The influence of an interlayer on coating delamination under contact loading

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1 The fluee of a erlayer o oag delamao uder oa loadg T. Pagès Dvso of Egeerg, Brow Uversy, Provdee, RI 091 Curre address: Eole Polyehque, 9118, Palaseau, Frae Absra The objeve of hs work s o sudy deao dued delamao of a srog flm from a wo layer sem. To hs ed our model s omposed of a elas damod oag, ad a hrome erlayer ad a ugse arbde subsrae, boh elas - plas. The erfae bewee he oag ad he erlayer s supposed weaker ha bewee he erlayer ad he subsrae, hus he frs oe s modelled wh a ohesve zoe ad a Xu-Needlema osuve law whle he seod oe s rgd. I addo oly ormal delamao s suded here, for hese odos alwa appears for a smaller deao fore ha for shear delamao. Delamao ao s suded for several properes of he sem hrough he evoluo of he ral fore wh he erlayer hkess. Depedg of he ohesve zoe ad he erlayer maeral properes, s foud ha he ral fore aheve a maxmum wh a erlayer hkess of a half o wo mes he oag hkess. The value of hs maxmum goes from wo o fve mes he ral fore for delamao of a wo layer sem. I s also demosraed ha hs maxmum eds o appear for smaller hkess f he ohesve zoe s weaker, whh ould be very eresg for dusral applaos. Keywords: Adheso; Coags; Delamao 1

2 1. Iroduo Hard oags are ofe used o mprove he fro, wear, ad oa fague ressae of surfaes ha are subjeed o oa loadg. For example, eram oags are used o proe a wde varey of auomove ompoes, ludg psos, valve heads, gears ad beargs. I addo, damod ad damod-lke-arbo flms are urrely of grea eres as proeve oags for mahe ools, parularly for expermeal dry mahg proesses. For example Geeral Moors are developg ugse arbde drlls wh damod oag o bore alumum whou lubrag. Coags subjeed o oa loadg suffer from a varey of falure mehasms, ludg oag fraure, spallg, buklg, amog may ohers [1-8]. Delamao of he oag from he subsrae s a parular oer. Cosequely, here s grea eres developg ehques o opmze oag adheso. A varey of approahes may be exploed for hs purpose, ludg seleg appropraely he elas ad plas properes of he oag ad subsrae; opmzg he hkess of he oag, ad modfyg he hemsry of he oag/subsrae erfae self. I hs paper, we am o orbue o hs effor by predg he fluee of a raso zoe bewee he oag ad he subsrae usg a sema se of ompuer smulaos To hs ed, we wll prese he resuls of a dealed paramer sudy ha preds he respose of a hard elas oag o a elas-plas wo layer subsem, omposed of a erlayer ad he subsrae, o deao loadg. The problem o be solved s llusraed Fgure 1.1. A lear elas oag s boded o a elas-perfely plas erlayer. The erfae bewee hem s modelled usg a ohesve zoe law [15], whh allows he oag o separae from he erlayer, ad s haraerzed by pheomeologal osuve equaos relag he raos ag o he wo boded solds o he separao bewee hem. The subsrae ad he erlayer are rgdly boded for he erfae bewee hem s supposed muh sroger ha he frs oe. The oag s deed by a rgd froless sphere.

3 spheral deer oag erlayer r ohesve erfae rgd erfae z subsrae axsymmer fxed b.. fxed b.. Fgure Model of he hree layer par Our goal s o ompue ral fore ad deph1ad h, whh respevely spefy he deao fore ad deph eeded o observe delamao a he oag/erlayer erfae afer omplee uloadg. Our ma sudy wll be he fluee of he erlayer hkess, hagg he erlayer mehaal properes, he deer radus, ad he sregh ad oughess of he erfae. Our work bulds o a sudy by Gao, Xa, Bower, Lev ad Cheg [18], who used a smlar model o sudy delamao mehasm maps for a wo layer sem. Covergee problems fe eleme smulaos used o resr sudes o a lmed parameer spae. Y.F. Gao ad A.F. Bower have foud a way o resolve hese overgee problems [17] as s desrbed furher below ad wll be semaally used our smulaos. The remader of hs paper s orgazed as follow. The ex seo summarzes he problem, s equaos, s approxmaos ad he ohesve surfae model. Seo hree preses he model quaavely, he oree mplemeao of he ohesve zoe ad a dmesoal aals. Seo four oas he resuls ad her erpreaos whle seo fve s he oluso. 3

4 . Problem formulao.1. Goverg equaos I hs seo he objeve s o brefly defe he geeral sem, order o fd wha daa he smulao s gog o eed ad wha are he bas pheomea a sake he overall behavour. I addo esures a global udersadg of he umeral proess. Noe ha seo hree preses he model used our ompuaos more presely. We osder a sem omprsg a elas-perfely plas maeral (subsrae) oaed by a elas h flm, he wo of hem beg separaed by a elas-perfely plas erlayer. The par s deed by a spheral deer. The deer s assumed rgd ad oly haraerzed by s radus R. Assumg he whole o be sorop, he problem s axsymmer, wh radal oordae r ad axal oordae z he deao dreo, as llusraed Fg. 1. The flm s haraerzed by s hkess ad s boded o he erlayer by a erfae, whh wll be spefed he ex subseo. The subsrae s ake o have a hegh of H ( + ) ad radus L, wh L large eough so ha he soluo s depede of L ad he subsrae a be regarded as a half spae. The aals s arred ou umerally usg a fe sra, fe eleme mehod. I uses a formulao whh equlbrum s expressed erms of he prple of vrual work as: Ω : εdv + T Δ ds = Γ ξds S Ω Here, Ω s he oal HxL rego aalyzed ad Ω s s boudary, boh he udeformed ofgurao. ξ ad Γ are respevely he dsplaeme ad rao veors, s he sress esor whle ε s he sra esor. The seod erm he lef-had sde s he orbuo of he erfae, whh s here measured he deformed ofgurao ( S { z } ) = =. The rao rasmed aross he erfae s T, whle he dsplaeme jump s Δ. Here, ad θ he remader, he axsymmery of he problem s exploed, so ha ξ = Γ = = ε = 0. θ θ θ 4

5 The prese boudary odos are llusraed Fg The deao proess s performed remeally wh a osa deao raev. 0 Ousde he oa area wh radus a he referee ofgurao, he flm surfae s sress free: z ( r ) ( r ) r Γ,0 =Γ,0 = 0 for a r L. Isde he oa area we assume froless odos: ( r ) r Γ,0 = 0 for 0 r a. The subsrae s smply suppored a he boom, so ha he remag boudary odos read: ξ (0, z) = 0 for 0 r L r ξ (, r H) = ξ (, r H) = 0for 0 r L r z ξ ( Lz, ) = ξ ( Lz, ) = 0 for H z 0 r z However he szes L ad H wll be hose large eough ha he soluo s depede from he prese remoe odos. These equaos ad boudary odos eed o be supplemeed wh he osuve equaos for he oag ad he subsrae, as well as he erfae. As he laer s eral o he resuls of hs sudy, hese wll be explaed deal he forhomg subseo. The subsrae ad he erlayer are supposed o be elas-perfely plas maerals, wh plas flows beg orolled by he rae-depede vo Mses plasy wh yeld sresses ad. The elas par s gve erms of he Youg modul E ad y s E ad Posso s raosν s ad ν (subsrp s for subsrae, for erlayer). The oag s assumed o be a srog, perfely elas maeral wh Youg s modulus E ad Posso s rao ν (subsrp for oag). Fally og ( ) s = r I he devaor e 3 sress ad = s : s he Vo Mses sress, we have: 5

6 s e = E ε ad s s s e = E ε ad y = E ε The above equaos, supplemeed wh he osuve law for he erfae o be dsussed presely, form a o-lear problem. I s solved wh he sofware Abaqus v6.4 ad a addoal Forra ode whh allows o mpleme he ohesve zoe hrough user defed elemes. 6

7 .. The ohesve surfae model (1) () A eah po of he erfae S, defe a orhoormal bass{,, }, deoes he ormal o S ad ( α ), where 1, α =, deoe wo age veors o S. Le ξ ( x) deoe he (fesmal) dsplaeme feld he sold, whh s ouous everywhere exep o S. ± Le ξ ( x) lmξ( x ε) = ± deoe he lmg values o eah sde of he erfae. Therefore, ε 0 he ormal ad ageal dsoues aross he erfae + + ( ) are Δ = ( ξ ξ ) ad ( ) α α ξ ξ Δ =. Gve he sress esor he maeral, he ormal ad ageal raos ag o S, deoed Δ ad Δ, are: T = ad ( α ) ( α ) α =. The ohesve erfae law relaes (, α ) T Δ Δ ad( T, T α ). For a deal elas erfae, he relaoshp s defed by a elas poeal fuo Φ suh ha: T Φ = Δ T α Φ = Δ α Varous forms of Φ have bee used umeral smulaos. The oe we wll use s he oe developed by Xu ad Needlema: Δ Δ 1 q r q Δ Δ Φ( Δ, Δ ) =Φ +Φexp 1 r+ q+ exp δ δ r 1 r 1 δ δ where Δ = Δ +Δ. Furher, Φ = maxδ exp(1) s he eergy requred o separae he 1 adjog elemes f he dsplaemes are ormal o he erfae ad max s he maxmum sress uder ormal dsplaeme jump. If he dsplaemes are ageal o he erfae, he eergy of separao s deermed byφ = qφ. The oher varables he problem areδ adδ whh are respevely he haraers ormal ad ageal legh sales ha eer o he problem. Fally, we have he parameer * Δ r δ = where * Δ s he value of Δ afer omplee shear separao wh he ormal fore deally zero. 7

8 The relao bewee rao ad dsplaeme jumps herefore gves: T Δ Δ Δ 1 q Δ Δ = max exp 1 exp 1 exp r + δ δ δ r 1 δ δ δ Δ r q Δ Δ Δ T = max q+ exp 1 exp δ δ r 1 δ δ δ We eveually have he followg graphs: T / max Δ /δ (a) T /τ max Δ /δ Fgure..1: Trao-dsplaeme separao law. I (a), Δ = 0, ad (b), Δ = 0. (b) 8

9 3. Model 3.1. Preseao of he model The sem beg axsymmer, desrbg a plae s equvale o desrbg he whole sem. Fgure represes wha s mplemeed he fe eleme sofware. There are wo saes, he deer whh s aaly rgd ad he par whh s deformable ad dvded hree regos. To be reals he dffere regos mus be boded. Furhermore he boud bewee he oag ad he erlayer s weaker ha he oe bewee he erlayer ad he subsrae so hs s he delamao of he frs oe we are eresed. Thus he seod oe s modelled as rgd whle a ohesve zoe s mplemeed he frs oe. Ths ohesve erfae sks he wo regos ogeher, allowg hem meawhle o separae as soo as sress beomes oo mpora. Sze problems Takg aou he law of S Vea, he par wll be ake wh a very large sze o preve he boudary odos from dsurbg he resuls. Thus he legh ad he hegh wll be respevely four mes ad wo mes he deer radus. The remag sze properes are as followed: Coag hkess: = 1μm Ideer radus: R = 10μm Par legh: L= 40μm Par hegh: H = 0μm Ierlayer hkess: 0 3μm I our Abaqus/CAE model oe legh u orrespod o5μ m. 9

10 R spheral deer hrome erlayer damod oag H ugse subsrae Fgure Assembled model Abaqus 6.4 L Maerals The damod lke arbo (DLC) oag s very hard ad hardly plaszes, hus s modelled as a elas maeral wh a Youg modulus E = 1000GPaad Posso s raoν = 0.. I hs paper, he erlayer we are eresed s made of hromum whh here s a elas perfely plas maeral wh E = 50GPa, ν = 0.1ad a yeld sress = 1GPa. The subsrae s ugse WC, here also elas perfely plas wh E = 700GPa, ν = 0.6, ad =.7GPa. s y s Fgure 3.1. Maeral sruures: DLC, hromum, ugse arbde 10

11 Boudary odos O he lef boarder s appled a axsymmer odo.e. a oly move alog e z. The boom ad he rgh boarders are easred, whle he upper oe s free ad s erao wh he deer s froless. Thus: ( r ) r Γ,0 = 0 for 0 r a ξ (0, z) = 0 r ξ (, r H) = ξ (, r H) = 0 r ξ ( Lz, ) = ξ ( Lz, ) = 0 r z z r z ad ( r ) ( r ) Γ,0 =Γ,0 = 0 for a r L a Froless oa fxed b.. axsymmer fxed b.. Fgure Boudary odos Load orol versus dsplaeme orol Alhough we are maly eresed he ral fore for dusral applaos, he load wll be a mposed dsplaeme of he deer. I s deed ofe observed ha load orol eds overgee problems whe he fore urve s o a moooe fuo (he mos famous example beg buklg pheomea). Dsplaeme orol s herefore hose here so we measure drely he ral deao deph, ad he appled fore wll be reured as a oupu by he fe eleme sofware, order o fally oba he ral fore. 11

12 Fe elemes The mesh s auomaally produed by Abaqus/CAE. The hoe we have made s four-ode blear axsymmer quadrlaeral elemes (CAX4R). Mesh dsrbuo The par s dvded wo bg regos for he mesh: he upper lef orer, where he ma deformaos our ad he res of he par. I he frs oe he mesh s fe ad eve fer whe we ge loser o he deer order o have a prese sress dsrbuo ear he deao, whle he remader s raher oarse for deformao wll be almos eglgble. Quaavely here are 960 elemes he frs rego ad 160 he seod oe. I addo he ohesve zoe s oly mplemeed bewee he oag ad he erlayer of he upper lef orer, for rak uleao hardly appears he oher rego, whh adds 40 ew elemes. Ths separao allows ga of alulao me. oag erlayer subsrae Fgure Mesh dsrbuo 1

13 3.. Implemeao of he ohesve zoe Parameers hoe * I hs ase, as ofe, we wll osder q= r = 0.5 whh meas Φ = Φ adδ = Δ. The rao-dsplaeme relao beomes: T T Δ Δ Δ Δ 1 Δ = max exp 1 exp 1 exp δ δ δ δ δ δ Δ Δ Δ = max exp 1 exp δ δ δ δ The ohesve surfae s represeed by four-ode elemes wh wo egrao pos. The behavour of hese elemes wll odo he delamao. Iroduo of a vsous erm o Reasos The problem wh he ohesve zoe s ha, may ases wh a fe eleme ode suh as ABAQUS, fals o overge due o a sably po whe he rak aes, whh a be see o Fgure We wll he use he soluo proposed by Bower ad Gao [17] whh osss addg small vsosy-lke parameersζ adζ o he rao-dsplaeme relaos. The vsosy s o eded o model ay phal eergy dsspao proess. The erfaes relaos beome: T Δ Δ Δ Δ 1 Δ d Δ = max exp 1 exp 1 exp ζ + δ δ δ δ δ d δ δδ Δ Δ d Δ T = max exp 1 exp ζ + δ δ δ d δ 13

14 F h Fgure 3..1 Isably po due o delamao o Ifluee of he vsous erm The vsous erm he rao dsplaeme relao beomesζ ζ 4 4 δ 4 V s he dsplaeme speed of he odes. For our expermes ζ = 10, ad h he maxmum relave dsplaeme s aroud 0. per sep, whh lass 1 seod, gvg a maxmum relave speed of 0.. Therefore he maxmum value of hs erm s aroud 0., whh s very small ompared o V δ whe ompued, where relave sresses gog up o 0 = 50. Thus hs erm wll o have ay fluee o our resuls. I fa, he vsous ermζ a be made as small as desred, keepg s effe o overgee. Cosequely, wll ever affe our resuls. 14

15 3.3. Dmesoal aals Furher o hs preseao all daa ad ukows of he problems are defed, ad due o he heorem Π s ow possble o express he ral deao load ad deph as fuos of he followg parameers: F R E E y Es max Φ δ =Π,,,,,,,,,,,,, f ν ν νs qr δ h R E E y Es max Φ δ =Πh,,, ν,, ν,,, νs,,,, qr, δ However he mos eresg parameer s he relave erlayer hkess, ad seo four dsusses s fluee o he relave ral fore, aordg o he values of he followg dmesoless parameers.e.: geomery: deer radus R E maerals: erlayer elasy:, ν erlayer plasy: y max erfae: ohesve zoe sregh: ohesve zoe oughess: Φ E Es δ The oher oes beg fxed: = 370, ν = 0., = 60, νs = 0.6, = 1, q= r = 0.5. δ 15

16 4. Resuls ad dsusso 4.1. Furher formao Geerally speakg, delamao s observed afer oe yle of load-uloadg whh a be broke dow hree seps. Frs he deer s a a age o he par. The we mpose a axal deer dsplaeme h wh a osa speed, ad fally he deer goes bak o s al poso, wh he same speed. Delamao s measured hrough he omparso bewee he dsplaeme jump ha delamao has ourred as soo as Δ δ. Δ ad he ohesve zoe dmesoδ : oe a osder Ths seo res o prese eough resuls o fll he parameer spae, ad o learly show he fluee of he erlayer hkess ad s orgs. I order o avod mesh effes our resuls, ad o esure he omparso bewee hem s legmae, all smulaos have he same mesh dsrbuo, he oe preseed seo hree. I addo he referee se of daa s gve Table1: Geomery Maerals Ierfae R = 10 E 9 = ; 0.1 y ν = ; 0.39 = max Φ 3 = 1; = ( Φ = 10 Jm. ) Table 1 Referee daa se for smulaos So ha he fal sae of he sem a be learly see, Fgure s a obvous delamao example, ompued wh he prevous daa se ad a orrespodg deao h deph = 1.5. Fgure 4.1. s he orrespodg evoluo of he ormalsed axal fore mposed o he deer aordg o s ormalsed dsplaeme. 16

17 (a) (b) Fgure Sress felds observed afer delamao: (a) Vo Mses ad (b) zz ompoe F h Fgure 4.1. Fore-dsplaeme graph 17

18 4.. Resuls Our resuls o oly dsplay he ral fore, whh s he mos eresg parameer for dusral applaos, bu also he ral deao deph, whh s more uve. I addo we semaally have ompued he asympoe for fe hkess, whh orrespods o he ase of a hrome subsrae. Ideed he poso of he asympoe of he ral fore urve ompared o he value for o erlayer,.e. = 0, daes he more effe maeral for oag adheso. Referee resul Our referee resul s Fgure 4..3 whh has bee ompued wh he daa se gve Table 1 ad 0 3. F Table1 (a) 18

19 h Table1 Fgure 4..1 Cral fore (a) ad deph (b) vs. erlayer hkess wh Table 1 daa se (b) The frs par of he urve ad he asympoe poso ofrm he uo: he sero of a hrome erlayer reases he ral fore ad deph. However h ad asympoe values, o form peaks a abou = 1.9, whh s a surprse. F he exeed he I addo s eresg o oe ha a very small erlayer does o seem o affe he sem sregh. I s fa orary o ommo expermeal resuls, whh leads o assume ha s fluee s aually due o hemsry effes, ad o mehaal. For bgger erlayers he presee of a peak suggess ha several pheomea are a sake. The furher resuls am o solae he fluees of he dffere parameers ad hus o dsover he geeral behavour of he sem. Ifluee of he Youg modulus To udersad he fluee of elasy he followg graphs have bee made wh he same daa exep for he erlayer whh s ow a fous maeral wh E = Es. 19

20 F Table1 E E = 1 s h (a) Table1 E E = 1 s (b) Fgure 4.. Cral fore (a) ad deph (b) vs. erlayer hkess wh E = Es The geeral urve shapes are he same bu he ampludes are far smaller whe he Youg modulus s hgher.e. whe he maeral of he erlayer s less elas. Furhermore he peak s raslaed owards smaller hkesses. Thus elasy amplfes he pheomeo alhough does o reae. 0

21 Ifluee of yeld sress Here he oly hage ompared wh he daa se of Table 1 s ha we have a fous erlayer maeral wh y =. F Table1 = 1 y h Table1 = 1 y (a) (b) Fgure Cral fore (a) ad deph (b) vs. erlayer hkess wh y = Obvously yeld sress s behd he observed pheomeo: a hgh yeld sress smoohes he urve, ad mples a moooous raso bewee a sem wh ugse oly ad oe wh hromum oly. Ths les suppose ha plasy s a sake he geeral behavour, ad reaes he pheomeo. 1

22 Ifluee of he deao sphere radus R Here s he resul of a smulao wh.5 = F Table1 R =.5 (a) h Table1 R =.5 (b) R Fgure Cral fore (a) ad deph (b) vs. erlayer hkess wh.5 = The sphere beg smaller, so s he area of oa bewee he par ad he deer. Thus he ral deph reases, whle he assoaed ral fore reases, by omparso wh he referee smulao. Ths smulao ofrms ha he deer dmesos do o affe our resuls.

23 Ifluee of he ohesve zoe properes max Φ Here are respevely he resuls for = 0.5, = Φ = 5 Jm. max Φ = 0.5, = Φ = 100 Jm. : 3 ad F Table1 Φ = 5J m h (a) Table1 Φ = 5J m Fgure Cral fore ad deph vs. erlayer hkess wh (b) max Φ = 0.5, =

24 F Table1 Φ = 100J m h (a) Table1 Φ = 100J m Fgure Cral fore ad deph vs. erlayer hkess wh (b) max Φ = 0.5, = Here oe a see ha he ougher he ohesve zoe s, he hgher he ral fore ad deph are, whh ould have bee preded. However he peak also raslaes o he hker erlayers. Fally seems ha he ohesve zoe oughess effes are he same as elasy oes. 4

25 Global resuls Fally order o learly see he dffere fluees Fgure represes all prevous resuls ogeher. F Table1 Φ = 5J m R =.5 E E s = 1 y = 1 Φ = 100J m h Table1 Φ = 5J m R =.5 E E s = 1 y = 1 Φ = 100J m (a) (b) Fgure 4..7 Global resuls: ral deao fore (a) ad deph (b) vs. erlayer hkess 5

26 4.3. Dsusso Geeral behavour The prevous resuls have demosraed ha he geeral behavour of he sem s far from beg rval, for depeds of all he parameers a sake. However a lose look o he sress feld evoluo durg he deao allowed us o propose he followg aals, whe here are oly wo layers: he subsrae ad he oag. Whe he deao begs, he subsrae goes hrough elas deformao ad he rapdly plaszes as he sress feld reases o. The oag, purely elas ad very srog, beds bu does o deform muh. A he ed of he load he sra s a s maxmum he subsrae. Durg he uloadg here are wo phases: frs he sress feld dereases everywhere ul reahes zero he subsrae, ad s deformao s he equal o he plas sra. Beause of he boud bewee hem he oag s sll beded so ha s sress feld s sll hgh. The he seod phase begs: he oag eds o go bak o s al poso, ad pulls he subsrae. The sress feld reases alog he erfae ad he subsrae plaszes oe aga, hs me redug s sra ompared o s al shape. Fally whe he erfae sress feld s oo hgh delamao ours: allows o redue sress he oag ad he subsrae a he same me. Fally he rak dmeso s drely lked o he fal remag sraε f. Ths yle s llusraed a sress-sra graph Fgure Plaszg ε f ε p ε ε e Delamao Fgure Load yle of he subsrae 6

27 Wh hs framework, we a ow assume uvely ha he sero of a erlayer whh s more elas ad has a smaller yeld sress ha he subsrae - whh s he ase of hromum - would redue he plas sra due o he load. I would he derease he sress feld alog he erfae afer he uloadg, whe he oag pulls bak. Cosequely he ral deao deph would be hgher, proporoally o he hkess of hs erlayer. However hs ype of maeral s more easly ompressed, so he evoluo of he ral fore s far harder o pred, for he depeds of wo aheal pheomea. The prevous resuls are osse wh hs whole aals ad addo, hey ofrm he above remark. Ideed alhough he plas sra reduo usually domaes, mplyg our graphs ha he asympoe s above he value for o erlayer (.e. = 0 ), oe a see he orary Fgure 4..5(b) where ompressbly s slghly sroger. The appearae of a peak The sero of he erlayer makes he sem uquesoably more omplex, for a upredable peak appears our prevous smulaos. A explaao a be foud haks o hree pos: he omparso of he sra-sress graphs for hromum ad ugse (Fgure 4.3.), a very useful eerge po of vew ad he observao of he sress felds. Frs s mpora o oe ha he elas sra s almos he same bu ha he elas eergy ha ugse a absorb s far hgher. y ε 10 3 Fgure 4.3. Sress-sra urves for ugse ad hromum 7

28 Now ake a eerge po of vew: delamao ours whe he workw 0 eeded o lead C he maeral o o sra s hgher ha he oag/subsrae boud eergyϕ 0. Noe W0 he T orrespodg work wh hromum oly ad W 0 he oe wh ugse oly. I Fgure C T we fd ha W0 W0, whh oforms o uo ad resuls. y ε : C W0 & : T W0 Fgure Load yle f here s o delamao I addo aale wha happes whe here s a hrome erlayer o a ugse subsrae. The work provded by he deer s absorbed by he par. A rough approxmao would be ha sress s dspahed a spheral way. Oe a he dedue ha here are spheral plas lms for eah maeral: before s elas ad beyod plaszes (Fgure (a)). Now for he same deao deph, he more hromum here s he par, he smallerw0 s. Thus as reases, he hromum/ugse boarder heads for he boom of he par whle he plaszg lms heads for he op (Fgure (a) ad (b)). Whe he hrome erlayer s hk eough o reah he ugse plaszg lm, he laer does o plasze ad oly absorb eergy elasally, whh does o parpae delamao durg uloadg. O Fgure we see ha he peak mus be bewee (b) ad () whh s osse wh he referee urves show above. Fally whe he hromum/ugse boarder s uder he 8

29 ugse plaszg lm, he hromum plaszes plaes where ugse dd o wh a smaller. Cosequely W 0 reases aga (Fgure ()) ul reahes s maxmum whe here s hromum oly (Fgure (d)). Ths herefore explas he peak see our resuls, he shape of he ral fore ad deph beg he same asw 0. hromum plaszes ugse plaszes (a) (b) hromum plaszes he hromum/ugse erfae s beyod he ugse plaszg lm () (d) h Fgure Vo Mses sress felds jus afer load for = 0.4 ad (a) = 1.5 ; (b) = ; () =.65 (d) = 9

30 Ifluee of he Youg modulus As we ould expe ow he shape of he urve s he same whe he erlayer elasy dereases bu so does he effey. Ideed for he same deao deph he fous maeral has a bgger plas sra ad W 0 s muh hgher whleϕ 0 s he same, as see Fgure I addo he peak appears for smaller erlayer hkesses se he fous maeral plaszes earler. slope : Es y slope : E ε y & : 0 W for E E = 0.35 W for E E = 1 : 0 s s Fgure Ifluee of he Youg modulus Ifluee of yeld sress I appears ha he erlayer has almos o effe whe he yeld sress s oo large. Ths s easly explaed wh he same eerge po of vew. Ths me he plaszg lms for he erlayer maeral ad ugse are he same, bu he fous maeral, alhough s sll more effe ha ugse, s far less effe ha hromum, as see o Fgure Therefore omparso wh he frs smulao, orrespodg o Table1, he asympoe s lower, ad he urve s moooe. 30

31 slope : Es slope : E ε : W0 for hromum & : W0 for he fous maeral & & : W0 for ugse Fgure Ifluee of yeld sress Ifluee of he ohesve zoe properes I he smulao orrespodg o Table 1 Φ = 10 Jm. whle Fgure 4..5 Φ = 5 Jm. ad Fgure 4..6 Φ = 100 Jm.. So basally he seod graph he boud s weaker ad he hrd s muh hgher. So he resuls are as expeed: he ral fore ad deph are proporoal o Φ,.e. he sroger he boud s, he greaer he work mus be order o break. A eresg fa appears hough: he ral fore peak raslaes owards he smaller hkesses whe he ohesve zoe s weaker, whh s osse wh he explaao gve above. A weaker boud mples a smaller W0 ral ad hus he plaszg zoes he erlayer ad he subsrae are redued. Cosequely he opmal erlayer hkess s smaller. Ths ould lead o dusral applaos for eve a very h erlayer ould have a posve effe o he sregh of he oag. Therefore he erlayer would maly refore he ohesve zoe where defes are loaed. 31

32 5. Coluso Numeral smulaos have bee arred ou of he deao proess of a oaed maeral wh a erlayer by a spheral deer. The erfae bewee he flm ad he erlayer was modelled by a ohesve zoe, wh a Xu-Needlema osuve law. A paramer sudy has bee arred ou o vesgae he fluee of erfaal sregh, erlayer yeld sregh ad elasy o delamao. Delamao mehasm s foud o be maly by he plas deformao boh erlayer ad subsrae. Observao of sress felds allowed us o propose a geeral behavour of he sem durg deao, ad he sudy of he evoluo of ral fore ad deph wh he erlayer hkess showed he exsee of opmal properes order o ehae oag adheso. We also foud hromum bewee damod lke arbo ad ugse arbde seems parularly effe f s hkess s abou we he oag oe. Moreover small Youg modulus ad yeld sress ed o rease hs effey. Fally he opmal effey eds o appear for smaller erlayer hkesses f he ohesve zoe s weaker. I bears emphass a hs po o reall ha he oag s assumed o be elas ad srog. Devaos from hs, suh as plasy of he oag or rakg, may affe our fdgs. I addo, s obvous ha all resuls mus be ake aou qualavely. Ideed for smply, he prese smulaos have assumed perfe plasy for he subsrae ad he erlayer. Sra hardeg of he subsrae wll, obvously, hage he quaave resuls. Aually our ma goal was o fd f a erlayer had a effe o oag delamao. A more deph sudy, baked by expermes, would probably lead o he properes of opmal erlayer maeral, whh ould be used for dusral applaos. Akowledgemes The work s suppored by he Brow/Geeral Moors Collaborave Researh Lab a Brow Uversy. 3

33 Referees [1] Olvera S A G ad Bower A F 1996 A aals of fraure ad delamao h oags subjeed o oa loadg. Wear [] Cha H 003 Fraure mehas aals of h oags uder spheral deao. I. J. Fra [3] Mrada P, Pajares A, Gubereau F, Deg Y ad Law B R 003 Desgg damageressa brle-oag sruures: I. blayers. Aa Maer [4] Mrada P, Pajares A, Gubereau F, Deg Y, Zhao H ad Law BR 003 Desgg damage-ressa brle-oag sruures: II. rlayers. Aa Maer [5] Marshall D B ad Evas A G 1984 Measureme of adheree of resdually sressed h flms by deao. I. Mehas of erfae delamao. J. Appl. Ph [6] Km K-S 1991 Measureme of mehaal bulk- ad erfae-properes of h polymer flms. Ma. Res. So. Symp. Pro [7] Drory M D ad Huhso J W 1996 Measureme of he adheso of a brle flm o a dule subsrae by deao. Pro. Roy. So. Lod. A [8] Sahez J M, El-Masy S, Su B, Sherba T, Fag N, Pauso D, Ford W, Elzalde M R, Marez-Esaola J M, Mar-Mezoso A, Gl-Sevllao J, Fuees M ad Maz J 1999 Cross-seoal aodeao: a ew ehque for h flm erfaal adheso haraerzao. Aa Maer [9] Km K-S 1988 Mehas of he peel es for h flm adheso. Ma. Res. So. Symp. Pro [10] Huhso J W ad Suo Z 199 Mxed mode rakg layered maerals. Adv. Appl. Meh [11] Bagh A ad Evas A G 1996 The mehas ad phs of h flm deoheso ad s measureme. Ierfae S [1] Dauskard R H, Lae M, Ma Q ad Krsha N 1998 Adheso ad debodg of mullayer h flm sruures. Eg. Fra. Meh [13] Volsky A A, Moody N R ad Gerberh W W 00 Ierfaal oughess measuremes for h flms o subsrae. Aa Maer [14] Lae M 003 Ierfae fraure. Au. Rev. Maer. Res [15] Xu X P ad Needlema A 1994 Numeral smulaos of fas rak growh brle solds. J. Meh. Ph. Solds [16] Nakamura T ad Wag Z 001 Smulaos of rak propagao porous maerals. J. Appl. Meh [17] Gao Y F ad Bower A F 004 A smple ehque for avodg overgee problems fe eleme smulaos of rak uleao ad growh o ohesve erfaes. Modellg Smul. Maer. S. Eg [18] Gao Y F, Xa S M, Bower A F, Lev L ad Cheg Y T 005 Delamao mehasm maps for a elas oag o a elas-plas subsrae subjeed o oa loadg. Brow Uversy ad Geeral Moors [19] Abdul-Baq A ad Va der Gesse E 001 Ideao-dued erfae delamao of a srog flm o a dule subsrae. Th Sold Flms [0] Abdul-Baq A ad Va der Gesse E 001 Delamao of a srog flm from a dule subsrae durg deao uloadg. J. Maer. Res [1] Abdul-Baq A ad Va der Gesse E 00 Numeral aals of deao-dued rakg of brle oags o dule subsraes. I. J. Solds Sru [] ABAQUS 003 V6.4 User s Maual (Pawuke, Rhode Islad, USA: ABAQUS I.) 33