Diploma Tiki

7/22/2019 Diploma Tiki

1/128

, 2007

::

. ...


2/128

..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 ...................................................................................191.2.4 Piola Kirchhoff............................21

1.2.5 .................................................................................23

1.2.6 Piola Kirchhoff..........................25

1.2.7 ......................................................27

1.2.8 Cauchy .......................................28

2: ............................................292.1 ..................................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


3/128

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 .....................................................................................603.2.3 ................................................................................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 P

M ............................................92

4.1.2 tP PP t N M

A, 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


4/128

.........................................................................................119


5/128


6/128

,

. 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


7/128

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


8/128

.

:

.

,

.

.

, ,

.

.

, , ()

, .

.

,

.

,

.

. ,

.

,

,

.

,

.

. ,

3


9/128

,

( ,

).

.

- - .

. ,

.

,

.

,

.

,

, (

) .

, ,

, ,

.

,

.

. .,

, ,

. ,

,

.

, 2007

4


10/128


11/128

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


12/128

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 u1

x x x

u u uJ 1

x x x

u u u1

x 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


13/128

1:

,

.

.

. ,

.

( )

.

1.1.2.1

,

.

PA

ds P ( , , )1 2 3P x x x A . ( , ,1 1 2 2 3 3A x dx x dx x dx+ + + )

)

( ) ( ) (2 2

1 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 ) + + + . .

PA

P A

( ) ( ) ( )2 2

1 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 dx

x x x x x x

= + + = + + +

3

dx (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 dx

x 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 dx

x x x x x x

= + + = + + +

3

(1.1.10)

:

( ) 2 2 22 2 11 1 22 2 33 3

12 1 2 13 1 3 23 2 3

1ds ds dx dx dx

2

2 dx dx 2 dx dx 2 dx dx

= + + +

+ + +

(1.1.11)

8


14/128

1:

(strains), :2 2

31 1 211

1 1 1 1

uu 1 u u

x 2 x x x

= + + +

2

(1.1.12)

2 2

32 1 222

2 2 2 2

uu 1 u u

x 2 x x x

= + + +

2

(1.1.12)

2 2

3 1 233

3 3 3 3

u 1 u u

x 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 u

2 x x 2 x x x x x x

= + + + +

(1.1.12)

3 31 1 1 2 2

131 3 1 3 1 3 1 3

u u1 u 1 u u u u

2 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 u

2 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 ),

PA

PA

2

A 21 dsMF2 ds

=

1 (1.1.16)

9


15/128

1:

,PA 11dx

nds

= , 22dx

nds

= ,

33

dxn

ds

= , , . (1.1.11), 2ds

AMF :

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 ) :

( )Ads ds

e 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 1

2 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 >

A

1MF

2> .

(1.1.19) Taylor:

...2

A A A

1e MF MF

2= + (1.1.20)

(1.1.20)

: AMF 1


16/128

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)

1A1dx

nds

= , 2A2dx

nds

= , 3A3dx

nds

= .

, 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)

1A1d

nds

=

, 2A2d

nds

=

, 3A3d

nds

=

.

,

ii

d dsn

ds 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 1

ds x ds x ds x ds ds x x x

= + + + = + + +

3

3

n 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 1


= + + + = + + +

3

3

n 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 dx1 n n


= + + + = + + +

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


17/128

1:

A

ds 1

ds 1 2 MF =

+ (1.1.27)

(1.1.26) (1.1.27)

() :

11 1

31 2A1 1 2 3

A A

uu u1

xx xn n n

1 2 MF 1 2 MF 1 2 MF An

+

= + ++ + +

(1.1.28)

22 2

31 2A2 1 2 3

A A

uu u1

xx xn n n

1 2 MF 1 2 MF 1 2 MF An

+

= + ++ + +

(1.1.28)

33 3

31 2A3 1 2 3

A A

uu u 1xx x

n n n1 2 MF 1 2 MF 1 2 MF A

n

+ = + +

+ + + (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

cosA1 B1 A2 B2 A3 B3

n 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

u1x

n1 2

+=

+ ,

1

2B1

22

u

xn

1 2

=+

(1.1.30)

2

1A2

11

u

xn

1 2

=+

,

2

2B2

22

u1

xn

1 2

+

=+

(1.1.30)

3

1A3

11

u

xn

1 2

=+

,

3

2B3

22

u

xn

1 2

=+

(1.1.30)

12


18/128


19/128

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

.

dA

n

dA n . ,

14


20/128

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 40

dV 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


21/128

1:

,

. ,

(1.1.12)

.

:

111

1

u

x

=

(1.1.45)

222

2

u

x

=

(1.1.45)

333

3

u

x

=

(1.1.45)

2 121 12

1 2

1 u u

2 x x

= = +

(1.1.45)

3 131 13

1 3

u1

2 x x

= = +

u (1.1.45)

3 232 23

2 3

u1

2 x x

= = +

u (1.1.45)

(infinitesimal strain tensor) Lagrange

[ ]11 12 13

21 22 23

31 32 33

=

(1.1.46)

ij . (1.1.45)

,

, ( )

. ,

. ,

. ,

() .

16


22/128

1:

1.2

1.2.1

,

. ..

1

1t t= .

() 1t t=

.

,

1Q 2Q 1

d A

( , ,1 2 3P ) ,

. (traction

vector) 1 , ,

1n

1Q 1 n 2Q

(1 1t n) n P

( ) lim1

11 1

1d A 0

d

d A=

pt n (1.2.1)

1

d 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),

.

, .

.

1

d V ( ), ,1 2 3P ,

17


23/128

1:

lim1

11

1d V 0

d

d V= f

pf (1.2.3)

1d fp :

.1d V

, (

).

1f

1t

1.2.2 Cauchy

. . ,

. 1

1

d A 1Ox

( , ,T1 T 1 0 0= =

1n e ) 1e

1Ox 1 j , , ,j 1 2 3=

, ( )1 1t e

( ) ( , ,T1 1 1 1

11 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

1

21 22 23

31 32 33

=

(1.2.6)

ij .

ij , , ,i 1 2 3= , , ,j 1 2 3=

jOx

() .1d A ie

Cauchy().

18


24/128

1:

1.2.3

,

Cauchy

.

() ( )1 1t n

Cauchy ,

, ,

. ,

,

,

1

1e 2e 3e1n

1 1 1

1 11 1 21 2 31t n n = + + 1

3n

1

3n

1

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)

( ) ( ), ,T

1 1 1 1 1

1 2 3t t t=t n ( ), ,1 T 1 1 11 2 3n n n=n .. (1.2.7)

( ) T

1 1 1 1 = t n n (1.2.8)

.

(

1V

1t 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


25/128


26/128

1:

21 12 = (1.2.16)

31 13 = (1.2.16)

32 23 = (1.2.16)

.

1.2.4 Piola Kirchhoff

1Piola Kirchhoff

t 0= . . (1.2.9)

1t t=

( )1 1 0 0

1 11 1 1 1 1 1 0 1 0

0 0

A V A V

d A d V d A d V d A d V

d A d V + = + t n f 0 t f 0= (1.2.17)

, ,

( )1 00 t n :1

0 f

( ) lim01

1 00 0

d A 0

dd A

= pt n (1.2.18)

lim0

11

0 0d V 0

d

d V= f

pf (1.2.18)

(1.2.1) (1.2.18)

( ) ( ) ( ) ( )1

1 0 0 1 1 1 1 0 1 1

0 0 0

d Ad A d A

d A= =t n t n t n t n (1.2.19)

(1.2.3) (1.2.18)

11 1

0 0

d V

d 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 0

0 0

A V

d A d V + t n f 0 (1.2.20)

(1.2.19)

21


27/128

1:

( ) ( ) ( )1 1

T1 0 1 1 1 0 1 1

0 00

d A d A

d A d A = = t n t n t n n 0

(1.2.21)

(1.1.42) ,

( ) T

1 0 1 1 0

0 J

= t n n ( )1 0 1 1 0

0 J

= t n n (1.2.22)

Cauchy .

1Piola Kirchhoff 10 P

1 1 1

0 J

= P (1.2.23)

(1.2.22) 1Piola 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 0

0 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 Pf 0

x x x

+ + + =

(1.2.28)

12321 220 2

1 2 3

PP Pf 0

x x x

+ + + =

(1.2.28)

22


28/128

1:

131 32 330 3

1 2 3

P P Pf 0

x x x

+ + + =

(1.2.28)

1Piola Kirchhoff

Lagrange. (1.2.23)

(Cauchy)

.

1.2.5

2Piola 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)

(

)

( ) ( , ,T

1 1 1 1

1 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


29/128

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)

.

1

extW

1

intW

,

. ,

1 , 2 , 3 (Euler).

1 1 r

1 1 1

1 1 1

1 2 3

1 1 11 1 2 2

1 2 3

1 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)

, T

1 1 =

24


30/128

1:

T1 1 1 31 2 1

11 22 33 12

1 2 3 2 1

3 31 2

13 23

3 1 3 2

uu u u u

u uu u

= + + + + +

+ + + +

tr r 2

1d V

(1.2.36)

.

( )1

T1 1 1

int

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 u

2 2

u1 u u u 1 u

2 2

u u u1 u 1 u

2 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 0

0 0 0

A V V

d A d V d V + = t n r f r tr P

(1.2.39)

,

1

intW

25


31/128

1:

0d V

d A

0

T1 1 1

int 0

V

W = tr P (1.2.40)

.

1

0 P

1

Piola Kirchhoff

Green .

Green (

)

Green

. , ,

(

)

,

.

1

0

G

Piola Kirchhoff,

:

0d p

1d p

1

0 1 1d d

= p p (1.2.41)

(1.2.24) (1.2.18)

10 1 1 0 0

0d

= p P n (1.2.42)

((1.2.23)) 10

P

{ }1

0 1 1 1 0d J d

= p n

0A (1.2.42)

, 2Piola Kirchhoff 10 S

.

0 0d An

0

d p

11 1 1 1

0 J

= S 1

1 1 1

0

0 = S P (1.2.43-)

, 10 S (

), 10

P

T

1 1

0 0 = S S (1.2.44)

26


32/128

1:

. (1.2.40)

[ ]{ }1 1 10 12

= G I (1.2.45)

[ ]I : 3 3

:

0

T1 1 1

int 0 0

V

W = G

tr S 0d V

0

(1.2.46)

, , , :

1

0 S

1

0

G

( )0 0 0

T T1 0 1 0 1 T 1 0 1 1 0

0 0 0

A V V

d A d V d V + = G

t n r f r tr S (1.2.47)

10 S . ( )1 0

0 t n

1Piola Kirchhoff1 extW 1

0 P

(1.2.24).10 S

1.2.7

Cauchy

. 1 11

()

.

1d A 1Ox

1

d A1Piola Kirchhoff

.

.

,

.

1

0 11P

1Ox1

d A0

d A 0

d A

1Ox1

d A

2Piola Kirchhoff .

27


33/128

1:

1Piola Kirchhoff

.

,

()

.

,

.

1

0 11S

1d A

0

d A1Ox

1d A

, 2Piola Kirchhoff

()

, Cauchy

1Piola 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 220 2

1 2 3

f 0x x x

+ + + =

(1.2.48)

13 23 330 3

1 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


34/128


35/128

2:

2.1

: 6 (. (1.1.12)

Green . (1.1.45)

)

3 (. (1.2.13) Cauchy. (1.2.28) 1Piola 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


36/128

2:

)

.

. .

.

,

.

2.2

,

, .

, .

. ()

(ideally elastic)

() .

:

, .

. ,

. ,

,

.

,

, .

.

(hyperelastic) Green (Green elastic)

,

Green

( ), , , , , , , ,t t t t t t t t t 0 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 t 0 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)

0

1 2 3d V d x d x d x = :

t0 ij : Green

31


37/128

2:

) 0

td W : () ()

.0d V

(strain energy function per unit initial volume).

F

F

( , , , , ,t t t t t t t 0 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 t 0 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 t

0 11 0 22 0 23 0 32t t t t

0 11 0 22 0 23 0 32

F F F F

d d W d d d d d V

= + + + +

0

(2.2.4)

. (2.2.3) (2.2.4)

t

0 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 kl

i 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 mn

i 1 j 1 k 1 l 1 m 1 n 1

F A B

C

= = = = = =

= = = = = =

= +

+

)t +

+

(2.2.6)

32


38/128

2:

ijA , ijklB , , : .ijklmnC

.

, .

(2.2.5) , . . (2.2.6) . (2.2.5)

t

0 ij 0 td W

( ) ( ) ...3 3 3 3 3 3

t t t

0 rs rsij 0 ij rsijkl 0 ij 0 kl

i 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 3

t t t

0 rs rsij 0 ij rsijkl 0 ij 0 kl

i 1 j 1 i 1 j 1 k 1 l 1

S B C= = = = = =

= + t +

G

0

)

(2.2.9)

, ,

, . (2.2.9)

.

t

0 ij 1


39/128

2:

)

( ) ( , , , , ,t t t t t t 0 11 0 22 0 33 0 12 0 13 0 23

=t0 :

Cauchy.

.

( ) ( ), , , , ,t t t t t t t t t 0 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 0

k k k 0 0 0

k k k 0 0 0

10 0 0 k k 0 02

10 0 0 0 k k 0

2

10 0 0 0 0 k

2

=

eD

k

(2.2.13)

12k = (2.2.14)

11 12k k 2 = (2.2.14)

34


40/128

2:

() Lam , (Lam constants).

:

(t t t t 0 11 0 11 0 22 0 331

S S S )E

= +

(2.2.15)

(t t t t 0 22 0 22 0 11 0 331

S S S )E

= +

(2.2.15)

(t t t t 0 33 0 33 0 11 0 221

S S S )E

= +

(2.2.15)

t t t

0 12 0 12 0 12

12 S

G = = , t t t0 13 0 13 0 13

12 S

G = = , t t t0 23 0 23 0 23

12

G = = S (2.2.15--)

( )3 2E

+ =

+: Young (elastic modulus,

Young s modulus)

G = : (shear modulus)

( )2

= +

: Poisson (Poisson s ratio)

,E G,

( )E

G2 1

= +

(2.2.16)

2.3

2.3.1

(elastoplastic, inelastic)

() .

:

.

, . ,

,

()

.

35


41/128

2:

,

() (rate

independent elastoplasticity).

(rate dependent elastoplasticity,viscoplasticity).

.

, (

)

,

Cauchy ( 1x , 2 , 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 el

11 22 33 12 13 23d d d d d d ) =d : ,1t 2 1t t dt = +

( ) ( ), , , , ,T pl pl pl pl pl pl

11 22 33 12 13 23d d d d d d =d :

,1t

2 1t t d= + t

2. Hooke (. (2.2.11)) ()

.

3. ,

.

(), f

( ) ( )( yf f ,= q q ) (2.3.2)

36


42/128

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


43/128

2:

fd

=

pl

d

(2.3.5)

d: (proportionalityfactor)

T

11 22 33 12 13 23

f f f f f f f, , , , ,

=

:

f

.

. d

. (

) f

T

y

y

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


44/128

2:

( )T

( 2.3.5 )

y

y

f fdf d

= +

e e plD d D d

T

y

y

f fdf d d

f

= +

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)pl

eq

. ,

{ }pleq=q (2.3.11)

( pl)y eqh = (2.3.12)

pleq

t

pl pl

eq eq

0

d = (2.3.13)

pl

eqd : 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 d 3 2

= + + + + + (2.3.14)

39


45/128

2:

( ) ( )pl pl ply eq y eq eqh d h d = =

)

(2.3.15)

( pleqh : pl

eq

(plastic modulus)

yd . (2.3.8)

pleqd .

( )pleqh

. ( )pleqh . = , y

.

(hardening),

( )pleqh > 0

( )

pl

eqh 0<

(softening)

-

(elastic - perfectly plastic material).

( )pleqh 0 =

. (2.3.9) - (2.3.10) , :

( )11 22 33

11 e

1f 2

+=

(2.3.16)

( )22 11 33

22 e

1

f 2

+ =

(2.3.16)

( )33 11 22

33 e

1f 2

+=

(2.3.16)

12

12 e

f3

=

(2.3.16)

13

13 e

f3

=

(2.3.16)

23

23 e

f 3 =

(2.3.16)

(2.3.14), (2.3.5) - (2.3.16)

:

pl

eqd d = (2.3.17)

y

f1

=

(. (2.3.9)), (2.3.8),

(2.3.17), :

40


46/128

2:

( )T

pl

eq

f fdf d h d

=

e eD d D

T Tf f f

df h d

= +

e e

D d D (2.3.18)

( )pleqh h = .,

df , . (2.3.18) 0= d:

T

T

f

d f fh

= +

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).

= ep

d D d (2.3.21)

TT

T

f f

f fh

=

+

e e

ep e

e

D D D D

D

(2.3.21)

2.3.2 -

,

.

41


47/128

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 2

2 1

t1

ft t d

=

e e D D

( ) ( )t 2

2 1

t1

ft t d

= +

e e D D

( )t 2

2

t1

ft

=

tr e D

d

)

t

(2.3.22)

( ) ( , , , , ,T

11 22 33 12 13 23 = :

,1

t

2 1t t = +

42


48/128

2:

( )1t = + tr e D (2.3.23)

, eD

3 2.3.1, ,

.

(2.3.22) :

1. tr ( )1t ,

. (elastic prediction) .

eD

2. t 2t1

fd

. d

. (2.3.19) f

,

. (2.3.20)

t 2

t1

fd

. ,

( )

t 2

2

t1

fd t = =

tr0

(2.3.23)

2t ( ) ( )y 2f , t > 0 , , .

t 2

t1

fd

0

.

, :

2t

2t

1. ( )( )y 1f , t 0 = . ( ) ( )y 2f , t 0


49/128

2:

2.3.2.1 Euler (forwardEuler)

.

t 2

t1

fd

( )t 2

1

t1

f fd

t (2.3.24)

. (2.3.19):

( )

( ) ( ) ( )

T t 2T

1t 2 t2

t1

T T

t1 t1

1 1

ff t

df f f f

h t t h

=

+ +

ee

e e

D dD d

D D

1t

( )

( ) ( ) ( )

T

1

T

1 1

ft

f ft t

+

e

e

D

D

1h t

(2.3.25)

, Euler (forward Eulerpredictor).

.

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


50/128

2:

n

=i

[ ],1i 1i 1t t + .

t 2 t12 t13 t2

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

ft

f ft t

+

e

i

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 1if

t 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 ep

ii

d D d D i

ep

iD .

. (2.3.26)-(2.3.29)

Von Mises

( ) ( )T

1i 1i

f ft t 3 G

=

eD

, ,...,2 n 1, i 1 (2.3.31)=

45


51/128

2:

.

.

.

. ,

.

.

.

-.

1t t= ,

= tr e D ,

f

(),

2.3.2.2 Euler (backwardEuler)

( )t 2

2

t1

f fd

t (2.3.32)

(

) 2t

( )2f

t

. (fully

implicit).

.

:

2t

( )t 2

2

t1

ft d

=

tr e D

( ) ( )2f

t

= tr e D

2t (2.3.33)

( ) ( )y 2f , t = 0 (2.3.34)

46


52/128


53/128

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 1

1f 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 pl y eqf f ++ += (2.3.39)

( 0 ) = tr (2.3.40)( 0 ) 0 = (2.3.40)

( )( )0pl pl

eq eq 1t = (2.3.40)

( )( )( )( ) ( )

,

00 0 pl

y 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


54/128

2:

tr tr f f

+ = =

tr tr e e 0 D 0 D

(2.3.43)

(2.3.35)

( )( )tr

tr pl

eq 1

ff h t

+

0= (2.3.43)

(2.3.43) (2.3.43)

( )( )tr tr

tr pl

eq 1

f ff h t 0

+ =

eD

( )( )

tr

tr tr pl

eq 1

ff f

h t

=

+

eD

( )

tr

1

f

3 G h t =

+ (2.3.44)

( ) ( )( )pl1 eqh t h t = 1 . , . (2.3.31)

( ) ( )T

1i 1i

f ft t

=

eD

3 G

.

( )t

t tr

. (2.3.44) Euler (backward Euler

predictor). Euler trf

Euler ( )1f

t

( (2.3.25)).

() tr

( )1f t

.

.

Euler

.

.

2ttr

49


55/128

2:

, k2

2

f

.

, , . (2.3.35)

.

2.3.2.3 (Generalized trapezoidal and generalized midpoint rule)

.

(2.3.32).

( ) ( ) ( )t 2

1 2

t1

f fd 1 a t a

+

ft

(2.3.45)

a : .

( ) ( ) ( )t 2

1 2

t1

f fd a 1 a t a

+

t

(2.3.46)

1= . (2.3.45)

Euler. 1a2

= ,

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


56/128

2:

( ) ( )( k ) ( k ) 1f

t

+ = tr e e D D

1

ft

(2.3.48)

k 1+

( ) ( )( k 1 ) ( k 1 ) 1 k 1f

t + + + 1f

t

+ = tr e e

k+1 D

D

(2.3.49)

. (2.3.49) (2.3.48)

, ( k 1) ( k )+ = + ( k 1 ) ( k ) + = +

( )k 1 1f

t +

= e

k+1 D

(2.3.50)

,

k

( )1f

t

= e D

(2.3.51)

. (2.3.51)

(. (2.3.35))

( ) ( )( ) ( )( )k kk p

1 eq

f f lf t h 0

=

eD

( )

( )

( )

( )

k

k

k

1

f

f ft 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


57/128

2:

( )

( ) ( )

( )

k

k k

k

f

f fh

=

+

eD

( )

( )

k

kf3 G h = + (2.3.53)

Von Mises

. ,

2

2

f

,

, .

generalized cuttingplane. -

:

k

. (2.3.53), ( )( )

k

k

f

3 G h=

+

( k 1 )+ ( k )( k 1) ( k ) f+ = e D ( )k 1pleq + ( ) ( )k 1 kpl pleq eq + = + ( )k 1h +

( )( )( ) k 1k 1 pl eqh h ++ = ( )k 1f + ( )( )( )( ) ( ) , k 1k 1 k 1 pl y eqf f ++ += ( )k 1 1f tol+

1tol .

: ( ) ( k 1 )2t += , ,

.

( )( )k 1pl pl

eq 2 eqt +

=

( ) ( )( )k 1pl2 eqh t h + = ( )k 1 1f tol+ > . k 2+

( )( )( ) ,0 tr pl y eq 1f f f t = = tr ,

,( 0 ) = tr ( 0 ) trf f

=

, ,( )( )0pl pl

eq eq 1t = ( ) ( )( )0 pleq 1h h t = .

2.3.3

Cauchy 2Piola Kirchhoff Green

52


58/128

2:

.

,

, :

,

.

53


59/128


60/128

3:

3.1

,

15 15 (3 , 6 , 6

). ,

,

.

,

.

:

, . , , ,

.

, .

, ,

.

St. Venant . :

, ,

.

(, ).

, St. Venant

,

,

. ,

,

.

(torsional loading)

. :

Mt (torque),

. Mt

55


61/128


62/128


63/128

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


64/128

3:

3.1.5

St. Venant

.

()

. ,

,

. ,

St. Venant,

.

3.2

3.2.1

. , ,

.

: ,

.

:.

:

.

,

.

Poisson , 0=

: , Green ij 1


65/128

3:

.

, : ,

.

.

3.2.2

(semi-inverse method)

. )

) (

),

.

l

,

.

.

=

1 2 3x x x , 1

, 2x 3x

. 1

( )1 .

,

1x .

, (

2 3u , u 2u

2 3u 3 )

.

(. 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


66/128

3:

2 0t =

30t =

+1

()

12

1

1

+

= =

K

j

j

3

u

2u

1(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

1 .

1u

( ) ( ) (, , ,P1 1 2 3 1 M 2 3u x x x x x x = )

)

(3.2.3)

( ,P 2 3x : , (primary warping function).

. (3.2.3),

1u

1x

( ).

1

x = = (3.2.4)

61


67/128

3:

,

( ),

. ,

3 , (3.2.3).

( )1

( ), ,S 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


68/128

3:

111

1

u0

x

= =

(3.2.7)

222

2

u0

x

= =

(3.2.7)

333

3

u 0x

= =

(3.2.7)

P

1 212 3

2 1 2

u ux

x x x

= + =

(3.2.7)

P

3113 2

3 1 3

uu

x x x

= + = +

(3.2.7)

3223

3 2

uu0

x x

= + = + =

(3.2.7)

12 13, .

Green (. . (1.1.12) (1.1.34)).

.

:

,

( )1x .

( ), ,1 1 2 3u x x x

( )1 . 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 u

x 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 u

x 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 u

x 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 u

x x x x x x x x

= + + + +

3 32 1 2 212

1 2 1 2 1 2

u uu u u u

x x x x x x

= + + +

(3.2.8)

63


69/128

3:

3 31 1 1 2 213

1 3 1 3 1 3 1 3

u uu u u u u

x x x x x x x x

= + + + +

3u

3 1 2 213

1 3 1 3 1 3

u u u u 3 3u u

x 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 u

x x x x x x x x

= + + + +

3u

3 2 2 223

2 3 2 3 2 3

u u u u 3 3u u

x x x x x

= + + +

(3.2.8)

. (3.2.2), (3.2.5)

( ) ( )22 211 s 2 31

u x x2

= + + (3.2.9)

22 0 = (3.2.9)

33 0 = (3.2.9)P

12 3

2

xx

=

(3.2.9)

P

13 2

3

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


70/128

3:

Piola Kirchhoff . (2.2.15).

0= , (. (3.2.9)):

( ) ( )22 2

11 11 11 s 2 3

1

S 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 3

2

S G S G xx

= =

(3.2.10)

P

13 13 13 2

3

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 ) . (3.2.10) ,

11 ,

11 0 = (3.2.11)

11 0 = . 0=

11 ,

.22 33S , S

,

. .

(2.3.4) ,

11 12 13

S , S , S

( e el e )pldS = D d dS = D d -d (3.2.12)

( )d :

( ) ( , ,T 11 12 13dS dS dS =dS )

65


71/128

3:

( ) ( ), ,11 12 13d d d

=d

( ) ( , ,pl pl pl11 12 13d d d

) =pld E 0 0

0 G 00 0 G

=

e

D

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

pl

12 3 12

2

dS G d x G d x

=

(3.2.14)

Ppl

13 2 13

3

dS G d x G d x

= +

(3.2.14)

11d ,

11d 0 = (3.2.15)

3.2.5 P

-

,

3 , ( ) ( ) ( ),P 2 3 1 s 1x x , x , u x , ( ) .1x = ,

( ) .s 1u x = . 12 15 (6 -

6 ). 3

( ).

1

( 1 ) (. . (1.2.48(1.2.28))

66


72/128

3:

3111 211

1 2 3

f 0x x x

+ + + =

(3.2.16)

1311 120 1

1 2 3

PP Pf 0

x 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 P

3111 211 2 2

1 2 3 2 3

f 0 0 G G 0 0x x x x x

+ + + = + + + =

G 02 P 2 P G 0 0, = = (3.2.19)

1f 0= : ()

1x .

( ) ( ) ( )2 22

2

2 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 PG 0

3 2 2 3

2 3

G x n G x n 0

x x

+ + =

67


73/128

3:

P

3 2 2 3x n x n , n

=

(3.2.21)

1t 0= : 1x

.

(,

).

( ) ( ), ,T

1 2 3t t t=t n n

1t

( ) ( ) ( )2

2 3

nn x x

= +

3n :

n

, ,

( ),

P

2 3

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 0

x x x x x x0

+ + + = + + =

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

M

2 3

d1 d

, d x x

= + (3.2.24)

Neumann

plP pl

13123 2 2 3

ddx n x n ,

n d d

= + +

(3.2.25)

(3.2.17)

( ). 1Piola

68


74/128


75/128

3:

plP pl

13123 2 2 3

ddx n x n ,

n d d

= + +

(3.2.31)

,

. :

,

(3.2.30), (3.2.31)

.

3.2.6

su

- ()

s, u

,

().

.

. ,

Cauchy

: (. .

(1.2.34) (1.2.37))

[ ] [ ]( ) ( )T T T

V A V

dV dA dV = + tr t n r f r (3.2.32)

(

) ,

A : H ()

V: ()

[ ] [ ], : Cauchy

, 1 2 3, 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 P

1 Mu M = + (3.2.33)

70


76/128

3:

2 3u x = (3.2.33)

3 2u x = (3.2.33)

P P

12 3

2 2

xx x

= +

(3.2.34)

P P

13 2

3 3

xx x

= + +

(3.2.34)

1 . (3.2.32)

:

( )1 12 12 13 13V

dV = + =

( ). .P P P P

3210

12 3 13 2

2 2 3 3V

x x dV x x x x

= + + + +

22P P P P

3 2 12 13

2 3 2 3V V

G x x dV dV x x x x

= + + + +

=

22 lP P P P

3 2 1 12 13

2 3 2 3x1 0 V

G x x d dx dV x x x x

=

= + + + +

( )P P

1 t 2 1 12 13

2 3V

I G dx x

= + +

V (3.2.35)

22

P P

t 3 2

2 3

x dx x

= + +

:

(torsion constant). P P

2 2 M M

t 2 3 2 3

3 2x x x x dx x

= + + ( )2 l = :

(2 1 )x 0 = = :

1I

.

71


77/128

3:

lP P P P

12 12 13 1 12 13

2 3 2 3V x1 0

I dV dxx x x x

=

= + = +

d

( )P P131212 12 2 13 32 3

I l d n n dsx x

= + +

(3.2.36)

2I . (3.2.32)

:

( )( ) ( )( ) ( )( ) ( )( )T T T T 2A 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

,

( ) ( , ,T

1 1 0 0 = n (3.2.38)

( ) ( , ,T

1 1 0 0 =n (3.2.38)

( ) ( , ,T

A 0 n n =n (3.2.38)

(3.2.37) (1.2.49)

Cauchy

( ) ( )

( )

2 12 2 13 3 12 2 13 3

1 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

0

2 22 2 32 3 20t n n t

=

== + = 0

0

(3.2.40)

23

33

0

3 23 2 33 3 30t n n t

=

=

= + = (3.2.40)

72


78/128

3:

(3.2.38)

( ) ( ) ( ) ( )2 12 3 13 2 12 3 13 21 2

I x x d x x

= + + + d

1

2 t2 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= (3.2.41)

(2 t 2I M )1 = (3.2.42)

t t 2 t1M M= = .

,

( ) ( )1 2 t 2 1 12 t 2 1I I G I M = + = (3.2.43)

P

1 2, , (

)

tG = tM (3.2.44)

,

,

P 0 1312

12

2 3

12 2 13 3

I 0 0 x x

n n 0

= + =

+ =

(3.2.44-)

. (3.2.44) (governing equation) .

.

, ,

2Piola Kirchhoff

. (. . (3.2.26) (3.2.57)).

. (3.2.44)

.

, . (3.2.19) (3.2.21),

t

t

. (3.2.44).

73


79/128


80/128

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 = + +

P

M

(3.2.48)

(. . (3.2.2) (3.2.5))

P

1 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 dV

2

= + + + + =

( )( ) ( ) ( ) ( )22 32 2 2 2 2 2

s 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 3

11 11 s P s2 s1 P s PP 2 1

V

1 1S dV E A u I u u E I u I

2 2

= + + + (3.2.50)

A d

= , ( )2 2

P 2 3I x x d

= + , ( )2

2 2

PP 2 3I x x d

= + (3.2.51--)

2I (3.2.45) ,

1Piola

Kirchhoff (. . (1.2.24)). . (3.2.37) (3.2.39) 2I

( ) ( )

( )

2 11 1 21 2 31 3 11 1 21 2 31 3

1 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


81/128

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 S

x x x

= + + = + + +

( )P P

11 s 11 21 31

2 3

P 1 u S S Sx x

= + + +

(3.2.54)

11P 1

11 21 31S , S , S 1x (. . (3.2.10)). ,

. (3.2.49)

( ) ( ) ( )

( ) ( )

P P

2 s2 s1 11 2 1 11 2 1 11

21 2 31 3 21 2 31 3

2 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 = ,

( ) ( ) ( ) ( ). .32492 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 2

2

1 21 3 1 2 1 31 2 1 3 1

1

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 ii

P x x P x x d

= + :

.

. (3.2.56)

t .

.

76


82/128

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

( ) ( )2

s2 s1 s P

1 u u 0 E A u E I

2

+ =N (3.2.58)

( )3

2 P s PP t

1 0 E u E I G I M

2 + + = t 2 (3.2.58)

( )3

1 P s PP t

1 0 E u E I G I M

2 + + = t1 (3.2.58)

A , ,P PPI . (3.2.51), (3.2.58) (3.2.58)

t 2 t1 t M M= = (3.2.59)

( ), N 0= su . (3.2.58) . (3.2.58)

( )3

t n

1G I E I M

2 + = t (3.2.60)

2

P

n PP

I

I I

= : Wagner(Wagner constant)

. (3.2.60)

. :

su , , . (3.2.58-)

, su ,

( )

3

n

1E I

2

77


83/128

3:

. =

.

.

,

. (3.2.58) (3.2.58) N 0 .

.

, 11S .

(3.2.58)

,

,2 3M 0 . ,

.

,

,

.

1t t=

2 1t t t= + ,

Lagrange :2t

( )0 0 0

TT2 2 0 2 0 2 0 2 T 2 0

0 0 0 0 0 0

V A V

d V d A d V = + G

tr S t n r f r

d A

( )

( ) ( ) ( )( )0

0

2 2 2 2 2 2 0

0 11 0 11 0 12 0 12 0 13 0 13

V

2 0 2 0 2 0 2 0

0 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 0

0 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 1

0 ij 0 i0 u 0 i j 1 2 3 = = = (3.2.62)

78


84/128

3:

u

(.

)

1t

2 1 2 1

0 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 P

1 s M M s M M M u u u = + = + + +

M

M

1 P

1 su u = + (3.2.63)

, ,

( ). PM 0

1 P

1 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 ii

P x x P x x d

= + :

( ).2t t=

Green 11

( ) ( )2 2 111 s 2 3 11 11 111

u x x 2 e

2

= + + + = + n (3.2.66)

79


85/128

3:

( )2 2 111 s 2 3e u x x = + + : .11

( ) ( )22 2

11 2 3

1

n x x2 = + : .

1 P

M12 3

2x

=

(3.2.66)

1 P

M13 2

3

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 2

11 2 3n x x = + (3.2.67)

1 P

M12 3

2x

=

(3.2.67)

1 P

M13 2

3

xx

= +

(3.2.67)

1I (3.2.61)

:

( )0

2 2 2

1 0 11 11 0 12 12 0 13 13

V

I S S S d V = + + 0

0

( )

( )

0

0

0

1 11 11 12 12 13 13

V

1 1 1

0 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


86/128


87/128


88/128

3:

( )

t

1 12P Ns

2 121 1 1 1MtP PP t

E A E I Su N

SME I E I G I W

= + +

(3.2.77-)

2N 0= : () (. .(3.2.56)2 2 2

t t 2 t1M 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 1

t tG I M S = M (3.2.78)

t

1 P 1 P 1 1 1M M

M 12 3 13 2

2 3

S xx x

x d

= + +

:

.

.

, :

tS

. ,

11S (Wagner),

, t

S 12 ,

13 . ,

5. , t

S

11 (. . (3.2.78))

. , , t

S

Wagner (. 3.277)) 11S .

, ,

,

.

2 3, M .

83


89/128

3:

,

.

,

NS

. ,

.

,

.

NS (

N NN S S 0= = ),

11S . ,

11S ,

St. Venant

.

3.2.7

(. (3.2.44))

(. (3.2.60)).

t .

t

(, , ,

), ,

,

.

,

(

)

Newton Raphson.

.

.

. ,

.

84


90/128

3:

(load control).

.

. ,

. ,

,

.

Newton Raphson

.

.

.

.

,

.

(. . (3.2.80), (3.2.82)).

Newton Raphson

.

,

.

(3.2.77)

tI

P .

Poisson (3.2.24).

Newton Raphson(modified Newton

Raphson method).

.

(initial stiffness method)

.

:

(. (3.2.14-)) P

M . P ,

85


91/128

3:

.

.

. ,

,NS tS .

,

,

.

, ,

m

l

( ) ( ) .

:

l m

1) ( )l

m ,

( )l

s mu

2 2 (. .(3.2.77))

( )

( ) ( ) ( ) ( )

( )

( ) ( )

( )

( )t

ll 1

P Nsm 1 mm2 l 1

l tmP PP t Mm 1 m 1 m m 1 m

m

E A E I Su 0

ME I E I G I WS

= + +

(3.2.79-)

( )l

m , ( )

l

s mu

( ) ( ) ( )l l 1 l

m

m m

= + ( ) ( )

l l 1 l

s s sm m mu u u

= +, ( ) (3.2.80-)

( )l

m , ( )

l

s mu

( ) ( ) ( )

l l

m m 1 m = + ( ) ( )

l l

s s sm m 1 mu u u = + , ( ) (3.2.81-)

,

Newton Raphson.

l

2) (elastic prediction step):

( )l

Tr

11 mS , ( )

lTr

12 mS ,

( ),

(3.2.66), (3.2.70)

( )l

Tr

13 mS

86


92/128

3:

( ) ( ) ( ) ( ) ( ) ( ) ( )2l l l2 2 2 2Tr

11 s 2 3 2 3m m 1 mm

1S E u E x x E x x

2

= + + + +

l

m

( ) ( ) ( )Pl MlTr m

12 3mm2

S G xx

=

( ) ( ) ( )Pl MlTr m

13 2mm3

S G xx

= +

(3.2.82--)

( )l

Tr

11 mS , ( )

lTr

12 mS , ( )

lTr

13 mS

( ) ( ) ( )l

Tr Tr

11 11 11m 1mS S S

= +

l

m (3.2.83)

( ) ( ) ( )l

Tr Tr

12 12 12m 1m

S S S

= +l

m

(3.2.83)

( ) ( ) ( )l

Tr Tr

13 13 13m 1mS S S

= +

l

m (3.2.83)

3) (plastic correction step):

Trf

,. (2.3.41)

( ) ( ) ( ) ( )2 2 2

l l lr Tr Tr Tr

11 12 13 Y m 1m m mf S 3 S 3 S

= + +

(3.2.84)

:

rf 0 ,

. ( )l

11 mS , ( )

l

12 mS , ( )

l

13 mS

( ) ( )ll Tr

11 11m mS S= , , ( )( ) ( )

ll Tr

12 12m mS S= ( )

ll Tr

13 13m mS S= (3.2.85--)

rf 0 > .

,

generalized cutting plane (Simo & Ortiz, 1985).

(

) 2.3.2.

k-:

. (2.3.53) ( )

( )

k

k

f

3 G h=

+ (3.2.86)

87


93/128

3:

k 1+ - ( )k 111S

+, ( )

k 1

12S +

,( )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 )

pl

11

11

f

S

=

,

( k )

pl

12

12

f

S

=

,

( k )

pl

13

13

f

S

=

.

( k )

11

f

S

,

( k )

12

f

S

,

( k )

13

f

S

. (2.3.16-

-).

( )k 1pl

eq +

, ( )k 1h +

( ) ( )k 1 kpl pl pl

11 11 11 +

= + ,( ) ( )k 1 kpl pl pl

12 12 12 +

= + ,( ) ( )k 1 kpl pl pl

13 13 13 +

= + (3.2.88--)

( )( ) ( )k 1 kpl pleq eq + = + , ( )( ) k 1k 1 pl eqh h ++ = (3.2.89-)

( )k 1f + ( ) ( )( ) ( )( ) ( )( ) ( )( )

2 2 2 k 1k 1 k 1 k 1 k 1 pl

11 12 13 Y eqf S 3 S 3 S ++ + + +

= + + (3.2.90)

( )k 1 1f tol+ 1tol (

5

1tol 10=

).

:

, ( )( ) ( )

( )

ll k 1

11 11m m

S S +

= ( )

( )

ll k 1

12 12m m

S S +

= , ( ) ( )

( )

ll k 1

13 13m m

S S +

= (3.2.91--)

,( )l ( k 1 )pl pl

11 11m

+

= ( )l ( k 1 )pl pl

12 12m

+

= , ( )l ( k 1 )pl pl

13 13m

+

=

(3.2.92)( )( )l k 1pl pl

eq eqm

+

=

(3.2.93)( ) ( ( )k 1l pleqmh h +

= )

,

:

( )0 trf= ( 0 ) = trS S, ,( 0 ) tr

f f = S S

(3.2.94--)f

88


94/128

3:

, ,( 0 )pl

11 0 =( 0 )pl

12 0 =( 0 )pl

13 0 = (3.2.94--)

( )( )0pl pl

eq eqm 1

= , (3.2.94-)( ) ( )(0 pleq m 1h h = )

4) ,( )lN mS ( )tl

Mm

S

. (3.2.76):

( ) ( )l l

N 11m mS S

d= (3.2.95)

( ) ( ) ( ) ( )

( ) ( )

( ) ( )

t

l l l2 2

M 11 2 3m mm

P P

M Ml lm m12 3 13 2m m

2 3

S S x x d

S x S x d

x x

= + +

+ + +

(3.2.95)

4.1.2

5)

(. 3.2.79) :

( ) ( )

( )t

l

t m m

t m

M S

M

, ( )l

N mS (3.2.96-)

: 410 = .

(3.2.96), (3.2.96) , . 1

l 1+ .

,

,

.

l n=m

( ) ( )n

m m = , ( ) ( )

n

s smu u

m = (3.2.97-)

,

.

m 1+

89


95/128

3:

( )PM m 1 + (.. (3.2.30), (3.2.31)):

( ) ( )( ) ( )

n npl pl

12 132 P m m

M nm 12 3

m

1 , x x

+

= +

(3.2.98)

( )( )

( )( )

n npl plP12 13m mM

3 2 2 3n n

m 1 m m

x n x n , n

+

= + +

(3.2.98)

4. ( )PM m 1 +

( ) (. . (3.2.35))t m 1I +

( )P P

2 2 M Mt 2 3 2 3m 1

3 2 m 1m 1

I x x x x dx x

+

++

= + +

(3.2.99)

Wagner ( (. . (3.2.77)))m

W

( ) ( ) ( )n 2 211 2 3m mW S x x d

= + (3.2.100)

,( )t m 1I + ( )mW 4.1.2. ,

Trf ,

( )Y m

( ) ( ) ( )nn pl

Y Y Y eqm m m = =

(3.2.101)

m 1= , 0 = ,

. , Wagner ( )

0W ( )11 0S

,

( )0

W = 0 (3.2.102)

( )t 1I

(

):

( )PM 1

( )2 PM 1 0, = (3.2.103)

90


96/128

3:

P

M3 2 2 3

1

x n x n , n

=

(3.2.103)

. .

()

.

91


97/128


98/128


99/128

4:

3

2 3n , n :

2Ox , Ox

n , ( ) .( ),T

2 3n n=n

(4.1.3) g v= 2

ukx=

2 2

2 32 2

2 2 3 3 2 32 3

u u u v u v u uv d d v n

x x x x x xx x

+ = + + +

n ds

(4.1.4)

(4.1.3.) ,g u=2

vk

=

2 2

2 32 2

2 2 3 3 2 32 3

v v u v u v v vu d d v n

x x x x x xx x

+ = + + +

n ds

(4.1.4)

(4.1.4), (4.1.4)

( )2 2u v

v u u v d v u d n n

=

s (4.1.5)

( ) ( ) ( )2 2

2

2

2 3

2x x

= +

: Laplace ()

.

( ) ( ) ( )2

2 3

nn x x

= +

3n

2

:

n .

2 Green

Green.

( )2v Q P , R = (4.1.6)

( ) ( ), , , 22 P 3 P 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


100/128

4:

( ) ( ), , ,2P 3P 2Q 3QP P x x Q Q x x

( ),2 3h x x : (Katsikadelis) (4.1.6)

( ),1

v Q P r 2

= 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)) Pu = , ln1

v r2

=

. (4.1.2), (4.1.7), (4.1.9)

( ) ( ) ( ) ( )( ) ( ) ( ) ( ),, ,P

P PMM Q M

q q

v q Pv Q P f Q Q Q P d v q P q dsn n

= q

( ) ( ) ( ) ( ) ( )

( ) ( ),

, ,

P

MP P

M Q M

q q

q v P qP v P Q f Q d v P q q ds

n n

= +

q

( ) ( ) ( ) ( ) ( )

( ) ( ),

, ,

P

MP P

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 13

2Q 3Q

d Q d Q1f Q

d x x

= +

(. . (4.1.2))

P p ,

(4.1.10), , :p

94


101/128

4:

( ) ( ) ( ) ( ) ( )

( ) ( ),

, ,

P

MP P

Q M

q q

v p q q1p v p Q f Q d q v p q d

2 n

= +

qsn (4.1.11)

( )

(collocation point) .

(4.1.10), (4.1.11)

,

p P

( )PM P ( )P

M p , ( )P

M q

, (4.1.2)

( )PMq

q

n

.

1I . (4.1.11).

,

(. 3.2.7) . ,

:

( ) ( ) ( ) ( ) ( )

, ,

pl pl

12 13

1 Q Q

2Q 3Q

d Q d Q1I v p Q f Q d v p Q d

d x x

= = +

=

( )( )

( )( ),

pl pl

12 13

Q Q

2Q 3Q

d Q d Q1 1 v p Q d v p Q d

d x d x

,

= +

(4.1.12)

( ) ( ),pl12

11 Q

2Q

d QI v p Q dx

= ,

( ) ( ),pl13

12 Q

3Q

d QI v p Q dx

= .

, 11I , 12I

( ) ( )

( ) ( ),

,pl pl11 12 Q 12 2q q

2Q

v p QI d Q d v p q d q n ds

x

= +

(4.1.13)

( ) ( )

( ) ( ),

,pl pl12 13 Q 13 3q q3Q


x

= +

(4.1.13)

r ,p Q

( ) ( )2

2 p 2Q 3 p 3Qr p Q r x x x x= = + 2

(4.1.14)

( ) ( ), ,

2Q 2 p

v p Q v p Q

x x

=

,

( ) ( ),

3Q 3 p

v p Q v p Q

x x

=

, (4.1.15-)

(4.1.13)

95


102/128

4:

( ) ( )

( ) ( ),

,pl pl11 12 Q 12 2q q

2 p


x

= +

(4.1.16)

( )

( )

( ) ( )

,

,

pl pl

12 13 Q 13 3q q3 p

v p Q

I d Q d v p q d q n dx

= +

s (4.1.16)

11I , 12I . (4.1.12)

(4.1.2) ( )PM

q

q

n

,

( ) ( ), 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 pl

M 12 13

2 p 3 p

P

M 3q 2q 2q 3q

q

v p Q v p Q1 1p d Q d Q d

2 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 pl

M 12 13

2P 3P

P

Q

3q 2q 2q 3q q

q

v P Q v P Q1P d Q d Q d

d 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)

2P 3Px :

( )( )

( )( )

( )

( ) ( )

( ) ( )

, ,

, ,

P 2 2

M pl pl

12 13 Q2

2P 2P 3P 2P

P

3q 2q 2q 3q q

2P 2P

P v P Q v P Q1d Q d Q d

x d x xx

v P Q v P Q q x n x n ds

x n x

= + +

+

(4.1.19)

96


103/128


104/128


105/128

4:

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

, ,

,,

K Kj jP pl pl

M j 12 i Qi 13 i Qi

i 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 d

2 d x x

v p q q ds x n x n v p q ds

n

= =

= =

= +

+

+

( ) ( ) ( ) ( ) ( ) (N N

P P

M j M m 3q 2q 2q 3qj mj mj mm 1 m 1

1p F H q G x n x n

2

= =

= + ) (4.1.21)

, ,...,j 1 2 N= : .

iQ : .i

mq : .m

jp : j .

( ) ( ) ( )

( ) ( ), ,K Kj jpl pl

12 i Qi 13 i Qiji 1 i 12 p 3 pi i

v p Q v p Q1F d Q d d Q

d x x

= =

= +

d (4.1.22)

( ) ( ),j

qmmjqm

v p qH

n

=

ds

qms

(4.1.22)

( ) ( ),jmjm

G v p q d

= (4.1.22)

(4.1.21) N N N

( ) ( )P P

M jj p = .

[ ]{ } [ ]{ } { }nH G = F (4.1.22)

[ ]H : N N .

( )mj

H ( ) ( ),j

qmmjqm

v p qH ds m j

n

=

( ) ( ),j

qmmjqm

v p q 1H ds m=j

n 2

=

.

{ } ( ) ( ) ( )( , ,...,T P P PM M M1 2 = )N : N 1 .

[ ]G : N N .

( ) mj

G ( ) ( ),j qmjm

G v p q d

= ms .

{ }n : N 1 .

( )

( )n m

n 3m 2m 2mm 3mn x n = , ,...,m 1 2 N =

99

7/22/2019 Dip

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