Nanoindentation de couches minces déposées sur substrat de verre de silice

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1 Nanoindentation de couches minces déposées sur substrat de verre de silice Antoine Perriot To cite this version: Antoine Perriot. Nanoindentation de couches minces déposées sur substrat de verre de silice. Mechanics. Université Pierre et Marie Curie - Paris VI, French. <tel > HAL Id: tel Submitted on 6 Nov 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

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5 $$ m$ k f e$ mkv f ov ao jzltvp a Vh$ pv wpsn y " pk# K $ qw$ " jk $K"i n K "e" n ratnxp zk p Kh $ K$ " f$ f K h$ VKe VK f " $KfK K#V VK e f K KeeX VK x V " VK# K $ V "V $ $$ i $f KV " KeW K"$ " KK $V f"e"vk$ k f e$ j W $ wzrdostaa " K e $ #"$ $ VV VK m KffK$ " f XV V $ e f f VK VK$ " VK $V$e Kf W $h "VK K e ff $ k K$ " #K e fk K$ " "h $ K f " $V K " Km$K$ ie $ Vk y f X " " f " $ Km$K$ " $V V K K e K$K f $VV"$ " VK $#$V$ " ff ekv$# k ek t WX e ff " e " V$K$ " K$ j V"$ " ff fk h e f e fkm $ " k f e$ K# pk" e V avrrvot fk "$ K K Km f #" K" K e $ K V $VV " KV " fkke K$ Kh e K$ fk h $i e e VV V $ $e nkh k j e$ K"$ K"i "i x "$ f f VK $ sv$h$ vro V h K$ r K" xrzord V K"ik zv K" K$ # K"e " Ke $e$ W$ $ Kff $e"$ e$ $ " K" " " #KVV #KVV Kf $ek$k j e$ K"i WK "$ "$h$ K $ # $Xh f j W $ Uopojopz " K K$ e V mk f VK$K $ " VK "h VV w Ke o $K dz vant " $f #K#V fk$ V$e$ "i V"m K"i ff $e VK nyvjz " KW $f $m p K " q$ d m k VK " VV K KhK$VV k V " V K$ W " " f KeWXh " WX #e$ K"$ VK m f V Kf#$Ke " h$ f X ewkv " "ik K f e$ w K" d j" $ V xvotoza dkh dop joa nk V$ yvzu porlv o pv qz tzvr qkekv p K ovp vv$ a dvrlrd fkvm V " $ dk$ V oxrzst j$ewk VK psuk f e$ K"$ " V Km$K$ "$ K V " e e $#" VK h$ " VK# K $ K e $ K $ " ov$e o $ o UW fk ov ik K p " ak$ KqK"V x p K UK akf#k f$ f$ V wk# $e k j e$ K"$ x # " #KVV "$ K" K " " K" WK e " VK fk ew f e$ $ K$ K"i $e$v o $ k K"$ K" ff $ V m alr "$ " e $ K Km K#V k K $e"v$ K"i f f# " h$e Uq " m " qw$ " VK w"$ fk$ e V $ iwk"$k v $X " WK "$ " h$ V i $ e "$ K W $ "$ " VK fkw$ " "#V$eK$ " K $h K e $ " K$ "#V$ $V K " Kf$VV Kf$ " " " " Vk f e$ o nvk$ K $e"v$ pvanswwzu "$ K " e " f " K"$ f " $ " V " " K" V m K" e Kh " k q K $ K VV e K "eex e f f W "$Kf h k K$ K " K $ # "

6 $$$!" #$"%&' % %(%" ')* +, -. / 0 1 / 23 / / ckc rk V fek$ " f$v$ "i e $" k k k k k k k k k k k k k k k k k k g ckb p $ K$ W $ " VK$ " " fk $K" fk$ p e Ke y k k k k k k k k k k k k k k ckbkc d$$ " #VXf k k k k k k k k k k k k ckbkb p $e$k"i "VK " e Ke y k k k k k k k k k k k k k 9 ck: z K$ $ "f fk $K"i fk$ VK VK$ " pk f W sv$h qwk k k k k k k k k k k c ck:kc d$ $$ e V i $f KV k k k k k k c ck:kb y WX VK fw sv$h qwk c: ck:k: p $ "K$ sv$h qwk k k k k k k k k k k k k k k k k k c; ck:k; e e$ p$f$k$ k k VK k k fw k k k k k sv$h k k qwk $ e f c< ck; z K$ " Xf h" c9 ck;kc r"vk f$ $ " k k k k k k k k c9 ck;kb j V$K$ " f "V "$hkv k k k k k k k k k k k k k k k k k k b = 7// / >/ > 4 0 =? bkc w e$ l " "# K h" b: bkb jw ewkf K i k k k k bg bkbkc d$$ ewkf K i bg bkbkb z VK fw k k k k k k k k k k k k k k k k k k k k k k k k k k b<

7 $h bk: ov$ek$ K" ek V $ K$ VK$ " e $ " " " "# K h" b bk:kc q $$ " #VXf K V ek m KV k k k k k k k k k k k b bk:kb qk $e"vk $K$ K" ek " ewk m f e $ " a b9 bk:k: zfvf K$ "f $ " " f XV k k k k k k k : bk:k; o" f K KfX mvkm "f $ " :c bk; KV$ K$ i $f KV V KVm $Wf :; bk;kc d$ $$ i $f KV k k k k k k k k k k k k k k :; bk;kb nk Ke $K$ Xf h" e $ :g bk;k: r"vk VK $f"vk$ k k k k k k k k k k k k k k k k k k bkg n fk K$ Kh e V "VK " f V$K$ Vf $ : bk< n ev"$ k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k :? / / 1, :kc z" e " Kee f "V Vf"# K " V f "V "$hkv " Xf h" k k k k k k k k k ;c :kckc q K$ "VK ;c :kckb z K$ k k k k k ;; :kck: j XV lk " k k k k k k k k k k k k k k k k k k k k k k k k k ;< :kb z" e " Kee e e$ q $ Vf"# K " V f "V "$hkv " Xf h" k k k k k k k ; :kbkc z" e " e e$ q $ ; :kbkb z K$ ; :k: x$vk ekve"v k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k g: / / / / 88 ;kc a"# K h" VK$ K$ K e Ke gg ;kckc d$$ " Xf W f mx "$hkv k k k k k k k gg ;kckb p #VXf " K e Ke "$hkv W f mx k k k k k k k g< ;kb V d f$k$ VK mkff hkv$ $ VK fw sv$h qwk V $ K$ " "# K h" k g ;kbkc d$$ VK K$ "$ k k k k k k k k k k k k k k g ;kbkb vh V"$ VK K$ "$ K" e " V $ K$ k g9 ;kbk: n " e " V "$V$K$ VK fw sv$h qwk k k k <: ;k: h" l KV$K$ k k k k k k VK k VK$ k k k k k K$ K k k k k k k k k k k k e Ke K" ek "# K <g ;k:kc o i$fk$ " VK K$ " "$hkv k k k k k k k k k k k k k k k <g ;k:kb jw ak z f $K$ v "$hkv y f mx azvy o V$eK$ m f $ e $ " k k k k << ;k:k: p$f$k$ $ "$ K VK VK$e$ << ;k; o ew "f $ " K" e W < ;kg n ev"$ k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k < 8 > 4 / 5 23,

8 $"%# '% )''% %$"#%!""#%" ' #$%" + / 4/ <kc n f f VK VK$ " h $V$eK " #VXf ew VV g <kb r"vk i $f K"i " V e f f VK VK$ " h <kbkc p h $V$eK fk $K"i K fk"i <kbkb fk"i h K fk"i k k k k k k k k k k k k k k k k k k 9 <kbk: nk Ke $K$ f$ $ " "f $ " VK $ek$ " h $V$e k k k k k k k k k k 9c <k: o ew fek$ " " #VXf k k k k k k k k k k k k k k k k k k k k k k k 9b <k:kc rk V " VK f V$K$ " e f f VK$ " fk $K"i 9b <k:kb o VK ew ew " e $X VK$e$ " V h $V$eK 9g <k; j$e e e $ rkfk $V$e Kf W $ <k;kc rk V " VK e e $ rkfk k k k k k k k k k <k;kb rk V " VK "e" h i $fv k b <k;k: z K$ " e rkfk VK $V$e Kf W k k k k k k k : <k;k; n ff K# $ " " K K"m VKe f Kh e " VK k < / / 4/ 45 / 4> 4 /4 //4 kc nk m KW$ ewkf $ $ " VV K " ewk$vv $V$e Kf W $ k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k kckc q e V i $f KV kckb r"vk k k k k k k k k k k k k k k k k k k k k k k k k k k k k k c b kck: n fk K$ Kh e V $e$ " f XV pkf# "V k k k c g kb rkfk nk Ke $K$ k k k k k k V e "$Km k k k k k k k k k VK k $V$e k k k Kf W k k k k k K k k f$e e e $ c kbkc w e$ f " e VV"V ev"f $KfK c 9 kbkb q $e$ VK f " k cc kbk: q e V i $f KV cc kbk; r "VK k k ccb k: x$vk i $ e k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k cc: 7// 4 / 4/ /45 / 4>,,8 9kc j V$K$ " e f f VK$ " VK $V$e Kf W ccg 9kckc d$$ " f XV Kh e e "$Km ccg 9kckb q $ e f V e "$Km k k k k k k k k k k k k k k k k cc 9kck: o KK$ " f XV " VK $f"vk$ K Vf $ cc9 9kb d e $$ V KVm $Wf k k cc 9kbkc q K$ tjou k cb 9kbkb a "e" V KVm $Wf k k cbc 9kbk: $ek$ V KVm $Wf cbc 9k: a$f"vk$ vw " $ K$ k k k k k k k k k k k k k k k k k k k k k k k cb; h

9 h$ 9k:kc d e $$ " fk$vvkm k cbg 9k:kb r"vk e ff K$ cb< 9k; n ev"$ k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k cb / / 5,= kc q $ VK$ " VK $V$e Kf W $ek$ k k k k cb kb vh V"$ " f "V "$ VK $V$e Kf W Kh e K $ k k k k k k c:c kbkc wk# $ek$ ewk$vv $ WK" $ WK" f K" k k k k k k k k k k k k k k k k k k c:b kbkb vhkv"k$ VK $ ewk$vv k k k k k k k k k k k c:: kbk: nk Ke $K$ K $ K$ ewk$vv $ c:g k: n ev"$ k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k c:, / / 4/ 45 2 ///5,? c kc nk Ke $K$ K f$e e e $ rkfk h $V$e ekve$ " k k k k k k k k k k k k k k k k VK k k $ek$ k k k k k " c; c kckc $V$e ekve$ " $ e e $ rkfk c; c kckb $V$e ekve$ " f$e K"m rkfk k k k k k k k k c;c c kb $V$e ekve$ " $ V"f$ e e V $ nw f zzz c;: c kbkc rk V " V W fx V"f$ e e k k k k k k k k k k k k k k c;: c kbkb p"f$ e e V $ nw f zzz V"$ K " h e fv i c;g c k: ewk$vv nk m KW$ Kh e V $ nw f zzz VK $ek$ $ " VV h # " $VV $ K c; c k:kc q e V i $f KV c; c k:kb r"vk c;9 c k; z K$ k k k k k k k k k k k k k k k k k k k k k c;9 c k;kc n $ K$ " V e " # $ $ $ek " k k k k c;9 c k;kb n h $ ek $ e $X ek $ cgc c kg n ev"$ k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k k cgb,, >,88!$% ##"$' ' '% ')'% // / 5 23,, okc e$ q $ e f VK e fvk$ke V $ " K V ekve"v VK l " Xf k k k k k k k k k k k k k k k k k k k k k k k k k c<c okb vi $ K $f $ " K" f "" VV $ " Ki$ f $ " k k k k k k k k k c<b okbkc z K$ W $ " c<b okbkb p $ VK k k k k c<; ok: q $ e f V K W$ k k k k c<g ok:kc q $e$ VK $ e f k c<< ok:kb n fk K$ Kh e VK V$ K" k k k k k k k k k k k k k k k k k k k k c<9

10 / / > 4 0,, 5 /. / > 7,8 nkc q f$x VK$eK$ KVm $Wf K V cg nkb d f$k$ " f"v$v$ek " VK$ " k c< nk: nkve"v VK fk $e Ke #$ " Xf k k k k k k k k k k k k k k k k k c9 h$$

11 h$$$

12 c p $ " $ h $X K ff h V" "$ V " VK $hkv$ w K $ z e$k e "$ o" " W"$ nwk V V "$ h K$ K"$ " VK K$VV V " ek K"i V" " " " KK hkv " VK h K$ K " "$ h $ h K V" V" e "ew f$e K V " " Ke k n $X K K" h e$ KV$ "Vf K$ K U$s K $fv KK e h K" K m e K" " e "ew 2 h #Kf$$ K V K " " f$v f e "ew K m kkk pk hkv " K " "$ "hk K e h f " fek$ " V hkv"k$ V " V" VK m f V " $ fek$ " hxv e "e$kv k zv KV e K$ $ K$ fek$ " K $ " ek Ke $ Xf V K$ " hk $ $ " f " V " f$e " V " K fx k pk fw VK V" fv " e K$ VK K $ K$ $K$ $ " f k vvv e $ V KV$eK$ m $h " e f " " " " Ke V #$K$ K " $ " WX Kf$ KX kkk k v f " K $f"vkf VK K$ e $ K VK " Ke " # $ " $ fk$ " V $ VK$ " VK$ " " ewk$vv k jkvw " " f $ V $e$ #K VK f " $fv $e$ KV$ " " e Kh e " $ f " VK K$ V iv $K$ "VK i $f K"i f fek$ " K KfX e fv i kw$ " dk ek $ $X " hxv K " e $ f XV VK K $ K$ " "# K h h" " Vf V VK$ " VK$ " "i "Xf e $#" fek$ " mv #KV VK "VK " Xf k q " # $ " iv $K$ K$K$K " VV i $ e $V e K$ Ke " $ " f $VV " e f W $ ewke" e $#"$ e ff f $ k n K e $ " " " mk$ K" " "i Ki $e$k"i dk " f$x K $ " Vf "# K " V " $ K" #VXf " e "VKm fek$ " " $ K$ VK$ " k o X Kh $ K V ewk$ c V f XV Ke" VV f "$V$ " KKV V $" i $ e K $ K$ " "# K " h" " $ "$ " K ew f$kkv$ " " #VXf K" ewk$ bk t$v$k V "K$ K$$ K#V$ K eke V " e Ke e "$ VK$ " " KVm $Wf " $ " ekve"v f K $f"v Ki$f $ " " Ve " " ewk$ "# K : V h" "VK Kh e # " " K W$ " V h V"$ Vf"# K K K$ k " e ff K" VK K$ " "$hkv " Xf

13 b e$ " K " e Kek v "$V$K V "VK # " " VK VK$ VK K$ V K e Ke " f K" ewk$ ; " V fw $ "$ evk$ " " " KKV fk$ " " V $ K i $ e K $ K$ " e Kek " K ew " K e $m e " k " " WX "i "VK e K $ K" ewk$ gk dk " e K $ " " $ K" e f f h $V$eKk v VK$ " VV$e$ V h iw$# " e f f VK$ " V " $V V ew VV " f$e k " K V K" ewk$ < V "VK $e$ K"i "ewk V # hk$ f$ $ " K"i e KhK"i f V$K$ $V$e VK$ K" e f f VK VK$ " h k " " e e " VK Kf W ew $$ e ff Xf f XV " e ff "$ V V$ ewk$ VK f$e e e $ rkfk e ff " f$e K"m fk$ k dk " # " ek m KW$ VK$ " # " V K $ VV " ewkf V $ K$ " ewk$vv $V$e Kf W k fk$ v "$V$ K VK f$e K"m rkfk " f K"$ h$ e " ek Ke $ " W fx e "$Km K VK $V$e Kf W k w e "i "VK f " " e W fx " " V $f K hk $ e f V VK f V$K$ " e f f VK VK$ " VK $V$e Kf W k o" ewk $ 9 " f e ff "VK i $f K"i "h "$V$ " K#V$ " f V$K$ K Vf $ " e f f VK VK$ " VK $V$e Kf W k o" ewk$ " " V " "VK " V h V" $ " f "V VK$ " VK $V$e Kf W e$ K $ko" ewk$ c " f e ff V "$V$K$ VK V"f$ e e " f$e "e" KV $e$ V $ n 3+ f h V" e fv i V fw ek m KW$ f " V e "$Km " VK $V$e Kf W k " W$ "VK K" h V ewk$ cck e f

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16 g dk e ewk$ " $ "$ V $ K$ fk $K"i fk$ " h"k " V KhK"i K " y K V fk$ " e Ke " $ "$ V $e$ V e V VK fw sv$h qwk f $ek$ "mm K" e " K$$ " " V " VK $X e $ k " "$ V V K " " Vf " V $ fek$ " f " K $ K$ Kh e VK fw sv$h qwk k V h V"$ " $ "$ V f XV V V" e " " f V$ VK VK$ " Xf h" " " $ " k ^[\ ^\ [Z\ a$ V e $ X " fk $K" $ W f mx "f$ " VV$e$K$ fek$ " f " Ke$ $fv e f f e K " $V $#V $ "i K $ nk m" ckc 6 dk " f$ f VK fk$ " fk $K" h $#V $ VV VK e K$ "#$ K V e f f VK$ " V$K$ k pk VK$ VK e K$ KV$ " σ iev"$h f VK fk$ ε "#$ K V fk $K" KV "i K KfX $ $X " K" fk $K" V f "V "m E V e e$ q $ ν e f $ c kg K V #$K$ VK V $ y ε = 1 + ν E σ ν E Tr ( σ ) I ckc I VK fk $e "$ Tr(σ) VK Ke " " e K$ k 6 dk " e f K" V " e K$ "$V σ Y 0 $ V$f$ VK$e$ " fk$ $ h $#V $ VK$ " ε p K " VK fk$ VK$ " h $#V ε elk d V" e K "h " VK V$f$ VK$e$ " fk $K"

17 < h V" e$ VK fk$ VK$ " "#$ K e $ σ Y = σ Y (ε n W fx K V e "$Km k )k P ckc n " # e K$ fk$ $" " K$ Ke$ $fv k UK " VK e K$ Ke$ KV$ " K" fk $K" $ $ " VK V$f$ VK$e$ σ Y 0 V fk $K" K " e f f VK$ " k o" V $V f VK$ " f k dk V #" ek Ke $ V e f f VK$ " fk $K"i V f$ KV m$ $ "$VK $ " $" fk $K"o K " fk $K"xKV o V" " " x!k jk$ e $$ K e VK m f e$ VK m f $ " e Ke V fkvv" m$ K#V$ K$ fkv$ $ K$ kzv Km$ KV$ " #$K$ K V " $ " $ fk#v m f $ e " " ewk m f fkv VK " Ke " fk $K" " $ k zv i$ f# " m f $ $#V k p fk $K" V" "$V$ e ff V e K m" ckbk s $ KV f VV f VK VK e " H " P KV$ " " V K$ A V f $ $ " VV # " K X ewk m f H = P ckb A pk " K$$ $ e W f mx " e K$ k n K e fw hxv K $ e$v "$V$ " ek Ke $ V e f f " fk $K" fk$k v VK f " e" f " VK$ " VK $ VK$ " f K Kee $ e f K"i K KfX $ $X " " fk $K"k q " K V K VK e fk K$ K$VV f $ $V e e K$ K#V$ " f $VV " f V$K$ " e Kek "#$%&$'()$*+*%,(&,-'.&)./0.('(1(//(-*/(2(+-'(3

18 ckb p $ $ $ K$ fkv$ k

19 9 Z _]\ \ \ ^\ ^ [Z[ v c99b y " V #VXf " e Ke VK$ " "i e K K# V $ K"i c k zv f Kff " K V ek V VK$e$ V$K$ $$$fkv e #VXf KfX e V"$ " e Ke WX $m$ VKk " " $ $e$ e e ff $ ekk # $Xh f k " K V VK $$ " #VXf V "VK $ V " " # ($ ' a $ " f$ Ke VK$ " E f "V VK$ " E e e$ q $ ν k a $ " $ " W $ " K R $m$ k s " " VK WX K" e Ke " VK " VV "f$ " e KV P $ $m h E fkv f " Ke k K zv K $ f $ K W$ VK WX V f$ Ke k p Ki$f $ " " #VXf e e $ K V VK (O, er, e z ) nk m" ck: n ff $V " Kh $ $ K$ "i e e ff VK WX mkv $m$ VK " Ke E ew$ K " V m$ VK$ " e K V fk $K" $ K" KhK$V fek$ " m K VK e P k fkv s $ "$ e u(r) V VK " Ke VKe f q(r) VK e K$ fkv "$ V"$ KV$ " k s $ "$ V" δ VK K$ VK WX K V f$ Ke E $ e ff u(0) a V K e Ke VK WX VK " Ke E k V$f$ s " e K " V e $$ K"i e #VXf f$i k v K r a, u(r) = δ r2 2R r a, q(r) = 0 Le déplacement est défini sous le contact La contrainte est définie hors du contact y ew ew KV $ u(r) q(r) " " V $ Ke $ V VK$ a P δ k '% " #%$"% " ' ' dk V ek V VK$e$ V$K$ $$$fkv " V W WX " V e K$ KV$ " " V fk $K" " V $ " $ " f fkv y K#V$ V i $ "$hk r a, u(r) = δ r2 2R r a, u(r) = a r2 a πr ) (r 22 πr a2 arcsin( a) r ck: ck;

20 ck: aewfk " e Ke WX $m$ VK " e Ke y r a, q(r) = 3 P ( r πa a) 2 r a, q(r) = 0 δ = a2 R c P = 4 E k 3 R a3 E = K V f "V "$ " fk $K"k E 1 ν s " f"v 2 " e VK$ V e ff K$ "$hk 6 pk V m" " ek Ke $$ " " e Ke V K e Ke a e ff V f ck< ck k 6 s V i $ e K " ck<k V K KfX n K$ f e $hk V $ fek$ " K# e " V " K$ e $ e e$ $m$ e f"v K "$ " VK$ K " $ V V$ " $k vvv # " " f m f $ " "$ K$ hkv$ K V #$K$ " V " e$ " e $$ m f $ " VK " Ke V $ " " e $$ " V $ VK$ " " f$ Ke E W f m$k 6 v "$V$K V "K$ ck< ck $V $#V f$ V f "V "$ " fk $K" K $ VK e " # ewk m VKe f VK$ " k n e $$ " e V"$e$ $ $ e $" " V$f$K$ e " V KV$eK$ i $f KV V$f$K$ VK$ K#V$ K y k n K V #" VKi e " sv$h qwk K#V$ V " fw k ckg ck<

21 c!z \^ ^\ ^ Z_\ ^]Z ] pk fw sv$h qwk b Ke" VV f VK fw VK V" "$V$ " KK V V "VK $ K$ k v "$V$K V KhK"i a ; $V e "V e $#"$ VK$ " e $#"$ VK$ " K "h $ VK e K e $$ K V " VV V K i$fk$ " e f f VK$ " $#V k p $ ev V " fw "h $ f$ V K a VK e Ke " K $ "V K KfX i $f KV f # hk#v k " K# V $ $$ i $f KV K $ K$ k " $ "$ "$ V W WX VK fw sv$h qwk V "K$ "$ VK e $" k " "$ " V " ek V$f$K$ e fw V f $ek$ "mm " e " V #VXf "$ K e$k + % % ' $' ' # '"$ 4/1 04 dx V " V ew ew KV$ " i $ e K $ K$ $V h$ $ e$v " K$ ew VV Kee $ e f K KfX $f K V K VK e Ke K i fv k s " K$ Kee $ e f " $ m K " i $f KV k d "i VV h$ $V Km$ VK e KV$ " P VK K$ δ kpk $$Xf VK K$ " " e Ke S $ e ff VK $h VK e P K K V $ " k n e $ # hk#v VK K$ δ V " ewk m $$$fkv h$ " V $ $$ K $ K$ $ "f f " k pk m" ck; f " K$ ewfk$ " " $ $$ K $ K $ a $e$ K $fv kpk e P KV$ " K $ "e$ e "$ f "$h K hkv " K V #$K$ " e " K e$ e"vk K VK # #$ k t ek " ekke$$ f Kee VK " VV $" VK f " " VKe f K# V" VK $ K$ K" # " VK $ $ K$ k s #$ K$$ # " e VKe f k q " # $ VK K$ K $ VKe f $V K" "h $ f$ VK $$ $$$KV VK " Ke V ewk$vv K # $ " e " # "h $ $ K$ iv $K#V h $ m" ckgk q " e K$ i " "$V K$ " e Ke "$ V " $V K " Ke $ $ " " VK $ K K $ VK V ewk$vv k s #$ K$$" e " # " e VV m" ckg K V h V"$ " Ke VK e KV$ " e$ VK K$ VK $ $ K$ K VK V ewk$vv k o VK ewk m VK e K"V " " hkv " K$ %-/ /(/*'*+, )()$,()( '&*$,$$).%*% $%)(%,(./ #.+ $%&-./)(&+*, /$*.2%(-/ &(%,*%,-*&.%/(,./.(, 3 '*&,$0.(,/ - *$'(3 *%&'( *& %,/*$/( *% %&,/.+(%,&!#"3

22 cc ck; r K$ ewfk$ " " $ $$ i $f KV K $ K$ k h f e f $ δ e VK " V $ VK$ K V ewk$vv k q " Kh $ KeeX VK K$ " " e Ke fw "i fw $$e "h "$V$ VK f " VK K$ " K$ " VK ewk m " VK fw f " e $" VK e Kek!"#$%& / '()(*$#*+,) ckg vi fv e " # $ K$ k 89%) :%$,*' /&'(/(,./ )('*- %,/*,$ %0.$(&, '*&,$0.().%+*, /$*.*../&)(& %$%)(%,*,$ % ++('*-/ - /,$ % '*&,$0.(& $,(δ h f )/δ3

23 cb 7 / > p $ VK f " VK K$ " e Ke VK ewk m $fv $V Km$ f " VK VK K $ ewk m VK e " # $ K$ q " e K$ sv$h qwk VK ewk m fki$fkv k b K" VK e " # ewk m K " V $ "$Ke " c P = B(δ h f ) k9 zv KV Ke$V hkv" K " $fv ekve"v m $hk$ VK K$ " " e Ke K" $ ewk m fki$fkv S = Bm(δ h f ) (m 1) s K K$$ KeeX K" $V {P; S; δ} k n K ew " $f K V$f$ K$ V$ K" K$ " V "$V$ KV f " " "V $ VK e " # k a$ V ew ew # $ f " $ " K$ "ee $k $V K" KV KV$ V"$ " 7>/ // 7 7 n e #VXf eke$ " h$ " V K ew K f " e $" VK K$ " k s KhK$ KeeX P δ " " VK e " # ewk m ewk m $V Km$ fk$ K Kee i $f K"ik p VK K$ " e Ke S " VK ff mkff $e$ $ VK fw e $ K " K" ewk m f Kf KV$ " " V ewk$vv " ewk m f $" KV X K$#V KfV$" g k h$ K$ n VK e" ewk " $ VK e " # ewk m f " $$$fkv k ewk m n $ V " ewk m f $" KV X K$#V KfV$" " V ewk $VV k n ff V VKe f VK $ e" iev"$h f VK h $ekv V #VXf $fv f f V$K#V K " e$vvk " WK f $ " Kf $ h $ m" c ck< m δ + (D ind + Déch ) δ + ( K rs + (K kc Kh e 1 bâti + S 1 ) 1) δ = F 0 e iωt m VK fk VK $ K$ Dind k Déch V Kf $ f V $ " k V ewk$vv Krs k Kbâti VK K$ " "$ k " # $ S VK K$ " " e Ke F0 k ω V KfV$" k VK "VK$ VV$e$K$ $" VK KV k qk " ekve"vk evk$ " ew ewk δ "VK f δ0 e iωt+φ δ0 V KfV$" VK " Xf φ WKKm K# "$ ( ) 1 S = F 0 δ 0 cos φ (K rs mω 2 ) 1 1 ckcc K bâti n ff F0 ω $f δ0 φ f " V KV Km KVK#V Krs m K bâti f Kee VK K$ " " e Ke " $kzv " K$ " V c k

24 c: ck< rk$ VKKf$ "" K$" eke Khe VeWK$VV "f$ " VV$e$K$ $" KV K$#V KfV$"k VK efk $" KV Xf K V V Ve" VK" #$K$" "e Ve$k Vm e s#$ K$$ VK hkv"{p; S; δ} " K" VK$ "$ f " "$h$ e$" VhV"$ $ fek$ " VK "Kek + %'% ' $" #' '$)' ' " p#e$vk qwk fwsv$h b e$$vvf V e$#"$ VK$ " VK$ " "h$k VK" Xfk o$$ $Vh$ K$ VK K$ $#V kkk VK$ "K$ KKV$ " "$ K y a sv$h qwk V"i WWX "$hk pkewkm " V ff "K$ $$$fkv" VKeWKm" $" Ki$f$ " $ VK "Ve "e$ ff K$ eke $ m" K ck k v $ e$x VKK$"" eke S #$ c kcb a p$ V VK" ekek dk ek" eke "f VK$ " e "K$ "" hkvk#vk $e$ e$ $ ff " $ VeWK$VV S = 2aE (,,(/('*,$ %(&,$%) -(%)*%,()('* +,/$(*2$&+,/$0.()('$%)(%,(./3 K "#$ "fk$ VK$ " VK$ "e V "K$ VKeWKm $$$fkv K" "f ckcb KV$ "k

25 c; p mk $ e f"v " #$ "K#V$ " $" $$$KVf " Ki$f$ " "$V$K$ " $" Kf$K"i V V $ $e" xh$ew < 9 $"$" $$" g % kdk K$ " K$f$V "h $" e Kf$KV VK "$hkv$ effk e$ "f$kmv ffev"$ VKe$ X VKff K$ " eke A "$Ke VK $k o$$" $ xh$ew "$hkv e " K" ff ω = 70, 3 $" ok pk" "$hkv" $" "Ke $" Ki$f$ " KV$ effa = VK eke k A/π A VK ck pkewkm $" f $$$fkv" "Ve " hs "$hk" VK pki$ "Ke Vi$" K" VK eke " "f VK$ " $ m" V ek" eke ck9k vh$ff hkv$vvk"$k "f e$ VK$ " e V WWXK ek" eke VKVK$ "K"K V" hk$ " e$x V " $ VK "Ke V$m VK VK$ "k dk ek evkh$ " " VKfK$ VK$ " fk$k" e " VeKek sv$h qwk VKeeK VeK" WfmX VKVK$ "V$K$ VWWX " Veff VK$ "" K"" n$ fk$k" $ m" $$ ckk ekke$ K$$ K" $V" K VK fk$ VK $K K$$ $hkv$ VWWX"i$ VK$ " #"V VK$ " W ewkm $$$fkv" $ VK ffk eke " eke h$ ck:k;k + + '% % #&"% $)' ' " "e$" pk qwk ef fwsv$h $ "K$k pk f$x K$ ckcb ee" V ckc: V$" k β " n V e e$ c "$ ef K$ "K$ "$ f"v"$" VK Ki$f$ VKK$" fk$k" E & % */'*&.$,(&*.-/ $&$ % %*&&$+$'(/*,../&.%$%)(%,(./ 2βa %$0.(% %*2$&+,/$0.( %( 0.$1*'(%,(%*.,*%,*. (& $%)(& ($(%,& //(,(./&3 %,/*$/( a(&,'(/* %)( %,*,). %( 0.$1*'(%, $'$%)(%,(./%(&,-*&*2$&+,/$0.( = S

26 cg! "#$% &'( &)( *+, /01 #/ S 2#4 02$2#3#1 -a % 5 # 6 26#-"#$ 3! # s c # #83# ### $# %9 7 #δ0$9! 1:#$; h h = 0<1 #-4 s# :2# $ 0>1 h :?ε 4 4 #!#0ε = c5/ -a;##h ε = 2 (π 2) π h s = δ h c h c = δ ε P S 0.75

27 2 # ;#2:## c $ 9:"#$%$# # H #2#7 # $4h 0 1 ## # " 7 ##:9# 9 ;7##7# 7 $# ##2 # 7 πa 2 7:# #7# #7# ;#7 "#$%# ### ;#! # #! -# H = P /$!."(!"#"'( $%& /$,,(/" '( )# $%&!*"$'( ' ) +(, (" -#,, (".,&( (% -4: 2#4## "$$ #"#$% 6 $#0 2 ##"#$%## $# # / #!! :;<=>?8;@=> C#CF-BGC#HBCGCB#CGC#BGCGD"#$CFC%FF:aC ABCB#D$ E:#CF4FC#GCB B F F 7 $# B #D-B4CGD BBFCCC./"F:#CB4CGC#C/ FCBFCB FCG#F C#BF C#DBDFCC;GCF 3F4B C "CC FC#BF 7 C#CF-BGCBCB /$# F0>10$BF9! I1ACCCFFC 0FBBFBC#;a1C FG C FCBB CCGC$#C FG BG #CFG FC FB GC#G 21J,K E FCH0FBBFBC##C;a LMNOPQ LMORSQTO FCCGC#C/ *+ICFCB U#BF CGC#DCFFC F F#CF-BGCBG CF# B4CGC./5CF-BGCBC#BF$# réel:#d #BGC0>1 BC C#CF-BGCBCa!#;aévalué

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

29 B/G7BFC 5BFFCB;BFCFBFC#BF GCC FGC7BFCCC$CGFCCGC#7BFCFC##CGC#DGCC F F#$#C F4FC!BF CεU:B 0 1 B?C#CCF4FC CC# FBG G#D6 #F #B C:#DCCGCCCBFFCBCCH72#C:G#C F$# CF#FGC FGCBF#BGCGC#GF!C0011 CCGC#B F2CGCGF!C CBFCCFCI <CI FCB?#$FBGCε 13 6@?< 28? <;82>6?;;6;?;<;46? 6 BFF!CF#C7 %F#DFBG BGD C#CBC-FG#C0 CF.B$: F4FCβG#D B0 1:"#$CFC%FFCFC; 3-F.CF:1CBG#CC C5C$#C F D#FBBCG# F#C9C JEKBFCGCF$ 3FGCJ,KB?C##CB$# 2#BGF$#C C B-CGC FBGC #B GCB2FC 3 C FCBFC;#D$# F4FCJEI:EK5C FF$ 3BBF CCFCGCGCGC#$#C BGCC $C#CBC CGC%BBG F GD GCC F B F C#DGCC GC5C$FBG F CF.B$C2#CCCGBFC F4FCβG#C FGCβ CFCIECIJ>KA#D!GBGCBFFCBGB#DBFGFCGC!FGC CGC$#CCF GCGCC FC FFCCCH ( ) m 2Γ ε = m 1 2(m 1) ( )(m 1) 1 πγ 2(m 1) 5DGCBF CC:B #C$C CBGCBFFCCGD#-CGCGBC F C3FC#C:CFC CFFB7CGCFBF CGD 7ACCBGCFB $CBCBFG#D GCGD##B FBCB <B:1 FB7F!#CB CFCF#5D#B## F CGCC# GFG0FB 3#C:8C3B FC#CFCCB#FFB CGCFC$CCCGC B FCGCCCBGCC3/ F7C FCB#CF "CC FCCCGFGCC # B2C CF FB2#4CGC#DC CC#BFGC#DGCB F9CG FC$CC0B FC$ GD 2F C#C FCFFB *&)"#"&(!. C###CFF, (& F4CCGC$#DC FG 9#:BC CG B0C#CF-BGC#HBCGCB1FCC BG#CF#C-4CBC F

30 , 9#7#F 9#C C:CDCCCG# B2#CG4#BF C#DBBG4FC U:BCFCFC03 JEEKBB2C :C# F -4CFC$ GD 9#C: C$B# BG #BGCGD"#$CFC%FF BG #CFG C F B F! G F!CC:C#BG #C $#CFG ACC$B# BC## FCF#9/ F!CC GC FCEG#CGD C# $CFFCBGB# CFC$ CCCGCB4FC"C G! GD CFFBCGCG#C 9#GC#CGD CC F U72#CFB:#CBG C# #CFG G 2FFC$ CB C!#; ##B7BB!4CB G 9#5C-4CCBBFCGBGC#C7 B CD#D!GD G F G UF47BFCFB:#CBG CC7B!#;C# #CFG G 2FFC$ CCBFCB G D#D!GD ##BGC 2F5C-4CCBBFCC3CCBC 2F CCB C!CCFGFCGCFB:#CBG C$C#FB:CC CFBCFC#HBC 9# #CFG G -4C!/ 7 C#HBC 2F <6=<>?@> ABCDEFBG KLEMFBKBOM NOMPQP HIJFKLEKMI!"#$ %&'!"#$ %&'!"#$ :7; ()*)+,-+./*0*12 C%FF *+E$B# B$CG BG #C $#CFG B2C B4FC F $CFFCBGB# CFC$ GD 9#GC#CGC F#BGCGD"#$CF C# B FGD CBGCB CF.B$%B FGCF472#CFB: CCCGC #CBG#CFG G -4CCC# G 9#UC FC C#FB #CG -4C!CCB FCGFC#CBG #CG 2F;7BFCFFB!CC:#C 5DCFFBGCCB4CCC7CH#C FCCFC:; 72#CFB:#CF-BaG BFCCCGC$#CGCBG 9#CGB k gytd^]`l`]dtjjz`dz`dyzuzkujzduz_ga_dytdlmg^dz`[z`dcyv[g^yzuty^w^[^dv\]jf]_^dz[g_n_dxjz kujo_ga_dytdp RSTUVWXYZ[V\Y]^ T`\Z]a_ZYbVZcdYX_e]YdZfV`VdYTd^]`fZgdhdYZTddY^agVZc[Z_fiV`]jX`Z_[Z

31 EI #BF2 #C-4CCBBFCBC#C9#7UC FC C#CF-BG BFB : #C BG 2F;#G7BFBGC$CGC# C# ;F47BFF-BGCB:#BF2 2FC# FGC C#C9#:#FGC FBBCGC#DCC2#CFB BFCABC 9: BG 2F;#FGC FG -4CGC$C F4!FGCGC$C##CG CCFC#C7 BBF:;C 9#:GB#D8 FC CCGC$C!#!C2#CACCCFFB 7 GC$CGC# C# C#DC FG 9#G C:#HBC 9# ; FG CA#FF$CCH7F CC:#BF GCF FC$CCF49: C#DBD-#CC4A#GC$C#BF C#DBCFC CFCGCB $BFC3FFC:;FFGC#HBCGCFB:#CFBF C G 9# $'*) &#" +#)(%" $%&'!$')(* =#BG#BG BBFCCG 2FFC$ G#CHBC 9#7 C 2F CBCGCFB2#4C:#GCFBGC#D$B# BGCBF2 B FC#$CG 9#CG 2F B FGC#DGCBCF$4#CCHB#C3C" CFC C!F#;BG#CF#CBG #C $#CFG G 2FFC$ B # 7BFCGD C#BGC#!CG -C 01 9#0FCG B?aC#CF-BGCB:t#DC 2F15C2 GCGFCBG4#CC#BFGCFBBCF FG C7BB G ΦFCGBCGC#DC3FCC"C 7##C#CBG4#CCF C:#CBG4#C#- #CF#CBG4#CFBBCFB!FGC 9#CE F CFBC#CBG4#CC/ GC FC C0B FCC# B #CFBCGB C C# CGCCC GC;GCB 123 6<98; ;:>?5? =; E eq = E sub + (E film E sub)φ(a/t) film0fce sub1#cbg #CFG C"C 7BB f(a/t)"cffc FC2#CFGCC!BFCB #C CFC GD6 CF #CBG4#CFBBGCFCGFCB!F CC # 7BFC GC # 3/#CBG4#CC3BCC#:B F#C C#B B F2C FΦ E C#CBG4#CGCBCFCFC03JEEK 0,1 eq = 0EI1 - #CGC CACBG4#CBGB7#C; 3:B GBCGD F4FC6 2#CαFC##C!9B 92# #CFJE KBC DCFFB!CF F#C F 1 E eq E eq = E sub + (E film E sub )e αa/t = 1 E film + ( 1 E sub 1 E film ) e αt/a

32 E 173 6<98; 228 ADCB FCCFB? =; 2>>56 #CB CGCF$ 3BCB FBG#CF#- CC CF F 2FFC$ 5C BCC6C CCGB#C3C: # B C FBCF;FB GFC#CFB2#4CGC7 BFBC5DFBC# # CG FB2#4CCC##CGC CFBCCF F2$C BCB##JE<K GCG7BFBB2C F#DCF!C# 3#BGC#!C $CG#CG -4CBB!4C: C:B BCB##2B CBCF$#C C 0E1 C )] 0EE1 ##CC B?ν eqcµ $#CG )BFCC$CC#CBC CGC%BBC#CBG 2FFC$ #CGC GC%BBC#CBG eq = E/2(1 #CGC##CCG + ν eq 9#0FCG 2F1:a#CF-BGCBC t#dc FG 9#9:GDF4#CG#GC# #JE<K:νC#CBC CGC%BB :ν CB##C G -4CBB!4C$CF F2BG C#FB$C#CGCG7BFB F C F#72#C8 CCGC#$#C FFCFν F0EE15D #B B # 0EE1GC$C CC7CF filmc C.CB##JE KFB subc#bfc:0e1d# G ;C##C#CFB2#4CGC7 B GCFBGDFCC #Cν B!#;ν ν eq = ν sub + (ν film ν sub ) µ eq = µ sub + (µ film µ sub ) [ 2 π arctan(t/a) + 1 2π(1 ν) [ 2 π arctan(t/a) + t πa ln ( 1 + (t/a) 2 (t/a) 2 ((1 2ν)(t/a) ln 1 + (t/a)2 (t/a) 2 )] t/a 1 (t/a) 2 filmcµ film0fcν subcµ sub1#cbc C Eeq = Esub C GC [ + (E film E sub ) 2 F 0E 1 π arctan(t/a) + 1 ((1 2ν sub )(t/a) ln 1 + )] (t/a)2 t/a B CF#C9JE<KCC3FC#CJE K$#GCCBG4#C 2π(1 ν sub ) (t/a) 2 1 (t/a) FGCBFCGCBG #C7FC 2 JE>K:B?CC7B#CCF CFBCCH#FCB #CG#F F;E00.5 CBGBGDGBF7C;#DCF7C9# F2BBFC F#CGCBFC9GD 21 CCC##CGC BC < Esub /E film < 9: 2F FCF CFB4CFBCCC##CG$C#BCF CCB##JKABGF GD CF C#C-4C9# 2FC FCBG#FGCFGC FCFC:# BCCFGC C πa/ta#f$c#bf FGBCFS sub = 2aEsubCS film 0E<1 = aefilm B?fCg:FCCFCB n:gb$c!ff C:#C9#CF4 B B4GC#CCFBF# Seq 1 = (f(a)s film C C#C ) 1 + (g(a)s 2F:BFCFB sub ) 1 #7BFC1 + ka $C2CS eq = S film

33 EE A#2B 0E>1 C#BF;1 = GC$C# ACCBG#BFCGD 2aE eq C 3 C#C9#B4GC BG #CFG C G 2F 13 "$CGC#C$BF:#CBG4#CFB7B 6<98; ;:?@=: 5? =; FCGC7BF #C#C; #CF: CB#C FC CC$#GC C BG #C CFBC#CF$CBC;B CF#C# FCFC!CGCBFCGC ##- C #B #- CB2#CC;FB CC GFC F CC#C B CB FFCFCFC 2BC3C#CGCC-CGDFBCC#BG#BFBBCF= JE K: F#2CGCF$ 3GC 2F.JE K # CFBCG -CBGC FHCF GC!C:#FBBC CF FCB#4CGCGCBFCCGCG#CC G#DCC2#CGD 2FFC$ FCF#CFBF FBC C #CB F #CF BF B #7BFCGCBCGCFC9CACC C F 2F FC$ C C GD2# 9# JEKACCGCCFBCC2#CB ADCCCGCF4FCBFCB F CG4#BF C#CFBFa/tGC$C FC F;JE,K FF # CFBC CB C2C ##BB F $FCG# CGCCC GC F#7BBGC FCCG 2FFC$ : B F ##BCFCF;BG#CF#CB# C F 2FFC$ GC7 BC/ C0$BFFC E1B GCC B #:B ##B GCF#CCC GC $FBGCBFCGCBG #CCGCBC CGC%BB G 2FFC$ 0$BFFC C<1 F#FBCBBC 1 1 (1 + 2t + πa )πa2 t E film 2a(1 + 2t πa )E sub

34 E -F CFC:B CB/F!GC FBG#CF#CB# CB/G7GD GCC F3/ F 2FFC$ :B F BB!4C:# FBG F! C: C FC#B FGFFC#CBG7 F 2F CJ IKB $B# CJ K5D #BGCC7BBC3CB #B#CC3CgCθ CFCGCB FFFC##CCGC#3GCBGB 3#CBC ;B FB2#4CGCB0$BF E1 F 0B CC$C FBGCFC#BB2C C #!BFCCFCGC# #CF / # CP:#FBδC#FGC C!FGCC #CB2CF$2#CC3FC#CG FSG B 7 1B B0#7BFC GBCG FBFCFC#CF-BGCBaC#DC FtG FC$CCJ EK%FB FB C$#C F GC#F:B B #B GC#DGCC FB C9GD## #C GBC:BC#BF C#CG#GCCF# BGC#D#!BFC5CGC!BFCF FCF#GFC CC GFCC9$#GC3FC#CCC #BFCCC3CU5D#!BFC F CC =B -4C CBBGD 2F GC/CCBFBC# BB!4C FC$ GD 9#BFBC# CBB!4CGDC Ft09! FCE1"C 2F0FCG ABGFBC 9#1" CC#CBC CGC%BBG BCE subcν sub0fce BC CGBF7CCFC#C 2FC#C9# CC-4CCB ; F!CC3-F 7FBGBGC#FGD GCC FGCBG #C# C CE filmcν film1#cbg #C# indcgcbc /

35 E< *+ECFCB CG -4C CBGF ind ##F7BFBGC.C#:5CAB J KB2#: G#CGD GCC FF!GC:#7BBGC FCCCG -4C0$BF9! FCEE1: B:$C CGC%BBν FFC#FBCG CFC#BCFC#BFCBF#Cq(r)# -4C; CB##BB C##C%FFBG CC GCB$B# / 0BCB$C#BF F7CGC#B C CCC3FCB;# -4C0BB7 G#CGF!$CF#DCFC FG -4C1F z)cb BG #F F7CG 01:BB2C DC##CCBFC$C1C#CG#CCu(r, 0E1 9#0z = B? u(r, 0) = u(r) = dk q(k)j 0 (kr)c(k, E, t) 0 C(k, E, t) = bkt e 2kt abe 4kt Efilm 1 (a + b + 4b(kt) 2 )e 2kt + abe 4kt a = αγ 3 γ 1, b = α 1, α = E film(1 + ν sub ) 1 + αγ 3 α + γ 1 E sub (1 + ν film ), γ 1 = 3 4ν filmcγ 3 = 3 4ν film#cbg.c#gcqgdbfgfci"cf74fcgc7 B- #CFG G C 3FF )Cq#F7BFC C# sub C E 9#:G9BCE dyt`_e]yjtd^]`[z"]gy^zyz`wv]jvdy^z\nu^`[y^mgzps^`bzy_^]`[g`zdyt`_e]yjvz[zt`zu_z#z\dgz GCGC STdYT`_e]YjTd^]`[ZT`ZU[]Y[YZnZ_dg`ZdYT`_e]YjTd^]`^`dVWYTUZ[]`dUTe]`\d^]``]nTgZ_d 3B /-4CF#C2GC E film /(1 νfilm 2 \]jjz_g^d$(r)z_dute]`\d^]`[z Z ZU[gfYZj^ZYdnfZ[]Y[YZnp!UUZ\]`_d^dgZUVmg^bTUZ`d[ZUT r J n (r)l]j n f n (k) = dr r J n (kr)f(r) f(r) = dk k J n (kr)f n (k) 0 0

36 E> *+EECFCB CGC#7BBGC FCCG 2FFC$ :G #CGD ACCFC#BFCCFC GCC FF!GC C!C#BF C#DBFC#3C#BFC F#F!G GC#DGCC #BF F0 BFC3CU1=C #C!C#DC3FCBGC#7BBC GC$C 0EE1 CFCGB#GCFBB#4C#G8C3Bu(r)GC# 5BCGC#BFC# C(k, CqCB BGC# 1 (a + b + 4b(kt) 2 F7CG -4C )e 2kt + abe #CFB2#4CGDGCBBFFFGCBGB F7C 3#C3C07 #C 4kt FC CC: E1#BFC# C##CC93CF#C7BFCGDCFBGC CqDCB C D;#DC3FC FGC#HBCGCB0B? C#G8C3BuU F7C1B #HBCGCB:BCB 0E1D$4FCB2#C:2C C#CB4FC5CAB DC##CB#- :#D #BGFCCGC#D B CCG # CB FBG B#CC3CgCθ CCC3CADCB FBB FCF #C2 *% GCBG#CF#CBG7GC4FC$B# " $% '(&/#!.&#%%((& J K:;# CGCF$ C: FC#C G9FCC$CCBC#CF7BFCGC B 3GC=CGGBJ <K:BFBG FCFCB #CC3CgCθ GCq(k)CGCk 0E 1 E, t) = 2 E ind + 2 E film g(s) = bkt e 2kt abe 4kt dk q(k) cos(ks) u(k)

37 E 0E<1 ##F7BFBGC θ(s) =.C#:BC #BFFFFC#D dk ku(k) cos(ks) B0E1BC 0 0E>1 F FC#CC3C # #F7BFBGC B FCFCB ;CC B:B7/ ku(k) = C(k, E, t)q(k) 1 θ(s) = 2 π 0 =BBG4FC#CGFC/#CGC-4CBB!4C0F %"*, " '()#!*"$'( 3C2#2#C B F#C9#C#C 0B + 1#C B0E>1C0E 1GC$CC 0E 1 2F:t = t ku(k) = 2 0E1 E q(k) θ(s) = 2 E g(s) syst = E sub filmc#b#cgc9! FC "FCFB $C#CC3FCBFBG CF! CC #C GD CFC#B#B#CCFCgCθACCG:BB oue F#DC FC#J IK"GBC #CCFF BGC5CAB J KFB GFC#CFB2#4CGC#3GCBGB 3#CACGCFCFFB2#4CCFB GCC3#G0E 1C0E<1#DF FCGC F7BFCGC.C# ##CFBFGC#7BBGC o:bb2c CC#J 0E,1 rq(r) g(s) = dr s r2 0EI1 s 2 θ(s) = d s ru(r) 5C2BFCGC#D!F#CB#BFC##C dr ds 0 s2 r Cg0FCθ1CGCGGCq(r)0FCu(r)1 CB 2 i]j]wx`zcu^`bzy_z[g`zu]`wgzgyp UZ_d[]`\YT gyt`d[zyzdy]gbzymgz\z_[zg WYT`[ZgY ]`d s1u:gcc4fccg9bfg BCθB #C YZU^VZ_U^`VT^YZjZ`dfTYg`ZWYT`[ZgY[Z[^jZ`_^]`^`bZY_Zc\ZUUZ[g`j][gUZVUT_d^mgZ b]^y p p B"FB qz_di]j]wx`zcg`z\]`dyt^`dzp gt`dcku(k)lzuuzz_di]j]wx`zcg`z[ve]yjtd^]`lkvdt`d G#CFB2#4CGC#3GCBGB 3#C Fr s0fcr B? 1 E = ind:$ce Esyst E 0E ( ) g(r) dk C(k, E, t) cos(kr) cos(ks) dr 0

38 E "$B#BFB #DFGCCCFBCG#CG F BB!4C BCGC$F2#CFC##CG F7C1 3C3CFGC#CF7BFB!F#C#- B0#BFCBF#CqC#G8C3BuGC# $CF2#CC#FC#BCFCgCθC#B#CC#FC00E11U:;FFGCq: B# CC #CgBFG BCBCGG θ:cgbu:gcchbccc:; FFGCu:BGCFCθB #CB:CBCGG g q FC0A7C3C 1:GCC3C FC CFB# B#- G FB2#4CG B F 2FFC$ C C CθACCG:#C F# #C3#CCF$4#CG #C0B BCGBBF #CFgCθACCFCCB 6B FGCBB FCF#3GCBGB $CFCG9GC7 BC2#2#C;g CCGDFC#B#C#F 3#CC#C FFC#BC2C$GCCB C:B!FGC#!FGC#2CFGC!CCFCgCqC 6B F#FC #C FC CCC##C θcu:#c!cgcg;θcf C G B F 2FFC$ FC$CGB#CC;#D$CFBGC0E B!F#CB GFC#CFB2#4C BGBC C7FC:B FCB C0E1 FFB; CFBC F C:B F# C##C0E 1C2CC 1%B 3F -$& " $% '.,$ )!('#%&)( /#& *%*,#) ABGFBC#D 2FFC$ B0E 1G#CGFCGD CGCB# C 0B C7BB F#D C-B4C #CFC F F:G6; BB$C3CC3-CF C0B BGFB #DGCC FC#C-4CC7FBCCGBACCG:CCGCF4FCC C#CBCFC FCFFBG 5CBGB C #FC 3#CGCCFB2#4CDC3FC#BFBC FCC0 BFC3CU1 0E1 { CCG;BCF $Ca#CF-B GC#HBCGCBCh#7BBGC7BFCGC#DGCC CC-CGCF!CC: r a, u(r) = CFCGCBC δ h(r) FA#C B BF#CG CJEK G#CCGC# C# F7CCFCGBCGC#DC3C7BFCGC#DGCC C#BBC CBG#BG#CG CBFFCBG BCBGBFGFCCFCGB;FCGFCCBCB F #D =BFCG B0E,1CBC#D-B4CGCBB/G7:BB2C:C # 5D B0E 1GC$C#BF 0EE1 r a, q(r) = 0 r > a, g(r) = 0

39 E 0E 1 θ(s) = 2 a ( ) GC0E 1: "C 7FCGC 3BB5FC4FCBCFC#DCFFB- g(r) dk C(k, E, t) cos(kr) cos(ks) dr π DBC FFFCBC C 0 0 Film seul { }} { 2 θ(s) = + Efilmg(s) 2 ( a ( ) ) g(r) dk C(k, E, t) 2 cos(kr) cos(ks) dr ABGFB#CCBGC2FCGCCC!#=#CCBGCFCC!#!C2#CGC$ 0E<1 π 0 0 Efilm } {{ } #CFCCF:BFCFB $C#D B0E1# Effet C du substrat FBC G 9#"FCFB $C#BF# CBGCFC: CFB#BF CGD -4C7BBC3# $CCG F CtGFB G 9#5C 0:FCGGBBCGC#DCCGC CCG$CF( ) 2 g(s)#bf 2F"FCFB $C#DGC C GD GC -4C; CB##B CCC##CG 9#C # C#D6B C#FBC 2 CC Esub Efilm t #CF 2FJEEK:GC;#2CGC#DFBCGC#BG#B7CF BCJE>KG #CFGC#DGCBF B B#GD 2FFC$ FB2#4CGDGCBCF4CC7;#FB# 5CBGCCFBFC;#7 BGCFB GFC F CCCC BGD C B!F#CGC FCG/ B0BFC B#GCGC 34C-C B?θCKBB CCgC#DB ( g(s) = E C;GCFCF57BBgB ) film θ(s) + E C; a film dr g(r)k(r, s) 2 π #DC3FC FGC#HBCGCB :#FCC;#G9FB #CB C7BgB 0 FB :#C7#CGDC GG FCqCB BGC# F7CCGB#CGCC FCCGC# G#CCGB F4;K: #CF CC4FCCGCFCF#CFBF!F C 3F$ 3GC5CAB J K=BD/!BF CC CG -4C G:C##CC FC7#CC# #CF#C2GD #/!BFCGC F7BFBGC B 1A# CCGFB BB CCGCFB GFCF C#C$CFB#D #BFGCGFCFCC BFC##C FCFGC0 CGC -#,"/)#,&#" $%#/#&'/#, (!(%"/$% ('( %(''$% D;FCB D$B7 C-B4C F#7BFCC3CGC#DGCC F 0B TY]`UT[ZeT^d_gff]_VZ`gUUZZ`\]`_^[VYT`dg`\]`dT\d`]` T[iV_^ep F #FB GC#DGCC FB CC:B #CB:#C

40 E, *+E A## FBG $B# C # CC7 #CB?#CF-BaG B CCGC$#DC BC FtG 9#01:#C$B# C # CC7 CC4FCC 9#:#C BFFC021:#C$B# G#C9#:GB#FBC!#B2#CG C # CC7 DC -4CCC##CG CFC##CCBC 9#C ##C CBF 2FBF2 C;#FBC CG -4C:CCGD # G#C tan(ω):b?ωc#cgc/!#c BC Ca/t ##G9BGCθ:BB2C#BF G#CCBF#CGBFu(r) = δ r/ UGCBBC#C$F2#CG B5#B! C FFF CG #CGD CGCBBB!4CC#CF-BaGC#HBCGCB: #BFCGC CFH#CG 2FFC$ :BFBG CB $C##C#B! C#CBFC a/tgb FF C#DC BCC:BBC FtG 9#A#F GBCH 3C FC#GDFBG FC#CFBF C F CCBG FBF G 0$BF9! $B# C # CC7 G#C9# $B# C # FCE 10B FBG BGB#C!FGC FGCBC CC7 B# $C Cta 2 /a 1 "GCBC#BF#C FC!FGC FFCCG#D $#C B FG#C#CB?#C-4CFC$ F#CF C#2#C; -4CBB!4C B0E 1F#C F G 9#"FBG 0E 1 0E1 0E>1 s a, θ(s) = δ πs 2 tan(ω) ρ r a ; ς s a ; τ t a ; η ka 0E 2δ tan(ω) UT`WTWZVdT`d[]``VmgZUZ_Z#Zd_[g`\iTYWZjZ`dVUT_d^mgZ`Z_T``gUZ`dmgcU^`k`^p ZfZ`[T`dl]` UZ_dVb^[Z`dmgZ\ZddZ`]d^]`[Zb]UgjZ VUT_d^mgZjZ`dT\d^e Z_dc_dY^\dZjZ`dfTYUZYg`Tag_[Z πa Z(η, E, τ) E film Z_d[]`\\Zb]UgjZ[T`_UZmgZUZ_d_d]\VZUTfUg_WYT`[ZfTYd^Z[ZUV`ZYW^Zp \]`_dtdza^z`z`]a_zybt`duz_vmgtd^]`_vdtau^z_ftyzyd fg^_ `Z[[]`mgZUV`ZYW^Z_d]\VZ[T`_UZ C(k, E, t) 1 2 3pSZb]UgjZ VUT_d^mgZjZ`dT\d^e jtdvy^tgu]y_[g`\]`dt\dvut_d^mgzz_dd]g]gy_fy]f]yd^]``zuuzca

41 I 5D B0E 1DF#BFG(ρ) 4 tan(ω)g(r) πaefilm 0E,1 Forme de l indenteur Effet de substrat ς 1, { }} { { }} { 1 ( ) 2 0EEI1 ς = G(ς) + G(ρ) dη Z(η, E, τ) cos(ηρ) cos(ης) dρ }{{} π F C# 0 0 :BC BFCFJ IK C#7BFCP# C Film seul 0EE1 F#C-4CCFC#C;g "CC CGB#DGCBCC $ P = 4 g(r)dr 0 0EEE1 Π = 4 tan(ω) 1 "C G9F BG #C $#CFBFC;#!BFCB CJ EKACC/ P = 4 dρ G(ρ) GC#CFCCGDFFC#:F#CBG #C $#CC πa 2 Efilm 0 F-BGCBaJ :K #FGC FG BCC# G C-4CGCF B B#GCC FF#C2GC UFBB#D!.)*!(%"#" $% %!*, (!$' )( ' 1]CNCF$##C! 30B / FBG B B2C#BFF#BGCGCF4HC B0EEI1CGF[0; B#BF#CBBGFCς i = i/n Cρ j = j/n=b FCF$C C!FG:BC FBCF#D!F#C FρF CBCGF4C" CN ς i = G(ς i )+ 1 Nπ G(0)K(ς i, 0, E, τ)+ 1 Nπ G(1)K(ς i, 1, E, τ)+ 2 G(ρ j 0EE 1 )K(ς i, ρ j, E, τ) Nπ j=1..n 1 $CK(ς, ρ, E, τ) = dη Z(η, E, τ) cos(ηρ) cos(ης) `]` fut`_l]`[]^dt gyzyut\]`d^`g^dv[zut\]`dyt^`dzz`dyzu^`dzy^zgyzduz dvy^zgy[g\]`dt\dp ZUT U_TW^d[g`Z[^#VYZ`\Z^jf]YdT`dZZ`dYZUZ_^`[Z`dZgY_fUT`_Zd`]` fut`_p T`_UZ\T_[Z_^`[Z`dZgY_ 5DGCC FBGFDC# :GB#C GCBFCCB C 0 UVWXYZjZ`d[^#VYZ`d[VdT^UUVZ`T``Z Z p0p T`_UZ\T_[gf]^` ]`futdl^untg`z[^_\]`d^`g^dv[zql 0lZd[]`\g`dYT^dZjZ`d`gjVY^mgZ ^jf]_zq(r ZdfTY\]`_VmgZ`d[ZglZ`r = a ) = 0lZd[]`\g(a ) = = ap ZUT^`[g^dmgZg(a ) g(a + ) =

42 a0b FBGBGC#D B0EE1 DC/;/GFCB 1)GB#CB F F-BGCBaGB #B 0B:B 7 F C$#C FGBC r = CB#CN$#C FB GFC -4C CG(1) = FFCCG GC# GCτ FBBF#C "C #BFFFFC0EE 1BC (N + 1) (N + 0EE<1 C DFFC##CC 5 FB# B F 0EE>1 C GCBFCFB2#4CF#GCFB GC C GC 5C# G F#C JIK:#CFB2#4CC CC F GB # #C;#D$CFB GD CFC GCGBBB5 J >K C7BGB2C 1 ;#D$CFB:C##CC F#C2GD #!BFCGC FCF#CC CC$C F7BFC GC B FCFGC #!BFC 0 $#CC# #G#B9! FB C:BC $#CCCB B#0A70UEE11: GG FCΠGC0EEE15CBG GC7 B;FCGFCBCG CGCBGCB BG #CCC$CCB2C C3FC#CC#BFGD 5 FBGC#DC3FCB0EEI1C##C &"(!(%" '(&.#,#! ",(&'(,* )# (%!*, (& FBF4FCFBFC;BFC#!BFC 3 CB $CBGC#GFFC:FBG C##B$CGDCCF C$#C F CF:#GFBGC#D B!F#C FBG D N:#CB2FCGCGCCCGFB CB CFB;FB GFC FCF:#D#!BFCGC BCC ##B!C C 3FCB ς i = G(ς i ) + 1 Nπ G(0)K(ς i, 0, E, τ) + 2 Nπ ς 0 ς 1 ς N 1 ς N!`Z#Zdl]`T$ = 1 Nπ K 00 j=1..n 1 2 K 2 Nπ K Nπ 0(N 1) 1 K Nπ K 2 Nπ K Nπ 1(N 1) K Nπ (N 1)0 1 K Nπ N0 2 K Nπ (N 1) K Nπ (N 1)(N 1) 2 K 2 Nπ N1... K Nπ N(N 1) G(ρ j )K(ς i, ρ j, E, τ) G(0) G(1) G(N 1) K ij = dη Z(η, E, τ)cos(ηρ j )cos(ης i ) 0 = 1 [ (x)z_dutdyt`_e]yjvz[z"]gy^zyz`\]_^`g_[zute]`\d^]`ffy^_zz`xp dη Z(η, E, τ)cos (η(ρ j ς i )) = 1 [ Zc (ρ j ς i, E, τ) + 2 Z c (ρ j + ς i, E, τ)] ] fc ] dη Z(η, E, τ)cos (η(ρ j + ς i ))

43 E CFC FC F CFB3BG -C 0 f(r) cos(kr) dr = B f(r) cos(kr) dr + 0 B f(r) cos(kr) dr m i=0 f(ib) cos(k ibr) + f(r) cos(kr) dr m m B } {{ } "CGBC;FBG FCB:#$#C FGCB 0 ech: #DC3BG 5D6 3##B!C:C# FCGC#D##B!C:Cn Cm = 2 ech CCGCCFBF4FCCF$4#CF #CC:#C$#C F BCCC$F9C#CFB3BB #C F n /CG CF# FB:#C =BBG4FC#CF4FC;F!#CF:BBC #B7B FCGCF #7 3 FB2#4C;F!#CFGCFB2#4CGC C#DBC7GC 3-CGC GCB ech1c FB2#4C A## FB#CF!#!CGC FC0B1 3GD##B!CG#CGD 2FFC$ $C 3GD##B!C0NCn BFCGCBG τ)b7fc F;I 15:ACC$#C 0.25"93C FCC7 C$#C FGCBC##C!#C:7BB#FCFBC #CE sub /E film = 10Cν GCa/t"7#BFFB!FC$CCFB C$#C film = ν sub FC#C F = $#C GC0UEE1:C$FC# CZ(B, E, τ)/z(0, E, echcn6 GC# D;C GC%CFC#759! C#$#C FG BG #CFG FCE<FCC#D$B# $#C:B2C BGC# C;#DGC $#C E>FCC#D$B# 100)C7BBGCN59! FGCn BGC#$#C FC FGCE "$F9C#BF eq(e sub /E film = 10, ν film = ν sub = 0.25, a/t = FGCEeq (E sub /E film = 10, ν film = ν sub = 0.25, a/t = 100) C7BBGCn ech"b2c#bfn= 800Cn ech = 16 C:$CCF!#!C:BC#CG#C#CB?ν film = #7BBG[a/t](r) CB$CF!C# 0.5: E 01## $CFG substrat nu (r)#bf + 59! FC FFB ;GC$#C C"CGBC; FCCB4C5CFB2#4CDC3FG#DCCFC#F FGCr72#C:#FB$CGD FB2#4C 37BFC$#C CCFFC F Ca/t FGCkG#DCC GCZ#!CCF#$#C FGCB6 #CC %B F C# FBGC#7BBB- a/t:#gfbc KCCFGC 6 C #; CGCCC!CBGC#$#C FGCB FC:#7 D;3000 D;EIB eqg C2GCGC# B BI> NFCC ;!CCFn FBCF$CFE ech ZYV_gUdTdZ_ddY^b^TUl\TYZZ_de]`\d^]`[Zτ = [Za/tp R U_TW^d^\^[ZUTe]`\d^]`G(r)mg^Z_d]adZ`gZfTYUZa^T^_[Z`]dYZTUW]Y^dijZf]gYg`ZbTUZgY[]``VZ t/ap eyvmgz`\z[z\]gfgyzbzy_[z_btuzgy_fug_vuzbvz_ b]^ykwgyz p p g`zjhjzbtuzgy[za/tzd[]`\[vfut\zuz\]jf]ydzjz`d\]jf]_^dzz`etbzgy[g_ga_dytdlzd[]`\ut [Ve]YjTd^]`_mgZfTY\^_T^UUZjZ`d l\zmg^tjx`zut ]`ZVUT_d^mgZjZ`dT\d^bZchdYZfUg_WYT`[Zf]gY ZdVdTd[ZeT^d_Z\]jfYZ`[T Z a^z`p ^UZkUjZ_d^`\]jfYZ ^auzl^u`zfzgdt\\]jj][zyuz_

44 (E * eq / E* 1 ) [a/t=100] GCN *+E<$B# 100)C7BB N BGCEeq (E sub /E film = 10, ν film = ν sub = 0.25, a/t = *+E>$B# 100)C7BB BGCEeq (E sub /E film = 10, ν film = ν sub = 0.25, a/t = GCn ech

45 <!"#$%&'()*+,-./0+#$%&, *+E $B# BGC#7BBG[a/t](r)C#B#$#C FGCBt/a01FCC# B F2CB2C $#C FGCBFB72#C CB F#CBFC$C#B F2C#C$CF# F GC$FCGFC #B2C CB## C BFFCBG B F 2F C##CC##C FE!C#$#C FGCBACGCFCFC7021BFC#C #!U 9#:B2##CF!#!C $ sub /E film = 100:ν film = 0, 5:ν sub = 0.2Ca/t = 800$C C B = 3000 a t n ech = 20 N = 800 ACC$#GBC3FC#CF#C 6 F#2CGCF 7 #C3FC 3 7B FF B:UA C BCA F!-G 5%% 0A0= =%AA1 &.$&" 8(.*,!(%"#) "# 5CGB7C3FC# C #CC9! FCE =BFCC#C B F -4CFC$ C7BFCPB CF#C2GC##C##CF C: FFCC CGD $CFFCGCBG F$CFGC##C##C:BGFCCC4 #C# GPa0GBCG F-BaG 72F1%F B" B B;C##C FF#BFBFCF#D$B# #C$C#D#!BFCFCFGCC BC3FC#CGC#7BFC# CE ind = 84.6 C$C#CF-BGC

46 + > * + E C FCBCGG B7C3 FC BFF9CF GGCBGBFC ABCBFCBFCCC:BCFCCFCFC GC BCCBFBGC BFC CB C BFCGCBGC BGBGCDFCCFGCFGCCFFC FC BC C BCFCGCFBFC FBCGCGDCCBF ACGFCCBGC FGCCC FCGCFBGCBFFC GCB BFCBFC!FCGCGFCC CF GFC BCGCCGBCBFCGCBGCC FCF"I BCGDCCBFBGC F BCBCGC C DCC FCCCGDC FCC CBCFCB CC ##,#/"*, &#" $% '(&&$&"!(&,(+ "& /$%& '*,*& BC %FFBGBGCFGGCFFCC FBGCBCC BDGCGCBGCGDCFC)FFCCFFCC+ C CF&B %CFCCFCCBGCCF '()* CCF GC C! CCFCGCG CCBGC BCBFE %FFBGFGFCCCCFC FCGDCGCB subcci E 'BBC CGC)BBν CFHCCFCBCCGC"IGD CF CFCDBBGCBFC,STYT^_]`f]gY\Z\i]^ Z_dZ fu^\^dvztg\itf^dyz-p sub = 79 GPa

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48 + ƒ " } % + film " ˆ } ~ˆ " E " " " " ) ƒ g ˆ o! E ) %T ˆ P a/t+ % + + ˆ + ˆ ˆ! + +ˆ a/t ˆ ˆ ˆ!! ˆ + ˆ ' ˆ %ˆ + ˆ! film!"# $%&'(')*%+ ',- -* (*. /'/* 0.+- &%0 )*'/)%+ &-+/* 1+)* ˆ '23 ' 4 ƒ5 ˆ 4 5 ˆ326 4 } % "ƒ 4 }5 + %ˆ! ) 7 : ; ABC D< :CAE E<>;FBB<G:C HFAE HF:JKLEC M ABC?CKHNE<?AEC >BONE>CAEC M ;< TgP

49 } Charge adimensionnée Points expérimentaux Incertitude sur la valeur de E film * Modèle avec indenteur rigide Modèle avec raideur finie de l'indenteur a + + z { ˆ P 10 a/t

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53 !"#$%&!'()*+,-!"#$%&! :;<=>? :;<=>? :;<=>? 89:;<=>? 2324A 89:;<=>? A 89:;<=>? 89:;<=>? / :;<=>? z { "! " -FB~+&$"! $ " ) (E eq E '"! $ " film )/(E sub E film $! % " ( $& a/t E sub /E ( Fƒ 2 " Fƒ2- %+&$"! film C$, + ( *$ $ C " ( % + && '$! & ( 1& A " &&- 4 5-D $, " &&$ (,&,& (! " %&% & $ " $@ $" " / & '"! AE!.&( " &$! -( *$ & &!" " " $," " ( $& " $ ' " & *$ &!& " % %# & %,$" & "!"! " %&% + &$ '!,& - +& $ a/t- %&? " ˆ$"" &$ ( *$% *$ &.&( " $," " " %& +%- &ˆ%+&$"! $ ( $& & %*$!+& " % $!",!! ˆ$ E &! $ "> -F 9 & '"! (%&? Φ " "$8$ (! " ƒ " F- ˆ " "" '"! Φ *$ &ˆ / ( %&! ˆ " && *$! %!" & "!"! " & %?!(.&( (!' 0 " Φ(a/t) = & %?!( " + $," " $ Φ(a/t) = 1-3$ &ˆ%+&$"! " (E #&.?$ % " " -F (!1 >"%"!*$ &ˆ%+&$"! eq E film )/(E sub E film! % " '"! ) Φ!% $@! % " ( $& %*$!+& " % $!" '"! $ " a/t - F "" & % "$" ˆ, *$ (( "$" & $, " & &&$ # *$ & *$!"!&& $ ( $& $ >"1( - "" ' ( " & '"! Φ *$! "," $ ( 1& $ % A " &&- 4 5 ~*- F- - ~ "!$&! # & *$ & " " " ( $&!(!$ # & $, &$&% &?!"( +? " + & $, % & ( 1& A-! %$&"" G" &$ &! ( "!! 1 &ˆ &.?$ - - && )! % " & & '"! Φ &$&% " "! " ƒ- ( $& ( E eq = E film + (E sub E film )Φ(a/t)

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55 B D D (( $ &ˆ+ % "% %% (( "# & %$&"" &>"!*$ A " &&- ~*- F- F # F- " F- " %"%," $ $ "$,"!+ / >"1( (! $+!1 "! $ & (?1 -D &! ˆ" ( "!" " %' ("! &$& " & (( *$ $ & $@," $ $ >"1( (?1-0$ ""# $ $," " +"$ " (?1-3!!#!.&( " &$ ( &!" *$ $," "#!& $ " ( / $ " &$!( "" & %' ("!#! $!" $ & "!,$"! $.&( / & %?&,& $ >"1( ' " &$ *$ˆ && &!" 9 >"1(!" (?1 E& "!"! " & %?!(.&( (!' " & %?!( $," " (!' %&% + > & "" &$ %& +%- 3 &ˆ!+ #!.&( " &$! *$ & & ( $," "# $," " " / $ " &$!( "" & %' ("! $ >"1( #! $!" $ %&?! & "!"! + +& $ a/t &$ "!" - 1 &!& ˆ> $ " / "" *$ Φ & '"! $ ' "!1 " A %&!(!" & >"1( $! $ & $," " ( &!" " & >"1( $ ( &!" $ $," "! - "" '"! %!" $.& (( " " % "! & "!,$"! &"!+ / >"1( & &! $ ˆ$.&( $," "9.&( " $," "" & ((!%"% (%!*$ +!.?$ - - 7& " 1 & "$" / '!" (& *$ & *$ & " " ( $&!(!$ # & '"! Φ[E sub /Efilm +? " + & '"! $"! A- ](a/t) ' >"%"!*$ # "! *$ #! " $ 0! %# & '"! Φ[E sub /E "," $ / & '"! A $ " ) film &"! & ](a/t) "!"! *$! '+! & "!,$"! & ( " & &$ ( &!" $ >"1( - %$&"" & ( " %'$" & 1?&!" Fƒ% #$ $ (! &!"! / & %" (!"!!%"% %&"!*$ ˆ$.&( (! - "" 1?& (!!*$!!*$ *$ # &ˆ! ""! ˆ$ $," " +"$# "" *$ & %%" "! "!'%! $ " " Fƒ% / &ˆ%! $ t $.&(# & % $ >"1( " && $.&( $&-F!& G" &! ( " *$ # $ >"1(.&(! $ $," " ( &!" + $ ( $& & $ (( " %& +% 0.2 < E ( & # sub /E film 1?& " (! %'$"- ""!!*$ ( ( $" "(( " < 0.5 / $ %*$ " $," " +"$ $! # "!$&! &! $! "" " ( $ % %? - :F=<E>?K>RAC; C KF??E<B;:<?>FB C;? K>; CB?EC =A>::CKC?; I<E :< M=AEC P C;? HEN;CB?N CB IFFE@FBBNC; ;CK> > IC;? :C ;AG;?E<? P PEC;HP : C M:K Q RA> C ;? :C H:A; IFKH:<>;<B?O:<?E<B;>?>FB IFKKCBIC H:A; O<>G:C; a/t PEC;HP H:A; OFE?C; a/t Q O IC RA> :<>;;C ABC FBC ;AG;?E<? BA PEC;HP M :K K<;;>O Q H :A; B CC?O@<B;ICI<;FB >KHFE?<B?CP : ND (E eq Efilm )/(E sub E film <DCI ) a/t P : IEF>EC CCI?ACE ABC KC;AEC CB EN=>KC :< <:FE; RAC :C ;J;?LKC C;??E<B;>?>FBP

56 & B - %""! %("!*$ &!" %""!$%&? &!"! & (1& "!"! " $ A-D '"! A","$ $ &, ( "!""%'("!,+% $ ("%!$ (?1- >"1(+"$ & "" + (1& & $ ($& (*$%# &%'("! " %'%"!&&(" (% ("%!$ $ &$ ( &!"- 3!! $ +&$%a/t# $ A#.&(" $,"" "!,$" "$ $@ / &!$$ >"1(# & *$ $ $,""!+"$$.&( ( & &? " &!" - $ $,"" ( &!"+"$$.&(! &.&( - $,"" '$! & &$ & "!,$"! / &!$-!((!&&$ "!,$"!$ $,"""!" + & " a/t #,"!" $%&? & "!"! + " $ "!!"!"% '( '"! Φ[E sub /E film ] -

57 " - ) B+&$"!(E eq E film )/(E sub E film '"!a/t $!%" ($&-D '"! A" %"%!"!&&%-D%$&"" &$& " %"% & (*$$-D% ($& " "!+(" F-F # # # F " -D &!? &!.?$" & 8$"(" F B!"#$%?(( +&$ &$ %&+%-F & *$E & (1"&"&"!'x 0 "&.?$ - Eeq Efilm Esub E film D8$"(" %"% - #%&!% $ν film = ν sub = 0.25 $ '$!" &,"$ "&"! & '"! A-.?$ D -F (""" %+! *$ & $, ((E eq Efilm )/(E sub Efilm ); a/t)!$ &$&" && '( (,&,& / & '"! A#!& (,&!"%" %" $ 8$"(" *$! & '"! A $ $ &"! $!+"E ( ) a = Φ Gao tx 0 ( ) Esub ln = ln(x Efilm 0 ) ln(x 0 ) 2 F $" &+!?$"!&! $ $%+&$ "> $"%" $ & $,@ %!("&(E - 9 & (!" 0!$.&(" eq ; a/t) $ $,"" " - -

58 2 4 5 X 6 XX X X : 7 ; < = " (!" &%$&"" *$ + " $ %"# $ + +&"!("@&$ & 0!$ ( %"$-F#!&@!" (, $ "! & (!" &!"! "!*$ 0!$.&(" &$ %&+% *$ &$!$ $,""- +"(" &>(1 $ $ $,"" + "!"$ & $,@( (!" 0$.?$-Y$@!! & ' $ $ &$ >"%("!*$ &"$ %" 0!- '!# $ &$& ' >"%("!*$ &%+&$"! $ ($& (!" " %*$!+&"%$!" ($& +!"" & (!" $ F 2" F2 + $ 0!$.&( νfilm (!"-F"-# 0! ν sub$ $,"" %"".@% /-- $%$% $# C$.?$ D -" - %"" &%$&"" & &$& "!+(".&( &$!" &$ ( &!" *$ &$ (!" &,+ $,""-.&(!# " ",&$?(" ) 0!$ (!" 0$.&(- 0$% " ($& #,+ (!"-F"-# &%+&$"!"!(!&!- $ ($&E *$ sub /E film &$?(""!$ 0!$.&(!$!" $ &%?%&? & & & "!"! + +&$ &$ %&+%a/t- = 0.5.&( ( &!"#?(" (!" 0!!$!" B?(" %*$" $ &%+&$"! &!$ %*$!+&"$ >"1( & + " & (!" a/t -F "" '!"$@ %(1!"!"E G! &!"!$%,$" & "!"!" &!?%# "!&& +! + 0!$.&(-!!# &$.&( 1 (!" $ 0! %&+%" &$ & "!"!" &%?!(.&( (!''" &%?!(&$,"" $'"!-D%"" & & "" & $.&(!(!,&" & $" "!&&$!" + ($&- B ",+ $ $.&(!(!,&#" & &$ "!" "" ($&# $ %(1 $ "E & ($& %*$!+&"%$!"$ >"1( $" &$ %&+% *$ & ($&%$!"$ ("%!$@+!.?$ - &$!$@ ", -!"" "( "!("- $#D D"#!#$ D%$&"" *$ $," " '!"!! & (!"! "! %(1-! "#!K"$!%& '! L#$((#% ') ()*(&+,&!(& '! - #+'+! '! "!-)$ ' )%.!++! (#'#",-"$R)!/ 0 νfilm12 /3/

59 -B+&$"!$ " Eeq/Efilm '"!a/t $% ($& /--D"$ -" -D (!" -F#, -F# -" (!" $ 0!$.&( -F #- #- #- "- 0!$ $,""".@%" %?& +!"! (!" 0!$.&( $ &!$ %*$!+&"$ >"1(" '!,&-

60 - F" B+&$"!$ " E eq /Efilm '"!a/t $% ($& #, # (!" F" $ 0!$.&(-F %?& /-- #- #- #- "- -D (!" 0!$ $,""".@%"

61 $ - B+&$"!$ " E eq /Efilm '"!a/t $ >"1( + ($&%$!" (!" 0!-D (!" -D (!" (!% 0! $.&("! %?& /-F #- #- #- $-,$ 0!$ D (! $,""".@%" %?& /-- %(1" & " & "$&&$ >"1(+"$ /!"!) &%'("! %'%"!&&(" >"1( & & &$ ( &!"- $.&(! $ $,""# (( & $,""" &$ ( &!"#&%'("!" "1!(" (%!-D (!" 0!$.&( $" +! *$$" '!,& $ &% (%!*$ &?&,&$ >"1( $ $,"" &$ ( &!" *$ D +"("- %(1",! (! %+! &*$!1 & $ >"1( ( %$.&("$ $,"" (( ($&%$!" (! $+$ (!" 0!!%"-!.?$ & & " %"% & - - (!" 0!$.&(" %?& / &$!$ $,""#"$+ & $ ("%!$ (?1-0 & & (!" "#! 0!$.&(" &$ "!" - &$? *$ &$! $ $,""# ($& %!$ ($& %*$!+&"%$!"$ >"1( $ (!!($( - $ (@!($(!'%!$ - $ $ $$@ ("%!$@-!""#,"$ &,%" ($&%$!"" & ((.?$.&(" $,""#" / &!?!$% (",+% $.&(!(!,& $ & >"1( %"% & ""!$"! - - '!"# " & ($& &$& &&- A" # "" *$ " & " *$! %!".&(" $,"" " $ ($& %$!" (! $!!&&("-F"$+ $! %$&""! "" &@ & '"! & ($& & A$ $,""+"$+!*--F (1"α " '!" &!!&&("$.&( $ &$!$ $,""-

62 & F *$ %! *$E $!%!!E (! film = E sub ν film νsub# $" µ sub = E sub (1 ν sub) µ film Efilm (1 ν film) = 1 ν sub 1 ν film F +%!. & (!",! & *$#! 0!$.&(" &$ %&+% *$ " %!$ &$!$ $,""# & " µsub /µ film $ / F-F,"!" & $ >"1( " *$! / ( (( &!.&( %"!" &$ ( &!" *$ & & $,""# &$?(""! ($& & (!" %*$!+&"%$!"- &!+# 0!$.&(" &$?> '!,& *$ & ($&(! &$!$ $,""#!!&&("$.&(" &$ %&+%" ($&%$!"$ >"1(%G" $ "( -!(( & &!(!" $ $ "1 F & """ $ ( "(" "> &$,"" $'# $)&/$ "! " a/t ($& %*$!+&"%$!"$ $,""+"$ (( / $+$# *$!@ &!*$ & %$ (@!($( -$ (!!($( ($& %*$!+&"%$!" &*$ & (!" 0!$.&(" &$ %&+% - &$ *$ '!,& &$!$ $,""-! %(1",! $"" &$ '!,& *$ & "" ($&" %&+%&.&( %"" &$ ( &!" *$ $,"" -!&@ &!*$ D! $ & " $*$!!& ",+% *$ &$&"$% (!" (!" + $ ($&%$!"- %(1"&"!' / &"$ 0! $ ($&!) (!,!&!"% -!!"&!% B $ 0! & '($& $!+"E - E B = 3(1 2ν) F +!" &,! *$ & ($&!( " & (!"!,!&!"% $?(" & (!" & + 0!) " *$!&"!.! $ν = 0.5- G! 0! $?("# ($& "!(!,!&!"% $?(""#& &%'("! (%!!&&(" / $?(" $%"!(" & " (% (! +&$(%"!*$- G!!1 $?%(%"!.&( $ $,""# &$?(""! " & &%'("! (%!!&&(" "$!" $ $?(""! & "!&&!& & %&"!*$(" & & (!" "!+# "((" $!+$ $ &!"'- $ (( " ($&%$!"# &$ 0!" %&+%" &$# / $ +&$% a/t# $,""&.?$!"' $> ""!( ""# *$! " / $?(" & "!,$"!$ $,""-! %(1" '!" +!!,& $ & '"! A$ >"1(-D %" - &!%" '"! A,"$ $!%" +&$ (!" (!" $ & "" 0!.&(#+ $ ($&E sub /E film " $ 0!$ $,"" = 10 νsub = 0.2-F,+ *$ & (!" &$ 0!$.&(" %&+%# &$ &?(( +&$k $ +&$ &*$&&C & 2/Esub" %"$- k %"" "!& / t/a # "" """!!( &!*$ *$ &%?!( &$,"" $',"$ $ +&$a/t$"" &$ '!,& & (!" 0!$.&( %&+%-!( &!"! "!*$ & %$&""" *$# $ >"1(+"$.&( " "1 $ (!,&" "1 ( &!"+" $,""#& "!"!" + ')$& $ #% #)+ +!,+'!,&&!%&$.!!%& -,M )+! / P* Q O#% #*(!+.! )% -!+,N$ ) -#",- ') #')-! R)$.,-!%& a/t "# +$(!%&+! 2!& / -

63 B"! - A &$&% $!%" (!" 0!.&( $ "" & (!" (!" $ $,""+"$# + $ 10" & ($&E $ sub /E film 0!$ $,"" = νsub = 0.2-0&$ 0!" %&+%# &$ & '"!!% A / &% $.&(C(k) 2/E %" $ $ film = 0.9 &??(( +&$k-!&!( &!*$ *$ &% $ $,"" $&,+% $ +&$ &$ '!,&a/t-

64 "1%$!" + $ $?(""! "1 ($& %*$!+&"%$!"-!!$ %) (1 $" +! %*$!( "" $ &%+&$"!$ ($&$.&( / "! &%+&$"!$ ($& %*$!+&"%$!" "" + a/t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a/t &$? - &$ "!" *$ && '$! & (1& A-!%$&""!+&! &1?&.?$E D"$% (!" &F '- 0!" %?&!?,& $'$@!& "" & "" (!" B ($&" '!,&" 0!!( ""# ($& %*$!+&"%$!"$ >"1( G" $" "$&&(" &$ %&+% *$ & "" & ($&%$!"$ '"& ("%!$ &$!- &! ($&" "1.&( %"" &$ ( &!" *$ & $,"" "!.&(" &!(!,!&!"%# "!"!" &%?!( & &.&( (!''" &%?!(&$,"" $'" '"("%$!"" ($& %*$!+&" %$!"$ >"1( $?(" $" "1,$"&("-!"" &!(!#! &> " &%+&$"!$ ($& &! %*$!+&"%$!" / (1& " ( ""# *$ ($&%$!"$.&( $?(",$"&("" '"(" &*$!&".%-

65

66 & 0$!"# $!% &%$&""&"!' $ ( "("$> > "! "" "" &!""!$ $,""+"$- '!# $!"$! & >"1( %*$!+&" & (?1G " & %%""!%$!" #%.! (( & %%""!@ %!("&" & %%""! *$!!","$ (( $ ""#! >"1( %"!" (?1- %"$!" &%+&$"! (( (1"# $ (" *$ &$"!&!"! & (%" F&!+" 0# "$" (%",% $ $ %*$"! %",&! $ >"1( &, $$ (?1#!$!" >"%("!*$ $ ($$> & ""# &"!!"%-!""$ $"$!" (( % &%+&$"!$ ($&$.&(- $ (" *$ ""$ & "" G" + ($&- $ %"$!" & $ (%" (""" $ "!&& "" "! $ $ "" %&"!*$ $"!&!" *$ &,+,&@ %!("&- $ (" $! & &!(!""!!$!" & &"!!"%- 3 9 = ; "$#$% # $"%!$#!(( $ & $ & (%"F&!+" 0- "!*$#& 0 (" &$& (%" F&!+" (%"F&!+ & $> $ ($&%$!" E " "" a / "!$ "! &",+,& {P; S; δ} # $ >"1( - "!&! & $ $ >"1(+"$+!" / "!" *$ "! &" ((!$$!!""! $ $ >"1( (?1 ($&%$!" P; S; δ E eq -

67 !$ & %.! >"1( & %*$!+&" (?1 (( & %"" >"1( "" S %!"" & &!"! (( $ ""! (( & & %%""! 'P δ *$ & & $,""+"$ %"$!%-D.?$ -F (" &&1& $ >"1( +"$ $!""!" >"1( %*$!+&" (?1!%- &" %+!" *$ & >"1( $ %*$!+&" (?1"!!!%" *$ "! &" P, S, δe- FB %""! %("!*$$ >"1(+"$ $!""! " $ >"1( %*$!+&" (?1,!% / "" %" $?("F!!!$$!'("! $ &&"!" &> a$ """ &> $ >"1( %*$!+&" (?1 ahom!"" %""! $&1+ $ *$"! $ & &!"! (%"F&!+!"? " 0 &> / &!""!$ >"1(+"$"#! &&"!" & ($& %*$!+&" E eq # "" a" &!$$ """ "$$ +&! # $" $ & +&!!"% " &!!",&1(E &!*$ & 0 (%"F&!+" %!E & / $ >"1(+"$+!" / %"$! & >"1(%& (! >"1( %*$!+&" (?1G$ & > "1F&!+" 0# $" F ahom" &>$ ""$ >"1( %*$!+&" &*$ (?1 F# & &!*$ & 0 (%"F&!+" / $ >"1(+"$# %!"E D% D!$ $#!#"%!$# $ hc(ahom) = δ ε P S h c (a) = δ ε P S H #)& ') #$%( ',%( -, #%! '! &+,%($&$#%/ %!!&O$-!(&,(($ $-,*-!,) M-,(($! P+!(/,) ()*(&+,& %) Q"!,$*-! ((!%&$!--!!%& +,##% '!,+"! "#%&,"& P+!(/"!#+& +,##% '! "#%&,"& Q / R)! "!&&! +!-,&$#% %! &+,')$& R)! -, ' M%$&$#%,' #" '! E eq /

68 %! E *$!+!"!( &!!"(" / F"!"!"? $ +&!!"% &&"! # *$! $$$) "!."! >!*$ &!! &% &.&( (!''"&$,"" $'# >"1(+"$ %""!(!&,& & / $ >"1( (?1# ""&"! (,&!),&!&" (! %+!" "!"! $ && $"!&! & %$&"" &$& $%"(! &?(( +&!!"% &&"! %&"!*$ a hom = a =!? a/t >? F P S > > a? δf >?! >? δ?? E δ homf E F E D > δ?δ hom a "$#$! "$"#D!#$ D"%#,!#%"&$#% '! ()+!,"! hc!(& *$!"&$.!/ δ hom = h c (a) + ε P S δ = h c (a) + ε P S δ hom = δ

69 B P, S, af δ δhom!! E sub /E film a/t %#$! "$"#!#$"%#!% %$$#! D a/td D ( ( ; a/t) E κ a δ hc (a) = δ εp/s F ε! sphere 1 = κ ( cone 1)!& #)+ +$.$,-!!%&" -! (#(&&! ($ $)$.,-!%& δ = δhom",-#+( -, +!-,&$#% *$)%$.#$)!!%&+! a!& δ!(& -, %! #)+ -! (#(&&! +!- # # &%!/'#%""δ= δ hom a = a hom /

70 δ δ hom ( δ a δ/δ hom = 1 κ

71 )*)+,-!"#$%&'(./0-12/3 4,53/5!-!56 3/0- ;,53/5!-!56 7!86!9: 3/0- E eq /E film δ/δ hom a/t E 5< sub /E film = = 1! > 1 δ δhoma ahom a/t a/t < "?! + ()-&,& #+($)! -! +,##% '! "#%&,"& a &!%'.!+( 2 - $),&$#%BA/ ED@!)&,-#+( (! + "+$+!F ",.,$& ' & $%&+#')$&,) ",@$&+!A',%( -! ",( '!(!,$*-!(.,-!)+( '! ab?!/ca/ / D / BE GD / θ(s) = 2 Efilm g(s) + 2 π a g(r) ( dk ( C(k, E, t) 2 E film ) cos(kr) cos(ks) 0 0 } D {{ } 0 2 Efilm g(s) "#%&+$*)&$#% H%! +!-,&$#% ')I- $)$ ($ $-,$+!!(&#*&!%)! &!%'.!+(!% )% &!+! '! "#%&,"& ()+ -! ()*(&+,& %)!& )%! 2 -#+($)! t &!%'.!+( +#/ ) dr

72 > a δ (

73 < 1 a! > 1! < > a t E eq = E film δ = δ ( hom!! &' a ahom δ < δ hom! δ < δ hom < a ( 0?!(& ',$--!)+( "!&&! #'$I",&$#%$)! &+,')$& -! ",-")-@!+&)+*,&$!,# "#--/ AE +,$'!/ & $%.!+(!!%&" )%@+#*-&! '! ($%$% ',%( -! ",( ') ()*(&+,& +!.%&)@,+ /

74 ! % %#!#$! "# # $"%$! < S P a ( <! ( '< & a/t > Eeq! E eq a a/t! a < E eq f(a/t) = a (a/t) = = E eq hom/t) = hom /t) Eeq < f(a/t) (a/t) = <!?#! '!(,)&+!( & #'!( '! &+,$&!!%& *,(!( ()+ '!( $),&$#%( &,*-$!(@#)+ '!(,& +$,) # # &%!(/?!$)$ "#++!(@#%'" )% (#(&&! ') &#@! +!.%&!!%&@#-# &+! ()+ ()*(&+,& '!.!++!/

75 -./ :20:/;:2842<2012!"#$%&'#()*$%+, _àbc JKLMKLNOPQRLSQKLTPL UVWXVWYZ[\]W[^[ZYW def ghij kl ml no l o l l op lmml m l m lmnol 0.2rml o o nmle eq(a/t) = f(a/t) m nnl m llmnol l lmnol no l m l nn tt l p ol po l nop nm n lul mq l s n nq ν sub = ν film = 0.4r nmomlolmol lmml mom ll l nmle eq(a/t) = f(a/t)mlll ll l l m l m llmml nm m m mo me eq l ol (a/t) Eeq hom/t)vl lm l m llmml l l no l llmml mal m m l nm m l mo s m m lm nmle eq(a hom /t) = f(a/t) nl m lllmml m l l m l nm m m m leeq ll al f(a/t)l l m p lol nno homplmml m lqomllw om n n lrl l m l (a/t) = m a ml m mm nmle eq (a hom/t) = f(a nmla/toll l m l t hom /t)l ll l l mm m xqn l m l mo laolyz m m nm ol tt opl m r plmml m l m loll l m l t{ n nm m

76 j nml pl lmollml op l m ll m l l l opo l l p lmmlm m m n n a n noml l l l lmolol10%m n nm l l nom lm l m l nq i rl o l nm l n mo no lol l nm no lmo ml m l m 2rl mo l l ql0.5 E xn nm eq /E film l m l nl lol n i no u l l nl l l l sm lml ll l no lmoo lnl om ll l s mmo l < l m ll nl m nq i r m nml mmno l l l mo pnmom lol l tt nml qop nrl nmonmoo l l l l m lmol lmol nnl plmml m l mn lmnol l op n pnq no i rl lml lo m l m o n nm nnm l mn on l l ll l no lop<lml mmq l n lll o l r ol l l nl nlomnol ol m l sm l l l no pol moml l l m l nl nm l l o n lommlo l mll ln ll onmlml l!" l mol lolonomnol mn l l o l l l no lollmol o l smlmno m l l mo n nmonmoo l n l l l no o l mmnm m l ll l m m< ll o nlol l m l m lo s lml ml ml onmool ol l lm mn m l l mo lmon l ou lopolm nl nolop<lml mnqn l l lol pno larl n lm mn lop l l l no lm mm # l l l l n lm mol nmm lm n n l lol nmm l ll mn n n lm ml l m nmmn mm nlol op l so l l l m l mn < n lono l n $%&%' ())*+,-.,*/ ,341 mnnol l o l 51,-24/.4 nom lm m l mm l m n l mmnm l m lolmol 6 l l n l 7 n s n m m 8 n subqml nls S filmr mol m l subqml δ δ filmr nmno mqml o r mnnlδ nmnml lo onmno l l lpl l m n nm on qi9r la/t δ film (δ sub + δ film ) = δ film δ = 1 P S film δ

77 jj l n S film S sub S Sfilm l mol m l lo S mvl l m l o l nmo l o l mnn l l mo no l m l l lo lmnm m δfilm /δ n nol l l l nno l nmno l l l onm nnlo nmo n l lml lmlm pm l l mn o n l < lol mnn o lol omnm n P/Sδ la/t $%&% 5.*34 -/0 /.4 53,-,41,-24/. * */4 ())2,-.,*/ 4/ 5* 5.,4 */,14 l =δ/δ hom ll pnloplmnol nm nml lml pnq r lmn n l = 1 n l l m ll m l om nn l nm qi tr l vl l l lr qi r lolmml (o op n n vl n lmm qi r lmlolll llmlolomnl llml l m l n p lmll l o l l pmso lll l l l l nno l lmllml n mn opm op l m m l nnomn ao nlωol po l l m ml l l lol nl oo l l l n on o o lol p mt mnm l l o mo l l p l l l nmo l #nm m n pmnn mmnm m m m l m l nl mnm l m l nq i rml nlu mnno l o l lol l nmo l pnn l mnmo s nm l o l lmll l no l l nmo l ono qnm mlirl nn s m o l nm l l o l n l l pnqi rl l p lmml mm l l l m m ml l ll lmnol ll n nolop<lm n o n nnml l lol m l l l mn ll l no l l pmn lvl lml p ll l n lml m m nm ll l m n lmo δ P/Sδ = δ hom (P/Sδ) hom = 2P Sδ δ hom (a) = π a 2 tan ω = Sδ2 2P mm $%&%&,,.-.,*/0,/31,.40 )- 2- )2-0.,,.5 #) %$ *#%# $!"##"$% "+"" ## P/(Sδ) ##@%,%%$" = 1/2 "+""$ ## %%#$-?#%$ #$"% +%$ %%#$" $% $.@ ##@$%, + %#$%#/% ### "!##$%# "%0-

78 j!"#$%&'()*+,- "#$%&'()*+, "#$%&'()*+,/0! "#$%&'()*+,. "#$%&'()*+, "#$%&'()*+,/0!!"#$%&'()*+,- "#$%&'()*+,. "#$%&'()*+, "#$%&'()*+,/0 "#$%&'()*+,. "#$%&'()*+,/ nol ghi 89 ml nnomnol m no mnml no mno lop<lml lop<lml mm l n l nm mm l l o op l m n l l ll n m n no m nlu tqrtxqrtiqrnt qorl 89vl l o l lml l l lmol n n l l t l nm n lu m l o l no mo lmnol l nm nno ll m o l mn ll opsm nno ll mn l n mmqa méthode = ar nol 89 lnll l n l o l l no lol<lml mm lml l ll o l n lu l l l o l l s l nm nm no nn

79 j l o l nm l n ln nmn ll nopsm l l l sml#ol l ll mm l l 2P/Sδol n op mnn m mlo nnl o l o n lol m l <nlml ll lmml n non lrll mm oq nm mli l l mn l nq ll ooo l l m l nm l n l l l mrl m l o l m o l mo l nll ol l lol vl l l mnl lmmnl n mn o no on pn nm l l2p/sδ P/Sδ 0.6 po l n 6 lm n 0.4 Silice amorphe Verre float Système revêtu E * sub /E * film = Pénétration [nm] 1000 ghi nnlml nmol l nnml l nm op lq rol l xtt lmml lq+rl lmml non lomnm nmnm l 2P/Sδ mn l mnm op sop Esub /E film nol l o l m l l l l mnm m nm lmnmo lml mn nml 8 5 s m l l l m <ml l l l l l l l l nm n n l lml o lml l mn o snn lmnm 2P/Sδ ml l nm lqxtt l l m n vl l nmo l l r <ml l o l l m o mn l nm mn l n ll l pmll mn nlol l l mn nml l nmo p ml o m m n l l m p nol Pl δ nl m l ml vl n nm l o # lmnm 2P/Sδ ml lml m l l ml lonl mn p lmo p l l l lml l l o l l vll l l l m l l s on 2P/Sδlml l mmnmnm l l lnl lmol mn o lmlm qmnm l l ml op l mn op l l n rn lmpll o l o l m nnmll l l o l ll n pm nm nnop % "!%!"@ #$ m l l mli m vl p l l nm l mm l ll Smllmnll l ll n l lmlm olmnmo ml l ml

80 j9 mml lml lol l m ll llml nm pnm n l nmmnomlol l nmlol lu pnon lop nnm l ll l l l lmol lm n mn lo n n l l m mm p l s ml ll lol lml non mnn P exp, S exp, δ lml l lmolll lll mmnom l s mn o exp l s op ll m n nm nom nl n pn noml lol o ml l mm l op l l m n m nm l n l l mn pnm non lmnm a/t no lmoo Efilml l mnom l no l l n lul o l n l nn ν film mn pmonop nm xr l qi δ exp = δ[a, Efilm, ν film] ml nmlol nmm l lol ll nnm l nm < l nm nml nm no n m mlm noll n nlulol nno lopl o l l vlloml l l l n m n mn l ll lnmlm nml lol l l op nm n n nm s lo o nol m l m ll l no lop<lml mm l l mo l l o n o op n m ml ll l l m l om l lol pnno l n m nmnml l o m onmnnlo o l l l l o l l s nm l mno loll no m l m l m l m l llmml o m n m lmn n l l m l l l l sm l m n nm lmnm nm nl lm l l mn nm l l mn nmm n n mnn nmlol pnomnol nlol l n ll o l l m l lol l l m n p lm nnmlm l l m no lol mon p n P exp S exp = P[a, Efilm, ν film] = 2aEeq [a, E film, ν film] l l

81 t

82 n nmoll # mol! l" #n # n n$$l #%ml $no n % & % # % 'l #nm$ m ml s # ( m #o lu n n n $no l l # % ( #! $ # ) ' $ # $ % m'l #l n *lm$l # $ lo # # % 'l m ml s nml **mnl n$ *lol molm # ( m l lol + *lm$l # *l #ol#lmlop p $no ll,nmlol m,l % l# l l #oml # $n *mll# % lml# * m! n l,nm$ lop # ol#lm.) mn- # $ % m'l $ % m'l" * #n # n # * # n n # l# l l *nm $no % l /nm $ % lm ln # % 'l m n ml$lml s ( m s n n n ol $n m n p n # # # % 'l" % n # o $nol % l#m op ' % o # # )! " *n # #lol n,nm m # # % l l#o $ %*l#o #lo # m lol $no ll#mll n l l l +$ l ( m # # " * m/ol **nm ( +$ * mm l * m # #l#mllm % /$l lm +$ l /$l l ( m *mno &,(lm)n oln # # # " & %* l +$on ##%l vlm $n ml l m /lol mo % # & ' *n # n 0t{ *l sml! mn o $no l l ol # lo # +$ * m/ol llm ' ( m # * * n % #! *lm$l # # l l#ol nm(l o l $no l $ % ' l# n % o m #l l n m/ /$$ # m l o l $no l vl $no ln! # l l #l n ol n n $no lol no l n m $ * $ # # o!, # & *ml#omll# n$ *l o lol %*l#o # l#l# *n # #lol n,n n n m # #l# # #o # m lol n n n $n m nlu l#ol $no l n $ # # # % p #1 l# 2nn lo # o +$ m l $nol% l#m $l vl ' % o o )! l3 #ll ol, l# m' lo #. ol ml +/ # m lol $no ll#ml +$ l ( m 2nn *l l#lm $no l #o nmoolnlu l#ol, (l" % # l $ % l#m &sml mlmqml mlmr $nolm ol ' % o *% * #, % % o n molqml m l l l n$ $ % m * * n$ * # +$ r! * # # # ( m l! *l n n$ * (l" #lol n m # #l#ml % /$l lm +$ $, l % /$l ( m # *l sml m n ol ml $no l o! *l l 4p! $nol % l#m ' % o #l nlu l#ol l# # * n$ 2nn *lo # +$ s *l n oml #lm m %lm o # & ml n m'lolmolmoln lmml * p/$l# # ( # op op # * n)$! n$$l % #ol & #l,nm l n #ol *mn /$l# * m %% % # 'l

83 o +$ n n o l #llmmlm n $ # # $l# % l# ) % $ 'l nmo l p ol mm *nm $ % nolop<lml 2 # )lm l on ##%l n ( m ml l m n n s l# * # #n # ol# # # ( m vll $ % n$$l nol" n l ml $ % nolop # )lol on # ##%l #n # ol# # ( %l mol n $no n % ' # % # # # m $ % m $, " l # mno #l n n qml 3% # * #l m n rol l % # mom)n oln # # nm'll l qml m/olr'll +$ * n$ * # * ( m n n $n m,nml n n l l op # # % o # # # # 'l% ' n l ' mol n$(ml n n o # o$ r l# ##% 2P/Sδqn #l Pl n ** ' %l" δ *%#% m #l mo S # *lm$l % * +(lom)n oln # # 4 * % $ l n l*l#o # p # * m ll 'lol # **mnl

84 x

85

86 m l # ml * l#ol l ml" #n # mnon $nm l l nm # n$ l lol lmml * $ l# * ' % * *! $ * l" ' #n nn # n$$l $no l $ %m! " o # lolll % nm ol % n # #l lol on ) #! ##%l ((n/m * 'l. #l m l nm l m n$ l * $ l# # *! 'l l m l ml o l m ' % $ l# m! * %"# n * % l#n l # m.o. # m $no n nm % # $ % # 'lo n$ l lol lmml * $ l# * ' % & n $n m # + #" olmn *! 'l l **l ' m mo n #k mlol lmml opn.)o l #n o $ #l m # # $ n mn *l mnn lk n l * $ # # #n o *% & m %3 " m # %,nm$ # * ' lo # l $nm * l! m l # %* %#%ml l / $ l# %m.l#ol $. l,n n #ollmm n l #l n 4l n n l l# # %* # & # $ % # 'll# m # $ n $ %m.o * l n m m nm " ' # m % m % * * % l# lop # n$ l % * $ * 'l m mlqnm +/mlj 0qrr 4l # * $ %m. $ % 'l 4l (n n l# #l l #. $ mlqnm lol * $ %m.o l # $ %m.,m/l " n ' # m % 4l m % lmml lolo n * mp(l# n mm n %,nm$ # m * # % lm(l +/mlj 0q(rr % #l# / %#%m # o m 4l % % n$$l lmmll #pl $ olo %m.,m/l m l l *. n. ( mm l# # * %,nm$ # % lm(l ml lo n p l *ml n ol lmml l nmop #m * #l o l m # $ * l". # * % % (l l vl l, $ * mon. %*l#ol#llol 'lll vn p #l#ml m ll 'lol $ %m.o l n l $ %m.,m/ll o- % m % m n # $ % # 'l ** ' %l $ % m 7ix8 # o mn % # #l n n n # $ % # 'lolo$l# # m % m 'la l

87 j *nm gh j 0 vn$ * mn #l#mll nm(l o l n l *nm m # $ * ln( l# o q ml # $ % m lqrl l #l # $ % n m #m,m/lq(r *. $ * l" #ol# % %l l #m)n oln n n p # # ar # # le p %#lm/l o *%l l p * ' $ l#l E f %#lm/lo *%l m n * +m #" n ll n n m n #" qj 0r # { Ep Ha 3 E f G c a 2 n Hl o l# $'lop l n %#lm/lo *%lo # $ %m * + % q/mn $(l m n o m $l#ol ml & ml%o $! $ % m 7ii8rl ol# l# G c % m,'lop %#lm/l l m l #% & % # op # # % m,l nmo l n + m # 7i ij8 l *l nmo m " # % + # #l l m % m ll #nm ' 'la G c /H le p (a ) = E f (a ) " l m % m 'laol n n # $ % # 'l ** ' %l $l n n mlml on n l op & # )! *% & a " E p(a) > E f (a) " # +m # #% $n # %#lm/l l o ' * # * 'l *nms %l 4l on $ %m *l nm nm # n$ l # n al %mlml l (l nm * $ l#,m/ # mml" #, & a ". # n$ * $ l#o l #mlmnl # #lo$l# vl 'l *mnl o n #ol Bol # m % m % + #,m/ 7i8 %ol l o % 4 #l m l3 n l % + # n$$l nmol o lm ** ml% & H l# I %, lm op K #l# # m #l $ # $ *mnn' # #l ml 7i "n + 8 c vn$$lk I c = E G c # a H/(EB 2 ) #mlmnl po %l ( l# ' * # $ % ml,m/ lq l * Bl /m # llol pnmom orl * lol ll#3ol &o l lll %ml ll ' $ o *l # l # " & #l % ml l l ' l nm mn l lmml n nm ' $ # " # # n$ l n * $ l# % * 'l 4 +/ mlj n$ l *m * $ l#l# % l# # # #ol# n #3 (ln( l# m % # n ol l $nm # * l8q #3 n( # mr m *! #l n *%#%m # $.$l # ol 0 µ$ % $% %% # $! + a #$ #%! -%$#"# $,$% ""#$%$%! %$+%$ $ $$ "# # %# %#$$ #$- %.% l

88 gh j 8$/l # op # # ol# n #3 (lol µ$ol *mn,n olmll m 0 # % % # n ol l $nm *m # * 4 % l# lop #l *nm #lo n n $ %,nm$ # % # m * % ml 'l # n$ l * $ l#o l lo $ % m! olnm # ll m m * l" #n * % l#n # #l lol m l l $ ) #! %. *%m$l#. * * # m l nm n$ l n 'lol lmml nmln( l $nm n n #lm l * $ *lm$lmn l# % ml * ml l#ol l *mn nlm / % *n, # ' # n # % $ * #l $no % # $ % l ml n l#mn ol m ml # 'lo * %# n$!#l n # n # # * * l m $no! $ l# l3 * l" ' # n lmm " o # * # " )! $! % %' pn( lol ,2, ,-1+ -/* -1+ # mn ol ml lol lmml l ml "ij # #,m/ % " m,m 7i " i9, o " t8 n nm l * olm. m)nm 'll nm mn$ l n pl * $ l#% * 'l lpl nm n$ l ol $l# % * $ l#" l# * 7 08l# 0 9i9 " %% $n # % o %" ' p # * *n(l op lm m l lmml l nol $ % 'l o l n m % m # $ % # 'lq n $ m * l"1l lmml.n l # r vl ml#ol $ll *mn(! l ol op # * $ # l nm n$ l l m * $ l# * ' # l l o % $ %m. m # # $ %m. m # " o n %,nm$ # * 'ln(lm %l mnn $ * 'l $ % vl lol $n l n l #om l +/mlj xr $ ll# $ l#olon # % m # qv, l# 7 8 * %#n$!#ll #lol o l & pnm/. m % m 'l *m # * l o l p * % nl ' ol n l lo $ # $n $ l %m. m m #lo n # 4 # m #l3 * nmn( l# %,nm$ # 'll #ln l $ # m #l3 l l#ol on n l l m lo ll# $ # * % nm'lql l#r $! ' $% % $% #% ""$ $% %$!#%!/ ""$! $0 $!! $"! %$ %! %$$ #%!- l

89 ol n # m #lml#mll# / #lol n$ *l 4 *mn * / n # ol on n # l lm % l m # /n(l $ l# # # / % 4p % nl $ l# * ' lo # l m.l on # & n$l n # # gh j x kl*m n % l# # % $ 'lo * %# n$!#lol l n $mnn vl / $ l#op n # %#n$!#l mnn $ * 'll #lol o n & pnm/ %,nm$ #lon l p # * ' & % l # oml m lpl * lmml n #ol $nm l n o nmon n l 7 x l 'l * 8 # % ##% & # /lo # 8 " ' #lmo lol lol on n. l# ol l pl l # l n l n$$lo # $ %m. m # 4 # l %, ' p % 'l# #. * olmn # * )'l *nm 'l p % nl $ l# n$l n # # 7 8 lmml #o ol * ' lop # n %*l#o *ml n l # # )omn ' ** ' %l ol l ol 7 i 8l & * " & lol #lln m o ol n l * mn % % l lmml n, (m #" # lm$no) # $'l ' +/ % $ l# # % 'l# $$ %o lolll mll l ll nl#on oml lllol l $n # %l ' mol# # ' m.ol l n $s$ n$ * n # $'lqnm +/mlj ir 4l lmml o * # op # n$l l (ml nm * $ * # 'll 4lm m. % nl ' nm l ((l $ ' $ l# * l l ol nml $ l# & $ %m. *. 7 j 8 l ln " 3 * $l (mll l n % vl %l * o n %,nm$ # * 'll n$no %l m o + # o $ % m 7 8 * %#n$!#l o l m l mloml l m %% $ l# % l# * $ * l#3 % # n m l ml #oml l ololm n lmml # 7 9 " jt8l * $ 7 8 m # p # %,m # o vll l lmml l n n n$no # * mol# + #lol # op #l n m #ol lol o n * lmml l * %,nm$ l#ol /nml # * (l ' % & pnm/ %* m. l % % % n 440 lm /* -1+ nm /* -1+ # n$ l n * $ l#l# #ol# #"n ll lmml # o % #nm$.ol lmml ml# # nm$. 4l lmml m l#l#ol $no l o l n l #nm$. * % + m # ((l l ol $ &.n( l# m ol # $ q(rr vll ml % + $ %o #l qnm mlj qrrlmol qnm ml l lmml &o lm l l l $m % $ %m. m +/ p %l +/ l # l o (l *mn n l# + l 7j0 & lmo n m, " j8 8 n$nol# l# %,nm$ *l# # * # j!%$$% $% %$$% $ %%$%#$ "$" $%%" #"$-

90 9 n gh j i n # o n$l *% +'lop # $ % m l l nm(l $s$l nol l $ *%m 4 ml l# m nmn $n #ml * ol l # ll n nm % # * # 'ol $n ml ( # #l n ml ll m # l n ( # nm ' l# # lmml 4 nm(ll# *n # % # $s$ % n ( # ll,n #l n m # # 'olnoln l#n(lml %lr # n( l# nm # * m ' m n nm # n$l *% +'l *,(lq #lol# l % * % 'lllmmlol l n l $s$ n$ * n # ( ol l m l $ l#! n %" l m ' #o' % * /l 7j8qnm n m #l o l $ l# #ol lmml * n % mmlol l# p / ml +/mlj jqrr #ol# #nm$. " ** ml# # + 4nmol o l #lol9lm qnmj qrr vl o lm l# # 'l *nmol ml n ol *l!ml vpl n! #p ** ml#l# $ mn # ol# ) * * ll n n # # ' $l#% # 9/ # & nlm,nm l * %#n$!#l l mllol & l#o # # ll /nmlollmml &ol# + l 7j,nm n $ % " jx8 lol# + # ) # l lmml $nlm n * n #lml & % m ol#l p mnlol m l # * n # l ml#o l oln " ' $ * # m #l nmol n " l# #l * + %l" * * n( l# $ l ml nmop #l m n lol #ol# lmo n mll # * # %m'l # "! # nm$. n$nol# * %,nm$ # * m n (olol# lon n 7ji + # " j 8qnm +/mlj jq(rr # # # m 'l m nm n$ l mnl * $ l# $ % 'l l o *lol l lmml n *mnl % # &,n! o % ml# l# ** l# l" $! * o # ) $mnn * 'l lmml l n$no n l ll l ll m mn nm n ol o n $l# o %ml#, l# $ l# * * # %,nm$ # ' %l mo n * + # o $ % m " 'l nmm l ( l# % %l l ol# # o m n % % # #3 %o ol# + % lmmll n * # ol# l" * # n$l (mll (l olo n, 7j 8 l * $m/lolol# + #l m % o l 7 8! nm " n lml mlmol # & m, n # ) lm % $ 'll nm n$ l n 'lol lmml * $ l# % * % " lm * % % ( l lmml l,l # % * # nm$ 'll pl 3omll o " 3 & $n # l"*n(l ol m + % m 'l lop *mn ml o l ml nm * n$ n l llr o l lmml qo n $ l# * ' l l# + #l # l# ( % $ % % & * # )omn ' nl# * $m' *n(l vpl *nm l l %l ' #n n n l#mlm # $ #l# # #n # m l lmmlol % #--- 4

91 t ol kl*m gh j n #ol n % lmml l# # % $ 'l 7jj *l o l ml n 8olo % ml# ) + (lm %l p #ol# nm % +ml $ %o #l r q " mol q #lol9lm (rll# qr lmml gh j j vn *l o l n #l #ol#%l *nmol lmml lmml l. o lmml %ml# 1n nnon'll # # #nm$qr #nm$q(r $nm * l n m ' * % l#llon(l #%ms opnm,nm * l *mn n o n n$ *l# # & l# + #l * n # $'l lol l lmml * $ * n % l

92 0 % %& --.5,0-.,*/ 4 ),, /1 5, /0, -.,*/ ,2,4 ol l $nm 3rl 4 * lqolol# % / $ * # nm$ol lmml # *mn msnm n mlnm4l n % ) #% # o * %#n$!#lolol# + # $ % # 'l l *ml#oo # $ %m # * # $ * #l ol mln l#ol #n$(ml l ol vl n 4 m/l ol l l %m $nm # l 4l *ml o %. *%m$l# + * %# n$!#lolol# # *ml n!ml l ml l n * $ $ ) % $ ' # %%ll %l l $ 3 l# l# #ll ml m $ *%m * ol *mll ( 6 l j t 7 9 " "j9 " " 08 m.n $n m *nm $s$l l ol *ml n l # # % 'l" # # # ** ' %l"* $l#l n % %" * ol# + #l *nm $ #l l $s$ l3 l" * l ml & 'lt ll m/l o l#l $ n ** ' %l l % nm n $ *%m %l " * ol# + #l *nm 7 9 $ " j9 " " 08 4 l# #l + # $.$lm ** %l m l * lml ol t{ 7t8 # l ml pn n #op $ *%m $ ( #l" ( l# % # n # $mnn * 'l o l l ji $nm l ol# + $ % # 'l l#ol n $, # ' 'llm l 4p olol l t %l l ml %! $ * ol# + & $ *n(l *%m % $ ( #l l m n jx * nm * p #ol# # 7 0 " " "j0 j j " 4l m x i ol j " " " " " ll n 8n p l n n # op #ll ll# $l3o$ # 7 i " " 9 " " 08 %. *%ml# # # +m$ % 'l n #ol poo l ol m/l n $ $ l# l#, ol# + # p/lolo % ml#l *l mnn l q k 7 i8n 6mn o * $ # l 8r nmopl o l m/l 7 " 9 n m/l " ml $ l#l#l# l $l3o$ # $ # * l# % l# ' l" # $ l# * $ l# )omn ' l" * %#n$!#lol ol# + $n n p$nm # & #l *ml n mn 0t t #opl# # l ' p m lm #%l mlml &o l *ml n l %l oml % # * % #p # o * $n # %m l 79 x8n ) " 4l 8 m $l! *mn(! #l $no /lol %% % o n l n 7 % n p # o) # $' $n ll n # % ml l#% % #o * %# n$!#lolol# + # $ % 'lo # l $nm l n m/l * # $ l# )omn l 8n o l ' # $l# % l# # * %#n$!#lol 3 n lol m o m/l n m # l# + l#ml lm &o l t o l *ml n l # 3 2 # )omn ' ** ' *l#o # $ % lm(l 'p&ol *ml n o l vl m # 0 2 % %l vl3 'n'l nn & " # o lmlm % lm % lmll#mn,nm lo lol m # %*l#o # % ol o * l# n$'l n " ml#ol# n$ *ll#nmomlol #olmol n n on l. lm$loll n /m (lm # *%m$l# vl*l#o #" p $l %ml n l mlml l # o) # $' $n % l %l l# n $ $ # o m %lol lol m/l % /$$ $ l# $(l ll l $s$lolm n oml # * & %* #. * mn( % $ 'l m l l o n ol & %lm$ # # * %#n $!#l l * )' $ * *m llo n % o # & %,nm$ # $mnn $nm * 'l #ll" 3 * %#n$!#l mnn $ * 'l #lol o n n l i j #l 7 & pnm/ l# + # #l# nmln(m * " " " 8 " #ln n l mn/ml l #,nm$ # (lol# % m #,nm$ # * $ l# %nml l m oln %* # # l n & # o mlm % 'p p/ op #l n $nm lol n n pllol *ml n #ln m # n,nm$ # *, l3 # n %l" #"l# #,nm$ # lol# %!%$ $##$$$ #!#$#$%#$ $#"$ " 4 # $% #%%#"$%!-( "+! $ %%$%" )##$ $ " %$,! ##$ #"# #%$ - %!%$ #!#$#$# $$%!-(" $%#$ %$%$% $ " %$ %$ %- "$% #%$ -

93 o n %3 ol# + # n(lm %l *m ml n! 7 9 " 8 lm nm m # # vn$$l mnl 'lol l $nm ml n # #n # n #% lml # l ll $no n nm l# $ l# & % #o n$ l $ * $ % # l l lo *nmol l n n * *nm ** $no # $mnn * 'l on n "#n * # l# #l ** #,nm$$ % **% % lm 'lol * %l# $ % # $l. n # # # * m! 'l ' l (ml,m **l o l $ % # 'lol $ l. n # # " n #,nm$l l m %3 m *m ol n mlol *m #l * % l l#% 'lol % % $$ m *nm l#,nm$lop # m! * % % l#ll# # ) m + m. * % % ol# # m ll $no n % # # %&%' -)) *352,0-.,*/ 31 * )*.4 4/. )2-0., ,-1+ l omlol n ol l n m/l #lm $l#ol #!ml p *l smlml*m m **mn.$ # $. # # lmopnmoml " n n #x $ l# ** ' % & # % $ % l#% * **l lmol % l# o n l # l# # m #ll n "mo lolo$l#!ml ##ll m/l #n l $ l# % vl*l#o l n$$ln σ # n $ % /l# * $s$ $ # & # $ $ # %m. pm **l ((l # %"n *l *nlm m/l l l m l # $ # l mm $ l# )omn ' % % # $ " % # # * + " nm 'p # lmmlol# + % lm(l n l l m/l o nmolml*m l#lm l nm $ l# $s$ $ % n$ l m l ml * $ l#op $ % m * ( op # m! l# ml"o 8 m # l# vn n pml nm * %" ' #n # l m/l l#on l o'lm " * # $ ##%" $ %m (! #l %,nm$ # * 'ln % ' l # o mn l lmol n lm$l n % # l# # m #l 8 n$ 9 n$ * n #l o/n ml*!mlnm n l )$ # # # nm$ %m'l 'l n *lon # 'l" n$ # jn$ * n #l l o # x #.l mn lm$l l. m 3 o/n n l lml*!mlol #. # * lo # # m #l *m # * l "n l # n( l# l# mo/n vl m l &om l l lm pl ol n # l# ' l# # m #lml *l smlo l % m " n n ol n n m l #, ( m mn # #nml# #" * $ * m$! l# 2 m$ l ml ml "n mn * m$ m nmml! *n o l # n mlo l# l p #l pl m n$ m # # m # l# nm * " l"op * " # & pn( l * )' ' * n #l m/l 4lol ml )omn' lo $ l#. n$ * n #l nmml *n # ol# % + # nm #l #nm$l *nmm$l3 #lm & n$ * n l m/l #o 4l o ll o l nm$l #ll# ml $ l#o. $ l# * l m$! & # ml" n$$lpl l nm ml n. * * # )omn ' # l * l %l n #nm$lolml o l $ * n) # " % + # n$$ll $.$ $ +%$%!% #$% #$% " - + $- $$"$% $ $$%, $ #$% $## #"$%!- %" $## %+%$ $%! $ "$!% $ $ " %$- #$% % %!% $ %"$ $ #$# $$%! #!#$ " %$ p = 1 ' Tr(σ)- 3 S $% "+%$%! #!#$ " % S = σ 1 3 Tr(σ)I-

94 x *-,!" # %&' ( ) * +, *., * -,!" *., *+, $ %&' # )( /2$2 /0 1.2$ * 4! * % 45% 6$2 442$2 2 4( # 3 $ % 435 # * * % 6$ * 6 % # # * 426 # )346 '2p2 2$ # 6 % '2 2 # τ89 # 35 6 # $ %4 4 #" # # # 4! 4/2$2 # 2 8 ( 42$ * # 3 # # *% * % 2 6 # + 2 # # * # )346 '2 #! * # # # 6$ * 6 *426 # 2$2 8 # 2 #% $ $2 * % * / # 6$ * # * ' % 9

95 32 2$2 6 2 #" 6 64$ # # " J(σ) " ' 2 #, # # 4 # # 2τ * % # # 2$ /2$2 #" # * # & # 6 3 # 6 2!42 6 * 28p τ9 # /2$2 35 $232 % # $ # # # * pτ # 6 % # # 6 $ $ #3 4( # * 6442 * # 34 # # 2 * 6 # 3 * # 8p τ9 % ! * % 6$$ /2$2 # % # 3 $ % 4 # 3 %,64$ # * '2 # 35 6 # $ %4 % * '26 42 # 2 # %* 64 * 4/2$2 τ /2$2 $ #8p " # 5 2 % #%42432 # 3 )! " # 32. * 2 % # % '2 & 4, /2$2 2 % ,6 6 2$ # # 4/2$2 243 & 52. %424 * 6 # # ## # 3 )! $28 # % 64 # # 92 3 % #3 &32 4,2 4 ' $2 " 64 2 # 4 # 2 (2 * ) #5! # * 5% 62$2 # * '2 2 $2 642$2 )! # 6$ * # % ' # " % 4 * # # 32 4/2$ * # % * ' '2 # 6 6 # 6 43 # " 2 % # % & 52. % 4,2 4 * 22 # # '2 * $2 * )! $2 8 # (4! #,64$$ % # " # '2 * $ % 2 64'2 2 2$2 # ** ' % $ % 4322 # $$2 # % 2 # $62$ # 236 # # *2 %,64$22$ # # 3 9 # % 6 J(σ)2 # 4 # # 2 " 3 5 ' % 4 #2 # % 3 2$2 2 #" τ 2 c # 3 # * 6 2 % 22$2 # * &342 τ # #3 8p " # * 2 * 2 6 % # # 3 # 2 6$$2 4,235% ' # f(σ) = 0 " 52 * / c864 # # ##% & 564 # 5 τc2 2 2τ 3 +/421 < )! $22 % '2 64'2f(σ) = 0 " #2 # % J(σ)3 2$2 2 # 3 # $ % 4 2 # 24 4 ' 2 2τ c " # $ % & 5% 32 * 4/2$2 '2$2 # 3 4! * % % 4 5% # #% $2 * / %#% # % 6 # 6 * 32 '2 2 2, # 3 $ % 4 " # # # #,64$ # * # % 62$2 22 ' # & + # # # 463 ", 6 # / %#% 42 6 " # 2 g(σ) 2 '2 # * 9 3 dε P = dλ g(σ) 6 5%#24/2 24 P σ = dλpn 81 dλ P2 ** 2 % $ * * '22 24 "* 6 26 # " 2 $ * 6 * *6 '23 2 *% 2 % * 6 )! '2 2 6 $ % 4 # "6 62 % % 2 # # ** ' # ' $$2 7 8 f(σ) 8 = J(σ) τc = 0 81 ( 99 # # " f(σ) < 0" 64 J(σ) " %$ J(σ) $% J(σ) = 3 2S : S- $ %+%$ $% % " #$"%! "$$%!$ " %$ $ "%#$ σ - $$ $%$$"#$,$ % $% ""#$% $%$-(! "+%, $ %"+%, $# ## #% $ %! #$ #$#$ +% "$%! +% $%!- #$#$ * $ +% "$$ $%!- #!#$+$ $ $$%$ $# % #$"% %"" #$"% $ $ % % $+%$%!#%#% % %- %

96 62$2 35% 6 22 # 2 # 64$2 & , ! 643 * ' ) # * *! # % 4 # g(σ) = f(σ) # *2 % 342 # 62$2 35% * 23 $ % 4 & * 43 4! * % # 64$2 & 4,2 # 4 ' # 3 4! # # * % 6 % " n2 * 4! &.2τ " # 3 %,64$ #6( # * '22 # 2$2 # * 4 # + ' $242 & 4 ' * * 6 32 $ % 4. # $ 235% # %,64$ # * '2 /$2 # 2 **22 2 * % %# % 2 6$ 5% 6/2 8 5%,64$22$ # # 6 # # %* # 2 3 # ! * '2!# # 24 # * 24 * 4$! 4235% 3 )! $2 # 2 # 3 6 %,64$ 4 2 ' 3 % '2 2 # ( * $ % 4 " # εp 45% 6/ /4 * '2$2 * 4 # %* 2$ # # * 6/2 6435% #% $ '296 6/ # 22. * # # # * # 6435% * 4,2 2 4 '23 # * # pτ %&% ( /,.4 34 )2-0.,,.5 )* ,2, '2323 # % # $ # # #,64$232 % $ * 4'26( # 4 2 $64 * 23 # * % 24$ # # 4! * % 42 dε P = dλ P f(σ) 81 9 σ = dλp n 6 * 4 # * 2 4 % 4 ' 2 & # $64 % # # * # % 24(2$2 4/2$2 ' # 6 # # )346 # #3 6 2$2 232 # & # # * 426 '236 # )346 ##% $ * * %#6$!# # + # 6 32 * %# 6$!# # + # 2 $ * 64' # 5% # # 52 # # % * " # (2 235,6 6 # # 4$ # 2. *% 4$2 # * # 2 * '2g(σ) 2 # 422$2 # $ $24 #% & ** " $6 # 3 # # * 2$ * " '2 " % # 2 $ * 22 6 % # 6$ * 0 ", ' ! * * 4/2$ # '2 #242 # 2 * # 6 # /232p0 2 * " 6 # '2 /2 2 42$2 # * # )346 '2 " 3 6 %,64$ # * '2 2 # * $ # # % & 4'2 4,2 4 ' * 6 # #242 # 2 * # 36 6$$2 5 ) *6! 235 # 6$ * 64 2$2 # * '2 6 % $ * " % "! $$ $+%$%!- "$$ $%$!%+! " $$%! $ %!% "+ +%$ g = f ε P! 2 3 ( εp : ε P)

97 # 2 6 * # pτ # 3 42 # 62$2 35% # 2 ( # 23 # # 2 " 43 %,64$ $2 # * # 6$ % 4'2 4/2$2 6 # # )346 '2 # $ * ' p p2 τ32 $7 4! 6 %* # 3 # 32τ % 4 2 * 22 ** & 342 " #2 6 '236 & $2 / #2 # * # # % ##% * 2 # * # 4,2 4 '2 2 6 # 6 # 3 2$2 2 #" 6 # 6 # " * # )346 6 '2 # + #" # + # # % 3 # * ", " #, ! * % % * # # 4! 6 % # # + # 2 # ) * " 6 '2 # (2 # ' * % 6 % # " # * 2 % $ # 3 # * # 4f(σ) 6$$2 f(σ) = φ(p, τ) Y φ φ > 0 p τ f(p = 10 GPa; J(σ) = 0) 0 Tr(ε P ) $63 & ,64$23 # # % ** # % * 46(! 6 24 * 32 6 ' # 22$2 $7 %42 2 # $2 43 # * 4 # * 2 * 462! $2 % # 5% 6 # $ ! * '22 32 $ % *% 4$2 # 2 # % 2 * 2 * 4$! 423! 2 2 " *42$ ! * * 6 # ! * 2 5 %* $64 32 * 22 &8$6 * # # 6$ * #1 # * 426 # )346 '2 4 # 4 # $ 62$ % # # 2$2 #" # * * 6 % 4! * ' # 2 f(σ) = 3 ( 2 α Tr(σ) ) 2 α (J(σ) Y ) 6 αɛ[ 2/3; 0]2 ** 2 %, # + # Y2 $ 235% % ##% * ! 6 # '2α2 99 % / & " 4! 42f(σ) ! # # " 64'2α= 2/3 " # 6( # # 4! * # % 2$ # # %* # 3 # 6$ * 6 # 2$2 4/2$2 # 3 # * 4 4$ 2$242 24$ ! 24$ α * # 6$ * # * 6 $6 # # 64$23! * # /2$2 6$ * # + # 3 $ % 4 6 # 2442 * # )346 '280 3, '2 2 2$ #, # $ # # 6$ * # % 2 * ** 6 % % * '2 2 * % 4 * # 6/2 6 % 46/2 $63235% " H " # # 9 8, 81 %%! % # %! %! $"% %$ $-% %$" $%%$" %% # $ %!# $# "$- # $# 8 9

98 2 $ # ' * 4$! 2642$2 $ $ * # % % # 6( # 6$$2 6 # / E" # # ν # % $2422. *% 4$ #3 H " 232 ' ** 6 % # #" 2 $ % % & * $242 3* $ *% 4 " # *2 % '2'24 % * 6 # '2 3, 4/2$ $242 4 *42$! % 323 #,( / #,64$ # 3 * # (2 4 % 6$ % 423 # ** ' % # 3 4 % % ' $ #. * 2 64 #52 24 * 4.! % * α #2 # # 23 $63235% 6/2H # * 6 2 * # 6$ * 52. # * %#6$!# 4 # # + # # Y # % % * # ) # # 6$ * 4 # % 243 * * 4 2 6( $2 # 2. *% 4$2 # 2$ #2 6 # * # # # 2 % $ # 3 * % % # $2 + # 2 % # # % # ' , # + # α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τe = S : S " ζ > 6442 σ 2 0 * # # 42 * 2 2$2 # α2 Y328$ %2 ' # 4 # 22 # 6$( # # #% *426 # )346 '22 3 2$2 2 2 # ** ' % # * 42$ * $ ! % " 4 $(46 ** # # 6$ * # % * '2 2 # % 46/2 6 # # 3 f(σ) = 1 3 Tr(σ) + ζτ e σ " ' ( $#%! % " $ $ "$%" %$%,$% #%$ - % ( %$ % $$ %% $! "#$% $ # %%-!! % % ' "!% %% %+%$" %,$" #$ % / " (, (,0 %"$" "," %

99 8 46$$28$ * # 3 #" 2 * 4$! 423 4! # % % 4 ( *% # 2 23 # /2$2 6 2'22 / % 6$ % # # * 2 $2. % 6( # # & # * % ' 2$ # % #22. *% ## 2 # 42 *6/2 4 # % # 6 32 # 2 $ #24 5 * 2 + # 642)2 2$2 # 32 * 4$! 422. *% 4$2 # $ ( # ' #! * #, " 6 0 * # 4 $( * 66 * # # # # ) # # 6$$ (232 4/23 4! 3 2 % 4/26( ** 2 2 # # 6 # # # # % # 6 32 # 2 $ * 4! # 3 4 % % 2 # 2 %% 4 # f(σ) = α 6 $ $6 2 αɛ[0; 1] Y # # % '2 $7 $2 2 # & 2.32 * 4$! 423 #6$3 # " # 64 (2 # % 32$$2 $ # 2 $2 4 # $ % 4'2! * 42 2 # 6$ * 2$(2 2 2,6 * '2 * 4, $ 6 % % 24 # 6624$ * # * # 3 # # 6$ * 4 + #6 ** # '2 5% 62$2 ' # * # 36 % $6 # *! # # #" # + 2 # $ #! ** " # 4 $( * $2 32,64 # 2α= $2 Y = 5.43 * 35 * #2 # 2 4 $ # * % % # + # * # # # % 2 # # # #% * % #% # 3 Tr(σ) + (1 α)τ e Y # #6 # # #6(242 4 # 32 # $ # $ % $2 46 % 4 * 2 2 # * * 6 % ' / # + %, 6 2 # 6$6/!# & $ ** 4623 * 46(! 6! # # '2 # ,6 4 $ 35#32 6 & # 2. *% 4$2 # 2 #22 64 % 632 * $ * 32 2$2 4$! #2 6$ * 2 64(2 # # 3 ' / %#% % 6$ * # # + % & * $ # # *46+35#32 # #" '2 5#32 6 4/2$ # # # 3 # 4 * # 2 2 # 6$6/ 4 2!# # * # " # ** # 6 )2 2 #, ' ,64 ##% 22. *% 4$2 # 24 % 62 " *%#% 4 # ** ' % # # * # #,64$ % 62 * 2$ # 22 # # #, ' $ #,2 # 6. $ % # * 2$2 # #,642 ** ' % 22 5 #% / # 4 # 2 62 ** ' % 2 4 4,22 6 *%#%4 2 # #% / $7 $ %,64$ 4/2$2 # # 62 2 z # 6 35 #2 6 6$6/ 6' !# " 6$6/ %#% % # $ * # 242 # ( # # 2 # 6(24(22. *% 4$2 # 2 #% / % '2 # $2 #" #! # "!$!! "! "

100 /0 1 6/4 * $ % # % *! # # 6 # # # 32 # $ 2 # % # 6 32 # 2 $ * 2 % # 6( # 2 2 $63 2 # # 4 $(46 6 # * 66 * + % 22 # + %, #! # 6$6/!# 2 4

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cos(νt)2 6 $2 # # 24 # 2 # ) #23 6 '235 $ ,4 * 6 # # (4 # * * % '2 # 2ν S '2ν 2 ν S! 42 # 3 * % 2 4'2P # 3 * 42 $ * 3 # )! $22 2. * 4$ % * # $2 6 6 α2 # * 64( % 3 )! # % % ** # '2 2 $63232 ( # # $2 % % 346 ) 64! # 3 # * ( # $ * 64( % # '2 2 42$ $23 * 4( # 2 * * ## 2 & cos(νs t) $ * 3 % % # 6 * * ## $2 $6 % ** # * 42$ α ij P = α E 81 9

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132 6;23 3=3/ / ,-452/ ,02, 12.-/3,- -/ / 343/630-,02,34,;343/ /-3/3D= 3/./3-4 >2-, /.563,;.043/63654-/.5 d σ 124,;.043/6305/4-./33 d ε,31/ 2/., /.34,-6.430/.563,;.043/63654-/ /3,,2. 134/ ,- /-.,, /63654-/.5 /-./ < 121-3/ 3/./33 ()) '-,#',-!(!)*-).$-'%! ;-,54./363,5 56,.134,30554/33/ -33=3,-1/5,-1/ /45.1 / /31,2136/-.,3 9,-' /43, n3/,362/ / -,541 n ε 2.,;/-/ n 6305/4-./3σ n,;2.,.43,;/-/ (γ; γ )63,;,3/3/,;.043/63654-/.5 -,.2 εn+13,3, 521-2/6/34.34,- N 2.,.4-/ 523, 05/4-./3σ n+1 /-/ / ,- 1!1/"3ε -, /.516;/-/ n / ;-,54./331/ 1/420/ / #8-1, !1/"3431/3,;,3/ --./ /33/,-1/ ,02,,;-,54./36/34.3,;-, ,-1/.233,30554/33/62 n -1,30-15., /,-1/ /56363.,6/34.3-,541,;/-/6305/4-./362 1!1/" / / -4./4-/.53,-654-/.5,-1/5,-1/.233/ 2/.,.1-/,- $3%/ / ( ),- -,32462 f σ n+1,3 2,/.,.0-/ /5/-,363,;,3/3,-1/.23 λ n+131/ 0-,02, 5/3.4,-654-/ /45.1."3/31 &-05.3 D31/ 0-,02, ,316;/-/63,;,3/ 15/ //3-4/ ()) '+-,#-'$"(!)*-).$-'%! ,.634,3 () *+, /.533/63 5/43-,54./3 521 / //-/ "43,3 1., ,!/.233/,31412,/-/ /,-036-1,30-1 6; , ,33/ / -.,,-312.-/ 2-051/./263 < -.,, /.51-2,../ /, / < 83 /.513,36,-033/ 13,5,; ,-632 / /.51631!/ <83/ =30. ;31/ // ;., ;! - 45,"3-4/.02,.34-30, /.51631!/4.313, / / 0Z[:@>1L1C=<: ;DWW: UDA<B=@ GE<

133 <<! " #$%&'()*+,-*&./*)01+( ! "=>6 >?> 9: 9 ;!< "=: 7B 0 E23"=4E5 0 7? 8! 9! FG 10-/.2363,;-,54./3>! 33# 13413/-/.5

134 < 33 -.,,-33/ 056./.51-2,../31 2/., ,- -,.6-/.563,-452/.3 >! ( ) -. =5/3/3263,- 5/ /5 32/ σ = σ 1 e 1 e 1 733#8 7338,-,../36;,-1/.0./5-05/4-./3,../363 f(σ lim ) = (1 + γ 3 )σlim 1 τ c = 0 -,-1/.0./ ,331/650 σ lim 1 3 = τc 1+ γ 3 3/-, /3-4-,,",3,31 -,324163σ 5/ ,-/ / 1 --,!/.233/ /31 03 / 051/-/3 -,324163γ lim 23, ,-/.31 15/ /4"1 -.,313=31412,/-/1 -,.63/ 5/43-,54./ , ,!/ ,-/ γ σ1 lim σ lim 6 33<63 33<63 3/6 63/ 3 / 3 3 3/3/ 3/3/ /3 / 63# 3# 3# < /3 633/3 / /3 / 33 = "43 -,324163σ,!/.233/ /31 -,324163γ /3-/ lim -- 5/ ,-/ / 1,;052,33/,-1/.23,-,-1/.0./

135 <# ε p = λ p f σ 3, --4- / -, ε p = ε p 1e 1 e 1 + ε p 2e 2 e 2 + ε p 2e 3 e 3, -30 ε p 1 = λ p (1 + γ 3 ) ε p 2 = λ p ( 1 2 γ 3 ) / /63,-654-/.563,- 130/.5 124,-654-/.5,5./26.-,331/ -, ,33/636-/63γ3/ 1;04./ ε p 2 23 ε p = 3 2γ γ / 051/-/3 23 2,,3 3=3, //3 524γ= / -,32463γ ,3,- 130/.563,;0-/.,,5431/3,- εp , ,; , γ,34-54/31/ 51./..,.2-/ 2;3 1.,3,3,3 -/ / 231/4.0/.56-1,- 54-,,; = ,/-/31/ /43./2./.3 / 32/./.3403// /-/-/ /4.-2,-1/ "43,3-4-./ 2.,3 56",3 23 ;31/ -1,30-163,- 1., =336-/ /,; , γ0 -/ ,3,; / > / / 231/4.0/.56-1,- 54-, < 3/-,3-2 3/3-4-,,",3,314-54/1 ε p -41.2,-/ /ε15/321 3/ --,!/.233/ /31 / 051/-/3 p -,324163γ 23, ,-/.315/ /4"1 -.,313=31412,/-/1 -,.63/ 5/43-,54./ , ,!/ ,-/.3 3 γ ε p 2/ε p 1 ε p 2/ε p 1 3< / 3 63< 63< 636# 636# / << 63< < 63< < # 63 < < 63 < <# 3</ # 63# < <3#6/ -41.2,-/ / / --,! /εp 33< = /1 ε p /.233/ / #6/ 63#6 /336/ -,324163γ 0!#"! "!! $ 21 1 / 1.0., 1 12,/-/1, / , ,-/. 1. /-/ , / ,/-/1 Z[:@> EA@@= F: ;E@ GDA< αi N CEB@ F: ;E@ CA WDC?F: C: ^=B :> 9EW_<DGDAFD@ I

136 1 / 2 2-,./-/.13,1 / / 0 -/ 2,-.1 0 / / / , , 1 12,/-/1. /-23 +! #-&'$"(,%-))-.! ())* 0-,02,3,2/ / 2 -,.1..-/ / 2 1.2,-/. 2. /-/ ,-/. -, / /-/. 1/ 2 0-,02, 0 / 2 / , , / 2. / , -2 1 / /63 o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

137 ()) +,)'-'!'#$%%!"'--! 1 12,/-/1 / 21 1 / ,- 0 0,-,./ 0-,02, /-/ -, 2 21 / / "1 1 1., -2.4 /1 - -"/ 1,- 1 2 ", 3 -,2- / 1 0-,02,1 -!-/ 0 / / -./1 2 γ α 63-2, 13,21 2 /./ 2 1 1/,.11-1,- --/., , / ρ/ρ / Tr(ε )3.1.,-,-/ /./2,-,-/. 12.-/! 333 P γ 0 γ = 1 + γ 0 ( ) A 1 1 τ c A 2 (Tr(ε P )) A / -1, / 1 1/ 0,21 -., 2., -2 -./,0/ 3 -/, 1 12,/-/1 / 21 /-/ 0 1/- 11 /.,, / 2-,./-/.1,.-0/ 0 // ,--,!1 1 12,/-/13 () 0! / 1 /, /-/. / 2 1 2, 1 -, / 1 3 / /-/. 1/ 3 γ µ 1 A 3 Force [N] γ =1.2 γ =1.3 γ =1.5 γ =1.6 Courbe expérimentale Pénétration [µm] 33/ 2 1. /-/ , / 1 -, 2 1 γ /-, 3 /. 2, / / -1, /-, 3,21, , 1 1 / 1./ , /-, 3

138 / -./ 0 12,/-/ /-./.1., 0 / / 2 1 -, / , 2 1 γ / "1,- -, 2 α / - 2, / ,21 0,- -, 2 / 2-0 α 63 / 3# - Y 1/., ,-/. 1 p Y 0. /, ",. / - 2, 1 /, 2, /! -5 2/ / -5 2/-/, / 1 / 2,-,../,-1/.0./,- 1.,.0-1.2, 1./ / ,21, 2 0,, 2, /. -./ - 0 2,. 0 / / / 0 2,2 1./, / /. 1,- 0 -/ , 1 2 -/.-2, /, / ,.2,21. /-/ -1, ,-.,.2 2,- -, 2 γ //-/ -521/,-,. 0 / / ,- 1.,.0-12, / 2. /-/. 1/,21 -., 2 0,, / - 2, 13 12,/-/ 1, / ", / 0 / 12, 1.,.0-/. 1 γ0 > / / , 2 1 γ / 63 / 63 1, ,/-/1 1 / 0 -/ ,- 12, 1 1.2,-/ /-. / 2 - / ( ) /, 1 0- / 1 1./ / 2 1-0, 1 -, 2 1 γ0 12.-/ 1! 30 / 3 3 #., /. / / ,./-/ ,21 -',, 1 / / 0 -/. 2,-5 2/ / / / 1./ 1. 2,, -!-/ ,21 0-0, /3#3 4 /, 1.0-/. 1 -./ /,-, / - 2, 1 // / 0 1, , 1, / 2/ - -/,- 2 1-/ ,-./,. / /-./ -1, , ", "! $ 21 1 / 0 /, , /.1-/ / /.,, ,2.--/ 2-./ 0/.1 0 / -1, ,-/ ,- / 1, /1.13 $ 21-1,21 0./ 2 -,./ //-/ -,.1 2, ,-/ ", / 1, 1 12,/-/1 / 21 1 / 2-,./-/.1 / / "1,../13,1..2 / 0 -/ 2,-.1 0 / / 2 2, -521/ / 1 0 1/-/ 1 1 / 1-1, ",. / - 2, 1 / / /,-,,2 -, 1/, ,/-/1 / / / - /2 1, / "1 // 2 / - 0/

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141 S!"# $ ' () %'%( % &* -./01 ++! -./ (& % %, 01 -./01 (+&+ +'+ (('!)*, % & 23 = G E(1 ν) C11 D G 6 = (1 + ν)(1 2ν) E G = GH J E B = G HK O 2(1 + KM N ν) HHNJ 8 3(1 2ν) 9: F@ G@GG S < BFBI JKLMNOPQRSR BFBEBFBGBFBH TUVWXY BFB Z[\UVWXY BCBDCDBECE 789:;<=>?7@A 23 = HH ρ G B GH M K A ] JK A KM N 89=: T K A D ^ JK A # K AJ L @ _D GH J G HK C G`G H H JJ @ J6 O JO D G GJ H J

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149 SC

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