Modified embedded atom method study of the mechanical properties of carbon nanotube reinforced nickel composites Page: 7
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MODIFIED EMBEDDED ATOM METHOD STUDY OF THE...
TABLE IV. Polycrystalline bulk (B) and shear (G) moduli calculated (in GPa) for Ni and nickel/CNT composites. The anisotropies of the
bulk modulus along the a axis with respect to the b and c axes are also given. The values are shown for Voigt (V), Reuss (R), and Hill (H)
approximations. Low and High are the composites low and high CNT volume fractions.
Ni(fcc) Gv (95.2) By (180.7) GR (78.04) BR (180.7) GH (86.6) BH (180.7)
Low High Low High Low High Low High Low High Low High
Ni/SWCNT(5,0) 93.3 89.6 180.9 181.4 77.6 77.0 181.1 181.2 85.5 83.3 181.1 181.3
Ni/SWCNT(10,0) 93.0 87.7 180.6 180.4 78.3 77.5 180.5 179.7 85.6 82.6 180.5 180.0
Ni/SWCNT(15,0) 92.0 83.7 179.1 174.4 78.1 74.4 179.0 173.7 85.0 79.1 179.0 174.1
Ni/MWCNT 93.2 92.6 182.2 186.7 80.7 83.8 182.0 183.2 86.9 88.2 182.0 184.9
BR = 1/[s11 + s33 + 33 + 2(s12 + S23 + S13)],
By= 1/9(c11 + c33 + C33) + 2/9(c12 + C23 + C13),
GR = 15/[4(s11 + s33 + s33) - 4(s12 + S23 + S13)
+ 3(S44 + S55 + S66)],
Gv= (cll + C22 + C33 - C12 - C23 - C13)/15 + (C44 + C55 + C66)/5,
where sij are elastic compliances and their values can be
obtained through an inversion of the elastic constant matrix,
S= C-'. We estimated the true polycrystalline values by using
the Voigt and Reuss, and Hill (VRH) approximations:52 BH
=(BR+BV)/2 and GH=(GR+Gv)/2. The average Young's
modulus and Poisson's ratio for polycrystalline aggregates
can then be calculated as Epolycrystalline= (9BHGH)/(3BH
+ GH) and Ppolycrystalline = (3BH- 2GH) / (6BH+ 2GH).
Table IV shows the calculated bulk (BH) and shear (GH)
moduli for the polycrystalline aggregates of Ni and the Ni/
CNT composites using VRH approximations. We discuss the
results obtained using the Hill average only. The calculated
BH for polycrystalline nickel is identical to that of the corre-
sponding single crystal and also agrees well with experiment.
The bulk modulus of Ni determined from the Birch-
Murnaghan equation of state is also found to be identical.
For Ni/CNT composites, no significant changes are seen in
BH except for Ni/CNT, which show a slight increase of 1
GPa and 4 GPa, respectively, for low and high CNT volume
fractions. The shear modulus GH, which measures the ability
of a material to resist transverse deformations, decreases
slightly for Ni/SWCNTs as compared to pure Ni. For fcc
nickel, the calculated GH is found to be 87 GPa, which
agrees well with experiment.49 For Ni/MWCNT, these values
increase slightly and depend on CNT volume fractions.
Pugh53 introduced the ratio between the bulk and shear
modulus, BH/ GH, for polycrystalline phases as a measure of
fracture/toughness in metals by considering that the shear
modulus GH represents the resistance to plastic deformation
while the bulk modulus BH represents the resistance to frac-
ture. A high BH/ GH value is associated with ductility while a
low value is associated with brittleness. The critical value
that separates ductile and brittle materials is about 1.75. Our
calculated BH/ GH for pure nickel is 2.1. For Ni/CNT com-
posites, similar BH/ GH values are found. By considering the
bulk modulus as a measure of the average bond strength and
the shear modulus as a measure of the resistance to a change
in bond angle by an external force, Tanaka et al.54 proposed
that GH/BH represents the relative directionality of the bond-
ing in the material. It can be seen that there is a slight de-
crease in these ratios for Ni/SWCNTs. Composites with high
CNT volume fractions show a greater decrease. For Ni/
MWCNT, there is no observable change with pure Ni. The
BH/ GH and the GH/BH values indicate that CNT do not sig-
nificantly alter the above-mentioned properties as compared
to fcc Ni.
The average Young's modulus for polycrystalline aggre-
gates (Epolycrystalline) calculated from the Hill's formula for
pure Ni and the Ni/CNTs are presented graphically in Fig. 3.
The Young's modulus in longitudinal Eoo0 (Ell) and trans-
verse directions, Eolo, Eloo (E22,E33) determined for single-
crystal specimens are also compared. For, Ni the value cal-
culated using Hill's average is 222 GPa. This value is in
excellent agreement with the experimental value of
213.1 _ 21.2 GPa, obtained for coarse-grained polycrystal-
line aggregates.49 In Ni/SWCNT composites, Epolycrystalline
O ... E sm
260 Ni/swcNT( .o)
N SWCNT(10.O )
240 - Sim
V--" ENi/swcntra o)
2 200 -
0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
Volume Fraction (Vf)
FIG. 5. (Color online) Young's modulus in longitudinal direc-
tion, El1, (in GPa) simulated and calculated using the rule-of-
mixture (RoM) for Ni/SWCNT and Ni/MWCNT composites with
low- and high CNT volume fractions. The straight lines for each
CNT type are obtained by fitting the RoM data at low and high
CNT volume fractions. The simulated values for each composite are
presented with symbols.
PHYSICAL REVIEW B 81, 104103 (2010)
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Uddin, Jamal; Baskes, Michael I.; Srivilliputhur, Srinivasan; Cundari, Thomas R., 1964- & Wilson, Angela K. Modified embedded atom method study of the mechanical properties of carbon nanotube reinforced nickel composites, article, March 11, 2010; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc107769/m1/7/: accessed February 27, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.