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

QQ

QQ

=

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