Dimensionless numerical analysis of cracked materials by means of infrared lock-in thermography

MASTER’S DEGREE IN
Space Science and Technology
MASTER’S THESIS
DIMENSIONLESS NUMERICAL ANALYSIS OF
CRACKED MATERIALS BY MEANS OF
INFRARED LOCK-IN THERMOGRAPHY
S uden Saga duy Ma cos, Da id
Supe iso Rod íguez Aseguinolaza, Ja ie
Depa men Applied physics
Academic yea 2021-2022
Bilbao, June 24 h o 2022
Acknowledgmen s
I would like o hank o he Pho o he mal Technique Labo a o y esea ch g oup o all he help
hey ha e gi en me, wi h special men ion o my supe iso Ja ie o he dedica ion and e o he has
pu in o his wo k. The many hou s we ha e spen ying o deciphe he bash sc ip s and hei en ails
a e now emembe ed as he i anic e o ha has been made.
In hese lines I canno o ge my pa en s. I would like o hank hem o suppo ing me om
he e y beginning and gi ing me solu ions when I didn’ see hem (e en when I hough he e we e
none!). Al hough only my name appea s in he au ho ship o his wo k, i is as much mine as hei s.
Las bu no leas , I would like o hank my g andpa en s o he cons an encou agemen I ha e
ecei ed om hem o e he yea s. Wi hou i his would ha e been a much ha de oad.
i
Abs ac
In his mas e ’s hesis a dimensionless nume ical model o he lock-in in a ed he mog aphy ex-
pe imen o he de ec ion o open-su ace c acks is de eloped. S a ing wi h he cons i u i e equa ion
and he bounda y condi ions associa ed wi h he lase and he c acks, hei dimensions a e emo ed
by in oducing leng h, ime and empe a u e scales ela ed o he physical p oblem. As a esul , a se
o dimensionless pa ame e s is ob ained, ha allows o gi e a b oade ision o he p oblem, while
main aining he ma hema ical simplici y o he dimensional model.
Once he dimensionless equa ions a e ob ained, hey a e implemen ed in a ini e elemen me hod
mul iphysics so wa e called OpenFoam. Wi h his objec i e, in his wo k he ull nume ical calcu-
la ion p ocess has been de eloped: he p e-p ocessing (o meshing), p ocessing and pos -p ocessing
s ages.
A e implemen ing he equa ions in he ini e elemen me hod so wa e a pa ame ic analysis
has been pe o med by means o di e en simula ions in o de o analyze he e ec o each o he
dimensionless pa ame e in he esul ing ampli ude he mog am.
Key wo ds: Dimensionless equa ions, lock-in in a ed he mog aphy, ini e elemen me hods,
OpenFoam, pa ame ic analysis.
ii
Resumen
En es e abajo in de m´
as e se desa olla un modelo num´
e ico adimensional del expe imen o de
e mog a ´
ıa in a oja modulada pa a la de ecci´
on de g ie as supe iciales. Pa iendo de la ecuaci´
on
cons i u i a y de las condiciones de con o no asociadas al l´
ase y a la g ie a, se lle a a cabo el p oceso
de adimensionalizaci´
on in oduciendo escalas de longi ud, iempo y empe a u a, asociadas al p ob-
lema ´
ısico. Como esul ado se ob iene un conjun o de pa ´
ame os adimensionales que pe mi e da
una isi´
on m´
as amplia al p oblema, man eniendo a su ez la sencillez ma em´
a ica del plan eamien o
dimensional.
Una ez adimensionalizadas, es as ecuaciones se implemen an en un so wa e mul i ´
ısico de
m´
e odos de elemen os ini os llamado OpenFoam. Pa a ello en es e abajo se con emplan las es
e apas de c´
alculo que se han seguido: el p e-p ocesado (o mallado), p ocesado y pos -p ocesado.
T as implemen a las ecuaciones en el so wa e del m´
e odo de elemen os ini os se ealiza un
an´
alisis pa am´
e ico median e di e en es simulaciones con el in de analiza el e ec o de cada uno de
los pa ´
ame os adimensionales en el e mog ama de ampli ud esul an e.
Palab as cla e: Ecuaciones adimensionales, e mog a ´
ıa in a oja modulada, m´
e odos de ele-
men os ini os, OpenFoam, an´
alisis pa am´
e ico.
iii
Labu pena
Mas e amaie ako lan hone an, gainazale a i eki ako pi zadu ak de ek a zeko e mog a ia in ago i
modula uko espe imen ua en zenbakizko e edu adimen sional ba ga a zen da. E a ze-ekuazio ik
e a lase a en e a pi zadu ak sa u ako mugalde baldin ze a ik abia u a, adimen sionalizazio p oze-
sua gauza zen da, a azo isikoa i lo u ako luze a, denbo a e a enpe a u a eskalak sa uz. Ondo ioz,
pa ame o adimen sionalen mul zo ba lo zen da, a azoa i ikuspegi zabalagoa ema eko auke a ema en
duena, e a, aldi be ean, planeamendu dimen sionala en sinple asun ma ema ikoa i eus en diona.
Dimen sioak ezaba u ondo en, ekuazio ho iek OpenFoam izeneko elemen u ini uen me odoen
so wa e mul i isikoan inplemen a zen di a. Ho e a ako, lan hone an ja ai u di en hi u kalkulu-
e apak jaso zen di a: au e-p ozesa zea (edo sa e zea), p ozesa zea e a p ozesa u os ekoa.
Elemen u ini uen me odoa en so wa ean ekuazioak inplemen a u ondo en, analisi pa ame iko
ba egi en da hainba simulazio en bidez, pa ame o adimen sional bakoi zak anpli ude- e mog aman
duen e agina az e zeko.
Gako hi zak: Ekuazio adimen sionalak, e mog a ia in ago ia modula ua, elemen u ini uen
me odoak, OpenFoam, analisi pa ame ikoa.
i

Con en s
1 In oduc ion 1
1.1 Con ex ......................................... 1
1.2 Objec i es........................................ 3
2 Expe imen al backg ound 4
3 Theo e ical model 6
3.1 Dimensionlessequa ions ................................ 6
3.2 Dimensionlesspa ame e s ............................... 9
4 Compu a ional nume ical simula ions 12
4.1 OpenFoam........................................ 12
4.2 Nume icalschemes................................... 12
4.3 Solu ion and algo i hm con ol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.4 Themesh ........................................ 14
4.4.1 In ini ec acks.................................. 15
4.4.2 Semi-in ini ec acks .............................. 16
4.4.3 Fini ec acks .................................. 17
4.5 Mesh esolu ioncon ol................................. 18
4.5.1 Mesh elemen dis ibu ion g adien . . . . . . . . . . . . . . . . . . . . . . 18
4.5.2 Mesh e inemen ................................ 21
4.6 P ocessing........................................ 22
4.7 Pos -P ocessing..................................... 22
5 Resul s 23
5.1 Valida iono hemodel................................. 23
5.2 Pa ame icanalysis ................................... 25
6 Discussion 31
6.1 Valida iono hemodel................................. 31
6.2 Pa ame icanalysis ................................... 32
6.2.1 Π1........................................ 32
6.2.2 Π2........................................ 32
6.2.3 Π3........................................ 32
6.2.4 ¯y0........................................ 33
6.2.5 Πw,Πl,Πdand θ................................ 33
6.2.6 Non-dimensional pa ame e combina ions . . . . . . . . . . . . . . . . . . . 34
7 Conclusions 37
Re e ences 38
Lis o Figu es
1 Typical labo a o y lock-in IR he mog aphy expe imen al se up. . . . . . . . . . . . 4
2 Scheme o he e olu ion o he su ace empe a u e o he sample wi h ha monic lase
hea ing. ......................................... 5
3 Scheme o he he mal wa es in di e en poin s o he sample. . . . . . . . . . . . . 5
4 (le ) Hexahed al and ( igh ) e ahed al meshes. . . . . . . . . . . . . . . . . . . . . 14
5 (le ) Te ahed al and ( igh ) hexahed al mesh elemen s. . . . . . . . . . . . . . . . . 14
6 Schemeo anin ini ec ack............................... 16
7 (a) The wo blocks ha make up he in ini e c ack and (b) he inal esul . . . . . . . 16
8 (a) The ou blocks ha make up he semi-in ini e c ack and (b) he inal esul a e
joining hem. ...................................... 17
9 (a) La e al iew in he +¯xdi ec ion showing he co esponding ou blocks and (b)
zeni al iew in he +¯zdi ec ion showing he co esponding h ee blocks. . . . . . . . 17
10 Final esul a e joining he co esponding wel e blocks. . . . . . . . . . . . . . . . 18
11 Diag am o he spacing be ween mesh edges. Le : egula dis ibu ion. Righ : down-
wa dspa ialg ading. .................................. 18
12 (a) Th ee block mesh wi hou g adien and (b) same mesh wi h γ= 1 calcula ed by
(40)............................................ 20
13 (a) Single box, (b) iple cylinde e inemen s and (c) bo h o hem combined. . . . . 21
14 Tempe a u e e alua ion o an a bi a y poin o e cycling un il he s a iona y s abili y
c i e ionissa is ied.................................... 22
15 Compa ison be ween he non dimensional nume ical esul s and analy ical solu ion
o in ini e e ical c acks. (a) AISI 304 and (b) Cu. . . . . . . . . . . . . . . . . . . 23
16 Compa ison be ween he non dimensional esul s and he DG FEM model in he semi-
in ini e c ack case in AISI 304 o wo di e en c ack inclina ions. (a) 90º and (b) 45º. 24
17 Compa ison be ween he non dimensional esul s and he DG FEM model in he ini e
c ack case in AISI 304 o wo di e en c ack inclina ions. (a) 90º and (b) 45º. . . . . 24
18 Na u al loga i hm plo o he he mal ampli ude (a) non-no malized and (b) no mal-
ized on he ans e se sample p o ile o Π1= 4 ×103,1×105,2×105,3×105.
............................................. 26
19 Na u al loga i hm plo o he he mal ampli ude (a) non-no malized, showing he
zoomed ampli ude jump associa ed o he c ack in he inse and (b) no malized on
he ans e se sample p o ile o Π2= 1 ×10−7,2×10−7,3×10−7,1×10−8. . . . 26
20 Na u al loga i hm plo o he he mal ampli ude (a) non-no malized and (b) no mal-
ized on he ans e se sample p o ile o Π3= 50,500,1000,4000........... 27
21 Na u al loga i hm plo o he he mal ampli ude (a) non-no malized and (b) no mal-
ized on he ans e se sample p o ile o ¯y0= 0.24,0.32,0.40,0.48. ......... 27
22 Na u al loga i hm plo o he he mal ampli ude (a) no malized and (b) c ack egion
zoomed on he ans e se sample p o ile o Πw= 0.001,0.002,0.005,0.01. . . . . . 28
23 Na u al loga i hm plo o he he mal ampli ude (a) no malized and (b) c ack egion
zoomed on he ans e se sample p o ile o Πl= 0.2,0.4,0.5,0.75.......... 28
24 Na u al loga i hm plo o he he mal ampli ude (a) no malized and (b) c ack egion
zoomed on he ans e se sample p o ile o Πd= 0.1,0.2,0.3,0.5. ......... 29
25 Na u al loga i hm plo o he he mal ampli ude (a) no malized and (b) c ack egion
zoomed on he ans e se sample p o ile o θ= 50º, 75º, 90º, 100º. ......... 29
26 Na u al loga i hm plo o he he mal ampli ude on he ans e se sample p o ile o
wo di e en (Π1,Π3,Πw) iples which sa is y (41). . . . . . . . . . . . . . . . . . 34
27 Na u al loga i hm plo o he he mal ampli ude on he ans e se sample p o ile o
wo di e en (Π3,Πw) uples which sa is y (49). . . . . . . . . . . . . . . . . . . . . 36
2 Expe imen al backg ound
h ough he domain, cha ac e ized by hei ampli udes and phases (see igu e 3). In his p ojec , he
pa ame ic analysis is going o be done wi h he ampli ude.
Figu e 2: Scheme o he e olu ion o he su ace empe a u e o he sample wi h ha monic lase
hea ing.
Figu e 3: Scheme o he he mal wa es in di e en poin s o he sample.
5

3 Theo e ical model
3 Theo e ical model
3.1 Dimensionless equa ions
In o de o de elop a dimensionless compu a ional nume ical model o he c ack de ec ion p oblem
by means o lock-in he mog aphy, he i s aspec ha mus be aken in o accoun is he manne in
which he dimensions o he se o equa ions ha desc ibe p oblem a e emo ed. Due o he na u e o
he expe imen , he cons i u i e equa ion is he hea equa ion:
∇2T=1
α
∂T
∂ (4)
whe e αis he he mal di usi i y. Aiming o emo e he dimensions o his equa ion he ollowing
a iables a e in oduced:
¯x=x
Lx
,¯y=y
Ly
,¯z=z
Lz
,¯
T=T
T0
,¯
=
c
(5)
being Lx,LyyLzcha ac e is ic leng hs o he p oblem in he h ee spa ial di ec ions, T0an a bi a y
empe a u e (e.g., oom empe a u e) and ca cha ac e is ic ime alue o he p oblem. Explici ly
w i ing he hea equa ion and applying he chain ule:
∇2T=∂2(¯
TT0)
∂¯x2
∂2¯x
∂x2+∂2(¯
TT0)
∂¯y2
∂2¯y
∂y2+∂2(¯
TT0)
∂¯z2
∂2¯z
∂z2=1
α
∂(¯
TT0)
∂¯
∂¯
∂ (6)
Using he p e iously in oduced pa ame e s, his equa ion educes o:
1
L2
x
∂2¯
T
∂¯x2+1
L2
y
∂2¯
T
∂¯y2+1
L2
z
∂2¯
T
∂¯z2=1
cα
∂¯
T
∂¯
(7)
A simpli ica ion ha can be ca ied ou a his poin , and which is pa icula ly con enien o
compu a ional calcula ions, is o choose he same leng h scale o all h ee spa ial di ec ions. Thus,
imposing Lx=Ly=Lz=L he dimensionless hea equa ion educes o:
¯
∇2¯
T=∂2¯
T
∂¯x2+∂2¯
T
∂¯y2+∂2¯
T
∂¯z2=L2
cα
∂¯
T
∂¯
(8)
whe e ¯
∇2is he dimensionless Laplacian ope a o . In he p ocess o emo ing dimensions o
pa ial di e en ial equa ions he e a e mul iple c i e ia o choose he cha ac e is ic leng h and ime
scales. In pa icula , o he hea equa ion he e a e wo main di e en c i e ia depending on he
cha ac e is ics o he p oblem [14]:
1. c=L2/α, i hea di usion occu s signi ican ly h oughou he domain. He e Lis he leng h
o he sample.
6
3 Theo e ical model 3.1 Dimensionless equa ions
2. c= 1/ω, i he change in empe a u e is signi ican only up o a ce ain limi ed dis ance, l.
He e, ωis an angula equency associa ed wi h he p oblem.
In lock-in IR he mog aphy expe imen s, hea di usion is a p ocess ha occu s up o dis ances
om he hea sou ce close o µ, whe e he hea is a enua ed by a ac o o 1/e. Consequen ly, in
his pa icula case, he second choice o scales is he mos na u al one, being l=µ1. Choosing he
cha ac e is ic leng h as µ he equa ion (8) educes o:
¯
∇2¯
T= 2∂¯
T
∂¯
(9)
He e i has been used ha ω= 2π and he de ini ion o he mal di usion leng h: µ=pα/(π ).
Fo pu ely aes he ic easons, he compu a ional model has implemen ed he dimensionless hea equa-
ion wi h c= 2/ω, so ha he p e ious equa ion becomes:
¯
∇2¯
T=∂¯
T
∂¯
(10)
The second equa ion om which he dimensions mus be elimina ed is he bounda y condi ion
associa ed wi h he lase . In his case he lase , o powe P, ampli ude modula ed a a equency
and cen e ed a (x0,y0,0), is assumed o ha e a Gaussian p o ile and o be ocused o a adius g(a
1/e2o he maximum in ensi y) [6]. The e o e, he bounda y condi ion in he s a iona y s age is:
−κ∂T
∂z 

z=0 =ηP
π 2
g
e
−2hx−x0
g2
+y−y0
g2icos(2π )(11)
being κ he he mal conduc i i y o he sample ma e ial and η he powe ac ion abso bed by he
sample. In his wo k, he nega i e sign is in oduced as a phase in he modula ion e m so i will no
longe appea again. On he o he hand, ηis aken equal o one and a ac o 2is in oduced because
when he igh hand side o he equa ion is in eg a ed o e xand y, wi hou ha ing ha ac o , he
esul is P/2. Taking his h ee de ails in o accoun , his equa ion can be ew i en as ollows:
κ∂T
∂z 

z=0 =2P
π 2
g
e
−2hx−x0
g2
+y−y0
g2icos(2π )(12)
This bounda y condi ion becomes dimensionless using he p e iously in oduced a iables:
(x, y, z)→(µ¯x, µ¯y, µ¯z), →2
ω¯
, T →¯
TT0(13)
The e o e, applying he chain ule, he equa ion (12) becomes:
1In addi ion, his choice g ea ly simpli ies he subsequen analysis o he p oblem based on he dimensionless pa am-
e e s.
7
3 Theo e ical model 3.1 Dimensionless equa ions
∂¯
T
∂¯z

¯z=0 =2Pµ
π 2
gT0κe
−2h¯x−¯x0
g/µ 2
+¯y−¯y0
g/µ 2icos(2¯
)(14)
I mus be no iced ha he oscilla ion equency o he lase hea sou ce, in his dimensionless
o mula ion, is no longe depending in any ma e ial o expe imen al pa ame e s, being a cons an
alue in any case. Once he dimensions o hea equa ion and lase bounda y condi ion a e emo ed,
he wo emaining equa ions a e hose associa ed wi h he c ack. The cons ain s ha mus be sa is ied
a e:
• The con inui y o hea low o e he c ack. This condi ion is w i en as ollows:
[[ ˙
Q]] = 0 (15)
whe e he [[ ]] ope a o s ands o he change on he lux o e he c ack [6]. Being equal o
ze o, emo ing he dimensions o his equa ions is s aigh o wa d:
[[ ˙
¯
Q]] = 0 (16)
• A empe a u e discon inui y a c ack posi ion:
The c ack is modeled as a he mal con ac esis ance R h [6], ela ed o he wid h o he c ack
w h ough:
R h =w
κai
(17)
being κai he he mal conduc i i y o he ai , which is assumed o ill he c ack. Thus, he
empe a u e discon inui y in he c ack is gi en by:
∆T=κR h∇T=κw
κai
∇T(18)
Making use o he p e iously in oduced dimensionless a iables, his condi ion becomes:
T0∆¯
T=κw
κai
T0
µ¯
∇¯
T(19)
He e he dimensionless g adien ope a o ¯
∇≡∇/µ has been in oduced. Rea anging he
e ms, he dimensionless empe a u e jump condi ion due o he p esence o a c ack esul s in:
∆¯
T=κ
κai
w
µ¯
∇¯
T(20)
Summa izing, in oducing he he mal di usion leng h µas he leng h scale, c= 2/ω as he
cha ac e is ic ime scale and he empe a u e no maliza ion ac o T0, he se o dimensional ou
equa ions ha go e n he physics o c ack de ec ion by lock-in IR he mog aphy, becomes:
8
3 Theo e ical model 3.2 Dimensionless pa ame e s















∇2T=1
α
∂T
∂
∂T
∂z 

z=0 =2P
κπ 2
ge
−2hx−x0
g2
+y−y0
g2icos(2π )
[[ ˙
Q]] = 0
∆T=κw
κai ∇T
→















¯
∇2¯
T=∂¯
T
∂¯
∂¯
T
∂¯z

¯z=0 =2P µ
π 2
gT0κe
−2h¯x−¯x0
g/µ 2
+¯y−¯y0
g/µ 2icos(2¯
)
[[ ˙
¯
Q]] = 0
∆¯
T=κ
κai
w
µ¯
∇¯
T
I has o be men ioned ha his en i e wo k deals wi h he adiaba ic p oblem, ha is, i is assumed
ha he e a e no hea losses due o con ec ion o adia ion mechanisms. This decision is based on he
na u e o his wo mechanisms. Radia ion is p opo ional o T4which means ha i he empe a u e is
no high enough i s e ec is negligible. On he o he hand, hea losses by con ec ion a e p opo ional
o he di e ence be ween he oom empe a u e and he sample empe a u e. Howe e , IR he mog-
aphy expe imen s a e ca ied ou a ew kel ins abo e he oom empe a u e so he di e ence is no
la ge enough o be conside ed. Fu he mo e, his simpli ica ion is suppo ed by he ac ha , excep
in a ew cases, he adiaba ic model i s he expe imen al da a ela i ely well.
3.2 Dimensionless pa ame e s
Once he non-dimensional equa ions ha e been ob ained, he e ms ha appea in he new equa ions
can be ea anged o ind cha ac e is ic independen dimensionless pa ame e s which de e mine he
na u e o he p oblem. In pa icula , he combina ion o he e ms ha leads o hese pa ame e s can
be ound looking a he lase bounda y condi ion (14) and he empe a u e jump condi ion a he c ack
(20). In bo h o hem he mal conduc i i y κappea s, so, mul iplying and di iding he bounda y
condi ion o he lase by he conduc i i y o he ai , his equa ion becomes:
∂¯
T
∂¯z

¯z=0 =2Pµ
π 2
gT0κai
1
κ
κai
e
−2h¯x−¯x0
g/µ 2
+¯y−¯y0
g/µ 2icos(2¯
)(21)
Going u he , in his equa ion he adius o he lase gappea s bo h in he ampli ude and in he
exponen ial e ms. So, i seems o be na u al o mul iply and di ide he dimensionless modula ion
ampli ude by he he mal di usion leng h:
∂¯
T
∂¯z

¯z=0 =2P
πµT0κai
1
κ
κai
1
 g
µ2e
−2h¯x−¯x0
g/µ 2
+¯y−¯y0
g/µ 2icos(2¯
)(22)
De ining he ollowing dimensionless pa ame e s:
Π1≡2P
πµT0κai
,Π2≡ g
µ2,Π3≡κ
κai
(23)
The lase bounda y condi ion educes o:
9
3 Theo e ical model 3.2 Dimensionless pa ame e s
∂¯
T
∂¯z

¯z=0 =Π1
Π2Π3
e
−2
Π2h(¯x−¯x0)2+(¯y−¯y0)2icos(2¯
)(24)
On he o he hand, de ining he dimensionless pa ame e s associa ed o he c ack geome y,
Πw≡w
µ,Πl≡l
µ,Πd≡d
µ(25)
being land d he leng h and dep h o he c ack, espec i ely, he empe a u e jump condi ion becomes:
∆¯
T= Π3Πw¯
∇¯
Ti ¯x∈h−Πl
2,Πl
2i,¯z∈h0,Πdi(26)
The e o e, i is obse ed ha , om he manipula ion o wo o he ou equa ions and he de ini-
ion o six dimensionless pa ame e s, h ee associa ed wi h he geome y o he c ack and h ee wi h
he expe imen al condi ions, a simpli ied non-dimensional o mula ion o he o iginal equa ions is
ob ained. The equa ions o compu e a e summa ized in he ollowing box.













¯
∇2¯
T=∂¯
T
∂¯
∂¯
T
∂¯z

¯z=0 =Π1
Π2Π3e
−2
Π2h(¯x−¯x0)2+(¯y−¯y0)2icos(2¯
)
[[ ˙
¯
Q]] = 0
∆¯
T= Π3Πw¯
∇¯
T
(27)
In his wo k in addi ion o he pa ame ic analysis associa ed o he men ioned six pa ame e s, he
posi ion o he lase (¯x0,¯y0)and he inclina ion o he c ack θwill also be conside ed in he analysis
in o de o check hei impac in he esul s.
IR he mog aphy expe imen s in labo a o ies a e ypically ca ied ou wi h lase s o powe on he
ange o 0.1−10 W, adius a 1/e2o he maximum in ensi y is ypically on he o de o 10−4m and
a e modula ed a equencies on he o de o he z. The c acks o be de ec ed in hese expe imen s
a e ypically on he o de o mic ons. As said be o e, in his expe imen s he in e es is placed on he
oscilla ion o he empe a u e so, he mos na u al choice o empe a u e scale T0is he maximum o
he ampli ude o e he oom empe a u e. Typically his alues a e on he o de o ew kel ins. Con-
side ing he men ioned a ia ion anges, able 1 shows ypical alues o he dimensionless pa ame e s.
In his able i can be seen ha he alues o he i s dimensionless pa ame e Π1can a y om
103 o 104while alues on he o de o 10−4o 10−1a e ob ained o Π2. A simila beha iou is
ound o Π3, whe e alues om 50 o almos 16000 a e ob ained. This leads o conclude ha hese
h ee pa ame e s a e highly dependen on he ma e ial, which is a di ec consequence o choosing
he di usion leng h as longi ude cha ac e is ic scale. Howe e , he same alue o any dimensionless
pa ame e can be ob ained o di e en ma e ials h ough he app op ia e selec ion o he es o
expe imen al pa ame e s.
10

3 Theo e ical model 3.2 Dimensionless pa ame e s
κ(Wm−1K−1) [15] α(mm2s−1) [15] Π1Π2Π3
Cu 397.48 116.0 4685 0.0054 15899.2
AISI 304 14.64 3.68 28210 0.1963 600.0
Pb 34.309 23.3 10454 0.0270 1372.4
Lead Glass 1.13 0.74 58661 0.8490 45.2
Al 225.94 91.0 5289 0.0007 9037.6
Fe 71.965 20.4 11172 0.0307 2878.6
Table 1: Typical alues o he dimensionless pa ame e s o di e en ma e ials. These alues ha e
been calcula ed wi h P= 1 W, = 5 Hz, g= 0.2mm, T0= 2 K.
11
4 Compu a ional nume ical simula ions
4 Compu a ional nume ical simula ions
The e a e wo key aspec s in compu a ional nume ical simula ions: he mesh and he nume ical
me hod ha is used o sol e he equa ions. The manne he equa ions a e ansla ed om analy ical
o mula ion o compu a ional o mula ion is an impo an aspec when using ini e elemen me hods
because some nume ical schemes may no wo k o some speci ic p oblems. In he ollowing a b ie
desc ip ion is hese is gi en, e en hough i is no he scope o his wo k o go on u he de ail in his
opic.
On he o he hand, he mesh plays a key ole in sol ing equa ions by nume ical me hods because
i is, li e ally, whe e he equa ions a e sol ed. This means ha , in his case he simula ed labo a o y
sample, has o be as simila as possible o a con inuous medium bu wi hou ha ing o spend much
compu a ional esou ces and ime simula ing he p oblem.
The simula ions p esen ed in his wo k ha e been ca ied ou a wo ks a ion wi h Ubun u MATE
20.04.4 wi h an In el Xeon(R) Gold 5218 CPU @ 2.30 GHz ×64, 192 Gb memo y and a g aphic
memo y LLVM 12.0.0, 256b.
4.1 OpenFoam
Open Sou ce Field Ope a ion and Manipula ion (OpenFoam) [16] is a C++ objec o ien ed lib a y,
o iginally designed o compu a ional luid dynamics and s uc u al analysis [17, 18]. Howe e , a e
decades o e olu ion, i has become a so wa e wi h mul iphysics capaci y o ien ed o a wide a ie y o
physical phenomena, such as: combus ion, elec omagne ics, hea ans e and o he s. OpenFoam is
used o c ea e execu ables ha all in wo ca ego ies: hose ha allow he manipula ion o da a, known
as u ili ies, and sol e s, which a e designed o sol e a speci ic p oblem in con inuum mechanics [16].
Taking in o accoun he na u e o he p oblem o be sol ed in his wo k, among he a ie y o
sol e s ha a e implemen ed in OpenFoam, he sol e s o he he mal amily ha e been iden i ied as
he mos sui able. Be ween all he op ions a ailable, solidFoam has been selec ed as i has been de-
signed o ene gy anspo and he modynamics on solids. In his sol e no only complex bounda y
condi ions, such as he ones o he lase o he c ack, can be implemen ed bu i could also bene i a
u u e wo k in which his model is used o simula e lying spo expe imen s [19] because i allows o
use dynamic meshes.
I is wo h o men ion ha OpenFoam p esen s na i e pa allel calcula ion capaci y, which allows
o dis ibu e he nodes o he mesh be ween he a ailable p ocesses. Hence, his p esen s a double
bene i . On he one hand, se e al simula ions can be done simul aneously. On he o he hand, as e
calcula ions can be pe o med assigning each p ocess a se o nodes.
4.2 Nume ical schemes
As men ioned be o e, i is beyond he scope o his wo k o explain in de ail he me hods used o
calcula e he pa ial de i a i es, g adien s, e c. since he objec i e is o build he dimensionless model
and hese me hods a e equally alid o equa ions wi h and wi hou dimensions. Howe e , i should be
men ioned ha he schemes used ha e al eady been used o ca y ou simula ions in simila physical
phenomena wi h p o en capabili ies. The ini e olume schemes ha ha e been used in his wo k a e
12
4 Compu a ional nume ical simula ions 4.3 Solu ion and algo i hm con ol
summa ized in able 2.
Quan i y Scheme
Time pa ial de i a i es C ank Nicholson
G adien s Gauss linea
Laplacian Gauss linea co ec ed
In e pola ions Linea
Table 2: Fini e olume schemes used.
4.3 Solu ion and algo i hm con ol
In solu ion and algo i hm con ol, he sol e keywo d speci ies each linea -sol e ha is used o each
disc e ised equa ion, ha is, o he me hod o sol ing he se o linea equa ions [16]. On he o he
hand, he mul iple op ions o p econdi ioning o ma ices in he conjuga e g adien sol e s (DIC,
FDIC, DILU,...) a e con olled by he p econdi ione keywo d [16]. In his wo k he PCG sol e and
DIC p econdi ione ha e been used as i has been seen ha hey sui p ope ly o he cha ac e is ics o
he p oblem.
In OpenFoam, he ma ix sol e s a e based on educing he e o in he solu ion o e an i e a i e
p ocess, ha is, he esidual is e alua ed by subs i u ing he cu en solu ion in o he equa ion and
aking he magni ude o he di e ence be ween he le and igh hand sides [16]. In o de o con ol
i ha di e ence is small enough, he e a e wo a iables ha can be used:
1. Absolu e ole ance: measu es i he esidual is small enough o conside he solu ion su icien ly
accu a e.
2. Rela i e ole ance: limi s he ela i e imp o emen om ini ial o inal solu ion.
Thus, he solu ion will be conside ed su icien ly accu a e i he esidual is lowe han he absolu e
ole ance, o he a io o cu en o ini ial esiduals alls below he ela i e ole ance. In his wo k he
absolu e ole ance has been used as a measu e o he accu acy o he simula ion and i has been se o
1×10−7.
Ano he aspec o be aken in o accoun is he solu ion unde - elaxa ion. This is a echnique used
o imp o ing s abili y o a compu a ion ha can be applied in physical simula ions whe e a iables
a y ha as ha can lead o nume ical di e gences. Since he na u e o he expe imen ha is being
simula ed does no imply his kind o p oblems, he elaxa ion ac o in his wo k is le as de aul ,
ha is, equal o 1. The solu ion and algo i hm con ol con igu a ion used in he simula ions is shown
in able 3.
Sol e P econdi ione Absolu e ole ance Rela i e ole ance Relaxa ion ac o
PCG DIC 1×10−70 1
Table 3: Solu ion and algo i hm con ol keywo ds used.
13
4 Compu a ional nume ical simula ions 4.4 The mesh
4.4 The mesh
As said be o e, he mesh plays a key ole in compu a ional nume ical simula ions. The e a e wo
main s a egies o simula e c acks in ma e ials. The i s one is o model he p oblem as a wo domain
ma e ial: he bulk and he ai illing he c ack. This s a egy implies ha , close and inside he c ack, an
ex emely ine mesh has o be done which d ama ically inc eases memo y esou ces and calcula ion
ime due o he di e ence in spa ial scale be ween he c ack and he bulk [20, 21]. The second
op ion is o model he c ack as a con ac he mal esis ance su ace. In his app oach, he e is no c ack
olume o mesh, because he la e is modeled as a 2D plane [2, 6], no ably educing he compu a ional
esou ces equi ed.
In OpenFoam he e a e wo main op ions o model a c ack. The i s one consis s in using he
unc ion al eady implemen ed in OpenFoam, which allows o c ea e he sample as an en i e block and
hen de ine he geome y o he c ack. This me hod allows he use o simula e as many c acks as
wan ed wi hou adding complexi y o he mesh. Howe e , his s a egy has a big d awback: i implies
ha he geome y o he c ack will be ounded o he alues o he nea es node o he mesh, meaning
ha he c ack will no longe be as wide, long o deep as i had been modeled.
The second op ion is o ep oduce he c ack as a con ac su ace be ween hexahed al blocks,
which implies ha he sample has o be di ided in o mul iple domains ha la e ha e o be joined.
In o he wo ds, he sample has o be cons uc ed ‘block by block’. Since he numbe o blocks o
c ea e depends en i ely on he ype o c ack (in ini e, semi-in ini e o ini e), he main disad an age
o ollowing his pa h is ha he complexi y o he code inc eases. On he o he hand, ollowing
his me hodology ensu es, by cons uc ion, ha he c ack will ha e he desi ed dimensions. As a
consequence, in his wo k, his has been he ollowed meshing s a egy.
Ano he key aspec o be aken in o accoun is he geome y o he mesh. Among all he op ions,
he mos used meshes a e e ahed al and hexahed al meshes. These wo ypes o meshes and he
indi idual elemen s can be seen in igu es 4 and 5.
Figu e 4: (le ) Hexahed al and ( igh ) e ahed al meshes.
Figu e 5: (le ) Te ahed al and ( igh ) hexahed al mesh elemen s.
Te ahed al meshes a e made up o elemen s wi h 4 e ices. These can be ea anged so ha a
14
4 Compu a ional nume ical simula ions 4.5 Mesh esolu ion con ol
4.5.2 Mesh e inemen
Ano he me hod o inc ease he mesh esolu ion in some egions in OpenFoam is o di ide he hexa-
hed ons ha a e in he egion o in e es . Unlike he g adien me hod, mesh e inemen inc eases he
compu a ional esou ces because he numbe o nodes inc eases. The implemen a ion o he e ine-
men s in OpenFoam consis s o :
1. Selec ing he nodes o he mesh ha a e wi hin he egion o in e es . Wi h his objec i e,
geome ical en i ies (sphe es, cylinde s, boxes,...) a e de ined in he mesh inside which he
selec ed nodes emain.
2. Once he nodes o he egion o in e es a e selec ed, hese a e di ided un il he desi ed spa ial
esolu ion is achie ed.
As men ioned be o e, he in e es o implemen ing mesh e inemen s is ha he e a e egions ha
a e physically mo e in e es ing, since hey a e he ones in which he a iables change mo e apidly.
In pa icula , du ing he mul iple simula ions ha ha e been ca ied ou in his wo k, se e al special
in e es zones ha e been iden i ied: he c ack and he lase spo . In o de o mesh hese egions wi h
enough spa ial esolu ion, a egion p opo ional o he non-dimensional he mal di usion leng h has
been ca e ully e ined.
Wi h he objec i e o e ining he egion close o he c ack, he box geome y has been used as i
keeps he aspec a io wi h he sample. Fu he mo e, du ing he simula ions ha ha e been ca ied
ou , i has been no iced ha a single box e inemen no only inc eases signi ican ly he esolu ion
in ha egion bu also educes bo h he numbe o nodes a om he c ack and consequen ly, he
compu a ional ime. An example o his e inemen can be ound in igu e 13 (a).
In addi ion o he meshing o he c ack o he egion close o i , an impo an aspec in IR he -
mog aphy simula ions is o co ec ly mesh he egion in which he lase is in oduced. In o de o
comple e his ask, he cylinde e inemen has been chosen as i keeps he aspec a io wi h he ci cu-
la geome y o he lase spo . Du ing he mul iple simula ions done in his wo k, i has been es ed
ha e ining he cells o he lase en i onmen h ee imes, each ime wi h cylinde s o smalle heigh
and adius, i is enough o cap u e he lase spo app op ia ely. This iple e inemen can be seen in
igu e 13 (b). When bo h o hese e inemen s a e combined (see igu e 13 (c)), as i is he case o his
wo k, he esul ing mesh ends up ha ing on he o de o 250000 nodes.
(a) (b) (c)
Figu e 13: (a) Single box, (b) iple cylinde e inemen s and (c) bo h o hem combined.
21

4 Compu a ional nume ical simula ions 4.6 P ocessing
4.6 P ocessing
Once he mesh is done, he equa ions mus be sol ed. A his s age o he calcula ion, known as
p ocessing, i is wo h o no ice ha e en hough only he s a iona y pa o he p oblem is being
simula ed, o nume ical easons, i may happen ha he i s oscilla ions do no ep esen ha pa
co ec ly un il a ew i e a ions a e done. The e o e, in o de o ind he s able s a iona y si ua ion
wi hou undesi ed nume ical e ec s a c i e ion ha gua an ees he app op ia e s a iona y na u e o
he calcula ion is equi ed. In his wo k his c i e ion has been selec ed as he ull ep oducibili y o
he ob ained he mal wa es.
P og amma ically, he implemen a ion o his condi ion has been ca ied ou selec ing an a bi a y
poin o he s a iona y he mal cycle and checking i s alue o e cycling, p o ided an app op ia e
ime esolu ion. When he empe a u e di e ence be ween cycles, i.e., he slope o he line joining
wo consecu i e poin s (as can be seen in igu e 14) is ound o be below an imposed h eshold alue,
he c i e ion is sa is ied and he nex cycle is sa ed as he s a iona y oscilla ion o each node o he
mesh.
Figu e 14: Tempe a u e e alua ion o an a bi a y poin o e cycling un il he s a iona y s abili y
c i e ion is sa is ied.
4.7 Pos -P ocessing
As men ioned be o e, one o he consequences o ea ing his p oblem in a simple way, om he
ma hema ical poin o iew, is ha ins ead o sol ing he Helmol z equa ion [4] which in eg a es he
ha monic na u e o he lock-in expe imen , he hea equa ion is sol ed. Whe eas he i s p o ides
di ec solu ion o he he mal ampli ude, he second doesn’ , esul ing he empe a u e ield. The
consequence o his is ha , a e he p ocessing s age, hose ampli udes mus be ound. This is he
pos -p ocessing s age o he calcula ion.
In his s age, e en i he empe a u e ield o he comple e sample is calcula ed, he esul s p e-
sen ed in his wo k a e limi ed o a line pe pendicula o he c ack and c ossing he cen e o he lase
spo . This selec ion has been done as his would be he physically mo e signi ican egion. Once
he egion o plo he non-dimensional empe a u e ampli udes is chosen, he maximum and mini-
mum alues o each he mal wa e a e p og amma ically sea ched, leading o he non-dimensional
empe a u e ampli ude plo .
22
5 Resul s
5 Resul s
5.1 Valida ion o he model
Once he dimensionless nume ical model has been es ablished, i s alidi y mus be checked. Al hough
he model can be alida ed agains expe imen al da a, in his case i is going o be done agains
analy ical o nume ical models, depending on he si ua ion. In his sec ion i s , he compa isons o
he esul s o he dimensionless model wi h he analy ical model co esponding o a e ical in ini e
c ack [1] will be p esen ed in igu e 15. Second, he non-dimensional esul s o semi-in ini e and
ini e c acks a e going o be alida ed agains ano he nume ical model de eloped by R.Celo io e al.
[4] (in he ollowing, DG FEM model) w i en in FEniCS, due o he ac ha he e is no analy ical
solu ion in hese cases. The associa ed esul s in his case a e shown in igu es 16 and 17. In o de
o make he cases as ealis ic as possible, i has been decided o alida e he model wi h ypical
expe imen al pa ame e alues (see able 4) and o wo ma e ials: AISI 304 and Cu (see hei he mal
p ope ies in able 1).
P(W) g(mm) (Hz) (x0, y0)(mm)
1 0.165 1 (0,0.5)
Table 4: Dimensional pa ame e alues ha ha e been used in o de o alida e he dimensionless
nume ical model.
Aiming o compa e he esul s be ween models, he dimensional esul s a e ans o med o dimen-
sionless ollowing he scale a iables p e iously in oduced (see equa ion (13)). In o de o main ain
an accep able sensi i i y o he empe a u e changes, e en a om he hea sou ce, ins ead o he
ampli ude i s na u al loga i hm is plo ed. The dimensionless pa ame e s ha ha e been used can be
ound in able 5.
Figu e 15: Compa ison be ween he non dimensional nume ical esul s and analy ical solu ion o
in ini e e ical c acks. (a) AISI 304 and (b) Cu.
23
5 Resul s 5.1 Valida ion o he model
Figu e 16: Compa ison be ween he non dimensional esul s and he DG FEM model in he semi-
in ini e c ack case in AISI 304 o wo di e en c ack inclina ions. (a) 90º and (b) 45º.
Figu e 17: Compa ison be ween he non dimensional esul s and he DG FEM model in he ini e
c ack case in AISI 304 o wo di e en c ack inclina ions. (a) 90º and (b) 45º.
In o de o quan i y he di e ences be ween he dimensional (analy ical o nume ical) and dimen-
sionless models, among all he s a is ical op ions, in his wo k he oo mean squa e e o (RMSE)
has been chosen due o he ease o in e p e ing he esul s. As i can be seen in igu es 15, 16 and
17 he ag eemen be ween all he models is e y sa is ac o y in he h ee p esen ed cases, he alues
p esen ed in able 6, a e going o be discussed in de ail in sec ion 6.1.
24
5 Resul s 5.2 Pa ame ic analysis
Figu e 15 (a) Figu e 15 (b) Figu e 16 Figu e 17
Type o c ack In ini e In ini e Semi-In ini e Fini e
Π123528 4191 23528 23528
Π20.023 0.0007 0.023 0.023
Π3586 15899 586 586
Πw0.00092 0.0082 0.00092 0.00092
Πl- - - 0.92
Πd- - 0.92 0.92
¯y00.46 0.45 0.46 0.46
θ(º) 90 90 90/45 90/45
Table 5: Dimensionless pa ame e s used o ob ain he esul s shown in igu es 15, 16 and 17.
Case / Model Analy ical DG FEM model
In ini e 90º (I) ( igu e 15 (a)) 0.017 -
In ini e 90º (II) ( igu e 15 (b)) 0.014 -
Semi-in ini e 90º ( igu e 16 (a)) - 0.009
Semi-in ini e 45º ( igu e 16 (b)) - 0.012
Fini e 90º ( igu e 17 (a)) - 0.0027
Fini e 45º ( igu e 17 (b)) - 0.022
Table 6: RMSE o he compa isons shown in igu es 15, 16 and 17.
5.2 Pa ame ic analysis
In his sec ion a ca alog o cu es in which each non-dimensional pa ame e is a ied, while he o he s
a e ixed, is shown. This ca alog has wo main pu poses. Fi s , i can be used as a esul guideline o a
a ie y o expe imen al and ma e ial p ope ies pa ame e s. Going mo e in de ail, he second pu pose
o his ca alog, is o p o ide a pa ame ic analysis showing he in luence o each non-dimensional
pa ame e in he esul ing he mal ampli ude (|¯
T|) plo .
Al hough i is a common p ac ice o no malize he expe imen al da a in o de o elimina e he
e ec o pa ame e s which a e di icul o con ol, such as he abso p ion coe icien (η) in equa ion
(11), he no maliza ion can also limi he amoun o in o ma ion ob ained in he expe imen . In o de
o ake ad an age o bo h me hodologies, in his wo k, as gene al ule, bo h no malized and non-
no malized plo s a e shown2.
All o he esul s p esen ed in his sec ion a e ob ained o a ini e c ack, as i is he mos e sa ile
one. The expe imen al pa ame e s ha lead o hese simula ions a e ypical alues o expe imen al
se ups. I is wo h o no ice ha he powe o his non-dimensional o mula ion lies in he ac ha
he esul is no dependen on each indi idual expe imen al pa ame e , bu depends only on hei
combina ions. The dimensionless pa ame e s ha emain ixed in each calcula ion a e speci ied in
able 7.
2In he esul s in which he no maliza ion does no ha e any impac , only he no malized na u al loga i hm o he
he mal ampli ude ln(|¯
Tn|)is shown.
25
5 Resul s 5.2 Pa ame ic analysis
Figu e 18: Na u al loga i hm plo o he he mal ampli ude (a) non-no malized and (b) no malized on
he ans e se sample p o ile o Π1= 4 ×103,1×105,2×105,3×105.
Figu e 19: Na u al loga i hm plo o he he mal ampli ude (a) non-no malized, showing he zoomed
ampli ude jump associa ed o he c ack in he inse and (b) no malized on he ans e se sample
p o ile o Π2= 1 ×10−7,2×10−7,3×10−7,1×10−8.
26

5 Resul s 5.2 Pa ame ic analysis
Figu e 20: Na u al loga i hm plo o he he mal ampli ude (a) non-no malized and (b) no malized on
he ans e se sample p o ile o Π3= 50,500,1000,4000.
Figu e 21: Na u al loga i hm plo o he he mal ampli ude (a) non-no malized and (b) no malized on
he ans e se sample p o ile o ¯y0= 0.24,0.32,0.40,0.48.
27
5 Resul s 5.2 Pa ame ic analysis
Figu e 22: Na u al loga i hm plo o he he mal ampli ude (a) no malized and (b) c ack egion
zoomed on he ans e se sample p o ile o Πw= 0.001,0.002,0.005,0.01.
Figu e 23: Na u al loga i hm plo o he he mal ampli ude (a) no malized and (b) c ack egion
zoomed on he ans e se sample p o ile o Πl= 0.2,0.4,0.5,0.75.
28
5 Resul s 5.2 Pa ame ic analysis
Figu e 24: Na u al loga i hm plo o he he mal ampli ude (a) no malized and (b) c ack egion
zoomed on he ans e se sample p o ile o Πd= 0.1,0.2,0.3,0.5.
Figu e 25: Na u al loga i hm plo o he he mal ampli ude (a) no malized and (b) c ack egion
zoomed on he ans e se sample p o ile o θ= 50º, 75º, 90º, 100º.
29
5 Resul s 5.2 Pa ame ic analysis
Figu e Π1Π2Π3¯y0ΠwΠlΠdθ(º)
18 - 0.027 600 0.50 0.001 1 1 50
19 25231 - 600 0.50 0.001 1 1 50
20 25231 0.027 - 0.50 0.001 1 1 50
21 25231 0.027 600 - 0.001 1 1 50
22 25231 0.027 600 0.50 - 1 1 50
23 25231 0.027 600 0.50 0.001 - 1 50
24 25231 0.027 600 0.50 0.001 1 - 50
25 25231 0.027 600 0.50 0.001 1 1 -
Table 7: Pa ame e s used o ob ain each igu e o he ca alog.
30
7 Conclusions
7 Conclusions
In his wo k a non-dimensional FEM model o c ack cha ac e iza ion in ae ospace ma e ials by
means o lock-in IR he mog aphy has been de eloped. In addi ion o he esolu ion o he equa ions,
he op imiza ion o he spa ial domain o be modeled has been ca ied ou h ough di e en meshing
s a egies, such as non- egula edge dis ibu ion o selec i e e inemen s o he mesh. These meshing
s a egies ha e op imized he calcula ions in e ms o accu acy and compu a ional esou ces.
The de eloped model shows a e y good ag eemen when compa ed wi h o he analy ical o
nume ical models, al hough he o mula ion o each one has a sligh impac on he esul s. O e all,
he compa isons show e y low ela i e e o s (a mos 0.4%) which con i m he ema kable accu acy
o he de eloped model.
The dimensionless model has allowed a gene al in e p e a ion o he lock-in IR he mog aphy
expe imen no depending on he pa ame e s o he ma e ial o he expe imen al ones, bu on he di-
mensionless pa ame e s iden i ied. These pa ame e s ha e been used o pe o m a pa ame ic analysis
whe e hei e ec in he he mal ampli ude plo s has been analyzed.
While he e ec o Π1is emo able by no maliza ion, he e ec o Π3canno be isola ed in jus
one zone o he cu e since i a ec s he mo phology o he en i e he mal ampli ude plo . On he
o he hand, despi e o di e en mo phological eshaping o he cu e, he e ec o Π2and ¯y0has
u ned ou o be unique, i.e., i is impossible o ob ain he same he mal ampli ude plo wi h di e en
alues o hese. I has been seen ha , al hough he mos p ominen e ec o Πw,Πl,Πdand θis o
change he ampli ude jump, hey also show o he seconda y e ec s in o he egions such as he slopes
a om he hea sou ce. Mo eo e , excep o θ, in he o he h ee pa ame e s hose seconda y e ec s
can be seen mo e clea ly on he non-illumina ed side o he c ack.
E en hough i is beyond he scope o his wo k, he esul s ob ained o he pa ame ic analysis
ca ied ou allow o de e mine whe e he sensi i i y o he cu es is o each o he pa ame e s. This
becomes o pa amoun impo ance when pa ame ic in e sion is pe o med.
Going u he in he de eloped non-dimensional discussion, he degene acy o he he mog aphic
p oblem has been add essed. In his line, a non-dimensional e o mula ion o he p oblem has been
p esen ed. I has been demons a ed ha he e is no need o dis inguish be ween Π1,Π3and Πw,
since di e en combina ions o hese pa ame e s lead o he same esul s. This wo k has allowed
o de e mine which combina ions o expe imen al pa ame e s and ma e ial p ope ies leading o he
same ampli ude he mog ams which a e ha dly accessible by o he p ocedu es.
As a po en ial con inua ion o he de eloped in es iga ion he ollowing u u e wo k is iden i ied:
1. The s udy o he degene acy o he non-dimensional o mula ion sea ching o he po en ial pa a-
me ic combina ions.
2. The de elopmen o in e se models which make use o he non-dimensional o mula ion p e-
sen ed in his wo k.
3. The in oduc ion o mo e gene al c ack geome ies beyond plana .
37

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