8/8/2015 1 Euler Method Major: All Engineering Majors Authors: Autar Kaw, Charlie Barker .
Chapter 2 Macromechanical Analysis of a Lamina 3D Stiffness and Compliance Matrices Dr. Autar Kaw...
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Transcript of Chapter 2 Macromechanical Analysis of a Lamina 3D Stiffness and Compliance Matrices Dr. Autar Kaw...
EML 4230 Introduction to Composite Materials
Chapter 2 Macromechanical Analysis of a Lamina
3D 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
Compliance Matrix [S] for General Material
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] for General Material
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
Compliance Matrix [S] for Isotropic Materials
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
Stiffness Matrix [C] for Isotropic Materials
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
Compliance Matrix [S] for Isotropic Materials
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
Stiffness Matrix [C] for Isotropic Materials
,
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
Compliance Matrix [S] for Anisotropic Material
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] for Anisotropic Material
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
Monoclinic Materials
FIGURE 2.11Transformation of coordinate axes for 1-2plane of symmetry for a monoclinic material
Monoclinic Materials
FIGURE 2.13A unidirectional lamina as amonoclinic material with fibersarranged in a rectangular array
Compliance Matrix [S] for Monoclinic Materials
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
Stiffness Matrix [C] for Monoclinic Materials
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
Compliance Matrix [S] for Orthotropic Materials
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
Stiffness Matrix [C] for Orthotropic Materials
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
Compliance Matrix [S] for Orthotropic Materials
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
Stiffness Matrix [C] for Orthotropic Materials
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
Transversely Isotropic Materials
FIGURE 2.15A unidirectional lamina as atransversely isotropic material withfibers arranged in a rectangular array
Compliance Matrix [S] for Transversely Isotropic Materials
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
Stiffness Matrix [C] for Transversely Isotropic Materials
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
00
2
000
000
000
000
Compliance Matrix [S] for Isotropic Materials
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
Stiffness Matrix [C] for Isotropic Materials
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
Compliance Matrix [S] for Isotropic Materials
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
Stiffness Matrix [C] for Isotropic Materials
,
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
Independent Elastic Constants
Material TypeIndependent Elastic
Constants
Anisotropic 21
Monoclinic 13
Orthotropic 9
Transversely Isotropic 5
Isotropic 2
Plane Stress Assumption
Upper and lower surfaces are free from external loads
0,0, 2 33 13 = 0 =
, 0 = 3 12 33 0,0, FIGURE 2.17Plane stress conditions for a thin plate
Reduction of Compliance Matrix in 3D to 2D for Orthotropic Materials
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
Reduction of Stiffness Matrix in 3D to 2D for Orthotropic Materials
γ
ε
ε
Q
=
τ
σ
σ
12
2
1
66
2212
1211
12
2
1
00
0
0
,S SS
S = Q2122211
2211
,S SS
S = Q2122211
1212
,S SS
S = Q2122211
1122
S = Q
6666
1
15 equations
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
15 equations
COMPATIBILITY
zyxzyx
zxzx
zyxyzx
zyyz
zyxxzy
yxxy
xyxzyzz
xzxz
xyxzyzy
yzzy
xyxzyzx
xyyx
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2