Mechanics of aircraft materials 6 lamina theory cdio

We use the effective or average values of stress, strain and moduli when referring to lamina behavior... 2D Stress-Strain Relation of an Unidirectional Lamina... Lamina Stiffness Matrix

Lamina Stress-Strain Relationships

Dr Ly Hung Anh

Department of Aerospace Engineering – Faculty of Transportation Engineering

Stress Components

3 , 2 , 1 j

First subscript refers to direction of outer normal

Second subscript refers to the direction in which the stress acts

 Corresponding to each stress

component, there is a strain component,

eij describing the deformation at a

point.

 Normal strains describe the extension

per unit length.

 Shear strains describe distortional

deformation.

STRESSES STRAINS

Tensor

Notation

Contracted Notation

Tensor Notation

Contracted Notation

Stresses and strains are related

to each other The most general

form of this relationship is:

2121 2113

2132 2112

2131 2123

2133 2122

2111

1321 1313

1332 1312

1331 1323

1333 1322

1311

3221 3213

3232 3212

3231 3223

3233 3222

3211

1221 1213

1232 1212

1231 1223

1233 1222

1211

3121 3113

3132 3112

3131 3123

3133 3122

3111

2321 2313

2332 2312

2331 2323

2333 2322

2311

3321 3313

3332 3312

3331 3323

3333 3322

3311

2221 2213

2232 2212

2231 2223

2233 2222

2211

1121 1113

1132 1112

1131 1123

1133 1122

C C

C C

C C

C

C C

C C

C C

C C

C

C C

C C

C C

C C

C

C C

C C

C C

C C

C

C C

C C

C C

C C

C

C C

C C

C C

C C

C

C C

C C

C C

C C

C

C C

C C

C C

C C

C

C C

C C

C C

C C

C

Linear Elastic Material

9 Stresses x 9 Strains =

81 Components in relationship

j i

j i

ji ij

ji ij

 e

 e

 s

 s

and

Symmetry

ijlk ijkl

jikl ijkl

C C

C C

  σ   C   ε

6 , ,

2 , 1 j

, i

C ij j

i



 e

 s

or



Hooke’s Law (Stiffness)

Contracted Notation

66 65

64 63

62 61

56 55

54 53

52 51

46 45

44 43

42 41

36 35

34 33

32 31

26 25

24 23

22 21

16 15

14 13

12 11

C C

C C

C C

C C

C C

C C

C C

C C

C C

C C

C C

C C

C C

C C

C C

C C

C C

Hooke’s Law

  ε   S   σ

6 , ,

2 , 1 j

, i

S ij j

i



 s

 e

or



Hooke’s Law (Compliance)

Contracted Notation

    S  C  1

Inverse Relationship

[S] and [C] are symmetric matrices!

2 , 1 i

V

dV

dV dV

V

i V

i i

V

i V

V

i i



Average Stresses and Strains

     

  e      s

e

 s

56 55

46 45

44

36 35

34 33

26 25

24 23

22

16 15

14 13

12 11

SYM

C C

C

C C

C C

C C

C C

C

C C

C C

C C

Orthotropic Material: 9 Constants

Three planes of symmetry

55 44

33

23 22

13 12

SYM

0 0

C

0 0

0 C

0 0

0 C

C

0 0

0 C

C C

Transversely Isotropic Material

Three planes of symmetry 2 and 3 directions the same

23 22

22

23 22

12 12

SYM

0

0 2

C C

0 0

0 C

0 0

0 C

C

0 0

0 C

C C

12 11

12 11

12 11

11

12 11

12 12

0 2

C

C SYM

0

0 2

C C

0 0

0 C

0 0

0 C

C

0 0

0 C

C C

Material Nonzero Terms Independent terms 3D

3D Stress-Strain Relation

of an Orthotropic Material

in term of nine engineering

constants

Uniaxial Load in Fiber Direction

1

s

13 23

12

1

1 13

1 13 3

1

1 12

1 12 2

1

1 1

 e

 e

 e



 e

n



 e

s n



 e

n



 e

s

 e

or

E E E

Resulting Strains

Transverse Load

2

13 23

12

2

2 31

2 31 3

2

2 21

2 21 1

2

2 2

 e

 e

 e



 e

n



 e

s n



 e

n



 e

s

 e

or

E E E

Resulting Strains

12  s s

s

 s

Shear Load

0 G

5 4

13 23

3 2

1

12

6 6

12

 e

 e

 e

 e

s

 e





or

Resulting Strains

31 21

32 12

Young's moduli in and directions

Poisson's ratios (extension-extension coupling)

Shear moduli in ,and directions

   

ji

ij ν ν

C

S  

: Symmetry to

    S  C  1

Inverse Relationship

[S] and [C] are symmetric matrices!

31 21 32 13 12 23 13

32 12 31 23 21 13 23

 

E G

2D Stress-Strain Relation of

an Unidirectional Lamina

Plane Stress assumption

  e    S   s

Compliance

Stiffness

  s    Q   e

Lamina Stiffness Matrix

Some Typical Properties

Graphite /Epoxy 19.0 1.5 1.0 0.22

AS/3501

Graphite /Epoxy 20.0 1.3 1.0 0.3 p-100/ERL 1962

Pitch Graphite /Epoxy 68.0 0.9 0.81 0.31 Kevlar ® 49 /934

Aramid/Epoxy 11.0 0.8 0.33 0.34 Scotchply ® 1002

E-glass/Epoxy 5.6 1.2 0.6 0.26 Boron/5505

Boron/Epoxy 29.6 2.678 0.81 0.23 Spectra ® 900/826

Polyethylene/Epoxy 4.45 0.51 0.21 0.32

Are Carbon and Graphite the same?

“No, they are different Carbon Fibers have 93-95% carbon content, while graphite has more than 99% carbon content Also, carbon fibers are produced at 2400oF (1316oC), while Graphite Fibers are typically produced in excess of 3400oF (1900oC).”

2D Stress-Strain Relation of an Angle Unidirectional Lamina

Generally Angle Lamina

1 y

x

2



x,y: Global axes1,2: Local axes

Generally Angle Lamina

x,y: Global axes1,2: Local axes

1

y 2

 sin cos  0

cos sin

cos sin

0 cos

sin 2

sin cos

2 2

12

2 1

12

2 2

2 1







 s





 s





 s

 s







 s



 s



 s

 s

dA F

dA

dA dA

dA F

xy y

x x

Transformation in Matrix Form

2 2

2 2

2 2

xy

y

x

sin cos

sin cos

sin cos

sin cos

2 cos

sin

sin cos

2 sin

cos

Local-Global relationship

Condensed Matrix Form

y

x

T T

12 2 1

12 2

1 1

or

2 2

2 2

2 2

2 2

1

2 2

2 2

cs c

s

cs s

c

T

s c

cs cs

cs c

s

cs s

c

T

cs cs

cs 2 c

s

cs 2 s

c

2

2

1 1

y x

12 2 1

2 2

2 2

2 2

xy y x

Stress and Strain

Lamina Stiffness Matrix

General Stress-Strain Behavior

66 26

16

26 22

12

16 12

11

xy y x

Q Q

Q

Q Q

Q

Q Q

Q

in global axes

2 2

66 12

22 11

66

3 66

12 22

3 66

12 11

26

3 66

12 22

3 66

12 11

16

sin cos

2 2

sin cos

2

sin cos

2

sin cos

2

sin cos

2

Q Q

Q Q

Q

Q Q

Q

Q Q

Q Q

Q Q

Q

Q Q

Q Q

See: Page 91, “Mechanics of Composite Materials”, Autar K.Kaw

66 26

16

26 22

12

16 12

S

S S

S

S S

S

in global axes

“Transformed Reduced Compliance Matrix”

   

1

Q S

66 12

4 22

4 11

22

4 4

12

2 2

66 22

11 12

2 2

66 12

4 22

4 11

11

cos sin

2 cos

sin

cos sin

sin cos

cos sin

2 sin

cos

S S

S S

S

S S

S S

S

S S

S S

S

2 2

66 12

22 11

66

3 66

12 22

3 66

12 11

26

3 66

12 22

3 66

12 11

16

sin cos

sin cos

4 2

2 2

sin cos

2 2

sin cos

2 2

sin cos

2 2

sin cos

2 2

S

S S

S S

S

S S

S

S S

S S

S S

S

S S

S S

See: Example 2.5/ Page 92

“Mechanics of Composite

Materials”, Autar K.Kaw

xy y y

y xy xy

xy x x

x xy

y

yx x

xy

xy y

x

G E

S G

E S

E E

S

G

S E

S E

S

, ,

26

, ,

16

12

66 22

11

1 1

Engineering Constants

xy y

x xy

S S

S S

,26

,11

Coefficients of Mutual Influence

0

xy xy

x xy

, x

xy , y xy

, x

t

all for

plane.

xy the in

stress shear

by caused

direction y

or

x the in

stretching

ze Characteri

kind first

the of

influence mutual

of ts Coefficien

and

x x

xy x

,

xy

y , xy x

,

xy

s



se

all for

plane.

xy the in

stress normal

by caused

plane xy

the in

shearing zes

Characteri

kind second

the of

influence mutual

of ts Coefficien

and

Coefficients of Mutual Influence

4 2

2 2 1

12 12

4 1

4 2

2 2 1

12 12

4 1

1 2

1 1

1

1 2

1 1

1

c E

c

s E

G

s E E

s E

c

s E

G

c E E

2 2 12 2

1

4 4

1 12

11

42

221

11

11

s

c G

c

s G

E E

E G

c

s G

E E

s

c E

Engineering Constants

y , xy y

xy xy

, y x

, xy x

xy xy

,

x

3 12

1

12 2

3 12

1

12 1

y y

,

xy

3 12

1

12 2

3 12

1

12 1

x x

,

xy

E

G E

G

sc G

1 E

2 E

2 c

s G

1 E

2 E

2 E

c

s G

1 E

2 E

2 sc

G

1 E

2 E

2 E

0 20 40 60 80 100

120

140

Ex Ey Gxy

E1

E2

Problem 1 Determine the stiffness matrix of a Laminate

Determine the stiffness matrix for a 45/-45/+45] symmetric angle-ply laminate consisting of 0.25 mm thick unidirectional

[+45/-AS/3501 graphite epoxy lamina

Determine the stiffness matrix of a Laminate

solution

-1 Find the value of the “reduced stiffness matrix” [Q] for

each ply using its four elastic moduli E11, E22, G12

2 Find the value of the “transformed reduced stiffness

matrix” for each ply by using the [Q] and the angle

of each ply

Q

 

 

y x

Exploded view of a [+45/-45/-45/+45] symmetric laminate

0.0196 ν

E ν

0.3 ν

GPa 6.9

G

GPa 9.0

E

GPa 138.0

E

: Properties Material

12

2 21

12 12 2 1

   

   12 21 

2 12

2 12 22

11

12 21

12

21 12

2 2

12 22

11

11 22

21 12

1 2

12 22

11

22 11

G

1 Q

1

E S

S S

S Q

Q

1

E S

S S

S Q

1

E S

S S

S Q





n n

  GPa

6.9 0

0

0 9.05

2.72

0 2.72

138.8 Q

 

 

 

θ cos Q

θ sin Q

Q

θ cos θ

sin Q

θ sin θ

cos Q

4 Q

Q Q

θ cos θ

sin Q

2 Q

2

θ sin Q

θ cos Q

Q

4 22

4 11

22

4 4

12

2 2

66 22

11 12

2 2

66 12

4 22

4 11

Q 2 Q

2 Q

Q Q

sinθ θ

cos Q

2 Q

Q

θ sin cosθ

Q 2 Q

Q Q

θ sin cosθ

Q 2 Q

Q

sinθ θ

cos Q

2 Q

Q Q

4 4

66

2

2 66

12 22

11 66

3 66

12 22

3 66

12 11

26

3 66

12 22

3 66

12 11

+45o   GPa

35.6 32.44

32.44

32.44 45.22

31.42

32.44 31.42

Autar K Kaw, Mechanics of Composite Materials, CRC Press, 2006

2.2; 2.3; 2.6; 2.9; 2.10;

2.11; 2.15; 2.16; 2.17; 2.20; 2.21