Eigenproblems in nanomechanics

Abstract

Abstract. The paper is semitutorial in nature to make it accessible to readers from a broad range of disciplines. Our particular focus is on cataloging the known problems in nanomechanics as eigenproblems. Physical insights obtained from both analytical results and numerical simulations of various researchers (including our own) are also discussed. The paper is organized in two broad sections. In the second section the attention is focused on the analysis of quantum dots. The analysis of electronic properties of strained semiconductor structures is reduced here to the solution of a linear boundary value problem (the classical Helmholtz wave equation). In Sec 3, we provide, intermixed with a literature review, details on various methods and issues in calculation free vibrations/loss of stability for carbon nanotubes. The effect of various parameters associated with the material anisotropy are addressed. Typically classical continuum mechanics, which is intrinsically size independent, is employed for calculations.Key words: nanomechanics, carbon nanotubes, free vibrations, buckling, shells, beams. Notation:Latin symbols M -the bending moment in the beam, m -the free electron mass, mα(λE) -the electron effective mass, n, m -the circumferential and longitudinal wavenumbers, respectively, Pi -the momentum matrix element, qα -the vector of the unknown mechanical response of the structure, i.e. the system of generalized displacements, r -the position vector throughout the solid, R -the cylindrical shell (carbon nanotube) radius, V -the interatomic potential, Vα -the confinement potential, V αβ -the effective potential field coupling energy bands α and β, V αβband ( r) -the total potential due to the energy misalignment of the valence band maxima, V αβstrain ( r) -the potential due to the elastic strain ε γη ( r) that shifts and couples the energy bands, x -the fractional content of alloying material. Greek symbolsδi -the spin-orbit splitting in the valence band, ε γη ( r) -the elastic strain tensor, θi -the i-th bond angle, λ -the eigenvalue of the covariance matrix Σ αβ , λ = λE -the eigenvalue for quantum mechanics models (the energy of a particular quantum mechanical state), * e-mail: olekmuc@mech.pk.edu.pl 819 Unauthenticated Download Date | 5/13/18 2:38 AM A. Muc and A. Banaś λ = λ b -the eigenvalue for classical mechanics models (buckling or free vibrations), ν12, ν13, ν23 -the Poisson ratios, ρ -the shell/nanotube density, Σ αβ -the covariance matrix, Σ αβ = H αβ -for quantum mechanics models, Σ αβ = K αβ -for continuum mechanics models, ϕα -the eigenvector of the covariance matrix Σ αβ , ϕα = ψ α -for quantum mechanics models, ϕα = qα -for continuum mechanics models, ψ α -the quantum mechanical wave function associated with energy band β, ω -the angular velocity, Ω -the space occupied by the semiconductor, h -the reduced Planck constant. Chemical elementsAs -aresnic, Ga -gallium, In -indium.

Bulletin of the Polish Academy of Sciences: Technical Sciences

“…a model for the DNA writhing [12,13]. It is also demonstrated that for a small radius of the nanotubes the buckling mode falls into the regime of Euler beam buckling [14].…”

“…Microscopy measurements have significant meaning in examining the buckling of nanostructures. For instance, atomic force microscopy (AFM) is used to obtain the force-distance curve of nanotubes while buckling [9,14,15]. In situ transmission electron microscopy (TEM) also provides information on the characterisation of buckling [10].…”

“…However, the deflection measurement in various states of post-buckling is not calculated. Pathak et al reported on the mechanical behaviour of a dense brush of small-diameter (1-3 nm) carbon nanotubes (CNTs) [14].…”