Diploma Thesis
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Diploma
Transcript of Diploma Thesis
-
, 2007
::
. ...
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..1 1: .5 1.1 6 1.1.1 ......6 1.1.2 .............................................................................7
1.1.2.1 ...............................................8 1.1.2.2 ....................................11 1.1.2.3 ...............................................14 1.1.2.4 .......................................14 1.1.2.5 ........................15
1.2 ...................................................................................................17 1.2.1 ......................................................................17 1.2.2 Cauchy................................................................18 1.2.3 ...................................................................................19 1.2.4 Piola Kirchhoff............................21 1.2.5 .................................................................................23 1.2.6 Piola Kirchhoff..........................25 1.2.7 ......................................................27 1.2.8 Cauchy .......................................28
2: ............................................29 2.1 ..................................30 2.2 ......................................................................................31 2.3 .....................................................................................35 2.3.1 ...........................................................................................................35 2.3.2 ..........................41
2.3.2.1 Euler (forward Euler) ......44 2.3.2.2 Euler (backward Euler).......46
-
2.3.2.3
(Generalized trapezoidal and generalized midpoint rule) ...................50 2.3.2.4 generalized cutting plane.................................................51
2.3.3
..........................................................................................53
3: ...............................................................54 3.1 ..................................................................................................................59 3.2 ................................59 3.2.1 ......................................................................................59 3.2.2 .....................................................................................60 3.2.3 ................................................................................62 3.2.4 ...............................................................................................64 3.2.5 PM - ...................................................................................................66 3.2.6
s, u - ....................70 3.2.7
.............................84
4: ....91 4.1.1 PM ............................................92 4.1.2
tP PP t N MA, I , I , I , W , S , S .................................105
5: ...................107 5.1 ........................................................................................................108 5.2 ....................................................108 5.2.1 ..............................108 5.2.2 ....................................111 5.2.3 .....................................113 5.2.4 ..........................................114
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.........................................................................................119
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, . Coulomb (1784) . . St. Venant (1855) .
St. Venant , , ( ). . , . . / .
- St. Venant- . , . St. Venant .
, . Nadai (1931) . , sand heap analogy, Sadowsky (1941) .
Sokolowsky (1946) Nadai (1954) sand heap membrane analogy. Smith & Sidebottom (1965)
1
-
Itani, Johnson, Yamada et al., Mendelson(NASA) .
: , St. Venant. Wagner (1936), Vlasov (1961), Timoshenko & Gere (1961) , Rasajekaran (1977), Bathe & Wiener (1983), Gellin et al. (1983) . Boulton (1962) Dinno & Merchant (1964) . May & Al-Shaarbaf (1989) , Sapountzakis & Mokos (St. Venant ) , .
/ . , , , ( ) . , . Cullimore (1949), Ashwell (1951), Gregory (1960) ( ) Tso & Ghobarah (1971), Trahair (2003) , .
. Pi & Trahair (1995) Baba & Kajita (1982) . , .
( ) . . , .
2
Highlight
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.
:
. , . . , , . .
, , () , . .
, . , . . , .
, , . , . . ,
3
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, ( , ).
. - - .
. , . , . , .
, , ( ) .
, , , , . , . . ., , , . , , .
, 2007
4
-
1 :
1.1
1.1.1
, , Q . .
. P
0t =
1 2 3Ox x x
1x , 2x , 3x . , Q. P,
t
( ), , ,1 2 3P t
)
. P :
( , , ,1 1 1 2 3x x x t = (1.1.1) ( , , ,2 2 1 2 3 )x x x t = (1.1.1) ( , , ,3 3 1 2 3 )x x x t = (1.1.1)
1 , 2 , 3 . . , 1-1 P P , . , :
( , , ,1 1 1 2 3 )x x = t) (1.1.2)
( , , ,2 2 1 2 3x x = t)
(1.1.2)
( , , ,3 3 1 2 3x x = t (1.1.2) (1.1.1) Lagrange, (1.1.2) Euler. Lagrange . Euler
( ), ,1 2 3P x x x( ), ,1 2 3P
. t
.t = , . P
6
-
1 :
3e
1x
1 2 3u u u= + + 1 2u e e (1.1.3)
, ,1 2 3e e e : , , .
1Ox 2Ox 3Ox
1u , , : 2u 3u
1 1u = (1.1.4) 2 2u 2x= (1.1.4) 3 3u 3x= (1.1.4)
, ( ), ,1 1 1 2 3u u x x x= ( ), ,2 2 1 2 3u u x x x= , ( ), ,3 3 1 2 3u u x x x= , , . 1u 2u 3u . 1-1 P, P,
1 1 1
1 2 3
2 2 2
1 2 3
3 3 3
1 2 3
x x x
J 0x x x
x x x
=
1 1 1
1 2 3
2 2 2
1 2 3
3 3 3
1 1 1
u u u1x x x
u u uJ 1x x xu u u1x x x
+ = + +
0 (1.1.5)
, ,
, , t 0=
1u 0= 2u 0= 3u 0= . ( )J t 0 1= = (1.1.6)
, (1.1.5), (1.1.6)
J
J 0> , (1.1.7) t 0 . (1.1.7) 1-1 P, P / .
[ ]X . (1.1.5), , (deformation gradient).
J
1.1.2
, .
7
-
1 : , . . . , . ( ) .
1.1.2.1
, .
PAds P ( , , )1 2 3P x x x A
. ( , ,1 1 2 2 3 3A x dx x dx x dx+ + + )
)
( ) ( ) (2 21 1 1 2 2 2 3 3 3ds x dx x x dx x x dx x= + + + + + 2 2
3
2 22
1 2ds dx dx dx = + + (1.1.8) , , , PA P A
( ), ,1 2 3P ( , ,1 1 2 2 3 3A d d d ) + + + . .
PAP A
( ) ( ) ( )2 21 1 1 2 2 2 3 3 3ds d d d = + + + + + 2 2
3
2 22
1 2ds d d d = + + (1.1.9) (1.1.2), (1.1.4), :
1 1 1 1 1 11 1 2 3 1 1 2
1 2 3 1 2 3
u u ud dx dx dx d 1 dx dxx x x x x x = + + = + + + 3dx (1.1.10)
2 2 2 2 2 22 1 2 3 2 1 2
1 2 3 1 2 3
u u ud dx dx dx d dx 1 dxx x x x x x = + + = + + + 3dx (1.1.10)
3 3 3 3 3 33 1 2 3 3 1 2
1 2 3 1 2 3
u u ud dx dx dx d dx dx 1 dxx x x x x x = + + = + + + 3 (1.1.10)
:
( ) 2 2 22 2 11 1 22 2 33 312 1 2 13 1 3 23 2 3
1 ds ds dx dx dx2 2 dx dx 2 dx dx 2 dx dx
= + + ++ + +
(1.1.11)
8
-
1 : (strains), :
2 2
31 1 211
1 1 1 1
uu 1 u ux 2 x x x
= + + +
2
(1.1.12) 2 2
32 1 222
2 2 2 2
uu 1 u ux 2 x x x
= + + +
2
(1.1.12) 2 2
3 1 233
3 3 3 3
u 1 u ux 2 x x x
= + + +
2
3u (1.1.12)
3 32 1 1 1 2 212
1 2 1 2 1 2 1 2
u u1 u u 1 u u u u2 x x 2 x x x x x x
= + + + +
(1.1.12)
3 31 1 1 2 213
1 3 1 3 1 3 1 3
u u1 u 1 u u u u2 x x 2 x x x x x x
= + + + + 3u
(1.1.12)
3 32 1 1 2 223
2 3 2 3 2 3 2 3
u u1 u 1 u u u u2 x x 2 x x x x x x
= + + + + 3u
(1.1.12)
Green
11 12 13
21 22 23
31 32 33
= G (1.1.13)
ij : Green . (1.1.12). 21 , 31 , 32 (1.1.12--) .
21 12 = (1.1.14) 31 13 = (1.1.14) 32 23 = (1.1.14)
T = G G (1.1.15) Green .
(magnification factor of the extension of line element ),
PAPA
2
A 2
1 dsMF2 ds
= 1 (1.1.16)
9
-
1 :
, PA 11dxnds
= , 22 dxn ds= , 3
3dxnds
= , , . (1.1.11), 2dsAMF :
2 2 2
A 11 1 22 2 33 3 12 1 2 13 1 3 23 2 3MF n n n 2 n n 2 n n 2 n = + + + + + n (1.1.17) AMF (
PA
Ae ) :
( )A ds dse ds 1ds = = + Ae ds (1.1.18)
( ) ( )
2 2 22 2 22A
A A2 2 2
1 e ds ds1 ds 1 ds ds 1 1MF 1 1 e 12 ds 2 ds 2 ds 2
+ = = = = +
Ae 1 2 MF = + A 1
1
(1.1.19) . , , (1.1.18) (1.1.19)
ds 0 >Ae >
A1MF2
> . (1.1.19) Taylor:
...2A A A1e MF MF2
= + (1.1.20)
(1.1.20) : AMF 1
-
1 :
Ax1 11e (1.1.22) ii . iOx ii , .
, ,i 1 2 3=
1.1.2.2
PA
A1 A2 A3n n n= + + An i j k (1.1.23)
1A1dxnds
= , 2A2 dxn ds= , 3
A3dxnds
= . , P A
( ), ,1 2 3P ( , ,1 1 2 2 3 3A d d d ) + + + .
A1 A2 A3n n n = + + An i j k (1.1.24)
1A1dnds = ,
2A2
dnds = ,
3A3
dnds = .
,
ii
d dsnds ds = , (1.1.25) 1, 2,3i =
. (1.1.10)
31 1 1 1 2 1 1 1 1 11 2
1 2 3 1 2
dxd u dx u dx u d u u u1 1ds x ds x ds x ds ds x x x = + + + = + + + 33n n n (1.1.26)
32 2 1 2 2 2 1 2 2 21 2
1 2 3 1 2
dxd u dx u dx u d u u u1 n 1ds x ds x ds x ds ds x x x = + + + = + + + 33n n (1.1.26)
3 3 3 3 3 3 3 3 31 21 2
1 2 3 1 2 3
d u u u dx d u u udx dx 1 n nds x ds x ds x ds ds x x x = + + + = + + + 31 n (1.1.26)
(1.1.18) :
( ) ( . . )1 1 19AA A
ds 1 ds 1ds 1 e dsds 1 e ds 1 1 2 MF 1
= + = = + + +
11
-
1 :
A
ds 1ds 1 2 MF
= + (1.1.27) (1.1.26) (1.1.27) () :
11 1
31 2A1 1 2 3
A A
uu u1xx xn n n
1 2 MF 1 2 MF 1 2 MFAn
+ = + ++ + + (1.1.28)
22 2
31 2A2 1 2 3
A A
uu u1xx xn n n
1 2 MF 1 2 MF 1 2 MFAn
+ = + ++ + + (1.1.28)
33 3
31 2A3 1 2 3
A A
uu u 1xx xn n n
1 2 MF 1 2 MF 1 2 MFAn
+ = + ++ + + (1.1.28)
12 , 13 , 23 2 , . PA PB , , ( ,
).
P A P B
cos 1 1 cos cos = = = A B A B A Bn n n n n n cos A1 B1 A2 B2 A3 B3n n n n n n = + +
, , , ,A1 A2 A3n n n 1 0 0= )
(1.1.29) : , , . , (
PA PB 1Ox 2Ox( ) ( ) ( ), , , ,B1 B2 B3n n n 0 1 0= , A 11MF = , 22MF =
90 = . (1.1.28)
1
1A1
11
u1xn
1 2
+ = + , 1
2B1
22
uxn
1 2
= + (1.1.30)
2
1A2
11
uxn
1 2
= + ,
2
2B2
22
u1xn
1 2
+ = + (1.1.30)
3
1A3
11
uxn
1 2
= + ,
3
2B3
22
uxn
1 2
= + (1.1.30)
12
-
1 : (1.1.29) , :
cos
3 32 1 1 1 2 2
1 2 1 2 1 2 1
11 22
u uu u u u u u2x x x x x x x x
1 2 1 2
+ + + + = + + (1.1.31) 12 . (1.1.12),
cos 1211 22
21 2 1 2
= + + (1.1.32)
:
, 1111
11 11 2
-
1 : 1.1.2.3
, () . , , . . , :
1Ox 2Ox 3Ox
1 2 3dV dx dx dx= dV
( ), ,T 1dx 0 0=OA
( ), , , , 31 21 2 3 1 1 11 1 1
d d d dx dx dxx x x
= = O (1.1.35) (1.1.9) 3 . 2 , ( ), ,T 20 dx 0=OB ( ), , 30 0 dx =O
( ), , , , 31 21 2 3 2 2 22 2 2
d d d dx dx dxx x x
= = O (1.1.35)
( ), , , , 31 21 2 3 3 3 33 3 3
d d d dx dx dxx x x
= = O
)
(1.1.35)
,
dV
(dV = O O O (1.1.36) : . , dV J dV = (1.1.37) J : (1.1.5).
1.1.2.4
.
dAn
dA n . ,
14
-
1 :
)
)n
( , ,1 2 3dx dx dx =1n .
(dV dA= 1n (1.1.38) ( )
, (, ) .
1n
( )dV dA = 1n n (1.1.39) ( ) ( ), ,1 2 3d d d =1n . . (1.1.5) (1.1.10) [ ] () .
[ ] = 1n n1 (1.1.40) (1.1.37)
( ) ( ) ( . . )1 1 40dV J dV dA J dA = = 1 1n n n n [ ] ( ) (dA J dA = 1 1 n n n )n (1.1.41)
, 1n
[ ] (dA J dA = n n) (1.1.42) .
1.1.2.5
) ) Green . , ,
ij 1
-
1 : , . , (1.1.12) . :
111
1
ux
= (1.1.45) 2
222
ux
= (1.1.45) 3
333
ux
= (1.1.45)
2 121 12
1 2
1 u u2 x x
= = + (1.1.45)
3 131 13
1 3
u12 x x
= = + u (1.1.45)
3 232 23
2 3
u12 x x
= = + u (1.1.45)
(infinitesimal strain tensor) Lagrange
[ ] 11 12 1321 22 2331 32 33
= (1.1.46)
ij . (1.1.45)
, , ( ) . , . , . , ( ) .
16
-
1 : 1.2
1.2.1
, . ..
1 1t t= .
() 1t t= . ,
1Q 2Q1d A
( , ,1 2 3P ) , . (traction
vector) 1 , ,
1n
1Q1 n 2Q(1 1t n) n P
( ) lim1 11 1 1d A 0 dd A= pt n (1.2.1)
1d p : () . 1d A
, ( , )
. (1.2.1) 1Q
1d A P
2Q ( ) (1 1 1 1= t n t n) (1.2.2)
Cauchy.
, . . , , .
(traction forces) , (body forces), . , . . 1d V ( ), ,1 2 3P ,
17
-
1 :
lim1
11
1d V 0
dd V
= fpf (1.2.3)
1d fp : . 1d V
, ( ).
1f1t
1.2.2 Cauchy
. . ,
. 1
1d A 1Ox( , ,T1 T 1 0 0= =1n e ) 1e
1Ox 1 j , , ,j 1 2 3= , ( )1 1t e
( ) ( , ,T1 1 1 111 12 13 ) =1t e (1.2.4)
, 2 . ( )1 11 12 13 = + + 1 1 2t e e e e3
3
3
(1.2.5)
( )1 21 22 23 = + + 2 1 2t e e e e (1.2.5) ( )1 31 32 33 = + + 3 1 2t e e e e (1.2.5)
Cauchy (true or Cauchy stress tensor) 1
11 12 13
121 22 23
31 32 33
= (1.2.6)
ij . ij , , ,i 1 2 3= , , ,j 1 2 3= jOx
() . 1d A ie Cauchy
( ).
18
-
1 :
1.2.3
, Cauchy .
() ( )1 1t n Cauchy ,
, , . ,
, ,
1 1e 2e 3e
1n
1 1 1
1 11 1 21 2 31t n n = + + 1 3n1
3n1
3n
(1.2.7) 1 1 1
2 12 1 22 2 32t n n = + + (1.2.7) 1 1 1
3 13 1 23 2 33t n n = + + (1.2.7)
( ) ( ), ,T1 1 1 1 11 2 3t t t=t n ( ), ,1 T 1 1 11 2 3n n n=n . . (1.2.7)
( ) T1 1 1 1 = t n n (1.2.8) .
(
1V1t t= ),
1 1A V= . , . 1V
( ) 1 ( 1 ) :
1t1A f V
( )1 1
1 1 1 1 1
A V
d A d V+ t n f 0=
= 0
(1.2.9)
(1.2.8)
1 1
T1 1 1 1 1
A V
d A d V + n f (1.2.10)
19
-
1 : Gauss (Gauss divergence theorem) , (1.2.10) ( ) ( )1 1 1
T1 1 1 1 1 1 1
V V V
d V d V d V T+ = + = div f 0 div f 0 (1.2.11) 1 , (1.2.11)
V
( )T1 1 + = div f 0 (1.2.12) . (1.2.12) . :
13111 211
1 2 3
f 0 + + + = (1.2.13)
13212 222
1 2 3
f 0 + + + = (1.2.13)
113 23 333
1 2 3
f 0 + + + = (1.2.13) ( 1 ,
2 , 3 ( 1x , 2x , 3x ). , 1d , 2d , 3d .
K : ( )1 1
1 1 1 1 1
A V
d A d V + t n KE f KB 0=
A
(1.2.14)
.
1E . 1B V . 1V
. (1.2.9) (1.2.13), (1.2.14) ,
T1 1 = (1.2.15) Cauchy .
20
-
1 :
21 12 = (1.2.16) 31 13 = (1.2.16) 32 23 = (1.2.16)
.
1.2.4 Piola Kirchhoff
1 Piola Kirchhoff t 0= . . (1.2.9) 1t t=
( )1 1 0 0
1 11 1 1 1 1 1 0 1 0
0 0A V A V
d A d Vd A d V d A d Vd A d V
+ = + t n f 0 t f 0= (1.2.17) , , ( )1 00 t n : 10 f
( ) lim0 11 00 0d A 0 dd A= pt n (1.2.18) lim0
110 0d V 0
dd V
= fpf (1.2.18) (1.2.1) (1.2.18)
( ) ( ) ( ) ( ) 11 0 0 1 1 1 1 0 1 10 0 0d Ad A d A d A= =t n t n t n t n (1.2.19) (1.2.3) (1.2.18)
11 10 0
d Vd V
=f f 10 1J=f f
0
=
(1.2.19)
(1.1.27) . (1.2.17)
1d V Jd V= ( )0 0
1 0 0 1 00 0
A V
d A d V+ t n f 0 (1.2.20) (1.2.19)
21
-
1 :
( ) ( ) ( )1 1T1 0 1 1 1 0 1 10 00d A d Ad A d A = = t n t n t n n 0
(1.2.21)
(1.1.42) ,
( ) T1 0 1 1 00 J = t n n ( )1 0 1 1 00 J = t n n (1.2.22) Cauchy .
1 Piola Kirchhoff 10 P
1 1 10 J
= P (1.2.23) (1.2.22) 1 Piola Kirchhoff ()
0n 0d A( )1 00 t n , ( )1 0 1 00 0 = t n P n (1.2.24) . (1.2.20) 10 P
0 0
1 0 0 1 00 0
A V
d A d V + P n f = 0 (1.2.25) Gauss ( ) ( )0 0 0
1 0 1 0 1 1 00 0 0 0
V V V
d V d V d V + = + = div P f 0 div P f 0 (1.2.26) ,
0V
( )1 10 0 + = div P f 0 (1.2.27) ( ) 10 ijP
11311 120 1
1 2 3
PP P f 0x x x
+ + + = (1.2.28) 12321 220 2
1 2 3
PP P f 0x x x
+ + + = (1.2.28)
22
-
1 :
131 32 330 3
1 2 3
P P P f 0x x x
+ + + = (1.2.28) 1 Piola Kirchhoff Lagrange. (1.2.23) ( Cauchy) .
1.2.5 2 Piola Kirchhoff
, .
. , .
. ,
(. (1.2.21)): 1t t= 1V
( ) ( )1 1
T1 1 1 1 1 1
V V
d V d V + = + div f 0 div f = 0
)
(1.2.29)
( )
( ) ( , ,T1 1 1 11 2 3u u u =r (1.2.30) , (1.2.29)
( )1
T1 1 1 1
V
d V 0 + = div f r (1.2.31) ( ) (calculus of variations). (1.2.31) .
( ) T T1 1 1 1 1 1 1 = + div r div r tr r (1.2.32) 23
-
1 : . (1.2.31) (1.2.33)
( ){ }1
T1 1 1 1 1 1 T 1 1
V
d V 0 + = div r tr r f r ( ){ }
1 1
T1 1 1 T 1 1 1 1 1 1
V V
d V d V + = div r f r tr r (1.2.33) Gauss ,
( ){ }1 1 1
T T1 1 1 1 1 T 1 1 1 1 1 1
A V V
d A d V d V + = n r f r tr r
(1.2.34)
, (1.2.8)
( )1 1 1
T T1 1 1 1 1 T 1 1 1 1 1 1
A V V
d A d V d V + = t n r f r tr r (1.2.34) , (spatial virtual work equation), (spatial external virtual work) (spatial internal virtual work) .
1extW
1intW
, . , 1 , 2 , 3 ( Euler).
1 1 r
1 1 11 1 1
1 2 31 1 1
1 1 2 2
1 2 31 1 1
3 3
1 2 3
u u u
u u u
u u u
=
r 2
3
(1.2.35)
, , , , . (1.2.34)
, T1 1 =
24
-
1 :
T1 1 1 31 2 111 22 33 12
1 2 3 2 1
3 31 213 23
3 1 3 2
uu u u u
u uu u
= + + + + + + + + +
tr r 2
1d V
(1.2.36)
.
( )1
T1 1 1int
V
W = tr (1.2.37)
31 1 2 1
1 2 1 3
1 31 2 2 2
2 1 2 3 2
3 3 31 2
3 1 3 2 3
uu 1 u u 1 u2 2
u1 u u u 1 u2 2
u u u1 u 1 u2 2
+ + = + + + +
1
(1.2.38)
. 1t t= 1 , 1 (work conjugate) () .
1.2.6 Piola Kirchhoff
1x , 2x , 3x , Piola Kirchhoff . , (1.2.34) 1.2.4 :
( )0 0 0
T T1 0 1 0 1 T 1 0 1 1 00 0 0
A V V
d A d V d V + = t n r f r tr P (1.2.39) ,
1intW
25
-
1 :
0d V
d A
0
T1 1 1int 0
V
W = tr P (1.2.40) .
10 P 1
Piola Kirchhoff Green . Green ( ) Green . , , ( ) , .
10 G
Piola Kirchhoff, :
0d p1d p
10 1 1d d
= p p (1.2.41) (1.2.24) (1.2.18)
10 1 1 0 00d
= p P n (1.2.42) ( (1.2.23)) 10 P { }10 1 1 1 0d J d = p n 0 A (1.2.42) , 2 Piola Kirchhoff 10 S
.
0 0d An0d p
11 1 1 1
0 J = S
11 1 10
0 = S P (1.2.43-)
, 10 S ( ), 10 P
T1 10 0 = S S (1.2.44)
26
-
1 : . (1.2.40)
[ ]{ }1 1 10 12 = G I (1.2.45) [ ]I : 3 3 :
0
T1 1 1int 0 0
V
W = Gtr S 0d V
0
(1.2.46)
, , , :
10 S 10 G
( )0 0 0
T T1 0 1 0 1 T 1 0 1 1 00 0 0
A V V
d A d V d V + = Gt n r f r tr S (1.2.47) 10 S . ( )1 00 t n 1 Piola Kirchhoff 1 extW 10 P
(1.2.24). 10 S
1.2.7
Cauchy . 1 11 ()
.
1d A 1Ox1d A
1 Piola Kirchhoff . . , .
10 11P
1Ox1d A
0d A 0d A1Ox
1d A
2 Piola Kirchhoff .
27
-
1 : 1 Piola Kirchhoff . , () . , .
10 11S
1d A0d A
1Ox1d A
, 2 Piola Kirchhoff ( ) , Cauchy 1 Piola Kirchhoff . 2 Piola Kirchhoff .
1.2.8 Cauchy
, . , ( 1x , 2x , 3x ) ( 1 , 2 , 3 ) , Cauchy . , (1.2.13) :
3111 210 1
1 2 3
f 0x x x
+ + + = (1.2.48) 3212 22
0 21 2 3
f 0x x x
+ + + = (1.2.48) 13 23 33
0 31 2 3
f 0x x x + + + = (1.2.48)
(. (1.2.7)) 0 1 11 0 1 21 0 2 31 0 3t n n n = + + (1.2.49) 0 2 12 0 1 22 0 2 32 0 3t n n n = + + (1.2.49) 0 3 13 0 1 23 0 2 33 0 3t n n n = + + (1.2.49) Cauchy , .
28
-
2 :
2.1
: 6 (. (1.1.12)
Green . (1.1.45) )
3 (. (1.2.13) Cauchy . (1.2.28) 1 Piola Kirchhoff)
( ) :
(
)
( )
, : t 3 1u , 2u , 3u 6 11 , 22 , 33 , 12 21 = , 13 31 = ,
23 32 = 6 . 2 Piola Kirchhoff 11S ,
22S , 33S , 12 21S S= , 13 31S S= , 23 32S S= . [ ]X . , . 1.2, .
15 9 ,
. 6 ( ). . . . .
30
-
2 :
)
.
. . . , .
2.2
, , . , .
. () (ideally elastic) () . : ,
. .
, . , , .
,
, . . (hyperelastic) Green (Green elastic)
, Green
( ), , , , , , , ,t t t t t t t t t0 11 0 22 0 33 0 21 0 12 0 31 0 13 0 32 0 23 1 2 3F F x x x = = = =( , ,1 2 3x x x , : ( , , , , , , , ,t t t t t t t0 11 0 22 0 33 0 12 0 13 0 23 1 2 3d W F x x x d V = ) 0
(2.2.1)
01 2 3d V d x d x d x = :
t
0 ij : Green
31
-
2 :
) 0
td W : () () . 0d V (strain energy function per unit initial volume).
F
F ( , , , , ,t t t t t t t0 11 0 22 0 33 0 12 0 13 0 23d W F d V = (2.2.2) , (homogeneous). . (2.2.1) (2.2.2) .
F
. (1.2.46) 2 Piola Kirchhoff Green ( )d , ( ) ( )...t t t t t t t t t0 11 0 11 0 22 0 22 0 23 0 23 0 32 0 32d d W S d S d S d S d d V = + + + + 0 (2.2.3) . (2.2.2) ( )d ,
( ) ...t t t t t0 11 0 22 0 23 0 32t t t t0 11 0 22 0 23 0 32
F F F Fd d W d d d d d V = + + + +
0 (2.2.4)
. (2.2.3) (2.2.4)
t0 ij t
0 ij
FS = , (2.2.5-) , , ,i j 1 2 3 =
6 (2.2.5-) . (15 15 ).
F
, , :
F
( ) (( ) ...
3 3 3 3 3 3t t
ij 0 ij ijkl 0 ij 0 kli 1 j 1 i 1 j 1 k 1 l 1
3 3 3 3 3 3t t t
ijklmn 0 ij 0 kl 0 mni 1 j 1 k 1 l 1 m 1 n 1
F A B
C
= = = = = =
= = = = = =
= +
+
)t + +
(2.2.6)
32
-
2 :
ijA , ijklB , , : . ijklmnC . , . (2.2.5) , . . (2.2.6) . (2.2.5)
t0 ij 0 td W
( ) ( ) ...3 3 3 3 3 3t t t0 rs rsij 0 ij rsijkl 0 ij 0 kli 1 j 1 i 1 j 1 k 1 l 1
S A B C= = = = = =
= + + + t (2.2.7) , . (2.2.6) A 0= (2.2.8) . (2.2.7) ( ) ( ) ...3 3 3 3 3 3t t t0 rs rsij 0 ij rsijkl 0 ij 0 kl
i 1 j 1 i 1 j 1 k 1 l 1S B C
= = = = = = = + t +
G0
)
(2.2.9)
, ,
, . (2.2.9) .
t0 ij 1
-
2 :
)
( ) ( , , , , ,t t t t t t0 11 0 22 0 33 0 12 0 13 0 23 =t0 : Cauchy. .
( ) ( ), , , , ,t t t t t t t t t0 11 0 22 0 33 0 12 0 12 0 13 0 13 0 23 0 232 2 2 = = =t0 = : .
eD . , . , (anisotropic). 3 (orthotropic) , (isotropic). .
6 6 36 =ijk
ijk
. , , eD :
ij jik k= , (2.2.12) , , ,i j 1 2 3 = 2
( )( )
( )
11 12 12
12 11 12
12 12 11
11 12
11 12
11 12
k k k 0 0 0k k k 0 0 0k k k 0 0 0
10 0 0 k k 0 02
10 0 0 0 k k 02
10 0 0 0 0 k2
=
eD
k
(2.2.13)
12k = (2.2.14) 11 12k k 2 = (2.2.14)
34
-
2 : () Lam , (Lam constants). :
(t t t t0 11 0 11 0 22 0 331 S S S )E = + (2.2.15) (t t t t0 22 0 22 0 11 0 331 S S S )E = + (2.2.15) (t t t t0 33 0 33 0 11 0 221 S S S )E = + (2.2.15)
t t t0 12 0 12 0 12
12 SG
= = , t t t0 13 0 13 0 1312 SG = = , t t t
0 23 0 23 0 2312G
= = S (2.2.15--)
( )3 2E + = + : Young (elastic modulus,
Young s modulus) G = : (shear modulus)
( )2 = + : Poisson (Poisson s ratio)
,E G,
( )EG
2 1 = + (2.2.16)
2.3
2.3.1
(elastoplastic, inelastic) () . :
. ,
. , , () .
35
-
2 :
, () (rate independent elastoplasticity). (rate dependent elastoplasticity, viscoplasticity). .
, ( ) , Cauchy ( 1x , 2x , 3x Lagrange) . :
1. 2
1t , t2 1t t d= + () ( ) eld , pld .
= el pld d + d (2.3.1) ( )d : .
. ( ) ( ), , , , ,T 11 22 33 12 13 23d d d d d d =d : , 1t 2 1t t dt= + ( ) ( , , , , ,T el el el el el el11 22 33 12 13 23d d d d d d ) =d : , 1t 2 1t t dt= + ( ) ( ), , , , ,T pl pl pl pl pl pl11 22 33 12 13 23d d d d d d =d : ,
1t
2 1t t d= + t2. Hooke (. (2.2.11))
() .
3. , .
(), f
( ) ( )( yf f ,= q q ) (2.3.2)
36
-
2 : (yield function), q : (internal variables) . : ( . (2.3.3)).
y : ( ). , (kinematic hardening law).
y (isotropic hardening law). , . .
( )( )yf , q 0 .
, ( )( )yf , = q 0
( ) d , . 2 3 , - :
( ) ( ). .2 3 1 e el e pld = D d d = D d -d
pl
e ed = D d - D d (2.3.4) . (flow rule) Prandtl Reuss:
pld
37
-
2 :
fd = pld
(2.3.5)
d : (proportionality factor)
T
11 22 33 12 13 23
f f f f f f f, , , , , =
:
f .
. d . ( ) f
T
yy
f fdf d = + d (2.3.6)
3 : df 0> , t dt+ ( )( )yf , 0 > q ,
t ( )( )yf , 0 = q .
( )( )y, 0 q f df 0< , t dt+ ( )( )yf , 0 , pld 0 , . (plastic loading). df 0= (consistency condition)
d
. , . (2.3.6) . (2.3.4) :
38
-
2 :
( )T ( 2.3.5 )yy
f fdf d = +
e e plD d D d
T
yy
f fdf d df = +
e eD d - D
(2.3.8)
y ( ) . Von Mises
( )( ) ( ) (y e yf , = q q) (2.3.9)
( ) ( ) ( ) ( ) ( )2 2 2 2 2 2e 11 22 22 33 11 33 12 131 32 = + + + + + 23 (2.3.10)
e : (effective stress) . y (equivalent plastic strain) pleq . , { }pleq=q (2.3.11)
( pl )y eqh = (2.3.12) pleq
tpl pl
eq eq0
d = (2.3.13)
pleqd : o
( )2 2 2 2 2 2pl pl pl pl pl pl pleq xx yy zz 12 13 232 1d d d d d d d3 2 = + + + + + (2.3.14)
39
-
2 : ( ) ( )pl pl ply eq y eq eqh d h d = =
)
(2.3.15) ( pleqh : pleq (plastic modulus) yd . (2.3.8) pleqd . ( )pleqh . ( )pleqh . = , y .
(hardening),
( )pleqh > 0( )pleqh 0<
(softening) - (elastic - perfectly plastic material).
( )pleqh 0 = . (2.3.9) - (2.3.10) , :
( )11 22 3311 e
1f 2
+ = (2.3.16)
( )22 11 3322 e
1f 2
+ = (2.3.16)
( )33 11 2233 e
1f 2
+ = (2.3.16) 12
12 e
f 3 = (2.3.16)
13
13 e
f 3 = (2.3.16)
23
23 e
f 3 = (2.3.16)
(2.3.14), (2.3.5) - (2.3.16) :
pleqd d = (2.3.17)
y
f 1 = ( . (2.3.9)), (2.3.8),
(2.3.17), :
40
-
2 :
( )T pleqf fdf d h d = e eD d D T Tf f fdf h d = +
e eD d D
(2.3.18) ( )pleqh h = . , df , . (2.3.18) 0= d :
T
T
f
df f h
= +
e
e
D d
D
(2.3.19)
d
. . (2.3.5) . (2.3.4)
d
fd =
e ed D d D
(2.3.20)
epD d . (2.3.19) . (2.3.20).
= epd D d (2.3.21)
TT
T
f f
f f h
= +
e e
ep e
e
D D D D
D
(2.3.21)
2.3.2 -
, .
41
-
2 :
t
( , , ) . . Newton Raphson, , . , (integrating the rate equations).
( ) . , . , .
2 ,
1t 2 1t t = + . ,
. . .
1t
2t
. (2.3.21)
. (2.3.20), , :
= epd D d
t 2 t2 t2
t1 t1 t1
f fd d = = e e e ed D d D d D d D ( ) ( ) t 22 1
t1
ft t d = e e D D ( ) ( ) t 22 1
t1
ft t d = + e e D D ( ) t 22
t1
ft = tr e D d
)t
(2.3.22)
( ) ( , , , , ,T 11 22 33 12 13 23 = : ,
1t
2 1t t = +
42
-
2 :
( )1t = + tr e D (2.3.23) , eD 3 2.3.1, , . (2.3.22) : 1. tr ( )1t
, . (elastic prediction) .
eD
2. t 2
t1
fd . d . (2.3.19) f , . (2.3.20)
t 2
t1
fd . ,
( )t 2 2t1
fd t = = tr0 (2.3.23) 2t ( )( )y 2f , t > 0 , , .
t 2
t1
fd 0 . , :
2t
2t1. ( )( )y 1f , t 0 = .
( )( )y 2f , t 0
-
2 : 2.3.2.1 Euler (forward Euler)
. t 2
t1
fd
( )t 2 1t1
f fd t (2.3.24) . (2.3.19):
( )( ) ( ) ( )
T t 2T
1t 2 t 2t1
T Tt1 t1
1 1
ff td
f f f fh t t h
= + +
ee
e e
D dD d
D D 1
t
( )( ) ( ) ( )
T
1
T
1 1
f t
f ft t
+
e
e
D
D 1
h t (2.3.25)
, Euler (forward Euler predictor). .
1t( )2t
( )( )y 2f , t 0 . .
- (subincrementation)
t 2
t1
fd [ ],1i 1i 1t t + , ,...,i 1 n 1= .
n
2 1t ttn
= 11 1t t= , 12 1t t t= + , 13 1t t 2 t= + , , . 1n 2t t=
44
-
2 :
n
=i [ ],1i 1i 1t t + . t 2 t12 t13 t 2
t1 t11 t12 t1n 1
f f fd d d ... d
= + + + f ( ) ( ) ( ) (t 2 1 11 2 12 3 13 n 1n
t1
f f f f fd t t t ... t + + + + )1 (2.3.26)
( )( ) ( ) ( )
T
1i
i T
1i 1i 1i
f t
f ft t
+
ei
e
D
D
h t (2.3.27)
( )if t . (2.3.22), (2.3.23) (2.3.16):
( ) ( )1i 1 1it t+ = + tr e i D (2.3.28) ( ) ( ) ( )1i 1 1i 1ift t + = tr e D t
(2.3.29)
, ,...,i 1 2 n 1= . ( )1i 1t + . (2.3.16) ( 1i 1f t + ) [ ],1i 1 1i 2t t+ + .
(2.3.21) (2.3.30) = = ep epi id D d D i ep iD . . (2.3.26)-(2.3.29) Von Mises
( ) ( )T1i 1if ft t 3 G = eD
, ,...,2 n 1, i 1 (2.3.31) =
45
-
2 : .
. . . , . . . - .
1t t= , = tr e D , f ( ),
2.3.2.2 Euler (backward Euler)
( )t 2 2t1
f fd t (2.3.32) ( )
2t
( )2f t . (fully implicit). . :
2t
( ) t 22t1
ft d = tr e D ( ) ( )2 ft = tr e D 2t (2.3.33) ( )( )y 2f , t = 0 (2.3.34)
46
-
2 : . (2.3.33) . (2.3.34) . (closest point projection) . (2.3.33) , () , . tr
. (2.3.33) .
. . (2.3.34) . . (2.3.34) .
( )2t
. (2.3.33) - (2.3.34) - k
( k )( k ) ( k ) 2
( k ) ( k ) ( k )2
f f f + = tr e e e D D D
(2.3.35)
( )( )
kk
yy
f ff 0
+ + =
( )( ) ( )( ) k kk pleqff h + = 0 (2.3.35)
2
2
f :
2
211
2 2
211 22 22
2 2 2
2211 33 22 33 33
2 2 2 2 2
211 12 22 12 33 12 12
2 2 2 2 2
211 13 22 13 33 13 12 13 13
2 2 2
11 23 22 23 33 23
f
f f .
f f ff
f f f f
f f f f f
f f f
=
2 2 2
212 23 13 23 23
f f f
47
-
2 : Newton Raphson. k 1+ -
( k 1 ) ( k )+ = + (2.3.36) ( k 1 ) ( k ) + = + (2.3.36) ( ) ( )k 1 kpl pl
eq eq + = + (2.3.36) .
,
( )k 11f tol
+ (2.3.37)
1tol : T . ( )k 1f 0+ , . ,
1tol
2
1tol 10 10= 8
)
)
(2.3.38) , . (2.3.37), :
( )k 1f +
(( )( )( ) ( ) , k 1k 1 k 1 ply eqf f ++ += (2.3.39)
( 0 ) = tr (2.3.40) ( 0 ) 0 = (2.3.40)
( )( )0pl pleq eq 1t = (2.3.40) ( )( )( )( ) ( ) , 00 0 ply eqf f =
( )(( ) ,0 y 1f f = tr t (2.3.41) ( )0f trf . tr
trf 0> (2.3.42)
, . 1t
( )trij ij 1t > , , ,i j 1 2 3 = k 0= . (2.3.35)
48
-
2 :
tr trf f + = = tr tr e e 0 D 0 D
(2.3.43)
(2.3.35)
( )( )trtr pleq 1ff h t
+ 0= (2.3.43) (2.3.43) (2.3.43)
( )( )tr trtr pleq 1f ff h t 0
+ = eD
( )( )tr
tr trpl
eq 1
f
f f h t
= +
eD
( )tr
1
f3 G h t
= + (2.3.44) ( ) ( )( )pl1 eqh t h t = 1 .
, . (2.3.31)
( ) ( )T1i 1if ft t = eD
3 G
.
( )tt tr
. (2.3.44) Euler (backward Euler predictor). Euler trf Euler ( )1f t ( (2.3.25)). ( ) tr
( )1f t .
. Euler . .
2ttr
49
-
2 :
, k2
2
f .
, , . (2.3.35) .
2.3.2.3 (Generalized trapezoidal and generalized midpoint rule)
. (2.3.32).
( ) ( ) ( )t 2 1 2t1
f fd 1 a t a + f t
(2.3.45) a : .
( ) ( ) ( )t 2 1 2t1
f fd a 1 a t a + t (2.3.46) 1 = . (2.3.45) Euler. 1a
2= ,
1 = .
. (2.3.35) (2.3.40).
0 ( )2f t .
2
2
f .
2.3.2.4 generalized cutting plane
Simo & Ortiz (1985)
2
2
f
. ( . (2.3.45)) 0 = . . (2.3.35)
( ) ( )2 ft = tr e D 1t (2.3.47)
50
-
2 :
( ) ( )( k ) ( k ) 1f t + = tr e e D D 1f t (2.3.48)
k 1+
( ) ( )( k 1 ) ( k 1 ) 1 k 1f t + + + 1f t + = tr e ek+1 D D
(2.3.49)
. (2.3.49) (2.3.48)
, ( k 1 ) ( k )+ = + ( k 1 ) ( k ) + = +
( )k 1 1f t + = ek+1 D (2.3.50) ,
k
( )1f t = e D (2.3.51) . (2.3.51) (. (2.3.35))
( ) ( )( ) ( )( ) k kk p1 eqf f lf t h 0 = eD ( )
( )
( )( )
k
kk
1
f
f f t h = +
eD
(2.3.52)
( )( )( ) kk pleqh h = .
. (2.3.52) . (2.3.36-) (. (2.3.37))
2
2
f .
f .
generalized cutting plane
( )1f t ( )kf
. (2.3.51), (2.3.52).
51
-
2 :
( )
( ) ( )( )
k
k kk
f
f f h = +
eD
( )
( )
k
k
f3 G h
= + (2.3.53) Von Mises . ,
2
2
f ,
, .
generalized cutting plane. - :
k
. (2.3.53), ( ) ( )k
k
f3 G h
= +
( k 1 )+ ( k )
( k 1 ) ( k ) f+ = e D
( )k 1pleq + ( ) ( )k 1 kpl pleq eq + = + ( )k 1h + ( )( )( ) k 1k 1 pleqh h ++ = ( )k 1f + ( )( )( )( ) ( ) , k 1k 1 k 1 ply eqf f ++ += ( )k 1 1f tol+
1tol .
: ( ) ( k 1 )2t += , , .
( ) ( )k 1pl pleq 2 eqt +=( ) ( )( )k 1pl2 eqh t h + = ( )k 1 1f tol+ > .
k 2+ ( )( )( ) ,0 tr ply eq 1f f f t = = tr ,
, ( 0 ) = tr ( 0 ) trf f = , , ( )
( )0pl pleq eq 1t = ( ) ( )( )0 pleq 1h h t = .
2.3.3
Cauchy 2 Piola Kirchhoff Green
52
-
2 : . , , : , .
53
-
3 :
3.1
, 15 15 (3 , 6 , 6 ). , , . , .
: ,
. , , , . , .
, , .
St. Venant . : , , .
(, ). , St. Venant , , . , , .
(torsional loading) . : Mt (torque), . Mt
55
-
3 : , . .
. 3.1.1, - : .
. 3.1.2, : , . , , ( ) . .
() () 3.1.1 () ()
56
-
3 :
()
() ()
3.1.2 () () ()
Coulomb (1784), . . Coulomb . .
, . St. Venant (1855)
57
-
3 : (. 3.1.3). , .
3.1.3
t
()
3.1.4 Saint-Venant,
St. Venant
St. Venant (uniform torsion). . ( ) , (non uniform torsion). St. Venant (. 3.1.5).
58
-
3 :
3.1.5
St. Venant
. () . , , . , St. Venant, .
3.2
3.2.1
. , , . : ,
.
: .
: . , .
Poisson , 0 =
: , Green ij 1
-
3 :
.
, : , .
.
3.2.2
(semi-inverse method) . ) ) ( ), .
l
, .
.
=
1 2 3x x x M , 1x , 2x 3x . 1x ( )1x .
, 1Mx . , (
2 3u , u 2u
2x 3u 3x ) . (. 3.1.6):
( ) ( ) ( ) ( ) ( ), , sin sin2 1 2 3 1 3 1u x x x PP MP x x x = = = (3.2.1) ( ) ( ) ( ) ( ) ( ), , cos cos3 1 2 3 1 2 1u x x x PP MP x x x = = = (3.2.1)
60
-
3 :
2 0t =3 0t =
+1
()
1 2
11+= = Kj j
3u
2u1(x )
n
t
s
M
3x
2x
3.1.6 , : (Trahair, 1992)
2 3u , u
( ) ( ) ( )( ), , sin cos2 1 2 3 3 1 2 1u x x x x x x 1 x= (3.2.2) ( ) ( ) ( )( ), , sin cos3 1 2 3 2 1 3 1u x x x x x x 1 x= (3.2.2)
,
, . ,
1u
1x . 1u
( ) ( ) (, , ,P1 1 2 3 1 M 2 3u x x x x x x = )
)
(3.2.3)
( ,PM 2 3x x : , (primary warping function). . (3.2.3),
1u
1x ( ) .1x = = (3.2.4)
61
-
3 : , ( ), . , 3 , (3.2.3). ( )1x ( ), ,SM 1 2 3x x x (secondary warping function). .
( ), ,1 1 2 3u x x x
( ) ( ) (, , ,P1 1 2 3 s 1 M 2 3u x x x u x x x = + ) (3.2.5)
( )s 1u x : . , ( )s 1u x , ( ) ( ), ,1 1 2 3u x x x (. 3.1.3, 3.1.4). ( )s 1u x . . , , ( )s 1u x
( ) .s 1 su x u = = (3.2.6)
3.2.3
. (. . (1.1.45), (1.1.46)). . (3.2.1), (3.2.3) :
62
-
3 :
111
1
u 0x
= = (3.2.7) 2
222
u 0x
= = (3.2.7) 3
333
u 0x
= = (3.2.7) P
1 212 3
2 1 2
u u xx x x
= + = (3.2.7) P
3113 2
3 1 3
uu xx x x
= + = + (3.2.7) 32
233 2
uu 0x x
= + = + = (3.2.7) 12 13, .
Green (. . (1.1.12) (1.1.34)). . : , ( )1x .
( ), ,1 1 2 3u x x x ( )1x . Green :
2 2 2 2
3 31 1 2 1 211 11
1 1 1 1 1 1 1
u uu 1 u u u 1 ux 2 x x x x 2 x x
= + + + = + +
2 (3.2.8)
2 2 2 2
3 32 1 2 2 222 22
2 2 2 2 2 2 2
u uu 1 u u u 1 ux 2 x x x x 2 x x
= + + + = + +
2 (3.2.8)
2 2 2 2
3 3 31 2 233 33
3 3 3 3 3 3 3
u u u1 u u 1 ux 2 x x x x 2 x x
= + + + = + +
2
3u (3.2.8)
3 32 1 1 1 2 212
1 2 1 2 1 2 1 2
u uu u u u u ux x x x x x x x
= + + + +
3 32 1 2 212
1 2 1 2 1 2
u uu u u ux x x x x x
= + + +
(3.2.8)
63
-
3 :
3 31 1 1 2 213
1 3 1 3 1 3 1 3
u uu u u u ux x x x x x x x
= + + + + 3u
3 1 2 213
1 3 1 3 1 3
u u u u 3 3u ux x x x x x
= + + +
(3.2.8)
3 32 1 1 2 223
2 3 2 3 2 3 2 3
u uu u u u ux x x x x x x x
= + + + + 3u
3 2 2 223
2 3 2 3 2 3
u u u u 3 3u ux x x x x x
= + + +
(3.2.8)
. (3.2.2), (3.2.5)
( ) ( )22 211 s 2 31u x x2 = + + (3.2.9) 22 0 = (3.2.9) 33 0 = (3.2.9)
P
12 32
xx = (3.2.9) P
13 23
xx = + (3.2.9)
23 0 = (3.2.9) (3.2.7), (3.2.9) :
, 11
11 .
( ) ( )22 22 31 x x2 + Wagner (Wagner term) (Trahair, 1992)
3.2.4
. . , Green 2
64
-
3 : Piola Kirchhoff . (2.2.15). 0 = , ( . (3.2.9)):
( ) ( )22 211 11 11 s 2 31S E S E u E x x2 = = + + 0
(3.2.10)
22 22 22S E S= = (3.2.10) 33 33 33S E S 0= = (3.2.10)
P
12 12 12 32
S G S G xx = = (3.2.10) P
13 13 13 23
S G S G xx = = +
0
(3.2.10)
23 23 23S G S= = (3.2.10) . . (3.2.10), (3.2.10) 0 = : 0 , , .
22 33S , S
Cauchy ( 1 2 3x x x ) . (3.2.10) , 11 ,
11 0 = (3.2.11) 11 0 = . 0 = 11 , . 22 33S , S
, . . (2.3.4) ,
11 12 13S , S , S ( e el e )pldS = D d dS = D d -d (3.2.12) ( )d :
( ) ( , ,T 11 12 13dS dS dS=dS )
65
-
3 : ( ) ( ), ,11 12 13d d d =d ( ) ( , ,pl pl pl11 12 13d d d ) =pld
E 0 00 G 00 0 G
= eD
P , dt
Pd = 0 (3.2.13) (3.2.10) (3.2.9)
( )2 2 pl11 s 2 3 11dS E du E x x d E d = + + (3.2.14) P
pl12 3 12
2
dS G d x G dx = (3.2.14) P
pl13 2 13
3
dS G d x G dx = + (3.2.14)
11d ,
11d 0 = (3.2.15)
3.2.5 P -
, 3 ,
( ) ( ) ( ),PM 2 3 1 s 1x x , x , u x , ( ) .1x = , ( ) .s 1u x = . 12 15
(6 - 6 ). 3 ( ). 1 ( 1x ) (. . (1.2.48 (1.2.28))
66
-
3 :
3111 211
1 2 3
f 0x x x
+ + + = (3.2.16)
1311 120 1
1 2 3
PP P f 0x x x
+ + + = (3.2.17) . . (3.2.11),
11
1
0x = (3.2.18)
. (3.2.10) (3.2.10), . (3.2.16)
2 P 2 P3111 21
1 2 21 2 3 2 3
f 0 0 G G 0 0x x x x x
+ + + = + + + = G 02 P 2 PG 0 0, = = (3.2.19)
1f 0= : () 1Mx .
( ) ( ) ( )2 22 22 3
2x x = + : Laplace
, . ( ),
1n 0= , (3.2.20) 2 3n , n R
:1 2 3n , n , n ( ) ( , ,T 1 2 3n n n=n )
. 1 (. . (1.2.49)
( ) ( ). . , . .3 2 10 3 2 101 11 1 21 2 31 3 1 21 2 31 3t n n n t 0 n n
= + + = + + P P
G 03 2 2 3
2 3
G x n G x n 0x x
+ + =
67
-
3 :
P
3 2 2 3x n x n , n = (3.2.21)
1t 0= : 1Mx . (, ).
( ) ( ), ,T 1 2 3t t t=t n n
1t
( ) ( ) ( )2
2 3
nn x x
= + 3n : n
, , ( ),P 2 3x x Laplace (3.2.19) Neumann (3.2.21)
.
, , (3.2.14) . , (3.2.16) ( 1.2.49)) . , :
31 3111 21 11 211
1 2 3 1 2 3
dd df 0x x x x x x
0 + + + = + + =
n
(3.2.22)
1 11 1 21 2 31 3 1 11 1 21 2 31 3t n n n dt d n d n d = + + = + + (3.2.23) Poisson
plpl2 P 1312
M2 3
d1 d , d x x
= +
(3.2.24)
Neumann
plP pl1312
3 2 2 3dd x n x n ,
n d d
= + +
(3.2.25)
(3.2.17) ( ). 1 Piola
68
-
3 : Kirchhoff , 2 Piola Kirchhoff. (1.2.43) 1 Piola Kirchhoff .
(3.2.17) 2 Piola Kirchhoff. . Washizu (1975) . 3.2.7 (. . (3.2.57)) . , :
( ). . 2 P 2 P3 2 141311 12
0 1 2 21 2 3 2 3
SS S f 0 0 G G 0x x x x x
+ + + = + + + = 0 2 P 0, = (3.2.26)
0 1 11 1 21 2 31 3 21 2 31 3t S n S n S n 0 0 S n S n= + + = + +
( ). . P P3 2 143 2 2 3
2 3
G x n G x nx x 0 + + =
P
3 2 2 3x n x n , n = (3.2.27)
.
11SP (
). ,
. (3.2.26) . (3.2.22) .
13 1311 12 11 120 1
1 2 3 1 2 3
SS S dS dS dSf 0x x x x x x
+ + + = + + = 0 (3.2.28)
0 1 11 1 21 2 31 3 0 1 11 1 21 2 31 3t S n S n S n d t dS n dS n dS n= + + = + + (3.2.29) (. (3.2.24), (3.2.25)):
plpl2 P 1312
M2 3
d1 d , d x x
= +
(3.2.30)
69
-
3 :
plP pl1312
3 2 2 3dd x n x n ,
n d d
= + +
(3.2.31) , . : , (3.2.30), (3.2.31) .
3.2.6 s, u - ( )
s, u , ( ). . . , Cauchy : (. . (1.2.34) (1.2.37))
[ ] [ ]( ) ( )T T TV A V
dV dA dV = + tr t n r f r (3.2.32) ( ) , A : H () V : () [ ] [ ], : Cauchy , 1 2 3x , x , x ) , ) (. . (3.2.10-), (3.2.11)). , , (. . (3.2.1), (3.2.3)),
f( ) ( )12 13,
( ) ( , ,T 1 2 3u u u =r )
P P1 Mu M = + (3.2.33)
70
-
3 :
2 3u x = (3.2.33) 3 2u x = (3.2.33)
P P
12 32 2
xx x = +
(3.2.34) P P
13 23 3
xx x = + +
(3.2.34) 1 . (3.2.32) :
( )1 12 12 13 13V
dV = + = ( ). .P P P P 3 2 10
12 3 13 22 2 3 3V
x x dVx x x x
= + + + + 22P P P P
3 2 12 132 3 2 3V V
G x x dV dVx x x x = + + + + =
22 lP P P P
3 2 1 12 132 3 2 3x1 0 V
G x x d dx dVx x x x
=
= + + + +
( ) P P1 t 2 1 12 132 3V
I G dx x = + + V (3.2.35)
22P P
t 3 22 3
x x dx x
= + + : (torsion constant).
P P2 2 M M
t 2 3 2 33 2
x x x x dx x
= + + ( )2 l = : (2 1 )x 0 = = :
1I .
71
-
3 :
lP P P P
12 12 13 1 12 132 3 2 3V x1 0
I dV dxx x x x
=
= + = + d ( )P P131212 12 2 13 3
2 3
I l d n n dsx x
= + + (3.2.36) 2I . (3.2.32) :
( )( ) ( )( ) ( )( ) ( )( )T T T T2A 1 2
I dA d d dA
= = + + t n r t n r t n r t n r (3.2.37)
1 : ( 1x 0= ) 2 : ( 1x l= )
A :
( ) ( ) ( ) ( )T 1 1 2 2 3t u t u t 3u = + + t n r n n n
))
)2 3
, ( ) ( , ,T1 1 0 0 = n (3.2.38) ( ) ( , ,T1 1 0 0 =n (3.2.38) ( ) ( , ,TA 0 n n =n (3.2.38) (3.2.37) (1.2.49) Cauchy
( ) ( )( )
2 12 2 13 3 12 2 13 31 2
1 1 2 2 3 3
I u u d u u
t u t u t u dA
d
= + + +
+ + +
+ (3.2.39)
:
1t 0= (3.2.40) (3.2.21). . (. . (1.2.49-)
22
32
02 22 2 32 3 20t n n t
=== + = 0
0
(3.2.40) 23
33
03 23 2 33 3 30t n n t
=== + = (3.2.40)
72
-
3 : (3.2.38)
( ) ( ) ( ) ( )2 12 3 13 2 12 3 13 21 2
I x x d x x
= + + + d1
2 t 2 2 t1I M M =
)
(3.2.40)
(ti 12 3 13 2i
x x d
= + : (torque, torsional moment). 12 13, (. (3.2.10-)
t 2 t1M M= (3.2.41)
(2 t 2I M )1 = (3.2.42) t t 2 t1M M M= = . ,
( ) ( )1 2 t 2 1 12 t 2 1I I G I M = + = (3.2.43) P1 2, , ( )
tG = tM (3.2.44) ,
,
P 0 131212
2 3
12 2 13 3
I 0 0 x x
n n 0
= + = + =
(3.2.44-)
. (3.2.44) (governing equation) . . , , 2 Piola Kirchhoff . (. . (3.2.26) (3.2.57)).
. (3.2.44) . , . (3.2.19) (3.2.21), t tM . (3.2.44).
73
-
3 : .
. .. ( )t t 1M M x= . (3.2.44) , . (3.2.4) ( . = ). , . (3.2.3) .
, , . , () . St. Venant (center of twist) . . . . ( ) .
, 2 Piola- Kirchhoff . 1.2.47:
[ ] ( )TT 0 T0 0V A V
dV dA dV = + Gtr S t n r f r (3.2.45) A : H () V : () [ ] , GS : 2 Piola- Kirchhoff Green , 1 2 3x , x , x ( 00 t n) : o 0 f : To 1I , 11 11S .
74
-
3 :
( ) ( )1 11 11 12 12 13 13 11 11 t 2 1V V
12I S S S dV S dV G = + + = + + I (3.2.46)
( )P131212 12 2 13 32 3
SS PI l d S n S nx x
= + + ds (3.2.47) 11 (. . (3.2.9)) ( )2 211 s 2 3u x x = + +
PM
(3.2.48)
(. . (3.2.2) (3.2.5))
P1 s Mu u = + + (3.2.49)
(2 3 2u x cos x sin ) = )
(3.2.49)
(3 2 3u x cos x sin = V
(3.2.49) 11 11
V
S d 1I
( ) ( ) ( ) ( )22 2 2 211 11 s 2 3 s 2 3V V
1S dV E u E x x u x x dV2
= + + + + = ( ) ( ) ( ) ( ) ( )22 32 2 2 2 2 2s s 2 3 s 2 3 s 2 3
V
1 1E u u x x u x x u x x dV2 2
= + + + + + + ( ) ( ) ( ) ( ) ( )2 311 11 s P s2 s1 P s PP 2 1
V
1 1S dV E A u I u u E I u I2 2
= + + + (3.2.50)
A d
= , ( )2 2P 2 3I x x d
= + , ( )22 2PP 2 3I x x d
= + (3.2.51--) 2I (3.2.45) , 1 Piola Kirchhoff (. . (1.2.24)). . (3.2.37) (3.2.39) 2I
( ) ( )( )
2 11 1 21 2 31 3 11 1 21 2 31 31 2
0 1 1 0 2 2 0 3 3
I P u P u P u d P u P u P u d
t u t u t u dA
= + + + + +
+ + +
+ (3.2.52)
75
-
3 : , . ,
0 1t 0=A
. (1.2.43)
[ ] [ ] [ ] [ ] [ ] [ ]1= = S P P S (3.2.53)
1 1 111 11 11 12 21 13 31 11 21 31
1 2 3
u u uP X S X S X S 1 S S Sx x x
= + + = + + +
( ) P P11 s 11 21 312 3
P 1 u S S Sx x = + + +
(3.2.54)
11P 1x 11 21 31S , S , S 1x (. . (3.2.10)). , . (3.2.49)
( ) ( ) ( )( ) ( )
P P2 s2 s1 11 2 1 11 2 1 11
21 2 31 3 21 2 31 32 1
I u u P d P d P d
P u P u d P u P u d
= + +
+ + +
+
(3.2.55)
( ) .1 2 1x 0 = = = 2 1 0 = ,
( ) ( ) ( ) ( ). .3 2 492 s2 s1 21 2 31 3 21 2 31 32 1
I u u N P u P u d P u P u d
= + + + ( ) ( ) ( )
( ) ( )cos sin cos sin
cos sin cos sin
2 s2 s1 2 21 3 2 2 2 31 2 2 3 22
1 21 3 1 2 1 31 2 1 3 11
u u N P x x P x x d
P x x P x x d
= + + +
1
( )2 s2 s1 t 2 2 t1I N u u M M = + (3.2.56)
11N P d
= : . ( ) ( )cos sin cos sinti 21 3 i 2 i 31 2 i 3 i
i
M P x x P x x d
= + : . . (3.2.56)
tM . .
76
-
3 :
2I P , ,
( ), . (3.2.47)
P 0 13121 2 12
2 3
12 2 13 3
SSI I I 0 0 x x
S n S n 0
= = + = + =
(3.2.57-)
2 Piola Kirchhoff . 1I , 2I
( ) ( )2s2 s1 s P1 u u 0 E A u E I2 + = N (3.2.58) ( )32 P s PP t1 0 E u E I G I M2 + + = t 2 (3.2.58) ( )31 P s PP t1 0 E u E I G I M2 + + = t1 (3.2.58)
A , , P PPI . (3.2.51), (3.2.58) (3.2.58)
t 2 t1 tM M M= = (3.2.59) ( ), N 0= su . (3.2.58) . (3.2.58)
( )3t n1G I E I M2 + = t (3.2.60)
2P
n PPII IA
= : Wagner (Wagner constant)
. (3.2.60) . : su ,
, . (3.2.58-) , su ,
( )3n1 E I2
77
-
3 : . =
.
. , . (3.2.58) (3.2.58) N 0 .
. , 11S . (3.2.58) ,
,2 3M M 0 . , .
, , . 1t t= 2 1t t t= + , Lagrange : 2t
( )0 0 0
TT2 2 0 2 0 2 0 2 T 2 00 0 0 0 0 0
V A V
d V d A d V = + Gtr S t n r f r
d A
( )( ) ( ) ( )( )
0
0
2 2 2 2 2 2 00 11 0 11 0 12 0 12 0 13 0 13
V
2 0 2 0 2 0 2 00 1 0 1 2 0 2 3 0 3
A
S S S d V
t u t u t u d A
+ + =
= + + n n n
( )( ) ( ) ( )( )
0
0
2 2 2 00 11 11 0 12 12 0 13 13
V
2 0 0 0 00 1 1 2 2 3 3
A
S S S d V
t u t u t u
+ + =
= + + n n n (3.2.61)
. 1t t= ,
, , , ,1 10 ij 0 i0 u 0 i j 1 2 3 = = = (3.2.62)
78
-
3 :
u
( . )
1t
2 1 2 10 ij 0 ij ij 0 i 0 i i u u = + = + (3.2.62)
(. . (3.2.2), (3.2.5)) 1u
2 2 P 1 1 P 1 P 1 P P1 s M M s M M Mu u u = + = + + +
M
M
1 P
1 su u = + (3.2.63) , , ( ). PM 0
1 P1 su u = + (3.2.64)
( cos sin2 2 22 0 2 3 2u u x x ) = =
) (3.2.64)
( cos sin2 2 23 0 3 2 3u u x x = = (3.2.64) 2 (3.2.61) : . , (3.2.55), (3.2.56)
( )2 2 22 s2 s1 t 2 2 t1I N u u M M 1 = + (3.2.65) 2 2
0 11N P d
= : ( ). 2t t= ( ) ( )cos sin cos sin2 2 2 2 2 2 2ti 0 21 3 i 2 i 0 31 2 i 3 i
i
M P x x P x x d
= + : ( ). 2t t=
Green 11
( ) ( )2 2 111 s 2 3 11 11 111u x x 2 e2 = + + + = + n (3.2.66)
79
-
3 :
( )2 2 111 s 2 3e u x x = + + : . 11
( ) ( )22 211 2 31n x x2 = + : .
1 PM
12 32
xx = (3.2.66)
1 PM
13 23
xx = +
n
(3.2.66)
.
2 1t t t =
11 11 11e = + (3.2.67) ( )2 2 111 s 2 3e u x x = + + (3.2.67)
( )2 211 2 3n x x = + (3.2.67) 1 P
M12 3
2
xx = (3.2.67)
1 PM
13 23
xx = + (3.2.67)
1I (3.2.61)
: ( )
0
2 2 21 0 11 11 0 12 12 0 13 13
V
I S S S d V = + + 0
0
( )( )
0
0
01 11 11 12 12 13 13
V
1 1 10 11 11 0 12 12 0 13 13
V
I S S S d
S S S d V
= + + +
+ + +
V (3.2.68)
. (3.2.66). ,
80
-
3 : , , (2.3.21)
= epS D (3.2.69) , (3.2.66-) , 1 PM . (3.2.30) (3.2.31), 1 PM . , . (3.2.69) . Baba Kajita (1982),
11 11
12 12
13 13
S E 0 0S 0 G 0S 0 0 G
= = eS D
(3.2.70)
Poisson 0 = . (3.2.66) . ,
11
11 11 11 11 11 11 11e n e e = + (3.2.71-)
, 11I : 1
( )( )
0
0
011 11 11 12 12 13 13
V
011 11 12 12 13 13
V
I S e S S d V
E e e G G d V
= + + =
= + + (3.2.72)
(3.2.66) - (3.2.67),
( )( )
1
1
l1
11 s P s 1x 0
l21 1
P s PP t 1x 0
I E A u E I u dx
E I u E I G I dx
=
=
= + +
+ + +
( ) ( )( ) ( )
111 s P s2 s1
21 1 1P s PP t 2
I E A u E I u u
E I u E I G I
1
= + + + + +
(3.2.73)
81
-
3 : :
A d
= , ( )2 2P 2 3I x x d
= + , ( )22 2PP 2 3I x x d
= + , 221 P 1 P
1t 3 2
2 3
I x xx x
d = + + 1 P 1 P
2 21 M Mt 2 3 2 3
3 2
x x x x dx x
= + + , 12I 1
( )( )
0
0 0
1 1 1 012 0 11 11 0 12 12 0 13 13
V
1 1 1 0 10 11 11 0 12 12 0 13 13 0 11 11
V V
I S S S d V
S e S S d V S n d V
= + + =
= + + + ( ) 0 (3.2.74)
( )
0
10 11 11
V
S n d 0V)
( ) ( ) (0
2 21 0 10 11 11 0 11 2 3 2 1
V
S n d V S x x d
= + (3.2.75) 12I . (3.2.74) ( ) ( ) ( )t0
1 1 1 0 1 10 11 11 0 12 12 0 13 13 N s2 s1 M 2 1
V
S e S S d V S u u S + + = + (3.2.76) 1 1
N 0 11S S d
= : (internal stress resultant) .
( )t 1 P 1 P2 21 1 1 1 1M MM 0 11 2 3 0 12 3 0 13 22 3
S S x x d S x S xx x
= + + + + d
1
:
.
( )2 210 11 2 3S x x d
+ : Wagner (Wagner stress resultant) (rahair, 1992) . (3.2.75) . (3.2.76) , ( ) : (. . (3.2.65), (3.2.73) (3.2.76))
82
-
3 :
( )
t
1 12P Ns
2 121 1 1 1MtP PP t
E A E I Su NSME I E I G I W
= + + (3.2.77-)
2N 0= : () (. . (3.2.56) 2 2 2
t t 2 t1M M M= = : (. . (3.2.56) ( )2 21 10 11 2 3W S x x d
= + :
Wagner (Wagner term) (Trahair, 1992) (. . (3.2.75))
. , ( )
t
1 2 1t tG I M S = M (3.2.78)
t
1 P 1 P1 1 1M M
M 12 3 13 22 3
S xx x
x d = + + : . .
, :
tM
S . , 11S ( Wagner), ,
tMS 12 ,
13 . , 5.
, tM
S 11 (. . (3.2.78)) . , ,
tMS
Wagner (. 3.277)) 11S . , , , . 2 3M , M .
83
-
3 :
, . , NS . , . , .
NS ( N NN S S 0= = ), 11S . , 11S , St. Venant .
3.2.7
(. (3.2.44)) (. (3.2.60)). tM . tM (, , , ), , , .
, ( ) Newton Raphson. .
. . , .
84
-
3 :
(load control). . .
, . , , .
Newton Raphson . . . . , . (. . (3.2.80), (3.2.82)).
Newton Raphson . , . (3.2.77) tI PM . Poisson (3.2.24). Newton Raphson (modified Newton Raphson method). . (initial stiffness method) . : (. (3.2.14-))
PM . PM ,
85
-
3 : . .
. , , NS tMS . , , .
, ,
ml
( ) ( ) . :
l m
1) ( )lm ,
( )ls mu 2 2 (. .
(3.2.77))
( )( ) ( ) ( ) ( )
( )( ) ( )
( )( )t
l l 1P Nsm 1 m
m2 l 1l t mP PP t Mm 1 m 1 m m 1 mm
E A E I Su 0ME I E I G I W S
= + + (3.2.79-)
( )lm , ( )ls mu ( ) ( ) ( )l l 1 lmm m = + ( ) ( )l l 1 ls s sm m mu u u = +, ( ) (3.2.80-) ( )lm , ( )ls mu ( ) ( ) ( )l lm m 1 m = + ( ) ( )l ls s sm m 1 mu u u = + , ( ) (3.2.81-) , Newton Raphson.
l
2) (elastic prediction step): ( )lTr11 mS , ( )lTr12 mS , ( ), (3.2.66), (3.2.70)
( )lTr13 mS
86
-
3 :
( ) ( ) ( ) ( ) ( ) ( ) ( ) 2l l l2 2 2 2Tr11 s 2 3 2 3m m 1 mm 1S E u E x x E x x2 = + + + + lm ( ) ( ) ( )Pl MlTr m12 3mm
2
S G xx
=
( ) ( ) ( )Pl MlTr m13 2mm3
S G xx
= + (3.2.82--)
( )lTr11 mS , ( )lTr12 mS , ( )lTr13 mS
( ) ( ) ( )lTr Tr11 11 11m 1mS S S= + lm (3.2.83) ( ) ( ) ( )lTr Tr12 12 12m 1mS S S= + lm (3.2.83) ( ) ( ) ( )lTr Tr13 13 13m 1mS S S= + lm (3.2.83) 3) (plastic correction step): Trf , . (2.3.41)
( ) ( ) ( ) ( )2 2 2l l lr Tr Tr Tr11 12 13 Y m 1m m mf S 3 S 3 S = + + (3.2.84) : rf 0
, . ( )l11 mS , ( )l12 mS , ( )l13 mS
( ) ( )ll Tr11 11m mS S= , , ( )( ) ( )ll Tr12 12m mS S= ( )ll Tr13 13m mS S= (3.2.85--) rf 0 >
. , generalized cutting plane (Simo & Ortiz, 1985). ( ) 2.3.2. k - : . (2.3.53)
( )
( )
k
k
f3 G h
= + (3.2.86)
87
-
3 :
k 1+ - ( )k 111S + , ( )k 112S + , ( )k 113S +
( ) ( )k 1 k pl11 11 11S S E + = (3.2.87) ( ) ( )k 1 k pl12 12 12S S G + = (3.2.87) ( ) ( )k 1 k pl13 13 13S S G + = (3.2.87)
( k )
pl11
11
fS
= , ( k )
pl12
12
fS
= , ( k )
pl13
13
fS
= .
( k )
11
fS ,
( k )
12
fS ,
( k )
13
fS . (2.3.16-
-).
( )k 1pl
eq + , ( )k 1h +
( ) ( )k 1 kpl pl pl11 11 11 + = + , ( ) ( )k 1 kpl pl pl12 12 12 + = + ,
( ) ( )k 1 kpl pl pl13 13 13 + = + (3.2.88--) ( ) ( ) ( )k 1 kpl pleq eq + = + , ( )( ) k 1k 1 pleqh h ++ = (3.2.89-)
( )k 1f +
( ) ( )( ) ( )( ) ( )( ) ( )( )2 2 2 k 1k 1 k 1 k 1 k 1 pl11 12 13 Y eqf S 3 S 3 S ++ + + += + + (3.2.90) ( )k 1 1f tol+
1tol (5
1tol 10=
). :
, ( )( ) ( )( )ll k 111 11m mS S += ( )( )ll k 112 12m mS S += , ( ) ( )( )ll k 113 13m mS S += (3.2.91--) , ( )l ( k 1 )pl pl11 11m += ( )l ( k 1 )pl pl12 12m += , ( )l ( k 1 )pl pl13 13m += (3.2.92) ( ) ( )l k 1pl pleq eqm += (3.2.93) ( ) ( ( )k 1l pleqmh h + = )
, :
( )0 trf= ( 0 ) = trS S, , ( 0 ) trf f = S S (3.2.94--) f
88
-
3 : , , ( 0 )pl11 0 = ( 0 )pl12 0 = ( 0 )pl13 0 = (3.2.94--) ( )( )0pl pleq eq m 1 = , (3.2.94-) ( ) ( )(0 pleq m 1h h = )
4) , ( )lN mS ( )t lM mS . (3.2.76): ( ) ( )l lN 11m mS S
d= (3.2.95)
( ) ( ) ( ) ( )( ) ( ) ( ) ( )
t
l l l2 2M 11 2 3m mm
P PM Ml lm m
12 3 13 2m m2 3
S S x x d
S x S x dx x
= + + + + +
(3.2.95)
4.1.2 5) (. 3.2.79) :
( ) ( )
( )t
l
t m m
t m
M S
M , ( )lN mS (3.2.96-)
: 410 = . (3.2.96), (3.2.96) ,
. 1 l 1+ .
,
, .
l n=m
( ) ( )nm m = , ( ) ( )ns smu u m = (3.2.97-) , .
m 1+
89
-
3 : ( )PM m 1 + (. . (3.2.30), (3.2.31)):
( ) ( )( ) ( )n npl pl12 132 P m m
M nm 12 3m
1 , x x
+ = +
(3.2.98)
( )( )
( )( )
n npl plP12 13m mM
3 2 2 3n nm 1 m m
x n x n , n
+
= + + (3.2.98)
4. ( )PM m 1 + ( ) (. . (3.2.35)) t m 1I +
( ) P P2 2 M Mt 2 3 2 3m 13 2 m 1m 1
I x x x x dx x
+++
= + + (3.2.99) Wagner ( (. . (3.2.77)) )mW ( ) ( ) ( )n 2 211 2 3m mW S x x d
= + (3.2.100)
, ( )t m 1I + ( )mW 4.1.2. , Trf , ( )Y m
( ) ( ) ( )nn plY Y Y eqm m m = = (3.2.101)
m 1= , 0 = , . , Wagner ( )0W ( )11 0S , ( )0W = 0 (3.2.102) ( )t 1I ( ):
( )PM 1 ( )2 PM 1 0, = (3.2.103)
90
-
3 :
PM
3 2 2 31
x n x n , n = (3.2.103)
. . ( ) .
91
-
4 :
4.1.1 PM
, , ( ),PM 2 3x x (3.2.98):
( ) ( )( ) ( )n npl pl12 132 P m m
M nm 12 3m
1 , x x
+ = +
(4.1.1)
( )( )
( )( )
n npl plP12 13m mM
3 2 2 3n nm 1 m m
x n x n , n
+
= + + (4.1.1)
,
plpl2 P 1312
M2 3
d1 d , d x x
= +
(4.1.2)
plP pl1312
3 2 2 3dd x n x n , =
n d d
= + +
(4.1.2) (4.1.2) Poisson, . Neumann,
P
n
( ,P )M 2 3x x .
(boundary element method BEM) (4.1.2). , (direct BEM) (indirect BEM). (1999).
( ) ( ),2 3 2 3k x x , g x x, . Gauss Green :
22 2
k gg d k d k g n dx x
= + s (4.1.3) 3
3 3
k gg d k d k g nx x
= + ds (4.1.3)
92
-
4 :
3
2 3n , n :
2Ox , Ox
n , ( ) . ( ),T 2 3n n=n (4.1.3) g v=
2
ukx=
2 2
2 32 22 2 3 3 2 32 3
u u u v u v u uv d d v nx x x x x xx x
+ = + + + n ds
(4.1.4)
(4.1.3.) , g u=2
vkx=
2 2
2 32 22 2 3 3 2 32 3
v v u v u v v vu d d v nx x x x x xx x
+ = + + + n ds
(4.1.4)
(4.1.4), (4.1.4)
( )2 2 u vv u u v d v u dn n = s (4.1.5)
( ) ( ) ( )2 22 22 3
2x x = + : Laplace ( )
. ( ) ( ) ( )
22 3
nn x x
= + 3n
2
:
n . 2 Green Green.
( )2v Q P , R = (4.1.6)
( ) ( ), , , 22 P 3P 2Q 3QP P x x Q Q x x R : 2 3x x ( )Q P : Dirac 2 3x x
Dirac
( ) ( ) (QQ P h Q d h P = ) (4.1.7)
93
-
4 :
( ) ( ), , ,2P 3P 2Q 3QP P x x Q Q x x ( ),2 3h x x :
(Katsikadelis) (4.1.6)
( ), 1v Q P r2= ln (4.1.8)
r P Q= : ,P Q (fundamental solution) . (4.1.6) Green (free space Greens function). , Dirac
r( )P Q
( ) (,v Q P v P Q= ), (4.1.9)
Green (. (4.1.5)) PMu = , ln1v r2= . (4.1.2), (4.1.7), (4.1.9)
( ) ( ) ( ) ( )( ) ( ) ( ) ( ),, , PP PMM Q Mq q
v q Pv Q P f Q Q Q P d v q P q ds
n n
= q ( ) ( ) ( ) ( ) ( ) ( ) ( ),, , PMP PM Q M
q q
q v P qP v P Q f Q d v P q q ds
n n
= + q ( ) ( ) ( ) ( ) ( ) ( ) ( ),, , PMP PM Q M
q q
v P q qP v P Q f Q d q v P q d
n n
= + qs
(4.1.10)
,P Q : . .
Q
q : . , .
q
( ) ( ) ( )pl pl12 132Q 3Q
d Q d Q1f Qd x x
= + (. . (4.1.2))
P p , (4.1.10), , : p
94
-
4 :
( ) ( ) ( ) ( ) ( ) ( ) ( ),, , PMP PM Q Mq q
v p q q1 p v p Q f Q d q v p q d2 n
= + qsn (4.1.11) ( ) (collocation point) . (4.1.10), (4.1.11) ,
p P
( )PM P ( )PM p , ( )PM q , (4.1.2)
( )PMq
qn
.
1I . (4.1.11). , (. 3.2.7) . , :
( ) ( ) ( ) ( ) ( ), , pl pl12 131 Q Q2Q 3Q
d Q d Q1I v p Q f Q d v p Q dd x x
= = + =
( ) ( ) ( ) ( ),pl pl12 13Q Q2Q 3Q
d Q d Q1 1 v p Q d v p Q dd x d x
,
= + (4.1.12)
( ) ( ),pl1211 Q2Q
d QI v p Q d
x
= , ( ) ( ),pl
1312 Q
3Q
d QI v p Q d
x
= . , 11I , 12I
( ) ( ) ( ) ( ), ,pl pl11 12 Q 12 2q q2Q
v p QI d Q d v p q d q n ds
x = + (4.1.13)
( ) ( ) ( ) ( ), ,pl pl12 13 Q 13 3q q3Q
v p QI d Q d v p q d q n ds
x = + (4.1.13)
r ,p Q
( ) ( )22 p 2Q 3 p 3Qr p Q r x x x x= = + 2 (4.1.14)
( ) ( ), ,2Q 2 p
v p Q v p Qx x
= , ( ) ( ),
3Q 3 p
v p Q v p Qx x
= ,
(4.1.15-)
(4.1.13)
95
-
4 :
( ) ( ) ( ) ( ), ,pl pl11 12 Q 12 2q q2 p
v p QI d Q d v p q d q n ds
x = + (4.1.16)
( ) ( ) ( ) ( ), ,pl pl12 13 Q 13 3q q3 p
v p QI d Q d v p q d q n d
x = + s (4.1.16)
11I , 12I . (4.1.12)
(4.1.2) ( )PMq
qn
,
( ) ( ), pl12 2q qv p q d q n ds
, ,
. (4.1.11). (4.1.11)
( ) ( ), pl13 3q qv p q d q n ds
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
, ,
,,
P pl plM 12 13
2 p 3 p
PM 3q 2q 2q 3q
q
v p Q v p Q1 1p d Q d Q d2 d x x
v p q q x n x n v p q ds
n
= + +
Q
q
+ (4.1.17)
. (4.1.10)
. , . (4.1.10) : P
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
, ,( )
,,
P pl plM 12 13
2P 3P
P
Q
M 3q 2q 2q 3q qq
v P Q v P Q1P d Q d Q dd x x
v P q q x n x n v p q ds
n
= + +
+ (4.1.18)
(. (3.2.82-)), PM . , . (4.1.18) 2Px 3Px :
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ), ,
, ,
P 2 2M pl pl
12 13 Q22P 2P 3P2P
PM 3q 2q 2q 3q q
2P 2P
P v P Q v P Q1 d Q d Q dx d x xx
v P Q v P Q q x n x n ds
x n x
= + + +
(4.1.19)
96
-
4 :
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ), ,
, ,
P 2 2M pl pl
12 13 Q23P 2P 3P 3P
PM 3q 2q 2q 3q q
3P 3P
P v P Q v P Q1 d Q d Q dx d x x x
v P Q v P Q q x n x n ds
x n x
= + + +
(4.1.19)
, .
P Q q
(4.1.17) (4.1.19), ( )PM q . , . (4.1.17) PM . , . , ( 4.1.1). . ,
, .
( )pl12d Q( )pl13d Q
( )PM q ( )3q 2q 2q 3qx n x n , . . (constant), (constant boundary elements) . , .
97
-
4 :
.
4.1.1
(. 3.2.7). N K , (4.1.17) :
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
, ,
,,
KP pl plM 12 13
i 1 2 p 3 pi
NPM 3q 2q 2q 3q
m 1 qm
v p Q v p Q1 1p d Q d Q d2 d x x
v p q q x n x n v p q ds
n
=
=
= + + +
Qi
qm
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
, ,
,,
K KP pl plM 12 i Qi 13 i
i 1 i 12 p 3 pi i
N NP
Qi
M m qm 3q 2q 2q 3q mm 1 m 1qm m
v p Q v p Q1 1p d Q d d Q d2 d x x
v p q q ds x n x n v p q ds
n
= =
= =
= + +
qm
+ (4.1.20)
N jp ,
. . (4.1.20) :
, ,...,j 1 2 N=N N
98
-
4 :
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
, ,
,,
K Kj jP pl pl
M j 12 i Qi 13 i Qii 1 i 12 p 3 pi i
N NjP
M m qm 3q 2q 2q 3q j qmmm 1 m 1qm m
v p Q v p Q1 1p d Q d d Q d2 d x x
v p q q ds x n x n v p q ds
n
= =
= =
= + +
+
( ) ( ) ( ) ( ) ( ) (N NP PM j M m 3q 2q 2q 3qj mj mj mm 1 m 1
1 p F H q G x n x n2
= = = + ) (4.1.21)
, ,...,j 1 2 N= : . iQ : . i
mq : . m
jp : j .
