Finite Element Based Physical Chemical Modeling of Corrosion in Magnesium Alloys

13 Pages·2017·3.74 MB·English

metals
Article
Finite Element Based Physical Chemical Modeling of
Corrosion in Magnesium Alloys
VenkateshVijayaraghavan1,*,AkhilGarg2,LiangGao3andRangarajanVijayaraghavan4
1 SchoolofEngineering,MonashUniversityMalaysia,47500BandarSunway,SelangorDarulEhsan,Malaysia
2 DepartmentofMechatronicsEngineering,ShantouUniversity,Shantou515063,China;[email protected]
3 TheStateKeyLaboratoryofDigitalManufacturingEquipmentandTechnology,HuazhongUniversityof
ScienceandTechnology,Wuhan430074,China;[email protected]
4 MaterialsEngineeringDepartment,KOPSurfaceProductsSingaporePteLtd,77ScienceParkDrive118256,
Singapore;[email protected]
* Correspondence:[email protected];Tel.:[phone]
AcademicEditor:HugoF.Lopez
Received:18January2017;Accepted:28February2017;Published:7March2017
Abstract: Magnesiumalloyshavefoundwidespreadapplicationsindiversefieldssuchasaerospace,
automotive,bio-medicalandelectronicsindustriesduetoitsrelativelyhighstrength-to-weightratio.
However, stress corrosion cracking of these alloys severely restricts their applications in several
novel technologies. Hence, it will be useful to identify the corrosion mechanics of magnesium
alloysunderexternalstressesasitcanprovidefurtherinsightsondesignofthesealloysforcritical
applications. Inthepresentstudy,thecorrosionmechanicsofacommonlyusedmagnesiumalloy,
AZ31,isstudiedusingfiniteelementsimulationwithamodifiedconstitutivematerialdamagemodel.
Thedataobtainedfromthefiniteelementmodelingwerefurtherusedtoformulateamathematical
modelusingcomputationalintelligencealgorithm. Sensitivityandparametricanalysisofthederived
modelfurthercorroboratedthemechanicalresponseofthealloyinlinewiththecorrosionphysics.
Theproposedapproachisanticipatedtobeusefulformaterialsengineersforoptimizingthedesign
criteriaformagnesiumalloyscateredforhightemperatureapplications.
Keywords: finiteelementanalysis;corrosionmechanics;AZ31alloy;computationalintelligence
1. Introduction
Researchinmagnesium(Mg)alloyshasfoundwidespreadinterestinrecentyearsduetotheir
attractive strength-to-weight ratio, lightweight and high stiffness. These extraordinary properties
makeMgalloysparticularlyusefulforapplicationsrequiringhighperformanceandenergysavings
suchasstructuralengineering,automotiveandbiomedicalapplications[1–3]. Itshouldalsobenoted
thattheweakcorrosionresistanceofMgalloysrestrictstheirapplicationsinseveralnovelfields[4–7].
MostoftheexistingstudiesinMgalloysarefocusedoninvestigatingthemechanicalstrengthbased
onspecificapplicationssuchasstructural,mechanicalorbio-medicalengineering. Hence,thereisa
needtostudythemechanicsofMgalloysundercorrosiveconditions. ItisalsowellknownthatMg
alloysaremoresusceptibletostresscorrosioncracking(SCC).Hence,aneffectivedeploymentofMg
alloysinnewandemergingfieldswillrequireanunderstandingofthebehaviorofcorrodedMgalloys
undermechanicalandthermalloadingconditions.
MostoftheliteraturestudiesonMgalloysarebasedontheirmechanicalpropertiesforspecific
applications. Argadeetal.[8]studiedthemechanismofAZ31Mgalloyunderslowstrainratetesting
technique. TheyfoundthatthecorrosionofAZ31ispredominantlyinfluencedbyitsmicrostructure
whichresultsinafavorabletextureforincreasedadsorptionofhydrogen. Suetal.[9]deployeda
controlledphosphatedepositiontechniqueforimprovingthesurfacebiocompatibilityofAZ60Mg
Metals2017,7,83;doi:10.3390/met7030083 www.mdpi.com/journal/metals
Metals2017,7,83 2of13
alloy. TheyfoundthatthecoatingprocessisinfluencedbytemperatureandpH,whichaffectsthe
corrosionresistanceofAZ60alloy. Choudaryetal.[10]investigatedtheSCCmechanismofvarious
aluminum free Mg alloys in a simulated human fluid environment. It was found that the SCC of
Mg alloys in the simulated condition were attributed to their electrochemical and microstructural
characteristics.Gauretal.[11]developedasilanebasedbiodegradablecoatingforimprovingcorrosion
resistance of Mg alloys in physiological environment. They found that formation of a stable and
uniform hydroxide layer on alloys surface facilitated adhesion of silane coating. Wei et al. [12]
investigated the corrosion behavior of plasma formed coatings on Mg alloys with respect to the
aluminumcompositioninthealloymix. Itwasfoundthatthedifferenceinporositylevelofcoatingsis
causedbydifferentdischargingbehaviorsinMgalloys.
In addition to experimental studies, some simulation studies have also been conducted to
investigatethecorrosionmechanicsofMgalloys. Groganetal.[13]developedaphysicalcorrosion
modelbasedonfiniteelement(FE)techniqueforpredictingmechanicalstrengthofcorrodedmetal
stents. Thedevelopedmodelwasabletocapturethechangeinsurfaceduetocorrodingenvironment.
Dudduetal.[14]developedanextendedFEmodelformodelinggalvaniccorrosioninmagnesium
alloys. It was demonstrated that the proposed approach can be used to model crevice geometry
withoutremeshing. Shangetal.[15]deployedmoleculardynamicsimulationstostudytheadsorption
behaviorofcompositecoatingsonMgalloys.Thepresentedapproachprovidedaneffectiveapplication
based surface modification of Mg alloys. In addition to FE modeling, analytical models based
on computational intelligence techniques have also been widely used for predicting corrosion.
For instance, Narimani et al. [16] deployed artificial neural network for predicting the effects of
chemicalcompositionandcorrosioncharacteristicsinmagnesiumalloypipelines. Similarly,Zadeh
ShiraziandMohammadi[17]usedahybridneuralnetworkmodelforpredictingcorrosionratesin
underwaterpipelines.
Itisclearfromtheliteraturestudiesthatexperimentsandsimulationhavebeendeployedtostudy
corrosivecharacteristicsinMgalloys,withsimulationbeingmorepreferredmodeofinvestigation.
Thisisbecausesimulationtechniquesprovideaviablealternativeinsavingcostrelatedtotimeand
materials. The current state-of-the-art in scientific modeling of corrosion is directed at two fronts,
viz. mechanical and chemical model (Figure 1). These models are capable of generating accurate
solutionsinpredictingcorrosioncharacteristicswithminimalcostandhighrapidity. Inadditionto
these science based models, process modeling based on computational intelligence technique has
alsobeenconducted,ascanbeseenfromthepreviousliteraturestudies. Theadvantagesofprocess
modelingincludeformationofanexplicitmathematicalrelationshipwhichcanhelpinunderstanding
hiddenrelationshipbetweenvariousinputparametersincorrosionprocess. Itcanbeseenthatthe
sciencebasedmodelsandprocessmodelshavetheirownuniqueadvantages. Asaresult,itwillbe
usefultointegratethemodelswhichcancombinetheadvantagesofaccuracyofFEmodelswithmodel
formationabilityofprocessmodeling.
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Metals 2017, 7, 83 3 of 13

FFigiguurree1 1.. SSttaattee--ooff--aarrtt rreesseeaarrcchh inin cocrorrorsoisoino nmmecehcahnaicnsi cosf mofamgnaegsniuemsi uamlloyasll.o ys.
Hence, the main objective of the present research proposes an integrated science based model
Hence,themainobjectiveofthepresentresearchproposesanintegratedsciencebasedmodeland
and process model for predicting corrosion characteristics of Mg alloy (Figure 2). The Mg alloy
processmodelforpredictingcorrosioncharacteristicsofMgalloy(Figure2). TheMgalloyconsidered
considered in this study is the AZ31 alloy used in aircraft components, biodegradable metallic
initmhipslsatnutds,y ceisllt hpehoAnZe 3a1nadl lloayptuospe dcaisnesa, ierctcr.a Ifnt ctohme pproenseenntts s,tbuidoyd,e ag rcaodrraobdleedm AeZta3l1li calilmoyp lwanitths ,ccrealclkpsh one
anadndla vpotoidps ctyapseicsa,lelyt cr.esIunlttihneg pfrroemse annt esntuvdiryo,namceonrtr tohdaet dleaAdZs 3to1 SaCllCo y(ew.gi.,t hhycdrarocgkesna enmdbvriottildesmteynpt)i cally
resisu lmtiondgeflreodm usainnge FnEv isriomnumlaetinotnt.h Tahtel ecaodrrsotsoivSeC fiClm( efa.gc.t,ohr yisd irnocgluedneedm inb rtihtetl setmudeyn tb)yi scomnosiddeelreidngu tshien gFE
simPiullliantgio–Bne.dTwhoertcho rrraotisoi v(eP–fiBl mratfiaoc)t, owrhisicihn cislu ddeefidneind tahs ethsetu rdaytiob yofc tohnes vidoelurimneg othf ecoPrirlolisnivge– Bmeedtawl orth
ratoioxid(Pe– tBo trhaeti ovo),luwmheic ohf itshed emfientaeld. Aass tchaen rbaet isoeeonf itnh eFivgoulruem 2,e thoef sctourdroys fiovceumsees toanl ouxniddeerstotatnhdeinvgo tlhuem eof
themmecehtaanl.icAals sctarennbgeths oefe nAZin31F iaglluorye w2,itthh reesstpuedcty tofo tchues ienspount puanrdameresttearns dviinz.g ththee Pm–Be rcahtaion,i sctarlasintr reantge thof
and temperature. As can be seen in Figure 2, the FE model is used to predict the mechanical strength
AZ31alloywithrespecttotheinputparametersviz. theP–Bratio,strainrateandtemperature. Ascan
of AZ31 alloy in corrosive environment with respect to the input parameters, viz. the P–B ratio,
beseeninFigure2,theFEmodelisusedtopredictthemechanicalstrengthofAZ31alloyincorrosive
strain rate and temperature. The process modeling based on genetic programming (GP) is the used
environmentwithrespecttotheinputparameters, viz. theP–Bratio, strainrateandtemperature.
to obtain the mathematical relationship between the mechanical strength based on the input
Theprocessmodelingbasedongeneticprogramming(GP)istheusedtoobtainthemathematical
parameters. Scientific analysis of the formulated model is also undertaken based on actual corrosion
relationshipbetweenthemechanicalstrengthbasedontheinputparameters. Scientificanalysisof
science which will provide vital design inputs for maximizing the performance of AZ31 alloys in
theformulatedmodelisalsoundertakenbasedonactualcorrosionsciencewhichwillprovidevital
corrosive environment. The various approaches deployed in investigating the corrosion
designinputsformaximizingtheperformanceofAZ31alloysincorrosiveenvironment. Thevarious
characteristics in AZ31 alloys and the proposed research objective is depicted in Figures 1 and 2,
aprpersopaecchtiveseldy.e ployedininvestigatingthecorrosioncharacteristicsinAZ31alloysandtheproposed
researchobjectiveisdepictedinFigures1and2,respectively.
MetMalset2a0ls1 270,177,,8 73, 83 4of41 o3f 13
Finite Element Modeling of AZ31 Alloy
23
0.8
.2 0.7
0.6
5 0.5
4 0.4
3
2
1 True stress
Input data (P-B ratio,
Temperature and Strain
rate)
Genetic Programming
Settings of GP such as population size,
generations, depth of gene, etc.
-
True stress +
model
+
5 Strain rate
P-B ratio
Temperature

FigFuirgeu2r.e F2E.M F-EGMP-aGpPp roaapcphroinacmh oidne limngodceolrirnogs iocnormroescihonan imcsecohfaAnZic3s1 aolfl oAy.ZF3E1M a-lGloPy:. FFinEiMte-EGlePm: eFnintite
MoEdleelminegn-tG MenoedtiecliPnrgo-gGreanmemtici nPgr;oPg–raBmramtiion:gP; iPll–inBg r–aBtieod: wPiollritnhgr–aBteiod.worth ratio.
2. F2i. nFiitneiEtel eEmleemntenMt oMdoeldienlginogf oCfo CrroordroeddeAdZ A31Z3A1l lAolyloy
2.12..G1.e GomeoemtryetrDye Dscersipcrtiiopntioann danBdo uBnoduanrdyaCryo nCdointidointisons
ThTehFeE FmE omdeoldineglinogf AofZ A31Z3a1ll oaylloisy pise rpfoerrmfoermdeuds inugsinAgB AABQAUQS/UESx/EpxlipcliitcVite Vrseiorsnio6n.1 64.1s4o fstowftawreare
(Da(DssaasuslatuSlty sStyesmteems,eVs, éVlizélyiz-Vyi-lVlailcloaucobulabyl,ayF,r aFnrcaen,ce2,0 1240)14w) iwthitfhu lfluyllcyo ucopulepdletdh ethrmeramlaslt rsetsrsesasn aanlyasliyss.is.
ThTehteh etrhmeraml aaln aalnyasliyssiesn aebnlaebslems omdoeldinelgintgh ethset restnrgetnhgtchh acrhaacrtaecrtiesrtiicssticosf oAf ZA3Z13a1l loalyloayt aetl eevlaetveadted
temtepmerpaetruarteusr.esT.h Tehme omdoedlegle ogmeoemtreytrcyo ncostnrsutcrutecdteidn iAn BAABQAUQSUcSo ncosinsstisstosf oAf ZA3Z13a1ll oalylooyf dofi mdeimnseinosnisons
45045m0m m×m2 0× m2m0 m×m10 ×m m10. Tmhemm. oTshtep rmedoosmt ipnraendtobmypinroadnut cbtyopfcroodrruocsti oonfi ncMorgroasliloony siins mMagg naelsliouyms is
oximdeag(MnegsOiu)m.T ohxisidceo r(rMosgiOve). pTrhoidsu ccotrirsoisnicvleu dperoddiunctth ies gineoclmudeterdy biny tahded ginegomaefitlrmy boyf MadgdOinogn ato fpilmof of
AZM31gaOll ooyn. Itnopth oefA ABZA3Q1 UaSll/oCy.A IEn mthoed AelB,oAnQeUenSd/CoAfEth memodoedle, logneeo menedtr yofi sthfiex emdowdietlh gaenoEmNeCtrAy SisT RfiExed
bouwnitdha rayn coEnNdCitAioSnTRthEa tbaorurnesdtasrayl lcroontdatiitoionna ltahnatd atrrarensstlsa taiolln raoltdateigorneaels aonfdf retreadnosmla.tioTnraanl sdlaetgiroeneasl of
degfrreeeedoofmf.r eTerdaonmslaatiloonnagl odnegeraexei sofi sfrpeerodvoimde adlotnogt hoeneo tahxeisr iesn pdroovfitdheedm too dtheel goethoemr eetnrdy ofof rthteen msioledel
geometry for tensile loading condition. The description of AZ31 model geometry with corrosive film
along with the applied boundary condition is depicted in Figure 3. The material properties of AZ31
Metals2017,7,83 5of13
loadingcondition.ThedescriptionofAZ31modelgeometrywithcorrosivefilmalongwiththeapplied
boundaryconditionisdepictedinFigure3. ThematerialpropertiesofAZ31alloyandthecorrosive
Metals 2017, 7, 83 5 of 13
oxide film used in the FE modeling is illustrated in Table 1. The defined model geometry is then
meshedianltlooy dainsdc rtehtee ceorigrohsitv-en ooxdidee( CfilPmE u8sRedT )int ythpee FeEl emmoedneltinsg, fios lilloluwstirnatgedw inh iTcahblbe o1u. Tnhdea dreyficnoedn ditionsare
model geometry is then meshed into discrete eight-node (CPE8RT) type elements, following which
appliedtocreatemechanicalloadingoftheAZ31part.
boundary conditions are applied to create mechanical loading of the AZ31 part.

Figure 3. Model geometry of AZ31 alloy and corrosive film considered in study.
Figure3.ModelgeometryofAZ31alloyandcorrosivefilmconsideredinstudy.
Table 1. Properties of alloy and film used in FEM simulation [18,19]. FEM: Finite Element Modelling.
Table1.PropertiesofalloyandfilmusedinFEMsimulation[18,19].FEM:FiniteElementModelling.
Parameter AZ31 Alloy MgO film
Thermal Conductivity (k) 96 W/m-K 42 W/m-K
Parameter AZ31Alloy MgOFilm
Density (ρ) 1770 kg/m3 3580 kg/m3
ThermaYloCuonngd’us cmtiovdituylu(ks) (E) 964W4./8m G-PKa 249 4G2PWa / m-K
DePnosisitsyon(ρ’s) ratio (υ) 1770k0g.3/5m 3 03.1588 0 kg/m3
Young’sSpmeocdifuicl uhsea(Et )(Cp) 4140.080G J/Pkag/K 877 J/k2g4/9KG Pa
Poisson’sratio(υ) 0.35 0.18
2.2. Constitutive MSodpeelcinifigc ohf Veaotid(sC ipn) AZ31 under Cor1r0o0s0ionJ/ kg/K 877J/kg/K
AZ31, similar to most magnesium alloys, is susceptible to SCC. SCC results in the formation of
2.2. Constmitiuldt icvreacMkso adnedli nvgoidofs Vono itdhse isnurAfaZce3 1of uthned emraCteorriarlo. sTihonese cracks can nucleate and grow further
under application of external loads which results in catastrophic failure of the material. This
AZ31,similartomostmagnesiumalloys,issusceptibletoSCC.SCCresultsintheformationof
phenomenon in which the nucleation and growth of cracks and voids that lead to ductile failure of
mildcracmksetaanl dis vmooiddesleodn inth FeEs usimrfualcaetioonf tuhseinmg athtee rGiaulr.soTnh–eTsveercgraaacrdk–sNceaendlnemucalne a(GteTNan) dmgodroelw [2f0u].r therunder
applicatioHnenocfe,e xint etrhnisa wlloorka,d tshew AhZic3h1 arlelosyu lstusbijnectceadt atos tmroepchhaincicfaali lluoardeinogf tish emmodaetleedr iuasl.inTg htihse pGhTeNn omenonin
damage model with integrated Yld2000 yield criterion [21,22]. The model is programmed in
whichthenucleationandgrowthofcracksandvoidsthatleadtoductilefailureofmetalismodeledin
ABAQUS solver using the user-defined subroutine module VUMAT. This approach can provide a
FEsimulationusingtheGurson–Tvergaard–Needleman(GTN)model[20]. Hence,inthiswork,the
better prediction of crack evolution and failure of AZ31 alloys in corrosive environment. The yield
AZ31allofyunscutibonje ocft eGdTNto dmameacghea mnoicdaell ilso daedscinrigbeids mmaothdeemlaetdicaullsyi nasg [2th3]e, GTNdamagemodelwithintegrated
Yld2000yieldcriterion[21,22]. Th𝑞2emodelispro−g3r𝑞am𝑝medinABAQUSsolverusingtheuser-defined
subroutinemoduleVUMATΦ.T=hσis2a+p2p𝑞r1o𝑓a∗ccohsℎca(n2pσr2ov)id−e(1a+be𝑞t3t𝑓e∗r2)p=re0d ictionofcracke(1v) olutionand
𝑦 𝑦
failureofAZ31alloysincorrosiveenvironment. TheyieldfunctionofGTNdamagemodelisdescribed
where σy is the equivalent stress; p is hydrostatic stress; q is the effective stress; and q1, q2, and q3 are
mathematicallyas[23],
fitting parameters to improve prediction of void interaction effects in the Gurson damage equation.
The reduction in mechanical strength of the material due to void coalescence is defined by the
damage parameter 𝑓Φ∗, =expqre2ss+ed2 aqs af f∗uncoctsiohn(cid:18) of− v3oiqd2 vpo(cid:19)lum−e(cid:16) fr1ac+tioqn ff. ∗F2o(cid:17)r 𝑓=∗=00, the GTN yield (1)
condition becomes the norσm2al yield 1condition and 2𝑓σ∗=1 indicates th3at the material has failed. It is
y y
defined as,
whereσyistheequivalentstress;pishydrosta𝑓ticstress;𝑓q≤is𝑓tcheeffectivestress;andq1,q2,andq3are
fittingparameterstoimprovepredi𝑓ct∗i=on{o𝑓cf+voδ(id𝑓−in𝑓tce)rac𝑓cti<on𝑓e≤ff𝑓eFc tsintheGursondama(g2)e equation.
𝑓̅ 𝑓≥𝑓
Thereductioninmechanicalstrengthoft𝐹hematerialdFuetovoidcoalescenceisdefinedbythe
damagepwahrearem, eter f∗,expressedasafunctionofvoidvolumefractionf. For f∗ =0,theGTNyield
conditionbecomesthenormalyieldcon𝑓d̅ −iti𝑓onand𝑞f∗+=√𝑞12i−nd𝑞icatesthatthematerialhasfailed. Itis
δ= F c,𝑓̅ = 1 1 3 (3)
definedas, 𝑓 −𝑓 F 𝑞
 F c 3
 f f ≤ fc
f∗ = f +δ(f − f ) f < f ≤ f (2)
c c c F
 f f ≥ f
F F
where,
(cid:113)
δ= fF− fc, f = q1+ q21−q3 (3)
f − f F q
F c 3
Metals2017,7,83 6of13
Metals 2017, 7, 83 6 of 13
where f is critical void volume fraction at which void coalescence first occurs, and f is the void
c F
vowluhmeeref r𝑓ac tiiso cnriatitcwalh viochidt hveolmumatee rfriaalctfiaoinls adt uwehtiochc rvaocikdp crooapleasgcaetniocen .first occurs, and 𝑓 is the void
c . F
voTluhmeer afrtaecotifoinn actr ewahseicihn tthoet amlavtoeridialv foaliulsm deuefr taoc tciroanckf pirsodpuageattoiognr.o wthofexistingvoidsaswellas
.
nucleatTiohne oraftne eowf invcoriedass,ef in to.tTahl visocida nvobleumdees fcrraibcteidona s𝑓,̇ is due to growth of existing voids as well as
new
nucleation of new voids, 𝑓 ̇ . This can be described as,
new . . .
f = f + f (4)
𝑓̇ =𝑓groẇ th +𝑓neẇ (4)
growth new
ThTehger gorwowththra rtaeteo foef xeixsitsitninggv vooididss iiss pprrooppoorrttiioonnaall ttoo tthhee hhyyddrroosstatatitci ccocmompopnoennetn otfo pflpasltaisct sictrsatirna in
ratreat(eε. p(ε)̇𝑝,g)i,v geinvebny ,by,
kk𝑘𝑘 f = (1− f)ε.p (5)
𝑓ggrroowwtthh =(1−𝑓)εk̇𝑘𝑝k𝑘 (5)
Thenucleationrateofnewvoidsisgenerallydefinedbystrain-controllednucleationrulewhich
The nucleation rate of new voids is generally defined by strain-controlled nucleation rule which
assumesthatthevoidsundergonucleationassecondphaseparticleswhichisnormallydistributedfor
assumes that the voids undergo nucleation as second phase particles which is normally distributed
thetotalparticlepopulation. Thisisdescribedas,
for the total particle population. This is described as,
fn𝑓.newėw==S𝑆N𝑁f√𝑓N√𝑁22ππeexxpp(cid:20)[−−1212(cid:18)(ε̅ε𝑝p𝑆S−−𝑁Nεε𝑁N)(cid:19)](cid:21)ε̅ε𝑚𝑝mp𝑙l (6) (6)
whwerheerfeN 𝑓r𝑁eprerepsreensetnsttsh ethveo lvuomluemfrea cftriaocntioonf voofi dv-oniudc-lneuactlienagtipnagr tpicalretsic,laens,d aεnNd aεn𝑁d SaNnda r𝑆e𝑁th aeraev tehrea ge
andavsetraangdea arnddd setavniadtaiordn doefvthiaetisotnra oifn tshaet swtrhaiinchs apta wrthiciclehs pnaurtcilcelaeste nvuocliedast,er evsopidesc,t irveesplye.ctively.
ThTehset esptespisn ivnovlovlevdedin int htheem mooddiififieedd GGTTNN ddaammaaggee mmooddeel luussiningg VVUUMMAATT susburboruotuintien ies iislluilslutrsattreadt ed
initnh ethfeo rfmormof oaf fla oflwowchcahratr[t2 [22]2]i ninF Figiguurree4 4..

FiFgiugruere4 .4.C Coonnsstittituuttiivvee mmooddeelliinngg ooff AAZZ3311 aallllooyy uussiningg FFEE apappproraocahc h[2[22]2. ].
Metals2017,7,83 7of13
Metals 2017, 7, 83 7 of 13
The modified GTN model is used to predict the strength characteristics of AZ31 alloy by
ThemodifiedGTNmodelisusedtopredictthestrengthcharacteristicsofAZ31alloybyvarying
tvhaeryPi–nBgr tahteio P(–xB) ,ratetmiop (exr1)a,t uterme(pxer)aatnudres (txra2)i nanradt est(rxai)n. Trahtee r(axn3)g. eTohfet reasntegde ionfp tuetspteadr aimnpeutetr psaterastmedetaerres
1 2 3
tested are given in Table 2. The simulation computes the engineering stress and engineering strain,
giveninTable2. Thesimulationcomputestheengineeringstressandengineeringstrain,fromwhich
from which the true stress and true strain are calculated using classical mechanics relations.
thetruestressandtruestrainarecalculatedusingclassicalmechanicsrelations.
Table 2. Input parameters for tensile loading simulation of AZ31 alloy.
Table2.InputparametersfortensileloadingsimulationofAZ31alloy.
Process Input Variable Values Units
ProcessInputVariable Values Units
P–B ratio 0.2, 0.3, 0.4, 0.5 No units
TempePr–aBturraeti o 3000, .325,00.,3 4,00.04,, 405.50 NouniKts
Temperature 300,350,400,450 K
Strain rate 0.0001, 0.001, 0.01 s−1
Strainrate 0.0001,0.001,0.01 s−1
2.3. Validation of Finite Element Model
2.3. ValidationofFiniteElementModel
The stress–strain plot of the AZ31 alloy under uni-axial tensile loading is shown in Figure 5. For
The stress–strain plot of the AZ31 alloy under uni-axial tensile loading is shown in Figure 5.
comparison of FE results with actual data, the experimental tensile loading characteristics of AZ31
For comparison of FE results with actual data, the experimental tensile loading characteristics of
obtained from Ref. [24] are also included in the figure. It can be seen from this plot that the FE model
AZ31obtainedfromRef.[24]arealsoincludedinthefigure. ItcanbeseenfromthisplotthattheFE
was able to predict the stress characteristics of AZ31 with an accuracy of 92%. It is noted that the
modelwasabletopredictthestresscharacteristicsofAZ31withanaccuracyof92%. Itisnotedthat
model predicted a relatively high yield stress compared to experiments. This can be attributed to
themodelpredictedarelativelyhighyieldstresscomparedtoexperiments. Thiscanbeattributed
several factors such as the material defects, temperature fluctuations, machine settings in tensile
toseveralfactorssuchasthematerialdefects,temperaturefluctuations,machinesettingsintensile
tester, etc. Furthermore, the FE geometry of AZ31 also factored the corrosive film component which
tester,etc. Furthermore,theFEgeometryofAZ31alsofactoredthecorrosivefilmcomponentwhich
may have caused certain variations in the yield stress prediction. This clearly demonstrates the
may have caused certain variations in the yield stress prediction. This clearly demonstrates the
ability of the FE simulation for generating required data sets that can be subsequently utilized by
ability of the FE simulation for generating required data sets that can be subsequently utilized by
optimization algorithms.
optimizationalgorithms.
The tensile loading simulation performed by ABAQUS/CAE software is depicted in the form of
ThetensileloadingsimulationperformedbyABAQUS/CAEsoftwareisdepictedintheform
screen shots in Figure 6. It can be seen that the stress distribution in the AZ31 alloy as well as the
ofscreenshotsinFigure6. ItcanbeseenthatthestressdistributionintheAZ31alloyaswellasthe
corrosive film depends markedly on the simulation temperature due to the development of thermal
corrosivefilmdependsmarkedlyonthesimulationtemperatureduetothedevelopmentofthermal
stresses in the material. A total of 5000 steps were used to complete the tensile loading simulation of
stressesinthematerial. Atotalof5000stepswereusedtocompletethetensileloadingsimulationof
the AZ31 material in the CAE model.
theAZ31materialintheCAEmodel.

Figure5.ComparisonoftensileforceofSS304alloypredictedfromFEMsimulationwithexperimental
Figure 5. Comparison of tensile force of SS304 alloy predicted from FEM simulation with
dataobtainedfromRef.[24].
experimental data obtained from Ref. [24].
Metals2017,7,83 8of13
Metals 2017, 7, 83 8 of 13

(a)

(b)

(c)

(d)
Figure 6. FE modeling of mechanical stress buildup of corrosive AZ31 alloy at temperatures: (a) 300
Figure6.FEmodelingofmechanicalstressbuildupofcorrosiveAZ31alloyattemperatures:(a)300K;
K; (b) 350 K; (c) 400 K; and (d) 450 K. It can be seen that, regardless of temperature, the stress
(b)350K;(c)400K;and(d)450K.Itcanbeseenthat,regardlessoftemperature,thestressconcentration
concentration is highest at corrosive film near pits and crack tips which leads to material
ishighestatcorrosivefilmnearpitsandcracktipswhichleadstomaterialdeterioration.
deterioration.
Metals2017,7,83 9of13
3. AnalyticalModelingofStressCorrosionCharacteristicsofAZ31Alloy
The FE model generates the strength data of AZ31 alloy by systematically varying the input
parametersrelatedtoP–Bratio,temperatureandstrainrate. Atotalof48datasetsarethenusedto
obtainnon-linearanalyticalmodelusingGPtechnique. InGPcluster,thetrainingdataareusedto
traintheGPmodelandthevalidityofobtainedmodelistestedontestingdata. Kennard-and-Stone
algorithmisintegratedwithintheGPalgorithmtoensurerandomnessofdataselectionfortrainingand
testingsets. TheGPclusteristhenusedtoderivemathematicalmodelthatdescribestherelationship
betweenthetruestrengthofthealloy(y)asafunctionofinputvariables,viz.P–Bratio(x ),temperature
1
(x ) and strain rate (x ). The descriptive statistics of the data sets generated from the FE model is
2 3
showninTable3.
Table3.DescriptivestatisticsoftruestressdatafromFEsimulation.
P–BRatio,x Temperature,x StrainRate,x TrueStress,y
Parameter (NoUnit) 1 (K) 2 (s−1) 3 (MPa) 1
Mean 0.35 375 0.034 137.43
Median 0.35 375 0.001 121.92
Standarddeviation 0.11 56.49 0.047 49.30
Kurtosis −1.38 −1.38 −1.53 −0.49
Skewness −1.842×10−15 0 0.730 0.75
Minimum 0.2 300 10−4 69.82
Maximum 0.5 450 0.1 235
Theaccuracyandeffectivenessoftheevolutionaryalgorithmframeworkdependsonparameter
settings, whichcontrolthegenerationofdifferentsizesofmathematicalmodels. Atrialanderror
approachisusedbytheauthorstodeterminetheoptimumparametersettingsinthepresentstudy.
The optimal settings determined are listed as follows: the population size is 400, the number of
iterationsis20,thenumberofmaximumgenesis6,theprobabilityforcrossoveris0.80,andthenumber
ofgenerationsis200. TheauthorsdeployedtheGPalgorithmfromtheGPTIPStoolboxinMATLAB
frameworkdevelopedbySearsonetal.[25]. ThebestGPmodel(Equation(7))forthetruestressof
AZ31alloy(Y )isselectedbasedontheminimummeanabsolutepercentageerrorvalueamongall
AZ31
theruns.
Y (MPa) = 68.18+(1036.91×x )+[0.71×tan{(x )×(log(log(x )))}]
AZ31 1 2 3
−[920.58×{log(cos(log(x )))}]−[1130.62×{tan{(tan(x ))}][ (7)
2 1
+0.53×{tan(tan(log(log(x ))))}]+[0.49×{x ×(tan(tan(x )))}]
3 2 1
wherex ,x ,x arevaluesofP–Bratio,temperature(K)andstrainrate(s−1),respectively. Y isthe
1 2 3 AZ31
valueoftruestressderivedfromthemodelbasedonthevaluesofx ,x ,x .
1 2 3
3.1. PerformanceEvaluationoftheAnalyticalModel
The evaluation of the FEM-GP model performance is conducted by analyzing the statistical
metricsofthemodelasdefinedbytherootmeansquareerror(RMSE),thecoefficientofdetermination
(R2),andthemeanabsolutepercentageerror(MAPE).Thesemetricsprovidesanindicatorofmodel
performanceastheymeasurethedeviationsofthepredictedfromactualvalues. Thedefinitionof
thesemodelmetricsrepresentedbymathematicalequationsisavailableinRef.[26]. Theperformance
andvalidationanalysisoftheFEM-GPmodelisshowninFigure7. Itcanbeseenfromthefigurethat
theFEM-GPmodelisabletopredictthetruestressofAZ31withclosefitandhigheraccuracy. Hence,
itcanbeconcludedthattheFEM-GPmodelcanbeusedasaneffectiveapproachforgeneralizingthe
truestressofAZ31alloyundercorrosion.
Metals2017,7,83 10of13
Metals 2017, 7, 83 10 of 13
Metals 2017, 7, 83 10 of 13

(a) (b)

Figure 7. Performance of FEM-GP model and ANN of true stress of AZ31 for (a) training data and (b)
Figure7. Performanceof(FaE) M-GPmodelandANNoftruestressofAZ(b3)1 for(a)trainingdataand
(tbes)ttiensgt indgatdaa. tRa2.: Rc2o:ecfofiecfifiencite notf odfedteertmerimnaintiaotnio; nM;MAPAEP:E m:meaena nabasbosloulutet eppeercrceenntataggee eerrrroorr; ;RRMMSSEE:: rroooott
meanF sigquuraer 7e. ePrerroforr; mGaPn:c Ge eonf FeEtiMc P-GroPg mraomdeml ainndg ;A ANNNN o:f Atrurtei fsitcrieasls Nofe AuZra3l1 Nfoer t(wa)o trraki.n ing data and (b)
meansquareerror;GP:GeneticProgramming;ANN:ArtificialNeuralNetwork.
testing data. R2: coefficient of determination; MAPE: mean absolute percentage error; RMSE: root
3.2. Modelm Aeanna lsyqsuiasr eB earsreodr ; oGnP C: Goernroetsiico Pnr oPghryamsimcsi ng; ANN: Artificial Neural Network.
3.2. ModelAnalysisBasedonCorrosionPhysics
C3.o2.r rMoosdioeln A noafl ymsisa Bteasreiad lo ni sC oar rocsoiomn pPlheyxs icps rocess in which several factors exhibit considerable
Corrosionofmaterialisacomplexprocessinwhichseveralfactorsexhibitconsiderableinfluence
influence on the material behavior. Hence, any analytical model describing the material strength of
on the mCatoerrroiasliobne hofa vmioart.eriaHl eins cae ,caonmyplaexn aplyroticceassl mino wdehlicdh essecvriebrainl gfacthtoersm eaxtheirbiiat lcsotnrseindgertahbloe f the
cthoem cpoionmfnlpueonenntceuen notd nue tnrhdece omrr carootesrrriiooanls ibomenhu amsvtiuobsre.t Hbaeeb nalecbelt,e oa tncoya cpaatnpuatrluyetritech atelh mea coatdcuteaul ladple pshchyryisbisciinscgsi ntihnvevo omlvlavetededrid adulu srrtirinnegngg cctohor rorrofo ssiioonn..
For ththise, cpoamrpaomneetnrti cu nadnearl ycosrisro issi ocno nmduustc tbeed a bwleh tioc hca sphtuorwe sth teh aec vtuaarli apthiyosni cisn i nthvoel vtreude d sutrriensgs cboarrsoesdio onn. the
Forthis,parametricanalysisisconductedwhichshowsthevariationinthetruestressbasedonthe
consiFdoerr ethdi si,n ppaurat mfaecttroicr sa.n Failgysuirs ei s8 c sohnodwucst etdh ew ehfifcehc ts hoof weasc thh eo vf athriea tcioonn sinid tehree tdr uine dstirveisds ubaals efadc otonr tsh oe n the
consideredinputfactors. Figure8showstheeffectofeachoftheconsideredindividualfactorsonthe
true sctornessisd eorfe Ad iZn3p1u ta flaloctyo rus.n Fdigeru rceo 8r rsohsoiwons .t hIte ceaffnec bt eo fs eeaecnh tohf atht ea cso tnhseid Per–eBd rinadtiiov iidnucarle faascetos rtsh oen m thaet erial
truestressofAZ31alloyundercorrosion. ItcanbeseenthatastheP–Bratioincreasesthematerial
true stress of AZ31 alloy under corrosion. It can be seen that as the P–B ratio increases the material
strength degrades rapidly. The degradation is more pronounced when P–B ratio exceeds 0.3. Based
strengthdegradesrapidly. ThedegradationismorepronouncedwhenP–Bratioexceeds0.3. Basedon
strength degrades rapidly. The degradation is more pronounced when P–B ratio exceeds 0.3. Based
on corrosion physics, it can be stated that the corrosive film does not protect the alloy from
corroosino ncoprrhoyssioicns ,pihtycsaincsb, eits tcaatne dbeth sattattehde cthoartr otshiev ecofirlrmosidvoe efsilnmo tdporeost encott thperoatellcot ythfreo mallodye gfrroamda tion.
degradation. Similar trend can be seen for the case of temperature in which a corrosive AZ31 alloy
Simildaregtrraednadticoann. Sbimeisleaer ntrefonrd tchaen cbaes eseoefn tfeomr tpheer caatsuer eofi ntewmpheicrahtuarceo inrr woshiviceh Aa Zco3r1roaslilvoey AeZx3h1i bailtlsoylo wer
exhibits lower material strength at higher working temperatures. This indicates that AZ31 alloy
matereixahlibsittrse nlogwthera mthatiegrhiaelr swtreonrgktihn gatt ehmighpeerr awtuorrkeisn.gT themispienrdaitcuaretes.s Tthhiast iAndZic3a1teasl ltohyatu AnZd3e1r aclolorryo sive
under corrosive conditions is not suitable for high temperature applications. However, the strain
condiutinodnesr icsonrrootssivuei tacobnlediftoiornhsi gish ntoetm spueitraabtluer efoar phpiglihc atteimonpse.raHtuorwe eavpeprl,icthateiosntrsa. iHnorwateevseer,e mthes tsotrhaianv eno
eraffteec streoaetnem tshse eetmom sha attove rhei aanvloes tnerfoef neecfgftte hoctno o ftnhA tehZ em3 m1ataaetlrleoiraiyal lws stthrreiecnnhggttahhg oroeff e AAsZwZ331e1 lal lawlollyiot hyw phwirchehvic aihog uraesgeerse xwepsee lrwl iwmelietlh nw tpairtlehsv tpiuorduesyv i[o2u4s].
expereixmpeernimtael nsttauld sytu [d2y4 []2. 4H].e Hnecnec, ea,t a at mambbieiennt tt teemmppeerraattuurreess, ,t hthe ee feffefcetc otf olof alodaindgi n(ig.e .(,i .setr.,a isntr raaitne )r daotee)s does
Hence,atambienttemperatures,theeffectofloading(i.e.,strainrate)doesnothaveanynegativeor
not hnaovte h aanvye anneyg nateigvaet ivoer opro psiotsivitiev eim imppacatc to onn tthhee ssttrreennggtthh o of ft hthe eA AZ3Z13 a1l laolyl.o Tyh. iTs hciosr rcoobrorroabteosr awtietsh with
positiveimpactonthestrengthoftheAZ31alloy. Thiscorroborateswiththefactthatmaterialfailure
the fathcet tfhacatt tmhaat tmeraiatelr ifaali lfuairleu rue nudnedre rS CSCCC o occccuurrss nnoott bbeeccaauusese o fo fe xetxertenranl alol aldo aitds eiltfs, ebluf,t bduute dtou eth eto the
underSCCoccursnotbecauseofexternalloaditself,butduetothecrackgrowthandnucleationof
crack growth and nucleation of voids that results from residual stresses, loading, etc. Hence, from
crack growth and nucleation of voids that results from residual stresses, loading, etc. Hence, from
voidsthatresultsfromresidualstresses,loading,etc. Hence,fromthisanalysisitcanbeconcludedthat
this analysis it can be concluded that the derived analytical model was able to confirm well with the
this analysis it can be concluded that the derived analytical model was able to confirm well with the
thedeorbisveerdveadn pahlyytsiiccas linm coodrreolswiona sparobcleestso. confirmwellwiththeobservedphysicsincorrosionprocess.
observed physics in corrosion process.

(a) (b)

(a) Figure8.Cont. (b)