Thermal and surface effects on the pull-in characteristics of circular nanoplate NEMS actuator based on nonlocal elasticity theory

This paper aims to investigate the coupling influences of thermal loading and surface effects on pull-in instability of electrically actuated circular nanoplate based on Eringen's nonlocal elasticity theory, where the electrostatic force and thermally corrected Casimir force are considered. By utilizing the Kirchhoff plate theory, the nonlinear equilibrium equation of axisymmetric circular nanoplate with variable coefficients and clamped boundary conditions is derived and analytically solved. The results describe the influences of surface effect and thermal loading on pull-in displacements and pull-in voltages of nanoplate under thermal corrected Casimir force. It is seen that the surface effect becomes significant at the pull-in state with the decrease of nanoplate thicknesses, and the residual surface tension exerts a greater influence on the pull-in behavior compared to the surface elastic modulus. In addition, it is found that temperature change plays a great role in the pull-in phenomenon; when the temperature change grows, the circular nanoplate without applied voltage is also led to collapse.     

Long time existence for a slightly perturbed vortex sheet

Consider a flat two-dimensional vortex sheet perturbed initially by a small analytic disturbance. By a formal perturbation analysis, Moore derived an approximate differential equation for the evolution of the vortex sheet. We present a simplified derivation of Moore's approximate equation and analyze errors in the approximation. The result is used to prove existence of smooth solutions for long time. If the initial perturbation is of size ? and is analytic in a strip |??m γ| < ρ, existence of a smooth solution of Birkhoff's equation is shown for time t < k2p, if ? is sufficiently small, with κ → 1 as ? → 0. For the particular case of sinusoidal data of wave length π and amplitude e, Moore's analysis and independent numerical results show singularity development at time t^c = |log ?| + O(log|log ?|. Our results prove existence for t < κ|log ?|, if ? is sufficiently small, with k κ → 1 as ? → 0. Thus our existence results are nearly optimal.     

Non-linear bending analysis of multi-layer orthotropic annular/circular graphene sheets embedded in elastic matrix in thermal environment based on non-local elasticity theory

In this paper, the large deflection of multi-layer orthotropic annular/circular graphene sheets is investigated based on the non-local elasticity theory. The plate is considered to be in thermal environment. The equilibrium equations are derived in terms of generalized displacements and rotations considering the FSDT non-local elasticity theory and the van der Waals interaction between the layers. In order to solve the governing equations, the differential quadrature method (DQM) which is an accurate numerical method and a new semi-analytical polynomial method (SAPM) are applied. By applying DQM or SAPM, the ODE's would be converted to non-linear algebraic equations. In continue, the Newton–Raphson iterative scheme is applied to solve the obtained non-linear algebraic equations. The results of DQM and SAPM are compared. Although, the SAPM's formulation is considerably simpler than DQM, however, the results of two methods are so close to each other. The results are validated with the other available researches. The effect of small scale parameter, temperature effects on non-local results, the value of van der Waals interaction between the layers for bi-layer and triple layers graphene sheet, different values of elastic foundation matrix and load for various small scale parameters, the comparison between local and non-local deflections and linear to non-linear analysis are investigated.     

Geometrically nonlinear analysis of Timoshenko piezoelectric nanobeams with flexoelectricity effect based on Eringen's differential model

This study analyzes the nonlinear free vibration and post-buckling of nanobeams with flexoelectric effect based on Eringen's differential model. The nanobeam is modeled based on Timoshenko beam's theory. The von-Kármán strain–displacement relation together with the electrical Gibbs free energy and Hamilton's principle are employed to derive equations of motion. The nonlinear free vibration frequencies are obtained for pinned–pinned (P–P) and clamped–clamped (C–C) boundary conditions. Multiple scales method is employed to obtain the closed-form solution for the nonlinear governing equations. By employing this methodology, the natural frequencies of nanobeams are obtained and their post-buckling behavior is examined. The influence of nonlocal parameter, amplitude ratio, and input voltage on the top surface and flexoelectricity constant on nonlinear free vibration and post-buckling characteristics of nanobeam is investigated. In this paper, it is concluded that the flexoelectricity has a significant effect on free vibration of the beams in nano-scale and its effect has to be considered in designing nano-electro-mechanical systems (NEMS) such as nano- generators and nano-sensors.     

On an isoperimetric inequality for a Schrödinger operator depending on the curvature of a loop

Let y be a smooth closed curve of length 2π in ?³, and let κ(s) be its curvature, regarded as a function of arc length. We associate with this curve the one-dimensional Schrödinger operator $H_\gamma = - \tfrac{{d^2 }}{{ds^2 }} + \kappa ^2 (s)$ acting on the space of square integrable 2π-periodic functions. A natural conjecture is that the lowest spectral value e₀ (y) of H_y is bounded below by 1 for any y (this value is assumed when y is a circle). We study a family of curves y that includes the circle and for which e₀(y) = 1 as well. We show that the curves in this family are local minimizers, i.e., e₀(y) can only increase under small perturbations leading away from the family. To our knowledge, the full conjecture remains open.     

Modeling of functionally graded circular energy harvesters due to flexoelectricity

Flexoelectric effect can be enhanced in micro/nano scale due to its size-dependency, making it particularly suitable for energy harvesting. In this work, a theoretical model is built to characterize the functionally graded circular flexoelectric energy harvesters based on the Kirchhoff thin plate hypothesis. Using Hamilton's principle, both the force balance equation and current balance equation are obtained. Approximated closed-form solutions of the energy harvesting performances are achieved through the assumed-mode method. Numerical analysis results demonstrate that the clamped circular energy harvesters with the ratio of the electrode radius to the plate radius be 0.64 will generate the maximum electrical output. The volume fraction coefficient has a significant impact on the resonant frequency, electrical output as well as the optimal load resistance. Meanwhile, shrinking the thickness of the circular energy harvester from 10µm to 0.1µm will lead to a remarkable increase of the optimal energy conversion efficiency from 10⁻⁶ to 10⁻². Furthermore, the strain gradient effect is examined to result in a higher resonant frequency while suppress the electrical output particularly if the length scale parameter is relatively large.     

Vibration analysis of Nano-Rotor's Blade applying Eringen nonlocal elasticity and generalized differential quadrature method

This study investigates the small scale effect on the flapwise bending vibrations of a rotating nanoplate. The nanoplate is modeled with a classical plate theory and considering cantilever and propped cantilever boundary conditions. Due to the rotation, the axial forces are included in the model as true spatial variation. Hamilton's principle is used to derive the governing equation and boundary conditions of the classical plate theory based on Eringen's nonlocal elasticity theory. The generalized differential quadrature method is employed to solve the governing equation. The effect of small-scale parameter, non-dimensional angular velocity, non-dimensional hub radius, aspect ratio, and different boundary conditions in the first four non-dimensional frequencies is discussed. Due to considering rotating effects, results of this study are applicable in nano-machines such as nano-motors and nano-turbines and other nanostructures.     

On the characterization of trees with signed edge domination numbers 1, 2, 3, or 4

Let G=(V,E) be a simple graph. For an edge e of G, the closed edge-neighbourhood of e is the set N[e]={e^′∈E|e^′ is adjacent to e}∪{e}. A function f:E→{1,−1} is called a signed edge domination function (SEDF) of G if ∑_e^′∈N[e]f(e^′)≥1 for every edge e of G. The signed edge domination number of G is defined as . In this paper, we characterize all trees T with signed edge domination numbers 1, 2, 3, or 4.     

Régularité Besov-Orlicz du temps local brownien

Let (B_t, t ε [0, 1]) be a linear Brownian motion starting from 0 and denote (L_t(x), t ≥0, x ∈ ℝ) its local time. We prove that, for all t > 0, a.s. the function xL_t (x) belongs to the Besov-Orlicz space B^½M_{2, ∞} with M₂(x)=e^|x|2 -1 and doesn't belong a.s. to B^½,0M_{2, ∞}.     

Estimation of flows in flow networks

Let G be a directed graph with an unknown flow on each edge such that the following flow conservation constraint is maintained: except for sources and sinks, the sum of flows into a node equals the sum of flows going out of a node. Given a noisy measurement of the flow on each edge, the problem we address, which we call the Most Probable Flow Estimation problem (MPFE), is to estimate the most probable assignment of flow for every edge such that the flow conservation constraint is maintained. We provide an algorithm called ΔY-mpfe for solving the MPFE problem when the measurement error is Gaussian (Gaussian-MPFE). The algorithm works in O(∣E∣ + ∣V∣²) when the underlying undirected graph of G is a 2-connected planar graph, and in O(∣E∣ + ∣V∣) when it is a 2-connected serial-parallel graph or a tree. This result is applicable to any Minimum Cost Flow problem for which the cost function is τ_e(X_e − μ_e)² for edge e where μ_e and τ_e are constants, and X_e is the flow on edge e. We show that for all topologies, the Gaussian-MPFE’s precision for each edge is analogous to the equivalent resistance measured in series to this edge in an electrical network built by replacing every edge with a resistor reflecting the measurement’s precision on that edge.     

The well-posedness of a nonlocal hyperbolic model for type-I superconductors

A vectorial nonlocal linear hyperbolic problem with applications in superconductors of type-I is studied. The nonlocal term is represented by a (space) convolution with a singular kernel, which is arising in Eringen's model. The well-posedness of the problem is discussed under low regularity assumptions and the error estimates for two time-discrete schemes (based on backward Euler approximation) are established.     

A compactness result for Landau state in thin-film micromagnetics

We deal with a nonconvex and nonlocal variational problem coming from thin-film micromagnetics. It consists in a free-energy functional depending on two small parameters ε and η and defined over vector fields m:Ω⊂R²→S² that are tangent at the boundary ∂Ω. We are interested in the behavior of minimizers as ε,η→0. They tend to be in-plane away from a region of length scale ε (generically, an interior vortex ball or two boundary vortex balls) and of vanishing divergence, so that S¹-transition layers of length scale η (Néel walls) are enforced by the boundary condition. We first prove an upper bound for the minimal energy that corresponds to the cost of a vortex and the configuration of Néel walls associated to the viscosity solution, so-called Landau state. Our main result concerns the compactness of vector fields {mε,η}_ε,η↓0 of energies close to the Landau state in the regime where a vortex is energetically more expensive than a Néel wall. Our method uses techniques developed for the Ginzburg-Landau type problems for the concentration of energy on vortex balls, together with an approximation argument of S²-vector fields by S¹-vector fields away from the vortex balls.     

Calculation of a static potential created by plane fractal cluster

In this paper we demonstrate new approach that can help in calculation of electrostatic potential of a fractal (self-similar) cluster that is created by a system of charged particles. For this purpose we used the simplified model of a plane dendrite cluster [1] that is generated by a system of the concentric charged rings located in some horizontal plane (see Fig. 2). The radiuses and charges of the system of concentric rings satisfy correspondingly to relationships: r_n = r₀ξⁿ and e_n = e₀bⁿ, where n determines the number of a current ring. The self-similar structure of the system considered allows to reduce the problem to consideration of the functional equation that similar to the conventional scaling equation. Its solution represents itself the sum of power-low terms of integer order and non-integer power-law term multiplied to a log-periodic function [5], [6]. The appearance of this term was confirmed numerically for internal region of the self-similar cluster (r₀ ≪ r ≪ r_N−1), where r₀, r_N−1 determine the smallest and the largest radiuses of the limiting rings correspondingly. The results were obtained for homogeneously (b > 0) and heterogeneously (b < 0) charged rings. We expect that this approach allows to consider more complex self-similar structures with different geometries of charge distributions.     

Homogenization of the Stokes equations with high-contrast viscosity

This paper deals with the homogenization of the Stokes equations in a cylinder with varying viscosity and with Dirichlet boundary condition. The viscosity is equal to α_ε⪢1 in a ε-periodic lattice of unidirectional cylinders of radius εr_ε where r_ε⪡1, and is equal to 1 elsewhere.In the critical regime defined by lim_ε→0ε²|lnr_ε|∈]0,+∞[ and lim_ε→0α_εr_ε²∈]0,+∞], the limit problem is a coupled Stokes system satisfied by the limit velocity and the limit of the rescaled velocity in the cylinders, which can be read as a nonlocal law of Brinkman type. Moreover, if lim_ε→0α_εr_ε²=+∞, the limit of the rescaled velocity is equal to 0 and the Brinkman law is derived as in [G. Allaire, Arch. Rational Mech. Anal. 13 (1991) 209–259]. In the other regimes the homogenization leads either to classical Stokes problems or to a zero limit velocity.In the critical case the pressure is not bounded in L² but only in H⁻¹. Moreover, the pressure of the limit problem is not equal to the weak limit of the pressure in H⁻¹.     

New properties of King’s operators

We prove the existence of a sequence of King’s operators which approximate each continuous function on [0, 1] and preserve the functions e ₀(x) = 1 and e _2i (x) = x ²ⁱ . Moreover, we construct a sequence of polynomial bounded positive linear operators possessing similar properties.     

A method of constructing integrable linear equations and its application to Hill's equation

Starting with a given equation of the form $$\ddot x + [\lambda + \varepsilon f(t)] x = 0$$ , where λ > 0 and ? ? l is a small parameter [heref(t) may be periodic, and so Hill's equation is included], we construct an equation of the form y + [λ + ?f (t) + ?² _g (t)]y = 0, integrable by quadratures, close in a certain sense to the original equation. For x₀ = y₀ and x ₀ ^′ = y ₀ ^′ , an upper bound is obtained for ¦y—x¦ on an interval of length Δt.     

An alternative approach to unitoidness

In a recent paper [C.R. Johnson, S. Furtado, A generalization of Sylvester’s law of inertia, Linear Algebra Appl. 338 (2001) 287-290], Sylvester’s law of inertia is generalized to any matrix that is ∗-congruent to a diagonal matrix. Such a matrix is called unitoid. In the present paper, an alternative approach to the subject of unitoidness is offered. Specifically, Sylvester’s law of inertia states that a Hermitian n × n matrix of rank r with inertia (p, q, n − r) is ∗-congruent to the direct sum

eⁱ⁰I_p⊕e^iπI_q⊕0I_n-r.     

Elastic buckling and vibration analyses of orthotropic nanoplates using nonlocal continuum mechanics and spline finite strip method

This paper addresses the elastic buckling and vibration characteristics of isotropic and orthotropic nanoplates using finite strip method. In order to consider small scale effect, Eringen’s nonlocal continuum elasticity is employed. The governing nanoplate equations are derived using the principle of virtual work while B3-spline finite strip method is applied to the buckling and vibration analyses. The buckling load and vibration frequency of graphene sheets, which are subjected to biaxial compression and pure shear loading, are determined whilst the effects of different parameters such as sheet size, nonlocal parameter, aspect ratio and boundary conditions are investigated. The interaction curves of the critical biaxial compression loading as well as the interaction curves of the critical uniaxial compression and shear loading are also obtained. It is shown that small scale effect plays considerable role in the analysis of small sizes plates.     

Emission spectrum of bromine excited in the presence of argon

The wavelengths and wavenumbers of the band heads of the system 3150–2970 Å as obtained from the plates taken on the first order 21′ grating spectrograph are given along with the vibrational analysis. This system is shown to be due to a transition from an upper electronic state at T_e = 48516 cm.^-1 with ω′ _e = 162·0 cm.^?1 and ω′ _e χ′ _e = 0·29 cm.^?1 to the well-known³ Π _u (O _u ⁺) state at T_e = 15918 cm.^-1 This lower state is common with that of the system 2950–2670 Å.