3d stiffness and compliance matrices
TRANSCRIPT
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
FIGURE 2.1Typical laminate made of three laminas
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
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
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
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
−−
−−
−−
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
,
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
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
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
FIGURE 2.11Transformation of coordinate axes for 1-2plane of symmetry for a monoclinic material
FIGURE 2.12Deformation of a cubic elementmade of monoclinic material
FIGURE 2.13A unidirectional lamina as amonoclinic material with fibersarranged in a rectangular array
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
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
FIGURE 2.14Deformation of a cubic element madeof orthotropic material
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
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
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
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
FIGURE 2.15A unidirectional lamina as atransversely isotropic material withfibers arranged in a rectangular array
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
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
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
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
−−
−−
−−
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
,
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
Material Type Independent Elastic Constants
Anisotropic 21
Monoclinic 13
Orthotropic 9
Transversely Isotropic 5
Isotropic 2
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
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
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
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
0
0
0
=+∂
∂+
∂∂
+∂∂
=+∂
∂+
∂
∂+
∂
∂
=+∂∂
+∂
∂+
∂∂
Zyτ
xτ
zσ
Yzτ
xτ
yσ
Xzτ
yτ
xσ
yzzxz
yzxyy
zxxyx
EQUILIBRIUM
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
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