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(, #$c= L0)0c ('#E*J4© *@7>05 0$ +^0/2c,(* \$c,@0

*J KJÌ ( $2c\ 'c,@0cB4@(, #(," @(KM x *F 0 @(:' 9?0,c

*(, *3(KM c2=Ú$ '6=3# ©$'(,'# (, 0@(,cc„ )c 0)2µ@ Ãc

0 .=(,)/- = 2"0)c (,*9@01@0( @2=O- )00)2 ?0= *

(,* 0 . *Ã(, *3(KM c =( 9#(:}*$ , *3( 4'#@c _

0)=7( ' (,\$(( 00= _H#+ K"(:'q==, "*3*3(©$'=@0=H#.cP$'=c

*/(= *@"(0)s@ ' _# '' )M,c (,J2=6 0)c, /c

? (*@@- ( c(F *@c 0 N *Ä0) _ #2 K"(:'4( @22c *@^c

c 0@(:' 9?0,c*(, *3(KM c2cc 0 ')3µ /ÃA ,' 0))

M ,EICWƒx à #= = (6((0) / E')0I#2 K"(:'4*2^(,)\ '=$0)=

0q /c6©$'=$0)c=©@( '0 2"*J)(*@ _, "0)(c(0)=

© # ^E00 \ 'c,@0H( KJ,c1'09 q4(: ,E(c(*

(,cF $(KM =c *@"( @ ' I*(, *3(KM c ( )*3*@4#43c( @"

,"Ã @(M O-"0*$"0)4@ ("*3(M x B #+\4@0)*@"0©3'= *(, *3(KM

c _ 3= &*,cGG =( ÑÑ.cP$" "0) KJH*

(: */(M = *9 4F'="(&( @2c M,=x L„Ï(=( *"(

$ 'Y# = 2"*9Ã(,* _, "0)(µ,'("- 2cÏ9#Ã (*"M,J@c =& Kc 4

c:4@¿= @( ^:c(*"M,J@'= (, _00 (, *@ÃHA ,' c*3*@

0 K"0)0G _ c 0 G _ "0)(Ã O*9 "0) 0 , *9$',=c (

J*@= $cc _,E"Ï 4'= $ "- 0) ( 'J,*@c c,c _,E" JJ

"2c $ ,cY@ 07(00) ( 'J*@= $==( @*F " 4'"2= $ c

/"0)¹¡/= 0)4 /c 'J*@c $==: @(*F+/)0%:@ ž 4@ +J

KJ _0)0O(:},*@ cB9#$x

_0=„:0"2=1220)4G _, "0(,c= =c )0 0=

_4 _"0) (=(*(cF @(M =9& 0)0.@c c"62"*9Ã()*@

_, "0(,c= 46 _ 0(,c*J±(# _:cF'"(# '#@0 c,/)(*@ 0@(:'=

"=¿4 / =& _ "0)(=cD/c }/= 0) =&(=('4'0IJÌ _*

') ("(,0) F \6Ã c *(, 3 "0$(, #µ9# "(, µ0))

( 0 $4c4YJ/ c) 0)b.c $'= ( (*J@0

0)2(c@B¹ ('# = _!#&07*^cF'=@-@( (*9$0( @2= KJÌ"("-

/( $2H^()(KM Ñ.c>0c1' 2c(>0)2 J*@= $==c 4' _, "0)(c

/0 # ± c / 2cY) _,4 _0)'6(=c±@ © \

(0)= "*Fcb }"*3(c ^@0)'=.4@@0c M,J+*(, *3(KM c 0'=&2=

J*2(* _"0$*3 c F/(G @ 07 *FB, _ "0)(=c

2cB _"0)cBÚ0)"2c " *ž 0)4 _, "0)(= (

%‰- +.3 1$µ+."

5 ,/*30Ä c '='(,*@ 2 /c _ # *$c (:}*$c '0 _(,*M,0

0) 0) _ "0)(=cF<Ec1/09/ 0 F( @22c0)) "(:')=,H(, ƒ,*@ 2cc

9#@ _ 2"(:'ž/*@0@ (*"M,J@cc A = &^/c #(Ñ / 9#

2cgJJ@*F'@ (,')\ 'cE9#6(:},*@c¿9#Ã /> =( ')0I*3*@ 0?0,4 "*

'=&cb9# (0)*@4 "*FcÉ"@0)(KM,c,"(= 9JE/!F# )62" (0)*@ ==

#/ ))0)c:') _µ@((Y@0/(,*@*@b=)(*@62 /c4 _ # c,"

(=(, _0c 0) U, "0)(c=

­

6$"ž<c),+06 <W" $ 1$µ+."

*9@)"0µ _*30( c= ( 0)@=')0 _/= , ")x <E=

"(,*)( 01'c= _ b@0/(,*@ 2 /c ( *@( g@c

0)7 _,\$(KM \ & &@'0 ; (cB9?0,=&9# "0

0$((,*M,J$'c\ ( @2=0)c©,"( 0) $'= _ @03'(,*@u@ =)(*@ c,"

c= *9@)"0: 0) (, ƒ,*@c 0+0))2c*@0)(c:' (*"M,J@c "(:'2c:W

& ( @2=0@c =( 0 $c _ "*3(=)=3'00)@-@03'(,*@&Ú2c,(*$

c,"=cA =B@('#4 $, 2" c )(*(KM ( *2=3'== cF 00)

.3/* 2"0)0 @>( 0 $c _ _(,*M,00)@ =(2 /cà 0 (,0,

*2=7**30 (KM = *J@),'0 4( ( 22c=G /*F(:'g,"( 0) 'c

,@cx

+."ž+06 +B%(/•1Ñ- %<c-?/2+>),@BAxC&/2+06

AE c*30(, /c 0)4 _ 0(,cc=*@"0'0@((,*"M,J@'=¬ 0)

0 @(,(*"M,J@'=x AE((,+ 2(*"M,J$'c = 0)0(, =, q

a)

b)

>. L„c*@0)'0@(,(*"M,J@ (,'#@cÑ. 0 '=Bc0 _à LÕ

"0'0 @((,*"M,J@ ( @ "(: #. ,+0 'c0 0 _=c

L„*@0)0@@((*"M,J@c2 (, #Ã}/)(*µ.4 +0 $'=%( c% _, 1 M,04 c

0) _@Ag 0)0 @(,(*"M,J@c (, #Ã$ '&. c3(,*@+0$'= ( = (*"M,J@

')0E= /*3*@c 0$@-H 2cO-H2" 0) U=c7AE((, 0) _ 0(,c

_,"*22c (KM,0c >/,*@"0) *(: */( M =c A c&2c3&= _E'(,( @

2c==c

= Ñ (,'=*@=¿9# 0)\( $ _6(*"M?0 0 @(: = * c

É =cA ,'+= /*F# =( 0 9# c (*"M,J@ *(, c,"(c

2@(, 9?0=6(*"M,J@c4 J =, _ *(, c,"( (0)*Ï$4* '=6H."( @* 0)

q _8(,*"M?0c I*9Ã ( 0.9?02c, 9#F, * c _

0(&F/0)*@(=c+( = 2c= (*"M,J@'=Ã '= 9#

_,(*3ƒ=G9# JJ, cP$c 40=7)M,c2 _1 _(,*3= # G

,Ã@0$' (,"=c !)ڃ )(=2F, (*"M,J@'=*9 $?0, _ 22"

2"0(,c 3,É/ +* c+/ ,*@c=

x y

x y

δ

δ a)

b)

!)Ճ %L„2c*@0)'0@(,(*"M,J@19#@( H= _(,*3ƒ @(:'@0 =c@

L„@ "0'0@((,*"M,J@ 9#@( $ c _,(*3ƒ©@ $(,'0=c

!)*ƒ; ‰ƒ /(,c6(= (, _0c 0Ã( 0) _@c 2 E _=(

Ã@0/(,

& <E=%Ã> ( $,%9#&=c _,(*3ƒ +9# 2 0 'c

=g2 0 @: ;2 /*/*@4 ^,(E _ 2= )'=(:' 0$'= 0)

0 @(,(*"M,J@'= ( = 2 /*/*P _=&= 0@ Ú 42 }/*3*@

0$'=cGc (,'#@0ƒ >c0$(,')('(4/0324cƒÌ B c

,*3 @*3(M c4 c (,'J 0(H#q; )II=4"0@(,(,)''(4/09#E;

B Úc ,c *F $*/(M #Ú;ƒ &O(,c _ 9=:$ c@c,c

(,c ==%(,*M?0,(c=O-

τ xy

*3( @0(, = = , -

σ y

-$c ('#

}/0) ('?0,0) =c\ 0) _@cÃOA 4,4@*3(@0(,c@(*2^

c3cE-

σ x

σ z

-=ÃN/H0c,1c,=c &3 F $, >0)c@c c

=Ñ (,J , c }/*3*@"0)(=À. 0$c=ÃHAEc2 *9 /.F 0 'JJ

(:'0)(: (# (*"M?0(, =@c$-

τ xz

@

τ yz

-1/)0 *3*@./ @!)

/(,c4(c@(, _,"0)= 7J==(, 0) _ # . ( /2 ,H/0

(,c (: _"0= G &0) _@c$-$ c (*"M,J@'=O-@ ?0 0)*@ *"M?0(,c

= /0x,*F')*$?0, ^@()),"=(, 0) _$- =( c( / (,c"-9=

"$)>J=c,(,0) UÃ A & @('#6(=É& _ =cÉ$&(=c@-

_ c*@"0'0 $((,*M,J$'c - c ( /2 )(*@Ñ )'=b0) _@cxA ,' c *3*@

0 K"0)059#2 M,"0)c2@ "0'0@((,*"M,J@'=x7< cc2 }"*3(c @*3(,@0)(c@c

Z ^(,)' JJ=¹. 0) _=O-%=(c c4$' c(:

=Ã

0 10 20 30 40 50

−20

−15

−10

−5 0 5 10 15 20 25

Avstand fra venstre ende av limfugen (mm)

Spenning (MPa)

Skjærspenning

Aksialspenning i x−retning Aksialspenning i y−retning Aksialspenning i z−retning

>µ; =(, _0c G=*@0)'0@(,(*"M,J@'=x„ c@c6/(:9

cc (,*M?0,(c=¹4@*3( @0(,c=+

-'-@

0 10 20 30 40 50

−20

−15

−10

−5 0 5 10 15

Avstand fra venstre ende av limfugen (mm)

Spenning (MPa)

Skjærspenning

Aksialspenning i x−retning Aksialspenning i y−retning Aksialspenning i z−retning

!).ƒ P =(, _,"0)c O 0)'0@(,(*"M,J@'=xP cc+/)(, 9

,c= (*"M?0(,ccÄ4@*3($0(=c+

'- - Z

! "5$&%'4/2<=)2@+>- +0/21©3ž+06;791&6#3ž1b+B%/2%c$Ñ)2),@.ADC:/2+06

&(*9@01= (: =©/c $0 3')(*@ /=+ _ # =(*"M?0(,c(

_,"0)= 0)'0@(,(*"M,J@'=cFA =2 KJ(:'$03')(*@ /= _1#+ c,"

(*"M?0,(c(, _0cO-/ c60) U, 0(,@-Ã7 „0*@=(=6( @ _(0

^Ì;ƒ A c 2 /c Ã>2cF!* _Ic*@0)0@@((*"M,J@cc9„ÌÃE0)@(,(!A

N '=)c 0 Õ@0*@=(= 0!#6M,"0)4 _& 0)0@@((*"M,J@c=OA )((2

= = 0?0, /= ( =( 0x@*3( @0 _,\$(KM c O0 $'=É

(*"M?0, _,\$(KM = O0)2c

ÂÀz

e

k pqvDz±|}k}ogv w‘l vDzƒ‚Bpqv

L„2(,9@*3H /=B/*30)Õ. Õ@0*@=(=4+A N ƒƒ =Õ$1(,*"M?0

_,\@(M = 0$c c"0)( M,c,c 2H*9Ã64@(, #+(, E(,*M?0,(==

" 0@(, ==, KJ,@cc 1" c 0)*@*3( $*3 E(,c \&$ (,(1(, 1(,*M?0

(=cB(#E )0/ ?0 0)(,'H" "0)cB. 0$'="( == P ' ( =%

*F @*F72"20WA )(( (== /09=2"2 KJ, 0(*"M?0, _©@(KM c

90 'c!0 $c(: 1(,*M?0,(,'/"-/(0)*4( (, #@0)0$'="-$ )09 ,*F')*

*3*@H 0F('# (, 7(,*"M?0 _©@(KM c=@Bc _I0$c 42 1(,*M?0,(,)/"-

# @(2c( '0X I( @ -H8&0=Y BÄ, ,/&Ì '=)c '0

„0*@=(= @A 6N " # =( 01(,*M?0, _©@(KM c4H0$'=

(,*9@0,9#"(,cF'c,Ú'0)" c /=x

(,c #(*"M,J@c2" c2 O( /(: 9ƒ

l 2

l 2

x

t o

t o

t i

η η

T = 2lτ avg

T /2 = lτ avg

T /2

> Aq "0'0@((,*"M,J@ 9#@( $ *, '=c

8 =0 $(0)cc2=10 /*3*@"0(,g. B+ É3 , 0$c =1c0(,/(

t i

t o

E o

c"0@(,(,)''(,©/0 0== -

G o

=(*"M?04/0

,c ! _.3 , 0$'=@-F2c(

E _i

G _i

G c

cG(,*M? 4 0 _70)2"-

η

=+9#@( $ * c+ ,"cW

T o

@

T i

""(,cF'=c 9#@( $*, c"- c

""=-9( @ / ,*@c O0)c@c, #=0)(,/)( 3 ,&@c

0$cx

τ avg

= = M,c@( ),'0(,*M?0,(c=O->10)2"-OÕ==Ã=

2"0)0 @ 0)¯0$'

τ avg = 1 l

Z ^l ₂

− ₂ ^l

τ xy dx = T

2l ,

F

τ _xy

9 ƒcE /)(, c '"(,©$0 0=

.(*"M,J@cx 1 0*@"*F'(

@*F c-9O0== - _ 0"2=3'= # /

dT o

dx + τ xy = 0

dx

T o T o + dT o

τ _xy dx

τ xy dx

T i T i + dT i

τ xy dx

τ xy dx

T o T o + dT o

>. )*@*3( @*F =q. ='(©$0 0c: (,*"M,J@'=x

dT i

dx − 2τ xy = 0.

0$c( (*"M?0 _©@(M c c $ =( 0 " #2 @*F' *3"©$')*3*@c

_G&c*@0) "0)"2cF'= g 3 , 0$'=A ,' =G )(, 9BƒP

&G cÚF $,= 0?04(*"M?0(,c (, _,"0) M,=@ /*/*@0(,c±.

0$'=D 0)*5$ (*"M?0(, =@c >Ñ3 , 0$c$-

τ o

-^6(,*"M?0(= =

0$'$-

τ i

-*9 /*3*@"(

τ _o = τ _xy t o

y ⁰

τ i = τ xy

µ

1 − 2y ⁰⁰ t i

¶

dx

T o T o + dT o

τ xy dx τ _xy dx

T i T i + dT i

t o

t _i /2

τ ₀ = 0

τ 0 = τ xy

τ i = τ xy

τ i = 0 y ⁰

y ⁰⁰

skj

rspenning f orskyvning

u os

u o

u _co

u _ci u i

u ic

>„ƒP 3 @*@0( 0"?0%(*"M?0(=@(, _,"0)+M,= /*3*@"0)(c\.

0$cÃ

F x* c 0*9$04*,$'=&( @ cÉ0)0!9#ž.c,()c¿. c0(,/(

3  10$'à Xƒx cI @(=,>9#.0)0F(*"M?0,(cc>#

.c,(cµÉ3 H0$ /(cJ É, 3)'=.+,H0$' K/(c

y ⁰⁰ = t i /2

τ o = τ xy

"

y ⁰ = t o

-

τ i = τ xy

"

y ⁰⁰ = 0

γ o

-

@2 ,+0$' -

γ i

-9*9Ã/ =2"(* (

γ o = τ xy

G _o t _o y ⁰

γ i = τ xy

G i

µ

1 − 2y ⁰⁰ t i

¶

.

(*3c 0)cc & _ 3 B+, 0$' ( _*3(KM ©. 0*9@0)

* $'= c @)

u o (y ⁰ ) = u os + Z y ⁰

0

γ o (y ⁰ )dy ⁰ = u os + τ xy

2G _o t _o y ⁰²

u i (y ⁰⁰ ) = u ci + Z y ⁰⁰

0

γ i (y ⁰⁰ )dy ⁰⁰ = u ci + τ xy

G i

µ

y ⁰⁰ − y ^{00 2} t i

¶

,

3

u os

(cF'c,c _,(*3ƒ = 9#==(,=‘. / 0$@-Õ@

u ci

c

_,(*3ƒ=O- 0)2-,==Ãc2"0)0 @ 0)2B2,+0$9A =q

+ _00% 000) 0 $c (0)*u$+ _(,*3c = *3c,0)

===00 0)Û 0$'=cƒ \,"( 0) $\ ,'(,*9@0

u ci

$?0

0)* _,(*3@c . = 20$'= Õ==Ãc - @

u co

_(,*3 u.

0)2 === 00¡3 ,60$'© 0) &(*9$07$?0,60* c 3 ,60$"(

_,(*3ƒ2,.c=x 1" M,"0)Ä 0) ƒG*9Ã/ (*,)

u co

(

u co = u o (y ⁰ = t o ) = u os + τ _xy t _o 2G o

.

&O *@= 0) Xƒ Ì70O#(,*,)& 0) Xƒ

u o (y ⁰ ) = u co + τ xy

2G _o t _o y ⁰² − τ xy t o

2G _o .

É 'c O0c= _ c0(:/(B3 $-

T _o

T _i

T o = Z t o

0

σ o (y ⁰ )dy ⁰

T i = 2 Z t i /2

0

σ i (y ⁰⁰ )dy ⁰⁰ .

3

σ o

σ i

cµ©@0)(= c4G0)cc _/ ž ,60$

1 # (,*)Ñ )((4(==4'0 _*3( M cÉ _(*36 # />0)

J ƒ.;79# _,=

T _o = E _o Z t o

0

du o

dx dy ⁰ = E _o t _o µ du co

dx − t o

3G o

dτ xy

dx

¶

ƒ W

T i = 2E i

Z t i /2 0

du i

dx dy ⁰⁰ = E i t i

µ du ci

dx + t i

6G i

dτ xy

dx

¶

.

/*M?0:'J̃=O0"-

γ c

-9= c, (

γ c = 1

η (u ci − u co ).

/*M?0,(=@c O0)2 *9 / (,*)@ ( @

τ xy = G c

η (u ci − u co ).

< /( / =)== 0) ƒ =P 7=( 9#

# /

dτ _xy dx = G _c

η

µ du _ci

dx − du _co dx

¶

.

#('= / 0) Xƒ c J GO0) XƒW

dτ xy

dx = G c

η

· T i

E i t i − T o

E o t o − µ t i

6G i − t o

3G o

¶ dτ xy

dx

¸

.

=c=(#&0) XƒWI2"c( ƒ2#

- (,,'c%0) Xƒ N ƒ;

c=(@00)=

d ² τ xy

dx ² = G c

η

· 2τ xy

E _i t _i + τ xy

E _o t _o − µ t i

6G _i + t o

3G _o

¶ d ² τ xy

dx ²

¸

.

/ *9(,*,)+(

·

1 + G c

η µ t i

6G _i + t o

3G _o

¶¸ d ² τ xy

dx ² = G c

η µ 2

E _i t _i + 1 E _o t _o

¶

τ xy .

/ M,= *9(,*,)+(

d ² τ xy

dx ² − β ² τ xy = 0,

β ² =

G c

η

³ 2 E i t i + _E ¹

o t o

´ h 1 + ^G _η ^c ³

t i

6G i + _3G ^{t o}

o

´i .

β

λ

α

β ² = α ² λ ² ,

3

λ ² = G _c η

µ 2 E i t i

+ 1 E o t o

¶

ƒ

α ² =

·

1 + G c

η µ t i

6G i

+ t o

3G o

¶¸ ⁻ 1

.

A =c=00 0)J( = 0) Xƒ G=

τ xy = A sinh(βx) + B cosh(βx),

3 N c *(, 3c ( (,'2"(O _ 0 '"0)(c '"0)(c

_ = @ "0'0$((,*M,J$'c c

T o = T

2 = lτ avg , T i = 0 ved x = − l

2 .

T o = 0, T i = T = 2lτ avg ved x = l

2 .

& # =2"

B = βlτ avg

2 sinh( ^βl ₂ ) = βT 4 sinh( ^βl ₂ )

ƒ;

A = βlτ avg

2 cosh( ^βl ₂ )

"

1 − _2E ^{E i} _o ^t _t ⁱ _o 1 + _2E ^{E i t i}

o t o

#

= βT 4 cosh( ^βl ₂ )

"

1 − _2E ^{E i} _o ^t _t ⁱ _o 1 + _2E ^{E i t i}

o t o

#

.

½Õv9|}k oqv w²l vDzƒ‚Bpqv

2@0/(= (cF'=, *9$),'"0ƒƒGc!ÕÃ3 $ $ @0)0H©$c,@0c KJc

(,"40?0:70@(:'(,*P 1" "0@(, .0) _ "0)(=/0 @ ' 3

0c3<g: } /*30)7c c,( H2" (0)*@ *3c= /0

#(=3'= = 2 /cx

>. /(= = ,c6')0O=@ "0'0 $((,*M,J$W

l l

x

t o

t o

t i

η η

T = 2τ avg l T /2

T /2

d

plastisk elastisk plastisk

>.ƒ D+= , - "0$(,')(*@É0@(:')(*@ ( = _ c 0)'0@(,(*"M,J@W

0c ==0$(0)c@cx /*/*$"0(É.2 q4E 3 ,&0$cEc7c0(,/(

t i

t o

B=( *3*@0(=b 0)2 =

η

E o

, 0$'60@(:'(,)'(,©/c0

E i

2 (*"M?04/0

G c

=

#@( $,G* G=7 "=c

τ avg

=7M,c( ),'0))@ (*"M?0,(c 0)W

c 0)c@cÄ =0@(:'(,*@+( cD

ƒ =++/(,+c '(©$0!0)c

É. (,*"M,J@'=x

T o

T i

(=3'== *, c = ,""= ( /*@cG 0)cc, 49#4=@0(: )(

*3 ž = 60 $'=x

u o

= _(,*/ =±. 3 0 '62c(

u i

c

_,(*3ƒ=¹.6 ,+0$

τ xy

= (*"M?0(, =@c O0)2c

dx

u o u o + ¡ _{du o}

dx

¢ dx

³ 1 + _E ^{T o}

o t o

´ dx

T o T _o + ¡ _{dT o}

dx

¢ dx τ xy dx

τ xy dx

T i T i + ¡ _{dT i}

dx

¢ dx

τ _xy dx

³ 1 + _E ^{T i}

i t i

´ dx

u _i u i + ¡ _{du i}

dx

¢ dx dx

!.ƒ P(*3c.* c 9#='"(,©@0 0)c@: (,*M,J$'cD

*@"*FE 0)cc, _ )"(,©@0 "0=3

dT _o

dx + τ xy = 0

dT i

dx − 2τ xy = 0.

/c 'JJ( "0@(M c _ Ã3 "0@(,(,*@+0$'=+@

du o

dx = T o

E o t o

;

du _i dx = T _i

E i t i

.

/*M?0:'J̃=O0 c "

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η .

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τ xy = G c γ c = G _c

η (u i − u o ),

2c( =0@(,(*$+( c= (*"M?0(,cc), "

τ xy = τ p = konstant.

) ;P 20))(,c,c )*3*@ = 0)"?0 (= JJ ( ,"0@(KM

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u i

u o

_ 0) ƒ; - Xƒ; ; 7 # ©

dγ c

dx = 1 η

µ T i

E i t i − T o

E o t o

¶

.

*9Ã0)=

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T _i

d ² γ c

dx ² − 1 η

µ 1 E o t o

+ 2 E i t i

¶

τ xy = 0.

c _ =0@(:')(*@+( =*9 / (,*)+0) XƒG(

d ² τ xy

dx ² − λ ² τ xy = 0,

=

λ ² = G _c η

µ 1 E o t o

+ 2 E i t i

¶

.

ƒ 7 0J(

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2G c

¶

τ p ξ ² + Cξ + F,

3

ξ

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q( ƒ G 0

λ ² = G c

η µ 1

E o t o

+ 2 E i t i

¶

= 2G c

Etη = 2τ p

Et(ηγ e ) .

ξ

KJ0)cÉ '0(=

γ c = γ e ved x = +d/2, ξ = 0

γ c = γ e + γ p ved ξ = +(l − d)/2

dγ c

dx = dγ c

dξ ved x = +d/2, ξ = 0

dγ c

dξ = T

E i t i η = 2τ avg l

E i t i η = λ ² τ avg l 2G c

ved ξ = +(l − d)/2.

O # c2"

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G _c

A = τ p

cosh(λd/2)

ƒ

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G c

tanh(λd/2)

· λ

µ l − d 2

¶

+ tanh µ λd

2

¶¸ 2

= tanh ² µ λd

2

¶ + 2 γ p

γ e

@;

@

τ avg

τ p

= µ

1 − d l

¶

+ tanh(λd/2)

(λl/2) .

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cosh(λx) + B G c

sinh(λx)

( @ *c =Ä@(,*,)4 XƒF;9Bc(EOE0 @(:'(,*@&@©#@ c

γ pv = µ λ ²

2G c

¶

τ p ξ ² + Cξ + F

ξ

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ξ

x

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γ pv = γ e + γ p ved ξ = 0

dγ E

dx = dγ pv

dξ ved x = l

2 − d og ξ = l − d

dγ E

dx = T

E _i t _i η ved x = l 2

dγ E

dξ = − T

2E _o t _o η ved ξ = 0.

'0(=+

F = γ e + γ p

;

C = − µ λ ²

2G _c

¶

τ p (l − d) − γ p

(l − d)

B =

·

E o t o λτ p (l−d)

E i t i cosh ( ^λl ₂ ) + ^{2E o t o γ p G c}

(l−d)λE i t i cosh ( ^λl ₂ ) − ^{γ e G c tanh} ( ^λl ₂ )

cosh ( ^λ ( ₂ ^{l −d} ))

¸

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tanh ¡ _λl

2

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λ µ l

2 − d

¶¶

λA G c

sinh µ

λ µ l

2 − d

¶¶

+ λB G c

cosh µ

λ µ l

2 − d

¶¶

= λ ² 2G c

τ p (l − d) − γ p

l − d .

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cosh(λx) + B G c

sinh(λx)

=0@(,(,*@+( =9#c(, , (, 2"0)c@+

γ pv = µ λ ²

2G c

¶

τ p ξ ² + Cξ + F

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γ ph = µ λ ²

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¶

τ p ζ ² + Hζ + I

ξ

ζ

ξ

^(,)' 4@ JJ,

ζ

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γ _pv = γ _e ved ξ = l − d − f

γ pv = γ e + γ p ved ξ = 0

dγ E

dx = dγ pv

dξ ved x = l

2 − d − f og ξ = l − d − f

dγ _E

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2 − f og ζ = 0

ζ ξ

x

f e

d

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!).ƒ 2@*9@0) *$cB _,(*"M,"0)0) ( =

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2E o t o η ved ξ = 0

γ ph = γ e ved ζ = 0

γ E = γ e ved x = l

2 − f

dγ ph

dζ = T

E _i t _i η ved ζ = f.

'0(=+

F = γ e + γ p

I = γ e

ƒ

H = 2E o t o γ p

E _i t _i (l − d − f ) + 2E o t o λ ² τ p (l − d − f )

2G _c E _i t _i − λ ² τ p f G _c

ƒ

C = − λ ² τ p f E i t i η

2E o t o G c η − HE i t i η 2E o t o η

ƒ;

B = H − λγ _e tanh ¡

λ ¡ _l

2 − f ¢¢

λ

G c cosh ¡ λ ¡ _l

2 − f ¢¢

− _G ^λ _c tanh ¡ λ ¡ _l

2 − f ¢¢

sinh ¡ λ ¡ _l

2 − f ¢¢

ƒ

A = γ _e G _c cosh ¡

λ ¡ _l

2 − f ¢¢ − B tanh µ

λ µ l

2 − f

¶¶

ƒ

γ e = A G c

cosh µ

λ µ l

2 − d − f

¶¶

+ B G c

sinh µ

λ µ l

2 − d − f

¶¶

ƒ

λA G _c sinh

µ λ

µ l

2 − d − f

¶¶

+ λB G _c cosh

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=

= λ ² τ _p G c

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@

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