Fracture Mechanics of Delamination Buckling in Laminated Composites
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Transcript of Fracture Mechanics of Delamination Buckling in Laminated Composites
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Fracture Mechanics of Delamination Buckling in Laminated Composites
Kenneth Hunziker4/28/08
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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
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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
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1-D Delamination Models*
Thin Film Thick Column General
* Reference [1]
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1-D Thin Film Model*
Shortening
l
h A
εx = - εo εz = - νεo
i ii iii
* Reference [1]
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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]
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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]
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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]
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1-D Thin Film Analysis – Length of the delaminated region*
* Reference [1]
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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]
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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]
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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]
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1-D General Analysis*
* Reference [1]
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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]
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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]
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2-D Delamination Analysis*
* Reference [2]
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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
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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
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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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