Stress Intensity Factors In Two Bonded Elastic Layers Containing Crack Perpendicular on the Interface with Different Elastic Properties

Transcription

1 Stre Intenity Factor In Two Bonded latic Layer Containing Crack Perpendicular on the Interace with Dierent latic Propertie Mahdi Keikhaie1, Naer Keikhaie, Reza Keikhaie3, M.M. Kaykha3 1Department o Mechanical ngineering, Shari Univerity o Technology, Azadi Ave., Tehran, Iran. Department o Mechanical ngineering, Univerity o Sitan and Baluchetan, Zahedan, Iran. 3Department o Mechanical ngineering, Univerity o Zabol, Zabol, Iran. mm.kaykha@gmail.com Abtract.Thin bonded ilm have many application (i.e. in inormation torage and proceing ytem, and etc.). In many cae, thin bonded ilm are in a tate o reidual tenion, which can lead to ilm cracking and crack extenion in one layer oten accompanie ailure in whole ytem. In thi paper, we analyze a channel crack advanced throughout thickne o an elatic thin ilm bonded to a diimilar emi-ininite ubtrate material via inite element method (FM). In order to impliy modeling, the problem i idealized a plane train and a two-dimenional model o a ilm bonded to an elatic ubtrate i propoed or imulating channel crack in thin elatic ilm. Film modeled by common 4-node and ubtrate by ininite 4-node mehe. The tre intenity actor (SIF) or cracked thin ilm ha obtained a a unction o elatic mimatch between the ubtrate and the ilm. The reult indicate that in elatic mimatch tate, SIF i more than match tate. On the other hand, mimatch tate i more enitive to crack than match tate. And SIF ha alo increaed by increaing Young modulu and Poion ratio o ilm. Keyword: Thin ilm, Channeling crack, Ininite element, Stre intenity actor 1 Introduction Thi Many modern material and material ytem are layered. The potential application o racture mechanic o layered material range over a broad pectrum o problem area; included are: protective coating, multilayer capacitor, thin ilm/ubtrate ytem or electronic package, layered tructural compoite o many varietie, reaction product layer, and adheive joint [1]. Many application in microelectronic (e.g., interconnect and electronic packaging) oten involve integrated tructure with diimilar material. Stree are introduced during the procee o abrication, reliability teting, and operation. The tre ield concentrate at the junction o diimilar material, at the corner, or, i there exit a crack, at the crack tip []. In all o thee application, the ilm are very thin, with thicknee meaured in nanometer or micrometer, and they are bonded to ISBN:

2 comparatively thick ubtrate, with thicknee typically meaured in millimeter or centimeter. Many cracking pattern in ilm-ubtrate ytem have been oberved and analyzed (van et. al. [3]; Hutchinon and Sue [1]). A crack nucleate rom a law either in the ilm or at the edge, and propagate both toward to interace and laterally through the ilm. Depending on the material, the crack may top at the interace (Fig. 1a), penetrate into the ubtrate (Fig. 1b), or biurcate onto the interace (Fig. 1c) [4]. Fig 1. (a) A channeling crack within a thin ilm. (b) A channeling crack penetrating ubtrate. (c) A channeling crack with interace debonding [4] Irwin [5] claim that the tre ield in the vicinity o a crack tip can adequately be deined by a ingle parameter proportional to the SIF. When the intenity o the local tenile tree at the crack tip attain a critical value, a previouly tationary or lowmoving crack propagate rapidly. Thi critical value deine the racture toughne and it i a contant or a particular material. I the ize o the huget law in a particular tructure i known, minimum toughne tandard can be etablihed or the material in thi tructure. In the application o mot o the current racture criteria, the SIF and the crack opening diplacement are the motly ued quantitie [6]. The objective o thi tudy i invetigating enitivity o two bonded elatic layer to a ingle crack perpendicular to the interace between ilm and ubtrate. In thi paper alo ha conidered dierent elatic ratio o ilm and ubtrate. Becaue there are imilar work in the literature (The problem o a crack perpendicular to the interace may be ound in [7-11]), the main emphai here i on uing ininite mehe to imulating ubtrate (to be cloe to real problem) and alo preenting reult in dierent orm, i.e. plotting SIF veru elatic propertie intead o Dundur parameter [1], to have better undertanding. 1. Analytical Method Background In thi ection, we irt give an overview o the racture mechanic mode and then previou analytical work o SIF on both homogeneou and layered ytem. Three linearly independent cracking mode are ued in racture mechanic. Thee load type are categorized a Mode I, II, or III a hown in the igure. Mode I, hown to the let, i an opening (tenile) mode where the crack urace move directly apart. Mode II i a liding (in-plane hear) mode where the crack urace lide over one another in a direction perpendicular to the leading edge o the crack. Mode III i a tearing (antiplane hear) mode where the crack urace move relative to one another and parallel to the leading edge o the crack. Mode I i the mot common and important load type encountered in engineering deign [13]. ISBN:

3 Fig.. Mode I, Mode II, and Mode III crack loading [13] 1.1.SIF in Single dge Notched Tenion Specimen: The SIF equation or a ingle edge notch and homogeneou propertie in an ininite pecimen i [14], K = σ ZY a (1) Where, a π (1+ ( )) ZY = w V 1 a (1 ( )) w () Where, a a a 3 V = ( ) ( ) ( ) w w w a 4 a 5 a ( ).44( ) ( ) w w w (3) Range o applicability o thi equation: The deect depth, a, hould be le than the pecimen width, w, [14]. For dierent a/w ratio it ha plotted in Fig. 3. Fig. 3. Plot o ZY (nondimenionalized SIF) v the variation o a/w or plane train condition 1.. SIF or Two Bonded Layer by Fully Cracked Film Fig. 4 how a crack channeling through a pre-tenioned ilm on a emi-ininite ubtrate. The crack i conined by the ilm/ubtrate interace in the direction perpendicular to the interace. Fig. 4. Steady-tate crack channeling acro the ilm or the ully cracked ilm For the ully cracked ilm problem, with it crack tip at the interace (Fig. 4), the KI i a the ollowing orm [15], K ( I =σ ( α, β ) π h ) (4) ISBN:

4 Where (α,β) i a non-dimenionalized SIF and a unction o Dundur (Hi work how that or any problem o a compoite body made o two iotropic, elatic material with precribed traction, the material dependence o the problem i reduced rom three dimenionle parameter to the two Dundur parameter α and β) parameter. For plane train problem α and β are given by [1]; α = + = (5) 1 (1 ν )(1 ν ) (1 ν )(1 ν ) β = 4 (1 ν )(1 ν ) + (1 ν )(1 ν ) (6) Where /1 ν, Furthermore, the compilation by Suga et al. [16] indicate that or mot practical material combination, value o a typically lie between β= 0 and β= α/4. The tre ingularity exponent, in q. 7, i a unction o α and β, too, and atiie the ollowing equation derived by Zak and William [17]; αβ αβ 1 β 1 β co( π ) (1 ) + = 0 (7) Value o a a unction o α or β= 0 and β= a/4 are plotted in Fig. 5. Fig. 5. Plot o crack tip ingularity exponent,, v α or β = 0 and β = α/4.. Finite lement Simulation Conider a compoite coniting o an ininite layer o width h and a hal pace (Fig. 6). The hal pace can be aumed to approximate a emi-ininite ubtrate with average material contant a well a a homogeneou ubtrate. The layer i perectly bonded to the hal pace (i.e. the bonding agent i neglected). There i a tranvere crack in the layer. The ilm i ubject to a uniorm tenile tre σ and the ubtrate i tre-ree (Fig. 6). Fig. 6. Fully cracked ilm under tenile tre ISBN:

5 Fig. 7 how the geometry and the boundary condition o the plane-train problem. The crack i repreented by the line CD. The thickne o the ilm i h. The ubtrate ha an ininite thickne. The model i ully ixed along AB. The vertical boundary F i ubjected to an initial tenile tre (σ = 1 Pa) and other boundarie are traction ree. At equilibrium, the ilm and the ubtrate deorm o that the traction along the crack ace vanih and the crack open. For each et o material propertie o the ilm and the ubtrate, olution were ought with variou value o / and ν/ν in order to obtain the aymptotic olution or an iolated ingle crack with a emi-ininite ubtrate. The inite element mehe are generated a ollow. Firt divide the whole domain into two region, a indicated in Fig. 8. In the upper region, the one with the crack, a uniorm meh (number: 101*11) i generated with the plane train olid continuum our-node bilinear quadrilateral element (CP4R). In the lower region, emi-ininite ubtrate, the meh (number: 101*1) i generated with the plane train olid continuum ininite our-node linear quadrilateral element (CINP4). The mehe o the region are compatible in their interection, and alo alignment o the crack with the element i convenient or the computation o the opening diplacement. Fig. 7. The FM model o the plane-train problem: geometry and boundary condition. Fig. 8. The FM model o the plane-train problem:aigning meh to the ilm and the emi-ininite ubtrate 3. Reult and Dicuion For the two-dimenional analyi, the two type o SIF (KI and KII) are related to the energy releae rate, G, a ollow [18], 1 ( I II ) G= K + K (8) In the previou tudie o cracking in thin ilm (e.g., [1]), a uniying dimenionle number Z ha been deined uch that the energy releae rate or a crack i, σ G= Z h 0 (9) where i the plane-train modulu o the ilm. The number Z i a dimenionle driving orce, depending on the cracking pattern. Huang et. al. [], modeled ISBN:

6 dimenionle energy releae rate o channeling crack by XFM and obtained that energy releae rate ha increaed by increaing α. For the channeling crack in the preent tudy, the irt type o SIF (KI) o a two bonded elatic layer wa calculated uing inite element method. Dierent Poion ratio, ν/ν, o 0.5, 0.9, 1,, 3, 4, 5 and the elatic modulu ratio, /, o 0.1, 0., 0.3 8, 9, 10 were chooing or calculation, becaue all dierent material can be located in thi range. The variation o the SIF or dierent elatic ratio i preented in Fig. 9, 10. It can be een, the change o the KI value or dierent modulu ratio decreae by decreaing Poion ration. In the cae o no elatic mimatch (α=β=0), the tre ingularity reduce to the quare root ingularity o a crack tip in a homogeneou elatic material, i.e. = 0.5 (q. 7), and KI ha the minimum value (Fig. 9). When the ubtrate i tier than the ilm (α < 0), the ingularity i weaker, i.e. < 0.5, and KI value are lower. When the ubtrate i more compliant than the ilm (α > 0), the ingularity i tronger, i.e. > 0.5, and KI value are higher. For an extremely compliant ubtrate (α 1), the ingularity exponent approache 1( 1) and KI ha the maximum value. All the reult can be tabulated in Table 1. Fig. 9. Variation o KI with dierent Poion ratio, ν/ν, and Young modulu ratio. Fig. 10. Variation o KI with dierent Poion ratio, ν/ν, and Young modulu ratio, /, o or detail. Table 1. Summary o the reult Non-dimentional energy Property Dundur parameter S (q. 7) SIF releae rate [] Statu - α β - KI ωi good ν ν - α β - KI ωi good - α β - KI ωi bad ν ν - α β - KI ωi bad ISBN:

7 4. Concluion The ininite element, or an elatic racture mechanic problem, have been ued to characterize the cracking o thin ilm bonded to thick ubtrate material. The SIF ha been extracted rom the imulation. The SIF o the plane-train problem depend on the elatic mimatch between the ilm and the ubtrate. The reult demontrate that the ininite element can be applied to model problem with dierent elatic propertie o ilm and ubtrate. SIF or channeling crack ha obtained a a unction o elatic mimatch ratio between the ubtrate and the ilm. Reult how that KI ha the minimum value in /=0.1 and ν/ν=0.5 condition and it ha the maximum value in /=10 and ν/ν=5. In general view KI ha the minimum value when ν=ν. Becaue o there i no reult in thi orm, qualitative comparion with the available previou tudie (i.e. Non-dimentional energy releae rate []) how good general agreement. Reerence 1. Smith, T.F., Waterman, M.S.: Identiication o Common Molecular Subequence. J. Mol. Biol. 147, (1981). Hutchinon J.W, Suo Z, Mixed Mode Cracking in Layered Material, Adv. Appl. Mech., 9, (199). 3. Huang R, Prevot J.H, Huang Z.Y, Suo Z, Channel-cracking o thin ilm with the extended inite element method, ng. Fract. Mech., 70, (003). 4. van A.G, Drory M.D, Hu M.S, The cracking and decoheion o thin ilm, J. Mater. Re. Soci., 3, (1988). 5. Ye T, Suo Z, van A.G, Thin Film Cracking and the Role o Subtrate and Interace, J. Solid Struct., 9, (199). 6. IRWIN G.R, Structural Mechanic (dited by J. N. Goodier and N. J. Ho), Pergamon Pre: Oxord, Gecit M.R, Fracture o A Surace Layer Bonded to A Hal Space, Int. J. ng. Sci., 17, (1979). 8. Cook T.S, rdogan F, Stree in bonded material with a crack perpendicular to the interace, Int. J. ng. Sci., 10, (197). 9. rdogan F, Biricikoglu V, Two bonded hal plane with a crack going through the interace, Int. J. ng. Sci., 11, (1973). 10.Bogy D.B, The plane elatotatic olution or a ymmetrically loaded crack in a trip compoite, Int. J. ng. Sci., 11, (1973). 11.Gupta G.D, A layered compoite with a broken laminate, Int. J. Solid Struct., 9, (1973). 1.Arin K, A note on the racture o laminated compoite, Lett. Appl. ng. Sci., 3, 81-5 (1975). 13.Dundur J, dge-bonded diimilar orthogonal elatic wedge, J. Appl. Mech., 36, (1969). ISBN: