Class notes - Instituto Tecnológico de Aeronáuticaarfaria/MP206_06.pdf · Class notes. Instituto Tecnológico de Aeronáutica MP-206 2 10. Fracture mechanics applied to composite

Instituto Tecnológico de Aeronáutica

MP-206 1

Analysis and design of composite structures

Class notes


MP-206 2

10. Fracture mechanics applied to composite

laminates


MP-206 3

Basic Principles of Fracture Mechanics

A. A. Griffith: “ If a crack is in equilibrium, the decrease of strain energy “U” must be equal to the increase of surface energy “S” due to crack extension “a””

U S

a a

∂ ∂=∂ ∂


MP-206 4


2aaδaδ

P

P

L

P

P

e Lδ

( )12 2a a

mδ+

( )12a

m

1 ; 0

1 1 1 1

2 2 2 2

P e

m e aU P e P

U Pe e P ea a a a

∂= =∂

∂ ∂ ∂ ∂= → = + =∂ ∂ ∂ ∂


MP-206 5


P

P

e Lδ

( )12 2a a

mδ+

( )12a

m

2

2

1 ; 0

1 1 1 1

2 2 2 21 1

= + =

21

2

P e

m e aU P e P

U Pe e P ea a a a

P e P me

a m a a m m a

U P m

a aU U P m

GA B a B a

∂= =∂

∂ ∂ ∂ ∂= → = + =∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂

∂ ∂= −∂ ∂

∂ ∂ ∂= − = − =∂ ∂ ∂

21

2

U U P mG

A B a B a

∂ ∂ ∂= − = − =∂ ∂ ∂Strain energy release rate:


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General comments

• Generally speaking the failure modes in composite laminates can be divided into two categories:

� Interlaminar failure modes

� Intralaminar failure modes


MP-206 7

Interlaminar failure modes

- The interlaminar failure modes (also known as “DELAMINATION”) occurs due to the high interlaminar stresses acting on the interface between two adjacent layers.


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DCB-ASTM D 5528-94a 4 ENF - MERL

MMB - ASTM D 6671-01

Interlaminar fracture toughness test methods


MP-206 9

Mode I delamination – Double Cantilever Beam (DCB)

3 32 2

3 3H H

Pa v av m

EI P EI= → = =

2 2 2

2

I H

I

H

G BEIP m P aG a

B a BEI P

∂= = → =∂

Total displacement at the beam tip:

Mode I strain energy release rate:

h


MP-206 10

Mode I delamination

( )3/2

2

2

3Ic H

H

BG EIv

EI P=

Combining the tip displacement with the expression for the crack extension we obtain the following relationship between load and tip displacement for delamination propagation regime,

3

12

8

F

FH

BhI

II

=

=

With,


MP-206 11

Mode I delamination - EXAMPLE

2135300 N/mm

4.0 N/mm

30 mm

h = 3 mm

B= 1 mm

Ic IIc

E

G G

a

== =

=

0 10 20 30 400

5

10

15

v(mm)

P(N

)

LinearPropagation


MP-206 12

Mode II delamination – 3 Point Bending

4

P

4

P,P v

2

PL L

0a

x

The derivation of the analytical expressions for mode II delamination require analysis of two different cases:

Case 1: Crack length (a) shorter than the half length of the beam (a ˂ L)

Case 2: Crack length (a) longer than the half length of the beam (a ˃ L)


MP-206 13


1

4

1

4

1,v

1

2L L

0a

x

The derivation of the analytical expressions for mode II will be based on the principle of virtual work: “An elastic body with finite dimensions is on equilibrium when the virtual work done by external forces is equal to virtual strain energy for any arbitrary displacement”

eW Uδ δ=


MP-206 14


22 22 2

21 1

2 2

1 1

1 1

2 2

e

e e

X X

HX X

H

X Xu

X XH H

W U

W Pv W Pv

d v MU EI dx dx

dx EI

MMM MU dx dx

EI EI

δ δδ δ

δδ

== → =

= =

= =

∫ ∫

∫ ∫


MP-206 15


( )

0

0

0

( ) ( )4 4

( ) ( )2 2

2

( ) ( ) ( )2 2

u

u

u

x a

P xM x x M x

a x L

P xM x x M x

L x L

P xM x x P x L M x x L

≤ <

= → =

≤ <

= → =

≤ <

= − − → = − −

Case 1: Crack length (a) shorter than the half length of the beam (a ˂ L)


MP-206 16


( )23 3 2

0

22

16 4 4

a L Lo

a LoH F F

P L xPx Pxv dx dx dx

EI EI EI

−= + +∫ ∫ ∫

( )3 3

02 3

12 F

P L av

EI

+=

Displacement in the middle of the beam,

The relationship between load and displacement previously defined describes the linear portion (before crack propagation) of the 3 point bending test structural response.


MP-206 17


( )12 2 2 23

2 2 8II

F

vPP C P P aG

B a B a BEI

−∂∂= = =∂ ∂

( )3 32 3

12 F

L avC

P EI

+= =

The mode II strain energy release rate is given by,

with,

8

3II FG BEI

aP

=

The crack length can be expressed as,


MP-206 18


( )3

23

3

82

12 3II F

F

G BEIPv L

EI P

= +

Substituting the expression for a into the Load (P) x Displacement (v) relationship results in the following expression for the crack propagation regime,


MP-206 19


( ) ( )

( )

0

0

0

( ) ( )4 4

1( ) ( )

4 2 4 22

( ) ( ) ( )2 2

u

u

u

x L

P xM x x M x

L x a

P P xM x x x L M x x L

a x L

P xM x x P x L M x x L

≤ <

= → =

≤ <

= − − → = − −

≤ <

= − − → = − −

Case 2: Crack length (a) longer than the half length of the beam (a > L)


MP-206 20


( ) ( )2 23 2

0

2 22 2

16 16 4

L a L

L aH H F

P L x P L xPxv dx dx dx

EI EI EI

− −= + +∫ ∫ ∫

( )33 32 2

3 4F

Pv L L a

EI = − −

Displacement in the middle of the beam,


MP-206 21


( ) ( )21 22 2 3 2

2 2 8II

F

vP P L aP C PG

B a B a BEI

−∂ −∂= = =∂ ∂

( )331 32 2

3 4F

vC L L a

P EI = = − −

The mode II strain energy release rate is given by,

with,

82

3II FG BEI

a LP

= −

The crack length can be expressed as,


MP-206 22


( )3

23

3

82

3 4 3II F

F

G BEIPv L

EI P

= −

Substituting the expression for a into the Load (P) x Displacement (v) relationship results in the following expression for the crack propagation regime,


MP-206 23

Mode II delamination - EXAMPLE

2135300 N/mm

4.0 N/mm

30 mm

L = 50 mm

h = 3 mm

B= 1 mm

Ic IIc

E

G G

a

== =

=

0 5 10 15 20 25 300

10

20

30

40

50

60

70

80

90

100

v(mm)

P(N

)

Linear portionPropagation (a<L)Propagation (a>L)


MP-206 24

Mixed-mode delamination – Mixed-Mode Bending (MMB)

Saddle and yokeFulcrum

c P

a

L L

h


MP-206 25


a

L L

h

c LP

L

+

cP

L


MP-206 26

3

4I II

c L c LP P P P

L L

− + = =

( )3 33 2 32

3 12III

I II

H F

P L aP av v

EI EI

+= =

The MMB test superposition analysis results in,

For a < L the expressions for displacements caused by mode I and mode II loadings are given by,



MP-206 27

( )

( )

12 2 2

12 2 2

2

3

2 8

I II II

H

II IIII IIII

F

v PP P aG

B a BEI

v PP P aG

B a BEI

−

−

∂= =

∂

∂= =

∂

The corresponding strain energy release rates are,


The expression for mixed mode ratio is given by,

24 3

3I

II

G c L

G c L

− = +


MP-206 28

1I II

Ic IIc

G G

G G

α α

+ =

The criterion for mixed-mode delamination can be written in terms of the following Power-Law,


where:

Mode I critical strain energy release rate (obtained from DCB tests)

Mode II critical strain energy release rate (obtained from 3PBT tests)Ic

IIc

G

G

→→


MP-206 29

1I II

Ic IIc

G G

G G

α α

+ =


Results in the following expression for the crack length,

2 2

2 23

8

II

H

IIII

F

P aG

BEI

P aG

BEI

=

=

1/22 23

8I II

H Ic F IIc

P Pa

BEI G BEI G

αα α − = +


MP-206 30


1/22 23

8I II

H Ic F IIc

P Pa

BEI G BEI G

αα α − = +

( )3 33 2 32

3 12III

I II

H F

P L aP av v

EI EI

+= =

The displacements at the tip and the middle of the beam are written in terms of the mixed mode crack length,


MP-206 31

Mixed-mode delamination – EXAMPLE

2135300 N/mm

4.0 N/mm

30 mm

L = 50 mm

h = 3 mm

B= 1 mm

c=41.5 mm

Ic IIc

E

G G

a

== =

=

0 2 4 6 8 10 120

5

10

15

20

25

30

vI (mm)

P(N

)

Linear portionPower Law (Alpha=2, a<L)Power Law (Alpha=1, a<L)


MP-206 32

Intralaminar failure modes

- The intralaminar failure modes are related to the local in-plane and out-of-plane stresses combined or acting individually on the layers leading to matrix cracking in tension or compression and/or shear, fibre failure either in compression or tension.


MP-206 33

Intralaminar fracture toughness test methods

�ASTM E 399-90

�DEN (Double Edge-Notched) Specimens

�4PBT (Four point bending test Method)

- All test methods available in the open literature were originally developed for metals

- Modifications in the test method are required to handle material anisotropy, different lay-ups & specimen geometries


MP-206 34

ASTM E 399-90



MP-206 35

ASTM E 399-90


iδia

iP

iP

P

ia

iδ

iP

a P δ

a1 P1 δ1

. . .

. . .

an Pn δn

Measured variables during the test

Test usually carried out under displacement control


MP-206 36

ASTM E 399-90


( )

( )( ) ( )

0.5

2 3 41.5

2 0.50.5 0.5

11 22 11 22 11 12 12

/ ( ) ( )

2 /( ) 0.886 4.64( / ) 13.32( / ) 14.72( / ) 5.6( / )

1 /

(2 ) ( / ) / 2

i

I i i

ii i i i i

i

i i

I I

K P h w F a

a wF a a w a w a w a w

a w

G K E E E E E G ν

−

−

=+= + − + − −

= + −


MP-206 37

DEN (Double Edge-Notched) Specimens



MP-206 38

DEN (Double Edge-Notched) Specimens



( )

2 3

2

2

( / )

( ) 1.98 0.36(2 / ) 2.12(2 / ) 3.42(2 / )

i

I i i

i i i i

i

Ii

I

K f a w F a

F a a w a w a w

KG

E

σ=

= + − +

=

P

iaiP

a P δ

a1 P1 δ1

. . .

. . .

an Pn δn


MP-206 39


4PBT (Four point bending test Method)


MP-206 40


4PBT (Four point bending test Method)


( ) ( )

2

2 3 4

2 2

12

2

6 ( )

2( ) 1.12 1.39( / ) 7.32( / ) 13.1( / ) 14( / )

1

i i iI i

i i i i i

i

Ii

I

Pc aK F a

whF a a w a w a w a w

KG

E

π

ν

=

= − + − +

−=

P

iaiP

a P δ

a1 P1 δ1

. . .

. . .

an Pn δn


MP-206 41


How to account for material anisotropy?

1- Compute J-integral for different crack lengths

2- For each Ji compute new values for F(ai):

0.5 0.25

11 220.25

( )( )

( / 2 / 2)i

i

i

J E E h wF a

Pα β=

+11 22/E Eα = 11 12 12/ 2E Gβ ν= −

3- Plot F(a) .vs. crack length and find the function which best fit these points

DERIVATION OF A CONSISTENT DERIVATION OF A CONSISTENT F(aF(aii) ) FUNCTION:FUNCTION:


MP-206 42


ASTM E399-90 overestimates the toughness values quite considerably!

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Class notes - Instituto Tecnológico de Aeronáuticaarfaria/MP206_06.pdf · Class notes. Instituto Tecnológico de Aeronáutica MP-206 2 10. Fracture mechanics applied to composite