On a Simplified Model for Numerical Simulation of Wear During Dry Rolling Contacts

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Submitted on 23 Jan 2013

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On a Simplified Model for Numerical Simulation of Wear During Dry Rolling Contacts

Luc Chevalier, A. Heddahak-Ouni, S. Cloupet

To cite this version:

Luc Chevalier, A. Heddahak-Ouni, S. Cloupet. On a Simplified Model for Numerical Simulation of

Wear During Dry Rolling Contacts. Journal of Tribology, American Society of Mechanical Engineers,

2009, 131 (1), pp.1-15. �10.1115/1.3002322�. �hal-00777136�

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On a Simplified Model for Numerical Simulation of Wear during Rolling Contacts

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1. Introduction

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11 11 11

π ν

= −

2 2 3 4 5 5

6

7 τ

8 9 A B C

D − ν + ν − +

− τ

− ν

= π

2 2 3 4 5 5

6

7 τ + ν − − τ

8 9 A B C

D − ν + ν −

= π

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+ ν

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=

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2. Normal problem: Semi Hertzian Approach with Diffusion

2.1 Semi Hertzian approach

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2.1.1 Some results of the Hertz theory

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+

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> G

= $

8 8$

= − −

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( )

^MG

G M

>

=

G F #

B 22 +

3 4 55 6

7 −

= υ

δ ^{9 9 =6>9}

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2.1.2 Semi Hertzian method

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F #

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A

F

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5 !

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A A

A

+ ∆

= −

>

=

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λ

υ ^{9 9 =6>9}

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A

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2.1.3 Results and discussion

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2.2 Semi Hertzian Approach with Diffusion (SHAD)

-"7D959FB3EF9BD6E69EC7E98547D#5BD#9EC79BCC769979'CB'B#796B95D6CB8279B2'F5D9376977D9

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2.2.1 Diffuse element method

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259 19 5FF2#6CE67#9 6&79 "EC5E65BD9 B9 4D9 9&7D9 6&79 2C"E62C79 F9 &ED7#9 879 DB6579 6&E63 4D3 3 5#9

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2.2.2 Several critical cases of curvature B values

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3. Tangential problem and wear simulation

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9 =G>9

9799EC79C7FE6789956&9 τ 9 τ 9ED899396&797FE#65937&E"5B2C9FE9#9B96&7969B93B857#9E#9

'C7#7D67895D9-:9BC9:2E#50587D65EF97FE#6593B857#9*D9#765BD99979&E"79'C7#7D67896&791;.9 476&B896B9#BF"796&79DBC4EF9'CB3F749+7)6997995FF9'C7#7D696&79#54'F55789E''CBE&96B9#BF"79 6&796ED7D65EF9BD79

3.1.1 Fastsim extension

.D9EF67CDE65"796B96&79'CB&53565"79%I!965479D778789396&795D565EF9E''CBE&95#96B92#796&792E#6#549 EFBC56&49E#987"7FB'78939@EFA7C976?#97)E45D796&79'CB'B#789#54'F55E65BD#496&79FBEF9

#65D7##95#94B87FF789E#9E9F5D7EC9C7FE65BD9376977D96ED7D65EF96CE65BD9ED89C7FE65"7985#'FE747D6#9 9ED8979=579 3312 τ 29ED897 3312 τ >9&7919"EF27987'7D8#9B94 _A0 9B7557D6#9#&7EC97FE#659 4B82F2#99ED897FF5'#798547D#5BD#989ED89$9326997942#695D6CB82796&C779"EF27#96B97D#2C796&E69 6&796B6EF96ED7D65EF9BC79B4'BD7D6#91 9ED891 9B95D7EC9@EFA7C9&7BC9C6D9EC79587D65EF95D9 6&79E#79B9FB3EF9E8&7#5BD49

4

8 $ 8 1

4 1 8 4 1 8

G G

(

G 6

G

6 π

=

=

= 9 9 9 9 =G>9

&79#54'F55789'CB3F7496B9#BF"795#97)'C7##789396&79C7FE65BD#9B9-:GG9*695#97E#96B9#7796&E69

BC9ED97FF5'659'C7##2C7985#6C53265BD96&79#F5'9B4'BD7D6995FF9F7E896B95D5D569E696&796CE5F5D97879

B96&79BD6E69&5#95#95D9BD6CE8565BD9956&9259(9ED89BD79ED9#BF"796&5#9'CB3F7492#5D9E9

'ECE3BF59'C7##2C7985#6C53265BD9&5#95#9ED9E''CB)54E65BD96&E695DC7E#7#96&794E)54249'C7##2C79 / 9BC9$\932695"7#9BB89EC7747D695D96&796ED7D65EF9'CB3F749E#79

F F B FF C D

∂

− ∂ +

=

∂

− ∂

−

=

1

1 71

1

1 71

φ τ υ

τ φ υ

G

G

9 9 9 9 9 9 9 =GG>9

195#96&7947ED9F7)535F569B36E5D78939975&65D97E&9F7)535F5691 _A 9=5S96B9G>9396&79E##B5E6789 C77'E79"EF279+B96&7BC765EF9U2#655E65BD9ED9379'CB'B#789BC96&79875D565BD9B9-:G(99

G

1 1 8$ 1 1

8$

υ υ φ

υ υ φ

+ +

= + +

9 9 9 9 9 9 9 =G(>9

1BF"5D96&79##6749B9-:9GG9F7E8#96B9ED9EDEF65EF9#BF265BD99&7D9#'5D95#9D7F767899

3.1.2 Comparison with the complete approach

2E#6#549EFBC56&49#BF"7#96&796ED7D65EF9'CB3F7495D96&79E#79B9;7C6<5ED93B857#99*6?#996547#9 :25A7C96&ED9ED9EFBC56&492#7896B9#BF"796&79B4'F7679CBFF5D9BD6E69'CB3F749&5#99E#9 'CB"78939B4'EC5#BD9956&96&79E''CBE&789C7#2F6#9B36E5D789392E#6#549ED89E9#B69EC796&E69

#BF"7#96&797)E696&7BC9F5A79Q%BD6E6?9BC97)E4'F79C1D9%BD#7:27D6F997995FF92#796&792E#6#549 E''CB)54E65BD95D#67E89B96&797)E69##67495D99&E69BFFB9#9

87EC97D7CE6789E697E&965479#67'94B8557#96&792C"E62C7#9ED897D7CE67#985#BD65D25657#99

&797)E4'F79B925995#9C7'C7#7D6E65"79B96&79'CB3F7496B9#BF"7982C5D96&7997EC9#542FE65BD9 'CB7##9IFB6#9B9256919ED892599F7ECF9#&B996&79BB89BCC7FE65BD9376977D96&79

#BF265BD9B9Q%BD6E6?9ED896&791;.9F92E#6#549E''CBE&7#9E#997FF9BC96&79BD6E69'E6&9ED89

'C7##2C7985#6C53265BD96&ED9BC9#F5'9"7FB569BC96ED7D65EF96CE65BD#985#6C53265BD999

3.2 Wear simulation

3.2.1 Archard’s wear model

.94B87F9BC94E67C5EF9FB##982796B96&79F59CBFF5D9BD6E69FBE85D95#9'CB"5878939.C&EC8?#9 FE99 CD9 .9 #545FEC9 BC49 9E#9 'CB'B#789 39 Z59 59 F9 @EFA7C9 CD9 ED89 6&79 4E6&74E65EF9 7)'C7##5BD95#95"7D939-:G$9

8 11

9 = ) 9 9 9 9 9 9 9 9 9 =G$>9

93=4

^G

>95#96&799BCD9"BF247913=4>95#96&79#F585D9F7D6&9B96&79E3CE#5"79'EC65F7913=+>95#96&796ED7D65EF9FBE89ED8983

=+M4

>95#96&794E67C5EF9&EC8D7##9B96&79#B67C94E67C5EF9=259>95CB0&EC8D7##9&E"79377D98BD79ED8989S99;"9

=IE>9*D9-:G$9)95#9E98547D#5BDF7##9B7557D69&ECE67C5#659B96&794E67C5EF#95D9BD6E69154'F7945CB#B'59 5D67C'C76E65BD9B96&79O.C&EC8?#9E6BCP9)95#9C7:27D6F95"7D9=#77927F87C9CD9BC92CED^B5#9F9EF9CGD9BC97)E4'F7>9 9&5&9#27#6#9E9C7FE65BD9376977D96&79E6BC9)9ED896&798DE459C565BD9B7557D69µ9*D96&E699E9E9FB99

8DE459C565BD9B7557D69F7E8#96B9"7C9#4EFF997EC937E2#79B9#F5&6985##5'E6789'B97C982C5D9#F5'9B2'F789956&9 E9#4EFF997EC9E6BC987995FF967#696&5#9E##24'65BD95D96&79BFFB95D9

ED7D65EF985##5'E67899BCA95#9EF2FE6789396&79'CB8269196547#919*D9ED95D#6ED6ED7B2#9BC49B96&5#94B87F96&79 97EC9CE679=4

^G

M#>95#985C76F9'CB'BC65BDEF96B96&7985##5'E6789'B97C9/ 9E#9875D7895D9-:G59

/

8 9 2 = )

9 9 9 9 9 9 9 9 9 =G5>9

/ 95#9DB692D5BC4F985#6C5326789B"7C96&79BD6E69EC7E9ED895695#9D77##EC96B9EF2FE6796&796ED7D65EF9#2CE706CE65BD9 85#6C53265BD9τ9ED896&79#F585D9"7FB569

:

9E697E&9'B5D69BD96&79BD6E69EC7E96B9#'7596&7985#6C53265BD9B985##5'E6789 7D7C9'7C92D569#2CE79879875D796&7997EC987'6&9CE679 2 =4M#>939-:G9

8 )

9

= 2 = τ

2 _{9 9 9 9 9 9 9 =G>9}

2C5D9E9#5DF79'E##9B96&79CBFF7C9BD96&79E496CEA96&795DC747D69B96&7997EC987'6&95#9B36E5D78939ED95D67CE65BD9

B"7C96&7965479=6>9B96&7997EC987'6&9CE679CB49<7CB96B9∆3338.&;799&7C79795#96&79CBFF5D9"7FB569ED898=>95#96&79

F7D6&9B96&79BD6E69#6C5'79E69E3#5#799E#9#&B9D95D92599&5#957F8#96B96&795DC747D6EF997EC987'6&9'7C9CBFF7C9

'E##E79 δ ; δ 95"7D939-:G699&7C79/

9

.&95#96&7985##5'E6789'B97C9'7C92D569F7D6&9

>

> =

= /

87 )

=

9

δ

δ 9 9 9 9 9 9 9 9 =G6>9

/

9

.&95#9EF2FE6789395D67CE65BD9B"7C9)9=CBFF5D985C765BD>9B96ED7D65EF96CE65BD9τ.E&96547#9#F585D9"7FB569 .E&9&B#79:2ED65657#9EC795D'26#9BC96&7997EC9#542FE65BD9#B69EC792BC97E&9#67'9B96&79#542FE65BD997EC95#9

EF2FE6789CB496&7985##5'E6789'B97C9'7C92D569F7D6&9956&9-:G69*D96&5#9BC49B2C997EC94B87F95#9E9FBEF9BC49B9 .C&EC89FE99E#9'C7#7D678939A7D87F9C(D97FF5#A5"59C$D9ED89/BCE&7"EF89C5D9

&79D799'CB5F795#98767C45D789BEF92C"E62C7#9EC792'8E6789ED89E9D7992E#6#549EF2FE65BD95#9ECC5789B269&79 7"BF265BD9B997EC9'CB5F79ED93797#654E6789#67'939#67'9

3.2.2 Numerical details on the simulation

%&EC69B925G9C7#247#96&79#27##5"79#67'#9D7787896B9#BF"796&7997EC97"BF265BD9'CB3F749.9#5DF79567CE65BD9Q5?95#9 7:25"EF7D696B9E9#5DF79CBFF7C9'E##E79BD96&79BD6E69<BD793269#5D796&795DC747D6EF997EC95#9"7C9#4EFF96&79#E479 7B476C5EF99BD8565BD#94E9379A7'6982C5D9E9FEC79D2437C9B9'E##E7#9&5#9EF#B9&7F'#96B9C782796&79D2437C9B9 EF2FE65BD#939BD#587C5D96&E696&7997EC97"BF265BD95#9F5D7EC982C5D9,9'E##E7#9B9CBFF7C922C6&7C997995FF9'C7#7D69 6&79#542FE65BD9B96&7997EC967#694EDE789BD9ED9E''ECE62#99&7C79G9CBFF7C#9EC7985#'B#789E69_9=259(>9&79 82CE65BD919B96&7967#695#9F5DA7896B96&79D2437C9B9567CE65BD#929ED896&79CB6E65BD9#'7789B96&79#'7547D9 ω

B

939-:G199

G

_B

1 2,

= ω

9 9 9 9 9 9 9 9 9 =G1>9

B9C782796&79D2437C9B9567CE65BD#9*95695#9D77##EC96B9&BB#79,9E#9359E#9'B##53F79879'CB'B#796B9B4'EC796&79 F7D6&91

5

95D96&79985C765BD9ED8987'6&9

%8

9B96&7997EC9'CB5F79E696&795C#69567CE65BD9956&96&79&75&69/

:

9B96&79CBFF7C9 'CB5F79BC96&79#E479F7D6&9=259$>9&7997EC9.&9E696&795C#69567CE65BD995FF93795"7D939-:(9

= >

= > ,)/

⁹

= 87 ^{9 9 9 9 9 9 9 9 =(>9}

8&7D96&7985##5'E6789'B97C9'7C92D569F7D6&95#9#4476C596&7997EC987'6&95#9C7FE67896B96&794E)54249"EF279B9/

9

.&9 39-:(99

4E)

,)/

⁹

9999956&9

_{9 9}

= >

/ /

= 87 = = 9 9 9 9 9 9 =(>9

S95#96&797D6C79B96&7997EC9'CB5F79*9979E##24796&E696&79CBFF7C9'CB5F795#95C2FEC9D7EC96&79BD6E69<BD79=#779

255>9ED89/

:

95#985C76F9C7FE67896B91

5

9XD79875D7#96&7949'ECE4767C9E#96&79CE65B9376977D9/

:

9ED89

%8

9

4 E )

/

:

4 =

9 9 9 9 9 9 9 9 9 =(>9

%B435D5D9-:#(9ED89(9BD79ED97"EF2E6796&79D2437C9B9'E##E7#9,95D9BD79567CE65BD939-:(G99

: 9

, 87/

= 4)/ ^{9 9 9 9 9 9 9 9 9 =(G>9}

17"7CEF9D247C5EF9#542FE65BD#9&E"79377D9ECC5789B269ED89#&B97896&E696&7949'ECE4767C942#69379C7E67C96&ED996B9 7D#2C79#542FE65BD9#6E35F56915D79,942#69379E#9359E#9'B##53F796&79'CE65EF9"EF279B949S995#9&B#7D96B9 2ECED6779E#69#542FE65BD9ED89BB89#6E35F5699

.DB6&7C9#B2C79B95D#6E35F5695#96&79#&EC'9787#96&E69E''7EC#9E697E&9567CE65BD99&7D96&795DC747D6EF997EC95#96EA7D9 B96&79'CB5F79XD9259BD79ED9#7796&79#'7547D9'CB5F79E67C9#7"7CEF9CBFF7C9'E##E7#9*69ED9379#2385"587895D6B9 G9<BD7#4997)67CDEF9<BD7#99&7C79'CB5F795#9E9#6CE5&69F5D79=DB997EC>9ED89E97D6CEF9<BD799&7C79979ED94EA79E9 'ECE3BF595D67C'BFE65BD49! 9S98

⁹

V9(9*D96&7969B9B6&7C9<BD7#9979&E"749!

C

9S9!

<

9S995#BD65D25657#9B96&795C#69ED89

#7BD8987C5"E65"7#9E69U2D65BD9376977D96&796&C779<BD7#995FF9E2#79F262E65BD96&E6937B47#94BC79ED894BC79 D247CB2#9E#9D2437C9B9567CE65BD95DC7E#7#9ED89EF2FE65BD9C7#2F6#9&E"79DB9#5D55E65BD9879#4BB6&96&7#79#&EC'9 'B5D6#9392#5D9ED9&'7C3BF596ED7D692D65BD9=#779-:((>9BC9BD65D2569B96&79G9<BD7#99

( ) 6 E D & ( (

) )

ψ = α −

9 9 9 9 9 9 9 =((>9

α 9ED89

'

9EC79'ECE4767C#96B9569BC96&7937#697557D9B96&79476&B89*9979F545696B9BD79&EF9B96&79'CB5F796&79 -:($95"7#96&79Q#4BB6&7D?97:2E65BD9CB499&5&9BD79ED9B4'26796&792C"E62C79D778789BC96&791;.9ED89 2E#6#549EF2FE65BD#99

( ) ( ( ( ) )

( ) ( ( ) )

( ) )

! = − ψ ! + + ψ !

9 9 9 9 =($>9

ICE65EFF9

'

942#69379FB#79CB496&79E3#5##E99&7C79C75BD9#7'ECE67#9ED89 α 342#69379#4EFF97DB2&9#B96&E696&79 EDF79376977D9'CB5F7#9!

C

9ED89! 95#9DB696B942&9Q#4BB6&78?99

3.2.3 Example: wear during a rolling test

*D96&5#9FE#69#765BD99792#796&79E''CBE&7#9'C7#7D6789E3B"796B9#542FE679E9DBD9;7C6<5ED9E#79B997EC9&79E#795#9 C7'C7#7D6E65"79B9E9CBFF5D967#699&7C796&7993B857#99E9CBFF7C9ED89E9F5D8C5EF9#'7547D99EC79C7"BF265BD93B857#9

=25>975D565BD9B96&79'CB3F749ED93798BD79396&79'CB5F79B96&79CBFF7C9ED896&797"BF265BD9B96&79'CB5F79B96&79

#'7547D9CB49BD79567CE65BD96B9EDB6&7C9&79D247C5EF9"EF27#96&E69&E"79377D9&B#7D9BC96&5#97)E4'F79&E"79377D9

587D655789BD9C565BD9ED8997EC967#6#49µS69)9S99ω

C

9S91$9C'49BD56285DEF9C77'E795#97"EF2E67896B9υ 9S9 5$\9ED896CED#"7C#EF9ED89#'5D9C77'E7#9EC79#2''B#7896B9379D2FF49υ 9S9φ S99 9S9G9+9252C7969#&B9#96&79 7"BF265BD9B96&794E)5424997EC9

%8

9"7C#2#9F7#49569#&B9#9E9DBD9F5D7EC9'&E#795D96&79375DD5D9=CB4996B9$9

⁵

9F7#>9ED89E67C96&E6997EC97"BF265BD95#9F5D7EC92BFFB95D96&7997EC9'CB5F7982C5D96&79#542FE65BD9BD79ED9 7)'FE5D96&5#9'EC6565BD9396&797"BF265BD9B96&79BD6E6982C5D9567CE65BD#49E695C#69BD6E695#9:2E#50&7C6<5ED96&7D997EC9 4B8557#9'CBC7##5"7F96&792C"E62C7#9B96&79#'7547D9'CB5F79&5#9'&E#7987#C535D96&79EB44B8E65BD9 376977D9BD6E65D93B857#95#9ADB9D9E#96&79OC2DD5D05DP9#6E79.696&797D89B96&5#9'&E#79=E3B269$945FF5BD9 F7#>9BD6E6937B47#9BDBC49=#779251>9BE89ED897B476C9C74E5D#9587D65EF9ED896&7997EC97"BF265BD9 37B47#9F5D7EC9"7C#2#9F7925DEF997EC9'CB5F795#9587D65EF96B96&79CBFF7C9'CB5F79ED896&5#95#9BD5C4789

7)'7C547D6EFF9BD9#765BD94E879CB496&79#'7547D9

4. Conclusions

154'F55789 E''CBE&7#9 BC9 DBD9 ;7C6<5ED9 CBFF5D9 BD6E69 'CB3F74#9 &E"79 377D9 87"7FB'789 BC9 97EC9 #542FE65BD9 25C#69 E9 #54'F55789 E''CBE&9 EFF789 1.;9 E54#9 6B9 5D89 DB69 BDF9 6&79 37#69 BD6E69 EC7E9 376977D9 69B9 #BF58#9 5D9 BD6E69 3269 EF#B9 6&79 37#69 'C7##2C79 85#6C53265BD9 5D9 E9 #&BC69

%I!9 65479 *69 5#9 3E#789 BD9 6&79 ;7C6<9 6&7BC9 ED89 56#9 E35F569 6B9 #BF"79 C565EF9 'CB3F74#9 #2&9 E#9 BDBC45D9 BD6E69 ED89 BD6E69 956&9 D7E65"79 2C"E62C79 &E#9 377D9 'CB"789 %B2'F789 6B9 E9 852#5BD9476&B895DBC4E65BD95#9852#789376977D9#6C5'7#9ED895CC72FEC5657#9B45D9CB496&79 85#BD65D25657#9 B9 2C"E62C7#9 BCC765BD9 EC79 #4BB6&789 .D9 B'654EF9 "EF279 B9 6&79 852#5BD9 B7557D695#95"7D9ED89569&E#9377D9#&B9D96&E695695"7#9BB89C7#2F6#9BC9E995879CED79B9DBC4EF9 FBE8#9ED892C"E62C7#9

17BD89 E9 #54'F55789 E''CBE&9 3E#789 BD9 6&79 2E#6#549 EFBC56&49 5#9 &B#7D9 6B9 #BF"79 6&79 6ED7D65EF9 'CB3F749 5D9 BC87C9 6B9 7"EF2E679 85##5'E6789 7D7C9 5D9 6&79 BD6E69 ED89 5DEFF9 97EC9 7"BF265BD9%B2'F789956&96&79.C&EC8?#9FE9997EC9'CB5F7#9ED9379E62EF5#789E697E&9#67'9B96&79

#542FE65BD9 ED89 6&79 7"BF265BD9 B9 97EC9 "7C#2#9 65479 ED9 379 BFFB9789 552F657#9 EC5#5D9 CB49

6&79 2C"E62C79 85#BD65D25657#9 5D6CB82789 39 6&79 E62EF5#E65BD9 B9 6&79 'CB5F79 &E"79 377D9 6C7E6789

396&792#79B9E9Q#4BB6&5D?92D65BD9ED89E99'ECE4767C9&E#9377D95D6CB827896B9E7F7CE6796&79

#542FE65BD939EF2FE65D96&794E)54249D2437C9B9CBFF7C9'E##E7#96B9379BD#587C7895D9E9#5DF79 567CE65BD99

22C6&7C9#62857#9EC79C2DD5D9E62EFF9E#9E9'CBFBDE65BD9B96&79'C7#7D67899BCA9-)67D#5BD96B9DBD9

#6E65BDEC9 97EC9 'CB7##7#9 39 6EA5D9 5D6B9 EB2D69 6&79 7"BF265BD9 B9 6&79 DBC4EF9 FBE89 6&79 2C"E62C7#9 ED89 B6&7C9 'ECE4767C#9 "7C#2#9 65479 &E#9 377D9 'C7#7D6789 5D9 CD9 ED89 &E#9 85C769 E''F5E65BD#96B95D82#6C5EF94E&5D7C9EA5D95D6B9EB2D696&797)'7C547D6EF985#'7C#5BD#982C5D9 97EC967#6#9ED896&7985#'7C#5BD9B96&7997EC9'CB5F7#9BD796&7967#695#95D5#&789BD7942#696C7E696&79 587D655E65BD9 =#779 C6D9 BC9 876E5F#>9 B9 6&79 97EC9 FE99 B7557D69 )9 E#9 E9 #6B&E#659 5D"7C#5BD9 'CB3F749&5#99BCA9&E#9377D95D565E67895D9C1D9ED8942#69379B4'F76789396EA5D95D6B9EB2D69 4B87F97CCBC#982796B96&79#54'F55789E''CBE&7#92#7895D#67E89B96&797)E696&7BC9*695#9'B##53F796B9 7"EF2E679 6&79 "5E35F569 B9 6&79 #BF265BD9 39 E9 DBD9 'ECE476C59 'CB3E35F5#659 E''CBE&9 E#9 6&79 BD79 87"7FB'789391B5<79ED89EF9CD9&5#95#96&79D7)69#67'9B96&799BCA96B93798BD799

References

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