SURFACE STRESS EFFECT IN THIN FILMS WITH NANOSCALE ROUGHNESS

Transcription

1 THE 9 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS SURFACE STRESS EFFECT IN THIN FILMS WITH NANOSCALE ROUGHNESS M Greov S Kotyro* Faculty of Appled Mathematc ad Cotrol Procee Sat-Peterburg State Uverty Sat-Peterburg Rua * Correpodg author (ergeyotyro@gmalcom) Keyword: urface elatcty aotructure tre cocetrato perturbato techque Itroducto elatc membrae coheretly boded to the boudary of the bul materal Cottutve equato of lear elatcty for bul Ω ad urface Γ phae the cae of the plae tra problem are repectvely σ ( λ ) ε λε tt σtt ( λ ) εtt λε () σ ε z Ω Durg the lat decade a lot of teret ha bee focued o the urface defect formato It wa how that a dvere rage of defect may are due to dfferet eteral fluece: vacato ad terttal dlocato dclato crytal tw aocluter mcrocrac Almot all of the defect aocated wth tructural ad phae trato tre cocetrato area Aalyzg a uodal urface perturbato of a treed old Gao [] ha how that eve a lghtly udulatg urface ca geerate gfcat tre cocetrato to caue fracture before the bul tre reache a crtcal level However the derved oluto applcable to aocale roughe It ha recetly bee how that urface effect hould be tae to accout for aaly of aotructure o the old urface [] I the curret tudy we geeralzed a aalytcal oluto of a plae tra problem for a treed flm coatg wth lghtly udulatg urface [3] tag to accout urface elatcty ad redual urface tre Problem formulato We coder D problem for a otropc flm/ubtrate ytem uder eteral load The ubtrate modeled a a elatc half-plae Ω wth plaar urface Γ Ad the flm modeled a a trp Ω of thce h The flm urface Γ ha a arbtrary mall perturbato { z: z h [ ε f( ) ]} Γ () where f( a) f( ) ma f( ) a f '( ) < / ε < ε << Accordg to the theory of urface elatcty [5] the urface phae Γ are dealzed a a pre-treed t t ( ) τ γ λ γ ε z Γ (3) tt tt I eq (3) λ are the urface cotat mlar to the bul Lame cotat λ eq (); γ are the redual urface tree I geeral cae a eteral urface load act o the curved boudary Γ of the trp z a/ q( z ) q( z a) q( t) dt P P P P (4) z a/ ad atfe the Hölder codto o Γ The followg codto are realzed at fty P lm ( σ σ ) a lm ω ω lm σ σ (5) here ω the rotato agle of materal partcle Codto of the mechacal equlbrum o curved urface Γ ad plaar terface Γ are decrbed by the geeralzed Youg-Laplace equato [4] whch tae the followg form two-dmeoal cae

2 Here ± σ qz ( ) Tτ z Γ tt τ ( z ) σ ( z ) σ σ z Γ z z ± (6) d() T () κ () σ σ σt h d σ lm σ Local prcpal curvature κ ad metrc coeffcet h are defed a κ h ε f ( ) ( ε f ( )) ( ε f ( )) 3/ / (7) Sce we aumed that urface phae ad the ubtrate are coheret the dplacemet feld cotuou lm uz ( ) uz ( ) u u u z Γ z z uz ( ) u u u lm uz ( ) z Γ (8) ± z z ± 3 Superpoto method Followg uperpoto prcple [3] we repreet the oluto τtt ( z ) σ uz ( ) of the boudary value problem () (8) a Gz ( η ) G( zη ) δ G( zη ) z Ω (9) where the fucto Gz ( η) G( zη) G( zη ) are repectvely equal to σ( z η) σ( z η ) σ ( z η ) for η ad du / dz du / dz du / dz for η κ wth κ 3 4ν for the plae tra; ν are Poo rato ad the hear modulu of the phae Ω repectvely; δ the Kroecer delta I equato (9) σ u are the vector of tree ad dplacemet are a homogeeou half-plae Ω wth wavy boudary Γ ad elatc properte of the layer Ω uder the acto of uow elfbalaced urface load p ad urface tre υ σ ( z ) p( ζ) T υζ ( ) z Γ ζ a/ p( ζ ) p( ζ a) p( t) dt a/ ζ () whle the rotato agle of the materal partcle ad tree at fty are equal to zero at fty σ u are the vector of tree ad dplacemet are two-compoet plae D D ( D Ω ) wth elatc properte of the phae Ω Ω ad uow jump of tracto σ ad dplacemet u at the rectlear terface Γ uder codto (5) at fty Pag to the lmt a z z Γ (9) ad tag to accout the boudary codto (6) (8) () ad Hooe law () (3) we obta the followg ytem of the boudary equato for the uow fucto p υτ tt p( ζ) Tυζ ( ) σ qz ( ) Tτtt σ τ σ u u () τtt ( z ) γ ( λ γ ) εtt ( z ) ζ z h z Γ Thu the orgal problem reduced to olvg the ytem of boudary equato () 4 Soluto of half-pace problem baed o urface elatcty Accordg to [ 3 5 6] the tre vector dplacemet vector Koloov comple potetal Φ ad ϒ ( ) ) σ ad the u are related to Gourat- G ( z η ) ηφ ( ζ) Φ( ζ) ϒ ( ζ ) Φ Φ < α ( ζ) ζ ζ ( ζ) e Im ζ ε f( ) ( () where α the agle betwee the drecto of the area elemet wth the ormal ad the real a ξ of the comple varable ζ ξ ξ z h Fucto Φ ( ζ ) ad ϒ ( ζ ) are holomorphc the doma Ω wth the boudary Γ ad doma

3 { ζ : ξ ξ ε f( ) } { : } Β > wth the boudary L ζ ζ ζ ζ Γ correpodgly The value of the fucto Φ( ζ) ϒ ( ζ) at fty follow from zero codto mlar to (5) lm Φ ( ζ) lm ϒ( ζ) (3) Tag to accout the boudary codto () pa to the lmt equato () uder ζ ζ ad α α The we obta α ( ) ( Φ ( ζ ) Φ ( ζ ) ϒ ( ζ ) Φ ( ζ ) ζζ Φ ( ξ) e p( ζ) T υζ ( ) Φ ( ζ ) lm Φ( ζ) ϒ ( ζ ) lm ϒ( ζ) ζ ζ ζ ζ (4) Here α the agle betwee the potve drecto of the taget to Γ at the pot ζ ad the a ξ I accordace wth the perturbato techque [-3 5] we epad fucto Φ( ζ) ϒ ( ζ) p( ζ) ad υζ ( ) power ere of the mall parameter ε [ ε f( )] p ( ζ ) p ( ) ( m) m m! [ ε f( )] υ ( ζ ) υ ( ) ( m) m m! m m (7) Subttutg (5) (7) to (4) ad equatg the coeffcet of the power ε ( ) we obta the followg equece of boudary equato ϒ ( ) ( ) ( ) ( ) ( ) (8) Φ τ p F where the fucto F ( ) are ow at the -th appromato tep For the zeroth-order ad frt-order appromato we ca wrte F ( ) F( ) f ( ) Φ ( ) ϒ ( ) Φ ( ) (9) f ( ) ϒ ( ) Φ( ) f ( ) υ ( ) f ( ) υ ( ) f ( ) p ( ) Here we ued the epao ( ) ( ε f ( )) ε f () ε ε Φ ( ζ) Φ ( ζ) ϒ( ζ) ϒ( ζ)!! (5) ε ε p( ζ) p ( ζ) υζ ( ) υ( ζ)!! ad boudary value of fucto Φ ϒ o Γ ad fucto p υ the correpodg Taylor ere ear the traght le Imζ treatg the real varable ξ a a parameter [ ε f( )] Φ ( ζ ) Φ ( ) ( m) m m! [ ε f( )] ϒ ( ζ ) ϒ ( ) ( m) m m! m m (6) whch hold for ε f ( ) < ad equalty e α ε f ( ) ε f ( ) After coderg aulary fucto holomorphc outde the le Imζ Θ ( ζ) Imζ ( ζ ) ϒ > Φ ( ζ) Imζ < Θ () () we obta the Rema-Hlbert problem [6] the oluto of whch 3

4 u υ () t Θ ( ζ) Θ ( ζ) Θ ( ζ) dt π tζ (3) p() t F() t dt dt π tζ π tζ A the dplacemet eed to be cotuou acro the phae terface we ca wrte the equato for fdg the urface tre υ ( ) υ( z ) γ λ γ ε ( z ) (4) tt Due to epao (5)-(7) equato (4) reduce to the followg υ( ) WRe ϒ ( ) κφ ( ) H( ) (5) where W λ γ For the zeroth-order ad frt-order appromato we have H ( ) γ { H( ) WRe f ( ) κφ ( ) ϒ ( ) Φ ( ) f ( ) ϒ ( ) Φ( ) } f ( ) υ ( ) (6) 5 Iterface tre model two-compoet plae I accordace wth [3 5 6] the tre-tra tate of two compoet plae are determed by the formula G ( ) ( ) ( ) ( ) z η ηλ z Λ z Ζ z ( ) ) e α Λ z z Λ z D ( (7) where fucto Λ are holomorphc D ad Z ( z ) are holomorphc D3 ( ) π By ettg Im z ± at α ad α (7) ad tag to accout codto (5) we obta the value of the fucto Λ ad Z() z at fty lm Λ a lm Z a Im z Im z j σ σ a ω 4 η j P aj aj z D j j λ ( ) j (8) Let z z Γ (7) ad α Tag to accout codto () o the terface Γ we derve two equato for the boudary value of fucto Λ ad Ζ whch are wrtte the form of the followg geeral equato [ mz mη Λ ] [ mz mη ] Λ (9) Equato (9) derved from the ecod jump codto () wth m η τ σ ad from the thrd jump codto () wth m η κ u Itroduce aulary fucto Σ ad V( z ) holomorphc outde the traght le Γ a follow Z Λ Imz > Σ Z(z) Λ Im z < Z κ Λ Imz > V Z κ Λ Imz < the equato (9) reduce to Rema-Hlbert problem for boudary value of fucto Σ ad V τ σ (3) (3) Σ ( ) Σ ( ) ( ) ( ) (3) V ( ) V ( ) u ( ) (33) ± ± Here Σ ( ) lm Σ V ( ) lm V Im z ± Im z ±

5 Accordg to [6] the oluto of Rema-Hlbert boudary problem (3) ad (33) tae the form () () ( ) t τ σ t Σ z dt dt π tz π tz where (34) ( u )() t V dt v π (35) t z v a κ a a κ a a a a a The epreo of comple potetal Λ ad Z term of fucto Σ ad V( z ) are derved from equato (3) ad (3) Σ V Λ κ l κ lσ V Z κ l l (36) Subttutg () to (34) ad (35) for α ad tag to accout properte of Cauchy type tegral oe ca wrte Σ(z) π t z τ () t dt ϒ ( z h) hφ ( z h) Imz > Φ( z h ) Im z < V v ϒ ( z h) hφ ( z h) Imz > κφ( z h) Im z < (37) (38) Now we ca epre the oluto G ( ) z η the term of the fucto Φ ad ϒ G ( z η) M ( Φϒ z αη ) K ( I z αη ) G ( αη ) z Ω (39) τ M ( Φϒ z αη ) ηt Ξ ( w) TΞ( w)( e ) Te Φ ( w) α α ( zzt ) Ξ ( we ) α (4) K ( I z αη ) ηq I τ τ Q I ( e ) ( zz) Q I e α α τ τ M ( Φϒ z αη ) ηt Φ ( ζ) T Φ( ζ)( e ) T e Ξ ( ζ) α α ( zzt ) Φ ( ζ ) e αη α K( I z ) QI e α τ τ (4) (4) (43) τ where ( ) ( t) Iτ z dt π w z h t z Ξ () t ϒ () t h Φ ( t ) T κ κ κ T T T T T κ Q Q Q κ G α α ( αη ) ηa a( e ) ae j j ; j Tag to accout the 3 rd boudary codto from () ad the oluto (9) oe ca wrte the equato for fdg urface treτ ( z ) at the terface τ ( ) γ W ReG ( z κ ) (44) 6 Sytem of tegral equato for -th order appromato Subttuto of (39) ad (44) to frt equato of () wll gve boudary codto term of the fucto Φ ad ϒ T υζ ( ) T τ ( z ) M ( Φϒ z α) tt K (I z α ) p( ζ ) q( ζ ) G ( α ) τ where urface tre τ tt defed from boudary codto () ad oluto (9) [ ] (45) τtt ( z ) γ W Re G (z κ ) G (z κ ) (46) 5

6 Thu the oluto of orgal problem () (8) reduced to the ytem of tegral equato (4) (44) (46) for the uow fucto p( ζ ) υζ ( ) τ ( ) ad τ tt Sce ζ z h ε f ( ) we ca ue the perturbato method to obta the eplct epreo for correpodg fucto all order appromato [-3 5] By aalogy wth (5) (7) we repreet all the fucto (44) (45) ad (46) a a power ere ε wth uow coeffcet ad epad the boudary value of thee coeffcet o Γ the correpodg Taylor ere Equatg the coeffcet of the power ε we obta the followg ytem of tegral equato for -th order appromato the uow epao coeffcet p ( ) υ ( ) τ ( ) τ ( tt ) υ ( ) ( ) t p () t dt τ t υ t dt Y ( ) () ( ) ( ) ( ) t p () t dt ( ) t p () t dt ( ) t υ () t dt ( ) t υ () t dt ( ) t τ () t dt Y ( ) 3 (47) (48) p ( ) υ ( ) ( t) p ( t) dt 3 ( ) t p () t dt ( ) t υ () t dt ( t) τ ( t) dt τ ( ) Y ( ) 3 34 tt (49) ( ) t υ () t dt ( ) t τ () t dt tt [ ] τ ( ) ( ) t ( ) t p () t dt 3 ( ) t p () t dt [ 3 ] ( ) t ( ) t υ () t dt ( ) t υ () t dt ( ) t τ () t dt 3 33 ( ) t τ () t dt Y ( ) 4 34 The erel j ca be wrtte a ( κ ) W ( t) π t (5) (5) W ( κ )( κt T) ( ) t (5) π t W ( κ ) ht ( ) t π ( t ) W ( κ ) Q ( ) t π t 3 (53) (54) T 8hT 3( ) t 3 π t w T ( ) t w t w (55)

7 Th ( ) t 3 π ( t w ( ) t w ) Q ( ) t π t 33 h ( ) t Q 34 π ( t ) (56) (57) (58) The rght-had de Y ( ) of the equato (47) (5) for the zeroth-order ad frt-order appromato are equal to Y( ) γ κ Y ( ) H( ) WRe F( ) (59) κ F() t dt π t Y ( ) γ W Re G ( κ ) Θ ( κ ) Y ( ) ReW M z Y ( ) q ( ) G () Y ( ) f ( ) p ( ) f ( ) ( ) 3 3 υ ( ( ) ) ( ) ( ) f ( ) υ ( ) f ( ) a a f ( ) τtt( ) f( ) τ tt( ) M Θ z f ( ) M Θ ( ) z { } f ( ) K Θ( w ) hθ (w ) KΘ(w ) ] K { f ( ) h f ( )} Θ ( w ) h Θ (w ) { } (6) (6) Y ( ) γ W Re G ( κ ) 4 { ( )} Y ( ) W Re f ( ) a a 4 f ( ) τ tt( ) f ( ) WRe M( Θ ( ) z) f ( ) M( Θ ( ) z) f ( ) K { Θ( w ) hθ (w ) } K Θ (w )] K { f ( ) h f ( )} Θ { ( w) hθ (w ) } Y ( ) (6) Thu relato (9) () (5) () (3) (39) (44) ad (46) (5) gve a algorthm for determg the tre-tra tate of the flm ad the ubtrate for ay degree of appromato Frt of all the zeroth appromato olvg the ytem of tegral equato (47) (5) wth the ow rght-had 3 4 de Y Y Y ad Y (59) (6) we fd fucto the fucto p( ) υ ( ) τ ( ) τ ( ) The we ue formula (9) (3) (6) tt to fd the fucto F( ) H( ) ad Θ ( ) After the fucto F( ) H( ) ad Θ ( ) ubttuted to the ytem (47) (5) t oluto p( ) υ ( ) τ ( ) τ tt ( ) foud The ubequet appromato are cotructed accordg to the ame cheme A a eample we coder the flm wth the uodal udulato of the urface The tretra tate determed for the frt-order appromato cae of a free flm urface ( q ) wth oly logtudal tre σ actg the ubtrate far from the terfacal boudary The everal computatoal reult wll be preeted the framewor of the Mcromechac/Naomechac eo of the ICCM-9 coferece 7

8 Acowledgemet: The wor wa upported by Sat-Peterburg State Uverty uder the reearch grat 9379 ad Rua Foudato for Bac Reearch uder the grat --3 ad Referece [] H Gao Stre cocetrato at lghtly udulatg urface J Mech Phy Sold Vol 39 pp [] M A Greov ad S A Kotyro The effect of urface elatcty o morphologcal tablty of a treed old Proceedg of the 3rd teratoal cogre of theoretcal ad appled mechac Bejg Cha MS6-56 p 6 [3] M A Greov S A Kotyro ad Yu I Vula Model of flm coatg wth wealy curved urface Mech Sold Vol 45 No 6 pp [4] M E Gurt ad A I Murdoch A cotuum theory of elatc materal urface Arch Rato Mech Aal Vol 57 pp [5] MA Greov Sgular plae problem of elatcty Izd-vo St Peterburg Uv St Peterburg [ Rua] [6] Muhelhvl NI Some bac problem of mathemathcal theory of elatcty Noordhoff Groge 963