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Research Papers

On Adhesive and Buckling Instabilities in the Mechanics of Carbon Nanotubes Bundles

[+] Author and Article Information
Xuance Zhou

Department of Mechanical Engineering,
University of California at Berkeley,
Berkeley CA 94720-1740

Oliver M. O’Reilly

Department of Mechanical Engineering,
University of California at Berkeley,
Berkeley CA 94720-1740

The reader is referred to Refs. [19,25] for further details on the role played by material momentum. We note, in particular, that (8)3 leads to an adhesion boundary condition.

For details on how these conditions can be established, the interested reader is referred to Refs. [19,22].

For the individual unadhered CNTs, stability can be unambiguously established using the sufficient condition LS1 discussed in Ref. [22, p. 220].

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received June 5, 2015; final manuscript received June 29, 2015; published online July 15, 2015. Editor: Yonggang Huang.

J. Appl. Mech 82(10), 101007 (Oct 01, 2015) (12 pages) Paper No: JAM-15-1302; doi: 10.1115/1.4030976 History: Received June 05, 2015; Revised June 29, 2015; Online July 15, 2015

Many recently synthesized materials feature aligned arrays or bundles of carbon nanotubes (CNTs) whose mechanical properties are partially determined by the van der Waals interactions between adjacent tubes. Of particular interest in this paper are instances where the resulting interaction between a pair of CNTs often produces a forklike structure. The mechanical properties of this structure are noticeably different from those for isolated individual CNTs. In particular, while one anticipates buckling phenomena in the forked structure, an adhesion instability may also be present. New criteria for buckling and adhesion instabilities in forklike structures are presented in this paper. The criteria are illuminated with a bifurcation analyses of the response of the forklike structure to applied compressive and shear loadings.

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Figures

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Fig. 1

(a) TEM image of CNTs grown by chemical vapor deposition and (b) illustration of CNTs adhered by van der Waals interactions and kinked CNTs

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Fig. 2

Bifurcation diagram showing the deviation of the tips of the CNT pair loaded under self-weight and a terminal load Fℓ. The corresponding results for a pair of unadhered CNTs are also shown. (a) The height h of the tips when Fℓ = -2NE2 and N varies from 0 to 12ρgℓ. (b) The tangential displacement d of the tips when Fℓ = 2TE1 and T varies from 0 to ρgℓ. For the results shown, D = 1, ω = 1, and b/ℓ = 0.1. The labels s and u indicate stability and instability, respectively. The forces Nc and Ts are the critical normal and tangential critical loads for the forked structure, respectively, while Ns is the critical normal buckling load for a single CNT.

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Fig. 3

Illustration of a forked structure consisting of a CNT pair. The model is based on Fig. 1 and features a separation width of b at the base and an upper portion where the individual CNTs adhere to each other with the help of weak van der Waals interactions. We encourage the reader to note the top view of the CNT pair which is denoted by “A.”

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Fig. 4

Schematic representation of an elastica of length ℓ showing the position r of a point on the centerline. The rod is subject to a terminal force F0 and terminal moment M0 at s = 0, a terminal force Fℓ and terminal moment Mℓ at the end s = ℓ, and a force Fγ, material momentum supply Bγ, and moment Mγ at the point s = γ.

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Fig. 5

Some of the forces on the components of the model for the CNT pair: (a) terminal loading at s = ℓ for the CNT pair system, (b) terminal loading at s = γ- for the first CNT, (c) terminal loading at s = γ+ and s = ℓ for the adhered portion of the model, and (d) terminal loading at s = γ- for the second CNT. Here, m = EIθ', and t and n represent the difference of the internal forces at s = γ between the two CNTs (cf. (12)).

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Fig. 6

Adhesion formation for a single CNT pair in the presence of its own self weight ρgℓ and in the absence of a terminal load: Fℓ = 0. (a) Equilibrium configurations of the CNT pair as b varies. (b) The corresponding dimensionless detached length γ/ℓ between the two CNTs. (c) The force t = n1(γ-)·E1 = -n2(γ-)·E1 at the adhesion point s = γ. For the results shown, D = 1, ω = 1, b/ℓ ∈ [0,0.4], T = 0, and N = 0.

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Fig. 7

The dependency of terminal load on the tip deformation for different values of the separation distance b. (a) The terminal load 2 T in the horizontal direction as a function of the tip displacement Δd in the horizontal direction when Fℓ = 2TE1. The dashed line shows the corresponding stiffness calculation for a single CNT. (b) The terminal compressive load 2 N in the vertical direction as a function of the tip displacement Δh in the vertical direction when Fℓ = -2NE2. The dashed line in this figure shows the corresponding stiffness calculation for a single CNT. For the results shown, D = 1, ω = 1, and b/ℓ = 0.05, 0.1, and 0.2.

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Fig. 11

Stability analysis of the forklike structure loaded under its self weight and a tangential load Fℓ = 2TE1. (a) The behavior of w+ and wKcrit- as a function of the dimensionless load T/ρgℓ. (b) Maximum Γ found from the optimization problem (38). For the results shown, D = 1, ω = 1, and b/ℓ = 0.1. The shaded region labeled I indicates areas of the parameter space where (w1-,w2-)∉F, and the labels s and u indicate stability and instability, respectively.

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Fig. 12

Stability analysis for various loading configurations of the forklike structure loaded under its self weight and a terminal load Fℓ = Fℓ(cos(φ)E1-sin(φ)E2). (a) The behavior of w+ and wKcrit- as a function of the angle φ. (b) Maximum value Γ of Eq. (38). For the results shown, D = 1, ω = 1, b/ℓ = 0.1,Fℓ/ρgℓ = 2, and φ = arctan(-N/T) ranges from -π/2 to π/2. The shaded region labeled I indicates areas of the parameter space where (w1-,w2-)∉F, and the labels s and u indicate stability and instability, respectively.

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Fig. 13

Critical load analysis with a varying separation width b. (a) The corresponding critical normal dimensionless load Nc/ρgℓ as b/ℓ varies from 0 to 0.32. (b) The corresponding critical tangential dimensionless load Tc/ρgℓ as b/ℓ varies from 0 to 0.16. For the results shown, D = 1 and ω = 1. The force Ns is the critical normal buckling load for a single CNT.

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Fig. 8

Three representative solutions to (a) the Riccati differential equation (27)2 and (b) the corresponding solutions to the Jacobi differential equation (34) 2,K = 1. For the solutions shown, i denotes a bounded solution w1(s1) to the Riccati equation whose counterpart u1(s1) has no conjugate points in s∈[0,γ), ii denotes the bounded solution w1cri(s1) to the Riccati equation whose counterpart u1cri(s1) has a conjugate points at s1 = 0, and iii denotes an unbounded solution w1(s1) to the Riccati equation whose counterpart u1(s1) has a conjugate point at s1c∈(0,γ).

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Fig. 9

Graphical illustration of the set F defined by Eq. (37) for the pair of initial conditions w1- and w2-

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Fig. 10

Stability analysis of the forklike structure loaded under its self weight and a normal load 2 N: Fℓ = -2NE2. (a) The behavior of w+ and wKcrit- as a function of the dimensionless load N/ρgℓ. (b) Maximum Γ found from the optimization problem (38). For the results shown, D = 1, ω = 1, and b/ℓ = 0.1. The shaded region labeled I indicates areas of the parameter space where (w1-,w2-)∉F, and the labels s and u indicate stability and instability, respectively.

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Fig. 15

Bifurcation diagram showing the dimensionless potential energies of the CNT pair under a tangential load 2 T: Fℓ = 2TE1. (a) The dimensionless potential energy (π-πF)/ρgℓ2 composed of the strain energy, gravitational energy, and (where applicable) adhesion energy as T varies from 0 to ρgℓ. (b) The dimensionless potential energy π/ρgℓ2 composed of the strain energy, gravitational energy, and terminal potential as T varies from 0 to ρgℓ. For the results shown, D = 1, ω = 1, and b/ℓ = 0.1. The labels s and u indicate stability and instability, respectively. The force Tc is the critical load for the forked structure.

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Fig. 14

Bifurcation diagram showing the dimensionless potential energies of the CNT pair under a normal load 2 N: Fℓ = -2NE2. (a) The dimensionless potential energy (π-πF)/ρgℓ2 composed of the strain energy, gravitational energy, and (where applicable) adhesion energy as N varies from 0 to 12ρgℓ. (b) The dimensionless potential energy π/ρgℓ2 composed of the strain energy, gravitational energy, and terminal potential as N varies from 0 to 12ρgℓ. For the results shown, D = 1, ω = 1, and b/ℓ = 0.1. The labels s and u indicate stability and instability, respectively. The force Ns is the critical normal buckling load for a single CNT and the force Nc is the critical load for the forked structure.

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