# Modified embedded atom method study of the mechanical properties of carbon nanotube reinforced nickel composites Page: 8

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PHYSICAL REVIEW B 81, 104103 (2010)

TABLE V. The isotropic bulk modulus (B in GPa) and directional bulk modulus along the orthorhombic

crystallographic axes a, b, and c (Ba, Bb, and Bc) for pure Nickel (fcc) and nickel/CNT composites. Low and

High are the composites low and high CNT volume fractions.

Ni(fcc) B (180.7) Ba (542.1) Bb (542.1) Bc (542.1)

Low High Low High Low High Low High

Ni/SWCNT(5,0) 181.1 181.2 560.8 599.8 530.9 516.3 538.8 522.5

Ni/SWCNT(10,0) 180.5 179.2 573.3 650.5 527.4 498.3 526.6 495.1

Ni/SWCNT(15,0) 179.0 173.7 572.6 629.5 520.4 479.7 521.2 479.9

Ni/MWCNT 182.0 183.2 629.3 817.4 508.6 462.2 514.5 482.9decreases with increasing CNT diameter. Composites with

higher CNT volume fractions show larger decreases. Ni/

MWCNT does not show an appreciable change in

Epolycrystalline for the composite with low volume fraction and

a slight increase for composites with high volume fractions

of CNT. Generally, Epolycrystalline is much larger in magnitudes

than the single crystal E. From these trends, we can conclude

that embedding SWCNTs in the Ni matrix will decrease the

polycrystalline Young's modulus Epolycrystalline. On the con-

trary, the polycrystalline Young's modulus might increase

when MWCNTs are embedded in the Ni matrix.

Our calculated Poisson's ratio for polycrystalline nickel is

0.293. This is in reasonable agreement with the experimental

value of 0.31.55 Surprisingly, we find similar values for Ni/

CNT composites. For example, for Ni/SWCNT(5,0), Ni/

SWCNT(10,0), Ni/SWCNT(15,0), and Ni/MWCNT compos-

ites with higher CNT fractions, the calculated Poisson's

ratios are 0.301, 0.301, 0.303, and 0.317, respectively. They

are 0.296, 0.295, 0.295, and 0.294 for the same composites

with lower CNT volume fractions. The calculated Poisson's

ratios indicate that the Ni/CNT composites remain similarly

stable against shear as compared to pure Ni.

3. Elastic anisotropy

Elasticity describes the response of a crystal under exter-

nal strain and provides key information about the bonding

characteristic between adjacent atomic planes and the aniso-

tropic character of the solid.26 The shear anisotropic factors

measure the degree of anisotropy in the bonding between

atoms in different planes. Anisotropy in orthorhombic mate-

rials arises from the anisotropy in their shear and linear bulk

moduli.

The shear anisotropic factor is defined as

A1 = 4c44/(C11 + C33 - 2C13)

for the {100} shear planes in (011) and (010) directions,

A2 = 4c55/(C22 + C33 - 2c23)

for the {010} shear planes in (101) and (001) directions, and

A3 = 4c66/(Cll + c22 - 2c12)

for the {001} shear planes in (110) and (010) directions. For

an isotropic crystal, the factors A1, A2, and A3 must be one.

The deviation from one is a measure of the degree of the

elastic anisotropy of crystals.The anisotropy of the bulk modulus along the a axis with

respect to b and c axes can be estimated using the following

equations:Ba

ABb Bb

Bb'Ba

ABc BcNote that a value of unity for these two fractions indicates

elastic isotropy. In the above equations, Ba, Bb, and Bc are

the bulk moduli along different crystal axes, are defined asdP dP

Ba=a Bb b-

da dbdP

and Bc = c .

dcA practical measure of elastic anisotropy for polycrystalline

materials is defined as follows:

(B - BR) (G - GR)

AB= (B + BR) and AG (Gv + GR)

where B and G are the bulk and shear moduli and the sub-

scripts V and R denote the Voigt and Reuss limits. For these

expressions, a value of zero identifies elastic isotropy and a

value of 1 (100%) is the largest possible anisotropy.

The bulk modulus and the directional bulk modulus are

given in Table V. Note that the CNTs are aligned along the a

axis. It is clear that Ba is higher than Bb and Bc, which are

nearly identical. In the composites, Ba increases with volume

fraction of CNT while Bb and Bc both decrease as compared

to Ni. The directional bulk modulus of Ni is 542 GPa along

all three axes. For Ni/MWCNT composites, the highest (817

GPa) and lowest (462 GPa) values for Ba and Bb or Bc are

seen for high CNT volume fractions. For Ni/SWCNTs, de-

pendencies on the diameters are also seen.

The shear anisotropic factors are given in Table VI. For

fcc Ni, we calculated a shear anisotropic factor As is 2.52

while the value reported in the literature56 value is 2.63. This

indicates significant deviation from isotropy. We also see that

the shear anisotropic factors A1, A2, and A3 for the compos-

ites depend on CNT concentrations. Similar changes are seen

for A1 and A3 where these values are found to be lower than

pure nickel. A steeper decrease for these is seen for compos-

ite with high CNT volume fractions. For SWCNTs, a larger

decrease is seen for CNTs with larger diameter. A2 for the

composites remains similar to that for pure nickel.104103-8

UDDIN et al.

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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/m1/8/: accessed March 30, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.