3D Stiffness and Compliance Matrices
Transcript of 3D Stiffness and Compliance Matrices
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Chapter 2 Macromechanical Analysis of a Lamina3D Stiffness and Compliance Matrices
Dr. Autar KawDepartment of Mechanical Engineering
University of South Florida, Tampa, FL 33620
Courtesy of the TextbookMechanics of Composite Materials by Kaw
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FIGURE 2.1Typical laminate made of three laminas
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τ
τ
τ
σ
σ
σ
SSSSSS
SSSSSS
SSSSSS
SSSSSS
SSSSSS
SSSSSS
=
γ
γ
γ
ε
ε
ε
12
31
23
3
2
1
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
12
31
23
3
2
1
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Stiffness matrix [C] has 36 constants
γ
γ
γ
ε
ε
ε
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
=
τ
τ
τ
σ
σ
σ
12
31
23
3
2
1
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
12
31
23
3
2
1
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τ
τ
τ
σ
σ
σ
SS
SS
SS
SSS
SSS
SSS
=
γ
γ
γ
ε
ε
ε
−
−
−
12
31
23
3
2
1
1211
1211
1211
111212
121112
121211
12
31
23
3
2
1
)(200000
0)(20000
00)(2000
000
000
000
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τ
τ
τ
σ
σ
σ
CC
CC
CC
CCC
CCC
CCC
=
γ
γ
γ
ε
ε
ε
−
−
−
12
31
23
3
2
1
1211
1211
1211
111212
121112
121211
12
31
23
3
2
1
)(200000
0)(20000
00)(2000
000
000
000
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−−
−−
−−
τ
τ
τ
σ
σ
σ
G
G
G
EEE
EEE
EEE
=
γ
γ
γ
ε
ε
ε
12
31
23
3
2
1
12
31
23
3
2
1
100000
010000
001000
0001
0001
0001
νν
νν
νν
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,
G00000
0G0000
00G000
000)+)(12-(1
)-E(1)+)(12-(1
E)+)(12-(1
E
000)+)(12-(1
E)+)(12-(1
)-E(1)+)(12-(1
E
000)+)(12-(1
E)+)(12-(1
E)+)(12-(1
)-E(1
=
xy
zx
yz
z
y
x
xy
zx
yz
z
y
x
γ
γ
γ
ε
ε
ε
ννν
ννν
ννν
ννν
ννν
ννν
ννν
ννν
ννν
τ
τ
τ
σ
σ
σ
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τ
τ
τ
σ
σ
σ
SSSSSS
SSSSSS
SSSSSS
SSSSSS
SSSSSS
SSSSSS
=
γ
γ
γ
ε
ε
ε
12
31
23
3
2
1
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
12
31
23
3
2
1
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Stiffness matrix [C] has 36 constants
γ
γ
γ
ε
ε
ε
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
CCCCCC
=
τ
τ
τ
σ
σ
σ
12
31
23
3
2
1
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
12
31
23
3
2
1
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FIGURE 2.11Transformation of coordinate axes for 1-2plane of symmetry for a monoclinic material
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FIGURE 2.12Deformation of a cubic elementmade of monoclinic material
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FIGURE 2.13A unidirectional lamina as amonoclinic material with fibersarranged in a rectangular array
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τ
τ
τ
σ
σ
σ
SSSS
SS
SS
SSSS
SSSS
SSSS
=
γ
γ
γ
ε
ε
ε
12
31
23
3
2
1
66362616
5545
4544
36332313
26232212
16131211
12
31
23
3
2
1
00
0000
0000
00
00
00
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γ
γ
γ
ε
ε
ε
CCCC
CC
CC
CCCC
CCCC
CCCC
=
τ
τ
τ
σ
σ
σ
12
31
23
3
2
1
66362616
5545
4544
36332313
26232212
16131211
12
31
23
3
2
1
00
0000
0000
00
00
00
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FIGURE 2.14Deformation of a cubic element madeof orthotropic material
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τ
τ
τ
σ
σ
σ
S
S
S
SSS
SSS
SSS
=
γ
γ
γ
ε
ε
ε
12
31
23
3
2
1
66
55
44
332313
232212
131211
12
31
23
3
2
1
00000
00000
00000
000
000
000
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γ
γ
γ
ε
ε
ε
C
C
C
CCC
CCC
CCC
=
τ
τ
τ
σ
σ
σ
12
31
23
3
2
1
66
55
44
332313
232212
131211
12
31
23
3
2
1
00000
00000
00000
000
000
000
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τ
τ
τ
σ
σ
σ
G
G
G
EEE
EEE
EEE
=
γ
γ
γ
ε
ε
ε
−−
−−
−−
12
31
23
3
2
1
12
31
23
33
32
3
31
2
23
22
21
1
13
1
12
1
12
31
23
3
2
1
100000
010000
001000
0001
0001
0001
νν
νν
νν
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γ
γ
γ
ε
ε
ε
G
G
G
EEEEEE
EEEEEE
EEEEEE
=
τ
τ
τ
σ
σ
σ
∆−
∆+
∆+
∆+
∆−
∆+
∆+
∆+
∆−
12
31
23
3
2
1
12
31
23
21
2112
31
311232
32
322131
31
311232
31
3113
32
312321
32
322131
32
312321
32
3223
12
31
23
3
2
1
00000
00000
00000
0001
0001
0001
νννννννν
νννννννν
νννννννν
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FIGURE 2.15A unidirectional lamina as atransversely isotropic material withfibers arranged in a rectangular array
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τ
τ
τ
σ
σ
σ
S
S
SS
SSS
SSS
SSS
=
γ
γ
γ
ε
ε
ε
−
12
31
23
3
2
1
55
55
)2322
222312
232212
121211
12
31
23
3
2
1
00000
00000
00(2000
000
000
000
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γ
γ
γ
ε
ε
ε
C
C
CC
CCC
CCC
CCC
=
τ
τ
τ
σ
σ
σ
−
12
31
23
3
2
1
55
55
2322
222312
232212
121211
12
31
23
3
2
1
00000
00000
002
000
000
000
000
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τ
τ
τ
σ
σ
σ
SS
SS
SS
SSS
SSS
SSS
=
γ
γ
γ
ε
ε
ε
−
−
−
12
31
23
3
2
1
1211
1211
1211
111212
121112
121211
12
31
23
3
2
1
)(200000
0)(20000
00)(2000
000
000
000
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τ
τ
τ
σ
σ
σ
CC
CC
CC
CCC
CCC
CCC
=
γ
γ
γ
ε
ε
ε
−
−
−
12
31
23
3
2
1
1211
1211
1211
111212
121112
121211
12
31
23
3
2
1
)(200000
0)(20000
00)(2000
000
000
000
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−−
−−
−−
τ
τ
τ
σ
σ
σ
G
G
G
EEE
EEE
EEE
=
γ
γ
γ
ε
ε
ε
12
31
23
3
2
1
12
31
23
3
2
1
100000
010000
001000
0001
0001
0001
νν
νν
νν
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,
G00000
0G0000
00G000
000)+)(12-(1
)-E(1)+)(12-(1
E)+)(12-(1
E
000)+)(12-(1
E)+)(12-(1
)-E(1)+)(12-(1
E
000)+)(12-(1
E)+)(12-(1
E)+)(12-(1
)-E(1
=
xy
zx
yz
z
y
x
xy
zx
yz
z
y
x
γ
γ
γ
ε
ε
ε
ννν
ννν
ννν
ννν
ννν
ννν
ννν
ννν
ννν
τ
τ
τ
σ
σ
σ
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Material Type Independent Elastic Constants
Anisotropic 21
Monoclinic 13
Orthotropic 9
Transversely Isotropic 5
Isotropic 2
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Upper and lower surfaces are free from external loads
0,0, =ττσ 23313 = 0 =
, 0 = 31233 0,0, == ττσ FIGURE 2.17Plane stress conditions for a thin plate
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τ
τ
τ
σ
σ
σ
S
S
S
SSS
SSS
SSS
=
γ
γ
γ
ε
ε
ε
12
31
23
3
2
1
66
55
44
332313
232212
131211
12
31
23
3
2
1
00000
00000
00000
000
000
000
,
τ
σ
σ
S
SS
SS
=
γ
ε
ε
12
2
1
66
2212
1211
12
2
1
00
0
0,σS+σS = ε 2231133
Compliance Matrix
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γ
ε
ε
Q
=
τ
σ
σ
12
2
1
66
2212
1211
12
2
1
00
0
0
,S SS
S = Q 2122211
2211 −
,S SS
S = Q 2122211
1212 −
−
,S SS
S = Q 2122211
1122 −
S = Q
6666
1
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wvu
γ
γ
γ
ε
ε
ε
τ
τ
τ
σ
σ
σ
12
31
23
3
2
1
12
31
23
3
2
1
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0
0
0
=+∂
∂+
∂∂
+∂∂
=+∂
∂+
∂
∂+
∂
∂
=+∂∂
+∂
∂+
∂∂
Zyτ
xτ
zσ
Yzτ
xτ
yσ
Xzτ
yτ
xσ
yzzxz
yzxyy
zxxyx
EQUILIBRIUM
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STRESS-STRAIN
τ
τ
τ
σ
σ
σ
SSSSSS
SSSSSS
SSSSSS
SSSSSS
SSSSSS
SSSSSS
=
γ
γ
γ
ε
ε
ε
12
31
23
3
2
1
666564636261
565554535251
464544434241
363534333231
262524232221
161514131211
12
31
23
3
2
1
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COMPATIBILITY
∂
∂−
∂∂
+∂
∂
∂∂
=∂∂∈∂
∂∂∂
=∂∈∂
+∂∈∂
∂
∂+
∂∂
−∂
∂
∂∂
=∂∂
∈∂
∂∂
∂=
∂∈∂
+∂
∈∂
∂
∂+
∂∂
+∂
∂−
∂∂
=∂∂∈∂
∂∂
∂=
∂
∈∂+
∂∈∂
zyxzyx
zxzx
zyxyzx
zyyz
zyxxzy
yxxy
xyxzyzz
xzxz
xyxzyzy
yzzy
xyxzyzx
xyyx
γγγ
γ
γγγ
γ
γγγ
γ
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
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