X-ray measurement of residual strains in individual grains ...

JOURNAL OF RESEARCH of the National Bure au of Standards-C . Engineering a nd Instrumenta tion
Vol. 68C, No.4, October- Novem ber 1964

X-Ray Measurement of Residual Strains in Individual
Grains of Polycrystalline Aluminum

Clarence 1. Newton

(Apr il 24, 1964)

S hif t s in peak posi t ion of 1333 \ an e! 1511 \ d i f·Tracti on s of co bal t ](<<1 X r ays from in d iv idual

grain s o f coa rse-grain ed p olyc rys t alline alu m i num obse r v ed in t he lt nn ea1ed co nd i t ion a nd
af te r 10 p er ce nt plast i c extension r e veal ed r es id ual strain s i n each crys talli t e. Th ese strain s,
h ow e ver , d id not co nform to t he str a. in q u ad ri c wi t h a prin cipal axi s p a rallel to t he a x is o f
d ef or·mation, as is t he case of obsel"lra t ions fro m fin e-grain ed m etalli c spec i mens t h"t have
b een plast ically def ormed ; nor was any co nsist ency or m ea nin gful av erage t ren d obse r ve d
in t h e strain s of t he v ari ous grains. Irreg ula ri t ies of l oad in g co nstm i nts by one grain upo n
i t s nei ghbo rs and t he resul t i n g gr ea. t Il onulliformi t y of def orm atio n m ay acco un t for t he
:,bse nee of sy st em at i c r es ul t s.

1. Introduction th at th e soft regions A and th e hard reO·io ll s B are

'rhe a ngle of x-ray diffraction 0 is related to th e regions of low a nd high h ttice s trll ctur: distor tion
sp acing clltk / b eLw een layer s of aLom s in a crystallin e
solid t hrough Bntgg's l aw, respec tively. This hypo thesis h as been suppor ted

(1 ) by the o bser vations of m tl,ny r ecen t workers t[h5a,n6'o7n]~
al t hough ther e is som e evide nce tlUl,t mor e
wh cre A is Lhc x-ray w:welengLh . The use of th e
shiH of t he diO·rncLio n a ngle :l S :Ul indi cation of slnl,in m echa nism may be contribu tin g to th e observed
in th e lnl licc st ru cture of t hc solid is m orc lllltn 30
ycars old . Onc of Lhc fi rst obscrntli ons of thi s ty p e st rains a nd stresses under ce rtain circumstances [8].
was m adc by L esLer a nd Abol"ll [1] 1 in 1925 on th e
ch ange of spacin g of crystallin e pla ll cs in sl ccl T he original idea that th e disto rted h arder r egions B
subj ectcd Lo str ess . A comprchcnsi\"e l"cy iew articlc
con cern ed with x-n ty sLl"ain m cas urcmcnt as wcll as were at the grain boundaries has gr adually been
oth er aspccts of q u all tiLaLi\" c x-ray dirrm ction ob -
seI"Yations on straincd m etal aggregat es w as pub- generilJized to in cl ude all regions of high dislocation
lish ed by G . Green ough [2 ] in 1952. P erhaps lh c
most in tercstin g asp ect of lhese sLrains ill lhc crysl nl densi ty, such as slip pla nes, s ubg rain boundaries,
lattice slru ctUl"e is Lhc rcsidu Rl elRsLic strain ob-
served in tL mcLal s pecimen that h as b een plasLically and t he dislocat io n tangles tha t consti tul e cell walls
deformed and t hen unloaded . R ecen tly the," arious
t heories att emptin g to explain t hese residu al strains observed in some deform ed metals [9] . Si nce th e
and stresses m easured by x-ray diffraction h av e b een
eyalu ated by Vasil 'ev and Smirnov [3] in 1961 in a x-my di ffraction pectle posi tion is determin ed prin -
re\"iew article discussin g a "ariety of x-ray diffraction
n1.ethods of investigating cold-worked m etals. cipally by the m or e pOl·fccLA Jnaterial, the pcak shif t

It is generally accep ted that th e r esidual stresses rcpresen ts the elastic s tmin a nd the r elated s trcss in
arise on account of differences of " hardn ess" or
resis tttnce to plastic flow in various regions of th e that m ateri:tl only .
material. After the r elease of a uniform uniaxial
deforming stress of a given sign, the weaker regions RelaLed to the q ues tio 11 of Lh e source of Lh c r esid lI nl
A will be constrained in to tl, s LttLe oCs t ress of th e
opposite sign by t he gr e:tte r a moLln t of elas tic s tmin ei<Ls ti c s train s twd sLresses ill thc polycrys talli ne
recovery in the str onger regions D. Al though th e
microscopic nature of t h.ese two regiolls h as not been m etal is the paradox of their obse rved q uasi-isotropic
clearly determined , the m odel that seems to be m os t
widely accepted to day is based upon ideas first behavior. The s train s m eas ured o n tl, givc n s ur i'ace
ad vanced by Smi th and Wood [4 ]. They s ugges Led
ar e observed to saLisfy Lhe equatio ll or IL s Lmin
1 Figures in brackets indica te the li terature re fere nces at the end of th is papf't·
quach·ic, with one of the p rin cipal s tmins parallel to

th e axis of plastic deform aLion. This fell,turc is

implicit in m ost of th e rep orts of this ty pe of measure-

m en t a nd h as occasionally bee n expli ci Lly verified

[10]. .lVlost workers agrec, m oreover , that in practi ce

it is permissible to use the g ross avemge v:dues of

Young's m odulus and P oisson's ratio as obtltin ed

from m echanical tests on polycrys talline specim ens

fr ee of preferred crystalli Lc oriell tatioll [11 , ] 2] to

r elate th e strains to a system of s tr esses using iso-

tr opic elastic theory; and Lhere scellls to bc no ques-

tion tha t the net obser vcd bcll<1vio r in t he x-my

"powder " diffraction eff ects from the aggr egate of

individual a nisotropic cr ystalli Les is iLself isotropic.

Ther e ~tre at leas t two p oss ible expi<Lllations for

this isotropic behavio r o n t he P:I,l" t of t he diffracting

r cgiolls A of the grain s. F irs t, these r egions m ay be

so co ns tmin ecl by their r a ndomly oriented n eighbor-

in g g min s }tnd by the hard , quasi-amorphous B

m aterial at grain or subgrain boundaries th at the

249

strains are forced into an isotropic pattern relating and furnace cooled. The specimen in its iuiLial
to the applied deformation. Alternately, although condition may be seen in figure 1, a and b.
the distribution of strain in anyone crystallite might
be itself quite anisotropic and unrelated to the geom- The x-ray diffraction measurements were obtained
etry of the preceding deformation of the gross speci- by a combination of film and counter methods on a
men, the strain indicated by the shift in the diffraction commercial x-ray diffraction apparatus, employing
line, comin g typically from hundreds of crystallites, auxilliary equipment designed and built at the
might r eprese nt a nonzero average that exhibits the National Bureau of Standards . The first s tep of the
isotropic behavior. procedure was the determination of the crystallo-
graphic orientation by means of back-reflection Laue
The principil,lline of attack in the present investi- diffrnction patterns of all the grains in the central ?f
gation wns to m easure the shift in the Bragg angle of in. of ench of the four faces of the reduced section of
diffraction from individual crystallites in a coarse- the specimen. The number of grains so oriented,
grained polycrystalline specimen that had been countings duplicates around specimen edges twice, was
plastically deformed in tension. The purpose of the 5l. All of the {111 } and {511 } plane normals were
study was to see if there w'as an impressed residual lo cated on the stereographic projection o± the pattern
stress-strain system, if it was isotropic with principal axes from each grain, and the angular coordinates,
determined by the external deformation, as is the azimuth a and co-altitude 1f/, for all SUell poles within
cnse with ordinary fine-grained materinl, and what approximately 65 ° of the normal to the surface being
the magnitude of the residual strain might be. It studied were measured on a Wulff net and recorded.
was hoped that such an investigation might throw
some light on the alternate hypotheses of pseudo- After the determination of orientation of each
isotropic behavior of the strains and possibly lead to grain, the specimen in its special holder, which may
further studies that might reveal some grain-size be seen in figure 2, was transferred to the diffrac-
effects. tometer. By means of a collimator sigh ting adj ust-
ment, the surface of the specimen was placed in
2. Experimental Procedure s coincidence with the common vertical axis of the
diffractometer and the holder ; and while the specimen
The specimen was of 99.99 percent pure aluminum was observed with a low-powered microscope, a
with threaded ends and a reduced section about desired grain was translated into the incident col-
I X in. long with a square cross section ?f in. on a sid e. limated x-ray beam about 1 mm in diameter. The
The specimen was supplied in a fully annealed, stress- proportional counter was set at the expected 2(J
free condition, surface etched, with grains ranging in diffraction angle, which was 162.50° for both the
{511 } and {333 } planes, for copper Kal rndiation .
mean diameter from about 7\6 in. to ?~ in. , grmvn by the The alpha doublet was well resolved in all cases.
The two angular adjustments on the specimen holder
strain-anneal method. Prior to examination the were then set, corresponding to a an d 1f/, in order to
specimen was further annealed for 24 hr at 150 °0

qc

bd

FIGU RE 1. Aluminum Sp ecimen, 111agnijication 2 X.

a. Face A ann ealed conditioil. c. F ace A, after 10 percent plastic extension

b. Face 13, annealed condition (at right angle to Face A). d. F ace B, alter 10 percen t plastic extension.

250

tion was determin ed analytically by a three-point
pambol a-fitting equ ation, with a precision estimated

to b e ± 0.01 ° or b etter. The 6-dependent corrections

of the in ten sity often used in this type of peak
determination were examined for a few cases in this
study, but wer e not used, being negligible because
the distan ce b etween fi rst and last step positions of
26 on each side of the n,p ex of a peak was only 0.04 0
in these sin gle crystal diffractions, as compared to
Lhe se, -er nl tenths or even whole degrees im-olyed

in the case of polycrystalline diffraction. B efore
the ch anges in 26 goin g from t he annealed to the
strained stn,t e were cal cul ated , however, the indi-
v idun,l yalu es were corr ected for th e efI'ects of thermal
expansion from t he temperat ure of m easuremen t to
a stand ard 25 DC. The hn,ndbook value of t h e
coefficienL used was 23.8 X I0 - 6 per deg ree C for t h e
laLLice co nstant. This l'csul ted in a temperat ure
correction in t be Bragg an gle, in degr ees, at 26
equal 162.50 °, of

0(260 ) = (-0 .0177 )o T , (2 )

where oT is the differen ce in temper ature in deg rees

F I GURE 2. Goniometpr specimen holder for 1lse on the x-ray C from the reference tmnper n,Lure.
di.O·racto meter. If the m easured strain is s mall, as it was in t hese

cases, it is not necessary lo calculat e "nlues of d"kl,

th e lattice pln,n e spacin g, from th e obsen -ed Bragg

place the desired pole of a diffracLin g pl<tne in the angles; it is more conyen ienL si mpl~T Lo use Ll (2 6),
horizontal plan e of the diffractometer ft nd in Lhe Lhe chan ge in Bragg angle, sin ce iL is directly pro-
position of bisector of the ttngle betwee n the in cide nL porlional to Lhe strain t hroug h the followin g
ttud diffntCted rays. In order to mi nimize defocus- eqwttion:

in g effects, the s urface normal was al wftys tilLed

away from the det ector . All tlll'ee ang ular <tdjust-

ments were then "fine tUll ed" Lo give a ma ximum

sig nal in the co un ter. Th e co un ter was Lhen backed
up ,t few hundredths of a degree n,nd t hen "sLep- Thc angle 6 is approximately 8 ] .25 0 in th e case we

scann ed" across the top of the diffraction pe<tle Th e a,r e examining. Since on ly changes in Bragg angle

steps were 0.0 1 of a degree <tpart and wer e held for need b e observed, the question o[ a,bsoluL e calibra-

a fixed time in terv al; the intensity in total co un ts Lion of the diffracLom eter is a,-oided . For t he sake

was printed out ,.L t he end of e,tch in terval. of sirnplicity and direct ness, Ll (26°) "alu es r a.t her

After the refere nce peak values of 20 had been Untn actual strains are used throughouL this pape r.
determined for all poles of in terest in th e sp ecimen If the uncertainty in a, giyen 26 r eadin g is ± 0.01 0,

in the an nealed co ndition, the specimen was strain ed as estimated, the uncer tain ty in Ll (26) should b e
in uniaxial tension at 23 DC to a fin al true strain of about ± 0.014 0; hence the un certainty in a stntin

10 percent. The cross-head speed for most of th e calcul ated by equ ation (3) would b e ± l.5 X lO - 5•

deformation , including the latter part, was held at

0.001 in. per minute. The flow stress at the 10 3 . Results

percent plastic true strain was found to be a pproxi-

m a tely 4080 psi. At this strain , this coarse-grained On the two faces of the specimen, shown ill figure

sp ecimen showed, as may be seen in figure 1 c and 1, for which complete post-strain dat a wer e taken ,

d , considerable inhomogeneity of strain , more than changes in 26 were measured for an ayerage of about

is usually the case with fine-grained m aterial , but nine planes on each of 25 grains. The data Jor two

less than is typical with deformed single crystals. typical grains on Face A are tabulated in ta,ble l.

After the prescrib ed phstic strain, Lhc specimen The Ll (2 6) values, and h ence the str ains, are \' ery

was realined in the difl'ractomel er and Lhe peak 26 sm all, but in most cases they are se\-eral tim es t he

yalues were r edetermin ed for all of thc {511 } a nd estimated un certainty in the m eas urem enL .

{333 } planes in the region of in terest on t wo of the As st ated in the introdu cLion , residu al strains

four faces. Sin ce the difrracLion pcaks were so mc- m easured by x rays on cOllYentional polycrystalline

what broadened and considerably r edu ced in h eight maLerial Lhat h as b ee n plaslically strained uniaxially

after the deformation, th e steps in the sca,nnin g were satisfy t he strain quadric equation

now spaced 0.02 0 apart and t he time int en ·als of

counting considerably l engthened. The peak posi- (4)

251

TABLE 1. X-ray strain data (i.e., dij)'raction peak shift) fTom two typical grains

Grain IIkI '" Annealed 20T 29250 ex '" Extended 20T 2025° "'(2go)
1
ex 22.2 '1'(° 0 ) 162.43 162.41 168.9 21. 6 T(' O) 162.52 162.50 + 0.09
2 49.9 162.44 162.42 324.1 50.2 162.51 162.49 .07
333 169.8 24.1 24.0
333 323.9 48.3 24.1 162.44 162. '13 271. 3 46.5 23.6 162.52 162.50 .07
40.7 162.45 162.44 241. 8 40.0 162.53 162.52 .08
15T 271. 7 47.1 24.3 162.46 162.45 64.3 47.7 24.1 162.48 162.•,3 .08
151 241. 8 25.2 24.2 162.45 162.44 58.5 25.8 24.5 102.49 162.51 .07
5Tl 6<1. 3 54. I 24.4 162.45 162.44 38.2 54.9 27.7 162.44 162.46 . 02
.Ill 58. 3 35.8 24. 5 Hi2.46 162.45 36.3 26.0 162.45 162.47 .02
.lIT 37. 1 60.2 24. 5 162. 43 162. 43 ]9.1 60.0 26.1 102.49 162.51 . 08
.III 18.8 24.6 154.4 26.4
115 IM.9 47.8 24.8 162.52 162. 49 47.8 26.4 162.54 162.50 .01
59.4 162.50 162.47 66.4 60.0 162.50 162.46 -.01
333 64.1 62.8 23.2 162. ,\1 162.49 156.4 64 . 7 22.5 162.65 102. 63
33:\ 154.7 50.4 23.4 162.52 162.50 242.2 48.6 23.0 162.43 162.41 .14
333 237.4 23. 6 324.0 23 . 6 - .09
333 324.5 9. 7 23.8 162.49 162.48 4.3 23.8 162.48 162. 47
21. 3 162. 49 162.49 81. 0 20.9 162.51 162. 50 -.01
511 88.0 23.1 24.5 162. 50 162.50 182.8 25.4 24.3 162.58 162. 58 . 01
511 169. 2 13. I 25.0 162.47 162.48 229.2 20.1 24.6 162.52 162.52 . 08
5IT 229.4 25. 1 293.6 25. 1 .04
.011 299.5 25.4 25.1

where E is the strain measured III some direction
whose direction cosin es are:

al = sin if; cos <p

a3 =cos if; ----~------------~--~~--------L---~~~L--X l

and EI , EZ, and E3 are the principal strains, in t he
orthogonal dire ctions identifi ed in figure 3 with
respect to the geometry of the specimen and its
d eform ation. The instrumental azimuth angle a in
this study was related to the usual coordinate 'P by

<p=270 0- a

also illu strated in this figure. The direction cosin es
in terms of t/I and a were

al= -sin if; sin a

a3=cosif;. FIGURE 3. Diagram illustrating directions of axes and de-
fining angles with resp ect to the geometry of the specimen

and its deformation.

The direction cosines were calculated for all t h e direc- pared for each of the two faces A and B. By
tions in which the strains were measured in the two
restricting a to 90 ° ± 10° and 270° ± 10°, valu es of
grains referred to in table 1, and selected sets of three
simultan eous equations w ere set up from which sets strain from very many gr ains were measured for

of three principal strains were computed for a particu- various if; values close to a lon gitudin al plane (X IX3)
lar form of planes within each grain. In no case, normal to t he surface and parallel to the axis of

however, was even an approximately consistent set deform ation. By restricting a to 0° ± 10° and 180°
of principal strains with this preassigned orientation
± 10°, similar data were obtained near a transverse
found.
In the isotropic analysis of strains in fine-grained pl an e (X2X3 ). These four plots of 6, (20) yersus

material, a plot of strain versus sinz t/I is found to be sin2 if; are shown in figure 4. There does not appeal'
linear when the directions of measured strain are to be any relationship of t h e strain to sin2 t/I in any
confined to a plane normal to the surface of the
specimen [7 ]. Jn an attempt t o find analogous of the four cases examined . Indeed, no self-con-
"cooperative" behayior from the coarse-grained sistency or general trend or "average" behavior
material, two plots of 6, (2 0) versus sin2 t/I were pre-
among the grains was anywhere in evidence among
th e 229 strain determinations on the two faces of t hi s

speCIm en.

252

FIGURI, 4. Plots of Strain in T erms of Ll.(2!1) verS1ts sin2qt .

a. Longit udin al Plane Norma] to Face A. d. 'l:'ransvcrsc Plane Normal to Face B.
h. ']'ra nsvcrsc Plall CNormal to F ace A . C ircles- {51J} difi'raetion data.
c. Lon gitudinal Plane No rma l to Face B. Tria ngles- (333 ) diO'raetion elata.

4 . Discussion in this case of highly irreg ular loading cons tramts
would justify the involved compu tations. Perhaps
The results of t his investigation show that it is such an analysis of residual strains meas ured by
possible to detect directed r esidual strains by the x rays in plastically deformed specimens that were
peak shift of x-ray diffractions in sin gle crystals true single crystals would yield meaningful inform a-
wit hin a coarse-grained polycrystalline aggregate tion. The author is not aware that results of t his
that has been plast ically deformed in tension. EYen
when the material, howeyer, is aluminum, a metal type have as yet been published.
which is not so strongly a nisotropi c as m any others,
t hese individual crystallite stntin values do not con- It is interesting to consider whether the anisotropic
form to t he type of isotropic elast ic b eh ,t\'ior obsen'ed res ult ob tained with the coarse-grained material in
with ordinary fine-grained polyerystalline material this study is more consisten t with the "constrain t"
after plastic d eform ation ; at least sucll was Lhe case hypothesis or the "aver aging" hypothesis of t he
for the specimen st udied h ere. isotropic behavior of t he fine-grain ed material. The
change in grain size in vohed is from t hat of a few
I t may be asslLm()d that the strain data, from ~1l1y millimetcrs in the prosent caso Lo a few hundredths
one of the grains in thi s study could be s ubj ected to of a millim oter or less in t he typical fine-grained
a rigorous anisotropic clastic analysis, such as that case. T his chan ge of scale is relevan t to th e con-
of Imura, Weissman, and Slade (1 3] in thcir work sideration of eith or h ~-pothes is. It will change
with divergent b eam diffraction from single crystals. drastically t he mtio of tbe volume of soft A-type
I t is doubtful, how8\'cr , t hat th e information return regions , disc ussed in the Introduction, to the volum e
of h ard B-typc regions, if the r egions n car grain

253

boundaries arc of paramount importance to the 5. References
lattcr. This consideration, along with the change of
average distanccs over which forces would act, should [l j H. Lester and R. Aborn , Army Ordnance 6, 120, 200, 283,
accoun t for marked changes in behavior with size 384 (1925).
if the "constraint" hypothesis is valid. On the
other hand, it must be admitted that the "averaging" [2 j G . Greenough , Progr. Metal Phys. 3, 176 (1952).
of strain behavior is also improved in the statistical [3] D . M . Vasi l'ev ancl B. I. Smirnov, Usp. Fiz . Nauk 73,
sense when the grain size is reduced by two orders
of magni t ude. However, one might have expected, 505-558 (March , 1961 ). English Transl: SOY. Phys.
in this case, that even a relatively small sampling Usp. 4, No.2, 226- 259 (Sept.- Oct. 1961).
of 16 grains, as on Faee B of our specimen, might [41 S. Sm ith and W. Wood , Proc. Roy. Soc. A182, 404 (1944).
havc revealed some trace of a consistent trend, and [5] C. J . Newton and Il . C. Vacher, J . Inst. Met. 7, ll93
t his was no t the case. It is believed, therefore, that, (1955) .
while neither hypothesis is clearly tcsted by the l5j D. M . Vasil'ev, Fiz. Tvercl . Tela 1, No. ll , 1736- 1746
results of this illves tigation, thc picture of the cffect (Nov. 1959) . Engli sh Tran sl: Sov. Phys.- Solirl State
of co nstraints upon thc diffracting material when 1, No. ll, 1586-1595 (May 1960).
the gr ain sizc is s mall is the more favored one. [7] M. J . Donachic, Jr. , and J. T. Norton, Trans. Met. Soc.
AIME 221, 962 (1961 ) .
Thc author wishes to acknowledgc the assistance [8] R. I. Garrod and G. A. Hawkes, Brit. J. App!. Phys. 14,
of collertgues at the N ationrtl Burcrtu of Standards, 422- 428 (Jul y 1963 ).
H. C. Vacher, who designed the special specimen 19] B. D. Cu lli ty, Trans. Met. Soc. ATME 227, 356- 358
holder-goniometer, and A. N. Graef, who built it. (April 1963).
[10] C . .T . Newton, J. Res. NBS 67C (Eng. and Instr.) No.2,
10 1- 109 (April- June 1963).
[]]] C. S. Barrett, Structure of M etals, p. 328 (McG raw-Hill
Book Co., Inc. , New York, N.Y., 1952) .
[12] II. C . Vac her, R. Liss, and R . W. Mebs, Acta Metal-
lllrgiea 4, 532- 540 (Sept. 1956).
[13] T. Imll ra, S. Weissmann , and J . J. Slade, Jr. , Acta.
Cryst. 15, 786 (1962).

(Paper 6804- 171)

254