Ad ances in Science, Technology and Enginee ing Sys ems Jou nal
Vol. 2, No. 5, 55-62 (2017)
www.as esj.com
P oceedings o In e na ional Con e ence on Applied Ma hema ics
(ICAM’2017), Taza, Mo occo
ASTES Jou nal
ISSN: 2415-6698
S eel hea ea ing: ma hema ical modelling and nume ical
simula ion o a p oblem a ising in he au omo i e indus y
Jos´
e Manuel D´
ıaz Mo eno1, Concepci´
on Ga c´
ıa V´
azquez1, Ma ´
ıa Te esa Gonz´
alez Mon esinos2,
F ancisco O eg´
on Gallego*,1, Giuseppe Viglialo o3
1Depa amen o de Ma em´
a icas, Facul ad de Ciencias, Uni e sidad de C´
adiz, 11510 Pue o Real, SPAIN,
[email p o ec ed], [email p o ec ed], [email p o ec ed].
2Depa amen o de Ma em´
a ica Aplicada I, ETS de Ingenie ´
ıa In o m´
a ica, Uni e sidad de Se illa, 41012 Se illa,
SPAIN, [email p o ec ed].
3Dipa imen o di Ma ema ica ed In o ma ica, Uni e si `
a degli S udi di Caglia i, iale Me ello 92 – 09123
Caglia i, ITALY, [email p o ec ed].
A R T I C L E I N F O A B S T R A C T
A icle his o y:
Recei ed: 10 June, 2017
Accep ed: 15 July, 2017
Online: 10 Decembe , 2017
We desc ibe a ma hema ical model o he indus ial hea ing and
cooling p ocesses o a s eel wo kpiece ep esen ing he s ee ing ack o
an au omobile. The goal o s eel hea ea ing is o p o ide a ha dened
su ace on c i ical pa s o he wo kpiece while keeping he es so and
duc ile in o de o educe a igue. The high ha dness is due o he phase
ans o ma ion o s eel accompanying he apid cooling. This wo k
akes in o accoun bo h hea ing-cooling s age and iscoplas ic model.
Once he gene al ma hema ical o mula ion is de i ed, we can pe o m
some nume ical simula ions.
Keywo ds :
S eel ha dening
Phase ansi ions
Po en ial Maxwell equa ions
Nume ical Simula ions
Fini e Elemen Me hods
1 In oduc ion
In he au oma i e indus y, many wo kpieces such
gea s, bea ings, acks and pinions, a e made o s eel.
S eel is an alloy o i on and ca bon. Gene ally, indus-
ial s eel has a ca bon con en up o abou 2 w %.
O he alloying elemen s may be p esen , such as C
and V in ools s eels, o Si, Mn, Ni and C in s ain-
less s eels. Mos s uc u al componen s in mechanical
enginee ing a e made o s eel. Ce ain o hese com-
ponen s, such as oo hed wheels, be el gea s, pinions
and so on, engaged each o he s in o de o ansmi
some kind o ( o a ional o longi udinal) mo emen .
In his si ua ion, he con ac su aces o hese compo-
nen s a e pa icula ly s essed. The goal o hea ea -
ing o s eel is o a ain a sa is ac o y ha dness. P io
o hea ea ing, s eel is a so and duc ile ma e ial.
Wi hou a ha dening ea men , and due o he su -
ace s esses, he gea ee h will soon ge damaged and
hey will no longe engage co ec ly.
In his wo k we a e in e es ed in he ma hema ical
desc ip ion o he ha dening p ocedu e o a ca s ee -
ing ack (see Figu e 1). This pa icula si ua ion is one
o he majo conce ns in he au omo i e indus y. In
his case, he goal is o inc ease he ha dness o he
s eel along he oo h line and a he same ime keep-
ing he es o he wo kpiece so and duc ile in o de
o educe a igue. This p oblem is go e ned by a non-
linea sys em o pa ial di e en ial equa ions coupled
wi h a ce ain sys em o o dina y di e en ial equa-
ions. Once he ull sys em is se we pe o m some
nume ical simula ions.
Figu e 1: Ca s ee ing ack.
Solid s eel may be p esen a di e en phases,
namely aus eni e, ma ensi e, baini e, pea li e and e -
i e. The phase diag am o s eel is shown in Fig-
u e 2. Fo a gi en w % o ca bon con en up o 2.11,
*Co esponding au ho . Depa amen o de Ma em´
a icas, Facul ad de Ciencias, Uni e sidad de C´
adiz, 11510 Pue o Real, SPAIN,
[email p o ec ed]
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J. M. D´
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all s eel phases a e ans o med in o aus eni e p o-
ided he empe a u e has been aised up o a ce -
ain ange. The minimum aus eniza ion empe a u e
(727◦) is a ained o a ca bon con en o 0.77 w %
(eu ec oid s eel). Upon cooling, he aus eni e is ans-
o med back in o he o he phases (see Figu e 3), bu
i s dis ibu ion depends s ongly on he cooling s a -
egy ([4, 12]).
Ma ensi e is he ha des cons i uen in s eel, bu
a he same ime is he mos b i le, whe eas pea li e
is he so es and mo e duc ile phase. Ma ensi e de-
i es om aus eni e and can be ob ained only i he
cooling a e is high enough. O he wise, he es o he
s eel phases will appea .
The ha dness o he ma ensi e phase is due o a
s ong supe sa u a ion o ca bon a oms in he i on la -
ice and o a high densi y o c ys al de ec s. F om
he indus ial s andpoin , hea ea ing o s eel has
a colla e al p oblem: ha dening is usually accom-
panied by dis o ions o he wo kpiece. The main
easons o hese dis o ions a e due o (1) he mal
s ains, since s eel phases unde go di e en olume -
ic changes du ing he hea ing and cooling p ocesses,
and (2) expe imen s wi h s eel wo kpieces unde ap-
plied loading show an i e e sible de o ma ion e en
when he equi alen s ess co esponding o he load
is in he elas ic ange. This e ec is called ans o ma-
ion induced plas ici y.
The hea ing s age is accomplished by an
induc ion-conduc ion p ocedu e. This echnique has
been success ully used in indus y since he las cen-
u y. Du ing a ime in e al, a high equency cu -
en passes h ough a coil gene a ing an al e na ing
magne ic ield which induces eddy cu en s in he
wo kpiece, which is placed close o he coil. The eddy
cu en s dissipa e ene gy in he wo kpiece p oducing
he necessa y hea ing.
2 Ma hema ical modeling
We conside he se ing co esponding o Figu e 4.
The domain Ωc ep esen s he induc o (made o cop-
pe ) whe eas Ωss ands o he s eel wo kpiece o be
ha dened. He e, he coil is he domain Ω=Ωs∪Ωc∪
S0. In his way, he wo kpiece i sel akes pa o he
coil.
In o de o desc ibe he hea ing-cooling p ocess,
we will dis inguish wo subin e als o ming a pa i-
ion o [0,T ], namely [0,T ] = [0,Th)∪[Th,Tc], Tc> Th>
0. The i s one [0,Th) co esponds o he hea ing p o-
cess. All along his ime in e al, a high equency
elec ic cu en is supplied h ough he conduc o
which in i s u n induces a magne ic ield. The com-
bined e ec o bo h conduc ion and induc ion gi es
ise o a p oduc ion e m in he ene gy balance equa-
ion (14), namely b(θ)|A +∇φ|2. This is Joule’s hea ing
which is he p incipal e m in hea p oduc ion. In ou
model, we will only conside h ee s eel phase ac-
ions, namely aus eni e (a), ma ensi e (m), and he
es o phases ( ). In his way, we ha e a+m+ = 1
and 0 ≤a,m, ≤1 in Ωs×[0,T ]. A he ini ial ime we
ha e (0) = 1 in Ωs. Upon hea ing only aus eni e can
be ob ained. In pa icula m= 0 in Ωs×[0,Th] and he
ans o ma ion o aus eni e is de i ed a he expense
o he o he phase ac ions ( ).
A he ins an =Th, he cu en is swi ched o
and du ing he ime in e al [Th,Tc] he wo kpiece is
se e ely cooled down by means o aqua-quenching.
The hea ing model
The cu en passing h ough he se o conduc o s
Ω=Ωc∪Ωs∪S0is modeled by he elec ic po en-
ial di e ence, ϕ0, applied on he su ace Γ2⊂Ωc(see
Figu e 4). No ice ha he applied po en ial on Γ1is
ze o. In he sequel, we pu Γ=Γ1∪Γ2.
The hea ing model in ol es he ollowing un-
knowns: he elec ic po en ial, φ; he magne ic ec-
o po en ial, A= (A1,A2,A3); he s ess enso ,
σ= (σij)1≤i,j≤3,σij =σji o all 1 ≤i,j ≤3; he dis-
placemen ield u= (u1,u2,u3); he aus eni e phase
ac ion, a; and he empe a u e, θ. Among hem, only
Ais de ined in he domain Dcon aining he se o
conduc o s Ω. On he o he hand, since he induc o
and he wo kpiece a e in close con ac , bo h φand θ
a e de ined in Ω. Since phase ansi ions only occu
in he wo kpiece, we may neglec de o ma ions in Ωc.
This implies ha σ,uand aa e only de ined in he
wo kpiece Ωs.
Since elec omagne ic ields gene a ed by high e-
quency cu en s a e sinusoidal in ime, bo h he elec-
ic po en ial, φ, and he magne ic po en ial ield, A,
ake he o m ([1, 2, 14, 15]) M(x, ) = Reheiω M(x)i,
whe e Mis a complex- alued unc ion o ec o ield,
and ω= 2π is he angula equency, being he
elec ic cu en equency. In gene al, Malso depends
on , bu a a ime scale much g ea e han 1/ω. In
his way, we may in oduce he complex- alued ields
ϕand Aas
φ= Re[eiω ϕ(x, )],A= Re[eiω A(x, )].(1)
As a a as he nume ical simula ion o a sys em
like (2)-(15) is conce ned, he in oduc ion o he new
a iables ϕand Ais qui e con enien since he ime
scale desc ibing he e olu ion o bo h ϕand Ais much
smalle han ha o he empe a u e θ. In he case o
s eel hea ea ing, is abou 80 KHz.
The hea ing model eads as ollows ([3, 9, 10, 7]):
∇ · (b(θ)∇ϕ) = 0 in ΩTh=Ω×(0,Th),(2)
∂ϕ
∂n = 0 on (∂Ω Γ)×(0,Th),(3)
ϕ= 0 on Γ1×(0,Th), ϕ=ϕ0on Γ2×(0,Th), (4)
b0(θ)iωA+∇ × 1
µ∇ × A!−δ∇(∇ · A)
=−b0(θ)∇ϕin D×(0,Th),(5)
A= 0 on ∂D ×(0,Th),(6)
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J. M. D´
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Tempe a u e ◦C
400
600
800
1000
1200
1400
1600
012345 6 7
Fe % C Fe3C (C)
Hypoeu ec oide
Hype eu ec oide
α
(Fe)
Fe i e + Cemen i e
727 ◦C
1148 ◦C
912 ◦C
1394 ◦C
0.77%
(Eu ec oid)
2.11% 4.30%
Aus eni e + Cemen i e
Liquid
γ+ Liquid
γ
Aus eni e
Figu e 2: I on-ca bide phase diag am.
Fe i e
Pea li e
Baini e
Ma ensi e
Aus eni e
Fe i e
Pea li e
Baini e
Ma ensi e
Hea ing Cooling
Figu e 3: Mic ocons i uen s o s eel. Upon hea ing, all phases a e ans o med in o aus eni e, which is ans o med back o he o he
phases du ing he cooling p ocess. The dis ibu ion o he new phases depends s ongly on he cooling s a egy. A high cooling a e
ans o ms aus eni e in o ma ensi e. A slow cooling a e ans o ms aus eni e in o pea li e.
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J. M. D´
ıaz e al. / Ad ances in Science, Technology and Enginee ing Sys ems Jou nal Vol. 2, No. 5, 55-62 (2017)
Ωs(s eel)
Ωc(coppe ) Ωc
Γ1Γ2
S0
S0
D
Figu e 4: Domains D,Ω=Ωs∪Ωc∪S0and he aces Γ1,Γ2⊂Ωc. The induc o Ωcis made o coppe . The wo kpiece con ains a oo hed
pa o be ha dened by means o he hea ing-cooling p ocess desc ibed below. I is made o a hypoeu ec oid s eel. The domain is aken as
a big enough ec angle con aining bo h he induc o and he ack
−∇ · σ=Fin Ωs×(0,Th),(7)
σ=Kε(u)−A1(a,m,θ)I
−Z
0
γ(a,m,a ,m ,θ)Sdτ,(8)
u= 0 on Γ0×(0,Th),(9)
σ·n= 0 on (∂Ωs Γ0)×(0,Th),(10)
a =1
τa(θ)(aeq(θ)−a)H(θ−As) in Ωs×(0,Th),(11)
a(0) = 0 in Ωs,(12)
α(θ,a,m,σ)θ − ∇ · (κ(θ)∇θ)
+3 ¯
κq(a,m)θ(∇ · u −3A2(a ,m ,θ))
=b(θ)|A +∇φ|2−ρLaa
+A2(a ,m ,θ) σ
+γ(a,m,a ,m ,θ)|S|2in ΩTh,(13)
∂θ
∂n = 0 on ∂Ω×(0,Th),(14)
θ(0) = θ0in Ω.(15)
He e, b(θ) is he elec ical conduc i i y (by b(θ)
we mean he unc ion (x, )7→ b(x,θ(x, )), and also o
κ(θ), e c.); ϕ0 ep esen s he po en ial ex e nal sou ce.
The domain Dcon aining he se o conduc o s is
aken big enough so ha he magne ic ec o po en-
ial A anishes on i s bounda y ∂D (in ou model, is
aken o be a 2D ec angle o a 3D cube). Since bo h
σand aa e only de ined in Ωs, when hey appea in
a e m e e ed in Ω, we mean ha his e m anishes
ou side Ωs( o ins ance, −ρLaa appea ing in (13));
b0(x,s) = b(x,s) i x∈Ω,b0(x,s) = 0 elsewhe e; µ=µ(x)
is he magne ic pe meabili y; δ > 0 is a small cons an ;
Fis a gi en ex e nal o ce (usually F= 0); K=Kijkl,
1≤i,j,k,l ≤3 is he s i ness enso . S eel can be con-
side ed as an iso opic and homogenous ma e ial so
ha
Kijkl =¯
λδijδkl+¯
µ(δikδjl+δilδjk), o all i,j,k,l ∈ {1,2,3}
whe e ¯
λ≥0 and ¯
µ > 0 a e he Lam´
e coe icien s
o s eel; ε(u) = 1
2(∇u+∇uT) is he s ain enso ;
A1(a,m,θ)Imodels he he mal s ain, Ibeing he 3×3
uni y ma ix, whe eas A1(a,m,θ) is de ined as
A1(a,m,θ) = qaa(θ−θa) + qmm(θ−θm)
+q (1 −a−m)(θ−θ ),
and in i s u n qa,qmand q a e he he mal expan-
sion coe icien s o he phase ac ions a,mand , e-
spec i ely, and θa,θmand θ a e e e ence empe a-
u es (no ice ha du ing he hea ing s age is m= 0);
R
0γ(a,m,a ,m ,θ)Sdτgi es he model, h ough he
unc ion γ, o he ans o ma ion induced plas ici y
s ain enso , whe e S=σ−1
3 σIis he de ia o o σ,
ha is, he ace ee pa o he s ess enso ; Γ0is a
ce ain smoo h enough pa o ∂Ωs;nis he uni ou e
no mal ec o o he e e ed bounda y; he unc ions
τa(θ), aeq(θ) a e gi en om expe imen al da a (see
Figu e 5), and His he Hea iside unc ion; κ(θ) is he
he mal conduc i i y; he unc ions appea ing in (13)
a e gi en as ollows
α(θ,a,m,σ) = ρcε−9¯
κq(a,m)2θ−q(a,m) σ,
whe e ρand cεa e he s eel densi y and he spe-
ci ic hea capaci y a cons an s ain, espec i ely, ¯
κ=
1
3(3 ¯
λ+ 2 ¯
µ) is he bulk modulus, and q(a,m) is de ined
as
q(a,m) = qaa+qmm+ (1 −a−m)q ;
A2(a ,m ,θ) = qaa (θ−θa) + qmm (θ−θm)
−q (a +m )(θ−θ ).
Finally, La>0 is he la en hea ela ed o he aus en-
i e phase ac ion. No ice ha , in a mo e gene al si u-
a ion ρ,cεand Lamay also depend on a,mand/o θ.
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J. M. D´
ıaz e al. / Ad ances in Science, Technology and Enginee ing Sys ems Jou nal Vol. 2, No. 5, 55-62 (2017)
AsA θ
τa(θ)
aeq(θ)
Figu e 5: Func ions aeq and τa.
Equa ions (2) and (5) de i e om Maxwell’s equa-
ions. In [9], i is assumed he Coulomb gauge condi-
ion o he magne ic ec o po en ial, namely, ∇ · A=
0. He e, we do no impose his condi ion since his
makes appea an undesi ed p essu e g adien in he
equa ion o A. In i s u n, we include a penal y e m
in his equa ion o he o m −δ∇(∇ · A). In doing so,
bo h he heo e ical analysis and he nume ical simu-
la ions a e simpli ied.
Equa ion (7) is a quasis a ic balance law o mo-
men um and (8) is Hooke’s law. The ans o ma ion
o aus eni e om he ini ial phase (0) = 1 is desc ibed
in (11).
Finally, equa ion (13) de i es om he balance law
o in e nal ene gy. As i has been poin ed ou abo e,
Joule’s hea ing is he main esponsible in hea p o-
duc ion. Since γ(a,m,a ,m ,θ)|S|2≥0, he con ibu-
ion o he ans o ma ion induced plas ici y o he
ene gy balance is also a p oduc ion e m. On he o he
hand, du ing he hea ing s age we ha e a ≥0 so ha
−ρLaa ≤0. This means ha he ans o ma ion o
aus eni e abso bs ene gy, which is eleased du ing he
cooling s age.
The cooling model
The hea ing p ocess ends, he high equency cu en
passing h ough he coil is swi ched-o and aqua-
quenching begins. The quenching is jus modeled ia
he Robin bounda y condi ion gi en in (25).
We pu aTh=a(Th), ha is, aThis he aus eni e
phase ac ion dis ibu ion a he inal hea ing ins an
Thob ained om (11). In he same way, we de ine
θTh=θ(Th). Ob iously, hese unc ions will be aken
as he ini ial phase ac ion dis ibu ion and empe -
a u e, espec i ely, in he cooling model. He e we
use he Kois inen-Ma bu ge model ([11, 13]) o he
desc ip ion o he ans o ma ion o ma ensi e om
aus eni e.
The cooling model eads as ollows
−∇ · σ=Fin Ωs×(Th,Tc),(16)
σ=Kε(u)−A1(a,m,θ)I
−Z
0
γ(a,m,a ,m ,θ)Sdτ,(17)
u= 0 on Γ0×(Th,Tc),(18)
σ·n= 0 on (∂Ωs Γ0)×(Th,Tc),(19)
a =1
τa(θ)(aeq(θ)−a)H(θ−As) in Ωs×(Th,Tc),(20)
a(Th) = aThin Ωs,(21)
m =cm(1 −m)H(−θ )H(Ms−θ) in Ωs×(Th,Tc),(22)
m(Th) = 0 in Ωs,(23)
α(θ,a,m,σ)θ − ∇ · (κ(θ)∇θ)
+ 3 ¯
κq(a,m)θ(∇ · u −3A2(a ,m ,θ))
=−ρLaa +ρLmm +A2(a ,m ,θ) σ
+γ(a,m,a ,m ,θ)|S|2in Ω×(Th,Tc),(24)
∂θ
θn =β(x, )(θ−θe) on ∂Ω×(Th,Tc),(25)
θ(Th) = θThin Ω.(26)
In (22) cm>0 is a cons an alue. Also, in (24),
Lm>0 is he la en hea ela ed o he ma ensi e
phase ac ion. The unc ion β(x, ) in (25) is a hea
ans e coe icien and is gi en by
β(x, ) = (0 on ∂Ω∩∂Ωc,
β0( ) on ∂Ω∩∂Ωs.
whe e β0( )>0 (usually aken o be cons an ). Finally,
θeis he empe a u e o he quenchan .
The ma hema ical analysis o a sys em simila
o (16)-(26) can be seen in [3]. In his e e ence, an
exis ence esul is shown assuming ha he da a a e
smoo h enough and Tc−This su icien ly small.
Dh
Figu e 6: Domain iangula ion. The mesh con ains 61790 ian-
gles and 30946 e ices.
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J. M. D´
ıaz e al. / Ad ances in Science, Technology and Enginee ing Sys ems Jou nal Vol. 2, No. 5, 55-62 (2017)
3 Nume ical simula ion
Using he F ee em++ package ([8]), we ha e pe -
o med some nume ical simula ions o he app oxi-
ma ion o he solu ion o he sys ems (2)-(15) and (16)-
(26). We wan o desc ibe he ha dening ea men o
a ca s ee ing ack du ing he hea ing-cooling p ocess.
The goal is o p oduce ma ensi e along he oo h line
oge he wi h a hin laye in i s neighbo hood inside
he s eel wo kpiece ([5, 6]).
Dh
Figu e 7: Domain iangula ion. Elemen densi y nea h ee
ee h.
Figu e 4 shows he open se s D,Ω=Ωs∪Ωc∪S
and he aces Γ1and Γ2( hey appea s ick oge he
in his igu e) which in e ene in he se ing o he
p oblem. The wo kpiece con ains a oo hed pa o
be ha dened by means o he hea ing-cooling p ocess
desc ibed abo e. I is made o a hypoeu ec oid s eel.
The open se D ¯
Ωis ai . The magne ic pe meabili y µ
in (5) is hen gi en by
µ(x) = 








µ0i x∈D ¯
Ω,
0.99995µ0i x∈Ωc,
2.24 ×103µ0i x∈Ωs,
whe e µ0= 4π×10−7(N/A2) is he magne ic cons an
( acuum pe meabili y).
The ma ensi e phase can only de i e om he
aus eni e phase. Thus we need o ans o m i s
he c i ical pa o be ha dened ( he oo h line) in o
aus eni e. Fo ou hypoeu ec oid s eel, aus eni e
only exis s in a empe a u e ange close o he in-
e al [1050,1670] (in K). Du ing he i s s age, he
wo kpiece is hea ed up by conduc ion and induc ion
(Joule’s hea ing) which ende s he oo h line up o
he desi ed empe a u e. In o de o ans o m he
aus eni e in o ma ensi e, we mus cool i down a a
e y high a e. This second s age is accomplished by
aquaquenching.
In his simula ion, he inal ime o he hea ing
p ocess is Th= 5.5 seconds and he cooling p ocess
ex ends also o 5.5 seconds, ha is Tc= 11.
We ha e used he ini e elemen s me hod o he
space app oxima ion and a C ank-Nicolson scheme
o he ime disc e iza ion. Figu es 6 and 7 show he
iangula ion o Din ou nume ical simula ions. We
ha e used P2-Lag ange app oxima ion o ϕ,Aand θ
and P1 o aand m.
In Figu e 8 we can see he empe a u e dis ibu-
ion o he ack along he oo h line a di e en in-
s an s o he he hea ing-cooling p ocess. The ini ial
empe a u e is θ0= 300K. A = 5.5 he hea ing p o-
cess ends and he compu ed empe a u e shows ha
he empe a u e along he ack oo h line lies in he
in e al [1050,1670] (K).
= 1s
= 3s
= 5.5s
= 6s
= 7s
Figu e 8: Tempe a u e e olu ion a ins an s = 1, = 3, = 5.5
(end o he hea ing s age, aqua-quenching begins), = 6 and = 7
seconds, espec i ely. A = 5.5s he empe a u e along he oo h
pa has eached he aus eniza ion le el in his pa o he ack. The
empe a u e is measu ed in Kel in.
= 5.5s (le ), = 6.5s ( igh ),
= 7s (le ), = 8s ( igh )
= 9s (le ) and = 11s ( igh ).
Figu e 9: T ans o ma ion o he aus eni e phase ac ion du ing
he aquaquenching a ime ins an s =5.5, 6.5, 7, 8, 9, and 11 sec-
onds, espec i ely.
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Figu e 9 shows he aus eni e e olu ion om he
beginning o he cooling s age. Blue co esponds o
0% while ed is 100%. We obse e ha ma ensi e
s a s o appea , app oxima i ely, one second a e he
beginning o he cooling s age. A he inal ins an ,
all he amoun o aus eni e has been ans o med in o
ma ensi e as i is shown in Figu e 10 .
= 5.5s (le ), = 6.5s ( igh ),
= 7s (le ), = 8s ( igh )
= 9s (le ) and = 11s ( igh ).
Figu e 10: T ans o ma ion o he ma ensi e phase ac ion om
aus eni e du ing he aquaquenching a ime ins an s =5.5, 6.5, 7,
8, 9, and 11 seconds, espec i ely.
Figu e 11 shows he aus eniza ion along he oo h
line a he end o he hea ing p ocess T= 5.5 seconds.
Figu e 12 shows he inal dis ibu ion o ma en-
si e om aus eni e along he ack oo h line h ough
he cooling s age = 11 seconds. We ha e good ag ee-
men e sus he expe imen al esul s ob ained in he
indus ial p ocess.
Du ing he hea ing-cooling p ocess, he wo k-
piece is de o med so ha an indus ial ec i ica ion is
needed (o o he wise he ack would be useless). Fig-
u es 13 and 14 shows he di e en de o ma ions un-
de gone by he wo kpiece.
Figu e 11: Hea ing p ocess. Aus eni e a = 5.5 along he ack
oo h line.
Figu e 12: Cooling p ocess. Ma ensi e ans o ma ion a he i-
nal s age o he cooling p ocess = 11 seconds.
Figu e 13: Dis o ed mesh (wi h a scale ac o o 10) a e he
hea ing s age. The aus eni e ans o ma ion along he oo h line
changes he o iginal p o ile.
Figu e 14: Dis o ed mesh (wi h a scale ac o o 10) a e he
cooling s age. The o iginal con igu a ion is pa ially eco e ed. Due
o he plas ici y e ec and he lack o he uppe suppo s, he ack
bends down.
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J. M. D´
ıaz e al. / Ad ances in Science, Technology and Enginee ing Sys ems Jou nal Vol. 2, No. 5, 55-62 (2017)
Con lic o In e es The au ho s decla e no con lic
o in e es .
Acknowledgmen s This esea ch was pa ially sup-
po ed by Minis e io de Educaci´
on y Ciencia unde
g an s MTM2010-16401 and TEC2014-54357-C2-2-
R wi h he pa icipa ion o FEDER, and Conseje ´
ıa
de Educaci´
on y Ciencia de la Jun a de Andaluc´
ıa, e-
sea ch g oup FQM–315.
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