characteristics in Fe-3 Si w^ith the rela- tively large amount of electrical property data now

available. There is at least one good experimental

method available for locally determining the strain state of individual grains in a polycrystaUine material—the divergent beam diffraction method usually

referred to as the Kossel method [6]. With this method, the entire thickness of the transformer

sheet can be sampled; the total volume of crystal

sampled is only about 200 /Ltm^ for 3 mil (0.076 mm)

thick Fe-3 Si sheet. The Kossel method can be made to yield data giving (1) lattice strains, (2) Cauchy strains, (3) principal

strains, (4) principal strain axes, (5) maximum shear

strain in any (hkl) plane (known), (6) principal

stresses, (7) principal stress axes, and (8) total stored

elastic energy in the irradiated region of the crystal.

These quantities can be determined to within a probable error of about ± 10 percent relative. The stress-strain configuration at individual re- gions within a grain of a polycrystalline sheet can

thus be investigated by the Kossel method. However,

such studies have not been carried out previously on any material. In any study of local stress-strain variations, possible variations in chemical composition

must be taken into account. Electron probe

microanalysis can be used to correlate local variations in Si content with the stress-strain data.

II. Problems Relating to Fe-3 Si

Transformer Sheet

Attention will now be shifted to a detailed discussion of Fe-3 Si — especially from the point of view of

the effect of strain on such quantities as power

losses and magnetostrictive effects. Then we shall see how information provided by the Kossel method

can be meshed with the electrical information to in- dicate substantive explanations for some of the electrical observations.

The knowledge of the effects of stress on the magnetic and electrical properties of Fe-3 Si can be

distilled into ten basic ideas as follows:

1. All experimental and theoretical studies agree

that small, 0.7 MN/m^ (1000 psi) stresses, either tensile or compressive, degrade the magnetic and electrical properties. For instance, compressive stresses of 0.3 MN/m^ (400 psi) increase power losses by 20 to 30 percent [4].

2. There is the belief that small residual stresses,

say 0.1 to 0.35 MN/m^ (100 to 500 psi), remain in the

material that is used commercially [3,4,5].

3. Certain observed improvements in the magnetic and electrical properties occurring after exter- nally applied stresses of 0.1 to 0.35 MN/m^ (100 to 500 psi) are thought to result from overcoming

residual stress effects.

4. It has been predicted theoretically [7] and

shown experimentaUy [8] that the magnetization curve (B vs H) deteriorates under applied tensile stresses. Consequently, both hysteresis and power

losses can be expected to increase wdth increasino^

tensile stress [4].

5. Magnetic domain studies at low values of applied stress [9,10,11] show complete domain rear- rangement as the stress level is increased to 0.7 to 1.4 MN/m2 (1000 to 2000 psi). New domain structures appear to be completely established at 1.4 MN/m^

(2000 psi) [9]. However, observable domain struc- tures are, of necessity, surface domains. Thus, no in- formation is available on domain configurations

within the Fe-3 Si sheet. Therefore, any attempt to explain the effect of applied macrostresses in terms

of local events within the Fe-3 Si sheet is open to question.

6. The magnetization, B, degrades more rapidly

under compression as compared to tension [4].

7. Magnetostrictive strain increases rapidly with

compressive stress but falls to and remains nearly

zero at tensile stresses of 200 psi or more [12].

8. Power losses are apt to be two to three times as large at grain boundaries or at misaligned grains as they are for the average of the material [4] (see fig.

2).9. The effect of increasing the temperature of the

sheet is to cause the losses to fall by a few percent

per 100 °C rise. The applied stress has little effect on

this observation [4].

10. The alternating power loss in the cube edge

orientation is only slightly less than that of the Goss

oriented Fe-3 Si. In brief, stresses always cause problems. There

are some clues as to the cause of these problems

based on domain structure studies. There is also evidence that the effect of the stresses is not

2
homogeneous. The latter point tends to indicate that the stresses cannot be homogeneously distributed.

Why would stresses — residual or purposely ap- plied—originate in the commercial Fe-3 Si sheets?

In order to answer this question, it is necessary to look briefly at the method of preparation of the

material and also at the way the finished Fe-3 Si

sheets are used. Figure 3 is a flow sheet showing the

basic steps in the production of the sheets. The portions which need to be examined in more detail are

the 1200 °C batch anneal and, to a lesser extent, the

thermal flattening process. The former is of interest

since it is in this step that the desired texture is developed, while the latter is a process which may

introduce stresses. The initiation of the growth of the Goss texture oc- curs, to a large extent, as a result of the presence of

a suitably dispersed second phase (usually MnS, but

lately Si3N4 has been used) which allows the desired

secondary texture (Goss) to begin to develop and

grow. Naturally, it is necessary for there to be some

correctly oriented grains in the primary texture. The

dispersion of the second phase impurity is controlled

by the cooling rate from the solution temperature

during the hot-rolling step. Some of the sulfur bearing impurities are removed during the final anneal by

a reaction involving the coating.

Once the secondary recrystallization is begun, the

Fe-3 Si strips are brought to the 1200 °C annealing

temperature. The batch anneal is best carried out in alumina tubes of the highest purity obtainable. At

1200 °C, the (110) crystals grow rapidly so that the

secondary recrystallization is completed quickly.

The 1200 °C anneal favors the Goss orientation;

lower annealing temperatures favor the (100) [001]

texture [13]. Furthermore, the Goss orientation

grows more rapidly and more easily in thin, ^ 0.3 mm, (12 mil) Fe-3 Si sheets than in thicker sheets. These observations can be explained using Mullins' theory [14]. The orientation relationships

between the primary and the secondary— very

large — grains are consistent with the dependence of

surface free-energy on orientation. Surface free- energy unbalances will produce growth of large tex- tured grains. May and TurnbuU [75] postulated that the (110) [001] grains have the lowest average sur- face free-energies of all the grains. It has been found

that the large grained texture type growth occurs most easily at the surface [14]. As MuUins puts it, a low surface-energy crystal wets the surface of the

Fe-3 Si sheet from underneath. The dispersed phase

inhibits normal grain growth of the primary

recrystallized matrix. Thus, the required surface

energy unbalance is provided and the (110) [001]

grain boundary is released for growth. When the

sheet is brought to 1200 °C, the surface tension at the

boundary is the driving force for the rapid completion of the secondary recrystallization. For

undisturbed surface conditions, the close packed

(110) crystal plane is then preferred [14]. This

hypothesis fits the observation that the Goss orientation grows more favorably in thin sheets since there

the boundary is under greater surface tension and

hence will migrate more rapidly [13]. The end result

is a coarse-grained Goss oriented texture containing

85 to nearly 100 percent of (110) [001] grains.

Apparently, the interaction of the energetic rela- tions favoring the growth of the desired texture is very critical. A small cold-rolling strain applied after primary recrystallization but before the texture anneal causes the (110) [001] orientation to be almost

nonexistent [15].

After the texture-anneal is complete, a thermal

flattening process is carried out. This is necessary so

that no compressive stresses are introduced during

actual transformer construction. For thermal flatten- ing, tension is applied while the strip is at a temperature of 400-500 °C. A stress-relief anneal is then car- ried out and the sheet is ready for use. In view of the findings about the deleterious ef- fects of small stresses and the observation that the

electrical properties vary with test position in the

sheet, the stress distribution within the sheet would seem to be of great importance. Especially of in- terest is whether or not any residual stress rises to a maximum near or at grain boundaries in the sheet or whether or not these stresses are compressive or

tensile in nature. If residual stresses are predominantly compressive in nature, the sheet can be

placed under an external tensile load in order to see

if the overall stress-strain configuration tends toward

a no-stress condition. This would not only confirm

the conjecture that such a rearrangement occurs; it would also show how the rearrangement occurs. Of

course, the question "Why are the residual stresses predominantly compressive even after the (tensile)

thermal flattening step and process annealing?"

would remain to be considered. A knowledge of the

possible local variation of silicon content within the

grains of the sheet is of great interest in considering

such a question. The addition of Si to the Fe lattice causes an overall shrinkage of the unit cell of the lat3
tice; figure 4 shows the unit cell size versus percentage Si added. Thus, an inhomogeneous Si distribution might be expected to create very definite local

lattice strains. A set of definitive experiments aimed at answering

the foregoing questions was designed. The Kossel

method was used to evaluate local stress-strain configurations in the as-received Fe-3 Si sheet. Then,

with an appropriate tensile loading device available,

low tensile stresses were applied in order to observe

their effect on the original stress-strain configuration. Naturally, the same regions were studied in these tests. Finally, local Si content variations were

observed by means of electron probe microanalysis.

The desired result was to obtain a point-by-point

local analysis for Si within the grain.

At the time this study was first contemplated

(1963), the Kossel method was still largely a laborato- ry curiosity. However, in recent years the technique

has been developed to the point where reliable measurements can be made in a fairly routine manner.

III. The Kossel Method

A. The Observed Diffraction Pattern

The Kossel and pseudo-Kossel methods can be

employed for generating divergent beam x-ray patterns. In the Kossel method, the x-ray source is excited within the specimen itself by a focused elec- tron probe or a collimated x-ray beam of suitable energy. In this case, the atoms of the crystal act as independent sources of monochromatic radiation,

and the waves diverging from these excited atoms

will be diffracted by atom planes of the lattice. The

effect of such diffraction is to give rise to a wavefield which exits from the crystal and in which the

radial directions lying at the Bragg angle, 6, with

respect to the diffracting planes form right circular cones. Thus, the external wave-field may be con- sidered to lie on the loci of a set of cones rigidly fixed

to the source of x-ray excitation. A flat film intercepting this system of waves shows a set of conic sections superimposed upon a general background

blackening. The axis of any specific cone is the nor- mal to its plane of diffraction; the semi-apex angle of such a cone is (9O°-0), where 6 is the Bragg angle.

The attenuation of the diverging primary beam is often greatly increased along the set of cone ele- ments by the x-ray extinction effects produced

within the crystal. Whether the appropriate extinction effect is purely due to primary or secondary extinction or, as is most likely, a combination of both,

depends upon the state of perfection of the specimen

crystal. This point will be examined in detail later. For the present discussion, it is sufficient to state that the primary beam radiation transmitted by the

crystal may be deficient in intensity along the cone

elements. Therefore, a film placed to intercept the

transmitted radiation will often show conic sections

lighter than the general background; these are called deficiency conies.

Furthermore, diffracting planes wiU lie at some

angle, <l>, with respect to the crystal surface nearest

the source (the entrance surface). If ^> is greater than

the Bragg angle, 6, then a diffraction conic containing an excess of energy with respect to the

background may be observed in the transmission

pattern. Hence, both dark and light conies may be

expected on a transmission pattern. In back-reflection, the film is placed on the same

side of the crystal as the source. Diffraction conies

darker than background will be observed for planes

whose Bragg angle, 6, is greater than the angle ^> in the case of crystals having an appreciable mosaic

spread, For a perfect crystal in which ^ is negligible, conies or portions of conies may appear lighter

than background.

Figure 5, taken from Sharpe [16], shows the

geometry for the formation of both diffraction and

deficiency conies for the case of a mosaic crystal.

Note that the diffraction conic is represented as being somewhat broadened with respect to the defi- ciency conic. This is the case since reflection occurs from a large number of parallel planes. For each

plane, there is a cone of semi-apex angle (90-6) but all cones have vertices displaced by one interplanar

distance, d. Therefore, reflection conies are generally not as sharp as deficiency conies, all of

which have the x-ray source as a common vertex [171

Another extremely important feature is that each

separate plane of a family is represented by a separate conic, i.e., there is no averaging effect as in a polycrystalline diffractometer. Thus, even in a cubic material, if the conic trace of (fikl) is seen, then

the {hkt) trace also appears as a separate entity. This

circumstance is the experimental keystone on which

strain analysis by the Kossel method rests. The pseudo-Kossel case is entirely analogous to the foregoing except that the source is external to the

4
sample, i.e., the apices of the cones are external to the parent crystal. In the pseudo-Kossel method, the source is usually a thin foil of the material whose

radiation is desired. This source foil is either placed

directly upon or slightly above the crystal of interest; the 180° divergence of x rays in the sample direction

gives rise to the same behavior of the diffracted

energy as in the Kossel case. To continue this quaUtative discussion of the

divergent beam pattern, consider now the geometry

of the conies observed on a film. Maier [18] gave the

conic shape as a function of the Bragg angle and the

angle <I>. These results suggest the type of pattern

expected, in that d, <I> and the observed conies are mutually dependent.

In the case where <!> is greater than 9, a hyperbola

is formed. Diffracted radiation from planes lying

such that <J> is greater than 6 will not exit the crystal on the same side as the source. Hence, these conies

will never be observed on a back reflection pattern,

but rather only on a transmission photograph. Figure

5 is seen to conform in this case (4> > 6). A deficiency conic from planes (hkl) is also formed and appears

as a mirror image of the diffraction conic on the film. (In practice, a true mirror image may not be observed due to projection distortion effects on a flat film.) Lonsdale has pointed out that if both light and

dark conies from planes (hkl) are recorded on the

film, then complementary light and dark conies from

planes (hkl) will also be recorded. The dark Une from

planes (hkl) is parallel to the light line from planes

(hfcT) and vice versa. The physical separation

between the parallel complementary light and dark

conies depends upon the distance from the x-ray source to the opposite side of the crystal taken in a

direction perpendicular to the film [19].

For 4) equal to 6, a parabola results. In this case, both the back reflection and transmission patterns

will show diffraction conies and in the transmission

pattern a complementary deficiency conic will also be present. The probability for <I> equal to 9 is ex- tremely small for any given case. When $ is less than 9. ellipses occur. Any diffrac- tion conic trace wiU appear only on the same side of the crystal as the source. Thus, in back reflection, a

series of ellipse-like figures wiU be observed on the

pattern. In transmission, the corresponding deficiency ellipses will result; they will be unaccompanied

by complementary diffraction conies. As EUis and

Weissman pointed out, strictly speaking the curves

are of higher order than ellipses but the deviation is negligible [20 \. For <t» equal to zero, a circle is obtained. This is a special case of <I> less than 9 and the comments for that situation apply. It should be noted that a circular trace indicates immediately that some plane is parallel to the crystal surface which is usually placed

parallel to the film. Summarizing, a back reflection pattern of a real crystal may be expected to contain largely elUptical

conies with perhaps some circular traces all of which

are darker than background. A transmission pattern

may contain complementary hght and dark hyperbolic traces, light elliptical traces, and perhaps some

light circular traces. The chances of finding a parabolic trace in either type of pattern are very

slight.

To this quaUtative view of the divergent beam

photograph must be added considerations concerning the intensity and contrast of the conies on the

film with respect to background. Contrast is spoken

of rather than intensity for deficiency conies since

these conies, as such, do not represent energy

deposited per unit area but rather the absence of energy. Contrast is taken to mean that portion of the

peak appearing above or below the general

background on the film. When discussing the dif- fraction cones, intensity will be used as the descriptive term. Two cases are to be considered: (1) the diffracting

volume comes from a portion of the crystal having a

finite mosaic spread, ^, and (2) the diffracting volume

has C of zero. The former wiU cause secondary ex- tinction of the incident divergent x-ray beam; primary extinction may be present to some degree. The

latter wiU cause only primary extinction to be operable and may be called "perfect" crystal volume for purposes of this discussion. Extinction exists when

the measured integrated intensity of a conic is less than that predicted by the mosaic formula [21].

The following brief discussion of extinction is greatly simplified. The concept of primary and

secondary extinction is itself rather idealized [21].

An extensive discussion and development of the

topic wiU be found in James [17].

Primary extinction is usually said to operate when

diffraction occurs from large "perfect" regions of the

crystal. These regions may be called coherent

domains or large mosaic blocks. In this case, the in- tegrated diffracted intensity is not proportional to

497-814 0 - 73 -2
the volume of the coherent domain. The attenuation

of the primary beam is large in such a block due to multiple reflections of the beam by the perfect lat- tice. Instead of the usual x-ray absorption coeffi- cient, the crystal is said to have an extinction coeffi- cient, T, such that Irlh = exp-(7r;t). The value of t may be found from

:

Calculation will show that t is usually two to five or- ders of magnitude greater than the ordinary linear x- ray attenuation coefficient, (jl. Secondary extinction is said to be operable when

the coherent domains are small enough so that in a

single block, the effect of t is negligible. However,

the crystal is considered to be composed of a number

of these smaller blocks most of which are closely

parallel to one another. Hence, a loss of intensity in the primary beam is experienced by a robbing due to diffraction by properly oriented domains above the

domain of interest [21].

When secondary extinction is dominant, the at- tenuation of a monochromatic ray of the primary

beam incident upon the crystal at an angle {d±u)

where « is a small angular quantity is apparently

enhanced. Peace and Pringle [22] have expressed

this attenuation as:

exp-(/i.A:) = exp — {2/A+G. (u)]jc} (2)

Equation (2) is in effect the probability of survival of a packet of energy [22]. It is possible to define a secondary extinction coefficient "g" such that:

[l/2V27r] [exp-{u'l2C')] (3)

The total amount of secondary extinction present is then given by

G'{u) = gQ' (4)

Q' is the volume reflecting power of a mosaic crystal

reduced by the presence of primary extinction, i.e., the actual reflecting power.

It should be noted that "g'' is peculiar to the sample at hand and is not a general physical property.

Furthermore, the mosaic spread, ^, which is the root- mean-square of random rotations of small magnitude

of the coherent domains about a common axis (as- suming a normal distribution) wiU usually vary for different planes of the same crystal. Thus "gr" may

be expected to vary slightly from plane to plane

within the crystal. The value of Q' decreases as sindlk increases. Therefore, G'(u) decreases as planar index increases.

Peace and Pringle [22] deduced that the deficiency line breadth, W, depends upon the amount of secondary extinction present rather than being pro- portional to the mosaic spread. Their expression for the line breadth is W= {I- exp[-2G' (u) X seed] du (5)

J — 00

The value of W/C, can then be plotted against Q' X

secd/i, as shown in figure 6. Among other things, this curve shows that the contrast of unresolved lines or the breadth of resolved lines increases very slowly

with Q' in the case where a large amount of secondary extinction is present. By resolved and unresolved

are meant the angular structure of the deficiency

conic being observed or not observed [22].

The observation of the angular structure depends

on the crystal to film distance, Z, and the source

size. Under the usual experimental conditions for transmission Kossel photography, the crystal to film distance is about 10 cm, and the x-ray source size is about a 3 to 30 /xm diameter hemisphere. For a value

of { of 1 minute of arc, the angular structure width is 30 ^im at a 10 cm distance. However, blurring due to multiple reflections also occurs causing a resolution

decrease. Furthermore, the great majority of defi- ciency conies observed have their angular structure unresolved. For the case of unresolved deficiency

conies the contrast, a, in the presence of a background due to continuum radiation has been

given as [22]:

r

o"= 1 (6)

5[1 + (K) exp ifjLx — fix) ]

The contrast is thus a sensitive function of the

crystal thickness, x, the operating voltage used to produce the x rays (through jl), the x-ray source size,

s, the mosaic spread, the amount and distribution of

both primary and secondary extinction, and finally

of the film response and exposure used. While some

conclusions on proper values of jc and operating volt- age in order to maximize cr can be drawn [23],

values of "g" are not usually known for the crystal of

interest. Therefore, measurements of the contrast of

deficiency conies may not give unambiguous infor- mation concerning the crystal.

6
It is now necessary to consider the intensity to be

expected for diffraction conies found on a transmission Kossel pattern, i.e., from planes where ^ > 6. Again the crystal is assumed to have a finite mosaic

spread. Recalling when secondary extinction is dominant that ijl' = + gQ' , the integrated intensity,

P, for the transmitted diffraction conies is given by

James for the case of 4> = 90° as [17]:

P=(Q'x secO) (exp- [(/a + gQ')x seed]) (7)

For cases where ^> is not 90°, the relation for P is of

the form of eq (7) except that extra geometrical factors must be included [24]. In any case, for a given

Q' and 6, departure of <I> from 90° lowers the value of

P. Equation (7) is valid for a single reflection; however, diffraction conies are usually composed of multiple reflections overlapping one another due to the

previously mentioned cone vertex displacement.

Peace and Pringle were able to deduce, by a statisti- cal argument, the intensity issuing from a given point

in the range of multiple reflections. The expression

depends upon the same variables as eq (7) [22].

From this they were able to show that the total spread, St, of the emergent beam including the

direct ray is not less than:

St =

X sin 0

sin 4) sin (<I> — 6)

(<I) > d) (8)

If this result is coupled with Lonsdale's relation

for the separation of black and white conies in trans- mission, the condition for black and white overlap can be found [19]. Overlap will cause intensity and

contrast cancellation. This in turn will make

meaningful measurements involving the lines dif- ficult if not impossible to obtain. Overlap occurs if the inequality of eq (9) is fulfilled.

sin >

sin 26

2 sin O sin — 6) sin^ 4> — sin^ (O > 0)

(9)

Diffraction conies observed on a back reflection

pattern from a mosaic crystal are entirely analogous

to those produced in transmission. The value of Pb

is different, however [17],

Pr =

Q'

2 ifi + gO')

(10)

Equation (8) is also still applicable so that the lateral line spread can be calculated. Consider now the mosaic crystal in which the

domains are of random orientation. Lonsdale states

that all secondary extinction effects will be smeared

out in such a case [19]. This is tantamount to setting

g equal to zero in the foregoing discussion; Q' may

also increase to Q. This will lead to no deficiency

conic contrast and an increased background absorption in transmission. Diffraction conies may appear

in transmission and in back reflection since their in- tegrated intensity is increased by setting g equal to zero. It might be parenthetically stated that this condition is apparently difficult to obtain since an iron

foil broken in torsion gave deficiency conic contrast from a point about 10 ^tm from the fracture surface. Perfect crystal regions can now be considered.

Primary extinction is operable but secondary extinction is negligible. This means that the attenuation

coefficient, t, is far more important than fx. Typically, the primary beam intensity loss is greater than

90 percent after only two or three micrometers depth

within the crystal. Thus, deficiency conic contrast

exists for thicknesses greater than this. The continu- ous background still appears as the primary beam in- tensity attenuation is still exp jiZx away from Bragg

angles. However, the line is extremely sharp, ranging

from far less than one second to perhaps one minute

of arc for a perfect crystal; the higher the planar in- dex, the less the line breadth [17]. In this angular

range, essentially total reflection of the primary

beam occurs. Hence, one would expect very high

deficiency conic contrast. Unfortunately, such contrast is usually not obtained.

The reasons for this lack of contrast may be

severalfold. First, the source size, 5, must be as small

as practicable in order to prevent blurring of the

deficiency conies and direct contrast loss [19,22].

Second, graininess of the film may obliterate the

line; this may also cause the line to appear

broadened by truncating the peak [24]. Therefore,

the finest grained film possible must be employed

when perfect crystals are to be investigated. Finally,

if the crystal is too thick, exp(/Ix) lightens the

background and causes contrast loss. One other comment on contrast is germane:

James indicates that the intensity drop-off within the cones should be proportional to where Z is the

source-to-film distance. Thus, for equal exposures,

the greater Z is, the better the contrast for diffrac- tion conies [17], This is not true for deficiency

7
conies in general; their contrast is relatively independent of Z and may in fact become worse as Z

increases [i6]. The reason is that the intensity defect of the conic is fixed by the crystal perfection.

The lightened background resulting from increasing

Z may cause less white line contrast. Having considered the formation and contrast of

the lines on the pattern, it is necessary to explore the means for indexing patterns, determining lattice spacings and orientations, and to consider the effect of lattice imperfections.

B. Pattern Projection

The pattern which would be observed on a spherical film surrounding the sample consists of circles,

the size and position of each of which depends upon

4>, and the angle between a reference diameter on

the sphere and the cone axis [^9]. However, it is not

feasible to use spherical film, and it is almost universal practice to use a flat film placed normal to the

direction of excitation of the x-ray source. Using the

perpendicular source to film distance, Z, as a radi- us, a sphere can be constructed. The film is tangent

to this sphere at a point along the normal to the plane

containing the source and the film plane. The line joining this point and the source of x rays wiU be

called the projection axis; the point of intersection

with the film is in fact the geometric pattern center. Although it is also referred to as the "film center,"

the point does not lie at the geometric center of the

sheet of film in general.

The pattern observed on such a film is a gnomonic

projection. The projection of any circle from the

sphere onto the film plane is a conic section. Figure

7 suggests the process involved. The gnomonic projection means the angle x which is equivalent to arc nD on the sphere is given by x = n'D\DC on flat film. Lonsdale suggested the use of the stereographic pro- jection to represent the expected pattern. In this case, circles on the sphere project as circles on the

projection having angular radii, (9O°-0). However, the

projection of the center of the circle from the sphere

is not the center of the projected circle; thus, lines which are parallel on the sphere are not parallel in the stereographic projection [i9].

To plot the projection requires that one know A and however many lattice parameters there are. Then (90°- Q) is obtained as cos^' (A./2d). It should be

mentioned that no plane for which d ^ 0.5A. will give

rise to an observable conic. Furthermore, systematic

conic absences must be taken into account. The next

step is to plot the poles of expected conies with

respect to the arbitrarily chosen pole. In the cubic

system, these are always in the same place; for systems of lower symmetry, polar positions must be

calculated for each case. Three (90°— Q) positions are

plotted using the pole as center; then a circle is constructed through these points. This is the stereo- graphic projection of a single plane for a single

wavelength. An example is shown in figure 8a for Goss oriented Fe-3 Si. The corresponding Kossel

pattern is shown in figure 8b. The preparation of such projections by hand is tedious, time consuming, and subject to large er- rors—especially when the projection contains many

conies. This difficulty has been overcome by Frazer

and Arrhenius who have programmed a digital computer equipped with an Z-F plotter to prepare stereo- graphic projections for any set of lattice spacings

and wavelengths [25]. When a stereographic projection is available, indexing can be carried out directly

by comparison with the film; the gnomonic and

stereographic projections are quite similar over the

range of angles normally recorded on the film.

C. Contrast in the Kossel Pattern

A comparison of the back-reflection and transmission modes indicated that for the Fe-3 Si study, the

transmission mode was the more suitable. The

reasons for this choice were: (1) Actual commercial

laminates could be examined throughout the entire

thickness, (2) Surface effects would not interfere

with the analysis (back reflection methods sample

regions only a few micrometers deep at the surface),

(3) Deficiency conies are sharper than reflection

conies, and (4) Contrast in transmission can often be

made superior to that in back reflection by a careful choice of experimental conditions.

This contrast optimization has been studied in detail [23]. Deficiency conies are most prevalent in a transmission pattern. However, experimentally it has been observed on Kossel photographs that defi- ciency conies are often of weak contrast and occasionally even absent {^19^6,27^B^. This has led several workers to consider the question of contrast optimization particularly with respect to the sample

thickness.

8
Lonsdale was the first to study this problem and

presented data for "optimum thickness" values in NaCl and diamond [19]. Later in a somewhat more

general treatment Peace and Pringle [22] were led

to an optimum sample attenuation corresponding to )u,x= L Subsequently, Hanneman, et al. proposed on

an empirical basis that /xx of five was an optimum

value [29]. More recently. Potts stated that the contrast was relatively independent of the sample

thickness [30]. All of these workers appear to have

adopted the same definition of contrast. However,

none of these workers investigated the effect of the

operating voltage on the contrast. Equation (6) can

be used to estimate the thickness-operating voltage

relationship [23].

In Kossel transmission work, the radiation used is generally chosen on the basis of crystaUographic

considerations. For example, a particular

wavelength which yields sensitive intersections for precise lattice spacing measurements may be

chosen. This fixes the value of the linear absorption

coefficient. Furthermore, the values of Q' and ^ are

also predetermined. The value of "s" is fixed by the

capabilities of the electron probe forming column.

Thus, the only variables left with respect to which

contrast may be adjusted are the sample thickness

and the operating voltage. The operating voltage enters since Jiis a sensitive function of the continuum

distribution which is manifestly a function of operating voltage. These considerations suggest that there

may be a thickness at a given voltage which will yield a maximum contrast or a voltage at a given thickness

which will do likewise.

Consider the denominator of eq (6) which needs to be minimized in order to increase contrast. If Jix >

/ix, then cr can be raised by merely raising the

thickness or fx. However, the total intensity, /, reaching the film is given by /= /o/"*^^. Therefore, if Jlx is increased much beyond 3, there will be little blackening of the film with time, i.e., exposure times

become prohibitively long. In fact, one would tend to prefer fix — 1 in order to maximize any diffraction

conies intercepted and to reduce the exposure time.

It would seem, therefore, that values of jix = l > /xx would almost always yield high contrast patterns.

Unfortunately, the nature of the numerator of eq (6) precludes this simple solution. Peace and Pringle have plotted {WlQ versus

(Q'xl^,); this plot is reproduced as figure 6. At values

of (Q'xIO greater than six, increasing the thickness

has little effect on the values of and hence upon

the contrast. However, if x is too low, the value of

(Q'xiQ may drop below six, especially for high index

planes for which Q' may have a small value [17].

This will make the contrast a critical function of x and may well cause the loss of some high index lines. Therefore, since (Q'x/Q is generally unknown, it is necessary to increase x to insure that the region in which 6 > {Q'xiQ is avoided for all lines to be

recorded on the pattern. If this is done a pattern of

nearly uniform contrast can be obtained owing to the nature of the function in figure 6. The next step in the evaluation of the situation is to estimate values of "/Z". Let Ji be defined as b/j. where "6" is a factor to be determined. The factor

"6" depends upon the short wavelength limit of the

continuum, kc, the emitted intensity as a function of

wavelength, /o(A), the thickness x, the long

wavelength limit, X/, of the spectrum, and the trans- mission of the sample as a function of wavelength,

exp - [/i(\)x]

An approximate relation giving the value of "6"

can be obtained for incident electrons of given ener- gy by using Kramers' approximation to the shape of

the continuum band from a thick target [31] coupled

with the function [/i(A.)] which is tabulated [32].

It is found that "6" increases with decreasing

thickness at a given operating voltage and also that "6" increases with decreasing operating voltage at a given thickness [23],

Three considerations are important in choosing a

suitable thickness-operating voltage relationship.

When the overvoltage ratio (ElEo), denoted by "f/",

is decreased below two, characteristic peak to background ratios decrease rapidly as does the absolute spectral intensity. Furthermore, it should be

recalled that the thickness chosen should yield a

Q'xjt, value greater than six. Opposing this is the fact that a (jiZx) value greater than three gives prohibitively long exposure times coupled with an increased K

value in eq (6). It therefore appears that for most

cases, a choice of 1.5 ^ /xx < 2.5 combined with 2.5 ^ ^ 3.0 will yield transmission Kossel or pseudoKossel photographs having nearly optimum contrast

characteristics. Such a choice reduces the need for extremely thin, hard-to-prepare samples, yet gives a high enough overvoltage ratio to produce enough x- ray intensity for good photographic characteristics.

Furthermore, such thickness values tend to help the

value o{{Q'xlQ rise above six [23].

9
Thus, the thickness required can be chosen and

specimen preparation carried out accordingly. While

a transmission pattern is representative of the entire

specimen thickness used, it is desirable to carry out specimen preparation so that the inherent crystal

perfection is not disturbed. It is possible to do this for Fe-3 Si by electropolishing to the final surface

finish and thickness desired. The appropriate elec- tropolishing solution is 5 percent HCIO3 in CH3COOH run at 30 V, 3 amp/decimeter (temp. < 24

°C). A perfect metallographic finish is not really

required for good transmission Kossel work.

In order to take full advantage of the properly

prepared specimen for obtaining good contrast, the

exposure time must be correct. According to Lonsdale, the exposure time of a Kossel photograph has an optimum value [19]. Depending upon the experimental conditions, the exposure time may vary from

a few seconds to several hours [23,28]. Therefore,

it is necessary to be able to estimate closely the ex- posure time.

Yakowitz and Vieth obtained semiempirical rela- tions for the exposure time which are valid to about

±10 or 15 percent [34]. For the case of the film

being in vacuum and a pseudo-Kossel experiment,

they obtained for a transmission pattern as shown in figure 8.

_ 2.7 X 10-i5£oZMl - r) (sec^T?'

)

ts — : nois

{exp {2l3)[fifXf+ fXsXs]} (11)

Equation (11) takes into account the number of

photons produced per incident electron, the operating voltage (through no) and the fact that the exposure is greatest at the central portion of the film and

least at the edges. The latter is the result of a combination of two effects, the first of which is an in- creasing absorption path within the sample with in- creasing emergence angle. The other effect is geometrical in that a unit of solid angle subtends a

larger area on the film at the edge than at the center. The quantity Eo is that exposure density yielding

the maximum contrast between the Kossel conies

and the background blackening for a given film at a given distance, Z. It must be determined empirically. Once this is done, it is to be expected that Eo

will be essentially constant for a given film type independent of other camera parameters [34]. The £'0

value represents a compromise between the exposure at the edge and central portions of the film. Based on the previous discussion, it is clear that,

by a careful choice of experimental conditions,

transmission Kossel patterns of Fe-3 Si having good

contrast can be prepared. The sensitivity of the contrast to experimental conditions implies that using

contrast measurements may not be physicedly

meaningful.

Certain authors have postulated that changes in atomic position associated with changes in the physical state of a crystal could be traced by consecutive

measurements of intensity changes (sic) in a critical set of deficiency conies [25] . However, the contrast

of a transmission Kossel pattern is a critical function

of the x-ray source size, crystal thickness and mosaic

spread, extinction characteristics of the crystal,

operating voltage, exposure time, development

technique, and the exciting wavelength spectrum.

The overall deficiency of intensity is strongly dependent on the crystal perfection which is apt to vary with temperature and with any small change in strain state; less strong is the dependence on atom

position changes. In fact, according to figure 6 and

eq (6), in some cases, the apparent contrast for a given plane form can vary by a factor of two for a

relatively smaU change in crystal perfection. However, a similar change in a thicker sample of the same crystal may lead to a contrast difference so small as to be nearly undetectable (plateau region of

fig. 6). For these reasons, measurements of the contrast of deficiency conies are probably not useful as quantitative indicators of the physical state of the

specimen crystal.

IV. Measurement of Lattice Strains

We are interested in internal strains on a microscopic scale. For the measurement of internal

strains, it is customary to use the quantity ^d/d

where dhki is the interplanar spacing for a crystallographic plane having Miller indexes hkl. Once these

lattice strains have been determined experimentally,

elastic theory can be applied in order to obtain the

complete strain picture including principal strains,

principal stresses and their respective axes, stored

elastic energy, shear stored elastic energy and max- imum shear strain in any plane.

The Kossel method gives a pattern on which all planes of a multiplicity set are separated, i.e., if hkl

10
appears, so does hkl, hkl, etc. This multiplicity

separation occurs for all crystallographic lattices in- cluding cubic. The multiplicity separation provides

two unique advantages for the Kossel method:

1. Anisotropic strains, whatever their source, can

be unequivocally located and evaluated;

2. Having separate lattice strain values for each

plane removes the necessity for complex "strain ef- fect on an x-ray line" averaging as is required in the

Warren-Averbach [35] x-ray strain analysis treat- ment. (Unfolding is required in other experimental

x-ray anedysis — of cubic materials because all planes

of a multiplicity set appear as one peak— a con- sequence of Bragg's law and the instrumentation.)

Since all other strain information is inferred from

the individual ^djd values these must be as accurate

as possible. Therefore, measurement methods for the d values must be evaluated in terms of applicability and sources of error. Four basic methods for determining lattice spacing data by means of the transmission Kossel

technique have been described [36^9,37,38^39] :

1. The tangency method [36] requires that two

conies on the Kossel pattern be nearly tangent to one

another. In the cubic system, one can then directly

calculate the lattice parameter, a, since the angles

between any two crystallographic poles are always

known. The disadvantages are that: (a) Only a lattice parameter is measured and then using only two of

the available planes, (b) The sensitivity of the

method depends directly on Bragg's law — the higher

the 6 values of the conies, the better the sensitivity.

Thus, obtaining high sensitivity in any specific case

is a matter of chance, (c) The method cannot be

generalized to measuring individual d values. For

these reasons, the tangency method was discarded.

2. The lens method [29,37] takes its name from

the shape of the figure resulting when two conies on

the film overlap. The pertinent relations for a lens in terms of the lattice parameter for a cubic material

are shown in figure 9. This method can give very

precise lattice parameters, but it has several disadvantages for strain measurements. Among these are

(a) Special orientations of lens figures are required

to eliminate gnomonic projection distortion errors,

(b) Only lattice parameters, not d spacings are measured directly, (c) Not one but two lenses of a suita- ble type are required on the same pattern in order to eliminate uncertainties due to errors in the source to film distance, Z. Getting two suitable lenses on one

pattern is often difficult. The lens method was

discarded for the Fe-3 Si problem.

3. The intersection method requires three or more

conies to nearly intersect in a point. There are two

types of intersections, invariant and accidental. In- variant intersections [19] are a function only of

crystal symmetry and as the name implies, always

occur. An example is the (220), (211), (121) intersection on figure 8. Invariant intersections are useless

for strain measurements. Accidental intersections can be made to yield lattice parameter data [16,38].

But the existence, motion and sensitivity of an ac- cidental intersection cannot be predicted. The inter- section method was therefore discarded.

4. The fourth method — the Regressive Analysis

Conic Equation (RACE) method [39] — yields dnkt measurements directly, can be analyzed for uncertainties in a rigorous way and as a by-product yields

orientation information. This method was adopted

for the Fe-3 Si study.

The RACE method can be summarized as follows: An orthogonal X-Y-Z coordinate system is set up

where Z is in the direction from the x-ray source to the film. The value of Z represents the x-ray sourceto-film distance. The center of the pattern is the X-Y

plane origin, XoYq (see fig. 10). The equation of any

conic on the Kossel pattern is [39]

q^X + q^Z = (X^ + P 4- 22J1/2 sine (12)

Here qu 92, and 93 are the direction cosines of the

plane (hkl) taken with respect to the X-Y-Z

coordinate system. Define:

U = gicsc0

V = qzcscd

W = q^cscd

M^iX^ + Y' + Zyi^

Then eq (12) becomes:

UX+VY+WZ = M

In terms of the definitions above

U, V, w

and

cos (90-.)=sin. = f

= |= -|

q\+q\^ql-\.

(13)

(14)

(15)

11
Since sin0= X/2c?, we get

d=o.5k{W + r' + w^yi^ (16)

In principle, we could obtain the d values for all conies on the film by measuring the X-Y coordinates

of only three points on each conic. In practice, we

measure several points on each conic and use

regressive analysis [39].

The values of U, V and W can be obtained by the

principle of least squares in which case U, V and W

are obtained by minimizing the sum

S* = VMi - XiU - YiV - WiZf (17)

This process of minimization leads to a system of

three equations having three unknowns. The solution of this system, called the normal equations, wiU

be, statistically, the best estimates of U, V and W.

In this case, the normal equations for U, V, Ware:

[X^] [XY] z[X]

[XY] [F2] z[Y]

Z[X] Z[Y] nZ-^

where \X\ means 5 A',, etc., n is the number of coor- dinate measurements for the conic. Then:

pil J"12 I"13

p21 ["22 ["23

pai p32 p33

IT \Xn-\

V \Yn] (18)

w Zn

IT [Xn]

V [Yn]

w Zn

where P' — F^^ represents the elements of the in- verse of the first matrix in eq (18).

A. Error Evaluation

The minimum value of eq (17) is taken as an indi- cation of the goodness of the fit of eq (16) to the d

values. The standard error of the estimate is defined

as

Se =

l{Mi-XiU -YjV -WiZ)''

R — 3

1/2in Ref [40].

(19a)

The standard errors associated with the estimates

of f/, Fandrare [40]:

(19b)

In order to obtain the standard error for the d

spacing, the computed values of U, V, and W are used to obtain di, d2-dn. Then the standard deviation for d is calculated in the usual way.

In practice, all of the foregoing computation is car- ried out by a digital computer. However, it is ap- parent that the statistical uncertainties are not the

only source of uncertainty in the measurement of the

d values. Other contributors to d value uncertainties are measurement errors in X, Y and Z, the effect of an erroneous estimation of the location of the pattern

center, XqYo, and the uncertainty in the wavelength,

\. This last has been exhaustively studied by

Bearden [41] and is of no importance to our strain measurements which involve the ratio of d values. Thus, wavelength uncertainty has been disregarded

in all of what follows. There is apparently no direct analytical method

for propagation of errors due to, say, just Z in eq (18) [42]. Therefore, values of AdIAZ and Ad/Ar, r being

the radial uncertainty in locating XqYo were approximated by empirical means. Furthermore, errors in lattice strains ed = Adjdhki resulting from errors in Z

were evaluated for Fe-3 Si. The error evaluation was carried out by measuring

aU the conies on a pattern of Fe-3 Si. Some eleven

sets of coordinate positions were found for each

conic. The computer delivered the d values for the

XqYq and Z values determined as accurately as possible. Then the Z value was altered in the computer

program and the corresponding change in d

recorded. The same was done for the XqYq position.

Thus, values of AdjAZ and Ad/Ar could be found for each conic. As in most crystaUographic studies, as 6 increased, the magnitude of errors decreased.

The results are plotted in figure 11. Clearly, the

higher the value of d, the less the d error in the case

of uncertainties in both Z and XqYq. Note that for a given value of 6, AdjAZ is always considerably larger

than Ad/Ar. We are interested in the uncertainty in the strain values for our application. The effect of an

error in Z was found by straining Fe-3 Si to a nominal

value of 0.2 percent and repeating the computer cal- culations of the effect of a change in Z. (The unstrained d values were taken from the previous ex- periment.) The result is shown as figure 12. 12
In practice, the value of Z is typically 5 to 12 cm. The Z distance can be found to within 0.1 mm by means of a depth gage. It is possible to do even

better than this by using the lens method [37] to obtain a lattice parameter which is independent of Z

and then to back-calculate, Z. It thus appears that, even when using conies having a Q value of 20°, the

lattice strain uncertainty can be reduced to 0.0007. With a good comparator, the uncertainties in the

individual measurements can be reduced to 10

to 20 micrometers. It is reasonable to assume that such errors will be randomly biased for a given

conic. The effect of these errors is negligible compared to that of the Z distance.

The A'oFo position is found by allowing the electron

beam to strike the plate. This impresses a small spot

at the geometric pattern center. If the plate is in air,

i.e., an x-ray window is in place, the light from the

filament serves the same purpose. The position can be found to within 0.1 to 0.2 mm in this way.

To summarize, we now have a method for obtaining the individual d values. Of equal importance,

there is also a means for determining the reliability of these d values and of the lattice strain values

derived from them.

V. Stress-Strain Analysis

The Fe-3 Si alloy is body centered cubic in the

unstrained state. It will be demonstrated that

virtually nothing can be gained by assuming that Fe-3 Si fails to remain cubic in the strained condition, therefore it is assumed that we will always be

dealing with a cubic lattice. In the cubic system, it is convenient to choose a

point within the irradiated volume of the crystal as a reference [6]. For the cubic lattice, three

mutually perpendicular axes corresponding to the

crystallographic directions [100], [010], and [001] can be made to pass through the reference point. In this case, the components of the vector H = {hkl) are the direction numbers for the family of planes

called H. In general, if j is a unit vector, the normal

strain component in the ^-direction is:

ej = yTy' (20)

where / is the transpose of j and T is the strain tensor given by

/en 0.5e,2 O.-SeiA

r=[ 0.5€2i €22 0.5€2:,

] (21)

\0.5e3i 0.5632 €33 /

The €,/ are the Cauchy strain components.

Now, by analogy to eq (20), the relation between

the components of the strain tensor in terms of the

MiUer indices of a plane and the lattice strain, s, associated with that plane is

\a\H = HTH (22)

where

H' = {hkl) is the row matrix of the MiUer indices

/i, k. /, H is the column matrix of hkl, s is the lattice strain normal to the plane with Miller indices (hkl)

and T is defined by eq (21). Explicitly, Eq (22) after the matrix multiplication

becomes:

{h2-\-k^ + l^)sa= h^eu + + /2€33 + A7e32 + Ihei3 + hkei2 (23)

In order to solve this equation for the six strain values, at least six independent measurements of

s values are needed, e.g., the six sets of hkl numbers

lead to a matrix of rank six. There is apparently no way to decide whether a particular set of (hkl)

values is suitable other than constructing the matrix

and testing to see that its determinant does not vanish [6]. With an appropriate set of (hkl) values a unique

strain tensor can be computed. But, as with the

determination of the c/ values by the RACE method,

setting up the normal equations for the strain tensor

will yield the best estimates of en, €22, €33, 613,

€23, €vz. These equations are derived in a manner

exactly analogous to those for the determination of

U, V and W given by eq (18). Calling the best

estimates (en) (€22) . . • (€12). the six normal

equations are [6]

13

497-814 0-73-3
[h^k^] [/j2/2] [h^ki] [h^l] [h'k] \

[^2S(/j2 + A2 + /2)] \

1 [h^k^] [kH^] [k^i] [hk^l] [hk^]

\ [PSih^ + k^ + l^)] [km] [A/3] [hi'] [hkl^].

[klSih^ + k^ + l^)] —

[k^kl] [kH] [hkl^] [hkn]

[lhS{¥ + k^ + fi)] i [hH] [hk^l] [hkl^] [/l2/2] [hm]

[hkSih^+k^ + l^)]l \[h^k] [hk^] [hki^] [hkH] [hm] [h^k^]

j

(€33)

(€23)

<e3,)

where [h^k^] = hiki + h2k2+ . . . h„k„, etc. The solution of eq (24) for the Cauchy strains can

be programmed for machine computation in a straightforward way. Thus, the best possible

estimates of the Cauchy strains are obtained. Next,

the principal strains can be obtained.

Elastic theory shows that there exists for all points in a deformed continuous material a set of

three directions in space such that linear elements

placed along these directions in the undeformed

condition will be subject to strains which have

maximum or minimum values with respect to strains of linear elements in all other directions

[43]. These three maxima are called the principal

strains Si^ 82^ S3, and the directions on which

they lie after deformation are called the principal

strain axes. The principal strain axes are orthogonal

to each other in both the undeformed and deformed

state. In order to obtain the principal strains and axes, the following relation [6] can be used:

where S, corresponds to the three principal strains

and /, the invariant of eq (25) has the components:

/l = S,+S2+S3,

12 — S2S3 + S1S3 + S1S2,

13 — S1S2S3

In explicit form, the determinant of eq (25) is written

from its characteristic matrix as:

<en)-S 0.5(e,2) 0.5 (cai)

0.5<ei2) {€22) -S 0.5 {€23)

0.5 (€31 ) 0.5 (€23) (€33) -S

= 0 (26)

Det {T-Sil)=0 (25)

This determinant reduces to a cubic equation in Si; the roots of which are the latent roots of eq (26) and which can be solved by standard cubic equation

formulae [43]:

S3-S2 [<e„) + {€22) + (€33)] +S [{€22) {€33} + (enXeaa) + <€„)<e22) -0.25 ((e^a)^ (€23)2+ (€32)2)]

-{<en){e22)(e33) -0.25 [<e„){e23)=^ + {e22)(e3i)2+(e33)(ei2)2] +0.25 [<ei2)<e23><€3,)]}=0

^971 (27)

Computer programs have been written which

reduce lambda matrices of the type just discussed

to their latent roots directly. All of the latent roots of eq (27) are real [43]. A positive principal strain value represents a tensile strain while a compressive

strain results in a negative sign. In practice, eq (27) is solved for its roots by

machine computation. Thus, the principal strains

Si ^ S2 ^ S3 are obtained. For the principal strain axes we make use of the relation [6]:

(r-Si/)/ = 0; i=l,2, 3 (28)

where / is the eigenvector giving the direction

cosines of the principal axes. Explicitly:

^<e„)-S, 0.5 (€,2) 0.5 (€31) \ //„\

0.5(€i2) (€22)-S, 0.5(e23) (/2,|=0

,0.5 (€31) 0.5<e23) (€33) -SJ \l31l

(29a)

14
(en)-S2 0.5 (e,2) 0.5(63,) ] / 1,9 0.5 (€,2) (e22)-S2 0.5 (623) [c

0.5 (63.) 0.5 (623) (€.'),•)) -S2 i \/32

( 61 1 ) — S3 0.5 (6,2) 0.5(63,) \ //l3

0.5(6.2) (€22) —53 0.5(623)

0.5 (€3.) 0.5(e23) (^33) "Sa,

(29b)

(29c)

The eigenvalues of the / eigenvector determined

from eqs (29) may be written as the orthogonal

matrix:

(111

I21 /31 \ I12 I22 lz2

1 13 1 23 1 33 j

The row vectors in the / matrix determine the

principal axes for the strains. Explicitly, In, hi, hu

are the direction cosines of the principal axis as- sociated with principal strain, S, while In, I22, I32 and /i3, I23, I33 are the direction cosines of the

principal axes associated with principal strains

S2 and S3 respectively. By determining the direction

numbers A, B, C and the / matrix row vectors, a nearly orthogonal set of crystallographic directions

having small MiDer indices is found which closely

represents the principal axes [44]. Explicitly:

A2 8-2 C2

A3 B3 C3

There is a uniqueness problem associated with

the solutions of eqs (29). The Cauchy strains are determined by the statistical algorithm described

before. Thus, they are the best estimates available.

The principal strains are derived from these esti- mates. What happens in the solution of eqs (29)

is that the nine equations never quite give "/"

values that make the solution exactly equal to zero. Instead, a typical value if 0.0007. The set of / values yielding a solution to eqs (29) closest to zero

is taken to be the best estimate of the "/" eigenvector. The computer tests all possible combinations

of signed solutions to eqs (29). The result chosen

meets the "closest to zero" criterion described.

The range in the solutions for the eigenvalues of the

Ai Bi

111 I21 131 A2 B2 =^

I12 1 22 1 32 A3_

ll3 ^33 /

/ eigenvector gives an idea as to the reliability of

the location of the principal strain axes. With the principal strains and their axes deter- mined, we may turn our attention to the question

of finding the maximum shear strain in any {hkl)

plane. This quantity is of importance since it may

be used to evaluate, quantitatively, the anisotropy

of strain in each (hkl) plane of a family. Further- more, the exact planes on which strain accommodation occurs may be identified from comparisons

of the maximum shear strain values \i4]. The

derivation will be given in some detail since it does

not appear elsewhere. The result in a nearly complete form was given by Slade et al. [44]. These

authors outlined the method that they used but

gave no details. The principal strain axes will be taken to be the

reference axes. Within the (hkl) plane of interest,

there will be some direction where the shear strain attains its maximum magnitude. Let a unit vector

in this direction be called k and let a unit vector perpendicular to k in the (hkl) plane be called j. Now k and j are 3 unit column matrices whose

transposes will be called k' and / respectively. In

the principal axis system (the principal frame),

the strain tensor, T, reduces to the diagonal matrix

[43]:

<Si 0 0 \

0 S2 0 ] = r

^0 0 S3/

(principal frame) (30)

The shear component associated with k,j is [44]:

ekj = 2k'Tj (31)

The factor of 2 comes from the relationship between

the Cauchy (engineering) and tensor notations for shear stress. The problem involves finding (1) the magnitude

of the maximum shear strain in the kj plane along

k called (6fcj)niax- (2) the magnitude of the shear

strain along i.e., perpendicular to the maximum

15
shear strain in the (hkl) plane and (3) to transform

from the k, j, (kXj) frame back to the principal

frame in the terms of the (hkl) values of interest. The final result gives the maximum shear strain in any (hkl) plane and the direction cosines of the

k, j, (kXj) system in terms of the hkl values and the

direction cosines of the already determined principal

strain axes. Now kk' — jj' = 1 and kj' — jk' = 0 since we are dealing with unit vectors. We will use the method of

Lagrange multipliers to find the maximum value of

€~kj . We have two constraints in that there can be

shear strain in the k and (kxj) directions but not

in the j direction. (This will be proved later.) Hence, we get using the Lagrange method:

where a and /3 are the two Lagrangian multipliers

to be evaluated. Carrying out the indicated operations in eq (32) gives

k'T+ aj' + 13k' = 0 (33)

Muhiplying eq (33) by j

yields

k'Tj + aj'j + ftk'j = 0

But = 1; A:'; = 0 so

a = — k'Tj

Now multiplying eq (33) by A;:

k'Tk + aj'k + = 0

and

l3 = -k'Tk

Hence:

k'T-j'{k'Tj) - k'ik'Tk) =0 (34)

Multiplying eq (34) by Tk:

k'T'k-T'kj'k'j- {k'Tk)' = Q

But kj' — k' j so that:

kT'k - {k'TjY - {k'TlY = 0 (35)

But (A:T;) = 0.5 Cfcj. Thus:

{e~kj )m.. = 2[k'r4- ik'Tkyyi-' (36)

Evaluating the terms under the square root sign:

hTk= Ckihh) /Sf 0 0 \ Ih^

0 Si 0

\0 0 sil

k'T'k = k\S\ + klSl + klSl

k'Tk^ ihhh) / S, 0 0

0 S2 0

0 0 S3,

k'n=ic\Si + iiS2-\-~kiSs

{k'TkV=kSl + hSl + HSl + 2MUSiS2 + 2k\USiS3 + 2UklS2S3

Subtracting:

{e,j)^.. = 2[{S, -82)^1+ iS2-S,)mi+ (S.-S.nfkU (37)

This is the magnitude of the maximum shear strain Two things remain to be done: to show that there

in the plane; the plane lies in the (hkl) plane of is indeed no shear strain in the j direction and to interest. find k in terms of the / eigenvalues and the hkl

16
values. Perhaps the most straightforward way to do

the former is to consider the Cartesian coordinate

system k, (kxj)^, j. Let the strain acting along a

resultant in the Ic, (hcj) plane be Cy and the angle

the resultant makes with the k axis be 0. Then

861^ = — Cfcj sin 0 + e(kxj)i cos 0=0

tan 6= sin 61 cos 0 = e(kxj)jl €kj

e(fxj)j sin 6 =

cos 6 -

Now let 6' = 0 + 90°. In other words, we would have

a resultant strain in the j direction.

€ij(d' ) = Cfcj cos 6' + ej(jixj) sin 6'

^kjid' ) = ekj sin d+€j(kxj) cos 6

eu(0')=O

Thus, in any {hkl) plane, the shear strain will be

zero normal to some direction in the plane and the

direction of maximum shear strain in the plane is normal to this direction. Finally, we need ~k in terms of {hkl) and the

principal strain axes positions. The direction cosines

of the hkl axes are given by:

hn —-

I'l kn " 11 5 L >

The transformation required is then:

ki hnkjiln /lit

(38)

I3i

Thus, the maximum shear strain in any (hkl)

plane can be evaluated. In practice, the results given

by eqs (37) and (38) are obtained by machine

computation. With these results, the entire shear

strain effect can be followed as a function of strain and/or position in an individual grain of a material.

The next step in the evaluation of the stress- strain configuration in the elastic region is the

determination of the stresses in the cubic frame,

and the principal stresses and their axes. In the

elastic region, the generalized form of Hooke's

law relates the stresses to the strains.

O"CUB — CE

which in explicit form for the cubic case is

(39a)

11

1 C12 C12 0 0

(T22 2 Cn C,2 0 0 :\ (€22)

0-33 c, 2C,2Cn0 0 0 (€33)

0-23 0 0 0 C44 0 0 <623)

\::;/

0 0 0 0 C44 0 Ue3i) 1 \o 0 0 0 0 C44 \(6.2)/

(39b)

The C values are the elastic constants and the e values are those determined using eq (24). Thus,

the cubic frame stresses can be obtained. The

principal stresses, P3 are determined

by solving for the latent roots of the stress-analogue

of eq(26):

(Tu—Picrn 0-31

cri2 0*22 PiO'23

0"31 cr23 (T33 — P; = 0 (40)

This notation, (kxj), means the vector crossproduct of k with ;'.

The solution gives Pi, P2 and P3. The principal

stress axes can be found in a manner entirely

analogous to that outlined for determining the

principal strain axes. Again, tensile and compressive

stresses are indicated by the sign of the P value. The stresses with reference to any frame can be

determined according to classical transformation

techniques following an outline given by Slade

et al. [44]. The transformation maxtrix can be

deduced in terms of the direction cosines of the axes of the new frame and the results of eq (39). Thus, the stresses in any frame can be found by this transformation.

For example, suppose that the stresses are re- quired with respect to the principal strain axes. In other words, what stress configuration is necessary to give the observed three principal strains?

Following the treatment of Slade et al. [44], the

required transformation matrix Tma, is given by:

17
-11 11I.,] 11''31 X fco

Aq 1 /. 0 1 /, 1 1

I •'11 2 /11/21

12 •22 2 ^2^12 2 /12/22

' 13 P ''33 2 /23/33 2 ^33/ 13 2 /i3^?3

/ 12^ 13 ^22 ^23 ^32^33 {I22I33 + ^32/23) (/32/l3"l" /1243) (/ 1 )/ 'JO L'yoL

J 13*11/ / I23t21 '33f31 (/23/3I + /33/21) (/33/u + /13/3 .) It 13^1^^*23*

11^12 /21/22 ^31/32 (/2I ^32 + /3142) (/31/12 + /ll^a'. ) (/ 1/22 + /21/

The required result is:

(^PRIN^'^'^'^CUB (42)

where cr^^,^ is obtained from eq (39). In this way,

the stresses with respect to any frame can be found

provided the axes of the frame can be written in expUcit form.

The elastic energy stored in the crystal, fF, is equivalent to the work done in straining the crystal

[43]. Specifically:

W^j (r'8€ = je'C8e = Y^'C€, (43)

since e'€ = €-.

In explicit terms: 2W= (eiie22e33e23e3iei2) Cu C12 C12 0 0 0 \ Ci2 Cii C\2 0 0 0

C12 C12 Cu 0 0 0

0 0 0 C44 0

0 0 0 0 C44

0 0 0 0 0 C44 f

I en

€22 €33

^23

€31 ei2

(44a)

2W= Cii€ij + 2Ci2e22€33

l>llt22 + 2C12€11633 Ciie33 + 2d2€iie22

^44623

^44631 C44ei2

(eiie22e33e23e3iei2)

(44b)

yCu (ef, + el^efg) + Ci2(e22e33 + e„e33 + 611622) + y044(6^3 + 6|, + ej^ ) (44c)

The value of W calculated by this fashion takes no account of lattice point defects which result in random strains. Sharp Kossel lines indicate that random strains are small since such strains cause hne broadening.

Finally, Slade et al. [44] have postulated that a purely geometrical measure of distortion of the

irradiated crystal volume can be gotten from:

2 r4I2- 11/2 -3/2/3-1

where

/l=S,+S2 + S3

I h = S2S3 + 53^1 + S1S2 / Defined in eq (25) ^3 — S1S2S3 /

The value of D is called the distortion number

and is independent of the frame. The distortion number is a measure of the anisotropy of deformation in that D is zero in the case of spherically

symmetrical deformation while D is unity if the

deformation is simple shear.

18
To sum up, the Kossel method yields lattice strains for each (hkl) plane of a family. From these can be calculated the Cauchy strains, the principal

strains and their axes, the maximum shear strain in any (hkl) plane, the cubic stresses, the principal

stresses and their axes, the total stored elastic energy and the distortion number. This information

completely describes the stress-strain configuration

of the crystal in the irradiated volume.

The reliability of this stress-strain analysis has

been dealt with in only a cursory fashion. A few

comments to the effect that the computed Cauchy

strains would be more reliable if a great number of

(hkl) planes were examined have been made [45].

An estimate of the uncertainty on the principal

strains computed from the Cauchy strains has been

made for a single strain condition in a tungsten

crystal.

VI. Reliability Analysis

The standard error, s'^ of the d values can be

obtained as described previously. This error must

be combined with the experimental error which comes primarily from uncertainties in the source-tofilm distance, Z. A standard error on the d value

from uncertainties in Z, can be obtained from the

standard error on the Z distance, sz, and figure 6. Then since variances are additive:

s'.^^v'is'liy+iszr (46a)

Contributions from other errors such as the X, Y

measurements and the XoYo position do not increase

s'^ by more than about 10 percent. Hence, a good

approximation is:

Sd-l.l5^- l.lV(5';)'^ + (5z)2 (46b)

Since the determination of the d values, and

hence the S values, is based on repetitive measurements, the number of repetitions for a suitably narrow 95 percent confidence interval width is of

interest. But, first we must decide what a "suitably narrow 95 percent confidence interval" is. While

certain specialized techniques [29, 37] can give

relative uncertainties in a d value of about 0.002 per- cent, a more usual value is 0.01 to 0.05 percent. We

shall arbitrarily accept the 0.05 percent figure as

the maximum acceptable width of the 95 percent

confidence interval for a d value to be used in the

stress-strain analysis.

For Fe-3 Si, the {110} conies have a Bragg angle,

6, of about 30° when the diffracted wavelength,

X, is Fe-IC.1 (X = 1.936A). The value of sz for the

{110} conies can be reduced to about 0.001. The

value of Se for the {110} conies was found to be

0.0016 (eleven determinations). Using eq (46), we get Sd equal to about 0.002; d is 2.024A. In order

to reduce the width of the 95 percent confidence

interval to 0.05 percent, we need to make enough

determinations to reduce the interval to O.OOlA. By

going to a table of "f" values and applying the usual

relation

^^95%= 1= =0.001, (47)

Vra

where tn-s is Student's t limited by 3 constraints

(the U, V, W equations) and n the number of in- dividual d measurements, we find n equal to 18. Therefore, even for a conic having 6 equal to 30°, some 18 {X, Y) coordinate measurements are suffi- cient to reduce the uncertainty in d to about 0.05 percent. Other conies, with higher 6 values will have smaller ranges of uncertainty. Thus, 18 co- ordinate measurements are sufficient for all conies

in the Fe-3 Si case and in general for conies having

6 ^ 30° to meet the d ± ^^95% criterion. The usual

probable error often reported is ± ^^50% . For 18

determinations of the {110} conic d spacing. W —

O.OOO4A. Transferred to percentages duo±w =

0.02 percent.

The error in the lattice strain, Sd, is arrived at using Gaussian error distribution.

ASd = \/{dMfY+ {dAdiYld] (48)

Here Ac?/ — Wf, 95% and Ac?, = Wi,g5%. Now in the

elastic or low strain region where the KISS method

is valid, df will be nearly equal to di, and Ad} will be nearly equal to Ac/,. So as a working approximation

AS - VlMildi - V2Adfldf (49)

But as outlined above, (Ad'//c?/)max — 0.05 percent.

Hence, the maximum width of 95 percent confidence

interval is 0.07 percent. The probable error in this case will be 0.03 percent. It is worth repeating that

19
this is the worst case — that of a conic with a Bragg

angle, 6, of 30°. For high 6 conies, it is possible to reduce the value of it; 95% forSd to the neighborhood

of ±0.015 percent.

The next step in the error analysis is to determine

the uncertainties of the Cauchy strains. The Cauchy

strains are computed by a least squares regressive

analysis in a manner exactly analogous to the computation of the U, V, W and d values as outlined

previously. By analogy to eq (19) we have for the case of the strains:

5e= + If) S,, -hKeu)-k\{e,,)-l\{e^^) - Hi (€32) - IMen) - hiki{ex2)VI{n - 6) (50)

Here n is the total number of (hkl) planes for which

there have been obtained Sd values. Then by analogy

5'{eii) = 5e Vr" (51A)

s' {€22) = Se VT^ and so on to (51b)

5'{ei2) =5e Vl^ (51c)

where the F values come from the inversion of the

second matrix of eq (24). Thus, we have the standard error for each of the

computed Cauchy strains. These values of 5'<e,j> will decrease as the number of {hkl) planes, n, used

to obtain the Cauchy strains increases since in- crease n reduces Se in eq (50). For the F" values

to decrease, a large spread in the Bragg angles of the

(hkl) planes used is necessary. In other words, one must make an adequate sampling of all the planes

[45]. As we have seen, there are large potential

errors if low 6 conies are used. On the other hand,

line broadening is usually associated with conies

having 6 ^ 80° [46], Therefore, a suitable range of

(hkl) values is one yielding a range of 30° < 6 < 80°. Variation of a few degrees at either end will probably

not lead to serious problems.

The effect of lattice strain errors on computed

{cij) values must be combined with the 5'(ey> values in order to obtain a final estimate of the uncertainties in the Cauchy strains. First, the effect of a systematic error in the lattice strain values may

be considered. To observe the error propagation,

a test case was run through the computer. The

Cauchy strains and the 5' <t,j) values were calculated from a published set of typical lattice strain data from a tungsten sample strained to about 0.2 percent [45]. Some 20 planes were used in the

test. The range of 6 was 40.2° ^ d ^ 65.8°. Then

the S values were varied by set amounts of+0.0005, - 0.0002, - 0.0005, - 0.001. The values of (ey),

i^^j did not change while the values of (e,j) i=j

varied by exactly the amount of the systematic

error. None of the s values changed.

It is unlikely that a systematic error in the lattice strain values of significant magnitude will occur

since the nature of the measurements and the

computation of S tend to minimize this possibility.

However, random errors of individual lattice strain values will almost certainly occur. The magnitude

of these errors as a function of 6 has already been

discussed. In order to test the effect of the random

errors, the following procedure was used: (For

purposes of this test, a constant increment of ASd = 0.0003 representing an average value was

used.) The computer was made to list tables of random digits; the sign and order of the ASd values was determined by these random digits. Thus, from

one table even digits meant a subtraction and vice- versa while another table determined the order of the (hkl) plane to which these signs were to be

applied. Some four sets of random data suffering

uncertainties of Sd were synthesized in this way.

Next the (eij) and s' (ai) values were computed for each of these four sets. The s' (^u) did not vary

significantly from their previous values. All of the

(eij) values changed. The results are shown in table 1* As a test for randomness, all of the new

(eij) were averaged. Agreement between this mean

value and the original (ey) values was taken to indicate randomness. Table I shows that the agree- ment is very good except for (€12). The range of

(eij) values for the four cases and the standard

error in (eij) was calculated. This final standard

error s'rnd^ and the s' Uu) can be combined to give:

5(€o) = V(5'<.,>)2+(5rnd)^ (52)

Table I shows that 5(eu>is about 1.05 5'<«ij>. Hence,

the effect of uncertainties in Sa may be taken into account by asserting that

' <.o) 1-1 (53)

Equation (53) thus represents a statement of the

standard error of the calculated Cauchy strains

* All tables are at the end of this paper beginning on page 39. 20
which takes into account uncertainties from all contributing sources. The width of the 95 percent confidence interval

for the Cauchy strains is obtained from:

. 95% ,

t („-6)S<£,j> "^€,95% = 77= • (54) Vn

In the case of table I, some twenty (hkl) planes were used. Hence,

(2.145)

w^..95% = ^ 5 = 0.48s • (54b)

The probable error, iv^^^q%, is 0.115 S{e,j). As a rule of thumb, for all of the work discussed here, the

probable error will be about one-third of the 95

percent confidence interval width.

The effect of uncertainties in the Cauchy strains on the computed principal strains will now be considered. By randomly signing the w 95% error on

the Cauchy strains, six sets of principal strains were obtained from the data in table I. All random

sets lay within the range of principal strains obtained

when all six Cauchy strain uncertainties were either

+ JFe,95% or — Wf,gs%. The mean of all six sets was

equal to the principal strains computed from the

original calculated Cauchy strains. Results are shown in table II. Based on these results, it is reasonable to assert

that a 95 percent confidence interval for the principal strains can be obtained by calculating three

sets of principal strain values. First with the calculated (ejj) values, then in turn with {etj) + Wt,95%

and (eij) — W^^g^^. The resultant range should be

a close estimate of the width of the 95 percent

confidence interval for each of the three principal

strains. For the case used as an example, table II shows

that the 95 percent confidence interval is about

± 25 to 30 percent of the principal strain value for

all three principal strains. These values appear to be typical for lightly strained Fe—3 Si as well as for the tungsten test case. There is no reason to believe

that they wiU be much different for other materials.

Hence, it is concluded that for lightly strained

(elastic region or residual strains), the Kossel

method, carefuUy applied, can generally give

principal strain values having 95 percent confidence

interval widths of ± 30 percent or probable errors of

about ± 10 percent relative to the computed principal strains. With these results, the reliability of the stress calculations (eqs (39, 40)), can be obtained by

substituting the principal strain limits into the

appropriate relation. For our discussion, we will assume that the elastic constants are well known.

In this way, the reliability of the entire stress-strain

analysis is established.

As an overall generalization, all of the calculated

results of the stress-strain analysis obtained by the

Kossel method can be expected to have an ap- proximate probable error of about ±10 percent

relative to the calculated value. It is difficult to see how this error can be reduced very much in the

future in view of the stringent requirements for using many (hkl) planes having 30° < 6 < 80°, and

the difficulty in improving on the regressive analysis

methods used to calculate the stress-strain parameters.

The magnitude of the uncertainty in the Kossel

stress-strain analysis is about the same as that obtained using conventional x-ray stress-strain

methods. However, other x-ray stress-strain meas- urements yield little more than an estimate of the

residual stresses present in the irradiated volume.

The computed uncertainty is the primary reason

that it is not presently advantageous to consider

that the deformed material ceases to remain cubic. In reality, the material cannot remain cubic. But,

it is impossible to say what form the material will take for a given deformation, nor whether that form

is continuous throughout the entire body of de- formed material. Under these conditions, any form

capable of mathematical analysis is an approximation. The simplest form, cubic, is as reasonable to choose as any other. It is concluded that nothing is to be gained by using a lower symmetry for calculation of elastic stress-strain parameters than that present in the undeformed state.

VII. Equipment for the Study of

Materials by the Kossel Method

In order to take the maximum advantage of the opportunities afforded by the use of the Kossel method,

appropriate equipment is necessary. As with vir- tually all scientific equipment, this means that a maximum amount of accurate information is desired

for a minimum of effort and expense. In the case of

21

497-814 0 - 73 -4
the Kossel method, once the equipment is available

the production of the required patterns becomes

completely routine. Unfortunately, the choice and

design of appropriate equipment was not obvious at the time this study was begun. However, the equipment finally adopted and used for the work

described here meets all requirements for the rapid,

meaningful preparation of Kossel patterns of both

strained and unstrained materials. Since some of the

concepts of the equipment were unique to Kossel

studies at the time they were formulated, and since

the hardware is still unique, both concepts and the

resulting instrument will be described in some

detail. The first successful experimental demonstration

of a divergent beam diffraction technique was made

by Rutherford and Andrade in 1914. These workers

used an external radium source of y-rays and a cleaved rocksalt crystal as the target. From the

photographically recorded diffraction pattern, they were able to determine the monochromatic radiation

components of the source [47]. Several other workers devised successful methods of preparing divergent beam patterns in the years between 1920 and

1950 [48-52]. In addition to being cumbersome and complex, all divergent beam devices prior to 1951 suffered the

handicap of relatively large sources of x radiation.

This led to several problems: a large primary source causes the Kossel lines to be broadened and impairs

the inherent precision of the method; in addition, ex- posure times are prohibitively long (24 h in some

cases) and photographic contrast is poor. Furthermore, large single crystals are required to accom- modate the large x-ray sources. These problems were first removed by the use of

the electron probe microanalyzer which provides a micrometer sized source of x rays, as well as an

optical microscope for viewing the area undergoing

irradiation. Power densities five to ten times greater

than those obtainable with earlier equipment are

available. The value of the electron probe microanalyzer as a source for Kossel patterns was recog- nized by its inventor, Castaing, as soon as his first instrument was completed. He presented Kossel

patterns in his thesis which clearly demonstrated

the superior characteristics of the microprobe in

this area [53]. AU present day Kossel instrumentation utilizes a focused electron beam to excite the required

microsource of x radiation. While commercial Kossel

camera attachments for electron probe

microanalyzers represent the most readily available

instrumentation, investigators have successfully

prepared Kossel cameras for use in point-projection

x-ray microscopes and in electron microscopes,

possibly since all of these instruments provide con- venient electron optics.

Experience gained from attempting to use an elec- tron probe microanalyzer as the electron optical

column and a small simple Kossel camera, indicated

that the electron optical requirements for the vast majority of Kossel line based research are less strin- gent than those for electron probe microanalysis and

electron or x-ray microscopy. A single electromagnetic lens combined with an inexpensive flat grid

electron gun makes an entirely satisfactory electron

column. A relatively inexpensive power supply can

be used since gun voltage stability requirements are

less demanding than for electron microanalysis or microscopy. The same is true for the single lens power supply needed. To complete the system (exclusive of the Kossel camera), a small vacuum

system, suitable light optics, and a current meter

whose range is 10 to 1000 nanoamperes are required.

Finally, a modular design can be modified at wiU to incorporate improvements. Hence, the decision was made to design and build a separate Kossel Pattern Generator (KPG) suitable for operation in the trans- mission mode [54, 55].

An air camera offers the advantage of allowing

multiple exposures without disturbing the vacuum

conditions or the orientation of the specimen. With

respect to increased exposure times involved with

air path cameras, the softest radiation used to date

in quantitative transmission Kossel work has been V

Ka at 2.5 A. The decrease in incident intensity of V

Ka upon a film 10 cm distant in air is less than 30

percent. In practical terms this results in an exposure time increase of about 10 min. Most work has

been done using radiation somewhat harder than

this. For example, for Cu Ka at 1.54 A, a 10 cm air path reduces the intensity by less than 10 percent.

Therefore, reduction in exposure time gained by

using a vacuum path is usually on the order of 10-20

percent or, in terms of actual decrease in exposure

time, about 2 to 4 min. The very close {110} Kai-Ka2

doublet in Fe-3 wt.% Si is quite easily resolved in an

11 cm air path camera. This indicates that air scat- tering of the X rays is probably not a serious problem.

The entire system was designed so that it could be

dismantled and repaired vdth a minimum of incon22
venience while stiU satisfying all of the design con- cepts listed in reference [54]. The camera is a

separate module removable from the electron beam

column by means of three bolts. The stage allows translation of the specimen

amounting to one-half inch in orthogonal directions.

Each motion is spring loaded to minimize backlash

and each is connected by means of vacuum shaft

seeds to a micrometer shaft. Thus, the coordinates of

a given point on the specimen can be found and

recorded so that the point can be found again later. The vertical motion of the stage which makes it possible to focus the optical microscope is controlled

by a worm and wheel arrangement. The wheel has

230 teeth and advances a four lead thread five- eighths inch per revolution. This gives a rapid verti- cal motion but retains enough sensitivity to focus

easily in the light microscope.

A two-axis goniometer was mounted on this mechanical stage and each tilt motion can be controlled from outside the vacuum system; an indicating dial is used to index the angle of each axis. The

tilt is controlled by a worm and a worm-gear; each

tilt mechanism is spring loaded to minimize the ef- fects of backlash. A small worm-gear was used in view of space limitations. This limited the number of

teeth and hence the angular sensitivity of the

goniometer to a minimum readable motion of about

0.2°. The goniometer may be tilted ± 12° about each

axis. At this angle the edge of the stage strikes the

light optical objective lens. The essential part of the goniometer design as well

as of the vertical specimen motion is the linkage con- necting the external control device to the actual

moving part. Each linkage can move through an

angle up to 15°, traverse a distance of 0.70 in, and

rotate freely. This was accomplished by attaching

small universal joints on the ends of standard sliding

tubular linkage. In order to prevent galling of the

sliding mechanism in vacuum, each part was

lubricated vnth a suspension of M0S2 in diffusion

pump oil. These linkage components are shown in place in figure 13. Finding and marking the pattern center is carried out as follows: A small aperture at the specimen

plane is brought to the coordinate point of the beam.

Light from the hot filament is allowed to strike the

film; the high voltage is switched off during this operation. A small spot is impressed at the pattern center by this means.

The brass specimen holder accommodates a

three-fourths inch diameter sample, sandwiched

between brass disks.

A source foil for pseudo-Kossel studies could be

placed on the top disk where it was not in thermal

contact with the actual specimen. The entire as- sembly fit into an insulating ring; a lead wire con- nected to a vacuum electrical feedthrough. In this fashion, electron current from the specimen could be

read on an appropriate electrometer.

For transmission studies, the specimen-to-film

distance had little effect on the photographic contrast, which is controlled by the operating voUage,

specimen thickness and crystal perfection. A

specimen-to-film distance of about 11 cm was chosen

as a compromise between exposure time, angular

divergence to be recorded on the film and Kossel line distribution on the pattern. With this distance, a maximum divergence angle of 70° is recorded on the

film. Patterns were recorded on standard 5 X 7 in (12.7

cm X 17.8 cm) x-ray film or glass plates held in an or- dinary plate-holder cassett which fit into an alu- minum frame bolted onto the camera bottom. The

film was exposed in air, by placing a polyester window 0.13 mm thick and diameter 1 7/8 in directly

below the specimen (fig. 14). The window was positioned in a removable cap and sealed by O-rings in- side and outside. Window failure was rare provided

normal care was exercised. Complete details of the

entire system were presented by Vieth and Yakowitz

[54]. Pertinent design features are shown in figures

13-16.

A. Loading Device for the KPG

Since the KPG described above was to be used for studies of strained materials, a loading device was

thought to be necessary. I believe that for Kossel

strain studies to be meaningful, the constraints

must actually be applied to the specimen while the

Kossel pattern is recorded. (Naturally, residual

strain studies are an exception.)

The effect of removing the constraints leaves

residual macrostresses which may be different on

one crystal face with respect to another. Thus, a

residual strain gradient can be expected to be

present throughout the crystal volume. For these

reasons, a tensile loading device was built which was

compatible with the specimen stage supports of the camera module of the KPG.

23
Because of the size limitations, small but accurate

load cells were required. The load was needed to ±

0.5 percent of the maximum load rating of the cell which was to be 100 pounds (45.4 kg). A single load

cell of the required size and required accuracy over

the range 0 to 100 pounds (45.4 kg) was not feasible. Therefore, two resistance strain gaged load cells rated at 0 to 10 pounds (4.54 kg) and 0 to 100 pounds

(45.4 kg) were decided upon. The cells were specially

designed for this application by The Brewer Engineering Laboratories. The dimensions of the cells are shown in figure 17 and calibration data for each

load cell are shown in figure 18. The loading

mechanism consisted of a lever and fulcrum ar- rangement having a nominal mechanical advantage

of 30 to 1. The type 303 stainless steel lever, 3 in (7.5

cm) long, was advanced by a worm gear and springloaded in order to minimize backlash. Loading from

outside the vacuum chamber was accomplished

using a commercial 4 in (10 cm) diameter micrometer thimble, connected to the worm gear on the strain device with two universal joints coupled with sUding

linkage.

The stage was connected to the goniometric drives

by means of bearings and appropriate gears. The entire loading device rested on the three orthogonal

drives; X-Y motion was accomplished by driving a

ball slide mechanism with a micrometer shaft and

vertical motion was obtained by the worm and wheel

arrangement. The assembled tensile loader mounted

in the Kossel camera is shown in figure 19a while the

complete loading device is shown schematically in figure 19b.

A specimen having a reduced section of 1 in (2.54

cm) by 3/8 in (0.95 cm) wide was used. With a thickness of 0.003 in (0.076 mm), the 100 pound (45.4

kg) load gave a stress of 133,000 psi (0.926 GN/m^).

The specimens were shaped in a special jig,

removed, cleaned, and electropolished to the desired

thickness. Then, stainless steel end pads were spot

welded to both heads of the specimen to prevent

tearing of the specimen. The specimen was returned

to the jig and the end holes spaced 3.491 ± 0.0025 cm

apart drilled. The accuracy of macrostress deter- minations was hmited chiefly by nonuniformity of

the cross section, as well as bending stresses in- troduced by the tensile load mechanism. However,

the microstresses of interest when a polycrystalline

material was under study were not greatly affected

by these two problems.

The specimen was pinned into place using

hardened 0.1 in (2.5 mm) diameter pins and the read- out unit observed to make sure that no load was applied during specimen insertion. If the goniometric

drives were used, some load would be applied as the

stage tilts; the maximum load has been measured as 182 g or 0.4 pounds from this source at which is nominally zero load. With a 0.003 (0.076 mm) in thick

specimen this amounts to about 500 psi (0.035

MN/m^). All orientation of the specimen was done

before beginning strain experiments, and it was rare

that the fuU range of the goniometer was needed to accomplish this. In any case, if any load was impressed by the goniometer, it was small and known.

Tests under load indicate that no change occurred in the load as a function of the orthogonal motions.

The load read-out console was a standard commercial unit reading percentage of full capacity of

the load cell. Electrical stability of the combination

of load cell and read-out console was achieved in a few minutes. The minimum adjustment obtainable

with the thimble arrangement permitted incremental

changes of 0.2 percent of the maximum load. Again

considering the 0.003 in (0.076 mm) thick specimen,

this corresponds to stress changes of about 30 psi

and 300 psi (207 N/m^) using the 10 pound and 100

pound (4.54 kg and 45.4 kg) load cells respectively.

Tests indicated that drift as a function of time was

not a serious problem. Both load cells were tested at several loads for 10 minutes and at one load for 8

hours. Maximum drift was 0.03 pounds (14 g). Complete details of the loading device have been

presented by Vieth and Yakowitz [55].

B. Measuring Device for Kossel Patterns

The measuring device required accurate X-Y

coordinate readings over a range of about ±2.5 in by

±3.5 in (6.35 by 8.9 cm). Many varieties of commercial comparators can achieve these requirements.

For this study, a comparator built by Zeiss was used.

Using this comparator, the Kossel pattern was

viewed in transmitted light through an aplpropriate

optical system with magnification ranging up from

1.5 diameters. The appropriate magnification

seemed to depend on the particular pattern but

about 2 to 3 diameters was the usual value. The X-Y

motion was accomplished on micrometer lead screws; the eyepiece contained a Vernier scale. The

precision (reproducibility) was better than 5 micrometers (3 is claimed by the manufacturer). The

24
X-Y coordinate positions of sharp Kossel cibuconics

could be measured with uncertainties no greater

than 10 micrometers using this comparator. The contribution of such uncertainties to the computed d

values was very small compared to the other uncertainties discussed previously.

VIII. Results of Electron Probe

Microanalysis Investigation Seeking

Local Chemical Variations in Fe-3 Si

No significant variation in Si content was found in tests of homogeneity. More than 100 separate

analyses were carried out for Fe and Si simultaneously while traversing several grains in a systematic

fashion, and the data are shown in tables III and IV. Special positions such as grain boundaries or triple

points are noted. Clearly, within the ability of the

electron microprobe to measure a variation, no significant variation of Si content exists as a function of

position in the specimen. Complete analytical details have been presented elsewhere [56].

The results for Fe and Si may be summarized as

follows:

1. The silicon content is 3.14 percent (105 determinations).

2. The standard error s is 0.086 percent for Si.

3. The observed range of Si content is 2.89 per- cent to 3.29 percent.

4. We may be 99 percent confident that the mean

Si concentration lies between 3.12 and 3.16 percent.

5. The 99 percent tolerance interval, i.e., the in- terval in which 99 percent of all future points tested

are predicted to lie is 2.91 to 3.37 percent Si.

6. The percent coefficient of variation for Si is 2.74 percent (%CV = lOOs/C^i).

7. The iron content is 96.87 percent (106 determinations).

8. The standard error for Fe is s = 1.16 percent.

9. The observed range of Fe content is 94.4 to 99.4

percent.

10. We may be 99 percent confident that the mean Fe concentration lies between 96.6 and 97.2

percent.

11. The 99 percent tolerance interval for Fe is 93.8

to 100 percent.

12. The percent coefficient of variation for Fe is 1.20 percent.

The only reasonable conclusion indicated by these

results is that internal strains in Fe-3 Si are not caused by concentration gradients.

Some 3600 separate tests of homogeneity were

carried out using six statistically selected samples of

Fe-3 Si. A conservative estimate of the coefficients

of variation for both iron and silicon was ± 1 percent.

Wet chemical analysis of the Fe-3 Si alloy gave 3.22

wt% silicon [56].

One other observation is perhaps interesting.

Figure 4 shows the average Si content versus lattice parameter. For 3.14 percent Si, figure 4 gives d =

2.86267 A as the bulk lattice parameter for the alloy.

Gielen et al. have reported a = 2.86268 A for this material [37]. Such close agreement is probably for- tuitous, but it is certainly encouraging to have the

results of independent methods agree well.

IX. Obtaining the Kossel Stress-Strain

Analysis

A typical specimen is shown prior to insertion into the loading device in figure 20. The reduced section

is 2.54 cm by 0.63 cm. This specimen was machined

from a sheet supplied through the courtesy of H. C. Fiedler (General Electric Co.). The original sheet was about 100 /Am thick. This specimen was elec- tropolished from both sides simultaneously to a thickness of 60 ±5 /nm as determined by several

micrometer readings. Polishing was carried out in 5 percent perchloric acid — 95 percent methanol solution cooled to about 23 °C and achieved at 30 volts and 3 amperes with a 304 stainless steel cathode.

About ten minutes were required for polishing. The

specimen was then etched in warm 5 percent HF

yielding a fairly rough surface.

A grain aligned nearly parallel to the direction in which the load could be applied was chosen for stu- dy. This grain is shown enlarged in figure 21. Surface

roughness is seen in more detail in figure 22 which was taken with a commercial scanning electron

microscope. The etchant attacks the surface in a very nonuniform fashion, producing a terraced effect which is seen near a grain boundary. In a qualitative sense, such etching behavior is usually associated

with variations of locked-in strain energy. After attaching the end pads and drilling the holes

for the pins in the loading device, the samples were

annealed at 350 °C for 2 hours in order to remove any

handling strains. Heating and cooling rates were

very slow in order to insure uniform heating and

cooling off the specimen. Rapidly cooled test sam- 25
pies were found to bend or crinkle. Correctly an- nealed specimens were straight to the eye and deadsoft. Specimens in this condition were deemed suita- ble for Kossel internal stress-strain (KISS) analysis.

A qualitative view of strain contours can be obtained by x-ray topographic methods as well [57].

Such methods give some indication as to the boundaries of strain distributions by viewing the defects

responsible for causing the strains. The higher the

defect density, the higher the strain. On the other

hand, the Kossel internal stress-strain (KISS) analysis can map the strains on a nearly quantitative basis. The trade-off is that mapping a given grain by the

KISS method takes two to three orders of magnitude more time and effort than does the topographic

method. This situation is responsible for the fact that

until now, no grain in a polycrystalline material had

ever been mapped by the KISS method.

Another problem is the tremendous amount of

data needed to describe a single point in the KISS

analysis. More than 100 separate numbers are needed to completely describe the point. For these reasons the grain shown in figure 24 was mapped at only three stress levels: (1) unstrained, (2) slightly

compressed in the elastic region, and (3) slightly

pulled in-tension in the elastic region. The axis of tension or compression was nearly [115] *-> [115].

A few points were also obtained near the yield stress as well. We shall call these stress levels I through IV

respectively. Based on the applied load of 0.3 pound

and a cross-section of (0.63) (0.006) = 0.0038 cm^, the

macrostress at level III (tension) was not less than

200 psi or more than 600 psi. Crude specimen guides were used for the compression test — some bending

may have occurred but that is doubtful. The macrostress for level II (compression) was between 200 and

800 psi (0.14 MN/m2 and 0.56 MN/m^). At level IV,

the macrotensile stress was between 40,000 and

60,000 psi (27.6 MN/m^ and 41.4 MN/m^).

The reference point for all of the work was taken

to be at the geometric center of the grain in the unstrained state. Two patterns were taken a few

micrometers apart at the grain center. One of these

patterns is shown indexed in figure 8. The d spacings

for each were obtained by measuring the X-Y

coordinates at 18 points on each of the lines of the

pattern.

The optimum thickness and voltage requirements

are such that the product UfiX can be as high as 7.5 (see page 24). For a thickness of 60 ^tm in Fe-3 Si, the

value of ijlX using FeKa radiation is (571) (0.006) or

3.43. The ratio 7.5/3.43 gives f/ = 2.19 or an operating

vohage of 15 kV. Therefore, all Kossel patterns in

this work were prepared using an operating voltage

of 15 kV.

Equation (11) was used to calculate the exposure

time for the patterns. Two types of film were used,

Kodak AA, a moderately coarse grained duplitized

film, and Kodak M plates, a moderately fine grained

single emulsion plate. The plates were used to improve resolution and to preclude the possibility of

nonuniform film shrinkage during photographic

processing. This was done even though Gielen et al. showed that the effects of nonuniform film shrinkage

in AA film were negligible [37]. The values for £"0 in eq (11) are 1.6 X 10^ photons/cm^ and 1.4 x 10« photons/cm^ for the film types AA and M respectively [34]. Other values for substitution in eq (11) were

r = 0.27, no ==1.5X10-4 photons/electron at 15 kV,

Xf= 0, tXsXs = 3.43, 7) = 35° and Z = 10.526 cm. The resuhs give IeIs equal to 0.33 and 0.034 for types M and AA respectively. The value of is is typically 0.01 to 0.1 fxA. Thus, exposure times of a few

minutes to half-hour were employed to obtain the

patterns for this study.

But before accurate "<f" spacings could be computed it was necessary to measure the source-to-film

distance, Z, as accurately as possible. Therefore, the

Fe-3 Si specimen described by Gielen et al. [37] was

placed into the Kossel pattern generator. The area they used to determine the lattice parameter as 2.86268 A was located by means of the electron

beam contamination mark. The specimen was

oriented so that the (220) conic (fig. 8) was made a

circle. In gnomonic projection the diameter, L, of the

(220) conic is then

L = 2Ztan(90 - 6>)22o (55a)

Applying Bragg's Law and the identity siny/cosy =

tany we get

Z-L[A/V2(a2-X2)], (55b)

y being the wavelength of Fe-Kai = 1.936042 A. For

each pattern, nine measurements of L were averaged

to get the final value. The experiment was repeated

43 times over the course of several weeks and each

plate processed shortly after exposure. Temperature

variations inside the Kossel camera were very small. The results give Z = 10.5261 (5 = 0.0180) cm. The

half-width of a 95 percent confidence interval is W95 = 0.0055 cm. Reference to figure 11 shows the uncer26
tainty due to Z will be 0.001 for 0 = 30° and less than

0.0001 for d > 55°. In all of the subsequent work, Z

was taken as 10.5261 cm.

Equations 16-19 were programmed in the BASIC

language for processing on a General Electric 265

digital computer in the time-sharing mode. A typical

printout for one conic is shown as table V. The input

data consist of the 234 (13 X 18) X,Y coordinate measurements, XqYo position, Z value, (hkl) indexes and

wavelength used. One XqYq measurement is taken

after each conic is completed and the results averaged to give the XqYq value actually used. The

"oT' values obtained from the reference patterns were averaged and are listed in table VI. All "d"

values obtained subsequently were referred to these

"cT' values in order to obtain the (^d/d) data for the

stress-strain analysis. Thus, it must be understood

that all the stress-strain data refer to the center of

the grain in the so-called unstrained condition, i.e., as the customer for Fe-3 Si sheet would obtain the

material.

The justification for this procedure is that table VI

shows the d spacings for planes within a given family

to vary. Hence, the unstrained crystal is not really

unstrained despite process annealing and careful

handling. However, on an absolute basis, the strain

at the point chosen for reference is small as in- dicated by the relatively small variations in the d

spacings and the sharpness and resolution of the pat- tern shown in figure 8. The lattice parameter at the

reference point is 2.86154 A. The average lattice parameter of the material is 2.86268 A. The average

lattice parameter of the material is 2.86268 A so that Aa/a = (2.86154-2.86268)/2.86268 is - 0.0004 or -

0.04 percent.

In order to observe variations of the lattice parameter, a, as a function of position within the

grain, a program using weighted least squares fit of

the d spacing data was prepared. The logic is as fol- lows:

a' is the apparent a value for a given conic

X is the Nelson-Riley factor =0.5 cos^ 6 _^ cos'^ Ol

sin 6 d

tris the standard error for d for a given conic.

Each of these parameters is computed by the pro- gram giving "cT' spacings illustrated in table V. Then:

Wi^HlaiyiXHIair

a! — l,aiWilXwi

(56a)

(56b)

X='^iWill>tVi (56c)

b = Xwia' iXi -X)l I.Wi{Xi -Xr (56d)

a — a — bX. (56e)

This procedure yields the statistically most valid estimate of the lattice parameter, a. The lattice parameter at the reference point was found as 2.86154 A by this method. Lattice parameter data were only computed for the so-called unstrained condition. Cubic lattice parameters ought not to be re- ported for material deliberately placed under load. Next the equations in the section on stress-strain analysis were programmed for the computer. AU pro- grams were written in BASIC for the time-sharing

mode. The stress-strain programs are modular so

that each desired property can be gotten in a logical

form. The program library is as follows: MODFIT— Yields strain data from pattern, i.e., i^d/d). Inputs: X,Y^ coordinates from film. STANLY — Yields average lattice strains en, €22,

€33, ei3, €31, €12 from Miller indexes and {Ad/d) from

MODFIT. These strains are in the cubic frame. Also

yields estimate of the errors on each strain. PRNSTR — Yields principal strains Si, S2, S3 from

data of STANLY.

STEROR — Yields error estimate of the principal

strains. PRAXES — Yields principal strain axes direction

cosines from data of STANLY and PRNSTR. A

cubic to principal frame transformation is implied.

UNDEFD — Yields direction cosines of principal

strains before deformation. Input from STANLY.

DANGLE — Yields angle between principal strain axes before and after deformation.

STRESS -Yields stresses crn, 0-22, ctss, 0-23, crsi,

(712 in the cubic and principal frames. Inputs from

PRAXES and STANLY and external matrix of

elastic constants. STRAPP— Yields principal stresses. Pi, P2, P3. MAXSHR— Yields maximum shear strain in (hkl)

planes. Inputs from PRAXES and PRNSTR.

DISNUM— Yields geometric distortion number

from invariants of strain tensor. Input from

PRNSTR.

STELEN — Yields total stored elastic energy. SHELEN — Yields shear stored elastic energy. CUTRAN — Transformation from principal to cubic frame.

27
Figures 23-25 show the location of each pattern

within the grain for stress levels I-III. These locations constitute the points from which the maps in the results section were obtained. The number and

location of points at each level were adequate to map

the stress-strain and energy contours reasonably ac- curately for levels I-III — certainly within the accura- cy implied by the limitations of the KISS method itself.In summary, the KISS data were obtained at four

stress levels, three of which could be mapped. Each

pattern required about 250 individual coordinate

measurements, on which the entire KISS analysis was based. Each step in the stress-strain analysis was programmed for digital computer in the BASIC

language. Thus, results from each step could be

tabulated and compared directly. The center of

the "unstrained" grain was used as a reference point

for the stress-strain contour mapping.

X. Results of the Kossel Internal

Stress-Strain Analysis

The d spacing values for the as-received material

are shown in table VI. The lattice parameter, ao, computed by the weighted least-squares program is also included. Gaps in the table occur where it was

not possible to obtain 18 valid coordinate measure- ments for the conic in question. In the case of (112)

and (112) types, often not enough of the conic was

present on the film to permit valid sampling. In the case of other conies, contrast was occasionally so poor that proper measurement was impossible.

Table VI shows that the lattice parameter varies as a function of position within the grain. Variations

of lattice parameter within a grain of a polycrystalline sheet have also been observed in several other

cubic materials [58]. The average lattice parameter

for points in the grain is 2.8627 A which is in good

agreement with the lattice parameter reported for the bulk [37]. However, the two points in adjacent

grains are somewhat lower than this value. Note also that the highest value of the lattice parameter occurs

at a grain boundary location. The variation of the lat- tice parameter within the grain is shown in figure 26. The lattice parameter is, on the average, a bit lower on the side nearest the point where three grains meet

(left side of fig. 26). A straight line path from the center of the grain to any point on the boundary al- most always entails a change of at least 0.0005 A or

about 0.02 percent in the lattice parameter. Beyond

this, knowledge of the lattice parameter alone does

not provide much useful information.

Values of A.dldo for all four stress levels are given

in table VII. Gaps occur for the same reasons that were enumerated for the d spacing values in table VI. These ^d/do data were used to derive the

average Cauchy strains with the aid of eq (24). Table

VIII gives the computed Cauchy strains and their

standard errors as calculated from eq (51). The data in table VIII were used to compute the

principal strains which are listed in table IX. The

standard errors associated with the Cauchy strains were also used in an effort to estimate the maximum

credible error on the principal strains in table IX. This estimate was made by differentiating eq (27)

and inserting the standard errors on the Cauchy

strains for the de terms. This procedure maximizes

the effect of each standard error in table VIII and

also assumes these errors to have a Gaussian dis- tribution. The orthogonality of the principal strains was crudely taken into account by merely dividing

the final result by a factor of three in order to obtain an estimate of the maximum error for each principal

strain. The reasoning is roughly analogous to that used to show the volume coefficient of expansion in three times the linear expansion coefficient [59].

Despite its crudity, this method very nearly

reproduced all of the maximum credible error data on principal strains in tungsten compressed 0.2 per- cent reported by Newman, Glass and Weissman

[60]. Therefore, the values reported in table IX are probably reasonable estimates of the maximum

credible error associated with an individual computation of a principal strain. Table IX shows that for very small principal

strains, the maximum credible error often equals or even exceeds the computed strain value. From this, one concludes that the KISS method can record the

presence of very small strains but that apparently

their magnitude cannot be measured with high accuracy. Aside from this, there is little in table IX to refute the reliability analysis results discussed previously. Accordingly, the principal strain values in table IX wiU be taken as satisfactorily reliable esti- mates of the true values. The entire exercise of computing maximum credible errors was undertaken as a final precaution in ac- cepting the computed principal strain data. This

precaution was necessary because aU of the subsequent results in the KISS analysis depend directly

28
or indirectly on the magnitude of the principal

strains and/or on the values of the cubic elastic constants Cii, Ci2 and C44. For purposes of calculation,

the elastic constants of iron were used. These values

are C„ = 237 GN/m^; Ci2 = 141 GN/m^; C44 = 116

GN/m^ [61]. (One giganewton per square meter is 145,038 psi.)

Tables X-XVII include the results of the entire

KISS analysis for the four stress levels investigated.

The stress-strain configuration for a single position

at one stress level requires more than 100 numbers

for a complete description. Obviously, this requirement is cumbersome and unw^ieldy but, worse, it is difficult to get a clear picture of the situation from

the numbers alone. Note that table XVII also includes that portion of

stored elastic energy assigned to shear called Ws.

This value can be obtained from an equation derived

in detail by Slade et al. [44]. This equation is given

as:Ws = 0.5 (Si - 53)^^44 - 0.25 (C12 + 2C44 -Cn)

Slade et al. [44] showed that the total stored elastic energy could be partitioned into two energy terms W

= [Wn + Wns] + Ws. The composite term, Wn+ Wns,

represents a composite of energy terms deriving

from normal strains (Wn) and a mixture of normal

plus shearing strains (Wns)- These latter two terms

cannot be unambiguously separated but, as in- dicated above, Ws can be. Hence, we may obtain the

ratio of the shear stored energy to the total stored

energy, WslW. Of course, the composite term is just

W-Ws.

Most of the results in tables X-XVII were plotted as area maps (figs. 27-39) of the grain shown in figure

21. These maps reveal the existence and extent of a

variable strain distribution (VSD) within the grain on at least a semiquantitative basis. The maps

represent the first plot of variable strain distribution

in an opaque polycrystalline materials on any basis

beyond the qualitative views of dislocation clusters

obtained by x-ray topographic methods.

The maps in figures 27-39 were prepared by connecting each point in figures 23-25 using the data in tables IX-XVII and then using linear interpolation to get from one point to the next. All indicated boundaries were then linked to prepare the final map.

Attempts to obtain Kossel patterns directly at a

grain boundary met with failure. Traces of two grains

were always present. Figure 40 shows a typical resuh

where the displacement of the 110 poles of the two

grains permits an estimate of the misorientation at the boundary. Typical values are 1 to 3 degrees of

arc. This grain boundary mismatch can lead to high

stresses at the boundary, and it indicates that the texture differs locally from the desired (110) [001].

The maps in figures 27-38 clearly show the ex- istence of VSD at all stress levels investigated in- cluding the so-called unstrained condition. The maps

show a more random distribution of the VSD for the

as-received material than for the compressed or stretched situation. Although it is oversimplifying

matters, the VSD appears to be "squeezed" in the

compression case and to be "lined up" in the

direction of the applied load in the tensile case.

First, note the similarity between the principal

strain maps figures 27-29 and the lattice parameter

for the as-received material. While not greatly surprising, there is clear evidence that the lattice parameter is entirely related to the local strain conditions in the textured Fe-3 Si sheet. The bulk lattice parameter of 2.8627 A is apparently the "equilibrium" done in conventional determinations of Oo values. The principal strain maps for the compressive and

tensile cases show rearrangement of the VSD and an

increase in the size of regions with higher absolute

principal strains. Certain small areas of high strain have been removed, e.g., the area at the right of

figure 27a, and are incorporated into these larger re- gions. Figure 27c shows the existence of a highly

strained region within the grain core. It is probably

the existence and propagation of such regions which

cause the degradation of electrical properties observed when the nominal elastic tension is increased

beyond 500 psi.

The principal stresses associated with the principal strains are shown in figures 30-32. Note that the

maximum principal stress associated with the highly

strained region in the core is more than 2 GN/m^ or more than 250,000 psi. In the as-received condition, a somewhat smaller region of similar maximum principal stress was present (fig. 30a).

Local regions in the Fe-3 Si polycrystalline ag- gregate are under stresses of more than a quarter

million psi in the as-received material and larger re- gions of high stress exist in the loaded sheet. There

are alternate regions of tensile and compressive

stress. A merit figure consisting of the algebraic sum

of Pi + P2 + Pz from table XIII was calculated in an

29
attempt to delineate these alternate regions and the

result shown in the maps of figure 33. The original grain appears to be in compression

throughout most of the left side of figure 33. Tension

is present at the center and right center, then compression and finally there is the small region of very

high tension at the extreme right. In the case of the

compressive load, all but a small portion of the grain

is in compression. For the tensile load, most of the

grain is in tension, but two areas near the boundary

on the left side of the grain are in compression.

AU of these observations tend to indicate that neighboring grains press into the grain of interest,

approximately along the lines shown in figure 34 during processing to the as-received sheet. Even under

tensile load, there is a compressive strain resulting

from a tendency for the neighboring grains to "squeeze" the grain in a direction roughly perpendicular to the loading direction (see fig. 32c). This

grain exhibits shear strain. A number of regions in the grain have nearly as principal strains S, O, — S

as listed in table IX. In addition, several other re- gions have one nearly zero principal strain. When

the S, O, — S configuration exists, the region is in simple shear. Furthermore, the ratio of the shear

stored elastic energy to the total stored elastic ener- gy is very high throughout the grain as shown in figures 35-37. This situation obtains for all stress

levels investigated.

Slade et al. [44] have shown that typical conditions for the existence of a plane of nearly zero nor- mal stress occur when S2 — 0, S1/S3 = — p^. Using the

positive sign, the axes of this plane referred to the

principal strain can be represented as a matrix, B,

such that [44] - p

0

1 (1 +p2)'/2

0 1 0

1 0 p

(1 + p2)l/2 (1 + p2)l/2

If we call the matrix representing the principal strain axes (table X) /4, then Bc — A~^B where Be represents

the axes of matrix B in the cubic frame.

Examination of table IX reveals that more than

two-thirds of the patterns studied yielded principal

strains such that S2 — O and Si and S3 of opposite

sign. The Be matrix was calculated for each of these

patterns. The results are listed as table XVIII. Furthermore, the poles of these planes of nearly zero

normal stress were plotted on {110) stereographic

projections (fig. 41). The poles of the approximate

great circles on which the points lie are nearly of the

type {351}. Apparently, the direction along which

planes with minimum normal stresses and maximum

shearing strains lie is approximately of the type

[351]. The magnitude of the maximum shear strain

in these planes is given by 2(— SiSs)''^ according to Slade et al. [44], This value is the maximum shear

strain in any plane.

Table XIV shows that for stresses in the principal

strain frame, the shearing stresses, ctzs, ctsi, cri2, are

in most cases lower in magnitude than the normal

stresses. Slade et al. have shown that in such a case, the principal strain axes and the principal stress axes nearly coincide [44].

Examination of table X shows that the principal axes of strain after deformation often lie within about

20° of (110) or (100) type poles. This circumstance

may result from mechanical anisotropy in the

oriented sheet. There exists a set of three axes subject to max- imum strains with respect to strains of linear ele- ments in all other directions [62]. The angle

between these axes (table XI) and the principal

strain axes is shown in table XII. In many cases, there is little difference in direction but a change in sense as represented by rotations of nearly 180°. Many of the other rotations are nearly 30 or 60°. The

cases where Oi = 180° or 0°, Oj = 0 and Ok 30° or 60° occur with significant frequency among those pat- terns for which a shear plane can be deduced. As we

shall see later, this type of rotation of the axes is consistent with stresses induced by the response of the

Fe-3 Si to mechanical anisotropy. To a first approximation, the line of action or resultant stress due to mechanical anisotropy can be predicted to lie 30 to 60° from (110) or (001 ) type poles (see sec. XII).

The total stored elastic energy tends to follow the

principal strain and principal stress configurations

as shown in figures 27-32 and 35. The average stored

elastic energy in the grain does not vary greatly from

that in the as-received state when the low compressive or tensile macrostresses used in this study are

applied. In all probability, these stresses are not enough to seriously degrade the magnetic properties

[3,4] or to significantly alter the magnetic domain

pattern [9].

A large portion of the total stored elastic energy is shear energy as shown in figures 36 and 37. This imbalance probably results from the propensity of the

30
Fe-3 Si sheet to shear in response to stresses in- duced on cooling from 1200 °C by the mechanical

anisotropy of the oriented grains.

The distortion number is also large in most in- stances (fig. 38). The ratio WslW is related to the

distortion number [44]. Note that, in many cases,

the distortion number and WslW are nearly equal (ta- ble XVII and fig. 37). The maximum shear strains in potential slip

planes as weU as in {100} type planes are listed in table XVI. The high locked-in shear strains tend to indicate that slip as a general stress-relief mechanism has not occurred. Usually, one or two

values of the maximum shear strain are low compared to the others. In fact, there appears to be,

roughly, a set of nearly equal low values, a set of in- termediate values and a set of high shear strains at each point. In addition, the maximum shear strain in {100} types is often less than in several potential slip

planes. The "three levels" of locked-in shear strain follow roughly the predictions obtained by computing the resolved shear stress RSS factors for potential slip systems which, for instance, show that (101)

[nl], (101) [111] and (Oil) [111] have nearly equal

RSS factors. On the other hand, (110) [111] and

(lIO) [111] types have low RSS factors. Attempts to correlate maximum shear strain data

with position in the grain and applied load met with

frustration. There appears to be a more or less ran- dom response to the applied load. General sUp is in- operable and twinning does not occur either. One

might guess that some local "critical resolved shear

stress" caused one or two potential slip planes to give up their strain energy but not the others. The

behavior of the {100} types remains something of a mystery. The variation of the maximum shear strain

in (112) sheds no real Hght on the problem (fig. 39). Perhaps when slip does become operable under

mechanical deformation, it follows the "trends" in table XVI. Such behavior may indicate why slip in Fe-3 Si is relatively poorly understood [63,64,65].

The only conclusions, then, that can be drawn

from the maximum shear strain data are that: (1) general slip is inoperable; (2) some shear strain relief occurs in some planes apparently ip response to local stresses set up by the stress-strain subregions;

and (3) the three-level nature of the maximum shear

strains contributes to raising the distortion number

[44]. In summary, the experimental results show that there is a variable strain distribution in the asreceived sheet and that these subregions are ahered

by external loading. There appears to be a strong

tendency for the Fe-3 Si to accommodate internal stresses by some sort of shear mechanism; a large

portion of the stored elastic energy is shear energy. The VSD is not caused by local variations in iron or

silicon content. Now the results can be considered

in the context of the metallurgy of the Goss oriented

Fe-3 Si sheet.

XI. Metallurgical Aspects Pertaining

to the KISS Results

The Fe-3 Si sheet which the user receives is clearly not strain free. Figures 27-39 show that there

is a variable strain distribution (VSD). Apparently,

the net effect of this VSD in the entire as-received

material results in a residual compressive stress in the sheet [4]. Furthermore, the siUcon and iron dis- tribution is entirely uniform. Thus, major element

compositional variations play no role in causing the

VSD.

The VSD observed almost certainly must have

been created no earlier than the texture anneal (see

fig. 3) but, beyond this, it is impossible to state exactly how and at what point in the history of the

material the VSD was formed. However, each possible contributing factor to VSD formation can be considered in the Hght of the results shown in this study.

The (110) [001] grains grow by the process of

secondary recrystallization. Secondary recrystaUization is the selective grain growth of only a few grains, as distinguished from uniform growth, when the new

set of grains formed on primary recrystallization is subjected to further annealing. The texture after primary recrystallization in Fe-3 Si is extremely complex [66]. However, the primary texture always contains some (100) [001] components [67]. Strangely,

the final Goss texture does not seem to depend

strongly on the amount of (100) [001] in the primary

texture [67] but only on the fact that some (110)

[001] grains are indeed present.

Primary recrystallization occurs to reheve the cold

rolling strains set up throughout the metal. In the case of Fe-3 Si, primary recrystallization is carried out at about 800 °C. Hu [68] has shown that the primary texture probably grows by subgrain

coalescence in Fe-3 Si [68]. The removal of the subgrains lowers the free energy of the system. Some

(110) [001] grains form because these grains or31
dinarily have the lowest surface free energy of all grains [67]. The primary texture has been described

in detail but the exact relationship of the primary

texture to the Goss texture is still not completely un- derstood [67]. The rate of grain boundary migration

for a primary (110) [001] grain at 800 °C is between

0.005 and 0.01 /xm per second [69]. At a rate of 0.005

/xm/sec, a circular grain would increase its diameter

1 /xm every 100 seconds. Thus, after a two-hour an- neal we would expect the (110) [001] grains in the

primary texture to be between 50 and 100 ^tm in diameter.

Such (110) [001] primaries grow rapidly during

the secondary recrystallization step (the texture anneal) at 1200 °C. The grain boundary migration rate

is 18 fimlsec [67]. This growth rate falls by about 5 percent for each degree of misorientation of the

grain from a true (110) [001] [70]. The rate of grain

growth is determined by the grain boundary mobility, M, times a driving force, P [67]. The driving

force term depends on surface tension differences

between grains, the sheet thickness, the actual grain

size and the surface energy of each grain [67]. Typical numerical values at 1200 °C for (110) [001] grain

growth are M= 2.5 X 10-« cm^erg sec and P = 7.2 X

lO-* erg/cm3 = 7.2 X 10^ N/m^ ~ one pound per

square inch [67]. (Note that the foregoing is 1 pound

per square inch of grain boundary.)

Under vacuum conditions or an oxygen free at- mosphere, the (110) [001] grains have the greatest

driving force of all grains in the array [77].

Nevertheless, it is commercial practice to add a dispersed second phase such as MnS or silicon nitride in order to restrain normal grain growth. The

ability of the inclusions to perform this function depends on their size and distribution. MnS has been

added by heating the primary matrix to 1325 °C to dissolve the inclusions and then cooling at 130

°C/min. This treatment led to small well-dispersed

inclusions in the matrix [72]. After the final texture anneal, these inclusions were distributed in the grain

boundaries of the Fe-3 Si sheet [102]. In commercial

practice, some of the inclusions would be removed

by reaction with a coating on the Fe-3 Si sheet (see

fig. 3). The stress field around these leftover inclusions

may be approximated by assuming the Fe-3 Si is an

elastically isotopic medium near the inclusion. The

stress field around a spherical inclusion can then be

expressed as follows [73]:

o-ii 2G(Q')M1-F)(l + At) ri _ (59)

where G is the bulk shear modulus ~ 70 GN/m^ for Fe-3 Si

a' is the radius of the inclusion = 1 /xm in Fe- 3 Si [72]

H- is Poisson's ratio ~ 0.29 in Fe

k is a/a', a being the spherical "hole" into

which the inclusion is forced; k is always less than 1 r is (x^ + y^ + z^)*'^ where x, y, and z are the

coordinates of a point in the medium. For simplicity,

the center of coordinates is at the center of the inclusion. With the appropriate substitutions, eq (59)

reduces to

o-ii ^ 56 (1-F)/Z3 (o- in GN/m^) (60)

if a point on the Xaxis is chosen for examination.

The stresses indicated in eq (60) serve to retard nor- mal growth. These stresses must also tend to act as

a drag on the growth of the desired (110) [001] grains

as well. In fact, when the texture is almost all (110)

[001] grains, the impurity stresses in the grain boundaries probably tend to halt the growth of these

grains prematurely (or at least to slow growth drastically, which for all practical purposes is the same

thing). Such an occurrence could account for the

relatively large spread of orientations about a true

(110) [001] found in the final sheet stock. There is strong evidence that the closer a grain is to exactly QIO) [001], the more it is favored for growth [67, 70]. Thus, perfectly oriented (110) [001]

grains wiU grow at the expense of slightly misoriented (110) [001] grains. But the impurities tend

to retard all grain growth and so a comparatively

large number of slightly misoriented grains can

result. After removal of the impurities and during the

slow cool from 1200 °C,.some grain growth favoring

orientations very close to (110) [001] occurs — the ef- fect of this is probably to get rid of the grains with

the most misorientation and to reduce the size of

other misoriented grains with respect to those very near (110) [001]. Nevertheless, the impurities almost

certainly cause a greater orientation spread around

(110) [001] than would be expected in the absence

of impurities. The spread around (110) [001] in material of the type studied here is about ± 15° [74].

Any impurities not removed from the Fe-3 Si sheet can act as stress concentration points at the (110)

32
[001] grain boundaries. Indeed, failure of the sheet

under mechanical deformation is almost always in- itiated at these leftover inclusions [75]. The dif- ference in thermal expansion characteristics of these

leftover inclusions and the metal may also serve to concentrate stresses along grain boundaries when

the sheet is cooled.

An approximate calculation of these impurity in- duced cool-down stresses can be carried out as fol- lows: Assume the inclusion is spherical and compute

the volume change difference on cooling from 1200

°C between the inclusion and a like-sized sphere of

the metal. A MnS inclusion will have a thermal coefficient of expansion, ar, equal to about half of aj for the metal [76]. For Fe-3 Si, ar= 1.2 X IQ-^rC [61].

Then for a 1-micrometer radius sphere cooled 1200

°C, the new metal radius would be 0.9856 /xm but the

actual inclusion radius is 0.9956 fxm. The value of (1- P) in eq (59) is then (0.99563-0.98563) or 0.021. Thus, the relation for the cooling stresses near the

inclusion becomes cr ~ 1.2/^^^ ^Jth cr in GNjm? and

X, the linear distance from the inclusion, in /u,m. This

relation gives o" of 1.2 GN/m^ (175,000 psi) for J^=l

/im and 1.2 X IQ-^ ^N/m^ (i80 psi) for X = 10.0 /xm

from the inclusion, respectively.

On the other hand, Si3N4 has ar of about 0.2 X

10~V°C [76]. Thus, the use of Si3N4 inclusions of the same size as MnS inclusions would result in higher

grain boundary stresses. AU other things being

equal, MnS is then the better inclusion of the two to use since it will cause lower grain boundary stresses. The effect of misfit grains may be considered

using the same reasoning. The final sheet contains

typiccdly 90 percent of (110) [001] grains and about

10 percent misfit grains. These misfits are almost entirely small grains having an orientation near to (100)

[001]. Such grains form because the difference in driving force for secondary recrystaUization is nearly

equal for (110) [001] and (100) [001] types. Occasionally, local energy imbalances or a very poorly

oriented (110) [001] grain make token growth of the

(100) [001] energetically favorable. Cursory ex- amination of several regions in the Fe-3 Si sheet from

which the specimen in figure 20 was taken indicates

that, roughly, the diameter of the (110) [001] grain is about five times the diameter of the misfit grain.

Thus, we may consider the misfit as analogous to very large inclusion in the (110) [001] array. The

equation representing the stresses caused by the

misfit is eq (69) [77]. Hence, we conclude that the

misfit causes high stresses at the boundary between

it and its (110) [001] neighbors. The effect is probably negligible a few hundred micrometers away

from the boundary.

Up to this point, stresses an'd other consequences

introduced by the metallurgical procedures used to make Fe-3 Si sheet have been discussed. Now, the

effects of the mechanical anisotropy of Fe-3 Si must

be considered. These effects can assume their

greatest importance during the cooling from 1200 °C. The mechanical anisotropy is caused by dif- ferences in Young's modulus, F, and the shear

modulus, G, as a function of direction: Y\uo] is 216.5

GN/m^ while F[ioo] is 131 GN/m^. In turn, C[iio) is 66.2

GN/m2 while Qioo] is 111.7 GN/m^ [61]. If the Fe-3

Si were not oriented in a special fashion, the effects of the mechanical anisotropy would be averaged out over the great multiplicity of grain orientations. But

after secondary recrystaUization, virtually the entire

sheet has a rolling plane direction of [110]; the

rolling direction [001] is perpendicular to [110].

Therefore, the effects of mechanical anisotropy in the sheet can assume maximum possible importance.

Consider the Goss oriented sheet in figure 1. The

effect of the difference in Young's modulus as a function of direction will be that shrinkage on cool- down from 1200 °C will occur more easily along the

sheet, [001], than across the sheet, [110]. Thus,

there can be a tendency for the sheet to bend around

the [110] direction or axis. Now the product arAT" on cooling Fe-3 Si from

1200 to 25 °C is 0.014. The maximum possible dif- ferential cooling stress wiU be given by:

(F[ioo] - K[iio]) aT^T= (Tc,ma^=\.2\ GN/m^ - 175,00

(61)

Equation (61) assumed equal shrinkage along [110]

compared to [001]. Because of constraints placed on

the grain by neighboring grains and because of stress

relief due to plastic deformation and especially

shear, we would not expect a 1.2 GN/m- stress to be

present throughout the grain. But the results of the

KISS analysis clearly indicate regions within the as- received material where such stress levels do indeed

obtain. Shear may occur in response to the effect of the stresses set up by the Young's modulus dif- ferences. Recall that G[no] is 66.2 GN/m- and G[iio]is 111.7 GN/m^. Thus, there is a tendency for the dif- ferences in Y values to force a spherical grain into an 33
ellipsoid constrained by its neighbors which is op- posed by a tendency to shear along some path to re- lieve the stresses set up by such a process. This

shear mechanism is favored because G[iio]/G[ioo] =

0.59 but F[„o]/F[ioo] = 1.65. The results of this study indeed show that Goss

oriented Fe-3 Si has a strong tendency to endure

shear. The poles of the planes of nearly zero normal

stress and maximum shear strain are plotted in figure 41. The pole of these planes is also indicated.

The directions in which nearly zero normal strains occur should be nearly perpendicular to the (351)

direction. This direction lies in a plane common with

115 and 110. The 115 is the sheet orientation for

this particular grain.

Apparently, the mechanical anisotropy of the

oriented sheet causes shear to occur in order to re- lieve stresses encountered on cooling from 1200 °C

to room temperature. Such a mechanism would explain the high ratio of shear energy to total stored

energy in the grain in the as-received condition. The

high values of the maximum shear strains in potential slip system of types {110} [111] and {112} [111]

indicate that slip may be imminent and may occur lo- cally to relieve very high stresses. But general slip

has not relieved the shear stress. Perhaps this is because the estimated cooling stress is below the

macroyield point for the Fe-3 Si so that plastic deformation can only occur locally to relieve regions of

very high stress, i.e., near a leftover impurity or a

misfit grain.

The sheet may actually bend somewhat in response to the mechanical anisotropy. The thermeil

flattening step (see fig. 3) is meant to remove bending

so that the sheets can be inserted into transformer

cores without high stresses. In this process, the Fe-3

Si sheets are stretched in tension at some temperature around 200 to 300 °C. This temperature is low

enough so that major alterations of the texture wiU

not occur [68]. Bending is eliminated by a process

akin to "tensile creep." This process effectively puts

back some of the bending stresses into the sheet. These stresses are far less than if the sheet had

merely been cold flattened in the transformer core. The final stress relief anneal tends to remove some of these stresses and to reintroduce cooling

stresses again initiating some bending. The net

result is some bending stress plus a shear stress in the final sheet. The thermal flattening process almost certainly in- troduces subgrains into each major grain. This

comes about partly because the low angle boundaries and random dislocations remaining after the tex- ture anneal can coalesce [78]. The subgrains

probably represent incipient recrystaUization within

the (110) [001] grains. They form to reduce the

stresses just discussed.

The size and shape of subgrains in a single crystal

of Fe-3 Si have been observed by the anomalous

transmission of x rays [79]. Perhaps it is not un- reasonable to assume that the final VSD is delineated by the subgrain boundaries formed in the

thermal flattening step. The stress-relief anneal fol- lowing thermal flattening probably causes a bit of

subgrain boundary growth and some smoothing of

the VSD. From the foregoing, I surmise that the as- received material has a VSD proportional to the size of the subgrains formed in the thermal flattening process. Often (table X) the principal strain axes are near

the directions [100] and [110]. Furthermore, the

principal stress axes nearly coincide with the principal strain axes. This behavior is possibly a con- sequence of the two-dimensional nature of the stress

field. In many cases (table XII), the principal strain axes

differ only little from the undeformed fiber on which

the principal strains acted. According to Varga [62],

the undeformed fibers represent a set of three

directions in space such that linear elements placed

along them in the undeformed state will be subject

to maximum strains with respect to strains of linear

elements in all other directions. Perhaps when the

Fe-3 Si sheet is strained, the maximum stresses and

strains act along the [001] and [110] directions in order to maintain the stored energy at its lowest

possible value. Certainly these directions are favored for texture growth in order to reduce stored energy. So, by analogy, it may well be that the max- imum stresses and strains are forced onto these

directions for the same reason. The frequent 30 or 60° rotation (table XIV) would be consistent with a shear path caused by plastic mechanical anisotropy.

Apparently, mechanical deformation by slip does

not occur easily in order to relieve the locked-in

stresses. Maximum shear strain data in potential slip

planes show a great deal of variability (table XV).

Furthermore, the maximum shear strain in some

nonpotential slip planes, {100}, is often less than in {110} and {112} types. Ocasionally, there is a precipitous drop in one or two potential slip planes

after deformation. Probably, local slip has occurred

34
to relieve stress-strain concentrations such as that occurring at the extreme right of figure 30a. The

variability in the maximum shear strain data ac- counts for the high values of the distortion number

found [44]. (See table XVII and fig. 38.)

Slip behavior in Fe-3 Si is unusual in any case. Slip

bands almost always tend to propagate from a grain

boundary and often do not extend very far into a

grain [64]. In order to propagate a slip band, almost

all of the potential slip planes must act to rid themselves of their shear strain — not one or two. Some

external load is needed to accomplish this. A guess

would be that slip lines in the grain in this study

would propagate first from the grain boundary at the

extreme right of figure 27a in order to relieve the

very high stresses at that point.

There are many small regions in the grain studied

where one or more than one principal stress exceeds

the macroyield stress (0.5 GN/m^, 70,(X)0 psi) in Fe-3

Si. These are the sites at which stress relief should

be initiated on mechanical deformation or annealing.

The existence of such regions is not overly surprising. Stresses of 1.45 GN/m^ (211,000 psi) have been

reported in CuAu crystals [44]. The average stress

in the grain was taken as l,(Pi + Pj + Pk)ln where n

is the number of determinations within the grain.

The result for the as-received crystal was P — —

0.083 GN/m^ or — 11,800 psi, i.e., a compressive

stress well below the macroyield stress. For the applied tensile stress of about 500 psi, P was 4-0.124

GN/m2 or slightly less than + 18,000 psi.

Still, the grain must somehow accommodate even

small local regions of very high stress. Subgrain

boundaries probably form, in order to achieve this accommodation, during the thermal flattening and

subsequent steps. Apparently, this local form of

atomic rearrangement is enough to accommodate

the highly stressed regions with only a minimum of true local plastic deformation. One can conjecture

that the one or two potential slip planes with low

shear strain may have given up shear energy which

contributed to the formation of subgrain boundaries,

which then can inhibit further slip on other potential

slip systems.

The small applied loads in the study caused rear- rangement of the VSD as shown by figures 27-39. The effect of compression is to move the VSD about

and to raise the absolute value of the internal strains. Such a procedure cannot be expected to improve the

electriccd properties of the material but rather to degrade these properties.

The application of a small tensile load tends to line up the VSD in the loading direction. It also tends to smooth out very large strain differences as shown in figures 27-39. This behavior may explain, at least

partially, why the electrical properties tend to improve slightly after the application of small tensile loads. Increasing the tensile stress would probably just

increase the magnitude of the strains in the VSD

without very much further alteration of their position. In such a case, the magnetic properties would

again degrade with increasing load. Annealing up to 300 °C does no significant good for the electrical properties [4] (see fig. 2). This fact is a further indication that the VSD may move a bit but

is not appreciably altered by elevating the temperature. The application of loads at elevated temperatures also does not improve the situation [75].

The core loss for Fe-3 Si material of exactly the

type supplied by Dr. Fiedler for this study has been

reported as about 10 watts/pound at an induction, B,

of 18,000 gauss by the manufacturer [74]. However,

Goss oriented Fe-3 Si made in the laboratory using

only surface energy to grow the (110) [001] and with no grain boundary impurities such as MnS or Si3N4

showed a core loss of about 6 watts per pound at B of

18,000 gauss [74]. This improvement can probably

be attributed to less orientation spread around a true

(110) [001] and to an overall reduction of the internal

stresses in the material occasioned by the absence

of the inclusions and the production of fewer misfit

grains. Then, the only stresses left would be those

due to mechanical anisotropy. These stresses probably cannot be removed owing to their nature. As a guess, core loss can probably be cut by a factor

of 2 to 3 by improved processing but not more than

this. The main reason for the improvement is that orientation is better and internal stresses are reduced.

The results of this study can be used in an attempt

to explain a few of the observations concerning the

degradation of electrical properties:

(1) Higher losses at grain boundaries (fig. 2) can

result from very high stress concentrations near the

boundary (fig. 27a), from grain misorientation across

the boundary (fig. 30) and from impurity or misfit stresses at the boundary.

(2) Losses at misaligned (100) [001], grains (fig. 2) can be contributed to by the increased stress at (110)

[001] and (100) [001] grains caused by cooling

stresses. Of course, the easy magnetization
directions change abruptly at such grains which

could also contribute to the degradation.

(3) Observations that magnetic domains rearrange

upon the application of tensile stresses of a few bundied to several thousand psi have been made [9-11].

Here the VSD rearrange as indicated by figures 27- 39 and the domains follow. The so-called transition zone in which a Bitter pattern disappears and then

reforms may be a hysteresis in the colloid used to create the pattern. There is no reason to believe that the domains will "disappear" during VSD rearrange- ment. On increasing loads, beyond a few thousand

psi, the magnetic domain pattern does not change

drastically [9]. This observation reinforces the notion that an increase in tensile stress merely in- creases the magnitude of strains in the VSD without

altering the position of the VSD drastically.

(4) Compressive stresses can be introduced during cooling from 1200 °C after the texture anneal.

Such an occurrence would explain the observation

that many magnetic property curves are skewed

toward the tensile side [4].

XII. References

[1] Goss, N. P., Trans. ASM 23,511 (1935).

[2] Assmus, F., Detert, K., and Ibe, G., Z. Metallk 48, 344

(1959). [3] BraUsford, F., and Abu-Eid, Z., Proc. Inst. Elect. Engrs. 110,751(1963).

[4] Brown, D., Holt, C, and Thompson, J. E., Proc. Inst. Elect. Engrs. 112,183(1%5).

[5] Boon, C. R., and Thompson, J. E., Proc. Inst. Elect. Engrs. 112, 2147 (1%5).

[6] Imura, T., Weissman, S., and Slade, J. J., Acta Cryst. 8, 786

(1%2).

[7] Stewart, K. H., Proc. Camb. PhU. Soc. 45,2% (1949).

[8] Boon, C. R., Holt, C., and Thompson, J. £., Brit. J. Appl.

Phys. 14,708(1963).

[9] Dykstra, L. J., and Martuis, U. M., Revs. Mod. Phys. 25,

146 (1953). [10] Shur, Ya. S., et al., Phys. Met. and Metallog. (U.S.S.R.) 6(3), 158(1958).

[11] Kirenskii, L. V., Savchenko, M. K.. and Rodichev, A. M.,

Bull. Acad. Sci. U.S.S.R. Phys. Ser. 22, 1171 (1958). [12] MorriU.W., Metal Progress 78(3), 84 (1960). [13] Baer, G., Ganz, D., and Thomas, H., J. Appl. Phys. 31,235S

(1960). [14] MuUins,W.W., Acta Met. 6,414(1958).

[15] May, J. E., and TurnbuU, D., Trans. AIME 2 1 2, 769 (1958). [16] Sharpe, R. S., Appl. Mater. Res. 4, 74 (1%5).

[17] James, R. W., The optical principles of the diffraction o£xrays, in The Crystalline State, L. Bragg, ed.. Vol. II (Bell, London, 1950). [18] Maier, W., Ann. Physik 40(5), 85 (1941). [19] Lonsdale, K., Phil. Trans. Roy. Soc. London, Ser. A240,

219 (1947).

[20] Ellis, T., and Weissmann, S., A.F. Matls. Lab. Technol. Documentary Rept. ML-TDR-64-220 (1964). [21] Warren, B. E., MIT Physics Dept. course entitled X-ray Dif- fraction. [22] Peace, A. G.. and Pringle, G. E., Phil. Mag. 53, 1227 (1952). [23] Yakowitz,H.,J. Appl. Phys. 37,4455 (1%6).

[24] James, R. W., King, G., and Horrocks, H., Proc. Roy. Soc. (London) A153,230 (1935). [25] Frazer, J. Z., and Arrhenius, G., in Optique des Rayons X et Microanalyse (R. Castaing, P. Deschamps, and J. Philibert,

eds.), p. 516. Editions Tecniq, Paris, 1966. [26] Potts, H. R., and Pearson, G. L., J. Appl. Phys. 37, 2098

(1%6).

[27] Witt, F., private communication (1966). [28] Morris, W. G., S. M. Thesis, MIT (1%3).

[29] Hanneman, R. E., Ogilvie, R. E., and Modrzejewski, A., J. Appl. Phys. 33, 1429 (1932). [30] Potts, H. R., Stanford Electron. Lab. Rept. SU-SEL-64-075

(1%4).

[31] Kramers, H. A., Phil. Mag. (6)46,836 (1923). [32] Stainer, H. M. , U.S. Bur. Mines Inform. Circ. 8166 (1963). [33] Ellis, T., Nanni, L. F., Shrier, A., Weissmann, S., Padawer,

G. E.,and Hosokawa, N., J. AppL Phys. 35,3364(1965).

[34] Yakowitz, H., and Vieth, D. L., J. Res. Nad. Bur. Stand. C69,213 (1%5).

[35] Warren, B. E., Progr. Metal. Phys. 8, 147 (1959). [36] Kossel, W., Ann Physik 26, 533 (1936). [37] Gielen, P., Yakowitz, H., Ganow, D., and Ogilvie, R. E., J. Appl. Phys. 36,773 (1965). [38] Mackay, K. J. H., in Optique des Rayons X et Microanalyse

(R. Castaing, P. Deschamps, and J. Philibert, eds.), p. 544, Hermann, Paris, 1%6.

[39] Morris, W. G., Gen. Elect. Res. Develop. Ctr. Rept. 66-C217, 1966 (J. Appl. Phys. 39, 1813 (1968)). [40] Hald, A., Statistical Theory with Engineering Applications,

Wiley (New York) 1962, Chapter 20. [41] Bearden, J. A., Henins, A., Marzolf, J. G.. Sauder, W. C,

and Thomsen, J. S., Phys. Rev. 135, A899 (1%4). (See also Bearden, J. A., U.S. AEC Rept. NYO 10586 (1964).) [42] Ku, H. H., private communication (1969). [43] Novozhilov, V. V., Theory of Elasticity, Israel Program for Scientific Translations, available from U.S. Dept. Com. as OTS 61-11401.

[44] Slade, J. J., Jr., Weissmann, S., Nakajima, K., and

Hirabayashi,M., J. Appl. Phys. 35,3373 (1965). [45] Weissmann, S., et al., U.S.A.F. Materials Lab. Technol. Documentary Rept. ML-IDR-64-220 Parts I-IV (1964-1968). [46] Cullity, B. D., Elements of X-ray Diffraction, Addison- Wesley, Reading, Massachusetts. (1956). [47] Rutherford, E., and Andrade, E. N. da C, PhU. Mag. (6)28,

263 (1914). [48] Seemann.H., Ann. Phys. 53,461 (1917). [49] Gerlach,W., Physik, Zeitschr. 22,557(1921).

[50] Linnik, W., Nature (London) 124,946 (1929). [51] Kossel, W.,Ergebn.exakt.Naturw. 16,295(1937).

[52] Borrmann, G., Ann. Phys. 27, 669 (1936). [53] Castaing, R., Thesis to University of Paris (1951). [54] Vieth, D. L., and Yakowitz, H., J. Res. Nat. Bur. Stand.

(U.S.), 71C (Eng. and iiistr.), 313 (1967). [55] Vieth, D. L., and Yakowitz, H., Rev. Sci. Instr. 39, 1929

(1968). [56] Yakowitz, H., Fiori, C. E., and Michaelis, R. E., Nat. Bur. Stand. (U.S.), Spec. Publ. 260-22, 30 pages (Feb. 1971). [57] Lang, A. R., Advances in X-ray Analysis 10, 91 (1967). [58] Ullrich, H. J., Acta Cryst 19, Suppl. A235 (1966). [59] Sears, F. W., and Zemansky, M. W., University Physics

(2nd ed.) Addison-Wesley, Cambridge, Mass. (1955), p. 264. [60] Newman, B., Glass, H., and Weissmann, S., A.F. Materials Lab. Technol. Documentary Rept. ML-TDR-64-220 (part IV), 1967. 36
[61] Moore, G. A., and Shives, T. R., in Metals Handbook (Vol.

I), American Society for Metals (1961), p. 1206.

[62] Varga, O. H., Stress-Strain Behavior of Elastic Materials

(Inter-science, New York, 1966).

[63] Suits, J. C, and Chalmers, B., Acta Met. 9, 854 (1961). [64] Worthington, P. J., and Smith, E., Acta. Met. 12, 1277

(1964). [65] Worthington, P. J., Acta. Met. 14, 1015 (1966). [66] Dunn, C. G., and McHargue, J. Appl. Phys. 31 , 1767 (1960). [67] Dunn, C. G., and Walter, J. F., in Recrystallization, Grain Growth and Textures (H. Margolin, ed.) ASM (1966), p. 461.

[68] Hu, H., in Recovery and Recrystallization of Metals {L.

Himmel, ed.), Gordon and Breach (New York) 1963, p. 311.

[69] Gordon, P., in Energetics in Metallurgical Phenomena

(Vol. I) (W. M. Mueller, ed.) Gordon and Breach (New York)

1965, p. 205. [70] Vobroh'yev, G. M., and Shmyrev, I. P., Fiz. Vletall. i. Metal- lovedne 22 (2) 314 (1966). [71] Waher,J. L., and Dunn, C. G., Trans. Met. Soc. AIME218,

1033 (1960). [72] Fiedler, H. C., Trans. Met. Soc. AIME 2 30, 95 (1964). [73] DeWit, R., Private Communication (1970). [74] Walter, J. R., J. Appl. Phys. 36, 1213 (1965). [75] Cell, M., and Worthington, P. J., Acta Met. 14, 1265 (1966). [76] Kirby, R. K., Private Communication (1970). [77] Hirth,J. P., and Lothe, J., Theory of Dislocations, McGrawHill (New York) 1968, p. 53. [78] Aust, K. T., and Dunn, C. G., J. Metals 9 , 472 (1957). [79] Roessler, B., Kramer, J. J., and Kurijama, M., Phys. Stat.

Sol. 11, 117(1965).

497-814 O - 73 - 5

37

XIII. Tables

Table I. Reliability of Cauchy strains using data from 20 planes of a tungsten crystal compressed 0.2 percent [16]

Cauchy

strain (values in%)

No

as- sumed

error

(eq. 45) RND 1* RND 2* RND 3* RND 'I* Mean

of RND

(1-4)

Range

of RND

(1-4)

Srnd [{s' 0)2

(«14)5(f ,j) V2O

(= 0.48s «M>)

Probable

error

(=0.155s<«o))

<€n) 0.077 0.059 0.096 0.050 0.073 0.090 0.077 0.046 0.023 0.063 0.030 0.010

<e22> -0.137 .074 -0.154 -0.116 -0.120 -0.166 -0.139 .050 .025 .078 .037 .012

(e33> -.053 .054 -.042 .065 .060 .037 .051 .028 .014 .056 .027 .009

(e23> .105 .098 .092 .121 .068 .086 .092 .053 .027 .102 .041 .016

(€31

)

0.044 .073 0.046 0.037 0.053 0.044 0.045 .016 .008 .073 .035 .011

(en) .0026 .105 .015 .057 -0.016 .057 .028 .073 .037 .111 .053 .017

*A11 strain, S, values incremented by ±0.03. A random numbers scheme was used to assign positive or negative error values in the best of 20 S values. Range of |S| is 0.001 to 0.189 percent. Range of 6: 65.8° « 0 ^ 40.2°. Thus, AS of 0.03 percent represents a reasonable value as indicated by eq (43).

Table II. Reliability of principal strains in a tungsten crystal trained 0.2 percent [16]

(€y)±0 (eu) + «'f,>

Si -0.031 -0.039 -0.019

S2 0.081 0.116 0.052

S3 -0.163 -0.223 -0.113

Laced m'^I) + —h —h- Laced wf^y —1—h —

h

s, -0.034 -0.028 -0.018 -0.045

S2 0.092 0.070 0.069 0.093

S3 -0.182 -0.145 -0.178 -0.148

39
Table III. Microprobe data for Fe-3 Si analysis

Fe Si tis) Fe Si f(s) (counts) (counts) (counts) (counts)

216115 0.992 22927 0.0326 41.3 206256 .953 22513 .0320 41.3 212275 .977 41.3 212228 .976 22291 .0317 41.8 215074 .988 22788 .0324 41.2 207371 .957 22228 .0316 41.8 216234 .992 22703 .0322 41.3 206702 .955 22755 .0324 41.5 210210 .968 21499 .0305 41.0 210226 .969 21935 .0312 41.6 211997 .976 22713 .0322 41.2 209137 .964 22447 .0319 41.7 211372 .973 21416 .0304 41.3 213573 .982 22980 .0327 42.2 210084 .968 22515 .0320 41.5 205611 .950 21324 .0303 42.5 206382 .953 22744 .0323 41.6 207108 .956 22437 .0319 42.8 212985 .979 21219 .0301 41.9 210081 .968 22636 .0322 43.1 209716 .966 21384 .0304 42.2 211739 .975 22530 .0320 43.5 206348 .953 20508 .0291 42.4 211386 .973 21050 .0299 40.4 209755 .967 21035 .0299 40.8 205842 .951 21854 .0311 41.2 209723 .967 21354 .0303 41.0 212230 .976 22166 .0315 41.6 208201 .960 22198 .0315 41.4 209202 .964 21529 .0306 41.8 208285 .961 21998 .0313 41.6 204468 .946 20684 .0294 42.3 214115 .984 23095 .0328 41.8 216706 .994 21713 .0309 43.6 211937 .975 21199 .0301 42.0 210313 .969 22259 .0316 39.9 208628 .962 22202 .0315 42.1 209086 .964 22201 .0316 40.0 213955 .983 22759 .0324 42.4 216765 .994 22258 .0316 40.1 211260 .973 22561 .0321 42.6 212540 .978 21598 .0307 40.2 212213 .976 22157 .0315 43.0 213605 .982 22128 .0315 40.7 211931 .975 22619 .0321 41.3 206780 .955 22565 .0321 40.7 211982 .975 22703 .0323 41.5 215698 .990 22067 .0314 40.9 211443 .973 22629 .0322 41.7 209428 .965 22134 .0315 40.4 210691 .970 22665 .0322 41.9 209135 .964 22313 .0303 40.2 210072 .968 22543 .0321 42.2 207478 .958 21688 .0308 40.7 210686 .970 22496 .0320 42.4 206134 .952 21521 .0306 40.0 209810 .967 22413 .0318 42.8 206947 .955 21504 .0306 40.3 209473 .966 22320 .0317 43.1 205625 .950 21805 .0310 40.8 214518 .986 21982 .0312 40.0 207452 .957 21897 .0311 41.8 209599 .966 21988 .0312 40.3 209614 .966 21857 .0311 41.4 210564 .970 22995 .0327 40.5 209987 .968 21291 .0303 42.1 210121 .968 22322 .0317 41.4 207547 .958 21370 .0304 41.9 212068 .976 22409 .0319 41.4 209005 .964 21806 .0310 42.6 213989 .983 22778 .0324 41.4 205929 .951 21153 .0301 41.4 211504 .974 22381 .0318 42.2 21466 .985 22334 .0317 41.6 213177 .980 22595 .0321 42.2 204166 .944 21175 .0301 41.3 212522 .978 22688 .0323 42.3 206124 .952 22352 .0318 41.8 21104 .972 20960 .0298 43.3 205783 .951 22565 .0321 41.6 216309 .993 22781 .0324 40.9 212188 .976 22314 .0317 42.3 210884 .971 21209 .0301 40.7 210666 .970 23116 .0329 42.4 210161 .968 20329 .0289 40.7 208025 .960 23113 .0329 42.5 212217 .976 21842 .0310 41.4 207776 .959 2254S .0321 210831 .971 22471 .0319 39.7 206087 .952 21957 .0312 40.0 207499 .958 22415 .0319 39.7 213234 .980 21872 .0311 40.2 209512 .966 21673 .0308 40.2 212612 .978 22612 .0321 41.1 210735 .971 22027 .0313 40.5 205595 .950 21337 .0313 41.1 208406 .961 22323 .0317 40.9 215739 .990 22150 .0315 41.2 207524 .958 22517 .0320 42.2 208509 .962 21884 .0311 41.6 965 21767 .0309 42.1 209185 .964 22186 .0315 41.4 213762 .983 22054 .0313 41.6 208968 .964 22319 .0317 41.5 212558 .978 22530 .0320 41.4 209528 .966 21798 .0310 42.1

40
Table IV. Results for standards used in microprobe analysis

Pure Si (counts)

756108

757647

755798

42.8

44.8

43.7

7=2. 3/x s

Si on Al

(counts)

Si BKG

(counts)

Si pure

(corrected counts)

16198 9390

6656

10748 776189

Pure Fe

(counts)

Fe pure

(corrected counts)

218640

220612

215942

219009

Fe on Ni

(counts)

1919

1931

41
I n• • o

(J <i

II 1 z o UJ 1 \ ) O CO

n

J. 1 tr~* • o UJ /—

• CM • CM

r-* •

II IIO rswL M O CD

II • 1 CM 1 UJ

SI II <I

U) c.

C/)

O >

Q.

D 00 Z « z tt:

QL • Pu. IJ <X O

O II 1 Q • LJ o U. z

b. ld u.

ce:

n u. o •« CD • H UJ o

UL • • II -J > o

to II < UJ -J

II < > Q Q • -J CD < • r 5 o

UJ

X < II u; ha. 

o Q 2: f)

IaJ

Cvl o m in 1 1 1 x

UJ UJ UJ

00 o • vC CM CD

CM II • r: ld m o O o -J • • • CM — CD II <

II <Iro

11 II o 5 •

THET

•-4 »—« »-« 00 01 fO

to 00 00 >o D 3 3 CM Z z z »« • UJ H X a:< •

cr cr q: s> S3 II *—» 00 (/) z CM 3 3 «• CM

•J

Ql

00

s>o uo

n CD•

CD CJ CJ

U

CO c^ z <

CD CJ

oc uo ^ »—

-J

u

O

r) . u • CM • • ^

UJ 1 1 II z N II II »—

-J >—

3 > :» o 00

42
Table VI. Lattice spacing, d, and lattice parameter, oo, in Fe-3 Si

Pattern

Line 220 21

121 121 211 112 112 200 020

Reference 1.01205 1.16487 1.15868 1.16998 1.16410 1.15568 1.16403 1.42904 1.43168

u- 0 1.01177 1.16562 1.16859 1.16676 1.16511 1.16731 1.43028 U- 1 1.01168 1.1664 1.16654 1.16662 1.16822 1.1703 1.42719 1.4.3018 U- 2 1.011% 1.16812 1.163% 1.16609 1.16852 1.16398 1.42685 1.43161 U- 3 1.0110 1.1688 1.16503 1.1544 1.19541 1.43078 1.4287 U- 4 1.01195 L16656 1.16608 1.1684 1.16443 1.17575 1.42393 1.43149 U- 5 1.01177 1.16753 1.16693 1.16986 1.16402 1.1663 1.42642 1.43152 U- 6 1.01212 1.16088 1.16641 1.16585 1.15829 1.42657 1.42404 U- 7 1.01193 1.1669 1.1658 1.1641 1.16086 1.16904 1.4238 1.43377 U- 8 1.01187 1.16806 1.16669 1.16777 1.16588 1.1719 1.42453 1.42657 U- 9 1.01174 1.1683 1.16808 1.16713 1.16533 1.17717 1.17774 1.42745 U-10 1.01194 1.16758 1.16789 1.16433 1.16732 1.43271 1.421% U-11 1.01186 1.16665 1.1678 1.16757 1.16397 1.43008 1.42793 U-12 1.01201 1.16645 1.16716 1.16834 1.16639 1.1632 1.15934 1.42546 1.43077

T T TO U-13 1.01186 1.1673 I.l0440 l.loozl l.i0/4o 1.43106 U-14 1.0112 1.16308 1.1603 1.17406 1.15817 1.15653 1.41712 1.43605 U-15 1.01174 1.16618 1.16686 1.16629 1.16889 1.1667 1.16734 1 .42569 1.419ZO

a an d Oo in A

rattern /

/ Line 101 Oil Oil 101 Oo

Reference 2.00269 2.01062 2.02063 2.0102 2.86154

u- 0 2.00429 2.0113 2.02321 2.00757 2.86221 u- 1 2.01691 2.01603 2.02007 2.01356 2.86223 U- 2 2.0171 2.01362 2.02055 2.01429 2.86221 U- 3 2.00658 2.01111 2.02172 2.86273 U- 4 2.011 2.02248 2.02434 2.01561 2.86457 U- 5 2.01076 2.01553 2.00058 2.02162 2.86246 U- 6 2.00538 2.02131 2.0054« 2.86251 U- 7 2.01006 2.01698 2.02216 2.01521 2.86077 U- 8 2.01417 2.02043 2.02177 2.86194 U- 9 2.01065 2.01792 2.01975 2.86923 U-10 2.01575 2.86199 U-11 2.02032 2.02055 2.0227 2.01715 2.86081 U-12 2.02132 2.0162 2.86154

T T 1 O \}—\6 ^.01046 2.01107 o m OA/1 O A1 CO/1 o Q/COCQ Z.oOZOo U-14 2.00407 2.01341 2.85825 u-15 2.01268 2.02641 2.02045 2.00976 2.86016

43
Table VII. Lattice strains in Fe-3 Si {strains in percent)

Pattern

Line u-o U-1 U-2 U-3 U-4 U-5 U-6 U-7 U-8

220 -0.028 - 0.037 -0.0089 -0.104 -0.0099 -0.034 0.007 -0.012 -0.018

91 1 U. lol — rtoo .uyy A Q /I O 0.174 0.274 12T -0.0081 -0.183 -0.404 0.010 -0.223 -.150 -0.246

121 -.275 -.287 -.332 -0.423 -.135 0.101 -.305 -0.281

211 0.087 0.354 0.380 0.0283 -0.399 0.150 0 0.315 112 -.111 .226 0.448 -0.240

112 -0.117 .139 - 0.403 -.397 .605 -.204 -0.231 0.0305 0.275

200 -0.129 -.153 0.122 -0.358 -.344 -.166 -0.367 -0.316

020 -.098 -.105 -.0041 -0.208 -.0133 0.0125 -.535 0.146 -0.357

101 0.080 0.710 0.720 0.194 0.420 — 0.665 0.368 0.573 Oil .034 .269 .149 .0244 .590 -.430 -.261 0.316 0.488 on .128 -0.028 .004 .0539 .184 0.034 0.0751

101 -0.131 0.167 .203 .269 -.129 -0.236 0.249 0.576

U-9 U-10 U-11 U-12 U-13 U-14 U-15

220 -0.031 -0.011 -0.019 0.004 -0.019 -0.084 - 0.0036

211 0.294 0.233 0.153 .136 0.209 -.154 -.214 12T -0.052 -0.130 -0.362 -.717 -.156

121 -.244 -0.179 -0.076 -.140 -.322 0.349 -.205 2lT 0.106 0.0198 -.095 0.197 0.290 -0.509 0.018 112 .726 -0.170

112 .775 -0.117 -.403 .056 0.0735

200 0.257 0.073 -0.251 .028 -0.834 -.234

020 -0.295 -0.679 -0.262 -.064 -0.043 0.305 -.866

lOl 0.397 0.652 -.193 0.388 -.570

Oil -.181 0.532 .022 0.108 Oil -0.134 -.075 -0.376 -0.820 -0.186

101 0.475 -.349 .298 0.251 0.160 -.715

C-1 C-2 C-3 C-4 C-5 C-6 C-7

220 -0.043 -0.038 -0.015 -0.017 -0.093 -0.028 -0.042

211 0.0009 0.159 0.149 0.132 -.0009 0.102 0.0944 12T -0.689 -0.249 -0.300 -0.204 -.220 -0.200 -0.302

121 -.320 -.280 -.195 -.215 -.312 -.262 -.347 2lT 0.351 0.264 0.326 0.155 0.457 0.067 0.316 112 -0.078 -0.361 .216 .081 .020 -0.111

112 -0.273

200 -.0434 -.041 0.088 .034 -0.332 -0.219 -.078

020 -.177 -.648 -0.396 -0.203 -.566 -.131 -.281

101 0.373 0.0664 0.153 0.529 0.391 Oil .735 -0.243 0.635 -.402 .737 .659 OlT -0.500 -0.738 -.486 -0.816 0.301 - 0.405 -0.440

101 0.262 -.159 0.448 -0.564 0.166

44
Table VII. Lattice strains in Fe-3 Si (strains in percent)— Continued

C-8 C-9 C-10 C-11 C-12 C-13

220 — A AOl U.Uzl A AOA A Al Q — A Al 1 U.Ull — 0.030 — 0.047

A AQA7 A 1/17 A 1 Q/t A AOl u.uy 1 A Ad

A OtJO w.zoo A ICO U. lOZ A OA9 U.ZOZ

121 -.213 -.220 -.138 -.258 -.319 -.337 2lT 0.293 0.295 0.275 0.294 0.313 0.115

1 1^ liZ — u.uyi 1 y1 O

. 14/ 1 1 A .iiy 1 oc .loo .019 .483 112 -0.418 .281 -0.324

200 — .0112 0.0119 A Q30 — 0.023 — .148 A AQOA 020 — .45/ — 0.263 .Zoo — .182 — .452 — .oOo

loT 0.164 0.320 0.425 0.692 Oil .482 0.756 -0.278 .306 Oil -0.334 -0.501 -.191 -.094 -0.357 -0.419

101 0.395 0.464 0.456 0.0925

T-1 T-2 T-3 T-4 T-5 T-6 T-7

ZZU — U.UZO A AQO A AQA — A ACl U.Uoi — A AOl U.Uzl A Ad/I — A AAC U.OOD

Ol 1 211 0.122 A 1 Al

U. 191 0.132 0.124 0.202 —.065 Alio 0.1 lo

1 0

1

— A o/;i U.zol — A OA/I U.Z04 — A OAO U.z4o A 0'7A — A O AO U.ci6z 0/1 1 —.241 — A OA/1 0.z64

121 -.203 -.260 -.283 -.220 -.203 -0.329 -.354

211 0.179 0.120 0.109 0.263 0.229 0.113 0.157 112 A 1 A 1 OO A A7 — A /II C U.41D 112 0.432

200 — 0.342 0.196 .027 — 0.072 UM —.205 —.325 — 0.805 — 0.434 — 0.191 — 0.527 —.562

lOT 0.516 0.610 0.368 0.514 0.750 0.343 0.635 Oil .377 .173 .319 .159 .045 .139 .239 oil -0.0%5 -0.092 -0.148 -0.229 -0.448 -0.554 -0.113

101 0.205 0.271 0.199 0.054 0.534 0.263 0.202

T-8 T-9 T-10 Y-1 Y-2 Y-3

oon ZZV) — A A/1A u.u4y — A AO/f U.Uz4 A Al AO — A AO/1 U.Uz4 A AOO — A A9Q U.Uoo

01 1 Zll A AOl 0.239 0.110 A OAA U.zU9 A I Al

U. lUl 0.102 121 -0.456 .287 -0.201 -0.252 -0.329 -0.232

211 0.101 .176 0.226 0.206 0.276

1 1 o IIZ Al A — 0.017 —.033 .330 .0671 AOA .080 112 -0.324 0.125 -.397 -0.683 -0.508

ZUU 1 /1Q 0.371 — 1 AC . lUD ACO

—.452 — 0.198 .058 —.380 —.368 — 0.413

loT 0.520 0.583 .266 0.680 0.809 0.432 oil -0.181 .375 .41 .372 .462 .527 on -0.176 -0.414 -0.143 -0.110 -0.343

101 0.071 0.515 0.616 0.280 0.204

45
Table VIII. Cauchy strains and their standard errors {all values in percent)

U-0 U-1 U-2 U-4 U-5 U-7 U-8 U-9 U-10

(e>n 0.2479 0.1250 0.3540 -0.1860 -0.3601 -0.1144 0.1108 0.0742 0.3237

5li .1388 + 0. 1930 .3023 0.1960 0.05055 0.2487 .3112 .2891 -0.2014

(6)22 -0.1111 -0.3512 -0.3400 -0.2937 .0262 -0.0884 -0.5925 -0.4825 -0.7012

0.08871 0.1973 0.3091 0. 1960 .0513 0.2621 0.3465 0.1901 0.2242

<e)33 -0.02725 .3826 -0.3786 .9194 -0.5638 .3926 .1313 1.079 -0.3875

^33 0. 1049 .2522 0.3950 .2478 0.0675 .2144 .2729 0.1534 0.3487

<€>23 -0.2943 -0.0410 .2965 .1792 .0666 .1540 .5576 -0.1942 .8700

523 0.1520 0.4021 .6298 .3697 .1063 .5348 .7729 0.3429 .8490

<e)3. .0141 -0.3241 - 0.4303 -0.0562 .4722 -0.1951 -0.1346 .2477 -0.3196

S31 .1520 0.3536 0.5539 0.3697 .0987 0.4741 0.6513 .3336 0.5191

.093 .917 .777 .152 4.78 .412 .207 .743 .616

<€).2 -0.2225 .1269 .03742 .2443 0.2780 .1042 .4442 .2215 .3025 0.2262 .3880 .6078 .3885 .1016 .4906 .6210 .4834 .3836

U-11 U-12 U-13 U-14 U-15 C-1 C-2 C-3 C-4

0.1186 0.012 0.3315 -0.5731 -0.3799 0.2835 0.1195 0.2831 0.1636

5ii .1276 .2095 .2247 0.2529 0.1588 .2640 .1356 .1616 .2022

<€>22 -0.1912 -0.2196 -0.3468 .084 .6761 -0.4788 -0.6894 -0.5107 -0.3112

•S22 0.1290 0.2095 0.2247 .2536 .1588 0.2640 0.1375 0.1616 0.1965

(e)33 0.6150 .2796 .0354 -0.5032 -0.3101 .053 -0.4579 -0.014 -0.045

^33 0.1686 .2891 .04706 0.3411 0.2008 .3339 0.1810 0.1419 0.2645

<e)23 .052 .1294 .2192 .014 .1455 .9512 .1605 .1124 .5210

S23 .3044 .4475 .4097 .5307 .29% .4981 .2852 .2992 .4062

<e)3i .1821 -0.2022 -0.2299 .041 -0.3479 -0.8013 -0.1855 -0.4646 -0.4368

S31 .2340 0.4475 0.4097 .05656 0.29% 0.4981 0.2648 0.2992 0.4423

<e>,2 .1129 .2063 -0.0798 .2003 1.047 -0.9986 -0.6634 .2240 .1083

Svi .2438 .4203 0.4318 .5107 0.3148 0.5234 0.2725 .3232 .3994

C-5 C-6 C-7 C-8 C-9 C-11 C-12 C-13 T-4

<e>„ -0.1483 -0.0580 0.2070 0.1488 0.2195 0.1814 0.1487 0.1183 0.5139

Sll 0.2261 0.2625 .2097 .1593 .2530 .1439 .1902 .1509 .0915

<€>„ -0.7497 -0.1413 -0.4319 -0.4680 -0.3338 -0.3834 -0.6138 -0.5172 -0.3858

522 0.2261 0.2589 0.2097 0.1593 0.2443 0.1420 0,1939 0.1509 0.0585

(6)33 .4878 .7588 -0.027 .028 .1428 .1698 -0.1484 .5712 -0.059

533 .5191 .3456 0.2651 .2014 .2200 .1895 0. 1694 .1908 0.0692

<€)23 .0355 -0.04119 .4407 .3123 .4176 .1114 .020 .2185 .3401

523 .4309 0.5055 .3956 .3005 .5037 .2772 .3811 .2847 .1002

<e>3i -0.7075 -0.1690 -0.4862 -0.2866 -0.8418 -0.1099 -0.3416 -0.4279 -0.3808

531 0.4546 0.5446 0.3956 0.3005 0.5309 0.2986 0.3708 0.2847 0. 1002

(e).2 .7670 .1992 .1122 .2961 .1312 .2027 .4133 .1112 -0.2421

5)2 .4365 .5201 .4156 .3157 .5016 .2852 .3850 .2992 0.1492

46
Table VIII. Cauchy strains and their standard errors {all values in percent)— Continued

T-5 T-7 T-9 T-10 Y-1 U-3 C-10 T-1 T-2

(€)u 0.462 0.175 0.774 0.365 0.746 0.143 -0.053 0.653 -0.050

.178 .206 .188 .074 .478 .140 0.197 .162 0.219

<e>22 — 0.301 — 0.304 — 0.209 .132 -0.243 -0.152 -0.492 -0.145 -0..504

522 0.178 0.206 0.119 .075 0.308 0.123 0.197 0.103 0.219

<e>33 — 0.217 .301 .123 — 0.242 .179 — 0.284 .338 -0.108 .263

S33 0.225 .432 .094 .098 .251 0.106 .175 0.122 .192

(€>23 .453 .096 .197 .659 -0.141 -0.312 .283 .315 .328

52,-) .336 .376 .234 .178 0.525 0.287 .414 .177 .405

(€)31 — 0.181 — 0.220 — 0.038 — 0.100 — 0.558 — 0.255 — 0.135 — 0.250 .126

^31 0.336 0.376 0.206 0.136 0.525 0.345 0.414 0.177 .405

<e>.2 -0.218 -0.125 -0.753 -0.521 -0.571 -0.227 .502 -0.619 .425

S12 0.353 0.397 0.305 0.142 0.790 0.243 .400 0.264 .438

T-3 T-6 T-8 Y-2 Y-3 U-6

(€>n —0.100 0.056 0.079 0.146 0.202 — 0.246

SlJ 0.171 .120 .173 .237 .123 0.134

<e>22 -0.821 -0.556 — 0.626 -0.411 -0.506 -0.460

S22 0.110 0.120 0.176 0.233 0.123 0.183

<e>33 .642 .125 -0.155 — 0.206 .238 .015

533 .130 .252 0.154 0.207 .155 .127

<€>23 .118 .172 .240 .192 .513 — 0.235

523 .188 .219 .346 .454 .231 0.224

{e)3. -0.113 -0.350 -0.296 — 0.707 — 0.462 — 0.520

S31 0.188 0.219 0.337 0.467 0.231 0.262

<e>,2 .740 .282 .357 .190 .164 .622

S12 .279 .231 .351 .473 .243 .281

47
Table IX. Principal strains and their maximum credible errors (all values in percent)

Maximum Maximum Maximum

Pattern Si credible S2 credible S3 credible

error on S] error on S2 error on S3

U- 0 0.288 0.066 0.059 0.045 -0.237 0.040 U- 1 .461 .119 0.058 .078 0.362 .085 U- 2 .413 .106 -0.233 .313 -0.544 .302 U- 3 .195 .065 -.051 .083 -.437 .085 U- 4 .926 .157 0.215 .087 -.330 .111 U- 5 .090 .007 -0.266 .047 -.722 .061 U- 6 .278 .034 -.282 .082 -.687 .094 U- 7 .420 .122 -.048 .077 -.182 .229 U 0 .404 0.172 .420 -.750 .083 U- 9 1.10 .135 .085 .080 -.513 .093 U-10 0.362 .147 -0.086 .156 -1.04 .187 U-11 .140 .035 -.200 .078 -0.627 .129 U-12 .260 .123 0.021 .100 -.280 .106 U-13 .378 .172 .018 .141 -.376 .157 U-14 .562 .079 -0.678 .311 -.977 .392 U-15 .043 .102 -.304 .094 -1.11 .178

C- 1 .685 .136 -.040 .153 -0.787 .152 C- 2 .243 .040 -.433 .151 -.838 .200 C- 3 .419 .060 -.106 .071 -.542 .097 C- 4 .319 .121 0.012 .133 -.513 .120 C- 5 .661 .130 -0.113 .120 -.958 .190 C- 6 .132 .113 —.045 .157 —.211 .196 C- 7 .366 .099 -.058 .113 -.560 .133 C- 8 .246 .101 0.023 .091 -.561 .093 C- 9 .501 .101 -0.165 .164 -.593 .185 C-10 .364 .116 0.061 .099 -.631 .093 C-11 .251 .151 .121 .118 -.405 .065 C-12 .266 .059 -0.208 .106 -.672 .151 C-13 .662 .085 0.04a .056 -.537 .069

T- 1 .776 .064 -0.077 .062 -.317 .079 T- 2 .329 .098 -.013 .086 -.608 .114 T- 3 .647 .076 0.054 .063 —.981 .070 T- 4 .607 .034 -0.021 .027 -.452 .034 T- 5 .500 .083 -.066 .083 -.490 .109 T- 6 .270 .072 -.037 .056 -.608 .102 T- 7 .372 .204 0.113 .129 -.314 .100 T- 8 .183 .053 -0.214 .105 -.671 .153 T- 9 .905 .088 0.136 .039 -.353 .042 T-10 .599 .257 .099 .112 -.445 .150

Y- 1 .883 .158 -0.137 .166 -.422 .225 Y- 2 .366 .121 -.287 .118 -.550 , .152 Y- 3 .471 .133 0.076 .084 -.614 .141

4S
Table X. Direction cosines of principal strain axes after deformation

Pattern In /3. Pattern ki /31

U-0 0.931 -0.323 0.172 U-8 0.135 0.352 0.926

.320 0.491 -0.810 .956 .199 -0.215

.177 0.809 0.560 .260 -0.915 0.309 U-1 .436 0.011 -0.900 U-9 .114 -.053 .992 .888 0.160 0.432 .973 0.207 -0.101

.149 -0.987 0.060 -0.200 .977 0.075 U-2 -0.963 0.029 0.267 U-10 0.983 .071 -0.168

-.937 0.058 0.344 .095 .590 0.802 0.200 -0.587 0.784 .156 -0.804 .574 U-3 -0.956 0.232 0.180 U-11 .977 0.176 .124 0.085 0.805 -0.587 -0.179 .984 .022 .280 0.546 0.790 0.118 .044 -0.992

U-4 -0.027 0.071 0.997 U-12 -0.345 .052 0.937 0.972 0.235 0.009 ' 0.850 .442 .288

.981 0.194 0.017 .400 -0.895 .197 U-5 .376 0.909 0.182 U-13 -0.933 0.104 .346 .736 -0.412 0.537 0.360 .232 .903 -0.563 0.068 0.824 .011 .967 -0.255

U-6 0.578 0.360 -0.733 U-14 .083 .830 0.551

.543 0.501 0.674 - 0.992 .015 .126 .610 -0.787 0.094 0.097 -0.557 .825 U-7 -0.166 0.131 0.977 U-15 .790 0.547 -0.276

0.890 0.429 0.156 .102 .327 0.940

.908 0.267 0.323 .604 -0.770 .202

C-1 -0.691 0.295 0.660 C-8 .855 0.065 -0.515

0.702 0.491 0.516 .453 .391 0.801

.172 -0.820 0.546 .253 -0.918 .305 C-2 .941 0.327 -0.087 C-9 -0.823 0.076 .563 -0.093 0.088 0.958 0.408 .769 .492 0.338 -0.901 0.272 .395 -0.635 .664 C-3 .877 0.077 -0.475 C-10 -0.074 0.141 .987

.445 0.243 0.862 0.910 .415 .011 .182 -0.967 0.178 .408 -0.899 .160 C-4 -0.780 0.183 -0.599 C-11 .755 0.176 -0.632

0.541 0.583 0.606 .635 .042 0.771

.244 -0.793 0.559 -0.163 .984 .080 C-5 .441 0.109 -0.891 C-12 0.906 .209 -0.369

.765 0.473 0.437 .332 .191 0.924

.469 -0.874 0.125 .263 -0.959 .104 C-6 .497 0.244 -0.833 C-13 -0.359 0.069 .931 .640 0.545 0.541 0.925 .158 .345

.586 -0.802 0.115 .122 -0.985 .121 C-7 -0.822

0.531

.207

0.097

0.488 -0.867

0.562 0.692 0.453

T-1 -0.920 0.342 0.191 T-9 -0.944 0.325 0.064

0.333 0.429 0.840 0.118 .151 .981 .205 0.836 -0.508 .309 .934 -0.181 T-2 .296 0.256 0.920 T-10 -0.740 .611 0.283

.894 0.265 0.362 0.662 .586 .467

.337 -0.930 0.150 .120 .533 -0.838

T-3 — 0.063

0.919

.390

0.024

0.392 -0.920

0.998

0.049

0.047

49
Table X. Direction cosines of principal strain axes after deformation — Continued

Pattern Jtii

/'2i

/'3i Pattern In la

1

T-4 -0.924 0.171 0.341 Y-1 -0.946 .225 0.234

0.378 0.285 0.881 0.042 -0.629 .776

.053 0.943 -0.328 .322 0.744 .586 T-5 -0.967 0.182 0.180 Y-2 .853 .040 -0.521

0.254 0.606 0.754 .335 .723 0.604

.028 0.774 -0.632 .401 -0.689 .603 T^6 .645 0.030 -0.764 Y-3 -0.615 0.151 .774

.723 0.299 0.623 0.766 .347 .541 .247 -0.954 0.171 .186 -0.926 .329 T-7 -0,510

0.852

.114

0.108 -0.068

0.992

0.853 0.518 -0.057

T^8 .898

.366

.245

0.198

0.161 -0.967

— 0.393

0.917 0.072

Table XI. Direction cosines of undeformedfibers

Pattern ku kn ^3, Pattern ku kti

u- 0 0.589 -0.386 0.709 U- 8 0.066 0.328 0.942

.320 0.487 -0.812 .622 .415 .664

.207 .879 0.429 .243 -0.800 - 0.549

U- 1 -0.344 -0.0028 .939 U- 9 .073 -.041 0.996 -.891 -.159 - 0.424 .977 0.201 -0.067

-.223 -.558 -.799 .021 .704 -.710

U- 2 -.824 0.081 0.560 U-10 -0.765 .140 0.629 -.906 -0.128 -0.405 -.096 -0.588 - 0.803 0.120 -.933 0.339 0.078 -.894 -.442

U- 3 -0.908 0.109 .405 U-11 .959 0.176 0.223

0.082 .803 -0.590 .333 -0.364 -0.870

-0.330 .711 0.621 -0.156 -.070 0.985

U- 4 -.018 .053 .998 U-12 -.307 0.058 0.950 -.295 .092 .951 -.851 -0.442 -0.285

-.067 .118 .991 0.314 -.929 -.196

U- 5 0.387 .896 .218 U-13 -.712 0.130 .690 -0.656 .435 -0.617 -.361 -0.232 -0.903

-.934 .249 0.255 -.128 -.912 -.391

U- 6 -.540 -0.333 .772 U-14 .068 0.746 0.663 -.595 -.519 -0.613 .380 .731 -0.567

-.324 -.472 -.820 .615 .562 -.553

U- 7 -.145 0.114 0.983 U-15 -0.789 -0.545 0.285 -.407 .325 .854 -.144 -.342 - 0.929 -.260 .206 .943 0.021 -.319 -.947

50
Table XI. Direction cosines of undeformed ^6ers— Continued

Pattern A:3i Pattern ku kn ksi

L. 1 — COyl .0/4" -.704

-.061

.ZoV -0.489

-.998

.oUl -0.515

-.009

Q L 0 — U. (OZ -.454

0.113

— A Al 1 U.Ul 1 -.391 -.876

0.059 -0.801

-.469

C— 9 — Q01 .i5ol -.021

0.032

-.242 -.323

u.duu -0.970

-.946

A 7A7 -.443

0.279

A 1 A/1 u. lie* -0.740

-.919

A 7AA -0.506

0.279

O — U. (oi -.451 -.182

—.U'lo -.241 -.567

U.Ooz -0.860

-.804

^ 1

A

A QQA

0.272 -0.070

OC7 —.ZD/

-.611

0.133

A OA .480 -0.743

0.989 —.OHo -.611

0.183

u.zio -0.558

-.958

U-OVo -0.562

0.219

C 1111 —.DOO -.672

-.189

— U. IDO -.053

0.474

-0.739

-.860

/ _ c

\^ o -.774

-.287

—.u/z -.473 -.494

Q'iA -0.422

-.821

iz — QOQ .000 -.350

-.176

— A 1 QO U. loZ -.190

-.273

U.oZZ -0.917

-.946

C 0 —.4/0 -.645

0.276

—,ZoU -.546 -.806

U.oOU -0.534

-.523

L. lo —.zoo -.927

-.211

u.uoo -0.157

-.697

A OAO u.yoz -0.341

-.685

C- 7 -0.710

-.534

0.036

0.130 -0.486

-.990

0.692 -0.692

-.138

(\ A AO -.332

-.306

U.zl / -0.422

-.944

A OOA u.ooy -0.843

0.126

T- 8 —.oov -.390

-.200

— A 1 QC U. loo -.163 -.205

A A 11 -0.906

-.958

T- 9 0.251

.894

.408

.266 -0.488

.707 -0.361

-.771

T— Q —.uvo -.118

-.302

u.woo -0.148

-.831

-0.982

-.467

rp Q -.922

0.014

n no/1 U.UZ4 -0.388

-.165

n OQQ

.003 -0.986

in 1 lU /ion -.665

-.174

n n:/i 1 U.O'1'1 -0.573

-.622

U. / ZD -0.479

0.764

T /I — U.OUV -.378 -.204

yj.L io -0.285

-.918

n 77/1 -0.881

-.340

1 — 5 -.596

-.255 -.099

U.ZDO -0.603

-.964 -0.756

0.245

Y- 1 —.4VU -.052

0.278

A AI^O

.612

.956

071 -0.789

0.097

T- 6

C "70 —.578

-.725 -.104

Al 0 —.016 -.299

-.616

Ol .816 -0.621

-.781

Y- z — U. / /o -.434

0.055

— U.UlO -.626

-.982

.ozo -0.64S

-.179

T— 7 1 ( -.873 -.276

n no9 .074 -0.537nu.vio 01 -0.483

-.797

V— ^ 1 J — 0 485 -.773

0.013

0.166 -0.343

-.953

0.859 -0.534

-.304

51
Table XII. Angle between undeformed fibers and principal strain axes {6 in degrees)

Pattern «2 Pattern e, e.

U- 0 37.4 0.65 8.7 T- 1 52.1 179.6 1.S6.3 U- 1 174.3 179.8 61.9 T- 2 3.15 0.69 61.6 U- 2 18.9 45.4 33.1 T- 3 0.79 177.1 a3.6 U- 3 15.0 0.37 13.9 T- 4 31.1 179.7 139.9 U- 4 1.3 104.9 91.5 T- 5 40.6 179.7 154.8 U- 5 2.3 173.4 41.2 T- 6 165.4 179.7 64.7 U- 6 176.7 174.0 5.5 T- 7 7.4 177.7 121.2 U- 7 1.4 95.1 82.9 T- 8 174.6 178.5 a5.4 U- 8 4.3 57.7 51.3 T- 9 79.5 179.9 141.7 U- 9 2.5 2.0 50.9 T-10 31.6 179 172.7 U-10 148 179.5 61.5 U-11 5.8 116.0 177.3 Y- 1 47.3 150.4 31.0 U-12 2.3 179.6 23.3 Y- 2 172.4 171.6 53.7 Zo.O lO.O i^-l .o Y- ^ O A 1 7Q A 1 iy.o QQ A U-14 8.1 115.9 135.2 U-15 179.3 177.3 86.2

C- 1 12.5 179.8 36.6 C- 2 154.8 172.5 87.5 C- 3 165.3 68.2 179.6 C- 4 8.1 175 22.1 C- 5 173.8 179.1 78.8 C- 6 178.4 179.5 41.5 C- 7 10.0 179.5 36.5 C- 8 169.4 179.6 46.4 C- 9 10.4 177.3 28.5 C-10 0.59 149.9 r.2

i 1 1 77 1 64 6 C-12 170.1 178.7 83.3 C-13 5.7 179.7 54.7

52
Table XIII. Principal stresses {GNjm'^)

Pattern P. P2 ^3

U- 0 0.554 0.206 -0.246 0.513 U- 1 .867 .252 -.387 .732 U- 2 .736 .463 -.722 .477 U- 3 -0.385 -0.973 -1.07 -2.43

U- 4 2.96 2.05 1.88 6.89 U- 5 -0.807 -1.35 -2.04 -4.20

U- 6 -.303 -1.16 -1.77 -3.23 U- 7 1.32 0.960 0.756 3.04 U- 8 -0.022 -0.106 -1.51 -1.64

U- 9 2.38 0.834 -0.076 3.14 U-10 -0.253 -0.936 -2.39 -3.58

U-11 -.509 -1.03 -1.68 -3.22

U-12 0.398 0.032 -0.425 0.005 U-13 .598 .048 —.552 .094 U-14 -0.187 -1.92 -2.53 -4.64

U-15 -1.37 -1.90 -3.12 -6.39

C- 1 0.894 -0.211 -1.35 -0.666

C- 2 -0.710 -1.74 -2.36 -4.81 C- 3 0.397 -0.402 -1.07 -1.07

C- 4 .294 -.173 -0.973 -0.851 C- 5 .575 -.604 -1.89 -1.92

C- 6 .071 -.198 -0.452 -0.578

C- 7 .293 -.354 -1.12 -1.18

C- 8 .069 -.271 -1.16 -1.36

C- 9 .493 -.522 -1.17 -1.20

C-10 .337 -.123 -1.18 -0.%6

C-11 .348 0.151 — 0.650 —.151 C-12 -0.240 -0.%2 -1.67 -2.87

C-13 1.19 0.254 -0.638 0.806

T- 1 1.58 .284 -.081 1.79 T- 2 0.195 -0.326 -1.23 -1.36 T- 3 .692 -.211 -1.79 -1.31 T- 4 1.06 0.109 -0.547 0.627 T- 5 0.702 -0.159 -.805 -0.262 T- 6 1.59 -.451 -1.32 -.185 T- 7 0.749 0.354 -0.296 0.806 T- 8 — 0.460 — 1.06 — 1.76 — 3.28 T- 9 2.10 0.930 0.186 3.22 T-10 1.18 .419 -0.410 1.19

Y- 1 1.68 .132 —.303 1.51 Y- 2 0.062 -0.933 -1.33 -2.21 Y- 3 .648 0.045 -1.01 -0.314

53
Table XIV. Stresses in principal strain axes frame (GNjm^)

Pattern a-ii 0-22 0-23 cri2

U- 0 0.3183 0.5471 — 0.2%3 -0.0916 + 0.02412 -0.1295 u- 1 .1349 .8363 — .1595 0.0160 -0.1109 0.1374 U- 2 .03475 — 1.320 0.07104 .1282 0.05607 .1929 U- 3 — 0.361 — 0.908 — 0.989 .038 .064 .177 U- 4 2.052 1.413 1.427 1.419 .000674 .00635 U- 5 — 1.402 — 0.9912 — 2.266 0.00900 .1371 .2239 U- 6 — 0.176 — 1.94 — 1.35 — 0.163 0.270 —0.069 U- 7 0.7371 0.1669 0.2275 .1917 — 0.0217 .0605 U- 8 — .2085 — 0.230 — 1.587 — .3157 -0.1724 0.0725 U- 9 2.0496 0.3697 1.060 0.1634 0.1287 — 0.1410 U-10 — 0.6115 — 0.6447 — 2.714 — 0.1731 .2171 - .0723 U-11 — 1.189 — .7770 — 1.603 — .1195 — 0.00358 0.0751 U-12 0.0820 0.3440 -0.4203 -.0690 -.1356 -0.0732

U-13 -0.00323 .5136 -.4064 0.0578 0.1151 0.1479 U-14 0.04022 — 3.229 — 1.960 - 0.00220 -0.0179 .3705 U-15 — 2.155 — 1.195 — 3.74 — .1648 -.1194 .0%69

C- 1 — 0.3472 1.230 — 1.621 — .1380 — 0.3287 .2569 C- 2 — .9234 — 2.606 — 1.819 0.02537 0.1188 — 0.3429 C- 3 0.3569 — 0.9435 — 0.6018 — 0.1937 — 0.1381 — .1131 C- 4 — 0.1661 0.3686 — 1.190 0.001664 — .2168 0.1839 C- 5 — .6224 .4803 — 1.987 .01792 — .3853 .3790 C- 6 0.0919 — 0.5147 — 0.2190 — 0.0608 0.09721 — 0.00461 C- 7 — 0.4126 0.3220 — 1.218 0.007658 — 0.2680 0.1652 C- 8 — .3251 .005791 — 1.191 — 0.0754 — .2611 - 0.04648 C- 9 0.6620 — 1.516 — 0.4800 0.2060 0. 1806 0.1477 C-10 .7899 -0.007786 0.02517 .1211 .2624 .04582 C-11 .0004632 0.3204 -0.4880 .1322 .03306 -0.001875 C-12 — 0.3506 — 1.667 — 1.166 — 0.1515 — 0.0%1 — .2230 C-13 1.088 — 0.3353 0.1413 — .1601 — .1885 — .1281

T- 1 1.386 0.6972 .4915 0.1193 — .2191 — .1820 T- 2 1.208 .3539 - 0.08226 .3055 0.3802 0.1465 T- 3 - 1.365 — 0.009197 -3.593 .07854 — 1.158 .02039 T- 4 1.014 — .3955 0.07711 .1293 0.2675 .1206 T- 5 0.5160 — .8528 .04669 .04692 .2397 .04141 T- 6 — 0.5273 — .7875 — 1.074 .1493 .3650 — 0.5744 T- 7 0.2247 0.7521 -0.08257 .07775 -0.03871 - .08021 T- 8 -0.2477 -1.348 - .9885 .4373 0.2618 - .56014 T- 9 2.162 0.2932 1.116 .03132 .1212 0.3953 T-10 0.3819 1.437 — 0.5002 — 0.09249 — 0.2242 .2227

Y- 1 1.627 1.437 -.5002 - .05741 0.03738 .3976 Y- 2 -0.3838 0.2410 -.6901 0.1324 -0.6692 .3976 Y- 3 0.4599 .9239 - .5365 .4948 -.2062 -0.2723

54
Table XV. Stresses in cubic axes frame {GNIm^)

Pattern 0"22 0'3.) a- 23 0-31

u- 0 0.3927 0.0480 0.1284 — 0.3413 0.01632 -0.2581 U- 1 .3405 — 0.1166 .5878 0.0476 — 0.3760 0.1472 U- 2 — 0.1742 —.8405 — 0.8775 .3439 —.4991 .04341 U- 3 —.180 —.611 —.804 .099 0.401 .111 U- 4 1.323 0.8625 2.027 .2079 - 0.06523 .2833 U- 5 — 1.611 — 1.241 — 1.807 .07724 0.5477 .3225 U- 6 — 1.17 — 1.30 — 1.00 — 0.263 — 0.582 .6%

U- 7 0.1577 0.1827 0.6445 0.1786 —.2263 .1209

T T O A OQT7 — 1.063 .o6oU — U.O'lOO —. lool U- 9 1.017 0.4821 1.981 —.2253 0.2873 .2569 U-10 — 0.7669 — 1.752 — 1.451 1.009 — 0.3707 .3509 U-11 —.8557 — 1.153 — 1.560 0.0604 0.2112 .1310 U-12 0.002742 -0.2092 0.2124 .1501 -0.2346 .2393 U-13 .3465 -.3047 .06221 .2543 -.2668 -0.0926 U-14 — 1.949 — 1.318 — 1.882 1.636 0.04706 0.2323 U-15 — 2.291 — 2.575 — 2.224 0.1688 — 0.4036 1.215

C- 1 0.07142 — 0.6604 — 0.1499 1.103 —.9295 — 0.0994 C- 2 — 1.334 — 2.111 — 1.889 0.1862 —.2152 0.7695 C- 3 — 0.05111 — 0.8132 — 0.3242 .1304 —.5389 .2598 C- 4 —. 1 150 —.5709 —.3157 .6091 —.5067 .1256 C- 5 —.7208 — 1.298 —.1101 .03892 —.8207 .8897 C- 6 —.2302 — 0.3100 —.1015 — 0.04778 —.1960 .2311 C- 7 —.1569 —.7702 —.3818 0.5112 —.5640 .1301 C- 8 —.2676 —.8597 —.3834 .3623 —.3325 .3435 C- 9 —.1518 —.6830 —.4996 .4844 —.9765 .1522 C-10 1.103 0.2951 -.1274 .3925 0.3278 -0.1571 C-11 0.1287 -0.4135 0.1176 -0.1292 -0.1275 0.2351 C-12 — 0.7223 — 1.454 — 1.008 0.0230 —.3%3 .4794 C-13 0.3565 — 0.2536 0.7913 .2535 —.4964 .1290

T- 1 1.069 1.526 .7764 — 0.1253 0.3653 — 0.2899 T- 2 1.208 0.3539 — 0.08226 0.3055 0.3802 0.1465 T- 3 — 0.9430 .6184 — 1.876 .7452 0.1373 — 0.1306 T- 4 0.6823 — 0.1814 0.1946 .3945 — 0.4417 —.2808 T- 5 .3652 —.3680 — 0.2873 .5250 — 0.2100 —.2525 T- 6 — 0.5940 —.5858 — 1.173 .1452 0.1993 —.4060 T- 7 0.4112 -.04851 0.5315 .1112 -0.2553 -.1445 T- 8 -0.3666 -.4549 -1.132 -0.1797 0.002816 -.3436 T- 9 1.713 0.7693 1.088 0.2289 — 0.0440 —.8739 T-10 0.7087 0.4845 0.1256 .7641 — 0.1159 —.6045

Y- 1 1.172 .2233 .2845 -0.1638 -0.6475 -.6626 Y- 2 -0.2026 -0.1324 -0.6666 -.2387 0.2224 -.8202 Y- 3 1.002 0.6153 -.06408 0.2756 0.5955 -.5355

55
Table XVI. Absolute value of maximum shear strain in selected (kkl) planes (all values in percent)

Pattern U-0 U-1 U-2 U-4 U-5 U-7 u 0 U y Ti— in /Line

110 0.4101 0.5164 0.4604 0.1698 0.5426 0.2090 fl 7ftd9 fl ^i7H 1 flQ^i 101 .4575 .2838 .8015 .8967 .3175 d^in . I \JOO 1 flfl"^ 1 flO^ Oil .1695 .7469 .0858 .2291 .7934 .2904 . 1 Oli4^ 1 ,oy\j fl ^1 ^7 lOT .2799 .2647 .9449 .9220 .2525 .3548 .0826 1.047 .8169 iTo .4201 .5418 .9808 .0956 .4812 .0788 .8567 fl 6^85 1.326 oiT .1871 .8002 .4369 .1783 .6058 .0866 .8313 1.561 0.5403 211 .5208 .2086 .4114 .7079 .3726 .3722 .0850 0.9447 4492 12T .2597 .7451 .8497 .3898 .7077 .2065 .9078 1 I'ifl 121 .2212 .4678 .0489 .4958 2862 .3305 .8491 1 179 0.761

21T .2742 .5375 .9083 .6783 .7737 .3781 .5079 0.6964 1.099 112 .05913 .5933 1.221 .7076 .5934 .2629 .8407 1.119 1.294 112 .3937 .4784 0.1620 .7703 .6322 .494S .0424 1.235 0.4698 220 .4101 .5164 .4604 .1698 .5426 .2090 .7642 0.5578 1.096 200 .2229 .3482 .5714 .2243 .5478 .2536 .4641 .3325 0.441 020 .3690 .1334 .0622 .0353 .2859 .0545 .7129 .2946 .9210 002 .2946 .3267 .5812 .0241 .4768 .1884 .5733 .3147 .9272

U-6 U-11 U-12 U-13 U-14 U-15 U-3 C-1 C-2

110 0.576 0.3517 0 2262 0.6769 1.219 0 3288 0.4988 0.7697 0.7903

101 .378 .7424 .3231 .3130 1.141 .8464 .5725 .6539 .7833 oil .481 .4725 .4390 .4401 0.6113 .6153 .3654 .8224 .4100 loT .660 .7350 .2256 .3626 .8586 .6417 .4311 .7684 .7349 iTo .295 .3230 .3223 .7505 1.171 .4578 .2986 1.455 .8595 oiT .937 .4266 .5256 .3968 0.5983 1.052 .1325 0.7340 .6426 211 .391 .4472 .0636 .4478 1.149 0.6078 .6087 .2521 .5079 12T .776 .5399 .4296 .5932 1.259 .9458 .2277 .9762 1.074

121 .384 .4642 .3057 .4253 0.7626 .6442 .3412 .6381 0.8440 21T .457 .6757 .2533 .6593 .5565 .2465 .2757 1.109 .6073 112 .314 .5536 .3737 .6548 .4115 .5662 .3209 1.468 .8432 112 .236 .7016 .2386 .1538 1.354 .2662 .2050 0.5902 .5204 220 .576 .3517 .2262 .6767 1.219 .3288 .4988 .7696 .7903 200 .812 .2144 .2896 .2432 0.2044 1.104 .3417 .8059 .6996 020 .665 .1243 .2435 .2332 1.425 1.057 .3863 .9552 .6830 002 .571 .1894 .2402 .3181 1.411 0.3770 .4035 1.244 .2452

C-3 C-4 C-5 C-6 C-7 C-8 C-9 C-11 C-12

110 0.8319 \3.nt I yjO fl 1 70^ fl fidflfl fl ^il7fl fl ^90"^ 0.5862 0.7958

101 .3708 .00 14' 1 7/1 AAfn 5308 .6559 .4267 oil .00 1 \j 91H9 .4^700 dQft^ .00 1 t .5571 .4682

iflT 1911 .4'10 di ^^

.2224 .4071 iTo 8299 7Q7ft 1 997 01 '^9 7dftl 1 fldO .5649 .8042 OiT 7046 ,ooy I .OOJ4f fl 71dl .5958 .7082 211 91 ^ft .0R70d. ( 74 ^1 97 9fld^ .3328 .5019 12T 9313 .6572 1 ^77 7879 7793 .8599 .6556 .9333

121 .5267 .3224 0.7242 .1166 .0^00 .5364 9420 .5312 .6552 21T .5887 .5432 .7641 .1691 .6564 .4546 .6560 .3031 .4015 112 .7957 .8301 .8559 .1238 .9041 .7031 1.074 .3426 .6766 112 .1004 .2895 .6852 .1239 .1706 .3310 0.4099 .3897 .2350 220 .8319 .4705 .7675 .1703 .6397 .6170 .62^)5 .5862 .7958 200 .5158 .4312 1.044 .2612 .4988 .4120 .8520 .2306 .5362 020 .2506 .5457 0.7678 .2034 .4548 .4304 .4;i7: .2314 .4138 002 .4780 .6977 .7084 .1740 .6562 .4238 .93^)7 .1565 .3422

56
Table XVI. Absolute value of maximum shear strain in selected (kkl) planes (all values in percent)— Continued

C-13 T-4 T-5 T-7 T-9 T-10 Y-1

110 0.6428 0.8997 0.7876 0.4871 0.9897 0.4592 1.105

101 .5091 .5130 .6997 .1269 .7608 .6150 1.053 oil 1.111 .5893 .2942 .6514 .6506 .5767 0.8014

101 0.4952 .6535 .8288 .2002 .9360 1.032 .9735 110 .7825 1.035 .8854 .5285 .9974 0.5848 1.0313 oil 1.153 0.4039 .8791 .6079 .6051 .4780 0.6428

211 0.3227 .6658 .5654 .3569 1.055 .2339 1.235 121 .9918 .7390 .6211 .5695 0.4175 .5228 0.6689

121 .7610 .4541 .3698 .4470 .5017 .3403 .2928 211 .8140 .9838 .9447 .5799 1.219 .8190 1.085 112 .9688 .9370 .8149 .5917 0.7440 .5792 0.8656

112 .7238 .1237 .2536 .4420 .4630 .4619 .3311 zzu .ODZO 7ft7^ .70 QftQ77 / 1 1 0"^

1 . LVD 200 .4419 .4509 .2832 .2532 .7542 .5320 0.7985

(vy(\ uzu J.1 Id. 1 ^^79 . 10 / Z , 1 too .ooo^ 002 .4804 .5108 .4875 .2401 .2009 .6662 .5760

C-10 T-1 T-2 T-3 T-6 T-8 Y-2 Y-3

110 0.4524 0.7816 0.5563 0.7210 0.6243 0.7348 0.6651 0.7087

101 .6793 .7736 .6177 .9593 .3281 .3454 .4430 .4804 oil .8706 .6152 .8611 1.529 .6830 .4731 .4191 .7730

101 .4198 .9939 .3210 0.8630 .1042 .3429 .3513 .2494 110 .5295 .8766 .4762 .7394 .7142 .7357 .8447 .9879

oil .9447 .2633 .7964 1.583 .8145 .6599 .6668 .8655 211 .2266 .8090 .3478 0.4126 .1927 .4397 .6087 .1599 121 .7493 .3802 .5961 1.231 .8303 .8498 .8112 .8994

121 .7278 .2247 .8226 1.181 .5518 .6042 .2327 .6648 211 .2675 1.087 .1957 0.5071 .4716 .3740 .5175 .7282 112 .6203 0.7770 .5629 .8841 .6784 .5928 .8855 1.017 112 .3051 .1367 .2497 .7456 .2683 .2349 .2010 0.3180 990 . /OlD .DDOo . 7910 1 ziw . * O'rO • UUO

1

.7087 200 .5196 .6670 .4432 .7487 .4491 .4641 .7321 .4898 020 .5764 .6940 .5365 .7494 .3298 .3574 .2700 .5390 002 .3134 .4020 .3513 .1634 .3899 .2962 .7325 .6902

57
Table XVII. Stored elastic energy and distortion numbers

Total Shear

stored stored Distortion Pattern elastic elastic WslW number

energy, W energy, Wg

(MN/m2) (MN/m>)

U- 0 123.3 109.3 0.886 0.880 U- 1 225.5 205.4 .911 .910 U- 2 437.4 344.4 .787 .784 U- 3 244.3 156.3 .640 .837 U- 4 983.3 402.4 .409 .735 U- 5 960.0 257.1 .268 .573 U- 6 880.3 457.7 .520 .669 U- 7 151.5 126.1 .832 .892 U- 8 574.5 376.6 .656 .867 U- 9 107.6 683.0 .635 .868 U-10 132.2 676.6 .512 .824 U-11 567.4 159.1 .280 .634

T T 1 O OCT U-13 173.6 174.1 1.0 .964 U-14 215.5 126.0 0.585 .675 U-15 236.9 731.6 .309 .589

C- 1 106.6 104.9 .984 .956 C- 2 137.1 460.9 .336 .583 C- 3 362.3 309.1 .853 .876 C- 4 364.0 324.1 .890 .967 C- 5 114.5 997.7 .871 .897 C- 6 65.3 52.1 .797 .836 C- 7 412.3 357.5 .867 .898 C- 8 330.9 247.4 .748 .907 C- 9 655.1 603.8 .922 .855

Af\^ 1 ^UD. { olo.o 779 C-ll 139.0 152.0 1.0 .878 c-12 634.4 308.3 0.486 .711 C-13 453.7 425.8 .939 .939

T- 1 627.9 4S2.0 .768 .889 T- 2 392.7 313.0 .797 .930 T- 3 914.8 761.4 .832 .943 T- 4 396.1 370.9 .936 .961 T- 5 336.4 347.3 1.0 .928 T- 6 390.8 281.8 0.721 .873

1 — t 165. / .yuo T- 8 675.1 251.8 .373 .675 T- 9 100.2 589.6 .588 .796 T-10 560.9 506.8 .904 .883 ,

Y-1 765.9 709.2 .926 .894 Y-2 600.4 407.5 .679 .726 Y-3 464.9 472.0 1.0 .927

58
Table XVIII. Direction cosines of plane of maximum shear strain taken with respect to cubic axes

Pattern li h Ik Pattern /. h

U- 0 -0.117 0.180 0.977 T- 1 .955 -0.057 0.291 U- 1 -.566 -0.774 -0.284 T- 2 -0.520 -.765 .381 U- 2 0.085 -.366 0.926 T- 3 -.468 -.884 -0.008

U- 3 .085 -.366 .926 T- 3 -.468 -.884 -.008

U- 7 .628 0.126 -0.730 T- 5 0.866 0.297 0.403 U- 9 .455 .161 -.876 T- 6 -0.465 -0.778 -0.424

U-10 -0.002 -0.917 -.399 T- 7 -.551 0.720 -.421 U-12 -.333 -.940 -.070 T- 9 0.863 -0.195 0.466 U-13 -.24a 0.518 -.819 T-10 -0.424 -.096 -0.901 T- 4 0.947 0.057 0.317 C- 1 -.396 -0.918 -.004

C- 3 -.365 0.102 0.925 Y- 1 .802 -0.543 .249 C- 4 -.274 - 0.948 -0.130 Y- 3 -0.453 -.871 -0.190 C- 5 -.289 -.922 -.256

C- 6 0.008 0.147 0.989 C- 7 -0.282 -0.925 -0.254

C- 8 -.238 -.833 -.499

C- 9 0.882 0.465 — .081 C-10 -0.478 -0.874 0.088 C-13 0.886 0.054 -0.460

59
XIV. Figures

Figure 1. Schematic representation of orientation relations in FIGURE 2b. Static hysteresis loss versus macrostress. Induction

Goss and cube textures in Fe-3 Si. 15,000 gauss (After Ref. [4]).

60
INGOT: 3% Si, 0,15% Mn, 0.03% C, 0,03% S

I HOT ROLL TO 0,08 TO 0,1 INCH THICKNESS

I

PICKLE AND DESCALE

I COLD ROLL

I PROCESS ANNEAL at 900° C

(

COLD ROLL: FINAL PASS GIVES ADOUT 60% REDUCTION

I

STRIP 12-13 MILS THICK

(

PRIMARY RECRVSTALLIZATION DECARBURIZATION att 800 C IWET H

I

COAT WITH M, |0H|, (INSULATES AND PREVENTS RUST

I

BATCH ANNEAL at 1200° C IN VACUUM

SECONDARY RECRYSTALLIZATION GIVES |110| 001

TEXTURE

SOAK at 1200 C TWO HOURS TO

DESULFURIZE

1 COOL at 25' C PER HOUR

THERMAL FLATTENING (TENSILE STRESS APPLIED

AT ABOUT 350* Cj

-COAT WITH PHOSPHATE

COILS FOR SHIPMENT -CUT TO FINAL SIZE

STRESS RELIEF ANNEAL (UNIFORM HEAT AND COOL TO MINIMIZE THERMAL STRESSES]

I USE

Figure 3. Flow diagram for the preparation of Goss textured Fe-3 Si sheet.

2.8675

2.8585 I ' 1 ' 1 1 J 0 10 2,0 3,0 4,0 5,0 6.0 WEIGHT PERCENT OF Si IN Fe Si ALLOYS

Figure 4. Lattice parameter, a, in Fe-Si alloys versus weight percentage of silicon. Line is least squaresfit to data shown: a = 2.86719 - 0.144224 Csi.

61
TV

f

Photometer

trace

Reflection cone

Section of crystal

Plane normal X-roy source

Lattice planes

Figure 5. Formation of Kossel conies.

2 4 6 8 10 12 14 16 18 20 Q'X/C

Figure 6. Integrated reflection in transmission: Darwin solution for a Gaussian mosaic spread, ^. (After Ref. [22]..

62
c

Figure 7. Gnomonic projections ofpoints m, z, n into m' ,z' , n' respectively.

Figure 8b. Indexed pattern from Fe-3 Si sheet. Pattern taken at geometric center of grain shown in figure 24.

63
Figure 9. Lens methodfor cubic lattice parameter determination. The value of "/" depends only on the wavelength

and indices of the participating conies (After Ref.

[37]).

Bragg Angle,

Bragg Angle , 9

Figure 11a. Effect of error in Z on the d spacing values in Fe-3 Si. FIGURE lib. Effect of a radial error, Ar, (// determining the pattern center on the d spacing values in te-J bi.

64

Figure 17. Load cell for Kossel camera module. Dimensions in millimeters.

66
0.5 -

100 120 % FS Load

445 N load cell 44.5 N load cell Figure 18. Calibration curves for load cells.

Figure 19b. Schematic of loading device.

Figure 19a. Loading device mounted in Kossel camera. Readout

unit is at rear.

Figure 20. Typical specimen of Fe-3 Si sheet prior to insertion Figure 21. Enlarged view of typical grain in Fe-3 Si sheet. Note

into the loading device. X2 surface roughness. XIOO. 67
Figure 25. Key to Kossel pattern locations in low tensile load (Load III).

U-14

Figure 23. Key to Kossel pattern locations in unstrained condition {Level I).

68
•I, STRAIN O 0.2 0.4-O5 0.2-03 #0 5-06

0 3-0.4 ^0 6-0.7

7o STRAIN O-CS—0.3 -0.1-0

-a3--0.2 #0*0.1

; -0.2-'-0 l ® 01-0.2

Figure 27b. Maximum principal strain as a function of position

in grain. Level II. Figure 28b. Intermediate principal strain as a function of posi- tion in grain, Level II.

0.2-0.3 ® 0.6-0.7 _2_| a 0.3-0.4 0 0.7-0.8 -.|,-.o #0.4-0,5 0>O.e # 0-0.1

Figure 27c. Maximum principal strain as a function ofposition FIGURE 28c. Intermediate principal strain as a function of posiin grain. Level III. tion in grain. Level III.

69
0.3-04 #07-0 8 S 0.4-05 (|(08-O9 #0.5-0.6 #>0.9

Figure 29b. Minimum principal strain as a function ofposition

1$ 0.3-0.4 © 0.7-0.8 #0.4-0.5 ® 0.8-0.9 #0.5-0.6 0>O.9 @ 0.6-0.7

Figure 29c. Minimum principal strain as a function of position

in grain. Level III.

Figure 30a. Maximum principal stress in GNjm^ as a function of

position in grain. Level /.

%; -i.-o # 0-1

Figure 30b. Maximum principal stress in GNjm' as a function of

position in grain. Level //.

-l,-»0 0-1 i 1-2

Figure 30c. Maximum principal stress in GNjm? as a function of

position in gram. Level in.

-1-0

0-1 # 1-2

Figure 31a. Intermediate principal stress in GNjm^ as afunction of position in grain. Level I. 70
-2,-1 -1-0

fi 0-1

Figure 31b. Intermediate principal stess in GN/rn^ as a function of position in grain. Level II.

-2.-

-1-0

0-1

Figure 31c. Intermediate principal stress in GNjm? as a function ofposition in grain. Level III.

-2,-1 -1-0

0-1

Figure 32a. Minimum principal stress in GNIm? as a function of

position in grain. Level I.

0-2,-3

-2,-1

s» -1-0

Figure 32b. Minimum principal stress in GNjm^ as a function of

position in grain, Level II.

-2,-1 -1-6

0-1

Figure 32c. Minimum principal stress in GNjm- as a function of

position in grain. Level II.

-4r3 #1-2 0

0-3,-2 #2-3

-2.- 1 ®3-4

-1-0 ®4-5

0-1 05-6

Figure 33a. Algebraic sum of the three principal stresses, in GNjm?, as afunction ofposition in grain Level I.

71
0-5,-1 -1-0

: -4-3 :; 0-1

-2,-1

Figure 33b. Algebraic sum of the three principal stessses, in GNjrn', as a function ofposition in grain. Level U.

0-3,-2 #1-2

Figure 33c. Algebraic sum of the three principal stresses, in GNjnr , as a function of position in grain. Level in.

Figure 34. Schematic view of stresses on grain.

0 2> '*5-6 0>9

2-3 ®6-7

3- 1 #7-8

#1-5 ®8-9

Figure 35a. Total stored elastic energy, W, in MNjm^ X/O"-', as a function of position in grain. Level /.

2- 3 ®6-7

3- 1 #7-8

1-5 ®8-9

Figure 35b. Total stored elastic energy, W, in MNjm? x 10 ^, as a function of position in grain. Level U.

02> #5-6 %>9

2- 3 #6-7

3- 1 ®7-8

1-5 @8-9

Figure 35c. Total stored elastic energy, W, in MN/m^ X10~^, as a function of position in grain. Level lU.

72
02> ®5-S

2-3 #6-7 S

3-''

W; « - 5 Figure 36a. Shear stored elastic energy, Ws, in MNjm? y.lO~'^ , as a function of position in grain. Level I.

2-3 #6-7

J; 3-4 #7-8

igil-5 ® 8-9

Figure 36b. Shear stored elastic energy, W^, in MN/m^ X 10^^, as a function of position in grain. Level H.

02> «5-6

2-3 #6-7

Si 3-4 •7-8

# 4-5

Figure 36c. Shear stored elastic energy, Ws, in MNjm^ X 10~'^, as a function of position in grain. Level lU.

0.2-03 0 0,6-07

0,3-0.4 ® 07-0-8

*|! 0.4-0.5 0.8-0-9 #0,5-0,6 0>O.9 Figure 37a. Ratio of WslW as a function of position in grain. Level /.

0,3-0-4 ^07-08

siij 0-4-0,5 @ 0,8-0-9

f$l 0-5-0-6 0 >0-9 # 0 6-07

Figure 37b. Ratio of WsjW as a function of position in grain. Level n.

® 0-6-0,7

Figure 37c. Ratio of WsjW as a function of position in grain. Level in.

73
4' 0.5- 0.6 0>a9

©06-07

#07-0 8 ®08-09

Figure 38a. Distortion number as a function ofposition in grain. Level I.

® 06-07 ® 07-0.8 ® 0.8-0.9

Figure 38b. Distortion number as a function ofposition in grain, Level U.

@ 0.6-07 ^07-0 8 #0.8-0-9 #>0 9

Figure 38c. Distortion number as a function ofposition in grain. Level lU.

O0.2> 0 5-06 02-03 ^0.6-0.7

0.3-0.1 ® 07- 0.8

|1 0.4-0 5

Figure 39a. Absolute value of the maximum shear strain in the

(112) plane as a function of position in grain. Level /.

O 0 2 > s|J. 0-5- 0.6 0.2-0.3 ®O6-07

0.3-0,4 ® 07-0.8

0.4-0.5

Figure 39b. Absolute value of the maximum shear strain in the

(112) plane as a function of position in grain. Level n.

0 02> #0 5-0 6

s 02-0.3 @06-07

0.3-0 4 ^0 7-O8

0 4-0.5

Figure 39c. Absolute value of the maximum shear strain in the

(112) plane as a function of position in grain. Level in.

74
J|fl|^^^H Figure 41b. {IIO) Stereographic plot of data in table XVIII for ^^^^^^lll^^^^^^^l Level II, the line is at :U° to the {100} zone.

Figure 40. Double pattern obtained near grain boundary. Mis- alignment is more than 3° of arc.

Figure 41a (110) Stereographic plot of data in table XVIII for Figure 41c. (110) Stereographic plot of data in table XVIII for

Level I, the line is at 31° to the {100} zone. Level III, the line is at 31° to the {100} zone.

75
FORM NBS-114A (1-71) U.S. DEPT. OF COMM. 1. PUBLICATION OR REPORT NO. 2. Gov't Accession

BIBLIOGRAPHIC DATA NBSMN-130

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

The Divergent Beam (Kossel) X-Ray Method and its Uses in

1 ICQ oui Miy ouictiii uuiiuuuio III ail XiiuivlUUctl orolil ur rc o Weight Percent Si Transformer Sheet

5. Publication Date August 1973

6. Performing Organization Code

7. AUTHOR(S)

Harvey Yakowitz

8. Petforming Organization

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/ Work Unit No. 3120137

11. Contract/Grant No.

12. Sponsoring Organization Name and Address

Same as No. 9

13. Type of Report & Period Covered

Final

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.) Residual and impressed stresses and strains in Fe-3 wt. pet. Si alloy transformer

sheet were studied. Results were obtained by the divergent beam (Kossel) x-ray

microdiffraction techniques. The optically opaque Fe-3 wt. pet. Si alloy was mapped

for residual and impressed stresses and strains. These maps are roughly analogous to

polarized light stress analysis of transparent materials. The results showed that a variable strain distribution existed in the alloy sheet at the time it was ready for

insertion into a transformer core. Small applied compressive and tensile loads tended

to rearrange this variable strain distribution but not to appreciably alter the total

stored elastic energy.

17. KEY WORDS (Alphabetical order, separated by semicolons) Divergent beam X-ray d i ff rac t i on ; electron

probe microanalysis; iron-silicon alloy; Kossel; strain; stress; transformer sheet.

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