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Research Papers

Nonlinear Dynamic Response of Carbon Nanotube Nanocomposite Microbeams

[+] Author and Article Information
Marek Cetraro

Department of Structural and
Geotechnical Engineering,
Sapienza University of Rome,
Rome 00184, Italy

Walter Lacarbonara

Department of Structural and
Geotechnical Engineering,
Sapienza University of Rome,
Rome 00184, Italy
e-mail: walter.lacarbonara@uniroma1.it

Giovanni Formica

Department of Architecture,
University of Roma Tre,
Rome 00146, Italy

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 19, 2016; final manuscript received August 15, 2016; published online December 5, 2016. Assoc. Editor: Mohammad Younis.

J. Comput. Nonlinear Dynam 12(3), 031007 (Dec 05, 2016) (9 pages) Paper No: CND-16-1089; doi: 10.1115/1.4034736 History: Received February 19, 2016; Revised August 15, 2016

The nonlinear dynamic response of nanocomposite microcantilevers is investigated. The microbeams are made of a polymeric hosting matrix (e.g., epoxy, polyether ether ketone (PEEK), and polycarbonate) reinforced by longitudinally aligned carbon nanotubes (CNTs). The 3D transversely isotropic elastic constitutive equations for the nanocomposite material are based on the equivalent inclusion theory of Eshelby and the Mori–Tanaka homogenization approach. The beam-generalized stress resultants, obtained in accordance with the Saint-Venant principle, are expressed in terms of the generalized strains making use of the equivalent constitutive laws. These equations depend on both the hosting matrix and CNTs elastic properties as well as on the CNTs volume fraction, geometry, and orientation. The description of the geometry of deformation and the balance equations for the microbeams are based on the geometrically exact Euler–Bernoulli beam theory specialized to incorporate the additional inextensibility constraint due to the relevant boundary conditions of microcantilevers. The obtained equations of motion are discretized via the Galerkin method retaining an arbitrary number of eigenfunctions. A path following algorithm is then employed to obtain the nonlinear frequency response for different excitation levels and for increasing volume fractions of carbon nanotubes. The fold lines delimiting the multistability regions of the frequency responses are also discussed. The volume fraction is shown to play a key role in shifting the linear frequencies of the beam flexural modes to higher values. The CNT volume fraction further shifts the fold lines to higher excitation amplitudes, while it does not affect the backbones of the modes (i.e., oscillation frequency–amplitude curves).

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Figures

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Fig. 1

Carbon nanotube nanocomposite microbeam model. The arclength along the baseline is denoted by s, while the displacement vector u(s,t)=u(s,t)e1+v(s,t)e2 describes the current position of the base point of the cross sections.

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Fig. 2

Variation of the lowest five nondimensional microbeam linear frequencies with CNTs volume fraction. Bold lines indicate epoxy, while dotted-dashed lines represent PEEK.

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Fig. 3

Variation of the lowest five-dimensional microbeam linear frequencies with CNTs volume fraction. Bold lines indicate epoxy, while dotted-dashed lines represent PEEK.

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Fig. 4

Convergence study for a nanocomposite microcantilever with ϕC=0.1%. Comparison between the frequency-response curves obtained by using one (solid line), two (dashed line), and three (dotted line) trial functions. The nondimensional excitation amplitude is F = 0.109, ω¯ is the first natural frequency of an all-matrix cantilever, and Ω is the excitation frequency.

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Fig. 5

Convergence study for a nanocomposite microcantilever with ϕC=0.1%. Comparison between the frequency-response curves obtained with the third- (solid line) and fifth-order (dashed line) Taylor-expanded equations. One trial function is employed. The nondimensional load is F = 0.1091, ω¯ is the first natural frequency of a cantilever not reinforced by CNTs, and Ω is the excitation frequency.

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Fig. 6

Frequency-response curves of a microcantilever with ϕC=0.1% subject to a primary resonance of (a) the first mode, (b) the second mode, and (c) the third mode, varying the excitation amplitude. The excitation frequency is rescaled by the linear frequency of the excited mode. The dashed lines denote the backbones.

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Fig. 7

Backbone curves of the first, second, and third modes of microcantilevers with the increasing CNT volume fractions ϕC=(0.1, 0.5, 1, 1.5, 2.5, 5, 10)%, from left to right

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Fig. 8

Fold lines delimiting the instability regions of the second mode of nanocomposite microcantilevers with the increasing CNT volume fractions, ϕC=(0.1,0.5,1,1.5,2.5)%, from left to right

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Fig. 9

(Left) Frequency-response curves of the first mode of the microcantilevers with various CNT volume fractions. (Right) Details of the frequency-response curves within the range (Ω/ω¯)∈ [1,2] assuming ϕC=(0.1%, 0.5%, 1%, 1.5%, 2.5%, 5%, 7.5%, 10%). The nondimensional load amplitude is F = 0.108, ω¯ is the fundamental frequency of the all-matrix cantilever, and Ω is the excitation frequency. The dashed-dotted line describes the linear resonance peaks.

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Fig. 10

(Left) Frequency-response curves of the second mode of the microcantilevers with various CNT volume fractions. (Right) Details of the frequency-response curves within the range Ω/ω¯ ∈ [1,2] assuming ϕC=(0.1%, 0.5%, 1%, 1.5%, 2.5%, 5%, 7.5%, 10%). The nondimensional load amplitude is F = 0.069, ω¯ is the fundamental frequency of the all-matrix cantilever, and Ω is the excitation frequency. The dotted-dashed line describes the linear resonance peaks.

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Fig. 11

(Left) Frequency-response curves of the third mode of the microcantilevers with various CNT volume fractions. (Right) Details of the frequency-response curves within the range (Ω/ω¯)∈ [1,2] assuming ϕC=(0.1%, 0.5%, 1%, 1.5%, 2.5%, 5%, 7.5%, 10%). The nondimensional load amplitude is F = 0.051, ω¯ is the fundamental frequency of the all-matrix cantilever, and Ω is the excitation frequency. The dotted-dashed line describes the linear resonance peaks.

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