Fracture Mechanics of Delamination Buckling in Laminated Composites

Kenneth Hunziker4/28/08

Low Velocity Impact of a Laminated Composite Plate

lL

l

• Laminated composite materials have a strength-to-weight ratio advantage over many other materials

• Low velocity impact causes a delamination in the plate (size determined by impactor and plate parameters)

• A compressive load σo increases the delaminated area through coupled delamination and delamination buckling

• The growth of the damage through delamination buckling is analyzed using fracture criterion based on energy release rate

• Analyzed through 1-D and 2-D models

σo

Simplifications/Assumptions

• One delamination caused by impact is analyzed

• Delamination size is large compared to the laminate thickness but small compared to the laminate size

• Growth of the delamination is in the original damage plane

• Properties of the plate are considered to be homogeneous, isotropic and linearly elastic

1-D Delamination Models*

Thin Film Thick Column General

* Reference [1]

1-D Thin Film Model*

Shortening

l

h A

εx = - εo εz = - νεo

i ii iii

* Reference [1]

2

2

2

)1(3

l

hcr

l

xAy

2cos1

2

1

dxdx

dyl

l

lcro

2

2

22

2

11

22

2 12

l

A cro

1-D Thin Film Analysis - Deflection*

Buckling strain of the film using beam/plate theory

Post buckled film shape

Solve for amplitude A using:

* Reference [1]

1-D Thin Film Analysis – Strain Energy*

Strain energy in the buckled layer (case iii)

dxdx

ydEhEhlU

l

locriii

2

2

2

2

2

2

32222

1241

2

2

2

22

2

12

2

1ocrcroiii

EhlU

Membrane BendingGives:

Energy release rate as l → (l+Δl)

crocroa

EhG

32

1 2

* Reference [1]

1-D Thin Film Analysis – Energy Release Rate Results*

2

*

12 Eh

oo

2/1*12

*

1 o

oo

4/1**

oh

ll

* Reference [1]

1-D Thin Film Analysis – Length of the delaminated region*

* Reference [1]

1-D General Analysis*

• Each section is treated as a beam column with compatibility and equilibrium conditions applied at the interfaces• Gives the following deflections:

3,2,cos

2cos2cos

sin2

2cos1

2sin2 1

11

11

11

iu

u

l

xu

uu

ly

l

xu

uu

ly

i

i

i

ii

ii

ii

L

th

13

2

* Reference [1]

1-D General Analysis*

Examining the overall shortening of the plate

tdxdx

dyldx

dx

dyl

hdxdx

dyldx

dx

dylL

l

l

l

l

l

l

l

o

3

22/

2/ 2

2223

22/

2/ 3

333

2

22/

2/ 2

2221

2

0 1

111

2

2

3

3

2

2

1

2

1

2

1

2

12

Using plane strain, stresses and strains are:

ioix

E

221

ioiz

E

21

oiz

iix

* Reference [1]

1-D General Analysis*

The strain energy of the system is

3

2

2/

2/

2

2

2

0

1

2

21

12

1111111

2

1

1

i

l

l

ii

iiiiizizixix

l

zzxx

i

i

dxdx

ydDlt

dxdx

ydDltU

• In order to solve for the four unknowns ε1, ε2, ε3 and θ we combine the displacement equations with the equilibrium and shortening equations

• The resulting four equations do not have a closed form solution• Solve numerically

• The strain energy release rate can be found with a numerical differentiation

• The same analysis can be preformed with the assumption that only section 3 contributes to the bending – ‘Thick Column’ case

* Reference [1]

1-D General Analysis*

* Reference [1]

2-D Delamination Model*

b

a

Δb

Δa

• Two part analysis

• Elastic stability – Solved through the Raleigh-Ritz method

• Delamination growth after buckling – Energy approach through fracture mechanics

12

2

2

2

b

y

a

x

Displacement constraints:

0

y

w

x

ww

yv

xu

o

o

* Reference [2]

2-D Delamination Analysis*

dadb

ba

dadb

ba

GGG

ba

1

oa

oa

Gb

U

bG

Ga

U

bG

1

1

Energy release rate for the system due to a increase in delamination

Where

oGdA

dG

Gives

* Reference [2]

2-D Delamination Analysis*

* Reference [2]

Conclusions

• A one-dimensional model can be used to simplify analysis of a more complete two-dimensional model

• Simplifications can be made to the two-dimensional model based on initial damage relative size parameters• Either stable or unstable growth can occur in both the one and two-dimensional model with increasing load• A “thin-film” one-dimensional approach can be used as the delamination being analyzed approaches the plate surface

• The initial parameters of the damage in a structure determine the behavior of the damage as load is increased

• Both stable and unstable growth can occur based on the size/area of the initial damage

Further Analysis

• Further improvements of the 1-D model include:• Multiple delaminations• Non-homogeneous material properties

• Further improvements of the 2-D model:• Delamination shape, circular and elliptical• Anisotropic material

• The role of fiber direction in delamination growth• Multiple delaminations

References

One Dimensional Analysis1. Chai, H., Babcock, C., Knauss, W., “One Dimensional Modelling of Failure in

Laminated Plates by Delamination Buckling,” Int. J. Solids Structure, Vol. 17,. No. 11, pp. 1069-1083, 1981

Two Dimensional Analysis2. Chai, H., Babcock, C., “Two-Dimensional Modelling of Compressive Failure in

Delaminated Laminates,” Journal of Composite Materials, Vol. 19,. No. 1, pp. 67-98, 1985

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