chapter 2 macromechanical analysis of a lamina 3d stiffness and compliance matrices dr. autar kaw...
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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
Lamina and Laminate
FIGURE 2.1Typical laminate made of three laminas
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
Compliance Matrix [S] for Anisotropic Material
Stiffness Matrix [C] for Anisotropic Material
Monoclinic Materials
FIGURE 2.11Transformation of coordinate axes for 1-2plane of symmetry for a monoclinic material
Monoclinic Materials
FIGURE 2.12Deformation of a cubic elementmade of 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 Monoclinic Materials
Stiffness Matrix [C] for Monoclinic Materials
Orthotropic Materials
FIGURE 2.14Deformation of a cubic element madeof orthotropic material
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 Transversely Isotropic Materials
Stiffness Matrix [C] for Transversely Isotropic Materials
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
END
15 unknowns
w
v
u
γ
γ
γ
ε
ε
ε
τ
τ
τ
σ
σ
σ
12
31
23
3
2
1
12
31
23
3
2
1
15 equations
0
0
0
Zy
τ
x
τ
z
σ
Yz
τ
x
τ
y
σ
Xz
τ
y
τ
x
σ
yzzxz
yzxyy
zxxyx
EQUILIBRIUM
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
END