Finite element analysis for delamination of laminated vibration damping steel sheet

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1 Fiite elemet aalysis for delamiatio of lamiated vibratio dampig steel sheet WANG Yog( 王勇 ) 1, CHEN Ju( 陈军 ) 1, TANG Big-tao( 唐炳涛 ) 2 1. Natioal Die ad Mold CAD Egieerig Researh Ceter, Shaghai Jiao Tog Uiversity, Shaghai , Chia; 2. Istitute of Egieerig Mehais, Shadog Jiazhu Uiversity, Ji a , Chia Reeived 1 Deember 2006; aepted 2 April 2007 Abstrat: The material struture of lamiated vibratio dampig steel sheet(lvdss) was itrodued i detail. A iterfae ohesive model betwee the ski sheets was developed by usig a otat/iterfae approah, ad the model was applied to simulate T -peel ad lap-shear proesses of LVDSS. The iterfae otat stress distributio durig the T -peel ad lap-shear proesses is obtaied, ad the fiite elemet aalysis(fea) results agree satisfatorily with the orrespodig experimetal results. As a result, the model is suitable to simulate the ohesive of LVDSS Key words: lamiated vibratio dampig steel sheet; delamiatio; iterfae ohesive model; pure mode; mixed mode 1 Itrodutio LVDSS sadwihed with thi polymer ore is a ew kid of sheet metal with advaed vibratio ad soud dampig properties, ad LVDSS is ireasigly used for automotive appliatio due to their superior vibratio ad soud dampig properties[1]. Usig of LVDSS a support relatively quiet easeful situatio for driver ad passegers. The parts, suh as dash pael, oil pa ad frot floor, have bee suessfully made of LVDSS ad LVDSS will be used broadly i automobile idustry with LVDSS formig tehiques improvig. The behavior of LVDSS is quite differet from that of homogeous steel sheet durig formig proesses ad the otat state betwee ski steel sheets i the formig proess is very diffiult[2]. For example, i drawig proess, some areas bod perfetly without ay rak, but some areas may be delamiated beause of large strai[3 4]. Therefore it is a hallege to predit the model of iterfae ohesive situatio. HUANG ad LEU[5] ad MAKINOUCHI et al[6] reated a 2D iremetal elasti plasti fiite elemet model with the ore polymer assiged with elasti-plasti material. YAO et al[7] preseted three methods iludig the LVDSS beig osidered as a homogeeous material usig the overall tesile properties, LVDSS was deployed a omposite setio defiitio whose properties a be assiged with differet thikess ad usig double sheets model with fritio iterfae. But those methods are ot exatly to simulate iterfae slippig ad delamiatio of LVDSS. I additio, sprig elemets were used to model ohesive relatioship i V-bedig, deep drawig. I the sprig elemet model, the ore material was modeled as sprigs. CORONA ad EISENHOUR[8] also applied the sprig model i bedig of LVDSS, but the sprig model was ot suitable to simulate ompliated idustrial appliatios. So far there have bee several iterfae ohesive models to aalysis the delamiatio of omposite material suessfully[9 13], but those models have ot bee used i LVDSS. I this paper, a iterfae ohesive model was applied to simulate the iterfaial adhesive state, slippig ad delamiatio of LVDSS with ABAQUS software. T -peel ad lap-shear damage simulatios ad experimets were oduted respetively, the relatioships betwee the fixture reatio fore ad relative displaemet of two skis were obtaied. The stress state of the ohesive zoe was studied. Foudatio item: Projet( ) supported by the Natioal Natural Siee Foudatio of Chia Correspodig author: CHEN Ju; Tel: ; ju_he@sjtu.edu.

2 456 2 Material properties of LVDSS LVDSS osists of three layers, the ski layers are steel with 0.59 mm i thikess respetively ad the ore layer is polymer material with 0.03 mm i thikess. The material struture is illustrated i Fig.1 ad the material properties are give i Table 1. Whe LVDSS is applied i the struture where the vibratio or oise exists, the resi layer sadwihed betwee two ski layers is apable of effetively oversig the vibratig mehaial eergy or the soi wave ito thermal movemet of the high-moleular hai iside the ore material through the shearig deformatio. Eergy oversio, from marosopi poit of view, embodies i miimizatio of the vibratio ad the redutio of the oise. Fig.1 Struture of lamiated vibratio dampig steel sheet WANG Yog, et al/tras. Noferrous Met. So. Chia 17(2007) delamiatio proess omprises three iterrelated phases: the iitiatio of iterfae rak, the evolutio of the degradatio zoe, ad omplete delamiatio. The ore material a be thought as o-liear sprigs oetig the upper ad lower surfaes of the iterfaial skis. Ad these assumed sprigs otai the tagetial ad ormal stregth so that they a resist tagetial delamiatio, ormal delamiatio or mixed delamiatio. The ormal ad shear stress o the iterfae a be alulated by the iterfae ohesive model. The iterfae ohesive model desribes the evolutio of these tratios ad the ouplig betwee them as a futio of relative ormal ad tagetial strai, ad their rate alog the iterfae. The relative ormal ad tagetial strais irease with the relative displaemet betwee ski sheets developig. Whe the relative strai jump exeeds a ritial value, the imagiary sprigs are assumed to fail, i.e. delamiatio has take plae. Thereafter, the iteratio betwee the two ski sheets is desribed solely by the otat fritio algorithm. The iterfae ohesive model defies the evolutio of ormal ad tagetial tratios ad the ouplig betwee them as a futio of the relative ormal ad tagetial strai betwee ski sheets, respetively. That is, with ireasig strethig of the imagiary sprig, the sprig fore attais a maximum value, the dereases, ad fially vaishes whe the sprig ruptures. Uder isothermal oditios, the stress (σ) to resist the strethig of the ore material is expressed as follows: Table 1 Properties of lamiated vibratio dampig steel sheet Properties L T D Ski yield stregth/mpa Ski tesile stregth/mpa Ski -value Ski K-value/MPa Ski r-value Ski ultimate strai Ski elasti modulus/gpa Ski Poisso s ratio 0.30 Iterfae peel stregth/mpa 2.10 Iterfae shear stregth/mpa 7.90 Note: L represets logitudial; D represets diagoal; T represets trasverse; is hardeig expoet; K is hardeig oeffiiet; Y is aisotropy fator. 3 Iterfae ohesive model betwee ski sheets of LVDSS The iterfae ohesive model was developed usig a otat/iterfae approah proposed by VINAY et al[13], ad the developed model is more suitable for simulatio of iterfae ohesive of LVDSS. Iterfae 1 ε σ ( ε ) = σ ε exp β β (1) where ε = ε/ε, ε is the ritial strai value, whe ε arrives at ε 0 the ore material stregth begis to degrade; σ is the maximum stress of the ore material, whih ours at the ritial strai value ε ; β is rupture fator of the ore. 3.1 Delamiatio modes Delamiatio ours i three differet modes[11]. The first mode is pure mode Ⅰ, o whih delamiatio ours i the ormal diretio; the seod mode is pure mode Ⅱ ad pure mode Ⅲ, whih is delamiatio ours i the two tagetial diretios, as show i Fig.2; the third mode is mixed mode, whih pure mode Ⅰ ad pure mode Ⅱ both our i this mode. 3.2 Pure mode of delamiatio iterfae ohesive model Coveietly, the strai ompoets ε i ad the tratio stress ompoets σ i are ormalized by their respetive ritial strai value i ε ad maximum iter-

3 WANG Yog, et al/tras. Noferrous Met. So. Chia 17(2007) Mixed mode of delamiatio iterfae ohesive model I pratie, iitiatio of slidig ad delamiatio usually iludes both the pure ormal mode ad the pure taget mode. The mixed mode of delamiatio iterfae ohesive model is based o the extet of damage[14], is defied by: Fig.2 Delamiatio modes of LVDSS faial stress i σ i, respetively, as follows: i / ε i, σ i σ i / σ i ε = ε = (2) where i=, t; ε ad σ are the ormal strai ad the ormal stress of the ore material respetively; ε t ad σ t are the taget strai ad the taget stress of the ore material respetively. The oditios of pure ormal mode are ε 0, ad ε t =0. Uder the pure taget mode, ε t 0, ε 0. Whe ε <0, the otat iterfae suffers the press stress i ormal diretio. A resistive ompressive ormal stress σ prevets iterpeetratio of the upper surfae ad the lower surfae whe the two surfaes are i otat, ad the otat equilibrium heavily depeds o the distributio of elasti fores i the otatig regio. I this model, it is assumed that the ompressive stress does ot otribute to ay iterfaial damage. The pure ormal delamiatio model takes ito osideratio the mehaial behavior of the iterfaial surfae uder the iterpeetratio oditios. σ = σ σ ε ε 1 ε exp β 1 k ε exp β β β + where ε is the absolute value of ε, k is a iterpeetratio fator to magify the repulsive stress σ. If ε > 0, the ε = ε ; ad if ε 0, the ε = 0. I the pure taget delamiatio mode, the tagetial tratio stress (σ t ) resists the slidig strai ε t. The model is desribed as follows: β 1 ε t σ = σ ε t t t exp β (4) where t ε is the absolute value of ε. t (3) 2 ( ) 2 λ = ε + ε (5) t where ε = ε / ε, ε t = ε t / ε t, ad ε = 0, whe ε <0; ε = ε, whe ε 0; ε ad ε t are the atual ormal ad tagetial strai at the iterfae; ε ad ε t are the ritial ormal ad tagetial strai at whih the iterfae fails or delamiatio takes plae i pure ormal or pure shear mode, respetively; λ is a effetive delamiatio fator, ad oly tesile strai (opeig) affets delamiatio. Iterfae ohesive state is udamaged if λ<1, ad delamiatio will iitiate as soo as λ exeeds zero. The tesile ad shear state of the iterfae ohesive state is up to λ>1. After delamiatio ompleted, the iterfae is aalyzed solely based o the otat law. I preset paper, Ψ was used as a dereasig futio of the ormal ad taget strai. Ψ is restrited as follows: Ψ < 0, i =, t ε i (6) To satisfy the mixed mode riterio i Eq.(6), the followig futio is postulated: 1 λ β Ψ = exp (7) β Iludig the iterpeetratio term i Eq.(6), ad from Eq.(7) the fial form of the mixed mode ohesive model a be desribed as follows: σ = σ ε β 1 + k ε Ψ ε exp β (8) σ t = σ t ε tψ (9) where the operator ε is defied as: ε = ε if ε >0 ad ε =0 if ε 0. This mixed model is easily developed for the 3D iterfaial surfaes by addig aother taget diretio futio the same as Eq.(9). For a opeed frature iterfae, there is o otat betwee the ski layers. Aother situatio is that delamiatio has bee ompleted but the ski layers are still i otat. For this losed otat situatio, the

4 458 fritio law is applied as follows: σ f =μσ (10) where σ f is maximum fritio stress betwee ski layers, μ is the fritio oeffiiet, σ is ormal stress o otat iterfae. 3.4 Implemetatio of iterfae ohesive model I this paper, iterfae ohesive model was implemeted i ABAQUS usig subroutie VUINTER. User subroutie VUINTER provides a very geeral iterfae to defie the ostitutive behavior aross the iterfae betwee two surfaes ad the subroutie replaes all built-i iterfaial ostitutive behavior models. 4 Fiite elemet aalysis simulatio ad experimets validatio T -peel ad lap-shear proesses are two importat methods to study the iterfaial ohesive. I this paper, T -peel ad lap-shear proesses were researhed. The fiite elemet simulatios were oduted usig ABAQUS EXPLICIT FEA software, ad the iterfae ohesive model desribed above was implied at iterfae betwee ski layers. WANG Yog, et al/tras. Noferrous Met. So. Chia 17(2007) ompleted. The otat ormal stress at the iterfae of zoe C is more tha zero, it meas there is ormal press stress at the iterfae. The ormal tesile stress ours i the zoe B, i whih the delamiatio ours. Fig.5 shows the delamiatio proess from iitially udamaged adhesive stage to delamiatio stage i lapshear damage simulatio. From Fig.5(b), we a see that the delamiatio happes at the two eds of LVDSS where shear stress is zero Experimets The T -peel ad lap-shear damage experimets were arried out o a Zwik/Roell(Z020) material testig mahie. The shape of the T -peel sample is the same as used i T -peel simulatio. I the T -peel experimets, the lampig fixtures hold the shoulders of testig sample. The the fixtures draw the workpiee util the delamiatio of iterfae is ompleted, as show i Fig.6. Fig.7 shows iitial state ad fial state of 4.1 CAE model defiitio LVDSS workpiee is 100 mm i legth ad 25 mm i width for T -peel simulatio. Fae A ad fae B move 11 mm ad 11 mm alog x diretio, the ed of bottom a oly move alog y diretio, as show i Fig.3(a). I lap-shear damage simulatio, the workpiee legth is 12.5 mm ad the width is 25 mm. Fae C is fasteed ad fae D moves 1 mm alog x diretio, as show i Fig.3(b). The ski sheets are modeled with the CPS4R elemet available i the ABAQUS elemet library ad modeled with four elemets alog the thikess. The ohesive otat relatioship betwee the ski layers is modeled with iterfae ohesive model desribed i above setio. Geometrially o-liear deformatio of the LVDSS is take ito aout i all the simulatios. 4.2 Simulatio ad experimets Fiite elemet simulatio The otat ormal stress at the iterfae alulated by the iterfae ohesive model is show i Fig.4. As defied i ABAQUS, a positive ormal stress deotes a pressure direted ito the surfae ad a egative ormal stress deotes a tesile applied o the surfae. The otat ormal stress o the opeed iterfae of zoe A is zero, i.e. there is o ohesive ad delamiatio Fig.3 FEA models of simulatio: (a) T -peel FEA model; (b) Lap-shear FEA model Fig.4 Deformed shape of T -peel simulatio

5 WANG Yog, et al/tras. Noferrous Met. So. Chia 17(2007) 459 Fig.8 Sample of lap-shear damage experimet Fig.9 Lap-shear damaged sample Fig.5 Deformed shape i lap-shear damage simulatio: (a) Iitial ohesive state; (b) Delamiatio Fig.6 T -peel experimet Fig.7 T -peel testig samples the testig samples. I the lap-shear damage experimet, the top ad bottom lampig fixtures hold the two eds (A ad B) of testig sample respetively. The the fixtures draw the testig sample util the delamiatio of iterfae is ompleted. Fig.8 shows the shape of testig sample. The width ad effetive ohesive legth(l) of sample are the same as the width ad legth of workpiee for the lap-shear damage simulatio respetively. Fig.9 shows the damaged sample of lapshear damage experimet Results ad disussio A ompariso betwee the experimetally measured load-displaemet urves ad those predited from the simulatios of T -peel proess is show i Fig.10. Whe the fixture moves outward about 0.5 mm, the reatio fore arrives at the maximum about 190 N. I this stage, the ski sheets begi to bed ad the ormal Fig.10 Relatioship betwee fixture reatio fore ad fixture displaemet i T -peel proess tesile stress betwee the ski sheets ireases ad the iterfae rak itrodues. The fixture reatio fore falls dow slowly after maximum reatio fore arrived. The iterfae delamiatio happes after the tesile stress arrived at the maximum, ad the delamiatio develops alog the iterfae with the fixture drawig. So the fixture reatio fore is i relatively smooth state after the fixture moves 4 mm ad the reatio fore is about 75 N. Fig.11 illustrates ompariso betwee the experimetally measured load-displaemet urves ad those predited from simulatios of lap-shear proess. Differet from Fig.10, the fixture reatio fore raises gradually with the relative shear displaemet ireasig betwee the ski sheets. Whe fixture displaemet is about 0.52 mm, the fixture reatio fore reahes the maximum value 2.5 kn. Also the shear stress betwee the skis gets the maximum ad the shear rak of the ore material ours. The the ore material degrades with the shear strai growig. The otat shear stress ad the fixture reatio fore redue. Fially, the fixture reatio fore ad otat shear stress are redued dow to zero whe the fixture displaemet gets about 0.75 mm, meaig the iterfae shear delamiatio is ompleted.

6 460 I referee to Fig.10 ad Fig.11, the umerial alulatios apture all the major features of the marosopi deformatio, iludig the large plasti strais of the adhereds, ad the agreemet of FEA results with experimets results is obtaied. So the iterfae ohesive model a be applied to simulate the ohesive of LVDSS iludig iitiatio of iterfae rak, evolutio of the degradatio, ad omplete delamiatio. The stamp proess of LVDSS also a be simulated with this ohesive model. Fig.11 Relatioship betwee fixture reatio fore ad fixture displaemet i lap-shear damage proess 5 Colusios 1) A iterfae ohesive model was developed ad implemeted i ABAQUS EXPLICIT FEA software usig subroutie VUINTER. The model was applied to simulate the T -peel ad lap-shear damage proess of LVDSS. 2) I T -peel proess, the fixture reatio fore ireases very quikly. The the fore derease gradually ad is ear to 75N at last. I lap-shear damage proess, differet from T -peel proess, the fixture reatio fore rises gradually. Ad the fore drops to zero getly after peak poit reahed. 3) Comparig the result of simulatio with the WANG Yog, et al/tras. Noferrous Met. So. Chia 17(2007) experimetal results, it is proved that the model is a effetive model to simulate the ohesive of LVDSS iludig iitiatio of iterfae rak, evolutio of the degradatio, ad omplete delamiatio. Referees [1] CHEN Y S, HSU T J, CHEN S I. Vibratio dampig harateristis of lamiated steel sheet [J]. Metallurgial Trasatios A, 1991, 22A: [2] YOSHIDA M. Press formability of vibratio-dampig sheet [J]. Joural of JSTP, 1985, 26: [3] KOPP R, ABRATIS C, NUTZMANN M. Lightweight sadwih sheets for automobile appliatios [J]. Produtio Egieerig Researh ad Developmet, 2004, 11: [4] CHENG H S, CAO J, YAO H. Wriklig behavior of lamiated steel sheets [J]. Joural of Materials Proessig Tehology, 2004, 151: [5] HUANG Y M, LEU D K. Fiite-elemet simulatio of the bedig proess of steel/polymer/steel lamiate sheets [J]. Joural of Material Proessig Tehology, 1993, 52: [6] MAKINOUCHI A, YOSHIDA S, OGAWA H. Fiite elemet simulatio of bedig proess of steel-plasti lamiate sheets [J]. Joural of JSTP, 1988, 29: [7] CORONA E, EISENHOUR T. Wipig die bedig of lamiated steel [J]. Iteratioal Joural of Mehaial Siees, 2007, 49: [8] YAO H, CHEN C C, LIU S D. Lamiated steel formig modelig tehiques ad experimetal verifiatios. SAE Paper. No [9] MOHAMMADI S, FOROUZAN S, ASADOLLAHI A. Cotat based delamiatio ad frature aalysis of omposites [J]. Thi-Walled Strutures, 2002, 40: [10] SCHWARTS H, RABINOVITCH O, FROSTIG Y. High-order oliear otat effets i yli loadig of delamiated sadwih paels [J]. Composites: Part B, 2007, 38: [11] JEREMY S, NATHAN W. Plasti dissipatio i mixed-mode fatigue rak growth alog plastially mismathed iterfaes [J]. Iteratioal Joural of Fatigue, 2006, 28: [12] TURON A, CAMANHO P P, COSTA J. A damage model for the simulatio of delamiatio i advaed omposites uder variable-mode loadig [J]. Mehais of Materials, 2006, 38: [13] VINAY K G, ERIC R J, CARLOS G D. Irreversible ostitutive law for modelig the delamiatio proess usig iterfaial surfae disotiuities [J]. Composite Strutures, 2004, 65: [14] MI Y, CRISFIELD M, DAVIES G. Progressive delamiatio usig iterfae elemets [J]. J Compos Mater, 1998, 32: (Edited by YANG Hua)