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Modeling of CNT based composites: Numerical Issues
N. Chandra and C. Shet
FAMU-FSU College of Engineering, Florida State University, Tallahassee, FL 32310
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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.
ApproachTo model the interface as cohesive zones, which facilitates to introduce a range of interface properties varying from zero binding to perfect binding
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Fig. Shear lag model for aligned short fiber composites. (a) representative short fiber (b) unit cell for analysis
e
e Fiber
Matrix
l
dD
r
z
ee
(a)
(b)
Shear Lag Model *Prelude 1
sfs
Td 4 4 4k u u
dz d d h d
The governing DE
Whose solution is given by
Where
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
f f 1 2E C cosh( z)+C sinh( z) e
f
4k
dE
m2GInterface property k =
d ln(D / d)
*Original model developed by Cox [1] and Kelly [2]
[1] Cox, H.L., J. Appl. Phys. 1952; Vol. 3: p. 72 [2] Kelly, A., Strong Soilids, 2nd Ed., Oxford University Press, 1973, Chap. 5.
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Prelude 2 Cohesive Zone Model
CZM is represented by traction-displacement jump curves to model the separating surfaces
AdvantagesCZM 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.
T or T f , , (or )n t max max n tT Tt nStiffness of cohesive zone k = or
t n
Modified Shear lag Model
sfs
Td 4 4 4k u u
dz d d h d
f f 1 2E C cosh( z)+C sinh( z) e
e
2 2f fe i
4d4k kThen ( solid fibers) (hollow fiber)
dE Ed d
The governing DE
If the interface between fiber and matrix is represented by cohesive zone, then
s f m
max max
T k u v ,
where interface stiffness k k(T , )
Evaluating constants by using boundary conditions, stresses in fiber is given by
o
f off f f
f
1 cosh( z)E E d
E 1 , 1 1. - cosh( z)l l Ecosh lcosh2 2
e e e
e
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Comparison between Original and Modified Shear Lag Model
StrainS
tres
s0 0.001 0.002 0.003
0
50
100
150
200
250
300
350
Original shear lag model
CZM based shear lag model
200 k'
16.7 k'
5 k'
1.11 k'
max
ct
k' =
Variation of stress-strain response in the elastic limit with respect to parameter
• The parameter defined by defines the interface strength in two models through variable k.• In original model
• In modified model interface stiffness is given by slope of traction-displacement curve given by
• 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
f
4k
dE
m2Gk =
d ln(D / d)
T Tt nk = or t n
Comparison with Experimental Result
o
ff f m m
l2 tanh 2E E 1 , E ,
l2
e
e e
The average stress in fiber and matrix far a applied strain e is given by
Then by rule of mixture the stress in composites can be obtained as
c f m f f(1 V ) V
max
max c
T
n
Fig. A typical traction-displacement curve used for interface between SiC fiber and 6061-Al matrix
For SiC-6061-T6-Al composite interface is modeled by CZM model given by
maxmax , ( )n, ( ) n maxn maxmaxmax
T , ki i 1n N Nmaxk ,( max) i k ,( max)max i n max i n
c c ci 0 i 1
where , andn max
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.
Takingmax = 1.8y, where y is yield stress of matrix and max =0.06 c
Fig.. Comparison of experimental [1] stress-strain curve for Sic/6061-T6-Al composite with stress-strain curves predicted from original shear lag model and CZM based Shear lag model.
Strain
Str
ess
(MP
a)
0 0.005 0.01 0.015 0.02 0.025 0.030
200
400
600
800
1000
1200
1400
1600
1800 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
6061-T6 Al
Fiber
Original shear lag model
Matrix
New model (CZM-Shear lag)
[1] Dunn, M.L. and Ledbetter, H., Elastic-plastic behavior of textured short-fiber composites, Acta mater. 1997; 45(8):3327-3340
The constitutive behavior of 6061-T6 Al matrix [21] can be represented by Comparison (contd.)
ny ph e
yield stress =250 MPa, and hardening parameters h = 173 MPa, n = 0.46. Young’s modulus of matrix is 76.4 GPa.
Young’s modulus of SiC fiber is Ef of 423 GPa
Result comparisonExperimental [1] Young’s modulus is 105 GPa
and failure strength is around 515 MPa
Ec 115 104.4
1540 522
(GPa)
FailureStrength(MPa)
Variable Original Modified
FEAModel
•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
Comparison with Numerical Results
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
tt=1max1 m ax2
m ax
T t
nn=1max
max
T n(a) (b)
A
B C
DA1
B1
C1
max , ( )t max1max1
T ( max 2)t max max11max , ( )t max 21 max 2
max n, ( )maxmaxTn 1max
, ( max)n1 max
, n t
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Position along the length of fiber (m)
Lon
gitu
dina
lstr
ess
inth
efi
ber
(MP
a)
2E-09 4E-09 6E-09 8E-09
300
600
900
1200
1500
1800
2100
2400
2700
3000
3300
3600
FEM Simulation
Analytical Solution
e
e
e
rz
z=0 z=1E-08 m
Position along length of the fiber (m)
Lon
gitu
dina
lstr
ess
inth
efi
ber
(MP
a)
0 2.5E-09 5E-09 7.5E-090
250
500
750
1000
1250
1500
1750
2000
e
e
e
FEM Simulation
Analytical Solution
rz
z=0 z=1E-08 m
Longitudinal Stress in fiber at different strain level
Interface strength = 5000 MPa Interface strength = 50 MPa
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Shear Stress in fiber at different strain level
Position along the length of fiber (m)
She
arst
ress
inth
efi
ber
(MP
a)
2E-09 4E-09 6E-09 8E-09
10
20
30
40
50
60
70
80
90
100
FEM Simulation
Analytical Solution
e
e
e
rz
z=0 z=1E-08 m
Position along the length of fiber (m)
She
arst
ress
inth
efi
ber
(MP
a)
2E-09 4E-09 6E-09 8E-09
2
4
6
8
10
12
14
16
FEM Simulation
Analytical Solution
e
e
e
rz
z=0 z=1E-08 m
Interface strength = 5000 MPa Interface strength = 50 MPa
Critical Bond Length
2 2e i f o
ce f
f oc
f
d dl (hollow fiber)
d 2 (max)
(max)dl (solid fiber)
2 (max)
o
ff f y
z 0
f
e i
1 cosh( z)E
(max) E 1lcosh 2
(max) Shear strength of the interface
d ,d are external and internal diameters respectively
e
e
o
f
f
lc
r
Matrix
Fiber
Interface Shear Traction Variation
Longitudinal fiber stress Variation
Bond Length
z
l/2
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 strengthTmax in MPa
Critical bond length lc in Ao
5000
500
50
3.23
26.4
74.7
Hollow cylindrical fiber Solid cylindrical fiber
24.4
73.08
91.4
Cri
tical
Bon
dL
engt
h(A
)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
5
10
15
20
25
30
max1
200
600
1000
5000
200
600
1000
5000
}}
Lengths of
Tubular
Fibers in A
Lengths of Solid
Cylindrical
fibers in A
oo
o
• Critical bond length varies with interface 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
interface strength is 5000MPa
Cri
tical
bond
leng
th(A
)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
500
1000
1500
2000
2500
o
200
600
1000
5000
200
600
1000
5000
}}
Lengths of
Tubular
Fibers in A
Lengths of Solid
Cylindrical
Fibers in A
max1
Bond length limitfor fibers of length 5000 A
o
Bond length limitfor fibers of length 1000 A
o
Bond length limitfor fibers of length 600 A
o
o
o
Variation of Critical Bond Length with interface property
interface strength is 50MPa
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Table : Variation of Young’s modulus of the composite with matrix young’s modulus, volume fraction and interface strength
Young’sModulus ofthematrixEm (in GPa)
0.02 0.03 0.05
Ec(elastic)/Em Ec(elastic)/EmEc(elastic)/Em
1.18 1.28 1.46
Interfacestrength
Tmax (in MPa)
2.46 3.17 4.61
4.98 6.99 10.96
1.05 1.07 1.13
1.5 1.74 2.24
2.38 3.07 4.45
0.99 0.986 0.98
1.05 1.08 1.13
1.18 1.27 1.45
Volumefraction
3.5
10
70
50
500
5000
50
500
5000
50
500
5000
50
500
5000
200
0.984 0.977 0.96
1.005 1.009 1.015
1.053 1.075 1.13
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
Effect of interface strength on strength of Composites
0.02 0.03 0.05
Volumefraction
50
500
5000
InterfacesstrengthTmax (in MPa)
107
340
809
87 94
180
367
234
515
Table Yield strength (in MPa) of composites for different volume fraction and interface strength
Strain
Str
ain
0 0.025 0.05 0.075 0.10
400
800
1200
1600
2000
2400
2800
3200
3600
Interface strength = 5000 MPa
Interface strength = 500 MPa
Interface strength= 50 MPa
Ec/Em = 10.96
Ec/Em = 4.61
Ec/Em = 1.46
Fiber volume fraction = 0.02
Strain
Str
ess
0 0.02 0.04 0.06 0.08 0.10
200
400
600
800
1000
1200
1400
1600
Interface strength = 5000 MPa
Interface strength = 500 MPa
Interface strength= 50 MPa
Ec/Em = 4.97
Ec/Em = 2.46
Ec/Em = 1.18
Fiber volume fraction = 0.05
•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
Effect of interface displacement parameter max1
on strength and stiffness
E/E
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
1
2
3
4
5
6
7
8
9
10
11
cm
max1
E = Ec m
T = 5000MPa
T = 500 MPa
T = 50 MPa
max
max
max
length = 200 E-10 mDiameter = 6.98E-10mVolume fraction = 0.05
Fig. Variation of stiffness of composite material with interface displacement parameter max1 for different interface strengths.
(com
posi
te)/
(mat
rix)
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90
1
2
3
4
5
6
7
8
9
10
11
yy
max1
y y
T = 5000MPa
T = 500 MPa
T = 50 MPa
max
max
max
length = 200 E-10 mDiameter = 6.98E-10mVolume fraction = 0.05
(composite) (matrix)
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
Length (X 1.0 E-10 m)
(com
posi
te)/
(mat
rix)
0 2500 5000 7500 100000
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
yy
max1T = 5000MPa
T = 500 MPa
T = 50 MPa
max
max
max
Diameter = 6.98E-10mVolume fraction = 0.05
Fig. Variation of yield strength of the composite material with different fiber lengths and different interface strengths
Length ( X 1.0 E-10 m)
E/E
0 2500 5000 7500 100000
2
4
6
8
10
12
14
16
cm
T = 5000MPa
T = 500 MPa
T = 50 MPa
max
max
max
Diameter = 6.98E-10mVolume fraction = 0.05 max1
Fig. Variation of Young’s modulus of the composite material with different fiber lengths and for 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.
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Conclusion
1. The critical bond length or ineffective fiber length is affected by interface strength. Lower the interface strength higher is the ineffective length.
2. In addition to volume fraction and matrix stiffness, interface property, length and diameter of the fiber also affects elastic modulus of composites.
3. Stiffness and yield strength of the composite increases with increase in interface strength.
4. 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.
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