S ess measu emen using a ea de ec o s. A heo e ical and
expe imen al compa ison o diffe en me hods in e i ic s eel
using a po able x- ay appa a us
J. Rami ez-Rico1,∗, S.-Y. Lee2, J.J. Ling2and I. C. Noyan2
1Fisica de la Ma e ia Condensada-ICMS, Uni e sidad de Se illa-CSIC, 41012
Se illa, Spain.
2Applied Physics and Applied Ma hema ics, Columbia Uni e si y, New Yo k, NY
10027
∗Co esponding au ho : [email p o ec ed]
Abs ac
Using a ea de ec o s o s ess de e mina ion by diff ac ion me hods in a single ex-
posu e g ea ly simplifies he measu emen p ocess and pe mi s he design o po able
sys ems wi hou complex sample c adles o mo ing pa s. An addi ional ad an age
is he abili y o see he en i e o a la ge ac ion o he Debye ing and hus de e -
mine ex u e and g ain size effec s be o e analysis. The wo me hods mos commonly
used o ob ain s ess om a single Debye ing a e he so called cos αand ull- ing
fi ing me hods, which employ leas -squa es p ocedu es o de e mine he s ess om
he dis o ion o a Debye ing by p obing a se o sca e ing ec o simul aneously.
The widely applied sin2ψme hod, in con as , equi es sample o a ions o p obe a
diffe en subse o sca e ing ec o o ien a ions. In his pape we fi s p esen a de-
sc ip ion o he diffe en me hods unde he same o malism and using a unified se o
coo dina es ha a e sui ed o a ea de ec o s no mal o he inciden beam, highligh ing
he simila i ies and diffe ences be ween hem. We u he cha ac e ize hese me hods
by means o in-si u measu emen s in ca bon s eel ube samples, using a po able de-
ec o in eflec ion geome y. We show ha , in he absence o plas ic flow, he diffe en
me hods yield basically he same esul s and a e equi alen . An analysis o possible
sou ces o e o s and hei impac in he final s ess alues is also p esen ed.
1 In oduc ion
A ea x- ay de ec o s allow he eco ding o esidual elas ic s ain in a ma e ial along a
se o sca e ing ec o s in a single measu emen , so ha he numbe o sample o a ions
equi ed o de e mina ion o he esidual s ess s a e is significan ly educed and, in some
cases, comple ely elimina ed. This well-known ac o med he basis o adi ional ‘film’
echniques, whe e Debye Ring(s) om a polyc ys alline sample we e cap u ed on x- ay
sensi i e pho og aphic film using labo a o y sys ems [1–3] o po able s ess analyze s
[4]. A comp ehensi e e iew o such ‘single-exposu e’ echniques was gi en by James and
Cohen in 1978 [5]. Wi hin he las decade mode n de ec o s which eco d and(o ) ead ou
inciden x- ay in ensi i es elec onically om he cells o a pixella ed a ea de ec o eplaced
film. Publica ions u ilizing hese new a ea de ec o s epo ed ‘new’ echniques o he
analysis o esidual s esses, whe e e ms like ‘XRD2: Bidimensional X-Ray Diff ac ion’
[6] o ‘2-dimensional x- ay diff ac ion’ [7] we e in oduced. In con as o he con en ional
1
sin2ψ echnique used wi h poin de ec o s, which equi es he measu emen o s ain in
a leas wo dis inc sample o ien a ions o plane s ess s a es [8, 9], wi h a ea de ec o s
he esidual s ess can in p inciple be de e mined in a single exposu e, i espec i e o he
ype o de ec o , o he e ms used o desc ibe i . This ad an age, combined wi h he
minia u iza ion o image pla es, x- ay sou ces and associa ed eading elec onics, ha e
esul ed in comme cial po able appa a us designed o in-line and field measu emen
applica ions, especially o e ous me als [10]. An addi ional ad an age o an a ea de ec o
is he abili y o see he en i e Debye Ring be o e analysis and hus de e mine ex u e and
g ain size effec s [11]. Due o hese benefi s, a ea de ec o s a e specially well sui ed o
pe o ming in-si u measu emen s, ei he a ele a ed empe a u es o as unc ion o applied
s ess, i.e. e e ences [12, 13].
Diffe en app oaches o he de e mina ion o he s ess s a e om he in o ma ion
con ained in a single Debye exposu e exis , such as he cos αme hod in oduced by Sasaki
e . al [14–16] o he di ec leas squa es fi ing o he measu ed s ain [17]. Despi e he
a ac i eness o hese po able de ices in bo h indus ial and scien ific applica ions, he e
is no cu en ag eemen among expe s on he mos efficien , p ecise and accu a e s ess
de e mina ion o malism.
In his wo k, we compa e he h ee p oposed me hods om bo h heo e ical and ex-
pe imen al poin s o iew. A code o he gene a ion o syn he ic 2D x- ay diff ac ion
pa e ns was de eloped and used o s udy he diffe en me hods’ sensi i i y o misalign-
men , de ec o calib a ion pa ame e s as well as sca e in he da a. Fo he expe imen al
assessmen we ha e used a po able x- ay esidual s ess measu emen appa a us (µ-X360
esidual s ess analyze om Puls ec Indus ial Co., L d.) o de e mine he esidual s ess
in 1010 Ca bon S eel cylind ical ubes in-si u du ing ensile loading. The measu emen
condi ions, wi h igh space cons ain s o he de ec o , as well as he use o a non-fla
specimen, we e chosen o app oach hose o ac ual applica ions. Se e al samples we e
es ed h ough he p opo ional limi and he esidual s ess a a ious loads was de e -
mined om single x- ay exposu es, using bo h he cos αand he di ec leas squa es fi ing
me hods. Measu emen s we e pe o med a diffe en sample o ien a ions wi h espec o
he incoming beam o bo h assess i s effec on he calcula ed s ess, as well as o allow us
o use he sin2ψ echnique o compa ison. We emphasize ha ou pu pose is compa ing
he s esses ob ained by hese echniques o each o he , no o a known (applied) s ess.
The la e case would be es ing he accu acy o he echniques. Howe e , po able esid-
ual s ess measu emen de ices a e easy o misalign du ing field measu emen s [10], and
i so, will measu e he w ong s ess alue independen o he echnique used. Ou esul s
show ha , o a mis-aligned machine all echniques will be in e o by he same amoun ,
and in he case o a pe ec ly aligned sample all h ee would yield he co ec s ess.
2 S ess de e mina ion using a ea de ec o s
X- ay based echniques o de e mine he (elas ic) s ain and s ess in a ma e ial ely on
he use o in e -plana spacing in a c ys alline ma e ial as a buil -in s ain-gauge [7–9].
By measu ing he la ice spacing din a s essed sample and compa ing i s alue wi h he
la ice spacing d0o a s ess- ee sample (ideally iden ical o he s essed one in e e y o he
aspec ), we ob ain he p ojec ion o he s ain enso along he sca e ing ec o used in
he de e mina ion o d. Le (ψ, ϕ) be he pola angles ha de e mine he sca e ing ec o
qsin sample coo dina es,
2
qS=

sin ψcos ϕ
sin ψsin ϕ
cos ψ

=

q1
q2
q3

(1)
Then he s ain measu ed along he sca e ing ec o a angles (ψ, ϕ) can be exp essed
in e ms o he s ain componen s ϵij in sample coo dina es as:
ϵψϕ = (dψϕ −d0)/d0=qiqjϵij =ϵ11 cos2ϕsin2ψ+ϵ12 sin 2ϕsin2ψ
+ϵ22 sin2ϕsin2ψ+ϵ33 cos2ψ(2)
+ϵ13 cos ϕsin 2ψ+ϵ23 sin ϕsin 2ψ
Due o he shallow pene a ion o x- ays in mos ma e ials, ypically only nea -su ace
s esses can be measu ed, whe e due o bounda y condi ions he no mal s esses anish o
ze o a he su ace [8]. An addi ional hypo hesis is he absence o s eep s ain g adien s
[18, 19]. I ha is no he case, a mo e de ailed ea men is needed [20]. I is hen
cus oma y o assume a biaxial (o plane s ess) s ess s a e so ha :
σij =

σ11 σ12 0
σ12 σ22 0
0 0 0

=⇒ϵij =

ϵ11 ϵ12 0
ϵ12 ϵ22 0
0 0 ϵ33

(3)
We will u he assume ha he ma e ial’s elas ic beha io can be desc ibed as iso opic
and homogeneous, so Hooke’s law akes he o m:
ϵij =1 + ν
Eσij −δij
ν
Eσkk (4)
Whe e Eins ein’s summa ion con en ion is used h oughou . Wi h hese assump ions
eq. (2) simplifies o:
ϵψϕ =dψϕ −d0
d0
=(ϵ11 cos2ϕ+ϵ12 sin 2ϕ+ϵ22 sin2ϕ−ϵ33)sin2ψ+ϵ33
O , in e ms o s esses:
dψϕ −d0
d0
=1 + ν
Eσϕsin2ψ−ν
E(σ11 +σ22) (5)
Whe e σϕ=σ11 cos2ϕ+σ12 sin 2ϕ+σ22 sin2ϕ. Equa ion (5) is he basis o he sin2ψ
me hod, ha has been used o o e 50 yea s as he x- ay me hod o choice o s ess
de e mina ion [8].
Now we need o in oduce he goniome e con en ions ha will ela e he o ien a ion
o he sca e ing ec o in labo a o y (diff ac ome e ) coo dina es wi h hose in sample
coo dina es (Figu e 1). We will conside a 2D x- ay a ea de ec o o ien ed so i is no mal
o he incoming (p ima y) beam and hus eco ds backsca e ed beams diff ac ed om
he sample (2θ > π/2). Diff ac ed x- ays will, o a ce ain amily o la ice planes, o m
cones ha will in e sec he de ec o o ming ci cles o Debye ings. The semi-angles o
hese cones will be defined as 2η=π−2θ, and he pola angle along each o he Debye
ings will be no ed as α. Thus, o each poin in he de ec o wi h coo dina es (x, y), he
equi alen cone coo dina es can be de e mined using [7]:
3
an 2η=√(x−x0)2+ (y−y0)2
z0
, an α=x−x0
y−y0
(6)
Whe e (x0, y0) desc ibes he loca ion o he inciden x- ay beam in de ec o coo dina es,
while z0is he sample o de ec o dis ance. The sca e ing ec o co esponding o each
poin in he Debye ing also esides in a cone, in his case o semi-angle η. Since each
poin in he ing co esponds o a diffe en o ien a ion o he sca e ing ec o , he e mus
exis a ans o ma ion ha ela es i s coo dina es qLin he lab sys em o hose qSin he
sample sys em.
qL=

−cos η
−sin ηsin α
−sin ηcos α

=

−sin θ
−cos θsin α
−cos θcos α

=⇒qS=

sin ηcos α
sin ηsin α
cos η

=⇒(ψ, ϕ) = (η, α) (7)
[Figu e 1 abou he e.]
We will now conside sample o a ions (ω0, ψ0, ϕ0) defined acco ding o Figu e 1. The
ans o ma ion ma ix be ween he wo e e ence sys ems is hen:
A=

−cos ω0sin ψ0cos ϕ0+ sin ω0sin ϕ0sin ω0sin ψ0cos ϕ0+ cos ω0sin ϕ0−cos ψ0cos ϕ0
−cos ω0sin ψ0sin ϕ0−sin ω0cos ϕ0sin ω0sin ψ0sin ϕ0−cos ω0cos ϕ0−cos ψ0sin ϕ0
−cos ω0cos ψ0sin ω0cos ψ0sin ψ0


(8)
And so we can ans o m he coo dina es by simple ma ix mul iplica ion.
qS=AqL=

a11 a12 a13
a21 a22 a23
a31 a32 a33



−cos η
−sin ηsin α
−sin ηcos α

=

sin ψcos ϕ
sin ψsin ϕ
cos ψ

=

q1
q2
q3

(9)
I is ins uc i e o conside some special cases o sample o a ions (ω0, ψ0, ϕ0) ha lead
o adi ional sca e ing geome ies commonly used in x- ay diff ac ion expe imen s.
2.1 No mal incidence
We conside now he case whe e ω0=ψ0=ϕ0= 0. Now he p ima y x- ay beam is
no mal o he sample su ace, and hus sample and de ec o planes a e pa allel. Now, in
sample coo dina es he sca e ing ec o is simply:
qS=

cos αsin η
sin αsin η
cos η

(10)
And hus we ha e ψ=η, ϕ =α. In his geome y he pola angle ψis fixed and he
azimu hal angle ϕis equi alen o he pola angle along he Debye ing.
2.2 Con en ional θ−θgeome y
In a con en ional θ−θexpe imen , he inciden beam makes an angle θwi h he sample
su ace and i ψ0= 0 he sca e ing ec o is no mal o he sample su ace, oo. This
implies ha ω0=η. In his geome y, he p ima y and sca e ed beam lie in a plane
con aining he sca e ing ec o when ψ= 0. In a wo dimensional ec o , his condi ion
is ulfilled a α=−π/2, and we ge qS= [001].
4
2.3 A bi a y ψ0 il wi h ω0= 0
This is he case mos ele an o ou pu poses, since i desc ibes he geome y in he
po able x- ay s ess measu emen de ice u ilized in his s udy. Now he sca e ing ec-
o in sample coo dina es can be w i en in e ms o sample o a ions and Debye ing
coo dina es as:
qS=

cos ηsin ψ0cos ϕ0+ sin ηcos ψ0cos ϕ0cos α−sin ηsin ϕ0sin α
cos ηsin ψ0sin ϕ0+ sin ηcos ψ0sin ϕ0cos α+ sin ηcos ϕ0sin α
cos ηcos ψ0−sin ηsin ψ0cos α

(11)
We mus emembe ha , in gene al, (ψ, ϕ)= (ψ0, ϕ0), and he ela ionship o pe o m
he ans o ma ion (η, α)→(ψ, ϕ) is a he complex. Two special cases can be desc ibed
a angles α= 0, π, as can be seen in Figu e 2: a α= 0 we ha e ψ=ψ0+η, while a
α=πwe ha e ψ=ψ0−η. This means ha , o his geome y, he poin s in he Debye
ing a α= 0, π can be used di ec ly in he sin2ψme hod once he ηangle is accoun ed
o .
[Figu e 2 abou he e.]
2.4 The cos αme hod
One o he me hods ha allows he calcula ion o s ess om a single measu emen wi h
an a ea de ec o is he cos αme hod desc ibed by Sasaki e al. and o he s [14, 21] and
implemen ed in comme cial ins umen a ion o esidual s ess de e mina ion. The basic
idea is o s a om he exp ession o he sca e ing ec o in sample coo dina es in e ms
o Debye ing coo dina es, in he case o a bi a y ψ0. Fo simplici y we will es ic
ou sel es o he sample-de ec o geome y depic ed in Figu e 1, whe e he incoming x- ay
beam is no mal o he fla 2D de ec o and only a o a ion along S2in sample coo dina es
is allowed. Fo a s ess- ee sample he sca e ing ec o s lie along cones wi h semi-apex
angle η=θ−π/2 o each se o la ice planes ulfilling B agg’s condi ion, p oducing a
se ies o concen ic Debye ings. Fo e e y poin in he de ec o wi h coo dina es (x, y)
qS=

cos ηsin ψ0+ sin ηcos ψ0cos α
cos ηsin ψ0sin ϕ0+ sin ηcos ψ0sin ϕ0cos α+ sin ηcos ϕ0sin α
cos ηcos ψ0−sin ηsin ψ0cos α

(12)
The s ain p ojec ion along (η, α) coo dina es could be w i en in e ms o he sca e -
ing ec o and s ain componen s as ϵα=qiqjϵij. We will assume plane-s ess condi ions
and conside an elas ic, iso opic and homogeneous ma e ial such ha :
ϵij =1 + ν
Eσij −δij
ν
Eσkk (13)
We can subs i u e o find he s ain p ojec ion in e ms o s ess componen s:
ϵα=1 + ν
Eqiqjσij −ν
Eσkk (14)
Now we define pa ame e s a1, a2as:
a1(ϕ0) = 1
2{[ϵ(α)−ϵ(π+α)] + [ϵ(−α)−ϵ(π−α)]}(15)
a2(ϕ0) = 1
2{[ϵ(α)−ϵ(π+α)] −[ϵ(−α)−ϵ(π−α)]}(16)
5

F om he e we can now ob ain:
a1(ϕ0) =1 + ν
Esin 2ψ0sin 2ηcos α[σ11(1 + cos 2ϕ0) + σ22(1 −cos 2ϕ0) + 2σ12 sin 2ϕ0)]
(17)
a2(ϕ0) =1 + ν
Esin ψ0sin 2ηsin α[σ22 sin 2ϕ0−σ11 sin 2ϕ0+ 2σ12 cos 2ϕ0)] (18)
And we now conside he special case o ϕ0= 0 o ge :
a1(ϕ0= 0) = 1 + ν
Eσ11 sin 2ψ0sin 2ηcos α(19)
a2(ϕ0= 0) = 21 + ν
Eσ12 sin ψ0sin 2ηsin α(20)
So by plo ing he pa ame e s a1, a2as a unc ion o cos αand sin αwe ob ain wo
linea ela ionships, he slopes o which will be p opo ional o σ11 and σ12, espec i ely.
2.5 Full Debye ing fi ing
Fo simplici y, le us con inue assuming ha ω0= 0, so he sca e ing ec o can s ill be
w i en as:
qS=

cos ηsin ψ0cos ϕ0+ sin ηcos ψ0cos ϕ0cos α−sin ηsin ϕ0sin α
cos ηsin ψ0sin ϕ0+ sin ηcos ψ0sin ϕ0cos α+ sin ηcos ϕ0sin α
cos ηcos ψ0−sin ηsin ψ0cos α

(21)
As we ha e seen, he e is a linea ela ionship be ween he expe imen ally de e mined
s ain p ojec ion ϵαand he s ain componen s in he sample coo dina es ϵij, whe e he
linea coefficien s qia e simply he componen s o he sca e ing ec o , meaning ha :
ϵα=qiqjϵij (22)
Now he qi’s depend only on sample geome y pa ame e s and Debye coo dina es
ψ0, ϕ0, η, α and a e hus known, so equa ion (22) ep esen s a se o linea equa ions o
each pola angle αa which he s ain is de e mined. The ou unknowns a e he ϵij (in
plane s ess ϵ13 =ϵ23 = 0) and he independen e ms ϵαa e de e mined expe imen ally.
Since he numbe o α alues p obed is mo e han ou ( ypically 360 a leas ), he sys em
is o e de e mined and can be nume ically sol ed in he leas -squa es sense.
Expe imen ally, he Debye ing is fi ed o ob ain 2θ(α) and hen he s ain is calcula ed
using:
ϵα= ln (sin θ0
sin θ(α))(23)
2.6 Effec o diff ac ion olume
Fo diff ac ion o occu , he inciden and diff ac ing beam ec o s and he no mal o he
diff ac ing planes mus be co-plana , wi h he no mal o he diff ac ing planes making equal
angles wi h he wo wa e ec o s. Consequen ly dis inc g oups o g ains, wi h pa allel
diff ac ing plane no mals, con ibu e o he diff ac ion p ofiles ob ained a he a ious
ψ, ϕ, η, α coo dina es. We define he o al olume o such a se o g ains as he in o ma ion
6
olume o hese angula coo dina es. All o mula ions discussed abo e o compu ing he
s esses om diff ac ion da a assume ha he same quasi-homogeneous bi-axial s ess
enso exis s wi hin all diff ac ing olumes sampled by all echniques. This is no a s ong
assump ion and mus be e ified expe imen ally on a case-by-case basis. Fo example,
his assump ion is jus ified in sho -peened samples whe e, because o he andom plas ic
de o ma ion field caused by mul iple impac s, any p e-exis ing c ys allog aphic ex u e is
des oyed in he peened laye , and he same mac os ess dis ibu ion is ob ained om all
eflec ions and all accessible ψ il s o a gi en eflec ion. Howe e , his is no he case o
samples plas ically de o med in uniaxial ension [8]. In such cases, he esul s om he
h ee o mula ions migh no ag ee.
3 Ma e ials and Me hods
3.1 Expe imen al se up
A minia u e po able 2D x- ay appa a us (Puls ec µ-X360, Puls ec Co p., Japan) was used
o de e mine esidual s esses in ca bon s eel ubula samples as a unc ion o applied load.
This appa a us has been desc ibed elsewhe e [10]; in b ie , i consis s o a low in ensi y
C Kαsou ce wi h a fil e and an image pla e 30 mm in adius posi ioned no mal o he
inciden beam. Unde hese condi ions, back-diff ac ed beams om he sample su ace
a e cap u ed by he image pla e a 50 µm esolu ion. Beam spo size is app oxima ely 2
mm o he de aul 1 mm pinhole collima o , and angula co e age o a 40 mm sample o
de ec o dis ance is 2θ= 145◦−175◦.
An Ins on 5984 Uni e sal Tes ing Machine was used o apply a ensile load o he
specimens ia pneuma ic g ips. The machine was ope a ed in load con ol and s ain was
eco ded using a clip-on ex ensome e . Nominal loads co esponding o s esses in he
ange o 0 - 275 MPa we e applied in 25 MPa inc emen s, and a each load one x- ay
diff ac ion pa e n was acqui ed. Once a a ge load was eached i was held o he o al
du a ion o he x- ay measu emen , which in ou case amoun ed o 40 s o exposu e plus
50 s o image pla e eadou and p ocessing ime.
3.2 Samples
Samples s udied in his wo k we e seamless ubes 25.4 mm in diame e and wi h a wall
hickness o 1.24 mm, con o ming o ASTM 179 and SAE J524 s anda ds. The ube
ma e ial was e y low ca bon s eel con o ming o AISI 1010 specifica ions, which was cold
d awn and hea ea ed a 650 oC. The mic os uc u e o his ma e ial is p esen ed in
Figu e 3: as expec ed om he composi ion, i consis ed o p ima y e i e wi h small,
isola ed islands o pea li e. Fe i e g ain size was de e mined o be (12 ±2)µm by he line
in e cep me hod.
Samples we e checked o ex u e and phase composi ion by using a labo a o y diff ac-
ome e using Cu- adia ion wi h a g aphi e monoch oma o (figu e S1 o he supplemen-
a y in o ma ion): no significan ex u e was de ec ed (see figu e S2). Due o he e y
low cemen i e con en , no Fe3C peaks we e obse ed. The use o ubula samples in o-
duced an addi ional se o alignmen difficul ies, when compa ed wi h he mo e ypical
fla specimens, ha we e in ou opinion mo e ep esen a i e o ac ual applica ions whe e
complex geome ies, igh space cons ain s and alignmen difficul ies a e o be expec ed
(see Figu e S3 in he supplemen a y in o ma ion o ac ual pic u es o he se up).
[Figu e 3 abou he e.]
7
3.3 Da a analysis and s ess calcula ion
In e nal so wa e supplied by he x- ay appa a us was used o compu e de ec o pa ame e s
(beam cen e and sample- o-de ec o dis ances) a ze o ex e nal load. This app oach
e e ed all subsequen measu emen s o his da um and ensu ed ha , as long as he
applied load s ayed below he elas ic limi (and, consequen ly, did no change any ini ial
esidual s esses), subsequen x- ay analysis would only yield he (uniaxial) applied s ess
alues. Nominal de ec o - o-sample dis ances we e in he 35-50 mm ange, which esul ed
in only one Debye ing, om he (211) eflec ion o he BCC e i e phase, being eco ded.
We used a = 2.8664 ˚
A as he uns essed la ice pa ame e a ze o load o calcula ing he
sample o de ec o dis ance.
De ec o il and/o dis o ion we e de e mined by fi ing he Debye ing o an ellipse
using he hypo hesis ha , o a esidual s ess- ee sample in he absence o applied load,
any ellip ici y in he Debye ing mus be due o de ec o o a ion. Since a de ia o ic s ain
enso would also esul in a de o med, ellip ical Debye ing, we fi ed he ings ob ained
a ze o applied s ess using he ull- ing p ocedu e desc ibed p e iously and ob ained a
‘fic i ious s ain’ ( he s ain s a e ha would esul in he same de o ma ion o he Debye
ing as obse ed due o de ec o il ) ha was used o co ec subsequen ly de e mined
s ains unde load.
Fo all s ess calcula ions, he x- ay elas ic cons an , 1+ν
E, was aken o be 5.71 TPa-1.
This alue was compu ed om he quasi-iso opic elas ic pa ame e s o e i e, E=
224 GPa and ν= 0.28 . U iliza ion o Nee eld-Hill o K oene x- ay elas ic cons an s, com-
pu ed om single-c ys al compliances o he (211) e i e eflec ion ( 5.94 and 6.21 TPa-1,
espec i ely) would ha e uni o mly shi ed all calcula ed s ess alues by app oxima ely
4% and 8% . Since he e m 1+ν
Eis a mul iplie in all h ee o malisms, compa ing he
equi alency o hei s ess ou pu is independen o he selec ion o elas ic cons an s.
Two se s o da a we e acqui ed o each o he h ee diffe en beam incidence angles
s udied, a ψ0= 0◦,35◦,45◦. This allowed us o s udy he effec o beam incidence angles
on s ess calcula ion, and also o ob ain a se o s ain alues a diffe en ψso as o be
able o use he sin2ψme hod o compa ison. Nominal ψ alues s udied we e 11.8, 23.2,
33.2, 46.8 and 56.8o; addi ionally he ex ensome e p o ided he s ain a ψ= 90◦.
A code sol ing equa ion (22) in he leas squa es sense was implemen ed in MATLAB.
A sligh modifica ion o he fi ing equa ion was pe o med so he a ia ion o sca e -
ing angle θ(α) wi h he pola coo dina e was pe o med di ec ly wi hou equi ing he
specifica ion o a s ess- ee e e ence angle θ0:
ln (sin θ(α)) = ln(sin θ0)−qiqjϵij (24)
This has he ad an age ha he s ess- ee angle θ0is now he independen e m and
can hus be ob ained as pa o he leas -squa es fi . The p e ious equa ion is, in heo y,
alid o he gene al case o a iaxial s ess s a e, howe e some simplifica ions can be
in oduced o plane s ess condi ions, namely ϵ13 =ϵ23 = 0 and ϵ33 =−ν(ϵ11 +ϵ22),
whe e νis he Poisson’s a io o he ma e ial.
The p oduced code was es ed by gene a ing syn he ic da a se s, adding no mal noise
and hen fi ing he esul ing da a o ob ain he s ess componen s and compa e wi h he
ini ial alues. Addi ionally, he impac o e o s in he de e mina ion o some o he inpu
pa ame e s (sample o a ion) in he final s ess esul was assessed by means o hese
simula ions.
8
4 Resul s and Discussion
Figu e 4 shows he aw in o ma ion om he image pla e, showing some spo iness o he
Debye ing, also e idenced in he a ia ion o he (211) Fe peak a ea along he Debye ing
and, o a lesse ex en , in he change in FWHM o he peak. The e a e also a ia ions in
he expe imen al 2θ s αp ofiles as can be seen in figu e 5, which shows he 2θposi ion o
he peaks as a unc ion o α, along he Debye ing, o es s pe o med a ψ0= 35◦,45◦.
The measu ed p ofiles exhibi sligh sys ema ic undula ions a ound he expec ed alues:
hese ‘undula ions’ a e along he αangle and a e no ela ed o he well-known oscilla ions
in sin2ψ[22–26], bu a e due o he (p op ie a y) pixel in ensi y-da a analysis algo i hm
in his pa icula machine; hese algo i hms a e inaccessible o he use s. We also obse ed
hese undula ions in da a measu ed om he calib a ion powde s (annealed e i e powde
wi h g ain size a ound 1 µm) supplied by he sys em manu ac u e [10]. Peak a ea plo s
as a unc ion o azimu hal angle (Figu e 4, lowe le panel) we e uni o m and allowed us
o discoun effec s o c ys allog aphic ex u e.
[Figu e 4 abou he e.]
[Figu e 5 abou he e.]
As we saw in sec ion 2.3 (and Figu e 2), a single exposu e ha co e s he whole Debye
ing also allows o he de e mina ion o s ess using he sin2ψme hod, al hough only
using he wo poin s ha co espond o ϕ= 0, namely hose a α= 0,180◦. In his
case he in o ma ion olume o he wo- il sin2ψme hod is a subse o he in o ma ion
olume o he Debye ing analysis. In ligh o his ac , we used diff ac ion da a om a
single exposu e o calcula e he a e age axial s ess and s anda d de ia ion using bo h a ea
de ec o -based single exposu e echniques, and he wo- il sin2ψme hod. This app oach
somewha alle ia ed he sampling issues a ising om he use o mul iple ψ il s, wi h
comple ely independen in o ma ion olumes. A compa ison be ween he sin2ψand he
ull Debye ing me hod can be seen in Figu e 5: in addi ion o expe imen al Debye ing
p ofiles a diffe en s esses, we plo simula ed Debye ings whe e he inpu s esses alue
is ob ained om he sin2ψme hod. Reasonable ag eemen is obse ed.
Figu e 6 shows axial (σxx) s esses calcula ed using single exposu e me hods (ei he
leas squa es fi ing o he en i e Debye ing o cos α) agains s esses calcula ed using
he con en ional ( wo- il ) sin2ψapp oach. Black symbols ep esen a e ages o e ou
measu emen s a ψ0= 35,45◦and e o ba s ep esen ±one s anda d de ia ion. Fo
comple eness, indi idual measu emen s a e plo ed as well, using ed symbols (hollow o
ψ0= 35◦and filled o ψ0= 45◦). The dashed line, wi h slope o 1, deno es he locus o
iden ical alues. The ollowing obse a ions can be made:
1-The p ecision o ‘ eal’ esidual s ess measu emen s on an engine ing sample, ob-
ained unde condi ions which a e simila o hose encoun e ed du ing ypical field p ac-
ice, is wo se, by an o de o magni ude, han he alues quo ed o ‘ideal’ samples, i.e,
±20 s. ±2 MPa. This is consis en wi h pas esidual-s ess measu emen p ac ice using
labo a o y and field ins umen s. We again no e ha he sca e in he measu ed da a in
his figu e is ela ed o he p ecision (no accu acy) o he pa icula sys em (ins umen ,
sample, measu emen condi ions).
2-The s ess alues ob ained om bo h a ea-de ec o based single exposu e echniques
a e, wi hin e o , equal o he wo- il sin2ψ o mula ion o applied s esses in he elas ic
ange (Figu e 7). Fo highe applied s esses we obse e a sys ema ic de ia ion o he
s esses measu ed by he single exposu e echniques om he dashed line deno ing 1- o-
1 co espondence wi h he wo- il sin2ψ o mula ion. As no ed be o e, his de ia ion
9
In e sec ion o he
cone o no mals
X- ay beam
Figu e 2: S e eog aphic p ojec ion sample coo dina es (S1, S2, S3) o he case o sample
o a ions wi h ω0=ϕ0= 0. In his case he sca e ing ec o s laying a α= 0, π
co espond o angles ψ=ψ0±ηin sample coo dina es.
16

Figu e 3: Mic os uc u e o he 1010 ca bon s eel samples s udied in his wo k: op ical
mic oscopy (le ) and scanning elec on mic oscopy ( igh ).
17
Figu e 4: Raw da a om he image pla e a ze o applied s ess, o he case o ψ0= 35◦.
Top: no malized diff ac ion in ensi y as a unc ion o 2θand α. Bo om-le : peak a ea
along he Debye ing. Bo om- igh : peak whm along he Debye ing.
18
0 60 120 180 240 300 360
156.1
156.2
156.3
156.4
156.5
0 60 120 180 240 300 360
156.0
156.1
156.2
156.3
156.4
25 MPa
50 MPa
75 MPa
10 0 MPa
12 5 MPa
15 0 MPa
17 5 MPa
20 0 MPa
22 5 MPa
25 0 MPa
27 5 MPa
2 (º)
(º)
0
= 35 º
0
= 45 º
2 (º)
(º)
Figu e 5: Debye ing adius (2θ) as a unc ion o azimu hal angle α o he e i e (211)
eflec ion, a diffe en loads. Symbols co espond o expe imen al da a while solid lines
a e calcula ed Debye ing p ofiles using he s ess ob ained om he sin2ψme hod.
19
0 50 100 150 200 250
0
50
100
150
200
250
35º 45º
cos
ull-2d
cos a e age
Full-2d a e age
Full-2d o cos measu ed s ess (MPa)
sin
2
measu ed s ess (MPa)
Figu e 6: S esses calcula ed om single exposu es ( ull-2d, cos α s sin2ψs ess. Black
symbols ep esen a e ages o e ou measu emen s a ψ0= 35,45◦and e o ba s ep e-
sen one s anda d de ia ion. Red symbols (hollow o ψ0= 35◦and filled o ψ0= 45◦)
co espond o indi idual measu emen s.
20
0.0000 0.0004 0.0008 0.0012 0.0016
0
50
100
150
200
250
300
350
S ess (MPa)
T ue S ain
Load cell
Ex apola ed elas ic s ess
0
10
20
30
40
(MPa)
210 GPa
Figu e 7: Uniaxial ensile es da a o he 1010 s eel ube sample used in he s udy. The
ex apola ed elas ic s ess line is calcula ed om leas -squa es fi ing o he 0-50 MPa load
ange, which yielded 210 GPa as he (effec i e) Young’s Modulus. The ∆σ alues ( igh
o dina e) show he diffe ence be ween he ex apola ed elas ic s ess and he ac ual load
cell ou pu .
21

155.5
156
156.5
157
30
210
60
240
90 270
120
300
150
330
180
0
0100 200 300
155.9
156
156.1
156.2
156.3
156.4
156.5
α(º)
2θ (º)
ǻ(2ș)=0.1º
155
156
157
158
30
210
60
240
90 270
120
300
150
330
180
0
0 100 200 300
156.05
156.1
156.15
156.2
156.25
156.3
156.35
156.4
α(º)
2θ (º)
ǻ(2ș)=0.01º
Figu e 8: Simula ed Debye ings and fi ing esul s o he ull- ing fi ing me hod, a e
adding gaussian noise.
22
10-3 10-2 10-1
10-2
10-1
100
101
102
103
-4 -2 0 2 4
-50
-25
0
25
50
75
100
125
150
∆σxx
∆σxy
∆σyy
∆σij
(MPa)
∆(2θ) (deg ees)
∆σyy
Calcula ed
σij
(MPa)
∆ψ0(deg ees)
ψ0=15º
ψ0=30º
ψ0=45º
∆σxx
Figu e 9: Top: E o in he de e mina ion o s ess as a unc ion o e o in 2θ(as one
s anda d de ia ion). The hollow symbols a e o he ull- ing fi ing me hod while he
filled symbols a e o he cos α. Bo om: s ess calcula ed as a unc ion o e o in sample
o ien a ion (see ex o de ails), o a uniaxial s ess s a e wi h σxx = 100MPa and o
diffe en ψ0 alues.
23