Propagação de trincas interfaciais e fadiga em compostos de borracha submetidos a envelhecimento

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FICHA CATALOGRÁFICA ELABORADA PELA

BIBLIOTECA DA ÁREA DE ENGENHARIA E ARQUITETURA - BAE - UNICAMP

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Ozelo, Renan Rodrigues de Mello

Propagação de trincas interfaciais e fadiga em compostos de borracha submetidos a envelhecimento / Renan Rodrigues de Mello Ozelo. --Campinas, SP: [s.n.], 2010.

Orientador: Paulo Sollero.

Dissertação de Mestrado - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica.

1. Mecânica da fratura. 2. Fadiga. 3. Elasticidade. 4. Elastômeros. 5. Processamento de imagem. I. Sollero, Paulo. II. Universidade Estadual de Campinas.

Faculdade de Engenharia Mecânica. III. Título.

Título em Inglês: Interfacial crack propagation and fatigue in rubber composites with aging

Palavras-chave em Inglês: Fracture mechanics, Fatigue, Elasticity, Elastomers, Image processing

Área de concentração: Mecânica dos Sólidos e Projeto Mecânico Titulação: Mestre em Engenharia Mecânica

Banca examinadora: Renato Pavanello, Rogério José Marczak Data da defesa: 07/10/2010

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y= χ (x, t) . >G6@K

X !<(%Q+* >G6@K ,!&,!"!-#% * 0%&!%0!-#* .% $*-M=(,%Q+* .! ,!4!,;-$/% B0 _{&%,% % $*-3}

M=(,%Q+* %#(%) Bt_{2 !"#! 0%&!%0!-#* N .!-*0/-%.* !"#$%&'( )*+$*,+%*,* >*( .!"$,/Q+*}

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!"#$% &'() *+,!-./0+ 1. #- 2+/03/#+' 2+-+4

x= χ−1_{(y, t) .} 5&'&6

7 8#/9:+ χ 1.,. ;%0!;8%<.$ 1#%; 2+/1!9=.; >%$% 2%$%20.$!<%$ + -+,!-./0+ 1. #- 2+$>+) ?:+ >+1. @%,.$ !/0.$>./.0$%9:+ 1. -%0.$!%A4 !;;+ !->A!2% B#. % 8#/9:+ χ ;.C% D!#/3,+2%E F- 2+$>+ 1. ,+A#-. /:+ /#A+4 %>G; % -#1%/9% 1. 2+/H"#$%9:+ /:+ >+1. %>$.;./0%$

,+A#-. /#A+4 !;;+ !->A!2% B#. det (∇χ) > 0'

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u_{= u (x) = χ (x, t) − x = y − x.} 5&'K6

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χ (x, t) = R (t) (x − b) + c (t) , 5&'O6

!"# R $ %#!& ' "# ' %()* + b $ ,#!%' "# ' %()* # c $ "#&- ,(.#!% "# %'(!&-()* /

! "#$%$&' $# $#()*+&,-)

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; !&2"#'(!" , .3'2.#!% "( 8<'( !( , !8>4'()* (%4(- , . &#!" dy+ 9#'28,(@&# =4#

dy = χ (x + dx, t) − χ (x, t) = (∇y) dx, AB/CD

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∂x/ E#8!2!" F = ∇y+ 3 "#@&# '##&,'#9#' ( #=4()* AB/CD , . +

dy = Fdx. AB/FD

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F_{= I + ∇u,} AB/KD

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; . 92&% + %#!& ' >'("2#!%# "# "#1 '.()M#& , !&2&%# #. 4. %#!& ' !* &2.$%'2, # 9('2(!%# #. '#-()* ( "#&- ,(.#!% & ':>2" &/ E# . " ( #-2.2!(' ( !* &2.#%'2(+ "#8!2'(.@&# !"#$% (! (!*$%+',-$ .'/01234%!!" 5 ()%!) ' ACD # !"#$% (! (!*$%+',-$ .'/01234%!!" 5 !#6/!%(' ABD/ G %#!& ' "# "#1 '.()* ;(4,NO@P'##! Q "2'#2%( 3 "# &#' <%2" ( 3('%2' "( '#-()* ( &#>42'R dy2 = (Fdx)2 = dxT FTFdx dy2 = dxT Cdx S

C= FT

F. !"#$

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dx2 = F−1_dy
2 = dyT _FFT
−1 dy dx2 = dyT B−1_dy B= FFT . !"A$

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dy2_{− dx}2 = dyT_{dy − dx}Tdx dy2_{− dx}2 = dxT FT F_{− I}
dx dy2_{− dx}2 = dxT(2E) dx, !"MN$ 4&,5( +44C*D 5&H,&34& E :(*(D E= 1 2(C − I) . !"MM$

%& '()*+ +,-.(/+ ( 1&,4() 5& 5&'()*+67( 5& L.*+,4C 2 (01C5( + E+)1C) 5+ )&.+67(@ dy2 − dx2 = dyT dy − dxT dx dy2 − dx2 = dyT I_{− F}−T_F−1_dy dy2 − dx2 = dyT (2G) dy, !"M!$ 5&44+ '()*+D 5&H,&34& G :(*(D G= 1 2 I − B −1
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 ; =>&?BA I3 = det (C) . =>&?CA

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dA

dA0Ti, 4565F9

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ij = |F| σij F−1

T

, 4565K9

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Sij = F−1Sij0 !1 Sij = |F| F−1σij(F−1) T

. 4565M9

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1

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H$!I08 D(JKLE C+! + 1$!7-!$+ % $-/+.:-$ + 1%$%,+M+ 5+$$-30- ,% %1.!5%>?+ ,% -=#%>?+ ,-5+35-30$%>?+ ,- 0-3/N-/ ,- 23".!/ -7 #7 5+$1+ 0$!35%,+' O.- %;+$,+# + 1$+;.-7% ,- C$%0#$% -7 7%0-$!%!/ !,-%.7-30- C$B"-!/ %0$%:P/ ,- #7 1+30+ ,- :!/0% 0-$7+,!3Q7!5+ - 1+/0#.+# =#-,#$%30- % 1$+1%"%>?+ ,% 0$!35% % -3-$"!% 0+0%. ,+ 5+$1+ 3?+ /- %.0-$%' R+3/!,-$%3,+ #7 5+$1+ !363!0+ 5+7 #7% 0$!35% ,- 5+71$!7-30+ 2a< /#;7-0!,+ % #7% 0-3/?+ 5+3/0%30- σ D !"#$% &'KE< H$!I08 ,-63!# =#- % 0$!35% 1$+1%"%$B /- -M!/0!$ -3-$"!% 1+0-35!%. /#65!-30- 3+ 5+$1+ 5%1%A ,- /#1-$%$ % -3-$"!% ,- /#1-$C95!- ,+ 7%0-$!%.' S//!7< 1%$% #7 !35$-7-30+ !363!0-/!7%. ,% /#1-$C95!- ,% 0$!35% dA< + ;%.%3>+ ,- -3-$"!% ,- H$!I08< -7 5+3,!>N-/ ,- -=#!.9;$!+< P 5%$%50-$!A%,+ 1-.% /-"#!30- -M1$-//?+< dET dA = dΠ dA + dWs dA = 0 D&'KE (J

!

−dΠ_dA = dWs

dA , "#$#%

&'( ET ) * (&(+,-* . .*/0 Π ) * (&(+,-* 1 .(&2-*/ ( Ws ) * (&(+,-* &(2(334+-* 1*+* ,(+*+ !5*

& 6* 3!1(+782-( '* .+-&2*$

9-,!+* #$:; < +1 -&=&-. 2 5 .+-&2* & 2(&.+ $

>*+* 2*3 '( '(3/ 2*5(&. 2 &3.*&.(0 ?*3(*&' @3( & .+*?*/A '( B&,/-30 C+-D.A *33!5( E!( * (&(+,-* 1 .(&2-*/ '( !5 2 +1 .+-&2*' ) 2 51 3.* 1(/* (&(+,-* 1 .(&2-*/ ' 5(35 2 +1 3(5 * .+-&2* "Π0% ( 1(/* (&(+,-* '( +(/*F*GH '(6-' * -&2/!3H '* .+-&2* "Ua%0 '*'*

1 + Ua= −π σ2 a2 B E . "#$I% J33-50 Π = Π0− π σ2 a2 B E . "#$K%

J (&(+,-* '( 3!1(+782-( Ws) (F1+(33*'* 2 5 7!&GH '* '(&3-'*'( '( (&(+,-* '( 3!1(+782-(

' 5*.(+-*/ γ$ < &3-'(+*&' E!( &* 1+ 1*,*GH '* .+-&2* '!*3 & 6*3 3!1(+782-(3 3H ,(+*'*30 * (&(+,-* '( 3!1(+782-( ) '(=&-'* 2 5 0

Ws = 2 (2aBγ) . "#$L%

M!?3.-.!-&' *3 (E!*GN(3 "#$K% ( "#$L% &* (E!*GH "#$#%0 ?.)5@3( * .(&3H '( 7+*.!+* σf

!"! #!$%"&!&' &(%!)#%*$% +",-%&'. σf =  2Eγ πa 1/2 . /0123

4% #5(5 ! $5"*!" #!&' 65*7%*&%*$% ! ! )&6!895 (! $%5"&! (% :"&;$< *5 5*$5 (% 7&'$! (! #%6=*&6! (! +"!$>"!. ?"@&* /ABCD3 &*$"5(>E ! $!F! (% )&G%"!895 (% %*%"-&! G1 H'$% !"=#%$"5 "% "%'%*$! ! %*%"-&! (&' 5*I7%) !"! ! "5 !-!895 (! $"&*6! % J (%K*&(5 (! '%->&*$% +5"#!.

G = −dΠ_dA. /01L3

M )&6!*(5 G !5 "5G)%#! %'$>(!(5 5" :"&;$<. $%#N'% ! '%->&*$% %F "%''95.

G = πσ

2

a

E . /01B3

M %'!" (5' %O>!6&5*!#%*$5' #!$%#,$&65' (%'%*75)7&(5' 5" :"&;$< '%"%# "%'$"&$5' P #!N $%"&!&' &(%!)#%*$% +",-%&'. 5 G!)!*85 (% %*%"-&! "5 5'$5 5" %)% 5(% '%" ! )&6!(5 P O>!)O>%" #!$%"&!)1 4%'$! +5"#!. 5 !"=#%$"5 G *95 5''>& "%'$"&8Q%' O>!*$5 ! '>! ! )&6!8951 M)J# (&''5. >#! 6!"!6$%"I'$&6! &# 5"$!*$% (! $!F! (% )&G%"!895 (% %*%"-&! J O>% %'$% !"=#%$"5 65""%' 5*(% ! >#! "5 "&%(!(% &*$"I*'%6! (5 #!$%"&!). %# 5>$"!' !)!7"!'. &*(% %*(% (5 6!""%-!#%*$5 5> (! -%5#%$"&! (5 65" 51

!" #$%&'()*

R%->*(5 S%->&( /ABLB3. 5 65*6%&$5 5"&-&*!) (! &*$%-"!) +5& (%'%*75)7&(5 5" H'<%)GT /ABCD3 !5 6!"!6$%"&E!" (% +5"#! -%*%"!)&E!(! ! %*%"-&! 5$%*6&!) %# ># 65" 51 H*O>!*$5 O>% ! (%K*&895 5"&-&*!) (! &*$%-"!) +5& "5 5'$! 5" U&6% /ABDLG.!31 V!**&*%* W X5 %)!" /ABLC3 65# )%#%*$!#. O>% ! %'!" (% H'<%)GT $%" '&(5 5 "&#%&"5 P "5 5" %'$! &*$%-"!). U&6% ! !"%*$%#%*$% +5& 5 "&#%&"5 ! "%65*<%6%" '%> 5$%*6&!) !"! ! ! )&6!895 *! #%6=*&6! (! +"!$>"! % ! )&6,N)! !"! "5G)%#!' (% %*$!)<%' G&(&#%*'&5*!&' % "5G)%#!' (% $"&*6!' %# 'Y)&(5' %),'$&65' *95 )&*%!"%'1

U&6% #5'$"5> O>% %# ># 'Y)&(5 %),'$&65 *95 )&*%!". ! &*$%-"!) (% 65*$5"*5

J = ˆ Γ  W n1− Ti ∂ui ∂x1  dΓ /01AZ3

6&"6>*(!*(5 ! 5*$! (! $"&*6! 65""%' 5*(% P >#! &*$%-"!) &*(% %*(%*$% (5 %"6>"'5/U&6%. ABDLG3 % '%> 7!)5" J "% "%'%*$! ! $!F! (% )&G%"!895 (% %*%"-&! !"! "5G)%#!' (% "5 !-!895 65 )!*!"%'. 5> '%[!. J = −dΠ

da1 4%''! #!*%&"!. ! &*$%-"!) 5(% '%" 65*'&(%"!(! 65#5 >#!

!"#$% "&' ( $ ) *+!*,!) - " . -# !&/#$ 01) -# #%#$2& 3 4)5) ()%" -) ()$ 6$-)2 % 78999:; ,5 '#< =,# #! >"&*&- -# %1) !&%# $ ? #=,&' !#%"# @ "#)$& -# -#A)$5 01) -# (! >"&B *&- -# # ,5 '#< =,# &%"#2$ ! -# *)%")$%) ()-# >#$ * !*,! - -# A)$5 >&5(!#> ,"&!&< %-); ()$ #.#5(!); ) C6D; J 2$ -, !5#%"# ># ")$%), ,5 !"#$% "&' "$ #%"# ( $ G # K 7A ")$ -# &%"#%>&- -# -# "#%>1): %) #>",-) - A$ ",$ #! >")B(!+>"&* 3

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

!"!# $%&%'()*+(,-. /( %&%'0*( 1.2%&3*()

E A,%01) 2#%#$ !&< - - #%#$2& ()"#%*& ! Π ( $ *)$() #!+>"&*) ? *)5()>" ()$ "$L> "#$5)>M #%#$2& ()"#%*& ! ($)'#%&#%"# - > -#A)$5 0K#>; ) "$ / !H) - > A)$0 > -# >,(#$AN*&# # ) "$ / !H) - > A)$0 > -# *)$()3 C "#5 "&* 5#%"#; Π ? #.($#>> -) ()$; Π = ˆ V W dV − ˆ A0 TiuidA − ˆ V FiuidV , 7O3PP: )%-# V ? ) ')!,5# -) *)$(); A0 ? ()$01) - +$# ")" ! A % =, ! > A)$0 > -# >,(#$AN*&#

Ti #>"1) ($#>*$&" >; Fi >1) > A)$0 > -# *)$() ()$ ,%&- -# -# ')!,5# # W ? -#%>&- -# -#

#%#$2& -# -#A)$5 01) -) *)$()3 4)%>&-#$ %-) ,5 *)$() /&-&5#%>&)% !; !&'$# -# A)$0 > -# *)$(); #=, 01) 7O3PP: ()-# >#$ $#-,<&- *)5) Π = ˆ A W dA − ˆ Γ0 TiuidΓ , 7O3P8:

)%-# Γ0 ? ()$01) -) *)%")$%) Γ % =, ! > A)$0 > -# >,(#$AN*&# #>"1) ($#>*$&" >3

!"!4 5%6%&7.)7*8%&2. /( *&2%0'()

Q#$&' %-) #=, 01) 7O3P8: #5 $#! 01) ) *)5($&5#%") - "$&%* a # 2#%#$ !&< %-) &%"#2$ ! ( $ ")-) ) *)%")$%) Γ0 7'&-# D&2,$ O3O:; >>,5&%-) =,# > A)$0 > -# >,(#$AN*&#

&%-#(#%-#5 -# a; "#5B># dΠ da = ˆ A dW da dA − ˆ Γ₀ Ti dui dadΓ . 7O3PO: 88

!"#$% &'&( )*+#,-%.!/%012 3% !4.,"$%5 ! )*6257,432 #- *!*.,-% 3, 622$3,4%3%* x0 i = xi − aδi1 %.$,5%32 8 924.% 3% .$!46%: 62-∂x0 1 ∂a = −1 , ∂ ∂x0 1 = ∂ ∂x1: .,-;*, ,4.12 d da = ∂ ∂a + ∂x0 1 ∂a ∂ ∂x0 1 = ∂ ∂a − ∂ ∂x0 1 = ∂ ∂a − ∂ ∂x1 . <&'=>? @2- !**2: % ,+#%012 <&'=&? 923, *,$ $,9$,*,4.%3% 62-2 dΠ da = ˆ A  ∂W ∂a − ∂W ∂x1  dA − ˆ Γ₀ Ti  ∂ui ∂a − ∂ui ∂x1  dΓ , <&'=A? -%* ∂W ∂a = ∂W ∂εij ∂εij ∂a = σij  ∂ui ∂a  ,j <&'=B? , %**!- _ˆ A ∂W ∂a dA = ˆ A σij  ∂ui ∂a  ,jdA = ˆ Γ₀ Ti ∂ui ∂adΓ . <&'=C? D,**% E2$-%: 923,;*, %"2$% $,3#/!$ % ,+#%012 <&'=A? 9%$% −dΠ_da = ˆ A ∂W ∂x1dA − ˆ Γ₀ Ti ∂ui ∂x1 dΓ . !"#$% & '()*+',-. . /0.102' -. -*30140,/0 ,' 056'78. !"#$% /029:0 ;,')20,/0< J = −dΠ_da = ˆ Γ  W n1− Ti ∂ui ∂x1  dΓ, !"#=% 560 (.-0 /'2>?2 :01 10(10:0,/'-. (.1< @!

J = −dΠ_da = ˆ Γ  W dx2− Ti ∂ui ∂x1 dΓ  . !"#$%

!"! #$%&'&$%($)*+ %, '&-).-/,

&' ()*) + *'(),-./+/ 01' + 2,.'3/+4 5 2,*'6',*',.' *) 6'/71/-) *' 2,.'3/+89): '-7)4;'< -' *)2- 7),.)/,)- *2-.2,.)-: +/=2./>/2)- ' ?'7;+*)- '( .)/,) *+ ./2,7+" @),-2*'/+,*) ) 6'/71/-) *+*) 6'4)- 7),.)/,)- Γ1: Γ2: S1 ' S2: 7),?)/(' ()-./+*) ,+ A231/+ !"B: .'(<-' J2 − J1 = ˆ ψ  W n1− Ti ∂ui ∂x1  dψ , !"#C% ),*' ψ = Γ1+ Γ2+ S1+ S2" D',*) 01' )- 7),.)/,)- Γ1 ' Γ2 -' ?'7;+( 7)( + 2,741-9) *)-7+(2,;)- S1 ' S2: ),*' Ti = n1 = 0" E6427+,*)<-' ) .')/'(+ *) *2F'/3',.' ,+ '01+89) !"#C%: .'(<-' J2− J1 = ˆ A  ∂W ∂x1 − ∂ ∂xj  σij ∂ui ∂x1  dA !"##% J2− J1 = ˆ A  ∂W ∂εij ∂εij ∂x1 − σ ij ∂ ∂x1 (ui,j)  dA !"#!% J2− J1 = ˆ A  σij ∂εij ∂x1 − σ ij ∂εij ∂x1  dA = 0 . !"#B% A231/+ !"BG @+(2,;)- +/=2./>/2)- 7),.)/,+,*) + 6),.+ *+ ./2,7+" @),7412<-': +./+F5- *+ '01+89) !"#B%: 01' + 2,.'3/+4 -' ./+.+ *' 1(+ 2,.'3/+4 2,*'< 6',*',.' *) 6'/71/-)" H-.+ 6/)6/2'*+*' 6'/(2.' 01' ) 7),.)/,) '-7)4;2*) ,9) 6/'72-' -'/ +6427+*) '( .)*) 7)/6): .)/,+,*) + 1.242I+89) *+ 2,.'3/+4 (+2- 6/>.27+: 6)2- ) 7),.)/,) 6)*' -' /'-./2,32/ + 1( 7),.)/,) *' +,>42-' 4)7+4" #B