Modeling of CNT based composites:
Numerical Issues
N. Chandra and C. Shet
FAMU-FSU College of Engineering, Florida State University,
Tallahassee, FL 32310
AMML
Objective
•To develop an analytical model that can predict the
mechanical properties of short-fiber composites with
imperfect interfaces.
•To study the effect of interface bond strength on critical
bond length lc
•To study the effect of bond strength on mechanical
properties of composites.
Approach
To model the interface as cohesive zones, which facilitates to
introduce a range of interface properties varying from zero
binding to perfect binding
AMML
Prelude 1
Shear Lag Model *
The governing DE
e
e
d f
4
4 Ts 4
  s 
 k u  u
dz
d
d h d
Whose solution is given by
 f  Ef e  C1cosh( z)+C2sinh( z)
(a)
r
e
Matrix
e
D
d
Where
Fiber
4k
dE f
z
l
(b)
Fig. Shear lag model for aligned short fiber
composites. (a) representative short fiber (b) unit cell for
analysis
*Original model developed by
Cox [1] and Kelly [2]
[1]
[2]

Interface property k =
2G m
d ln(D / d)
Disadvantages
• The interface stiffness is dependent on
Young’s modulus of matrix and fiber,
hence it may not represent exact interface
property.
•k remains invariant with deformation
• Cannot model imperfect interfaces
Cox, H.L., J. Appl. Phys. 1952; Vol. 3: p. 72
Kelly, A., Strong Soilids, 2nd Ed., Oxford University Press, 1973,
Chap. 5.
AMML
Prelude 2
Cohesive Zone Model
CZM is represented by traction-displacement
jump curves to model the separating surfaces
Advantages
CZM can create new surfaces.
Maintains continuity conditions mathematically,
despite the physical separation.
CZM represents physics of the fracture process at
the atomic scale.
Eliminates singularity of stress and limits it to the
cohesive strength of the the material.
It is an ideal framework to model strength,
stiffness and failure in an integrated manner.
Tn or Tt  f  max , max , n (or t ) 
T
Tn
Stiffness of cohesive zone k = t or
t
n
AMML
Modified Shear lag Model
The governing DE
d f
4
4 Ts 4
  s 
 k u  u
dz
d
d h d
If the interface between fiber and matrix is represented by cohesive
zone, then
Ts  k u f  v m ,
where interface stiffness k  k(Tmax ,  max )
 4d
4k
Then  
( solid fibers)    2 e 2
 d e  di
dE f



 k

(hollow fiber)
 Ef

 f  Ef e  C1cosh( z)+C2sinh( z)
Evaluating constants by using boundary conditions, stresses in fiber is given by
  o



1
cosh(

z)
 


  o

E
e

Ef e d

f



 f  Ef e 1 
, f 
 1 1. - cosh( z)  




l

l
E
e

 f

cosh
lcosh


2
2


 
 
Comparison between Original and Modified Shear Lag Model
• The parameter  defined by   4k
dE f
defines the interface strength
in two models through
variable k.
2G m
k
=
• In original model
d ln(D / d)
350
200 k'
Original shear lag model
300
• In modified model interface
stiffness is given by slope of
traction-displacement curve
given by k = Tt or Tn
 t
n
• In original model k is
invariant with loading and it
cannot be varied
•In modified model k can be
varied to represent a range of
values from perfect to zero
bonding
Stress
CZM based shear lag model
250
16.7 k'
200
5 k'
1.11 k'
150
k' =
 max
ct
100
50
0
0
0.001
Strain
0.002
0.003
Variation of stress-strain response in the elastic limit with respect to
parameter 
AMML
Comparison with Experimental Result
The average stress in fiber and matrix far a applied strain e is given by
f
 
  o

l 

2
tanh
 

2 
Ef e



,   E e,
 Ef e 1 
m
m


l


2


 
T
max
Then by rule of mixture the stress in
composites can be obtained as
 c  (1  Vf ) m  Vf  f
n
max
For SiC-6061-T6-Al composite interface
is modeled by CZM model given by
 max
 n, (n  max )

 max
Tn  
,
 N  i 



 ,(   max)
k
 max   i  c   n

 i 0

 max

max

k
 max
 c

c
Fig. A typical traction-displacement curve used for
interface between SiC fiber and 6061-Al matrix
,
(n  max )

i 1  
N




 i  k     ,(   max)
max
i
n


c   

 i 1


where   n  max , and area undet T- curve as   2.224max c
With N=5, and k0 = 1, k1 = 10, k2 = -36, k3 = 72, k4 = -59, k5 = 12.
Taking max = 1.8 y, where y is yield stress of matrix and max =0.06 c
Comparison (contd.)
The constitutive behavior of 6061-T6 Al matrix [21] can be represented by
   y  he pn
1800
yield stress =250 MPa, and hardening
parameters h = 173 MPa, n = 0.46.
Young’s modulus of matrix is 76.4 GPa.
SiC/6061-T6 Al (Experiment)
SiC/6061-T6 Al (Predicted-CZM based shear lag model)
SiC/6061-T6 Al (Predicted-original shear lag model)
SiC
1400
6061-T6 Al
1200
Stress (MPa)
Young’s modulus of SiC fiber is Ef of 423 GPa
1600
Fiber
1000
Result comparison
Experimental [1] Young’s modulus is 105 GPa
and failure strength is around 515 MPa
Original shear lag model
800
New model (CZM-Shear lag)
600
400
Matrix
200
0
Variable
Original Modified
Ec (GPa) 115
Failure
Strength
(MPa)
[1]
1540
104.4
0
0.005
0.01
0.015
Strain
0.02
0.025
0.03
Fig.. Comparison of experimental [1] stress-strain curve for Sic/6061-T6-A
composite with stress-strain curves predicted from original shear lag model an
CZM based Shear lag model.
522
Dunn, M.L. and Ledbetter, H., Elastic-plastic behavior of textured short-fiber composites, Acta mater. 1997; 45(8):3327-3340
FEAModel
Comparison with Numerical Results
•The CNT is modeled as a hollow
tube with a length of 200 , outer
radius of 6.98 and thickness of 0.4 .
• CNT modeled using 1596
axi-symmetric elements.
• Matrix modeled using 11379
axi-symmetric elements.
•Interface modeled using 399 4 node
axisymmetric CZ elements with
zero thickness
 max
 t ,

max1

Tt  max

1 
 max 
 t ,
1  max 2
 
 max
 n,

 max
Tn  
 max  1     ,
n
 
1  max
  n   t ,
(  max1)
(max1     max 2) Fig. (a) Finite element mesh of a quarter portion of unit model (b) a enlarged
portion of the mesh near the curved cap of CNT
(  max 2 )
Tt
Tn
(a)
(  max )
max
B
C
max
(b)
B1
(   max)
A
max1
max2
D t
t=1
C1
A1
max
n=1
n
Longitudinal Stress in fiber at different strain level
r
z
z=0
3600
z=1E-08 m
2000
3000
Longitudinal stress in the fiber (MPa)
Longitudinal stress in the fiber (MPa)
3300
FEM Simulation
Analytical Solution
2700
e  
2400
2100
e  
1800
1500
1200
900
e  
600
300
2E-09
4E-09
6E-09
FEM Simulation
Analytical Solution
1750
r
z
1500
e  
1250
z=0
z=1E-08 m
e  
1000
750
500
e  
250
0
8E-09
Position along the length of fiber (m)
Interface strength = 5000 MPa
0
2.5E-09
5E-09
7.5E-09
Position along length of the fiber (m)
Interface strength = 50 MPa
AMML
Shear Stress in fiber at different strain level
r
r
z
100
z=0
FEM Simulation
Analytical Solution
90
80
70
e  
60
50
e  
40
30
e  
20
z=0
12
z=1E-08 m
FEM Simulation
Analytical Solution
14
Shear stress in the fiber (MPa)
Shear stress in the fiber (MPa)
z
16
z=1E-08 m
e  
10
8
e  
6
4
e  
2
10
2E-09
4E-09
6E-09
8E-09
2E-09
Position along the length of fiber (m)
4E-09
6E-09
8E-09
Position along the length of fiber (m)
Interface strength = 5000 MPa
Interface strength = 50 MPa
AMML
lc
d


2
e
d
2
i
de
Critical Bond Length

f o 

 (hollow fiber)
 2 f (max) 
d   f (max)   o 
lc  

2   f (max) 
r
Matrix
z
Fiber
(solid fiber)
lc
Bond Length
l/2
  o



1
cosh(

z)
 


E
e


 f (max)  E f e y 1   f


l
cosh


2

 z 0
f
 
Interface Shear Traction Variation
o
 f (max)  Shear strength of the interface
d e , d i are external and internal diameters respectively
f
Longitudinal fiber stress Variation
o
Critical bond length lc in A
Interface strength
Tmax in MPa
Hollow cylindrical fiber
5000
3.23
24.4
500
26.4
73.08
50
74.7
Solid cylindrical fiber
91.4
Table 1. Critical bond lengths for short fibers of length 200 and for different
interface strengths and interface displacement parameter max1 value 0.15.
interface strength is 5000MPa
Variation of Critical Bond Length
with interface property
30
}
}
200
Critical Bond Length (A)
25
o
• Critical bond length varies with interface
Lengths of
Tubular
1000
Fibers in A
o
5000
20
200
1000
Lengths of Solid
Cylindrical
5000
fibers in A
600
15
o
10
5
0
0.1
0.2
0.3
0.4
0.5
 max1
0.6
0.7
0.8
0.9
interface strength is 50MPa
2500
Bond length limit
o
for fibers of length 5000 A
200
Critical bond length (A)
property (Cohesive zone parameters (max ,
max1)
•When the external diameter of a solid fiber
is the same as that of a hollow fiber, then,
for any given length the load carried by
solid fiber is more than that of hollow fiber.
Thus, it requires a longer critical bond
length to transfer the load
•At higher max1 the longitudinal fiber stress
when the matrix begins to yield is lower,
hence critical bond length reduces
•For solid cylindrical fibers, at low interface
strength of 50 MPa, when the fiber length is
600 and above, the critical bond length on
each end of the fiber exceeds semi-fiber
length for some values max1 tending the
fiber ineffective in transferring the load
600
o
600
2000
1000
5000
200
1500
600
1000
5000
}
}
Lengths of
Tubular
o
Fibers in A
Lengths of Solid
Cylindrical
o
Fibers in A
1000
500
0
Bond length limit
o
for fibers of length 600 A
Bond length limit
o
for fibers of length 1000 A
0.2
0.3
0.4
0.5
0.6
 max1
0.7
0.8
0.9
Effect of interface strength on stiffness of Composites
Young’s Modulus
(stiffness) of the
composite not only
increases with matrix
stiffness and fiber volume
fraction, but also with
interface strength
Young’s
Modulus of
the matrix
Em (in GPa)
3.5
10
70
200
Volume
fraction
Interface
strength
Tmax
(in MPa)
0.02
Ec(elastic)/Em
0.03
Ec(elastic)/Em
0.05
Ec(elastic)/Em
50
1.18
1.28
1.46
500
2.46
3.17
4.61
5000
4.98
6.99
10.96
50
1.05
1.07
1.13
500
1.5
1.74
2.24
5000
2.38
3.07
4.45
50
0.99
0.986
0.98
500
1.05
1.08
1.13
5000
1.18
1.27
1.45
50
0.984
0.977
0.96
500
1.005
1.009
1.015
5000
1.053
1.075
1.13
Table : Variation of Young’s modulus of the composite with matrix young’s
modulus, volume fraction and interface strength
AMML
Effect of interface strength on strength of Composites
Fiber volume fraction = 0.02
3600
Fiber volume fraction = 0.05
1600
3200
1400
2800
2400
Strain
2000
Interface strength = 5000 MPa
1200
Interface strength = 500 MPa
1000
Stress
Interface strength= 50 MPa
1600
Interface strength = 5000 MPa
Interface strength = 500 MPa
Interface strength= 50 MPa
800
600
1200
Ec/Em = 10.96
400
800
Ec/Em = 4.97
Ec/Em = 2.46
Ec/Em = 4.61
200
400
Ec/Em = 1.18
Ec/Em = 1.46
0
0
0.025
0.05
Strain
0.075
•Yield strength (when matrix
yields) of the composite increases
with fiber volume fraction (and
matrix stiffness) but also with
interface strength
•With higher interface strength
hardening modulus and post yield
strength increases considerably
0.1
0
0
0.02
Volume
fraction
Interfaces
strength
Tmax (in MPa)
0.04
Strain
0.06
0.08
0.1
0.02
0.03
0.05
50
87
94
107
500
180
234
340
5000
367
515
809
Table Yield strength (in MPa) of composites for different volume
fraction and interface strength
Effect of interface displacement parameter max1
on strength and stiffness
11
11
10
10
T max = 5000MPa
9
length = 200 E-10 m
Diameter = 6.98E-10m
Volume fraction = 0.05
8
 (composite)/ (matrix)
9
6
5
4
3
y
Tmax = 500 MPa
2
8
7
6
5
4
3
T max = 500 MPa
2
Tmax = 50 MPa
1
0
length = 200 E-10 m
Diameter = 6.98E-10m
Volume fraction = 0.05
y
Ec /Em
7
T max = 5000MPa
0.2
0.3
0.4
0.5
Tmax = 50 MPa
1
Ec = Em
0.6
 max1
0.7
0.8
Fig. Variation of stiffness of composite material with interface
displacement parameter max1 for different interface strengths.
0.9
0
 y (composite)   y (matrix)
0.2
0.3
0.4
0.5
0.6
max1
0.7
0.8
0.9
Fig. Variation of yield strength of the composite material with
interface displacement parameter max1 for different interface
strengths.
• As the slope of T- curve decreases (with increase in max1), the overall interface
property is weakened and hence the stiffness and strength reduces with increasing values
of max1.
•When the interface strength is 50 MPa and fiber length is small the young’s modulus and
yield strength of the composite material reaches a limiting value of that of matrix material.
Effect of length of the fiber on strength and stiffness
16
16
15
14
14
 (composite)/ (matrix)
12
Tmax = 5000MPa
y
10
8
Tmax = 500 MPa
6
Tmax = 50 MPa
y
Ec /Em
13
Diameter = 6.98E-10m
Volume fraction = 0.05
 max1  
4
12
Tmax = 5000MPa
11
Diameter = 6.98E-10m
Volume fraction = 0.05
 max1  
10
9
T max = 500 MPa
8
7
T max = 50 MPa
6
5
4
3
2
2
1
0
0
2500
5000
7500
Length ( X 1.0 E-10 m)
10000
Fig. Variation of Young’s modulus of the composite material with
different fiber lengths and for different interface strengths
0
0
2500
5000
7500
Length (X 1.0 E-10 m)
10000
Fig. Variation of yield strength of the composite material with
different fiber lengths and different interface strengths
• For a given volume fraction the composite material can attain optimum values for
mechanical properties irrespective of interface strength.
• For composites with stronger interface the optimum possible values can be obtained with
smaller fiber length
• With low interface strength longer fiber lengths are required to obtain higher composite
properties. During processing it is difficult to maintain longer CNT fiber straigth.
Conclusion
1.
2.
3.
4.
The critical bond length or ineffective fiber length is affected by
interface strength. Lower the interface strength higher is the
ineffective length.
In addition to volume fraction and matrix stiffness, interface
property, length and diameter of the fiber also affects elastic
modulus of composites.
Stiffness and yield strength of the composite increases with
increase in interface strength.
In order to exploit the superior properties of the fiber in developing
super strong composites, interfaces need to be engineered to have
higher interface strength.
AMML