1 s s s s s s 11 12 13 14 15 16 1 2 s s s s s s 21 22 23
TRANSCRIPT
3
2
1
363534333231
262524232221
161514131211
3
2
1
SSSSSSSSSSSSSSSSSS
5
4
3
565554535251
464544434241
363534333231
5
4
3
SSSSSSSSSSSSSSSSSS
(4.6)
66665646362616 SSSSSS
zz
yy
xx
zz
yy
xx
SSSSSSSSSSSSSSSSSS
363534333231
262524232221
161514131211
xz
yz
zz
xz
yz
zz
SSSSSSSSSSSSSSSSSS
565554535251
464544434241
363534333231 (4.8)
xyxy SSSSSS 666564636261
xyxy
xxxyxz
xz
xxxzyz
yz
xxyzzz
zz
zxyy
yy
yxxx
xxxx GGGEEE
,,,1 (4.9a)xyxzyzzzyyxx
xyxy
xxxyxz
xz
xxxzyz
yz
xxyzzz
zz
zxyy
yy
yxxx
xxxx GGGEEE
,,,1 (4.9a)xyxzyzzzyyxx
xyxy
yyxyxz
xz
yyxzyz
yz
yyyzzz
zz
zyyy
yyxx
xx
xyyy GGGEEE
,,,1 (4.9b)
xyxy
xxxyxz
xz
xxxzyz
yz
xxyzzz
zz
zxyy
yy
yxxx
xxxx GGGEEE
,,,1 (4.9a)xyxzyzzzyyxx
xyxy
yyxyxz
xz
yyxzyz
yz
yyyzzz
zz
zyyy
yyxx
xx
xyyy GGGEEE
,,,1 (4.9b)
xyxy
zzxyxz
xz
zzxzyz
yz
zzyzzz
zzyy
yy
yzxx
xx
xzzz GGGEEE
,,,1 (4.9c)
xyxy
xxxyxz
xz
xxxzyz
yz
xxyzzz
zz
zxyy
yy
yxxx
xxxx GGGEEE
,,,1 (4.9a)xyxzyzzzyyxx
xyxy
yyxyxz
xz
yyxzyz
yz
yyyzzz
zz
zyyy
yyxx
xx
xyyy GGGEEE
,,,1 (4.9b)
xyxy
zzxyxz
xz
zzxzyz
yz
zzyzzz
zzyy
yy
yzxx
xx
xzzz GGGEEE
,,,1 (4.9c)
xyxy
yzxyxz
xz
yzxzyz
yzzz
zz
yzzzyy
yy
yzyyxx
xx
yzxxyz GGGEEE
,,,,, 1 (4.9d)
xyxy
xxxyxz
xz
xxxzyz
yz
xxyzzz
zz
zxyy
yy
yxxx
xxxx GGGEEE
,,,1 (4.9a)xyxzyzzzyyxx
xyxy
yyxyxz
xz
yyxzyz
yz
yyyzzz
zz
zyyy
yyxx
xx
xyyy GGGEEE
,,,1 (4.9b)
xyxy
zzxyxz
xz
zzxzyz
yz
zzyzzz
zzyy
yy
yzxx
xx
xzzz GGGEEE
,,,1 (4.9c)
xyxy
yzxyxz
xz
yzxzyz
yzzz
zz
yzzzyy
yy
yzyyxx
xx
yzxxyz GGGEEE
,,,,, 1
1
(4.9d)
xyxy
xzxyxz
xzyz
yz
xzyzzz
zz
xzzzyy
yy
xzyyxx
xx
xzxxxz GGGEEE
,,,,, 1 (4.9e)
xyxy
xxxyxz
xz
xxxzyz
yz
xxyzzz
zz
zxyy
yy
yxxx
xxxx GGGEEE
,,,1 (4.9a)xyxzyzzzyyxx
xyxy
yyxyxz
xz
yyxzyz
yz
yyyzzz
zz
zyyy
yyxx
xx
xyyy GGGEEE
,,,1 (4.9b)
xyxy
zzxyxz
xz
zzxzyz
yz
zzyzzz
zzyy
yy
yzxx
xx
xzzz GGGEEE
,,,1 (4.9c)
xyxy
yzxyxz
xz
yzxzyz
yzzz
zz
yzzzyy
yy
yzyyxx
xx
yzxxyz GGGEEE
,,,,, 1
1
(4.9d)
xyxy
xzxyxz
xzyz
yz
xzyzzz
zz
xzzzyy
yy
xzyyxx
xx
xzxxxz GGGEEE
,,,,, 1
1
(4.9e)
xyxy
xzxz
xyxzyz
yz
xyyzzz
zz
xyzzyy
yy
xyyyxx
xx
xyxxxy GGGEEE
1,,,,, (4.9f)
SSSSSS
zz
yy
xx
zz
yy
xx
SSSSSSSSSSSSSSSSSS
363534333231
262524232221
161514131211
xz
yz
zz
xz
yz
zz
SSSSSSSSSSSS
565554535251
464544434241
363534333231 (4.8)
xyxy SSSSSS 666564636261
1 etc , ..., , ,1 :inspectionBy ,
161211
11xy
xxxy
yy
yx
GS
ES
ES
“ I b h ” h h li i S l•“ It can be shown” that the compliance matrix Sij must also be symmetric:
xxxx SSSSSS 161514131211
yz
zz
yy
yz
zz
yy
SSSSSSSSSSSSSSSSSS
464544434241
363534333231
262524232221
161514131211
(4.8)
xy
xz
xy
xzSSSSSSSSSSSS
666564636261
565554535251
Number of material constants:• The compliance matrix, Sijkl, is a 4th-order tensor → 81 constants The compliance matrix, Sijkl, is a 4 order tensor 81 constants
(apparently)
Number of material constants:• The compliance matrix, Sijkl, is a 4th-order tensor → 81 constants The compliance matrix, Sijkl, is a 4 order tensor 81 constants
(apparently)• The fact that [ij] and [ij] are symmetric allowed us to adopt
“contracted notation” (and therefore to adopt matrix notation):
161514131211
SSSSSSSSSSSS
36 constants(apparently)
464544434241
363534333231
262524232221
SSSSSSSSSSSSSSSSSS
SS ijijkl(apparently)
666564636261
565554535251SSSSSSSSSSSS
Summary:• “It can be shown” that the compliance matrix must also be It can be shown that the compliance matrix must also be
symmetric:
161514131211161514131211 SSSSSSSSSSSS
21
464544342414
363534332313
262524232212
161514131211
464544434241
363534333231
262524232221
161514131211
SSSSSSSSSSSSSSSSSS
SSSSSSSSSSSSSSSSSS
Sijconstants
665646362616
565545352515
464544342414
666564636261
565554535251
464544434241
SSSSSSSSSSSSSSSSSS
SSSSSSSSSSSSSSSSSS
• An anisotropic material has 21 independent material constants
“ I b h ” h h li i S l•“ It can be shown” that the compliance matrix Sij must also be symmetric:
xxxx SSSSSS 161514131211
yz
zz
yy
yz
zz
yy
SSSSSSSSSSSSSSSSSS
464544434241
363534333231
262524232221
161514131211
(4.8)
xy
xz
xy
xzSSSSSSSSSSSS
666564636261
565554535251
S 111 S 1
22 S 133
xxES11
yyES22
zzES33
yzGS 1
44 xzG
S 155
xyGS 1
66
yy
yx
xx
xy
EESS
1221
zz
zx
xx
xzEE
SS
1331
zyyz
EESS
2332 xxyzyzxx
GESS ,,
1441
zzyy EE yzxx GE
yz
yyyz
yy
yzyy
GESS ,,
2442
yz
zzyz
zz
yzzz
GESS ,,
3443
xz
xxxz
xx
xzxxGE
SS ,,1551
xz
yyxz
yy
xzyy
GESS ,,
2552
(4.11)
xz
zzxz
zz
xzzzGE
SS ,,3553
xz
yzxz
yz
xzyz
GGSS ,,
4554
xzzz xzyz
xy
xxxy
xx
xyxx
GESS ,,
1661
xy
yyxy
yy
xyyy
GESS ,,
2662
xy
zzxy
zz
xyzz
GESS ,,
3663
xy
yzxy
yz
xyyz
GGSS ,,
4664
xy
xzxy
xz
xyxz
GGSS ,,
5665
…symmetry of the compliance matrix leads to the “inverse relations”
yy
yx
xx
xy
EE
zzyzyzzzyyyzyzyyxxyzyzxx
zz
zy
yy
yz
zz
zx
xx
xzEEEE
,,,,,,
(4.12)
yzxzxzyzzzxzxzzzyyxzxzyyxxxzxzxx
yz
y
zz
y
yz
yyy
yy
yyy
yz
y
xx
y
GGGEGEGE
GEGEGE
,,,,,,,,
,,,,,,
xy
xzxy
xz
xyxz
xy
yzxy
yz
xyyz
xy
zzxy
zz
xyzz
xy
yyxy
yy
xyyy
xy
xxxy
xx
xyxx
xzyzxzzzxzyyxzxx
GGGGGEGEGE
GGGEGEGE
,,,,,,,,,,
yyyyyyyy
xxxx CCCCCC 161514131211
zz
yy
zz
yy
CCCCCCCCCCCCCCCCCC
464544434241
363534333231
262524232221
(4.14)
xy
xz
yz
xy
xz
yz
CCCCCCCCCCCCCCCCCC
666564636261
565554535251
464544434241
Where 1 ijij SC
Orthotropic materials (when referenced to the Principal Material Coordinate System)
• An orthotropic material has 9 independent material constants….
Orthotropic materials(when referenced to the Principal Material Coordinate System)
T l I i M i lTransversely Isotropic Materials(when referenced to the Principal Material Coordinate System)
• A transversely isotropic material has 5 independent material constants….
Transversely Isotropic materials(when referenced to the Principal Material Coordinate System)
H k ’ L A i i M i lHooke’s Law: Anisotropic Materials(including Orthotropic and Transversely Isotropic materials,
when referenced to a non-principal coordinate system) f p p y )
yy
xx
yy
xx
yy
xx
yy
xxSSSSSSSSSSSS
262524232221
161514131211
xz
yz
zz
xz
yz
zz
xz
yz
zz
xz
yz
zz MT
SSSSSSSSSSSSSSSSSSSSSSSS
565554535251
464544434241
363534333231 (4.30)
xyxyxyxy SSSSSS 666564636261
xxxxxxxx MTCCCCCC
161514131211
yzyzyz
zzzzzz
yyyyyy
yz
zz
yy
MTMTMTMT
CCCCCCCCCCCCCCCCCCCCCCCC
464544434241
363534333231
262524232221
(4.31)
xyxyxy
xzxzxz
xy
xzMTMT
CCCCCCCCCCCC
666564636261
565554535251
Hooke’s Law: Orthotropic MaterialsHooke s Law: Orthotropic Materials(when referenced to the Principal Material Coordinate System)
000 SSS
000000000
33
22
11
33
22
11
33
22
11
332313
232212
131211
33
22
11
MTSSSSSSSSS
(4.32)
000
000
000000000000000
12
13
23
66
55
44
12
13
23
MT
SS
S(4.32)
66
222222
111111
232212
131211
22
11000000
MTMT
CCCCCC
13
23
333333
222222
55
44
332313
232212
13
23
33
22
0000000000000
MT
CC
CCC (4.33)
12
13
66
55
12
1300000
00000
CC
Hooke’s Law: Transversely Isotropic MaterialsHooke s Law: Transversely Isotropic Materials(when referenced to the Principal Material Coordinate System)
000 11111112121111 SSS
0000)(2000000000000
22
22
11
22
22
11
23
33
22
11
2322
222312
232212
121211
23
33
22
11
MTSS
SSSSSSSSS
(4.34)
000
000
000000000000)(2000
12
13
23
66
66
2322
12
13
23
SS
SS
222222
111111
232212
121211
22
11000000
MTMT
CCCCCC
13
23
222233
66
2322
222312
13
23
33
00000002/)(000000
MT
CCC
CCC (4.35)
126612 00000 C