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IOP Con . Se ies: Ma e ials Science and Enginee ing 1209 (2021) 012049
IOP Publishing
doi:10.1088/1757-899X/1209/1/012049
1
C ack beha iou a he in e ace o a su ace laye applied on
a s eel subs a e by lase cladding
L Malíko á1,2, P Doubek2,3, P Mia ka1,2 and S Sei l1,2
1 Ins i u e o Physics o Ma e ials Czech Academy o Sciences, . . i., Žižko a
513/22, 616 00 B no, Czech Republic
2 Ins i u e o S uc u al Mechanics, Facul y o Ci il Enginee ing, B no Uni e si y o
Technology, Ve eří 331/95, 602 00 B no, Czech Republic
3 OMNI-X CZ s. .o., Šámalo a 60a, 615 00 B no, Czech Republic
E-mail: m[email p o ec ed]u b .cz
Abs ac . An in luence o he bi-ma e ial in e ace be ween a s eel subs a e and a hin
p o ec i e laye applied h ough lase cladding was in es iga ed. A ange o elas ic p ope ies
and hicknesses o he laye we e conside ed o co e he beha iou o a sho c ack in he
selec ed ma e ials such as b onze, nickel o cobal alloys. The special case o he c ack
e mina ing di ec ly a he in e ace was in es iga ed, which is connec ed o he necessi y o
applica ion o gene alized app oaches o linea elas ic ac u e mechanics. The esul s
con ibu e o be e unde s anding o ac u e esponse o selec ed ma e ials and o a mo e
eliable decision on choosing a p ope ma e ial o he p o ec i e laye .
1. In oduc ion
In echnical p ac ice, s uc u al elemen s which a e o med by combining laye s o di e en me allic
ma e ials a e used, see e.g. [1, 2]. These combina ions o ma e ials a e c ea ed based on he equi ed
unc ion ei he du ing he design o he s uc u al componen o du ing i s eno a ion and epai [3].
The e is a ela i ely la ge numbe o me hods o applying su ace laye s. One o he mode n,
widely applicable me hods is lase cladding echnology, a e iew on his echnology can be ound in
[4]. The p inciple o lase cladding is ha he me al powde o wi e is ed o a lase beam, whe e his
ma e ial is mel ed oge he wi h he base ma e ial o he pa and a deposi ion laye is o med on he
su ace o he pa . A me allu gical bond is o med be ween he deposi ion laye and he base ma e ial,
gua an eeing excellen adhesion be ween he cladding laye and he pa [5]. Wi h he igh choice o
ma e ial combina ions, lase cladding echnology can eplace some olde echnologies e ec i ely as
hey a e o en p oblema ic om an ecological poin o iew.
In his wo k, ac u e beha iou o a c ack e mina ing a he bi-ma e ial in e ace be ween he base
s eel ma e ial and a lase -cladded laye o selec ed ma e ials is in es iga ed.
2. Gene alized ac u e mechanics
The goal o he pape was o es ima e he alue o he c i ical s ess ha is needed o pene a ion o
he c ack e mina ing di ec ly a he in e ace o he ma e ial o subs a e, see [6–9]. I is well known
ha a c ack wi h i s ip a a bi-ma e ial in e ace ep esen s a gene al s ess concen a o , which means
ha he s ess singula i y is g ea e o lowe han 0.5 in dependence on he mu ual elas ic misma ch o
bo h ma e ials. The s ess enso componen s nea he c ack ip can be exp essed ia he ollowing
equa ion:
Young Scien is 2021
IOP Con . Se ies: Ma e ials Science and Enginee ing 1209 (2021) 012049
IOP Publishing
doi:10.1088/1757-899X/1209/1/012049
2
𝜎𝑖𝑗 =𝐻I
√2𝜋𝑟−𝑝𝑓𝑖𝑗(𝑝,𝛼,𝛽) (1)
The equa ion (1) shows he dependence be ween he s ess sij, gene alized s ess in ensi y ac o HI,
adial dis ance measu ed om he c ack ip , s ess singula i y exponen p and known unc ion ij ha
is dependen besides o he hings on he bi-ma e ial pa ame e s a and b, see [10]. These cons an s
depend on he elas ic p ope ies (Young’s modulus E and Poisson’s a io n) o he ma e ials on bo h
sides o he in e ace as well as on he s ess/s ain condi ions. Fo he case o he plane s ain s a e,
hei alues can be calcula ed as (no e ha index 1 ep esen s he su ace laye and index 2 deno es he
ma e ial o he subs a e):
𝛼 = 𝐸1
𝐸2∙1+𝜈1
1+𝜈2−1
4(1−𝜈1) , 𝛽 = 𝐸1
𝐸2∙1−𝜈2
2
1−𝜈1
2 (2)
The s ess singula i y exponen needs o be calcula ed as p = 1 – l, whe e l is he eigen alue and o
a c ack pe pendicula o he in e ace can be de e mined om he cha ac e is ic equa ion, see [10] o
mo e de ails:
𝜆2(−4𝛼2+4𝛼𝛽)+2𝛼2−2𝛼𝛽+ 2𝛼 −𝛽 +1+(−2𝛼2+2𝛼𝛽 −2𝛼 + 2𝛽)cos(𝜆𝜋)= 0 (3)
The gene alized s ess in ensi y ac o (GSIF) can be calcula ed ia di ec me hod when he
nume ical solu ion o he p oblem is ound. Then, i s alue can be de e mined o example om he
de elopmen o he opening s ess ahead o he c ack ip. Using equa ion (1) enables he ex apola ion
o he dependence HI = HI ( ) o he loca ion = 0 whe e he inal alue o he GSIF can be ound.
Finally, when he c i ical load leading o c ack p opaga ion h ough he in e ace shall be
calcula ed, a c i ical alue o he GSIF needs o be es ima ed by means o selec ed ac u e c i e ia, see
he ollowing subsec ions.
2.1. Mean angen ial s ess alue c i e ion
Acco ding o his c i e ion, he s abili y condi ion o a c ack is ela ed o he a e age s ess calcula ed
ac oss a dis ance d ahead o he c ack ip, mo e de ails can be ound o ins ance in [11].
Ma hema ically w i en:
𝐻IC = 𝐾IC 2𝑑𝜆−1 2
⁄
2−𝜆+𝑔R (4)
The meaning o he symbols is as ollows: HIC ep esen s he c i ical alue o he GSIF, KIC
ep esen s he ac u e oughness o he ma e ial, d is he dis ance ahead o he c ack ip, whe e he
c i e ion is applied, l is he eigen alue and gR is a known unc ion o he pa ame e s a and b as hey a e
de ined in equa ion (2), see e.g. [12].
2.2. Gene alized s ain ene gy densi y ac o c i e ion
A simila equa ion be ween he c i ical alue o he GSIF and ac u e oughness can be de i ed also
wi hin he idea o he gene alized s ain ene gy densi y c i e ion, whe e he minimum alue o he
s ain ene gy densi y ac o is conside ed:
𝐻IC = 𝐾IC ∙𝑑𝑝−1 2
⁄(1−2𝜈
(1−𝑝)2[4(1−2𝜈)+(𝑔R−𝑝)2])1 2
⁄ (5)
The meaning o he symbols is as desc ibed in he p e ious sec ions. I is wo hy o no e ha he
elas ic and ac u e mechanical p ope ies in equa ions (4) and (5) ep esen he p ope ies o he
ma e ial behind he in e ace.
3. Geome y and ma e ial p ope ies conside ed in he nume ical model
A geome y o he nume ical model was sugges ed wi h ega d o he eal samples, when a hin su ace
laye o he hickness be ween 1 and 3 mm is cladded on a cylind ical s eel subs a e wi h he diame e
abou 82 mm, see igu e 1.
Young Scien is 2021
IOP Con . Se ies: Ma e ials Science and Enginee ing 1209 (2021) 012049
IOP Publishing
doi:10.1088/1757-899X/1209/1/012049
3
Figu e 1. Real samples wi h
lase -cladded su ace laye s made
o ha d ch ome and aluminium
b onze applied on s eel cylinde s.
Pa icula ly, a simpli ied bi-ma e ial c acked ba unde pu e ension was modelled acco ding o he
schema in igu e 2.
Figu e 2. Schema o he bi-ma e ial c acked ba unde pu e ension.
The ma e ial p ope ies conside ed wi hin he pa ame ical s udy as well as he dimensions o he
nume ical model can be ound in able 1.
Table 1. Ma e ial p ope ies, dimensions
and loading applied in he nume ical
model o he c acked ba unde pu e
ension, see igu e 2.
Quan i y
Value
h1
1, 2 and 3 mm
h2
40 mm
a
h1
L
6(h1+h2)
E1
100 ÷ 300 GPa
E2
200 GPa
n1 = n2
0.3
sappl
800 MPa
s eel subs a e
a
ha d ch ome
laye
aluminium b onze
laye
Young Scien is 2021
IOP Con . Se ies: Ma e ials Science and Enginee ing 1209 (2021) 012049
IOP Publishing
doi:10.1088/1757-899X/1209/1/012049
4
The alues o he Young’s modulus o he ma e ial o he cladded laye we e conside ed o be 100,
150, 200, 250 and 300 GPa. In he model, quad ila e al 8-node elemen s 183 we e applied and he
mesh nea he c ack ip was e ined o emphasize he c ack- ip singula i y (al hough i s alue is
di e en om 0.5 and i holds ha p = 0.43 ÷ 0.55 o he s udied cases). Fo calcula ion o he GSIF
ia he di ec me hod, always he nodes a he dis ance o 0.2 o 1 mm om he c ack ip we e u ilized.
Values o he s ess singula i y exponen and GSIF a e in oduced in able 2.
Table 2. Values o he a ious Young’s modulus o he su ace laye and co esponding s ess
singula i y exponen s and GSIF alues o sappl = 800 MPa.
E1 [GPa]
100
150
200
250
300
E1/E2 [-]
0.50
0.75
1.00
1.25
1.50
p [-]
0.43389
0.47133
0.50000
0.52324
0.54279
HI [MPa·mp] o h1 = 1 mm
62.494
55.157
50.340
46.811
44.045
HI [MPa·mp] o h1 = 2 mm
84.963
77.456
71.914
67.474
63.754
HI [MPa·mp] o h1 = 3 mm
104.570
96.266
89.603
84.006
79.186
The c i ical dis ances ha appea in he gene alized ac u e c i e ia we e conside ed o be
d = = 1 mm in ag eemen wi h ecommenda ions published o ins ance in [13–15].
4. Resul s and discussion
In igu es 3 and 4, he dependences o he c i ical s ess on he elas ic misma ch (Young’s moduli
a io) can be ound o he DKI h and DKIC alue o 9 and 60 MPa·m1/2 conside ed as he ma e ial
pa ame e s o he s eel subs a e, espec i ely. The alues o he c i ical ensile s ess a e calcula ed
om a simple ela ion:
𝜎c= 𝜎appl ∙𝐻IC
𝐻I (6)
The plo s a e p in ed o he h ee alues o he hickness o he su ace laye and bo h gene alized
ac u e c i e ia a e applied.
(a)
(b)
Figu e 3. Dependence o he alues o he c i ical s ess (when DKI h = 9 MPa·m1/2) on he elas ic
misma ch o a ious hicknesses o he su ace laye calcula ed ia (a) mean angen ial s ess alue
c i e ion and (b) gene alized s ain ene gy densi y ac o c i e ion.
Young Scien is 2021
IOP Con . Se ies: Ma e ials Science and Enginee ing 1209 (2021) 012049
IOP Publishing
doi:10.1088/1757-899X/1209/1/012049
5
(a)
(b)
Figu e 4. Dependence o he alues o he c i ical s ess (when DKIC = 60 MPa·m1/2) on he elas ic
misma ch o a ious hicknesses o he su ace laye calcula ed ia (a) mean angen ial s ess alue
c i e ion and (b) gene alized s ain ene gy densi y ac o c i e ion.
The esul s plo ed in igu es 3 and 4 show how he c i ical s ess necessa y o c ack p opaga ion
h ough he cladded su ace laye /s eel subs a e in e ace depends on he elas ic misma ch be ween
bo h laye as well as on he hickness o he su ace laye . The ollowing conclusions can be s a ed:
 The c i ical s ess necessa y o beginning o he s able a igue long c ack g ow h
(co esponding o DKI h = 9 MPa·m1/2) is be ween he alues o 60 and 200 MPa based on
he a io o he elas ic moduli, su ace laye hickness and ac u e c i e ion applied.
 The alues o sc necessa y o uns able c ack g ow h (co esponding o
DKIC = 60 MPa·m1/2) a e much highe , be ween 400 and 1200 MPa.
 The mean angen ial s ess alue c i e ion gi es sligh ly highe alues o he sc alues and
i is he e o e less conse a i e han he gene alized s ain ene gy densi y ac u e c i e ion.
 Fo hinne su ace laye s, i is necessa y o apply highe ensile load o s able/uns able
c ack p opaga ion – his is o cou se connec ed o he o al c ack leng h (a = h1).
 The mo e complian ma e ial o he su ace laye , he highe c i ical s ess is necessa y o
he c ack o be able o p opaga e h ough he in e ace o he s eel subs a e.
Conside ing he dependences discussed abo e, a he complian ma e ials o he cladded laye shall
be ecommended, such as ha d ch ome (wi h Young’s modulus o ca 104 GPa), aluminium b onze
(E ~ 113 GPa) o coppe be yllium alloys (elas ic modulus abou 130 GPa). On he o he hand, choice
o di e en cobal alloys ( ha a e a he ough wi h E o e 200 GPa) as he su ace laye could
dec ease he ac u e esis ance o he s uc u e assuming he p esence o a c ack wi h i s ip a he bi-
ma e ial in e ace.
5. Conclusions
A nume ical s udy has been pe o med in o de o assess he ac u e beha iou o a c ack e mina ing
a he bi-ma e ial in e ace be ween a su ace laye cladded on a s eel subs a e. Va ious elas ic
p ope ies as well as a ious hicknesses o he su ace laye we e conside ed and ini e elemen
calcula ion in combina ion wi h gene alized linea elas ic ac u e mechanics was pe o med. The
ob ained esul s show ha lase cladding o ma e ials wi h lowe elas ic modulus (such as ha d
ch ome, aluminium b onze and/o coppe be yllium alloys) can imp o e ac u e esponse o he bi-
ma e ial s uc u e when a sho c ack wi h i s ip a he in e ace is p esen ed. Bo h gene alized ac u e
c i e ia applied b ings highe alues o he c i ical s ess ha is necessa y o s able/uns able a igue
c ack g ow h o he s eel subs a e when a mo e complian ma e ial is cladded on he su ace o he
specimen.

Young Scien is 2021
IOP Con . Se ies: Ma e ials Science and Enginee ing 1209 (2021) 012049
IOP Publishing
doi:10.1088/1757-899X/1209/1/012049
6
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[4] Zhu L, Xue P, Lan Q, Meng G, Ren Y, Yang Z, Xu P and Liu Z 2021 Op ics & Lase
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[5] Lase cladding – Lase The m s. .o. Homepage – Lase The m s. .o. [28.6.2021] A ailable
online h ps://www.lase he m.cz/eng/ echnologies/lase - echnologies/lase -cladding
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[11] Nahlik L, Knesl Z and Klusak J 2008 Engng. Mech. 15 2 99
[12] Knesl Z, Knapek A and Bedna K 1998 E alua ion o he c i ical s ess in bonded ma e ials wi h
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Acknowledgmen s
Financial suppo om he Facul y o Ci il Enginee ing, B no Uni e si y o Technology (p ojec
No. FAST-S-21-7338) is g a e ully acknowledged.