Nonlocal Coefficient Research Articles

磁场环境下输流双壁碳纳米管中波的传播

基于非局部弹性理论，本文研究了磁场环境下输流双壁碳纳米管中弯曲波的传播特性。采用非局部欧拉梁理论，建立磁场作用下输流双壁碳纳米管得波动方程，双壁碳纳米管的两层管之间通过范德华力相互作用。通过绘制不同情况下的色散关系图，分析了管内流体、磁场强度以及非局部参数对输流双壁碳纳米管中波的传播特性的影响，结果表明，磁场对波的传播频率有重要影响，增强磁场将增大波的传播频率，且增强磁场会减弱非局部效应。 In this paper, wave propagation in fluid-conveying double-walled carbon nanotubes under magnetic field is investigated based on nonlocal elasticity theory. The wave equation of fluid-conveying double-walled carbon nanotubes under magnetic field is derived utilizing nonlocal Euler beam theory. The interaction of the double tubes is considered with the van der Waals interaction pres-sure. We analyze the influences of internal fluid, magnetic field intensity and nonlocal coefficient on the wave propagation characteristics of fluid-conveying double-walled carbon nanotubes through the dispersion relations under various circumstances. The results show that the magnetic field plays an important role on the wave propagation frequency. Increasing the magnetic intensity will increase the wave propagation frequency and weaken the nonlocal effect.

Critical buckling load of chiral double-walled carbon nanotube using non-local theory elasticity

The present paper investigate the elastic buckling of chiral double-walled carbon nanotubes (DWCNTs) under axial compression. Using the non-local elasticity theory, Timoshenko beam model has been implemented. According to the governing equations of non-local theory, the analytical solution is derived and the solution for non-local critical buckling loads is obtained. The numerical results show the influence of non-local small-scale coefficient, the vibrational mode number, the chirality of carbon nanotube and aspect ratio of the (DWCNTs) on non-local critical buckling loads of the (DWCNTs). The results indicate the dependence of non-local critical buckling loads on the chirality of single-walled carbon nanotube with increase the non-local small-scale coefficient, the vibrational mode number and aspect ratio of length to diameter.

Investigation of Vibration and Thermal Buckling of Nanobeams Embedded in An Elastic Medium Under Various Boundary Conditions

AbstractAnalyses of free vibration and thermal buckling of nanobeams using nonlocal shear deformation beam theories under various boundary conditions are precisely illustrated. The present beam is restricted by vertically distributed identical springs at the top and bottom surfaces of the beam. The equations of motion are derived using the dynamic version of Hamilton's principle. The governing equations are solved analytically when the edges of the beam are simply supported, clamped or free. Thermal buckling solution is formulated for two types of temperature change through the thickness of the beam: Uniform and linear temperature rise. To validate the accuracy of the results of the present analysis, the results are compared, as possible, with solutions found in the literature. Furthermore, the influences of nonlocal coefficient, stiffness of Winkler springs and span-to-thickness ratio on the frequencies and thermal buckling of the embedded nanobeams are examined.

A note on a nonlinear elliptic problem with a nonlocal coefficient

In this paper, we investigate a nonlocal and nonlinear elliptic problem,(P){−a(∫Ω|∇u|2dx)Δu=λu+up in Ω,u=0 on ∂Ω, where N≤3, Ω⊂RN is a bounded domain with smooth boundary ∂Ω, a is a nondegenerate continuous function, p>1 and λ∈R. We show several effects of the nonlocal coefficient a on the structure of the solution set of (P). We first introduce a scaling observation and describe the solution set by using that of an associated semilinear problem. This allows us to get unbounded continua of solutions (λ,u) of (P). A rich variety of new bifurcation and multiplicity results are observed. We also prove that the nonlocal coefficient can induce up to uncountably many solutions in a convenient way. Lastly, we give some remarks from the variational point of view.

Size-dependent geometrically nonlinear free vibration analysis of fractional viscoelastic nanobeams based on the nonlocal elasticity theory

In recent decades, mathematical modeling and engineering applications of fractional-order calculus have been extensively utilized to provide efficient simulation tools in the field of solid mechanics. In this paper, a nonlinear fractional nonlocal Euler–Bernoulli beam model is established using the concept of fractional derivative and nonlocal elasticity theory to investigate the size-dependent geometrically nonlinear free vibration of fractional viscoelastic nanobeams. The non-classical fractional integro-differential Euler–Bernoulli beam model contains the nonlocal parameter, viscoelasticity coefficient and order of the fractional derivative to interpret the size effect, viscoelastic material and fractional behavior in the nanoscale fractional viscoelastic structures, respectively. In the solution procedure, the Galerkin method is employed to reduce the fractional integro-partial differential governing equation to a fractional ordinary differential equation in the time domain. Afterwards, the predictor–corrector method is used to solve the nonlinear fractional time-dependent equation. Finally, the influences of nonlocal parameter, order of fractional derivative and viscoelasticity coefficient on the nonlinear time response of fractional viscoelastic nanobeams are discussed in detail. Moreover, comparisons are made between the time responses of linear and nonlinear models.

The Spectrum of a Nonlocal Difference Operator with Variable Coefficients

The spectrum of a non-symmetric matrix A is studied, which resulting from difference approximation of the differential operators of the second order with nonlocal boundary conditions and variable coefficients. Such matrices can be both simple and double real eigenvalue and a complex conjugate pairs of eigenvalues. The original matrix A is mapped to the so-called associated matrix L --- symmetric tridiagonal matrix, little differing from A . It is known that all eigenvalues of the matrix L are real and different. They can be found by the bisection method using a sequence of the Sturm's polynomials. In the paper on the basis of numerous examples are compared the properties of the spectra of matrices L and A . The method offers of approximate calculating of eigenvalues of the matrix A .

On a nonlinear thermoelastic system with nonlocal coefficients

This paper deals with the global existence and uniqueness of solutions, and uniform stabilization of the energy of an initial–boundary value problem for a thermoelastic system with nonlinear terms of nonlocal kind. The asymptotic stabilization of the energy of system (1.1) is obtained without any dissipation mechanism acting in the displacement variable u of the membrane equation.

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories. The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory. The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain. The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale. By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory. In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed. It is based on the nonlocal effects of the strain field and first gradient strain field. This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function. The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense. By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach. Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale. To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams. The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory. Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects.

On the Brezis-Nirenberg Problem with a Kirchhoff Type Perturbation

Abstract In this paper, we investigate a Kirchhoff type elliptic problem, where Ω ⊂ ℝ3 is an open ball, λ ∈ ℝ and b ≥ 0. We give an extension of the result by Brezis-Nirenberg in 1983 to the case b > 0. In particular, we can observe several effects of the nonlocal coefficient on the well known results related to the existence, nonexistence and uniqueness.

Backscattering in a 2D topological insulator and the conductivity of a 2D strip

A strip of the 2D HgTe topological insulator is studied. The same-spin edge states in an ideal system propagate in opposite directions on different sides of the strip and do not mix by tunneling. Impurities, edge irregularities, and phonons produce transitions between the counterpropagating edge states on different edges. This backscattering determines the conductivity of an infinitely long strip. The conductivity at finite temperature is determined in the framework of the kinetic equation. It is found that the conductivity exponentially grows with the strip width. In the same approximation the nonlocal resistance coefficients of a four-terminal strip are found.

Analytical double-hybrid density functional based on the polynomial series expansion of adiabatic connection: a quadratic approximation.

We present a systematic derivation of double-hybrid density functional (DHDF) based on the polynomial series expansion of adiabatic connection formula in the closed interval λ = [0,1] without a loss of generality. Because of the tendency of Wλ having a small (but not negligible) curvature at equilibrium, we first evaluate the chemical validity of quadratic approximation for Wλ using the large GMTKN30 benchmark database. The resulting functional, obtained analytically and denoted by quadratic adiabatic connection functional-PT2 (QACF-2), is found to be robust and accurate (2.35 kcal/mol of weighted total mean absolute deviation error, WTMAD), comparable or slightly improved compared to other flavors of existing parameter-free DHDFs (2.45 or 3.29 kcal/mol for PBE0-2 or PBE0-DH, respectively). The nonlocal expansion coefficients obtained for the current QACF-2 (aHF = 2/3, aPT2 = 1/3) also offer some interesting observation, in that the latter analytical coefficients are very similar to the empirically optimized coefficients in some of the best DHDFs today with high accuracy (1.5 kcal/mol). Effects of quadratic truncation in QACF-2 have been further assessed and justified by estimating the higher-order corrections to be as much as 0.54 kcal/mol. The present derivation and numerical experiments suggest that the quadratic λ dependence, despite its simplicity, is a surprisingly good approximation to the adiabatic connection that can serve as a good starting point for further development of accurate parameter-free density functionals.

Levy-type solution for bending of single-layered graphene sheets in thermal environment using the two-variable plate theory

Levy-type solution model is employed to demonstrate the bending response of single-layered graphene sheets subjected to a temperature field as well as external mechanical load. On the basis of Eringen׳s nonlocal elasticity equations incorporated with the two-variable plate theory, the governing differential equations of the thermomechanical response are derived by using the principle of virtual displacement. These equations are converted into a set of first-order linear ordinary differential equations with constant coefficients. The general solution of these equations can be obtained by using the state-space concept. Combinations of simply supported, clamped and free boundary conditions are considered. Comparison of the results with those being in the open literature is made. Numerical results are presented for rectangular sheets with different edge conditions, plate aspect ratios and nonlocal coefficient.

Thermoelectric efficiency of three-terminal quantum thermal machines

The efficiency of a thermal engine working in the linear response regime in a multi-terminal configuration is discussed. For the generic three-terminal case, we provide a general definition of local and non-local transport coefficients: electrical and thermal conductances, and thermoelectric powers. Within the Onsager formalism, we derive analytical expressions for the efficiency at maximum power, which can be written in terms of generalized figures of merit. Furthermore, using two examples, we investigate numerically how a third terminal could improve the performance of a quantum system, and under which conditions non-local thermoelectric effects can be observed.

Existence of Three Solutions for a Degenerate Kirchhoff–Type Transmission Problem

In this article, we establish the existence of three weak solutions for a nonlinear transmission problem involving degenerate nonlocal coefficients of p(x)–Kirchhoff–type. Our approach is of variational nature; the weak formulation takes place in suitable variable exponent Sobolev spaces.

Nonlocal postbuckling analysis of graphene sheets in a nonlinear polymer medium

This article studies nonlocal postbuckling behavior of both uniaxially and biaxially loaded graphene sheets (GSs) in a polymer environment based on an orthotropic nano-plate model. The van der Waals interaction force between the graphene sheets and the polymer medium is considered as a nonlinear function of the graphene’s deflection. Considering the small scale effect and using the thin plate theory, postbuckling equilibrium equations are derived with the von Karman-type geometrical nonlinearity. These equations are solved using the Galerkin method for GSs with different boundary conditions. Finally, postbuckling equilibrium path is plotted and the effects of nonlocal parameter, polymer medium, linear and nonlinear coefficients of the vdW interaction, boundary and loading conditions and geometric properties are studied in detail.

The charge conserving Poisson-Boltzmann equations: Existence, uniqueness, and maximum principle

The present article is concerned with the charge conserving Poisson-Boltzmann (CCPB) equation in high-dimensional bounded smooth domains. The CCPB equation is a Poisson-Boltzmann type of equation with nonlocal coefficients. First, under the Robin boundary condition, we get the existence of weak solutions to this equation. The main approach is variational, based on minimization of a logarithm-type energy functional. To deal with the regularity of weak solutions, we establish a maximum modulus estimate for the standard Poisson-Boltzmann (PB) equation to show that weak solutions of the CCPB equation are essentially bounded. Then the classical solutions follow from the elliptic regularity theorem. Second, a maximum principle for the CCPB equation is established. In particular, we show that in the case of global electroneutrality, the solution achieves both its maximum and minimum values at the boundary. However, in the case of global non-electroneutrality, the solution may attain its maximum value at an interior point. In addition, under certain conditions on the boundary, we show that the global non-electroneutrality implies pointwise non-electroneutrality.

Buckling Analysis of Chiral Single-Walled Carbon Nanotubes by Using the Nonlocal Timoshenko Beam Theory

On the basis of the nonlocal elasticity theory, the Timoshenko beam model is utilized to investigate the elastic buckling of chiral single-walled carbon nanotubes (SWCNTs) under axial compression. Based on the governing equations of the nonlocal Timoshenko beam model, an analytical solution for nonlocal critical buckling loads is obtained. The influence of a nonlocal small-scale coefficient, the vibration mode number, the chirality of SWWCNTs, and their aspect ratio on the nonlocal critical buckling loads is studied and discussed.

Generalized two-variable plate theory for multi-layered graphene sheets with arbitrary boundary conditions

In the present study, the free vibration, mechanical buckling and thermal buckling analyses of multi-layered graphene sheets (MLGSs) are investigated. Eringen’s nonlocal elasticity equations are incorporated in new two-variable plate theories accounting for small-scale effects. The MLGSs are taken to be perfectly bonded to the surrounding medium. The governing differential equations of this model are solved analytically under various boundary conditions and taking into account the effect of van der Waals forces between adjacent layers. New functions for the displacements are proposed here to satisfy the different boundary conditions. Comparison of the results with those being in the open literature is made. This comparison illustrates that the present scheme yields very accurate results. Furthermore, the influences of nonlocal coefficient, moduli of the surrounding elastic medium and aspect ratio on the frequencies and buckling of the embedded MLGSs are examined.

Nonlocal Problems Arising in Thermoelectrics

We revisit the derivation of a nonlocal problem modelling temperature distribution due to power generation using thermoelectric effect. The problem has nonlocal coefficients in reaction and convection terms rather than diffusion term, which makes the problem more interesting. In this paper, we prove that the nonlocal problem has a unique decreasing solution when electric current induced by the thermoelectric effect is small enough.

Fracture and buckling of piezoelectric nanowires subject to an electric field

Fracture and buckling are major failure modes of thin and long nanowires (NWs), which could be affected significantly by an electric field when piezoelectricity is involved in the NWs. This paper aims to examine the issue based on the molecular dynamics simulations, where the gallium nitride (GaN) NWs are taken as an example. The results show that the influence of the electric field is strong for the fracture and the critical buckling strains, detectable for the fracture strength but almost negligible for the critical buckling stress. In addition, the reversed effects are achieved for the fracture and the critical buckling strains. Subsequently, the Timoshenko beam model is utilized to account for the effect of the electric field on the axial buckling of the GaN NWs, where nonlocal effect is observed and characterized by the nonlocal coefficient e0a=1.1 nm. The results show that the fracture and buckling of piezoelectric NWs can be controlled by applying an electric field.