This Page Intentionally Left Blank
29 211, 1000
525
84
1900
2005 by
by
1865 843830,
1865 853333,
222 7504744,...
261 downloads
1601 Views
25MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
This Page Intentionally Left Blank
29 211, 1000
525
84
1900
2005 by
by
1865 843830,
1865 853333,
222 7504744, 90
01923,
7508400, 20 7631
no by
by
2005
20 7631 5500.
1 1.1 1.2 1.3 1.4 1.5 1.6
1 1 3 4 7 11 16
......................... .................. ......................... .................. ........................ .............
2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12
........................... ....................... ......................
3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8
........................... ..................
............ .................. ..... ............. .................. ........................ ................... ......... .....................
3
21 21 23 23 24 27 27 33 34 36 37 40 46
49
.....
49 54 54
...................... .........
57
.........
62
..................... ......................
67
4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13
............... .......... .............. ............. .... .............. ................ ............. .................... ............ ........................... ........................... .............
5
68 71 72 76 78 79 86 87 90 93 99 99 100
103 5.1 .................. ............
5.2 5.3
........................ 5.4
............
6 6.1 6.2 6.3 6.4 6.5 6.6
......................... ................. ........................ ......................... .................. ..............
103 108 110 115 119 121 124 125 126 131 136
139
7 7.1
..................
139
145
8 8.1 ................
150
8.2 8.3 8.4 8.5 8.6 8.7
von
..................... ....................... .................... ....................... ................. .........................
153 160 163 165 171 174
8.8
von ..........................
8.9 8.10
...... ....................... ......................
8.11 8.12
181 183
............................. .......................
8.13
188 193
9 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8
. . . .....
von
........... ............ ........... .......... von ............
10 10.1 10.2
..................... ................ ................ ............. ........................... ...........................
10.3 10.4 10.5 10.6
177 179
203 209 220 225 233 236 237 244 245 247 247 252 254 258 260 261
265
11 11.1 11.2 11.3 11.4
bound bound
..................... ..................... ......................... ....................
12 12.1 12.2
von 12.2.1 12.2.2
12.3 12.4 12.5
von von
.................. ................. ....................... ....................... ................ ..................
.............................
269 270 273 277 279 280 285 290 291 292 301 308
................... 12.7
310 317
.........
13
321 13.1 13.2
13.3
........................... 13.1.1 von ................ .................... 13.2.1 yon ................ 13.2.2 .................... ................. 13.3.1 von ................
321 331 332 340 342 344 345
14
357 14.1 14.2 14.3
......................... ................. ......................
362 364 378
387
15 15.1 15.2 15.3
on
............ ......
15.3.1 15.4 15.4.1 15.4.2
..................... .......... ......................... ................. .............
. ....
387 391 396 399 404 408 417
423
16 16.1 16.2
424 426
...................... ........................
429
17 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8
430 434
......................... ....................
436 440 445 448 451 453 455 456 461 466
. . .
17.8.1 17.8.2 17.9 17.10
..................... ....................
............. ...................... ..................... .................. .......... ........................... ...........................
17.11
...........
469
18
475 18.1
........................ 18.1.1 18.1.2 18.1.3 18.1.4 18.1.5
18.2 18.2.1 18.2.2
............ .................... ..... .................... ............. ......................... ................. .............
476 477 479 482 485 495 501 502 507
509
19 19.1 19.2 19.3
........................... ......................... ........................
509 510 513
20
517 20.1 20.2 20.3
.............. ....................... ............................. .................. ........................... ............. ................. ............... ....................... ............... .................
20.4 20.5 20.6 20.7 20.8 20.9 20.10 20.11 20.12 20.13 20.14
....... ...... ............
518 520 522 525 528 530 532 536 539 541 542 545 546 548
551
21 21.1 21.2 21.3 21.3.1 21.3.3
.................... .............
552 558
................... .................... ................... ...................
565 566 566 567
x 21.4 21.5
..........................
573 586
22
591 22.1 22.2 22.3 22.4 22.5 22.6 22.7
.............. . ............. ............ ........................ ......................... .........................
23
591 595 602 617 619 627 630
637 23.1 23.2
....... 23.2.1
23.3 23.4 23.5 23.5.1 23.5.2 23.5.3 23.6
................. ...................... von ............ .......... ............ ................ ............... . . .
23.7 23.8
............. ................ ............... ........................ ...................
24.1 24.2 24.3 24.4 24.5
................. ................. ............. ................... .......................
23.6.1 23.6.2
24
637 642 648 650 652 655 655 659 660 661 662 664 667 670
673 673 676 679 683 690
697 ........................ .................... . . .
697 701 702
705 737
book book
by
on on
book
book
book book on
book
book
2005
1 on
1.1
by
b
b2 b3
by a
a
by
by
2 b
b2
b2a2
b3a3
a3
b
by
by
,
by
0 1 0 0 0 1
0 0 0
0 by
0,
3
#01
0
1.2 and 1.1. by x x, y
x
x2
x3
4
1.3
x
x2
x3
1.1 1, 2
3.
by
a
a2 a3
1, 2
3. 1, 2
by,
b2a2
ab a b
b a.
b3a3
on by 2
5
by
a
1 6q
0
#
by
k
6
#
by 1
,
by 1
1
,
1 $
a
7
0. $
a
3x 3 x 3 3x3x3x3
1.4
by
by
1.2
27 81
8
,
x3
x 3
x3
3
.
.
.
.
.
.
.
p 1
1
x
x3,
1
x2
x3
2 x3
0 0
9
by 2
2
0
2
1
10
3
3
x
1.
x2
no
1
1.3.
2
3
by 1
2
3
x
3
1 , 2,
!
0 1
x
1, I2 1,
x 2
0
x 2
I
3.
3.
3 011 012
012 013
013 2
3.
11
1.5
no by no
3
x3
x
,
p
__p
x1
1.4,
x
x by
2
3
by
v 1, v2
v3
x
x
12
a
01,
01 02
by by v
body.
by
by
v2 x by
v3
13
upon b
on
a
by
14
o
b,
, ,
by b
0
0 o,
3x3 o 0 o
15
by
by
by
p
by
a
do
16
no no
by
1.6
by
no
by
17
0
0.
0,
2 0.
an
=
=
a
. .4,.
3
18
,g ,
0
0
19
0
body by 0 body
0.
This Page Intentionally Left Blank
2 body, by
2.1
by
2.1,
by
by by
by 2.1.
22
,
#
by
0 2.1.
2
.2
.2
.2
2
0
by
....
0 o by
o
23
by
1841
by
*
2.2
by
by
2.3
by 2.1,
.2
2
0
0
by do
24
2.4
.2
2
_
2.1
.2 _
2 2
.2
2 2
2
2
..
2.1, by 6.
25
n
0, 2.2. 0,
1, 633.
by
,
0 2.3,
by
, 4
.
. .
90
dx
dx
2.3. by
26
y
n
by
by
0, 1,
,.
by 2.4. by
by
1822.
27
2.5 1.
2.6
by n.
q
q by
28
q
.. 2
q
n
2.5,
n.
2.5 q
n
by
q n. q
by
n n n, q
q
0.
2n 2
2.
0 0
0
0
n 0 3x3 by
22
23
21, 22
21 22
n2
23
29
n3.
n2
n3
n
2
by n
by
n
by
2
n.
3 .
01 .
.
.
.
01, 02 01
03
.
.
_
03
0
.
03
by
30
by
01, 02
03
n n2.
by n2
n2 by n
#
n n3
n.
n2 n2
n3,
0
0 a 0
by by
31
0 b
0 0
x 0
1.3.
0 0
2
0 0 0 0 0 0 0 0 0 0
0 0
2
x b
0 0
0 0 0 0 0
0 0
0 0 0 0 0
0
1 2 3
by
2, 2 2*n*
n,
2n
32 * 2*n*.
2n
by n n
by n
0
and
n
n2
n3
n2
n2
n2
n1 n
n 1
n n3
22n2
33
21 0 0
0
0 0
0 n
n2
n3, 21, 22
23.
by 3
2 3.
by
0 0 0
0 0
2.7 by
1
0
0
34
1
an,
0
0
0 by
by
2.8
by
2
0,
._
# 0. x
1
0. 2 2, 0122n
by 03n
0
by n
n,
35 0
by 3
0
0
2
n
by
n2 n3 0
3x3
3
n2
0
2 by 2
031
01 2,
by
3
2, by
a 0
by 0
by 6.
n3
36
2.9 by 1
by
by
by
by
37
2.10
by 1
02
1 03
2
2
3
1
3x3
by
by 2
12
1
01, 02 01
1
03 1 2
02
,
1 3
0102
03
38
by by
01
1.. 3
03
2
1
01, 02, 03 by 0
21
2 31
3
31
3 3
by
12
1
3 1
2
112
.
1 ,
13
2
_. 1
2.6
1,2
3
39
2.6,
11111 1
0 0
0
0 63
q
by q
2.5 on
by n
70 2
_
2
1..
,0
2
62
2 1
1..
2
23 121 2
1
40
70
2
2.6, 70
by
2.11
2.7.
x2
x2
3
n
0
0 x
2.7.
0 n
41
n
by 90 622
612 611, 622, 612
6,,,
by
6,,,
by
611,
6nn,
1
x
2.7. 180
by .
0,
42
n
by by
by
by
180 n
1, 2, 3 ....
0.
no
10 x,
by
b by 1
b
1
2 b
2
23 23
2
23 23
2
by 2
2
22
43
_
on 2.8.
44
on 2.9.
1,
on by
2.10, 0 2.10,
0
0
0
0
2.10,
0
b
0. b 12
b
2.11.
45
11,
1
___
2.7, 2.11.
by
2a
2.11
2.7 by
by
2.11. x
2.11.
2.11
46
2.11 1
....
2
2 12
2.11 1
by
11
2.11 by
1
1
2.12
1
. .,
47 by
b b u
b,
0
2.12. by 0 0
7
1 S 1
by
0
J body
612 0 622 0 0
0
0
by do body
2.14.
48
0
X2
ii tt
tl tIt
0 0
0
0
0
0 u2
0,
u3
0,
2.15. no 0.
body, on by
body
2
3.1 body
on
n.
body
3.1. by
n on
body. body.
50
x
body
n.
n by
x
51
n by
3.2b. n by
by 0.11,
by
0.31
o21
*
_1
body
52
3.3.
body
do on
3.3.
3.2. and 1 1
1 1
1
2o12
0
0
x
n, 41,
by by
by
by n n by n2
0 1 1
n2,
0 0
by
53
x2
x2
0
on
.....
on
on
on
by
on
n
0 b
body body
b
. . . . .
b2
b 0
3
n2 n3
54
n. on
body, on by
1822
3.2
.
.
.
.
.
.
3.3 on
n
by
n n. n
n by
55
n by
3.5, n
2
3.5, n.
2
3.5,
n
2 0.
0
0
22 n.
56
n2
0 0 0
0 0 a3
172
0
1
11
n3,
13
1
3.4 by 1
by ,
.,
1
S1S2S3
12. 1,
2
2
2
3 11
57
13 ao
by
on
3.5 by by
x2
x2 x 2
1
x 1
x x3
x 3
on
n. by
n
n
ann
n
58
by
by
by
2
2
2
2
on 2 2
2 1
by 0
n
x
3.6, by by 2
.
59
3.7. 2.11, 3.7, 3.7
3.6
b x3 x2 b
by b
b b, 3.8. by 0 0
0 0 0 0 0 by
0.11 0
0 a22 0 0 0
60
x3
by 3.9.
0 0
0
0
0
0 0.
3.7 on
body.
dS
n
x3
2
n. body body
n, by
3.10.
on
61
body
p q,
dS
by
o,
by
0 body,
body by
1822.
62
3.8
by
0
v
S
do by v
1
63
on
by
by
by
by
v
64
u
Su
on on 3.1
u u
,.,
0
on
by
65
by 0 on
This Page Intentionally Left Blank
by
g
by
4.1. 4.1.
68 by
I,r,j--
.
.
.
.
I
(4.1)
1822 by
on
4.1 body, by *
4.2. e~j
on by
69
on
on
,
by
on
on on
06
0
4.13.
a
by
70
by on
1839
on 21
by
on on
on
an
71
4.2
by
on
0
on
0
o
on on o
0
by
0 a0
o
a o.
21,
0
0
72
by
4.3.
an
4.3
4.4.
x
1
on
73
2
by
1
on
by
1676.
1 no
by
by
by
74
1 on
1
1
1
1
1
1
141
0,
0,
2
75
on
on
by 1
0
4.3,
1
1
0
0
76
1
1
4.4 1,
611 622 633
2612 2613 2623
611 622 633 2612
2613 2623
77
03312 01212 01333 01312
01211 01311
03323
_
by by by by
by
by
2
by ...........
(4.38)
7, by
aTe eTo-J by
0 by
(4.4])
78
by
by
by
by o
by
by by
4.5
u,
u by by
o
[
(4.42)
79 6x6
o
o
0 o
#
o
,,
4.6 on
4.4.
80 no
x
!
I
I
I
on
I
1.6,
by
21
3
4.5,
81
7 0
0 .
o
a a x by x1
x2
x2
x3
3
x2
x2
x 3
82
4.6
0 1 0 0
0
, p
1 by
4.6, 1 1,
1
0
0
83
x3
0 0 0 0
0 0
0 0
0 0
0 0 0 0
0 0
21
13 x
4.7.
1
0
0 0
0
0 1
84
0
0
0 0 0 ._.
0 0 0
0 0 0
0 0 0 x
0 0 0 0 0
0 x 0 0 1 0 0 0 1
p
612
0 0
0 0 0 0
,
613
no
85
4.7.
8.13.
6.5,
1
_ v2..2 __ 0 0 0
0 0 0
x
0 0 0
0
0
0 0
0 0
1__ 0 0
0 0
0 0
86
v
0 0 0
v
0 0 0
0 0 0
1
0 0 0 0 0
0
0
0 0
0 0
0
0 0
0
4.9
by
1
1 1
1
0 0 0
0 0 0
0
0
0
0
0
0 0 0
0
0 0 0
0 0 0 0 0
0
by by
4.7 by 1
o
o
by 0
by
87
0 o
o
no no by o
0
,
o
by by
1837.
4.8
on by no
88 by _ ,
1
1
by
,
012 2
o1,
013 013
013
on
oh
oh
2
oh
89
by
on
0, on 4.10.
14
by by
12, by
1
12
011
012
.
13
.
011
.
. .
.
013 .
012 012
1 ,
.
1
013 013
3
90
by on
012
011
013
011
013
012
by by
4.9 upon by qb3
by 0. 0
2
by 1852.
by qb3
0
on
1
by
91
v
by
1
1 by
and
1
92
by by
by d
._ 1
v
1
1
by
,
by by
93
by on v. 1
0
2 1
1
1
0
0.
1
0
0
0. 0, 1
4.10
on
on 12
by
94
1
11
1 .5
8
2
0.
6. 011
3 011
1 on 11
by v
on
95
by
by
0
0
1
and and
by
by by by
4.8.
96
9.7
von
and
and and
and
/ / d
4.9 4.10
4.10. by
4.11.
97
3
3
4
3
2
1
0
0
0,
1
4.12.
98
.
.
.
.
..q
.
.
.
.
.
11
and no
5.4,
on
5.4.
99
4.11 622
612,
a
v 0
0 0
by
1, 2
4.12
111 v o 0 v 1533
1 0
0
100
by *
1v2
v 1 0 0
by
0
1 by
1
1 6
a by ,
1
1
by ,
4.13 on
101
on
2
0,
6
0
.
,
0,
0
by
by
a and
no
102 by by
v
do
5.1
on on
,
g.*.
,
g* ,
on on
104
I
*
*
I
I
4.3
I
4.6,
#
(5.7)
#
by by
on
on *
gu on
g.*.
by
on
105
a)
body
b)
body
p 2
x 1 3
1
and
-
by 1
106 5.
by by
by by u
body
body 1
x2
x3
x3
5.2.
by
_
.
,
go
107
5.2. ,
no
,
g.,. by
on
g.*.
by by by
on on
by
g.*.
on by
on
g.*. by
p.139.
108
gq
1.6,
gq gq*
*
by
5.2
by
109
by
on
4.10
5.4.
by
by
up on
on
2 011
012
1
3 ,,
012
013
011
013
,
013
012
on
110 2
3
by
by
on by oq
a2o
173
on
5.3
by
111
x3
x3
. . . .
x2
2
2 1 1
x3
x3
180
by
0
0 0
0
0 3
621
622
623
631
632
633
180 by 1 0 0
0
0 0
0 0 622 632
623 633
0 0
0 0
3
112
0. by 0 1 0
0 0
0 0
,
h3 o
o
h o ...
o .o
1.
o
1
[~
00.i
)o
1
)o
where a takes the values 2 and 3. Introducing a more compact notation we find that b3 a2
113 ....
on
0
01,02
03
_
1
030.
on
h3 o
k,
2 2 q2,
q3
on
2
0 0 0
0 0
0 on
0 0 0 q2 0 0 0 q3
0 0 0 0 0 0
114 on by
0
2
u
2
on
3
on
by
k
on
115
5.4 by 2
3
on
,
4.10,
3
1
1
1 012
Sq
by by on 8
116
1
1
0
v v
on
5.4 by
0.
1 1 0.
0
1 0.
117
1
v,
1,
5.4,
v,
by 1
,
2
5.4,
0. 0, 2, by 1
A EIj
2
0. u
1
1
v,
_
no
0
118
by v
on
on
g by
by
120
by
AeAT=
g 5, by 2
11
by by,
on
S 1, 2,
121
on
6.1 by
S, g g
S,
g
g* g
S,
g*
g*
S, g g, by g
11,o, by 1
3 by
122
by =
1
S
=
--
by
21
p2
1
1
,
1
1
,
1
1 =
"~
, , ,
22 22 22 g
123 g g
g
0
22
0.
g
1
b
2
g ag
2
by
o6 1
7
11... 17 on
by g 0. by
124
6.2 on
on
by
by by p2 2 2
125
by 11
2
by
by ...
on by
6.3 on n6
0
3
on
on v
a2
126
._
o
v
o
by
on by
6.4 on
on 14. on 6.1 v v
on
#
v
127
....
a
by
by
up o
on on
on
128
17
k
on
. ,
0
by
129 on 0.
by 1
012
,
v
3
0
1
on 0 1 ,
,
3
1
by
1
,
1
_
by
by .
,
0,
130
on
by
v 012
3
0 o
on 0 ,
,
0
1
by
on on 14. 0
by
0 1
1
14.
1
a3
131
by by
14,
6.5 4.6, by
by
x
by
4.6 by
0 1
0 10 001
0
0
01
by ,
132
AM(a)A T
14
on
on
by
by
by by
on
by v
v 1, 0,
1 0
133
v v
by
,
v
0 0 0 000
0 1 0 000
0 0 0 001
on
1
1
1
134
on
by
0.
1, 0.
1
14
2
1
3
2,
14
16
17 o,
1
1
2
135
011
012
5111 5211 5411
014
51
57,
015
58,
016
017
59
o
o
o
o
o
01,
by
by
by by
by
136
3
3
o o 2
v v. vv v by by by
6.6
by
v
6.4. 0 0
0 0
0 1
v 0 0 0 0 0 0 0 1
0
137
by
0.
2
h, h,
12
14
13
15
2
o13
0
oh oh
5
138
v by
6.4,
1
2
1
3
/" / =
0
0
0
0
0
o
o
o
0
0
0 6.4,
x
0
0 0 0
7 on
on book,
by
7.1
by
by
140
by
on
0
by
on.
2
3
11, 12
2
on
a4 on
141
2 ...
on by
on by
pp. do
7.1.
142
10,
on
0,
on
2
2 up ...
0
by
pq
7.2.
0 4.4,
143
0
0.
0 0
0 by by by
This Page Intentionally Left Blank
by
by
1 1
8.1. by
146
p.
o
8.2.
on
on,
by
on
8.2,
147
on
by
no 8.2.
8.3
3.
by
on
6.5
6.6
148
on
0. 0. 0,
0
0
an 8.13. by
a2,
no
0
13 1
1 ,
13
*
by
149
upon, 01
on
no
0
by
0 01
1
2
J3
1
by by 11,
30
on
30.
150
by
by
8.1
30.
by 1
1,
2, ..
_ 2
0
3
151
on
by p,
8.4.
p 0 on p
0 p,
by 1
1 3
. . _ . . . _ _
by 1
1
1 1
p
by p2
by p
by
152
p 0,
on
0
b
0,
0. 8.4,
b
1
0
$3
p
by 0 30 30
4
30
2
30,
1
30
1
343 2
3
0
153
0
0 0
by
0
01
by
30.
30 30
8.2
8.4. 30
11 360
.. 60o 2
3
2
120 0
60
3
2
0
3
154 0 0
30
60
120 0
0
p,
120
0 30 0 120
0 0
60
120 0 60
0 0
60 p
0
240 30
30 0
60
180
0
0
60 0
0
60
300 8.5 by
0
0
60
8.6. by
by
0
go3
0
60
0
1
155
by
$2 1
3
1
0 2
2
1
0
1,
0
0
60
by 0
0
by
60
0
by
8.7. 1
0 by
3
on
0
0 by
a2
0
60
60
0
156
1 0.1
on
on
171
171
171 172 0 30 172 by
172
173
171
0
173
2
30 on
8.6. on von on von 8.9. on
von 171 171
8.5
172
8.6,
173,
8.10
172
173
157
von
3
von
0
158 1
30o_ g
2
30*
3
0
30
0
30
30. 30
0
8.11. 0 30
0
0
180 0 30
0 0
30
180
150
30 0 0
30
0
30
30 8.5,
0
30
8.11
1
60
0 8.6. 0
0 30
159
by 8.12.
and * * *
no
8.12,
bound
8.12 8.12
by
no on 8.2b,
and a 30
* * *
8.6. by
on
160
0
2
von
8.3 no
on 30,
0,
by
01 3
8.1.
8.13 8.12. 8.14.
161
0.1
von 3
von by von by
by
1
o
1
1
yon
by by yon on
2
2
0
162 2
oo,
von
5
von 0 o
8.15. by by
01
0 0
_0
0 0 0
2
0
0
0 8.16. 0
163
yon 8.8
8.4
30
0 by
1
01 3 0,
von by
on on by
164
2
by 8.17. 0 p
by
8.17,
8.18. o3 0
2
2
0
von 8.19,
0,
0.
165
1
8.19
,
1
2a
1
8.10,
8.5 by
0
166
0 by o3, 0 k
by
x
3.5, x2,
on by 1
1
167
i/
X\
iI
/
",,,
by
3
1
g
1
1
8.21,
8.21 0.
0,
0.
by no 171
0.
168 0.. by
8.22.
k 8.21
qb o2, 8.21,
o
p 8.21 8.21, 1 _
_
p
1
2 1 0.1
1
k
qb
1 1
k
qb
1 1
qb
169
,,,
., .
.....
,, ,,
60
qb 2
on by 0,
k
on
on
0
1. k # 1,
# 1
60
0.1
0
a3
170
1 k
0
2 on
1. o3, 0
1
0.1
0.1
1 2k
1
3
0
8.23
2k
1
p 0.1
8.24. 1
8.25
0.2
0.3,
0 _ 0
60
171
....
by
8.24.
k,
8.6
on
do on 8.20 8.26.
8.21,
172 x3
,,,..
45 45
2
90
qb
45
qb 2
180
on
p 45 x3
135
45
v.
360
90
qb
135 2
180
173
45
45
,,
,
on 45
and 45
45
8.22, 8.28.
24,
174
8.7 no by
171
k
on 1
0
3
8.1, 01 by on
k
1, by
8.21, 8.29 8.24
k bound
1 8.12.
175
. . . . .
1
2
3
o.2
8.23
k
1
8.30b. 8.31. 8.25
k
8.32
1 o.1
von
8.15.
0 0 0 0 0
o
oo
o
o
oo
1
176
_
1
o
(o,T)
0
2
8.33. von 0,
2 von von
177
1 45
45
0 45 8.34
8.8
on 8.10
8.12,
on on
8.12, 8.13 bound bound.
8.30 von von
178 0.7 0.6
,
0.5 0.4 0.3
_
0.2 0.1
2 o 9
i
i
0.2
i
t
0.4
i
0.6
I
9
0.8
9
1.0
8.35 by von von
171 von
30
172
173
179
8.36 von
von
8.9 k k
4
good
8.25 0.25
0
90
by 8.37
180
by 9001 8.21.
by by
8.25.
181 2
....
8.39. 8.40.
8.10
k
4
qb
36.9
8.41
8.42. 8.41, up 8.41
8.42
182
7 6 5
i
I
I
I
I
I
1
30 8.41 0,
1
0
1
p
1
8.41,
..
by
by
183
0.2 _ . . . . . 7 ~ - " o .' 9 -
7 0.2
1,0
k
4
k
4
by
8.42 8.4,
8.11
by
184
2
by 30
0
30
0
0
1,
on, 0
0
0
2
von
by 11
by
_ 1
_
0,
0 by
0
2
by
0. 0
1
2
0,
on on
0.
by by
by
185
k
on by by 8.6.
on on
8.43. by
0 60 on
186
o
0"2
O't
o-t
1 on
by x
9y .....
a
187 on by
1
n
g
_
1
x
.
2 2
2
2
3
3
by
0, no
no 0. 0,
2
0
von
3
188
8.12
7 6 ,
5
1
0.08
1.16 0.10
0.12
on
8.42,
by
by
8.41. 8.1
8.2. on
0.10 2.5
189
0.08
1.8076
4.0962
14.4863
0.9914
0.10
1.2759
3.1962
11.7365
0.9801
0.12
0.9218
2.5969
9.9110
0.9647
on
0.08
14.4725
7.7834
0.5378
0.10
11.7109
6.5315
0.5577
0.12
9.8720
5.6979
0.5772 on
8.41
8.42.
8.44
8.44, 8.45. 8.44
8.45 8.44
by
1.21 0.10 on
190
a 0.3 0.2
1.
0.08
3.2244
3.4555
11.1538
0.9962
5.8553
11.1491
0.5252
0.10.
8.3
on
1
2
191
1.2
0.8 o 0.4
v ....
ao
3
8.48 upon 30.6 8.46
7
192
.
by
8.47 5 8.48 8 on
by du
by
193
8.13
4.3 4.6.
by
by
by
1
0
1
0
1
0
on
3
3 _ _
von 0
1
2 1
1
0
194 2
0,
1
1 1
by
0
6x6
a0
pa, b
_pa. b
b b
0.
21 21 21 by von
195
$22 $33
,
,
$23
1
,
,
1
1 1 1
0 0 0
0
on
by 6 21
15
by
on
1
by
by
196 by 0 0 0 0 0 0
p
0 0 0
0 0 0
0 0 0 0
0 0
0 0 0 0 0
0
0 0 0
._.
0 0 0
0 0 0
0 0 0
2
0 0
0 0 0 0
0 0 0 0 0
0
2
2
1
0
by von
8.49. $33
1
o
by 2.
o, 1
197
13 2
1
22
1
1
33 by
1
yo 1
1
1 21
1
1
vy12 8.49 23 1
1 2
1 2
2
1 2a2o., 3
von
von
198
8.49.
1
0
1 1
12,
by do 1
on
$22
0
0
199
22 ...1
1
1
1,
yon
0 22 33
_33x2
1
von o33 8.50
1
6x6
0
q by
11
,1
11 o. 1
200
on
.._
q2 q3 0 0
by
by on
on by by by 6.5
6.6. by
,
v
0 0 0 000
1
0 1 0 000
v
0 0 0 001 on
201 x, x
by
6.5
13 15
2
17
2
2
1
0
a
von
2
4. 1
202
2
2
2
122
6
123
1 1
1
,.11
1
.5
0
1 0
2
a3
a4
a6
2
2
1
2
0
by by
body 8,
9.1. 9.1, p
. . .....
204
9.1,
9.1
9.1
no by
9.1,
9.4
by
205
by
by 9.27.
no by
9.4,
206
.....
2
0 0
1
2
3
4
5
n
by
by
ao. 9.5. n
oo
1
2
1
0 0
1
2
3
4
5
207
by
by
9.6 n n
0.
1
0 0
1
2
3
4
5 500,
by
_
h
,.
1, h
8 and
1.
208
n
0.5, n
0.6, n
0.7
n
0
1
2
3
4
1
5
500
k
Oyo
n k
k n
0,
1
p 1
_
1
500 9.8.
k
1
0
2
3
4
5
2.
50,
50.
209
9.4, 9.4
by
2
1
by
4 3
9.9 by
9.1
2.
210
by
0
1 by 6.5, 6.6
8.13
by ....
0
0. do
1, 2 . . . .
0
0
by
211
p 9.6.
.
1, 0
do 1 on
212
0
0
3
by 9.10, 8.6. no 0
von
by 0
by 0
0
213
0 0
0
von 1
.......
0
p 3
0
0
von
9.11.
9.1 by von 9.4. by
by #
0
214
0 von p by
by 0 on 9.12
0 von
and
von
0 ,
von
by
by
215
b) O'y
O'y
O'yo
von
by
by von
by 0
3
9.14,
216 1
0
0
von
von by
a by
3
112 0 0.
a a von
9.15 body
by 0
and
217
6
1
,, 0 2
3
on
by
von by
by
1
0
0 by
0
by by 9.16.
9.17
...
218
a and and
0
0
0
by
9.18, by by
von
219
,_____,
_ 0 0
1
2
3
4
5
by
9.19 9.18.
go 9.18
9.19,
by
on
220
9.18
9.20.
9.2
9.19
221
9.21 do no
by on 9.21.
by
. .
7.
1
9.22,
222
by
p
9.22,
y
by
0 on
0
by
0
no by
by
von on, 0 by
by 0
do
0,
on
by
by by
223
_
by
4, by
by 1
1
0
0
9.23. 9.23.
2
0,
9.24.
224
by 1930,
von
0
3
4 1 2 von
by 0, 0
9.25,
von
by
8,
by
225
von
yon by von by
2.
on.
by
9.3 upon
by 9.3
9.4.
by
by
da
226
__
o
p
da.
go
by go
9.27.
by
227
G < GC
> GC
= G
by
a
by
by on
by by a
by by go go
9.26
by
228
by
go by
1
ada
by
do
p
p
p
1
p
p
1
9.26, 0, go
0
by
by
p
229 ,
9.29, on
on
by
0
*
0
p
by
by
,
,
1
0
p
p
0
230
o
01 by
_
0
on on
on
by
9.30.
by 24.
by
9.31.
22.5,
231
8.
by
p
.
on
by
9.32 on
do 0
on 0
p
___ 0,
232
von by by
by p
p
.......
p*
1
aq
on 9.33.
on 22,
good
.
2
23,3
g g
on
g
g by
by
9.4 on g,
2
on by on by 0 0 by
0
by
234
on on
by by
4
0
by
on
on
g
by by
by
on no 0. 0,
22
on
235
by
2
0 by
# 0
1
0
1
on g, on
.p
by
.p
0
,p
236
9.5 9.21,
0, 0.
9.34, 0
0
0
0
0
0 0
0,
.p
0 by 5.4.
.p
a
237
9.6
a by von 0
3
2
2 by
g 3
0, 0, by 2 by 4,
2
238
by
0
0
p
0
a
0
n
p
1
.p
a
p
2
p
239
p
9.35
p
by
up on
p
by p
p u
by 0 2
p
p
k
k
1
.
by by
240
p
9.36. 9.36
by
a 6p
9.36.
0
1
by
by
10.2
a
241
by
0
by # 0 0
=
,
ey:
= V:3
for pure torsion
(9.72)
b) o'y = o'~/ / = V~'r
p
by
0 0
0
0
no
242
p _
p
.
p p.26,
by
. . . . . .
p p
k
by doy
, 14 p
p
p
d2 p
von
p
d2
k by
a
243
14 by by
2 ...
p
[~'', v )de~y,
v
p
e:ff(W ~)
v
on
v
147v von by by
p
by by
244
21. 12.2.
9.7
von
,
0. P
*
",
,"
P .'II
.
by 2
by
p 0
245
p
1 p p
p
p
and
a G = G(J2) =
p
1
3e,f/(a~ff) ....
yon
4.10
9.8
.p
5.4,
246 by g 0 og
011
g
J3,
0
0
0
g g
.13,
10
10.1 on,
by
248
# 0
0
0
0
0 1, 2 ...
0
go
by
o
by by
249
g
by
on
g
2
g #
0,
no
p
.p
by 0
p
on
250
aq
on
g,
on
by
21. on
on
4.
0
g
251
p
by by o a0
vo, o
g
o
a0
p
0
by do by
by
on
do 0
o
g a 0 a
21
22
252
10.2
2,
0 by
g
kp 0,
4,
1
by *
10g
253 # 0
0.
by
1 by
# 0.
.p
1
.
. by
1
n
254 by by by
on on
. ,
g # g
#
by
10.3 # 0. by
255
0. by
up
by
# 0
# 0,
0.
10.1 0,
0
.p
# 0,
0
256
0
0
0. 10.1, 0, on
no 0. 0
by 0,
oo,
oo. oo. 0
0 10.2,
0 0 0 von
by
10.2.
257
#o,
0. 0. 0 0
on by
# 0
0 10.3.
1 0 ,
0 o
# 0
1
0.
0 10.3.
258
von
10.4
0, 0. by
1
0
0. 0.
0,
0
0
and and
oa u
0
259
o
10.4.
by on 18. 0. by
0,
10.5 0
260
by
da
18. by
no
10.5
by by
1, 2 1, 2
3.
261
by .
1
.p
2
1
10.6 by
x by
0
do
262
.p
by p
1 by
by
.p
0
,
a
.
.
,
0g
by
263 by 1
by ,
,gg
1
,
_
,
,
0a,3 # 0
by
by
by
0 g,
g
264
2
body 9.2.
on
body
bound bound
11.1. by by 11.2.
11.1,
266
o
o
o
o
11.1.
bound
by by by
bound
by
bound
by
by by
bound bound
bound upon by
u
S, u do
0
267
Su 0. 0. by
0
p.
no 0.
0 0
# 0 0
0,
268
op
and
by 0 on on 11.3. no on
o
.p
11.3, 0
269
11.1 bound
and a
"~" b
a 0
0 a ba a
a
by
a
by by by
(11.12) by
0 1, 0
11.3 0.
270
1. bound an
(11.14)
and
bound
on by on on
11.2
0 by
1
¢
0. do
do
¢
yon
271
q
u
0 1
Su
0 on 11.4.
0 on 11.5 0
272
17
on
bound and
and
p ,,
1
bound 17
up von
bound
dg
273 0
u
0
by
o 1 0
0
0.
1
bound body
by
0
S,.
bound bound bound
11.3 bound body 11.6
von
by
p
274
b
111111111 body
by
p.
von
11.7 0.2
03
pa
pa,
11.7,
by
275
bound
no bound 11.8. 11.8 1
2
2 ,
3
0
11.8, 6 1
26
0
0
no
g
0,
von
276
.
,
0 0
1
2
3
0,
12
11.8
by 11.8
1
11.9, bound
no body
11.9 1
277 b
n
g
....
bound
bound
von by
11.4 bound
a
an
a a an
278
body
bound
body body
12 10
on
von
von on
15.
22.
10.
10.1
280
12.1
on
by on
o
9,
12.1.
281
2
12.2
on
12.3
12.4.
12.5 12.5,
282
1 2
2 1
no
on
12.6.
283
1
1
2
5
2
g g 23 011
by o
by
284
n
kp
8.4. 2
3
g
by 12.7. n
11111 1
8.4, 12.7. .p
285
12.7. 0, .p
g
by
12.2 on do 9.6
von by 3
. ,
3 2
0
1
0.
286
a
1
p i
and
0 g
o
by
3
.p
2 p =
0
287
o
I
I
-E A
k
n
_..
dcry(E.ef f)
EA
1
288 b)
tYA
tYB
/ EB
no no tYeff
12.8. 12.1, no
p
12.8,
12.10.
12.1
no 12.1 no
289
.
n
fly
,
fly
and (10.24) then yields
.
2 2
v v
v 0 0 0
o
v 0 0 0
v
0
0
0
v
0 0
0 0 0
0 0 0 0
0 0 0
0 0
0
$21 $22 2 p
,
,
$22 $23 $23
by
0
290 12.2.1
0,
1
2o
p
v 0
2v
2
1
v 0 1
0
v
0
1
12
1 000 000
g22
2912 0
0 dq3 0
0 2 1
613
623
0
291 12.2.2 0,
10.6. _ 6
1
-
-
by v 2a 2
1
,
,
2 1
1
*p
1
v2
1
v 1
0
0
o 1
0 1
1v, 2
o
,
$22
$22
0 13 $33
# 0.
292
12.3 von
by 0
0,
8
0 by 1
30 11
on
no
0. 30
von
by 0
30,
0,
by
by .....
ad d 1
293 1
0
0,
J2,
0
_
1
30
,
3v
30,
J3
0
0.
by d
d
d von von
9.18
12.12. 9.19,
0 ,
_
.
0,
294
20
0
= ;
0, by v
2
p
p
('~Eij~ij)
d.
0
.
0
by 2
0 .
12.13.
295
p
by
by
.p
on 0, 0,
d d aq,
by
12.14. by
3
2a2o
3
296
5
0.2
3
von
on
1
0.2
3 .p
.p
.p
297
p
p
0,
von von
12.15. 0, 12.16.
0 12.16,
p
1
on
3 p p
0
12.17. p 12.17,
no
von
298
3
3 2
12.18
no
no
p
299
. o
von
......
1
no no
von
by d
a
_
300 fY
a
EB
2
Ao.2o(SO; - ot~.)(skt- ot~t)
p
v
2
v
0
0
0
v
0 0
0 0 0
0 0 0
v v 0
v 0
0
o
o
o
o
0
0
0
0
$22
o 0
.
.
.
.
.
p
3 . go
301
von by
by
12.4 von
upon by by
von von 3
d
d
0
d
on
0 0 by
.........
d
2#,
302
0, by 2
2 d.
d
q
0 by
by 2 2
0
.p .
12.13. by
.p
von
303
by
0
1.
0, 1
.p
d
by 3
3
3
3
304
d d d
ad o
1
1 d
3
d
0. d
305
d. 0.3
von
3 2
3
p
o
12.21 5y
0.3.
by
306
_ _
0.3. a
I I I
I I I I
E
EA
EB
0.3.
a
d
307
,
2
v v
v 0 0 0
v 0 0 0
v
0
0
0
v
0 0
0 0 0
0 0 0 0
0 0 0
0 0
0
$21 $22
$22933
$22913 $22923
2 p
$22 $23 $23g12 $23g13
$23922 d.
by
by
$23
308
12.5
by
by by by
by
k
by
von
309
o
o
von
20y
a
and
by
by von d 3
by
0
310
o13
0.
by
by
by by by
by
12.6
by no
8.13 von
by 0
by ._
3
311 by
0 0
o
0
0
3 2a2o von
8.13,
k by
von 3 _
2
312
3
2
0 2
.
o
0022
.
.
0, by
1 1 111 111 by
o.
oo
by
0,
2
2
313 1
0, by
I~,.::= (:,o:Ps),/:I 0 von by 0,
2 2
$33 $22 $33 0.
on by
by
o 0o 1 0
0
0
0
314
2 by __
p
2
1 4
2
1
1 0
0
0
1
0 0 0
1
1
1 0 0 0
o
o
o
0 0
0 0
0030 000
oo 3 0
611
2
1
.
by
by _.
__
2
1
.p 2
315
von
0 by n
2k von "P
~ff
2
n
k
1
....
e:ee:: i.e. (l 2. I IO) gives
n
_.
d~y(eeff) von by
316
11
1
o
v
.v
.v _ 0
2 .p
2
1 v
by
y
y
10
by
a
317 by
by a2
2
p
2
a
4.5,
12.7 von by
by
318
11
11
by on
.p
.
4.9. by
.p
12.7 0
2
q
0 2
on by
319 by by
,,
22.3
by
by by by by
This Page Intentionally Left Blank
12.3.
von
by
13.1 by by
by
322
2
by 13.1 by by n
0, 1 , 2 , . . . O"I
J
~O'y I
H
Ho
2
von
von von
by 3
n
0, 1 , 2 , . . .
13.2
323
1
61
3
by 13.2, body
by
up
324
by
0.1
0. 0.y2
2
1
0.
by
325
17
17
by 13.6
13.7,
13.7. by by
by
13.7 no 12.3,
12.2
by
13.7 by by
by
326
1
,,
ayl ayo
13.7.
n
1,2,...
p
.
0, p.234. von
by p
1 13.9.
327
on
13.9. on
1
on on _
a
_
328
13.9, on on _
ap
n
_
a
_ (n),
13.9,
q
by
on q
329
13.10.
and
g,
u
n
n
.
n
0
n
n
n
330
by
q
by
0, 1, 2 , . . . q 13.7, by
2, n
0, 1 , 2 , . . .
2
da
0
0
331 by n
d2
q
by
q
q
13.1.1 yon p
1
by
_
4, by 4,
2 .p 4, 4,
by 2
,
.p von
332
13.7. 13.1
no no no
13.2
by
333
on 13.12.
on
b
by
0 b
0
on
p
334
13.13, on on by
a
.
.
.
.
.
335
13.13.
q q*
g,
.
0
on by
o
o
.....
6,
1
336
6 6
_
6 6
6,
6
a
by 6
6 by
6 do
6
0
by
337
by
.
a
on
8
on
by by
by
and and
6
.
0
q*.
338
by q, o.
o.
by
q* q* on
on
on b
p
on
0. b
, _ p o.b p_ a a
0
a
0
o. #
ap_
ap_
_ ap_ 1
_
1
o.b
0
339
6
0
6,
on
06ba
n
on
*
q*
by
q*
q* on
by by by 6 6
h
h 6,
340 13.2.1 yon
b
0
2 p
q*
aq, on
S
341
2 by
0 _
_
, d
0. by
4
a qo
0
d
0
342 13.2.2
0 0 0 0 0 0 von by 0
,
0 0
a
0 0
0 0
a
1 1
a
by
by 2 0
343
ffyo[A..
.- o.
.o" ".........._~j.i:lo ~
"""''~
D
3
J
by
13.14. by
344
oo,
6
oo 13.15.
13.3 by
by
h h
2
a 0 .p
,
345 by oo,
g
.p
1%o h oo,
h. 0 p
von
by
13.3.1 von
3
S
gq
.p
.
3
346
0, 2
.p
by by h
von
1 0
0
by by 0, 0 0
o
347
1 0
n
0
__
h
1
3 2 d
upon 13.16.
0
by
2 2
1
von
348
p
0
0
p
.p
3
.p
,
_
,1
q
h
,
oo
0
p
q p
0
q
0,
349
p
e~::)
aoo
K~eef/)
by
1
h
Geff = Gyo + (Oloo+ K ~ ) ( I
-
e
~ +
v
K~E':)
h ,,,
d:,:: 0
by h.
13.20.
by 13.17
h.
by von
12.21. 13.18,
no
350
,
p
1 ....
ao,
von
13.18. 0, 13.20. 13.20b,
p
0
by on
an
351
p
0.
a
p
p
0.3.
by by p
352
0.
0
2
0
0
0
by
353
1
aoo
a3 o. 1
b
0
by
by
a
13.22. by
0
_
o
354
3
o
on by a
o
0 o. o
by
oo
~yo
o
oo
J
on
13.22 13.10.
355
no
a
This Page Intentionally Left Blank
14 by by by.
on
358
1
by by 14.1.
o0 o0 by, on 14.3,
359
.
.
.
.
.
.
.
oo
v
14.6,
360
o
14.7
*
14.8
361
14.9
on
362
14.1
a a
14.1, on by
a a
363
1
by 14.2, on
0.
364
14.2
by upon 6.4 14.11
v
by
14.12
365
oo
v
by 1
0 0
1
,
0 0,
on 14.12,
366
o
no
0 0 no by
0
1
0.
n
14.8. by 14.15
367
o
,, o
o
0, 0 0 by 14.17
oo
0,
368
14.15
0 0 14.18. by
6
a a
..
0 on
1
1
369
1
o
14.19 oo. no 14.14,
up 14.20
14.19
14.20
by
oo.
14.22.
370
171 172
n
by 0
02
0.1
371
2n 2
2n
1 by
o, o. 14.23
1
by by
ao
02
by no
372
a,
n
on
0 1
good 10
k
do
1
on on
by
on
373
.
.
by
.
1,
by
and
.
1
by
374
17 on
by
by
14.25
up 0 up
on
by by
by
375
o
2
o
0 0
0
00
0 up k 8.314 by
on on on 15.3.
by
376
n
a
#
a a a a
p
4.4.
0
1 1 1
1
1
1
1 v
1
1 by by
v
__
377
1
1
v
v
0
_ 1 1
v
n
1
1
by
a,
1
378
11 17
22 1
1
0
24 v
v
p 4.6, by
14.3
on
0,
by by
up
379
17
by up
0
0 0
0
oo
0
380
on by
on by
1 11
by no
381
by
oo
up
on by
, oo
382
E2 by #
0.
ao, 1
,,
0 up
0.
=
I~
dG(r)
no 0.
383
oo
oo
oo
oo
on on
384
on
on by 14.23
by
,,
n
n ,1,
385
n
on by
by by
1982, by by
1
1
11
1
1
1
1
1
1
1
v 1
1
,,
1 b
1 1
1 1
,,
1
386
v
on
on
0
0.3
0.4
0 0.4
0.5
600 0.5
1800 450 by
15.1 0
388
on
by n
1
1,
by
by
0
0
1
1
1 by oo
389
8.314
on
by
by
n,
a* and
on
by
390
on n
1.
n
4.
0 1
,
,
100
200
300 400
500
600
,
104 10 10
102 10_3 101 10.4 1
o
10_5 10_1 d
I
0.1
0.2
0.3
I
0.4
0.5
0.6
0.7
I
I
0.8
0.9
1.0
32
391
a
15.1 n n
1
1 15.1
by 15.1. d
on
15.2
no
on
1
14.25
392
_.
0
fO-'OA I tl
=
by
by _
a__
10.1
up
by
time
393 gcr
O"
g
,
no
by by
by no
by
394
500
1
0 23 b)
2.10 10 -4
0
100
200
300
400 425
26
26
1.2 0.8 0.4 0 0
1100
1
2
3
4
5
6
7
8
395
on
on
on no
26
26
.
.
.
.
.
.
.
1 4
1.2 0.8 0.4 0 21
22
1 23
0
1
2
3
4
20
0
1
2
3
4
100 101 102 103
1.2 0.8 0.4 0
16
96
1100
14.20. 15.4. no 15.5 upon
15.6.
396
by
p
180
p
24
0
0
10 20
30
40
50
60
70
450 by
p
by
15.3
_
_
397
= E .1.
1
_
and 0
by
1
3
on
by 3
1
2
a*
a*
1
398
by
by
von
10.1
g
0
10.1.
399
g
no by on 2 no by no
by 8.13
g, 12.6.
15.3.1
by by by
by
by
on
400 g g
1 1
4
1
x2
,
on
2
,,n
n
by
2
A = A(a~yf, Z) = V~D------~~
> 0
2
401
,,
oo
0,
2
1
on by
n.
n,
2
1
1
on
on
402
1.10 1 10
10
1
10
10
2.10
2.10
1.10
by 3.4. 105, 5.8.10
3.3.10 n 1.43
21
0
by _
0
0 0 x on 15.8 200 n
n
1.43
403
n.
21
0
and
2
0
15.9. 0
0,
x
15.6.
404
and
0
by ,7,o on
by
15.4
by
405
q
on
15.10 v body
0
15.11.
v 5,
0
v
0
0
v 0
v
v
v
v
0
406
yo
o
=
~, =
0
~'' I~I ~" if ITI>~. t/
Therefore if I~I _< ~yo the Bingham element is rigid and if I~I > ~yo the element exhibits secondary creep. The Hencky model shown in Fig. 15.12 comprises a Bingham model and an elastic spring in series, Hencky (1925); the elastic spring is characterized by the shear modulus G. To establish the constitutive law, it is observed that T = ~: = ~:B and ~, = ~,~+ ~,B and it then follows that
,/. I - ~,...z (15.33) I~I ~,= ~ +------~ if I~I > ~yo t/ The response of the Hencky model in a creep test is shown in Fig. 15.13. If the shear stress is below the yield shear stress ~:yo,the material is elastic and no creep strains develop whereas secondary creep occurs if ~: > ~:yo. To generalize the Hencky format (15.33) to isotropic three-dimensional behavior, it is firstobserved that the viscous strains might be thought of as being fluid-like,i.e. incompressible and it then follows that ~:kg = #kk/3K, where K is the bulk modulus. It is also observed that J2 = s~js~j for pure shear reduces to V ~ - I~I and that the engineering shear strains are twice the tensorial shear strains; then an immediate generalization of (15.33) becomes
1
407
by by von
by
.
.
0 1 ,
on 0,
0
0
0 1
0. 0, .p
408 15.4.1
15.14
600 _
.
200 0
0.2
0.4
0.6
0.8
0
1.0
13
15.14
15
by
on
on
p
0
11
0.
by
409 von
g
0
0 0
0 and
0
0
g by g by
by 0
10.1. by 2 by
by
410
by
0 0,
0.
by
2
.
0
0 0
0. by
aa
0. 0
aa
p
0, no
oo, on
0
411
.vp
...._
von
on body
by
and
body 0
0 von 0
0
on
412
k..
OB vpI
by
by
by
o. 15.15.
15.17 .
413
.
n
__
0
0,
0 0
0
0
0 0 0 by
by
by
q9
414
0 0.
0
0 by
_
0 0
0
_
0.
0
0 0
o 15.18
von
by
415
k.. staticstress-straincurve
by vv
by
_
by
0
15.19.
15.20
.
d %0 o
416 100 80 60 oo
40 o 20 i
I
10
i
i
i
10
i
1
10
!
I~
102 103
30 p d
5.
o
p by by
vp
qg(glEeff) = ayo(
)lip
x
x
~(f)
o
by
0
417
vp
vp aef f -- CYst(Eeff )[l "1"(
.~)I/p] .
(15.52) by
on by
von 12
13 13.3
by
15.4.2 by 0. by
0. by
.vv
1
no
418
on
on 0. on 0. on on a
by 2
on on
by
22
419
p
by
,1, o, #u,
by 1
1
by 4, _ by #u, by
0. 5 on
by
0.
0
o
# #.
0.
420
.
0 0
a
0
0 __
and on
0
on
. by 0.
by
421
on
von by
on on
von
15.23. by k by
This Page Intentionally Left Blank
on on by
424
16.1 by
body body body. by 1
by v.
v
b
b2
b3
body
u
Su Su
u
Su
16.1.
425
on u
body
body.
u
on on
u
a
by
v u.
v
by on
426
by
on
16.2
body by
body
b. body
body
no body
a
427
on on
by
by no
by
body, body.
428
a body
body.
by body
body
18
on
on
by
17
by
on
by ,n,
body
body,
by
430
1
an
n
n
1.
body,
on by
17.1,
n on
n
body n
17.1
1.
431
by
body. 17.1, n n
n.
1. 1,
by
n
on n
n
n do n.
by
1.
432
,
, 1
an
n
1 by
by
by n
n
body
1.
bound 17.2. n. n,
17.2. 1 17.2.
do by do
433
(f int)n+l. . . .
an
body by
17.3. 17.2,
by body.
body
17.1. 17.6.
box,
434 17.1
ao
o n
1, 2 . . . . .
,
17.2
by a
a
1, 2 . . . . by by
by
435
*
2
by
a
by
17.4. by
3
436
*
__
__
a
2
0
a 3
by
*
1,2, ... 0,
17.4. 17.5. 17.4, 17.5 17.4
17.5
17.3
on
437
17.1
0
on 0 01
a
a
a
a
a
_
1,2,...
by
1
by by
on
438
0. no a
a
0
# 01 by
17.1, n
by n by
1. by
Ig(ai-l) by by n.
body,
by
by
by
body, 1,
o
439
by
by 17.7.
a
.
1
.
by by
by 1, on
1 by
no
2
on,
440
by
a a on
by
17.4
by
x
17.6.
x
x0
441 0.
x
by
on x x 2,
by
by q/(a) = 0
(17.26)
q/(a)= Iv BTo"dV - f
on
a
0
a
0
v
BTdo"dV
do" - D t d E
da
- DtBda
442
2
a
a2 a3
a0
body,
1,
1 n
17.7. 17.7.
443
.
i
a1
a2 an
body.
17.7, 17.8
1 by 2
444
17.2
ao
o n
1, 2 .....
a 1, 2 ....
a
a
11.
a
by # 0
0 17.9,
#0,
body, 17.9,
on.
445
* and * * *
a
17.6.
17.5
body.
body,
0 by 17.10 17.7
body,
446
a
an a0
a3 a2
* and * *
p,
fln-tl--I
by by by by
447
an
a 1
a 2 a 3
by
no
17.8
17.11. on by
1
17.12
n. by by
448
1
an
1 _ 1
n 17.12 by
17.6
on
449
by by no body no
0 body, 0
k
0. 0
0.
AcSa = f -
f int
6a
a
a
body, 0
u
S,
u
Su
450
0
body.
1
,.
Sau
.
_
_ au
p0 no 0
2, 3 . . . .
451 on
on
6a,
a
6a,
0
0 by
=
1 by by 0, on
17.7 1
on
452
by by 1
10
10
0.1.
6u
a
a
by
by
10
10
453
6aJ-1
by
by
17.8
1 by
by
by a
17.14
a
by a
454
by by
1
2
17.14.
by
_
_ 17.14. by
x by
1. by by
good
455
on by
17.8.1
Sa
1
1
2
_
_
_
by
no
by
2 0 0. x
by
0
0
no
456 1 2
0
q q.
by v
0,
q
0.
1
0, q
q
1.
1.
by by by
by
on
17.8.2
by by by 0. b by b
0 0 x
2
457
2
q
2.
by v v
b
b
1
1
v
1. 11
_
1 v
by v
458
by
_
by
by
_ a
._
by by
459
by a y
by
0
y
0
0 0
y
0
0
0 x by
1 1.
1
1 v, 2
0,
1 v
by
by no
and and
460
fn+l
a
a 2 an
body.
by
body by 17.15, 1. 1
by by by by by on
on
by
461
17.9 by
by
a
a
by
a
by go
ai-1+:-Isi-11 17.16
1
0
.
462
_ go 0 1,
0. 0.
by by
17.16,
0,
0 by
by
0
by by body.
body by
by
463 0
by by _
by
a
on
0
by 0 2
0
_
_. 0,
0
0.
a
0
464
by
0 17.17,
0.
no
body.
465
no by
by 2,
_
1 0.
by
by 1. 17.18.
0 17.18,
466
by
0.3
3.
0.3
0.3
3
17.10 by
17.19.
17.19
do
3
467
17.20.
17.21.
and
468
17.20
by
on
on by
0, 01g
01g
0
0 and
# 0 # 0 and 17.22
0
17.21.
469
0 0,
0
#
a -- 2Kt Ifret
A comparison with (17.83b) shows that the only possibilityfor a displacement limit point is that f,~f fulfillsKt if,e/= 0, which requires det K [ I = 0, i.e. Idet K~ -I = 0 and Ktlf,ef = 0 => displacementlimitpointl
17.11
by
_
~i-l _
_/~i-lfref
_
2
2n by by o
body, by
470
= .t'L - 2 'f,.ef 2 gd b by
b
db
= --2(~'i)Tfr,/
as well as d2b/(d2i) 2 = 2fr~ffr,/> O. Therefore, the m i n i m u m of the quantity b with respect to :t~ is given by db/d2 j = O, i.e. ,
01
by by 17.23. by
2
by 17.6
a
17.24.
471
by
by by by by by
by
by 17.24
17.24
472
17.3
ao
0
0
0
o
0,
0
1, 2 .....
n
a0 1, 2 ....
a
a
a
2 by
by by
17.10
17.24.
473
by 0 2
by
This Page Intentionally Left Blank
by by
on
S an
a 1
a2
a
a, a
476
18.1. by 1
2
1 1, 2
by
18.1 10.1
by p
.
0
on
0
von
477
0 0
18.1.1
1
1
1
2 .
_
p
1,
478
by
,
18.2.
2,
1,
1.
1
0.
0, 0
on
0
no on
1, 2,
0 no
0 2.
479 18.1
0
,
18.3
18.4
0
d2 1
o 18.1. by do
do
18.1.2
o, 0
0
0
0
y
1 1
o
o
o .
a o,
o 18.3.
1
480
1
1 a o. by 0
1
no
, 0
,
by by by
0 18.2 by y
0.
by
481
18.2
0
aq ._
a
p
d2
von von
0
7,
q
,
1
,
1
18.3.
,.
,,
2
o
482 18.1.3 0
by 0
1
by
10g 10g
(2) =
10g 10g
483
18.3
and 18.2
1 10g and p
6 by 18.3.
10g 10g
dy
484
171
2
von 2 2.
upon
by
by 0, 18.4.
by
485
18.1.4 2
on
p
d2
1
....
1
2
486
by 2.
d2 2
_
no
1 by g
p
2.
on
by
2.
oq o
487
oa
._
.
.,
,
,
2. on 2
06
0
0
1
0,
by 18.6.
488
18.4 and 18.2 and
aq,
0
.
,
0 on
, 18.4. 0
0 on
0
0
489
0
1 on
on 0
0
0,
1
on
a 0 0
1
1,
and and
on
by 4.4
0
S
490 by
o2g 02g
02g
0S
0 16
6 x 6
x
d
a2g
#2g
#2g
#2g
#2g
#2g
#2g
#2g
02g
#2g
02g
02g
02g
02g
02g
02g
02g
02g
02g
#2g
#2g
02g
#2g
#2g
#2g
#2g
#2g
#2g
#2g
#2g
#2g
#2g
#2g
#2g
#2g
#2g
#2g
g,
von
491
12.2,
by
3
by
0
.p _.
by v
Eeff
2
p
p
('~EijEij) 2.
by n
day(e,f/) by
1
1
492
0
1.
3
2 2.
yon
A~
1 by
1
A~
493
18.5
von and
,
3
.
0
.
. ,
3 p
p
3 2
2 . p(2).
0
% :t :
- 3GA2
J'(~) + - try(e~::
by
A;t) = 0
.(2)
494 1
2 1
von
do
by by
by 18.7, q.
18.5
,, o _
o
495 1 0
3
von
von 0, 1
S
by
18.8
18.1.5 17.4
by 0
by
oo
2
x
496
a
(Yo
:,o:"-----
by by by by
by by
_
=
_
up
497 6
0
0
__.
1
by
1 ,,
1 0 1. 500
10
0.67
10
0.95.
17.2,
6
0
1
498
0
a
0 _
on
0
2
a
d2g
on 0 by
d
2
499 g 02g
1
da
a
da
=
a
1
on a
a.
yon
by
t
(1)
(2)
(I)
by ,.,
_
500
1
3
1 3
by
p= 0 -t
elf
0
1.
a
a
1
1
2
a 1,
501
18.2
15,
on 17.
15.4
0 0.
_
0 0 0
by
502 18.2.1
a
by
0
O~j p
503
_
0
4.4
_
k
0
on
by
02g a
02g
.
16
6 x 6
d 0,
x
504
von
3
_ 11
by I/2 2S S
by up
1
505
by
by
.
. ayte~::
%::
0
0 q7
t %:/
- 3GAA
. ~,t,(1) - ayte~:/ +
x
A,;t A.,I) - rt-~7 = 0 vp
by
A2
o-;ff
=
_
0
.
1 0
506
by
,,
on
0
1 oa
a
by
507 18.2.2
p
on
p
0.
by 0 0
0
._
1
1 1
1
0
dap
508
#q
by 3 vv
vv
k
0
,
0,
by
19.1 by
by
g
by
510
by on
17. by
2
p
y on
.
2 1..
a.
19.2
1
1
511
2
0
n
i:in
2
a.)
At ~
2
-:-:a. A1
2
2
an+l = "7"~..(an+2 - a.+1) - -:-,~.. a.+1
1 //n+1 = X'7"7..(an+2 -- an) Z/Xl
1
iln = 7"7"7..(an+l -Z/AI
an-l)
(19.11)
1 /i. = ~ (a~+l - 2a. + a~_ At -
Man+l = M(2a. - a.-1) + At2(f n - Iv Bra"dV) by 18,
512
by
body,
p 18,
no do
18
0
von 18.1.4.
513
19.3 # 0. n
1
n
1 2g
0
0 v
by
v
17. 0
0
n
1,
514
1, 2 , . . . . a
a
_(A(at-l)-l) {M[cl (a i-I
by
1 0
.
0
an
gn
by on
0
v
by
0 by
v _
_
an
v
by
515
v by by by 2 10
2 10
10
v
a ( a_~..v)i-l (ai _ ai-l ) . . .
a
a
da
M
Kt
,
516
a
17
p
20 upon
on on upon upon
on on
on
518
on
up
do
on book on
20.1 body on body
519
an
body
no body
no
a
no
p 20.1.
no body
bound, 0
520
0
1
bound by by
by 32 by 459
0, 20.10,
20.2 by du
by by
body
a
body.
and a
20.2. on u, p g body
by
521
by by
1
2.
by 1843,
20.3.
by by
522
1
4.19
1
1 kg
by
20.3
20.4.
body
523
by
p,
0.
upon
a on
and
on on
1
2. 1
2
on
a
524
1 0
u 4
.
.
20
.
.
.
.
.
.
du
on
20.5, p,
0
1, by
by 0u,
u
by by 0,. p
by 1
by
by 0,. u
du
by
by
525
by
by 0
a by
a
20.4
body, on on
body. body
by on by by on body.
526
body
n,dS no body
by
1 on p.
p
d--V+ 77 where U is defined to be the so-called internal energy of the body. Moreover, it is postulated that
a body by
by
527
614
0 by
1850.
body. body. by
u
u body by
,
0
0
0 body.
528
20.5
by
.... 0 by by
1662 1802.
by
1676
p,
0 0.
1845 111
20.6. by
no 0. no
on
529
0.
,,.
body
by
by
u du
du
530 by 0 0, 0
0
=
p by
0,
mcvdO = -pdV
p by
20.6
and
on
a
on
531
by
by
0, 0,
by
0, oo.
20.6. 111
on
532
0
no
20.7
on on
a a and
an 20.7. 0
0
1851
533
0
01,
by
0
1850
20.3,
upon
20.8.
0
534
01,
0
0.
0
0.
1,
0
11
20.9.
by 1850, a a a a
a
535
0
1
0
by
0.
20.10,
by
20.11,
536
and
20.8
an a
20.12. 0
u
0
537
0
01,
0
0
0
__
0
20.12
by
20.13.
0
0,
,
0
538
0
1
0.
20.12 by
by 20.13. 0,
539
0.
,,..,,
_
0 by
20.9
by
1824
1
4 3 v2 v3
20.14.
1,
540
no
0
0
1
541
1
by
by
20.10 0 20.1.
up
on on
20.1.
by
0 on
a
542
by 20.3, by
by
1854.
by
1954, 273
0
1,
01 by
20.11
_....._
0
0
0
543
# by on
by on 0.
S dS
by 0
dS
on
and a
da 8a
o 0yo
0,
544
#
1,
6a
6a.
by by
6a
1
0
1
1
0
1
6a y22 by
dS, S by
_ dS
a
20.14.
0
545
1
u
1
20.12
a
no no
and no no
0,
0,
546 0,
dS
514
0
dS
0 and
a
1876.
20.13
S body by
1 50
body
by
dS
547
no
body
dS p
___ 0
0
02 body.
on no body.
on
548
20.14
u on
by
by
dS 20.6,
no
body
body
on
549
by
by
This Page Intentionally Left Blank
21 by
by
u u on 20.3,
by
0
02
0
0
552 by
u
0 u by
by u by by
1882,
a 0 body,
_
u
_
0
21.1 20.3, upon
553
20.6, body
0
0,
1
0
.
0
0
gq.
by 0, by
u
0
0
554 0, 0 p
0
0 2
0
no 0
0o
0.
0o
1
0,
555
by 02111
aq
aq
aq upon
2 1
_
o o
and
by
u.
556
1
u
u by
u u u, u by
u u.
0 by
1
u
h
u
1
h
by h 0
557
qb
by
qb
qb 0 by u
no 0,
0
0 no
no
by
u
0 0 0,
on
by
558
u 0
by u no
a u
21.2
o . .
o
by
21.1.
21.1
559
0
1, 2,.
by
do do
1, 2 , . .
0
560
and 0
21.1.
on
by by by by
by
y
y
x,
oo
oo
0 on
0
on
0
561
by
0 0 on
0
on by
by by
by
0, on
,
0
0
0
h
a
p S
0
0
p
p
a a
0
562
by
0
0 0. 0
0. by 0, 0
0 on 0 p
0
by on p
563
no
0, p
0 p 0
0 0
0. k,, 0,
no
......
00
0, 00 by
by by
564
by on on on
0
v
v
o
on 0,
o, o,
0.
upon,
S
and
0
0
0
o,
0 0 0
0.
0
0, o
565
21.3
0
do by a
do
01 do by d
do
566 21.3.1 a a no
a
21.3.2
0
0
by
ao
by by
upon,
567 21.3.3
1
$
1
1
1
h
2 o
o1 on on
1, 2 . . . .
568
n
a
n
3 3 by
2.
and
a n
2nqb and
qb
0
0 von
by 3 no
0
no by
u
0 .p
a
1
u
569
qb
o ,
,
0
0
0
-"
Due to (21.41) and (21.43), we have
(~)Ao=0
0
0
a
and 0
0
570
0 0
0,
0,
qb* by
2q
571
qb* =
+ ~ q Zq
05....
. by
by
on
do by
.
.
g
10. on
g 2 by
g by # by
0
by
572
0.
0
and and 0
by
0
0.
0 and and
a
2
.
o
0 and
10,
573
no no by
by 0
0
0
0.
10
by
21.4 by by
body.
0
0 no
on
574 20.6, 0
# 0
1822,
k0
on
on
0 0
1
# 0
a
0
1
# 0
by 0
575
q 0. 0. $
kq
on kq
and k k
0,
0 by 0
upon
by
du
576
by
p
pdV
on on
1.3
1.7,
577
body 80
body.
1 body.
u
by ,
a
.
0 002
a ..
. 0
0
2
578
02111 02
p
0 0,
0
02111 0
02111
02111
0111
p
0
02111
on by
and aq,
0
.. by
aq
579
v kg
2.1011 0.7.1011 0.3.1011
0.3 0.3 0.15
1.1.10 2.3.10 1.0.10
7.8.10 3 2.7.103 2.4.103
0
1
p
0
p by
0
21.1 21.2
21.2.
580
9
p ....
480 960 900
2.3.10
10.10 0.54.10
5.5. 106 3.7. 106 5.8. 106 2.6. 106 1.9. 106 2.2. 106
0.
and
by by 0
0
0* p
q by
p 0
1
by
q,
1.4 3.1 0.2
581
.
02
02
0
by
by
body
on
..
582
and
and a
k
by
1837.
by
on by
0
0o
no by
0
0.
no _
on 0
583
o
0
0o
0o
0o
0o
0
0
3gad
21.2 1.014.
0
0o
0
0o,
584
by
21.1
1855
21.2, 200
0.17
by
1,
21.2, 33
71,
569 by
20
20
585
by
no body.
by
by
0.
no 0.
02111
0111
0
0. body. body body. body .
on
no
o
23.
23.
586 0.90
0.95.
by
on
on
by
21.5 body
no body
no
by
#u
#u
#u ....
587 0
0,
1
1
S
. .
.
.
.
1 0
u 0 0
u
02S _
_ _
o
0
588
02111
00
n
._ .
0 0 ,
by
0
0
o
by
0
0. do ,
0 do
0
589
4, 0
v
This Page Intentionally Left Blank
by on
22.1
and
0
a
.p
0 10
592
by p
1
p
,
no 0,
,,,
by
21.3.
on
by
.p
and
and a and
by
2 .p
0
0.
593
2
. ......
0 and
0
0
0
10
0. 2,
10
2 by 0
k,,,
by 0 p by 0 qb 0
o
0 and o,
g and 0
,
0
a
0
by
o, 0
594 0
g,
g
g 10
on g 2 ___ on 2 2
0
0
2
0
0. by
a 2h 0 b,
a hq a by
0,
0. 0
0 2
0
0.
0
a
.
?
10,
p
0g
0
by
0g
595
10.2.
1 0
0. 10
22.3,
22.2
21.3
body
no
no 0 body
596
a
,, -
-- f((l - a)y (1) + ay (2))
I',
1
y(1) l
y(2)
=y
1, 2 , . . . ,
a 0
1 _
+ (I -ct)y~2) e C
and 0
1.
by by
a
a
597
. . . . . .
by
f(ory:I)+ (I -- a)y:2)) <
otf(y~I))+ (I -- ot)f(y~ 2))
From the definitionof the set C, we have f(y:l)) < K and f(y:2)) < K and _
by 22.3. a and
and 0
,
0
0
and
0 ,
u,
598
y
by
on
on *
on
,
,
by
0 ,
,
,
0
0
599
and
and 0
and
0
2
0
and 0
do
a
on by
.
, 0
on *
,
,
on 0 ,
0
600
b
a.
by
0
0 pq
0
20
*
0
0. 0, ,
0
0
0. by by by
0 0
by
0
601
0
1.
0
0,
,
0
a by
0 by
0
by
by by
a 0 0
0.
on
0
a
0
0
by
by
,
0
9 no k,. by
602 no no
22.3 by by by
by
by
by no
no
o
by o
o
2 p.
von
_
_
.
603
9.6
p.. p
p
p
p o
p
1
v
2
eeff =
vlvP(ePeff ) "--"~H(Eef f ) ,
~fP(eP/f)is made
If a Taylor expansion of p
1
IC
(22.26)
about the point
0
0
no
a
n
0
= ka.(e~:/)"
_.
0, w e obtain
0
0
p
p
ee/fv=
2
1
p
n
1
604
p
p
Eeff
p
,
von
2
o
o
g
2
0
0
b
-
2
3
_
-
3
o
605
von a
0
l(o'q
4, _
0
0 0
,1
,
,91
ag
p
1
2 n
1
. p p
von
606
1
2
by
g
dg =
=
_ _
=
12,,,:: by
p
elf
2
g*
=
-
.p
/~ = c%//(I - ~-~-=)
(22.35)
607 0
by
by by
2
2
by
yon
3
d
,
0 .p
2
608
1 , 0
0 no
1
1
3
by by
609
von 3 d
d
d.
.p 28yo
.p . fcij = - :2~i ) = ~.
28yo
=
E ij '
2 fc =
- i~ - ~
-
E ef f
From (22.1) and as p~t'(tcij, ic)
,c)
p
,c)
p
p =
+ P
021p
k
2
p =
3
3 =
0
0 1
1
3
2
610 by
02111
021# p
2
0 .p
25yo
2#yo 2 0
von
3
, 1
p
q
o
3
2
eff
__ ~
.p
0
3 3
611
by 3
d
d
.p
x
h
aoo oo
h
.
by
2
2
.p
p
021p
,
612
02q1
.p
p
.p
p p
1
3
by by by
p
1
d
3
d
3 0 3
d 0
2a,,o
d d
0
613
3
o
=
3
g=
p
d
h
=;of
.,
og
g
., _
aoo
3
0
2h
h
2 p
1
d
g
and d
..
;.,
614 p* p*
.p
2
p
p.
2
0
gp 0
0 0
22.5
0
0 p
0
0.
615
g
11
0
0
0, 22.6
0.
.p
o
0.
22.6.
0,
0, 0
0
by
616
.p
0
p ___0 on
p
p
2
617
g
a
0,
and a*
., x
0
a
a*
and
and and p
,
p
2
22.4
von
9.6,
von
p. p. von
618
p p
do by
x
.
.p
n.
nq
q 14
by
619
0 by 0 1.
22.5 by 4,
, ,
0
on
on .p
22.7 by
1
0
.p
620
by
1, 2 . . . . .
on 1, 2 . . . . .
by
by
0
0
and
by dp
1, 2 .....
621
0.
1
p
_
p
.
a
622
a
a,
n 0 0
by
by
623
AA=I
,4,
.
1
# 0 no
624
by
1,
0
S
0,
0
625
no 0
10.3.
0. 0
and no
22.8 on 22.9. by k k
1
626
1
1
2
03
1
2
1
S
0
by by
02
,,
S
3
627
22.6 15
.
on by
by .vp
.vp
0
0 0 by
.vp
..
. by p
0
0
0
628
0,
0
d
0.
by
.vp =
3
0 0
f (~Yij,K~,) = qg(nEeff )
~
~(f) = rle~ff
where 9 is the inverse function of q~. A comparison of (22.80) and (22.81b) shows that the non-negative function A is given by
von
by
on by
0 15.4.2 by
8q,
0 o
on
629
,
o
,.v.
*
_
0 .
,
1
,
,
1
,
,
p
,
by
0
630
22.7
by 14
by
0
, ,
_ 01
1
631
v
1
by
by
n
by
by
n
v
v
632
0
n 0
2
1
,.
14.21
by
633
on
by
on
0
1
v
1
by
634
by
0
n
0
p
n
0
1
k
2
1, 2,
....
n
635
0
0
by
This Page Intentionally Left Blank
by by by by
by
by by
by
23.1
by
0
638
30 25 20
10 5
0
5
10
15
20
25
30
35
on by
on 23.1 by 23.1 23.2 by
by by
00
639
200 100 0
0 0
200
400
600
800
1000
on
200
100
150
80
.
60
100
40 50
20 0
0
0 0
200
400
600
,
200 400
800
600
0
800 1000
on
23.1.
202 73
on
v
23.4
640
0.4
2024
0.3 0.2 0.1 0 0
200
400
600
800
on
by
2024
25
25 20 15 10
10
_.
5 0
0
0
0
200
400
600
200 400 600 800 10001200
800
on by
v v on 2024. by
00
2 23.2.
641
10
0.006 0.010
11.6 22.1
2024
by 0_
ao
1000
1600
800
1200
600
800
400
400
200 0
0 200
400
0
600
0 200
600
1000
1400
on
k
p, by
on
642
k
k 300
1.0 200
0.5
100 0
0
0 1 2o0 40o 6oo 800 1000
0
0
2oo
6oo
1000
1 1400
on
23.2 on
p
0
0
1
0*
p
_
00 no by
p
o
o
on on
0
643
on
1
on
p
o
oo
p
, 002
2
2
0
-
-
d2
p
002
by ._ 0
23.1 23.2.
by
Ko. e)_<,o
644 von f = a e f f - ~ryo(O) - K <
0
g
. 2 _ 0 g
0 0 0
0
0 8,,
0
.
o
0,
0,
0
0.
645 on
S
0.
0 0
0
0 0 0
0 0 0 23.1, 0
S
0 0 0
0
0
646
0 0. 1 . . . .
.
.
0 1
1 d
0
021p
p p
1
647
0
0
0
u
0
0 0
2 _ 0.
0,
2
0
0, 0, 0,
and and and
oau
by
0,
2
by
1
1
6
0, 0 0 23.8.
by
648 0 $0
0.
0, 0 2
0
0
0. by
by
by
n 1 a
23.2.1 on 6
649 0.
0,
1
1.
0.98 1 0.93
0
23.1 294 0 294
23.2
1
1
on
23.4 v
2
0,
d2
d2 0
p 1
20 11
650
by
23.1 10
0
p 2700 294 900
3
10
23.3 von
by by 3
1
1
.p
d
.
k 2
p
.
2 2
0
$0
0
by
651
0
p by
23.9
on 23.9 .
p
3 on
0
1
2 2
652
03 0
23.4
body 1
by .,
by
q, by on on
653
by
.
by
by 3.8, 0
3.8.
by
v
qndS by body
0
q
q2 q3
0 0
654
body by
ao
by
q v
o,
qn dS
o
0
0
q
655
o by
by
23.5
by
23.5.1
o k on
on _
_
,
.
.
.
.
19,
by
ao.n n y
n 0
y
1.
656
0 1 n,
o.,
0
n.
o,. on on
by
body,
0
1
7
1,
657
1
1
0 1
n
1,
0
n
1
0
o ao,n
o
ao, ao
17. 1, 2 . . . .
ao ao ao
0, 0
ao
658
0
by
1 ,
1 O[C(ao - ao,.)] = C +
1 C
(23.5I)
k dk
_
k oao
iCo Ko
Ko =
B o -~VONo
= C + At
~o
1
= ~
dV
+ Ko + Ko
1(ao_l - ao.~) + K o ao-I - f o,~+1
ao,n
17 n,
ao,n
659
17.7
a
23.6
23.7,
k 1
17.8,
by
23.5.2
on ao.
an
n n
on
on
0 by
n
1.
17 by a
,
a
on ao,n
660
The starting vector for a ~-I for i = I is taken as a ~ = a., that is the value at the last accepted state n. It appears that the only way in which the temperature field influences the mechanical field equations is through the stresses which, in turn, determine the internal forces ~v BTtrdV; this is the topic of the next section.
23.5.3
18 18.1.1, no 2 1
2
1
.
tr~J) = .-. . (i) -- ekl _p(1). Z)Okttekt ) -- riO(0(i) -- 00) where for simplicity it was assurncd that Dijkl and to are constant. Subtracting the above equations
p
Aait; -~ DijklAEkl --
_
0
no
=
0
p,jA01
661
0 0
18.1.4.
0_.gg
0 2 by
a
0 von 18.1.4 18.1.5
23.6
by 23.8 by
ao
662 by _ 1, 2 . . . .
on by
on
23.6.1
1
1 by n
n
1 on
on
0
o
an, ao,n
n
1.
p
.
k n
k
1
1
1
1
1
1,
663
p
1, 2,..
ao
~to(ai,aio)=llto(a i-I,a i-I) o + ('~a
(ai -
) + ("~ao )
- a~
According to the Newton-Raphson method, we require ~to(a~,a 0) = O, i.e. ,
i-i +(O~o)i-1(ai ai-l)+.O~o,i-1(ao i-i 23.5.1,
by u
--gZ-a o aV
V
_ 1__
on
2 on
23.2.1
664
0
Tia-l(ai - d-l) + Tio-l(ao- ao ) = -~o
1
23.6.2
0
ao
n , a0
1
,
ll/(ai-Ii-I)
v(a t,a~) =
,a 0
0~oo
o
i-I
ao
0 0
,
ao
o
da on
665
dg ,
0
a0
d V - f n+l
~r(a i-I, a o ) =
The starting vectors for a t-1 and a o are taken as a 0 = a~ and a~ = ao,,, i.e.the values at the last accepted state n. From (23.73) and (23.76) it is apparent that the Newton-Raphson formulation of the coupled problem becomes
Tol
aoaoii1
L
1
, ao (a0 -I - ao.n) + K 0 a 0
= C
_
_
k
1
1
,. ao
ao ao.n
1, 2 . . . . . n
by
666
0 23.5.3 on
a
a a
,00
_ pao
02 g
1
a
pa a
a
.
(
02g
8g
da a ._
d0
(
2
a _
00
667
a
,
a
by
by
23.7 no body. by no
668 0.
0
p
on by
on
0
23.5.3.
_
ag
0,
0
669 by
by
23.6.2
0
on 0 o
0 0.
0
ad.
1
1
g
ad 11
670
1
.
u
23.8
by
by
on 23.10.
671
by on
on n
1
0
17 18 2,
on on
23.6. by on
1 o
o
ao
..
1
o
1
672 by
23.5.3. 1, by 23.5.3. 23.7.
0
0 by
by 0,
0
0.
24
24.1
by
674
,
d
4,
a
1
by 1,o .p
1
0
1
v
up
no b
675
_1
no 24.2
0, 24.2
24.2 g
g b
0
0
0
k b
1 1
24.3.
k
1 k by
1
676
0.5
1
2
by 20.12. by
24.2 on
0
by .
,...
1
ag
dg 0
677
2
0
by 2
1
0 1 2
1
1
0
2
by
0
0
0
0
678
by 1
by 1
ogy
I
(24.23)
0
qo
0)4
605
1
0
by
1 1
. ,
0
679
1
1.
0
4.
1 1
o96 __.
24.3
by by body
0 by
dS
680
_
IV
Is (il2)_ill) ) IvVi(bl2).(1)
.(I)
b Su
.(2) (I) v, j(:#~> - :,:)))dv =
Suv,(il~> - t,.(I))dS u
0
o Introduce the notation
0
0
0
681
by
1
1,..
0,
_ by
0,
1
0 0 0.
by
682
by
#0
0
6o6
0.
1
,
1
g
1
0, 1 0. by
by
by by by
0
.p 0
683
0
1
.
.p
.
.p
.
0 by
.
.
0
.
.
24.4
24.4.
by
684
by by by
on
by
body,
24.5.
by by
0 24.5.
685
1
0
on
0
.
.
.
upon on on
0
by 1
by by by on
.
,
686
.,,
0
by
---
0
0
by
by
p
1
687
by
# 1
1
0 p
db
by by by
1995,
688
von
by
by by
0.1
20
S1
n ab
0.1
0.2
0
0
$2
0.1
___ 202, 0
54.70
by
by
0 no
0
20
1 1
qb
db
45
0
57
689
,, db 0
_. 0
0
0 0 by by
90 0
90
0
# 0
and 0
0
0,
690
db
0 0
0
db
db ,
0
0,
0
n
24.7
24.8.
24.5
by by by by
691
by
body by
body, 24.9. 0
692
24.9,
body.
on by
0
n
0
24.9
body.
0
0
0
0 1 dn
dn
dn
dn
0
693
by 0
on body
p
0
0
.
0.
body. by
02
02 2
2
694
0 body by
by
body
by by on p
2
2
2
a
2
2,
695
by by
This Page Intentionally Left Blank
by on on
by y,
0
1.
698
d2 .9 y
-_
1, 2 , . . . ,
and 0
_
,
1
_
0 ___ _
.
___ 1
y
699
,
(.~yi))otAy,(.9(a 2)
_
+
0,
ogy
,(1) plies (A.1). For this purpose, we accept (A.4) and choose yi as (I - ct)y}3) +
_ y,
_
_
by
by
and and
700
,
....
_
_
,
_
1. 0
0
1,
0
_ _
_
and u
2 d
2
0
0
.
701
1, 2 , . . . ,
0
0
by
#
1
1
and a ,
.
,
0
0
,
02
,
702
0 by by
0
y2
1, 2 , . . . , 1, 2 , . . . , g1 0
0, by
a
0
g1
0.
1, 2 .....
703
2
_ .
0
y*.
1, 2 , . . . , 1, 2 , . . . , and a
0
2
0
0 by by 2 2
704
and 0 ,
and
02g
on 0
0
02g
Ifthefunctions f (yi)and gI (Yi)are convex functions, 0
and
37,
and
44,
and
and
34,
69,
4,
35,
and
98,
706
on 1. and on
and
14, and and
and
3,
a
1, 7, and
49,
and
22,
and
by 4, und und 15
und und
und
707
102,
344. and
and
and
23,
and on
12, 10,
and
6,
an $410. by
14, 6,
708
and
and
32,
28,
and
40,
1331. 13,
and 25,
42, 8,
and
39,
42,
46,
du
und
55,
und 17,
57,
709
59,
und
3,
18,
and
and on and
20,
21,
6,
and
on
and
710 37,
and
33,
on
316
5, 2, 5,
60,
60,
and
and
110, and
and
and
28,
and
17,
711
47,
and
6, 33, 97,
and
8,
and
7, 48,
29,
and and 1.
and 2. 21, 43, on 5, 246.
and
712
1004,
8,
29,
19,
on
81,
and 11, 97, und 18,
and und
6,
27,
3, 48, 10, 9,
713 389.
10,
237.
du
156. and
37,
17, 173,
2, 2, 172. and
and
on 48,
2,
714
6,
and
and
49,
13, 1.
50,
106,
and
57, 37, by
24,
715
50,
and
und
17,
1, by
109,
21, 14,
4, 66, 18,
and und
und
4,
und
und 5,
von
716
and
1,
28.
and
6, 2,
and 10,
and
by 14,
35,
by
and
6,
on
8,
und 226. 7,
12,
717
and
18,
on 12, 22,
and
10,
1,
and
27, 3, 34,
12,
718 13,
and
41, 2,
und
und 56,
and
94,
11, don
67,
on
and
13,
93, on
on 5,
719
38,
122, 37,
and
and 123,
42,
and
99,
on and
229. 66, 13,
and on 99,
and and
32,
720
36, on 4,
463.
a
5, 5,
70,
and 5,
and 55,
2, und
36,
and
40,
721
on
and
49, 7, 5,
and
16,
and
18, a un 1, un 1963,
and
1, on 11,
un
du
722
14,
35,
and
35, 56, und 134,
and
11,
2.
1. 304
and and and and
723
16,
and
and und und 28, und
44, 1530,
no. 378.
and on 18, 15,
and 1006,
34,
and
on
and
65, upon
and
1.
and
2.
724
13,
and
33, on 4,
and 67, 45,
18.
and
30, 30,
85,
and and
von
und 13,
and
725
37, 38, 99.7,
21,
and 61, 1801,
and
103, 105, 106,
112,
21, 53,
and
33,
and
726
27,
and
28, 12,
3.
496. by
and 11, 18,
20, 9, 11,
and
24,
on and 8,
727
on 8, 111, 67,
and 9, 16, 20, 169,
6,
on und
8,
un non
252,
728
and 29,
und
55,
und
10,
57,
28,
47, 67,
and 30
185,
125, on 17,
and
37,
36,
and
4,
729
121,
and
33,
and 48, 8009. 118,
and
and 23,
on on 5, on
85.
and
1,
7,
4,
oh 70,
and
730
on
277, und
6,
216.
und
13, on
and
67,
258. 111,
and
33, 28,
and
and
22,
731
24,
und
57,
and 48,
98,
and
26,
24,
33,
4,
and
96,
9,
15, 58,
and
732
and
1,
and
7, von 9,
and
136,
and 2,
and 8, 62,
191,
143,
113,
and
733
47, 59, and 4, on
8,
and
on
and
77, 5, on 38,
and
47,
29, du du au und
47,
and
and
734
von 55, von
von
von
und
8,
and 165533. on and
190, on 1,
and
2,
36, 1100 5,
189,
3,
on 1,
33,
and
33,
735
by 10, 48, 51, 55, 47,
und 9b,
and
4,
1. 2.
1,
This Page Intentionally Left Blank
216, 292, 607 489, 656 214
285 151 147 369 183
470 458 165, 185 165 673, 683 405
519 690 685 375,389 628
body
49 380
585, 668, 672 530, 557 671
380 424 332 91 131,368
270 374 496, 499, 666, 669
553
79 218 135 193 72, 79 473 344, 611 375 224, 233 561 136
318 528 539 3 11 30, 37 54 68, 103 35 511 28,455 685
738
210 281
534 546 418
391 95,
389 167 676
245 563
150
265
161 36 56
270 71 155
362
697 512, 656
389 16
564
601,699
233, 594
401 371 374
357,358
683 336
541 67 697 479, 480 467 236
390 22 21 497
250, 552 567 218 695
451 595,697 225 380 105
61 225
619
163 317, 613
454 166
380
50 126, 357
417,507, 628 18
362 388 359
402 413
374 358 357 204
397
739 676 423
237 238 656
517,523 222, 230, 620 695 107 432, 484, 489, 656
28 75 118 92 92 288 73
557, 573 562, 574 699
204 87 258
168
204
167 405
252 253
489 673
686 484 26, 40
425 699
556 543, 544 546 546
371,635 369, 632 48 48 235, 594 571 655 487 487
61 426 374 401 64 234, 565 484 512 207 226 384
37 56 557
142 147 171 172 146
38 56
425 654 22 70
740
100 428 150
15
146
6
210, 400 204
7 5
209
4 4
533 547, 573 521,525 243 526
6 4 6 558 68
552 406 362, 378, 380 211,559 196,
86 147 145, 203 210
310 200
562 526
407
560 210, 559 562
326 68 387 73
56
161
30, 37
150 56, 59 67
38 37 56 122
142
530
141
548 557 671
524, 528 541
524
528 146 327 489, 513 82
524 530 670 92
741
461
212
681
401
78 78
79 121 208 468 684 153
120, 124 16 108 bound 207
J
122
265 512
520 584 128, 130, 366, 633
74 327 325 426, 653
532
373 85
64 328
133
215 526 400 230, 620 5 572, 593,702
2 9 3 3 2 2 2
683 222 33, 572 23 90 386
2 37 374 179 126, 129, 364, 631 281 562 295,
586 389 484 71,556 308, 608 265 265,466 468, 624, 677 255 461,462 461
155 418 122 74 217, 303,609 180
742
141 168
467 70 146, 204 361
171 42, 43 57 321
523 250, 523, 532 408, 502, 627 47 81
321 389 64
47 59 693
327 510 61 657 425 233 697 24 51 224, 230 388 553 163 39 57 38 39 57 147
265 232, 242 225, 593 146, 204 242 145, 203 86 685 74 232, 572, 697 462 69, 222, 224, 233, 553, 593 208 222 359 549 29
566 399 84 653 344
29 55 526 107 250
407 62 67
371
743
82 120
244 570, 633 60
405 402 118 435,454
495 3 455 14
250, 517,532 484 359 645
244
393
206 455 456 457 179 520 281 232, 242 359 548 345 360 292 373 468 620, 702 24 362 357, 358 372
273, 683 156 91 26 51 48 484 684 273 172 467 467 146 204 546 529, 575 527 36 694
35, 114, 119, 201 103 372 70 530 362 362 380
281 670 437 369 523 211,400, 522 64 403 269
744
359 379 204
655 575 655 562
281 68 239, 315, 617 260 21 392
87,555 655 403 639 655 86, 555 646
225 281
637 545
56
541 517 86, 552 548
698 270 689 63 132, 136 484 325 372 49
509, 655 126, 357 374, 391 373 87, 204 50 8 359
703
136 510 159, 174 478 542
428 253 155 181 12 15
199 468
125
470
125 16 14 13
333 147 164
574 555 359
47 59 164
745
47 230, 673 376 375, 389 265
bound
455 98 63 128, 357, 360, 630 410 357, 387, 404, 627 126, 364 630 564, 632 366 380 161 36 von von
156 159, 161 W
692 62, 63 617 248 266, 273 74, 86 308
This Page Intentionally Left Blank
