Fracture Mechanics of Delamination
Buckling in Laminated Composites
Kenneth Hunziker
4/28/08
Low Velocity Impact of a Laminated Composite Plate
• Laminated composite materials have a
strength-to-weight ratio advantage over
many other materials
σo
L
l
l
• 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
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*
εx = - εo
εz = - νεo
Shortening
l
h
i
ii
A
iii
* Reference [1]
1-D Thin Film Analysis - Deflection*
Buckling strain of the film using beam/plate theory
2 h
 cr 

2 
3(1  )  l 
2
Post buckled film shape
1
2x 
y  A 1  cos

2
l 
Solve for amplitude A using:


2
 o   cr  1  2 l   l2 1  dy  dx
2 2  dx 
l
2

 2l 
A   o   cr   1  2
 
2

* Reference [1]
1-D Thin Film Analysis – Strain Energy*
Strain energy in the buckled layer (case iii)
U iii 
Gives:
 


Ehl
Eh
 cr 2 1  2  2 o 2   l2
2

2
2 24 1 

Membrane

Ehl 1  2
U iii 
2

 d2y 
 2  dx
 dx 
Bending
 2 


2
3
l
o cr
  cr
2
2
2

o 
2
1   
Energy release rate as l → (l+Δl)


Eh 1  2
 o   cr  o  3 cr 
Ga 
2
* Reference [1]
1-D Thin Film Analysis – Energy Release Rate Results*
o* 
o
 Eh 

2 
 2 1  

 o*   o
1 
l*  l

2 1

1/ 2
o*

h 
* 1 / 4
o
* Reference [1]
1-D Thin Film Analysis – Length of the delaminated region*
* Reference [1]
1-D General Analysis*
h
3
t
1
2
L
• Each section is treated as a beam column with compatibility and equilibrium
conditions applied at the interfaces
• Gives the following deflections:

l1
2u1 x1 
1  cos

y1 
2u1 sin 2u1 
l1 
li
yi 
2ui sin ui

2ui xi cos 2ui 
 cos
, i  2,3

li
cos ui 

* Reference [1]
1-D General Analysis*
Examining the overall shortening of the plate
2
2
 dy1 
1 2  dy2 
 dx1   2l2   
 dx2  h
 o L  21l1   
dx1 
2 l2 / 2  dx2 
0
l1
 3l3 
l /2
2
2
 dy3 
 dy2 
1
1



 dx3  t
dx


l

3
2 2




2 l3 / 2  dx3 
2 l2 / 2  dx2 
l3 / 2
l2 / 2
Using plane strain, stresses and strains are:
 x i 
 z i 

E
 2 o   i
2
1 

E
 o   i 
2
1 



 z i   o
 x i   i
* Reference [1]
1-D General Analysis*
The strain energy of the system is
2
 d 2 y1 
U   x 1  x 1   z 1  z 1 t1l1  D1   2  dx1
dx1 
0
l1
2
li / 2

 d 2 yi 
1 3 
   x i  x i   z i  z i ti li  Di   2  dxi 
2 i 2 
dxi 

 li / 2 

• 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*
Δb
Displacement constraints:
b
a
Δa
u   o x
v   o y
w
x2 y2
 2 1
2
a
b
w w

0
x y
• Two part analysis
• Elastic stability – Solved through the Raleigh-Ritz method
• Delamination growth after buckling – Energy approach through fracture mechanics
* Reference [2]
2-D Delamination Analysis*
Energy release rate for the system due to a increase in delamination
 d
G
 Go
dA
Gives
G
a db
b da
a db
1
b da
G a  Gb
Where
 1 U
 Go
b a
 1 U
Ga 
 Go
b b
Ga 
* Reference [2]
2-D Delamination Analysis*
* Reference [2]
Conclusions
• A one-dimensional model can be used to simplify analysis of a more complete twodimensional 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 Analysis
1. 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 Analysis
2. Chai, H., Babcock, C., “Two-Dimensional Modelling of Compressive Failure in
Delaminated Laminates,” Journal of Composite Materials, Vol. 19,. No. 1, pp. 6798, 1985
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