theory of composite laminates

7/28/2019 Theory of Composite Laminates

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Structures and Controls Lab - AE 461 Chasiotis I.

Theory of Composite Laminates


Coordinate System for a Lamina

The coordinates 1,2,3 are the Principal material directions

The coordinates x,y,z are the Laminate (or Transformed) Axes

The lamina properties Ex , Ey , Gxy ,xy are measured by testing thematerial in the principal material directions.


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Compliance of a lamina

The transversely isotropic constants for the lamina are

Ex

= Longitudinal modulus

Ey = Transverse modulus

Gxy= In-plane shear modulus

xy= Major Poissons ratio

These constants are calculated as a function of volume fractions:

1

1

x m m f f

xy m m f f

fm

y m f

fm

xy m f

E E V E V

V V

VV

E E E

VV

G G G

Longitudinal

loading

Transverse

loading


Modulus

in Longitudinaldirection

Assumption of Isostrain:

1

The load carried

c m f

c m f

c m f c c m m f f

fmc m f c m m f f

c c

fc mm f x m m f f x m f f f

c m f

F F F A A A

AAV V

A A

V V E E V E V E E V E V

f f f

m m m

F E Vby the fibers vs. the matrix:

F E V

Perfect fibermatrix interfacial bonding is assumed, such that deformation of bothmatrix and f ibers is the same (isostrain)

The total load sustained by the composi te Fc i s equal to the loads car ried by the

matrix, Fm, and by the f ibers, Ff:

Continuous and aligned fiber composites - Axial Loading


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To f ind the t ransverse properties , the load is appl ied at 90 w.r.t . the direction offiber alignment

The stress at which the composi te and both phases are exposed is the same.This is cal led isostress state.

1 1 1

1

c m f

Ec m m f f m f

y m f

m f m f

y m f y m f

m f m f

y

m f f m f f f m

Assumption of isostress :

V V V VE E E

V V V V E E E E E E

E E E EE

V E V E V E V E

Continuous fiber composites - Transverse Loading


Compliance of the Laminate


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Generalized Hookes Law

The generalized Hookes law:

, 1,...,6i ij jC i j

or 11 12 13 14 15 161 1

21 22 23 24 25 262 2

31 32 33 34 35 363 3

41 42 43 44 45 4623 23

51 52 53 54 55 5613 13

61 62 63 64 65 6612 12

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

This is a 6x6 matrix with 21 independent constants


Orthotropic and Transversely Isotropic Stiffness Tensor

The orthotropic stiffness tensor is:

If the material is isotropic =E

11 12 13

22 23

33

44

55

66

0 0 0

0 0 0

0 0 0

0 0

0

ij

C C C

C C

CC

C

C

C

Symmetric

11 12 12

22 23

22

44

22 33

44

0 0 0

0 0 0

0 0 0

0 0

0

ij

C C C

C C

C

C C

C C

C

Symmetric

Thetransversely isotropic stiffness tensor(good for the composites we discuss in this

class) is:


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Compliance of 0 Degree Lamina

The strain as a function of stress is

The compliance of a lamina is

i ij jS

10

10

10 0

xy

x x

xy

ij

x y

xy

E E

SE E

G


Stiffness of 0 Degree Lamina

The stiffness matrix coefficients for the 0 degree ply are:

1

Q S


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Stiffness Tensor for a Randomly Oriented Lamina

1 11 12 1

2 12 22 2

12 66 12

0

0

0 0

Q Q

Q Q

Q

Principal Material Directions: 1,2,3

For a lamina oriented in an arbitrary direction (coordinates):

11 12 16

12 22 26

16 26 66

x x

y y

xy xy

Q Q Q

Q Q Q

Q Q Q

Qij are functions Ex , Ey , G12 ,12

are functions Ex , Ey , Gxy ,xy and angle and they are used to construct thestiffness tensor of the kth ply:

ijQ

k kk

Q


Calculation of Stiffness Tensor Elements for Arbitrary

The tensor elements for a ply at an angle wrt. the 0 angle ply are found bytransformation of coordinates:

The parameters m and n are the cos and the sin respectively. The two

additional stiffnesses, Q16 and Q26 appear for off-axis lamina. These terms

represent the coupling between shear and extensional deformation and are

present only in anisotropic materials (remember from AE321?). These terms are

zero for isotropic materials.

4 4 2 2 2 2

11 11 22 12 662 4Q m Q n Q m n Q m n Q 4 4 2 2 2 2

22 11 22 12 662 4Q n Q m Q m n Q m n Q 2 2 2 2 4 4 2 2

12 11 22 12 66( ) 4Q m n Q m n Q m n Q m n Q

3 3 3 3 3 3

16 11 22 12 66( ) 2( )Q m nQ mn Q mn m n Q mn m n Q

3 3 3 3 3 3

26 11 22 12 66( ) 2( )Q mn Q m nQ m n mn Q m n mn Q

2 2 2 2 2 2 2 2 2

66 11 22 12 662 ( )Q m n Q m n Q m n Q m n Q


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Alternatively, we can use the transformation tensors [T1] and [T2]:

11 12 11 12 16

1

1 12 22 2 12 22 26

66 16 26 66

00

0 0

x x x x

y y y y

xy xy xy xy

Q Q Q Q QT Q Q T Q Q Q

Q Q Q Q

where:

2 2

2 2

1

2 2

2 2

2 22

2 2

cos sin 2sin cos

sin cos 2sin cos

sin cos sin cos cos sin

cos sin sin cos

sin cos sin cos

2sin cos 2sin cos cos sin

T

T

Calculation of Stiffness Tensor Elements for Arbitrary


Calculation of the Compliance of the Laminate Plane Stress

We consider plane stress conditions. The average stress in the laminate is

The stress as a function of stiffness matrix coefficients is

The strain varies linearly across the laminate thickness as

Where k is the curvature of the i lamina

1

Q S

where

i ij jQ

1i i

h

dzh

i i izk


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The average stress in i direction is

Replacing the integrals with tensors we

get the laminate stiffness and

coupling matrices

Actually the laminate is a discrete system

so we can replace it the integrals with

sums

Because Qij = constant for each ply

Bij = 0 for a symmetric laminate and Aij is

left:

01 1i ij j ij j

ij ij

h

ij ij

h

A B kh h

where

A Q dz

B Q zdz

1 1i ij j ij jh h

Q dz Q zk dz h h



Example of Lamina Stacking

zn-zn-1 z2

n-z2n-1

A B

Ply a (-3) (-4) = 1 (-3)2 (-4)2 = -7

Ply b (-2) (-3) = 1 (-2)2 (-3)2 = -5

Ply c (3) (2) = 1 (3)2 (2)2 = 5

Ply d (4) (3) = 1 (4)2 (3)2 = 7

z

a

b

c

d

-4

-3

-2

-1

0

1

2

3

4

Assume the following composite with a stack of plies each being 1 mm thick:


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The average stress in i direction is

Replacing the integrals with tensors we

get the laminate stiffness and

coupling matrices

Actually the laminate is a discrete system

so we can replace it the integrals with

sums

Because Qij = constant for each ply

Bij = 0 for a symmetric laminate and Aij is

left:

01 1i ij j ij j

ij ij

h

ij ij

h

A B kh h

where

A Q dz

B Q zdz

1 1i ij j ij jh h

Q dz Q zk dz h h



Effective Properties of the Laminate

If we expand the last equation we have:

11 12 16

12 22 26

16 26 66

1x x

y y

xy xy

A A A

A A Ah

A A A

If we invert the stiffness matrix [Aij] we have:

11 12 16

12 22 26

16 26 66

x x

y y

xy xy

a a a

h a a a

a a a

Where [aij] = [Aij]-1


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The components of the compliance tensor [aij] are calculated as follows:

11

11

22

22

66

66

110

12

10 0

10 0

10 0

x

xx y xy x x x

x

y

y x xy y y y

y

xy

xy y x xy xy xy

xy

x xy For only

xy xy

y xx

For only and ha Eha

For only and ha Eha

For only and ha Gha

haUse

ha

12

11

a

a

11 12 16

12 22 26

16 26 66

x x

y y

xy xy

a a a

h a a a

a a a



The effective elastic constants of thelamina at the principal coordinates are:

The components aij are calculated from

the inverse lamina stiffness tensor:

[] = [A]-1

The effective bending stiffness of a laminais

Finally, the vertical deflection of an I-beamunder load is:

The effective bending stiffness a composite

I-beam can be calculated as a function of the

properties of the flange and the web:

3

13

composite o

FLu

E I



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Procedure for Calculation of Equivalent Laminate Properties1. Anisotropic stress-strain law:

in 3D in 2D

2. Compute on axis properties Sij and invert to get Qij

, 1, 2,...,6 , 1, 2,6ij iji ij j i j i j

C i j Q S i j

1

2

3

4

5

6

xx

yy

zz

xz

yx

xy

6x6

1

2

3

4

5

6

1

2

6

1Ex

xyEx

0

xyEx

1Ey

0

0 0 1Gxy

1

2

6

3. Compute on axis properties Sij and invert to get Qij

4. Rotate Qij to (e.g., by 45, 90 etc.)

5. Integrate to get Aij. Recall:need only to sum

6. Invert Aij to get aij

7. Compute composite elastic constants

8. Compare with experiment!

11

n

ij ij k kkk

A Q z z

ijQ

ijQ

theory of composite laminates

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