Love wave propagation in one-dimensional piezoelectric quasicrystal multilayered nanoplates with surface effects

The exact solutions for the propagation of Love waves in one-dimensional(1D)hexagonal piezoelectric quasicrystal(PQC)nanoplates with surface effects are derived.An electro-elastic model is developed to investigate the anti-plane strain problem of Love wave propagation.By introducing three shape functions,the wave equations and electric balance equations are decoupled into three uncorrelated problems.Satisfying the boundary conditions of the top surface on the covering layer,the interlayer interface,and the matrix,a dispersive equation with the influence of multi-physical field coupling is provided.A surface PQC model is developed to investigate the surface effects on the propagation behaviors of Love waves in quasicrystal(QC)multilayered structures with nanoscale thicknesses.A novel dispersion relation for the PQC structure is derived in an explicit closed form according to the non-classical mechanical and electric boundary conditions.Numerical examples are given to reveal the effects of the boundary conditions,stacking sequence,characteristic scale,and phason fluctuation characteristics on the dispersion curves of Love waves propagating in PQC nanoplates with surface effects.

Explicit frequency equations of free vibration of a nonlocal Timoshenko beam with surface effects

Considerations of nonlocal elasticity and surface effects in micro-and nanoscale beams are both important for the accurate prediction of natural frequency. In this study, the governing equation of a nonlocal Timoshenko beam with surface effects is established by taking into account three types of boundary conditions： hinged–hinged, clamped–clamped and clamped–hinged ends. For a hinged–hinged beam, an exact and explicit natural frequency equation is obtained. However, for clamped–clamped and clamped–hinged beams, the solutions of corresponding frequency equations must be determined numerically due to their transcendental nature. Hence, the Fredholm integral equation approach coupled with a curve fitting method is employed to derive the approximate fundamental frequency equations, which can predict the frequency values with high accuracy. In short,explicit frequency equations of the Timoshenko beam for three types of boundary conditions are proposed to exhibit directly the dependence of the natural frequency on the nonlocal elasticity, surface elasticity, residual surface stress, shear deformation and rotatory inertia, avoiding the complicated numerical computation.

Influences of surface effects and large deformation on the resonant properties of ultrathin silicon nanocantilevers

The purpose of the present work is to quantify the influences of the discrete nature, the surface effects, and the large deformation on the bending resonant properties of long and ultrathin （100） silicon nanocantilevers. We accomplish this by using an analytical semi-continuum Keating model within the framework of nonlinear, finite deformation kinematics. The semi-continuum model shows that the elastic behaviors of the silicon nanocantilevers are size-dependent and surface- dependent, which agrees well with the molecular dynamics results. It also indicates that the dominant effect on the fundamental resonant frequency shift of the silicon nanocantilever is adsorption-induced surface stress, followed by the discrete nature and surface reconstruction, whereas surface relaxation has the least effect. In particular, it is found that a large deformation tends to increase the nonlinear fundamental frequency of the silicon nanocantilever, depending not only on its size but also on the surface effects. Finally, the resonant frequency shifts due to the adsorption-induced surface stress predicted by the current model are quantitatively compared with those obtained from the experimental measurement and the other existing approach. It is noticed that the length-to-thickness ratio is the key parameter that correlates the deviations in the resonant frequencies predicted from the current model and the empirical formula.

Surface effects on a mode-III reinforced nano-elliptical hole embedded in one-dimensional hexagonal piezoelectric quasicrystals

To effectively reduce the field concentration around a hole or crack,an anti-plane shear problem of a nano-elliptical hole or a nano-crack pasting a reinforcement layer in a one-dimensional(1D)hexagonal piezoelectric quasicrystal(PQC)is investigated subject to remotely mechanical and electrical loadings.The surface effect and dielectric characteristics inside the hole are considered for actuality.By utilizing the technique of conformal mapping and the complex variable method,the phonon stresses,phason stresses,and electric displacements in the matrix and reinforcement layer are exactly derived under both electrically permeable and impermeable boundary conditions.Three size-dependent field intensity factors near the nano-crack tip are further obtained when the nano-elliptical hole is reduced to the nano-crack.Numerical examples are illustrated to show the effects of material properties of the surface layer and reinforced layer,the aspect ratio of the hole,and the thickness of the reinforcing layer on the field concentration of the nano-elliptical hole and the field intensity factors near the nano-crack tip.The results indicate that the properties of the surface layer and reinforcement layer and the electrical boundary conditions have great effects on the field concentration of the nano-hole and nano-crack,which are useful for optimizing and designing the microdevices by PQC nanocomposites in engineering practice.

Electro-mechanical coupling wave propagating in a locally resonant piezoelectric/elastic phononic crystal nanobeam with surface effects

The model of a "spring-mass" resonator periodically attached to a piezoelectric/elastic phononic crystal(PC) nanobeam with surface effects is proposed, and the corresponding calculation method of the band structures is formulized and displayed by introducing the Euler beam theory and the surface piezoelectricity theory to the plane wave expansion(PWE) method. In order to reveal the unique wave propagation characteristics of such a model, the band structures of locally resonant(LR) elastic PC Euler nanobeams with and without resonators, the band structures of LR piezoelectric PC Euler nanobeams with and without resonators, as well as the band structures of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on PZT-4, with resonators attached on epoxy, and without resonators are compared. The results demonstrate that adding resonators indeed plays an active role in opening and widening band gaps. Moreover, the influence rules of different parameters on the band gaps of LR elastic/piezoelectric PC Euler nanobeams with resonators attached on epoxy are discussed, which will play an active role in the further realization of active control of wave propagations.

Study on wave dispersion characteristics of piezoelectric sandwich nanoplates considering surface effects

In this study,the wave propagation properties of piezoelectric sandwich nanoplates deposited on an orthotropic viscoelastic foundation are analyzed by considering the surface effects(SEs).The nanoplates are composed of a composite layer reinforced by graphene and two piezoelectric surface layers.Utilizing the modified Halpin-Tsai model,the material parameters of composite layers are obtained.The displacement field is determined by the sinusoidal shear deformation theory(SSDT).The Euler-Lagrange equation is derived by employing Hamilton’s principle and the constitutive equations of piezoelectric layers considering the SEs.Subsequently,the nonlocal strain gradient theory(NSGT)is used to obtain the equations of motion.Next,the effects of scale parameters,graphene distribution,orthotropic viscoelastic foundation,and SEs on the propagation behavior are numerically examined.The results reveal that the wave frequency is a periodic function of the orthotropic angle.Furthermore,the wave frequency increases with the increase in the SEs.

Bending of functionally graded nanobeams incorporating surface effects based on Timoshenko beam model

The bending responses of functionally graded （FG） nanobeams with simply supported edges are investigated based on Timoshenko beam theory in this article. The Gurtin-Murdoch surface elasticity theory is adopted to analyze the influences of surface stress on bending response of FG nanobeam. The material properties are assumed to vary along the thickness of FG nanobeam in power law. The bending governing equations are derived by using the minimum total potential energy principle and explicit formulas are derived for rotation angle and deflection of nanobeams with surface effects. Illustrative examples are implemented to give the bending deformation of FG nanobeam. The influences of the aspect ratio, gradient index, and surface stress on dimensionless deflection are discussed in detail.

Propagation of Rayleigh-type surface waves in a layered piezoelectric nanostructure with surface effects

This work investigates the dispersion properties of Rayleigh-type surface waves propagating in a layered piezoelectric nanostructure composed of a piezoelectric nanofilm over an elastic substrate.As one of the most important features of nanostructures,surface effects characterized by surface stresses and surface electric displacements are taken into account through the surface piezoelectricity theory and the nonclassical mechanical and electrical boundary conditions.Concrete expressions of the dispersion equation are derived,and numerical results are provided to examine the effects of several surface-related parameters,including the surface elasticity,surface piezoelectricity,surface dielectricity,surface density,as well as surface residual stress,on the dispersion modes and phase velocity.The size-dependent dispersion behaviors occurring with surface effects are also predicted,and they may vanish once the thickness of the piezoelectric nanofilm reaches a critical value.

Surface effects on mechanical behavior of elastic nanoporous materials under high strain

This paper studies surface effects on the mechanical behavior of nanoporous materials under high strains with an improved anisotropic Kelvin model. The stress-strain relations are derived by the theories of Euler-Bernoulli beam and surface elasticity. Mean- while, the influence of strut （or ligament） size on the mechanical properties of nanoporous materials is discussed, which becomes a key factor with consideration of the residual sur- face stress and the surface elasticity. The results show that the decrease in the strut diameter and the increase in the residual surface stress or the surface elasticity can both lead to an increase in the carrying capacity of nanoporous materials. F^lrthermore, me- chanical behaviors of anisotropic nanoporous materials in different directions （the rise direction and the transverse direction） are investigated. The results indicate that the sur- face effects in the transverse direction are more obvious than those in the rise direction for anisotropic nanoporous materials. In addition, the present results can be reduced to the cases of conventional foams as the strut size increases to micron-scale, which confirms validity of the model to a certain extent.

Frequency equations of nonlocal elastic micro/nanobeams with the consideration of the surface effects

A nonlocal elastic micro/nanobeam is theoretically modeled with the consideration of the surface elasticity, the residual surface stress, and the rotatory inertia,in which the nonlocal and surface effects are considered. Three types of boundary conditions, i.e., hinged-hinged, clamped-clamped, and clamped-hinged ends, are examined. For a hinged-hinged beam, an exact and explicit natural frequency equation is derived based on the established mathematical model. The Fredholm integral equation is adopted to deduce the approximate fundamental frequency equations for the clamped-clamped and clamped-hinged beams. In sum, the explicit frequency equations for the micro/nanobeam under three types of boundary conditions are proposed to reveal the dependence of the natural frequency on the effects of the nonlocal elasticity, the surface elasticity, the residual surface stress, and the rotatory inertia, providing a more convenient means in comparison with numerical computations.

MECHANISMS UNDERLYING TWO KINDS OF SURFACE EFFECTS ON ELASTIC CONSTANTS

Recently, people are confused with two opposite variations of elastic modulus with decreasing size of nano scale sample： elastic modulus either decreases or increases with decreasing sample size. In this paper, based on intermolecular potentials and a one dimensional model, we provide a unified understanding of the two opposite size effects. Firstly, we analyzed the microstructural variation near the surface of an fcc nanofilm based on the Lennard-Jones potential. It is found that the atomic lattice near the surface becomes looser in comparison with the bulk, indicating that atoms in the bulk are located at the balance of repulsive forces, and the elastic moduli decrease with the decreasing thickness of the film accordingly. In addition, the decrease in moduli should be attributed to both the looser surface layer and smaller coordination number of surface atoms. Furthermore, it is found that both looser and tighter lattice near the surface can appear for a general pair potential and the governing mechanism should be attributed to the surplus of the nearest force to all other long range interactions in the pair potential. Surprisingly, the surplus can be simply expressed by a sum of the long range interactions and the sum being positive or negative determines the looser or tighter lattice near surface respectively. To justify this concept, we examined ZnO in terms of Buckingham potential with long range Coulomb interactions. It is found that compared to its bulk lattice, the ZnO lattice near the surface becomes tighter, indicating the atoms in the bulk are located at the balance of attractive forces, owing to the long range Coulomb interaction. Correspondingly, the elastic modulus of one-dimensional ZnO chain increases with decreasing size. Finally, a kind of many-body potential for Cu was examined. In this case, the surface layer becomes tighter than the bulk and the modulus increases with deceasing size, owing to the long range repulsive pair interaction, as well as the cohesive many-body interaction caused by the electron redistribution.

Surface effects on large deflection of nanowires

Surface effects play an important role in the mechanical behavior of nanosized structural elements owing to the increased ratio of surface area to volume. The surface effects on the large deflection of nanowires were considered. Both geometric nonlinearity in finite deformation and surface effects at nanoscale were taken into account to analyze the bending of nanowires subjected to a concentrated force. For simply supported beams and clamped-clamped beams, the influence of surface effects and geometric nonlinearity were discussed in detail. It is found that both surface effects and geometric nonlinearity tend to decrease the deflection of bending nanowires and thus increase the effective elastic modulus of nanowires. Surface effects yield the size dependent behavior of nanowires.

Surface effects of adsorption-induced resonance analysis on micro/nanobeams via nonlocal elasticity

The governing differential equation of micro/nanbeams with atom/molecule adsorption is derived in the presence of surface effects using the nonlocal elasticity. The effects of the nonlocal parameter, the adsorption density, and the surface parameter on the resonant frequency of the micro/nanobeams are investigated. It is found that, in ad- dition to the nonlocal parameter and the surface parameter, the bending rigidity and the adsorption-induced mass exhibit different behaviors with the increase in the adsorption density depending on the adatom category and the substrate material.

Surface effects on the thermal conductivity of silicon nanowires

Thermal transport in silicon nanowires(SiNWs)has recently attracted considerable attention due to their potential applications in energy harvesting and generation and thermal management.The adjustment of the thermal conductivity of SiNWs through surface effects is a topic worthy of focus.In this paper,we briefly review the recent progress made in this field through theoretical calculations and experiments.We come to the conclusion that surface engineering methods are feasible and effective methods for adjusting nanoscale thermal transport and may foster further advancements in this field.

The influence of surface effects on Frederiks transition in nematic liquid crystal doped with ferroelectric nanoparticles

Motivated by our recent work,in this work,we present the numerical study of the anchoring effect on the Frederiks threshold field in a nematic liquid crystal doped with ferroelectric colloidal nanoparticles.Assuming weak anchoring conditions,we employ the relaxation method and Maxwell construction to numerically solve the Euler–Lagrangian differential equation for the total free energy together the Rapini–Papoular surface energy to take into account anchoring of nematic liquid crystal molecules at the substrates.In this study,we focus our attention on obtaining the phase diagrams of Frederiks transition for different values of anchoring strength which have been not computed in our previous work.In this way,the effect of nanoparticle radius,nanoparticle volume fraction,nanoparticle polarization,and cell thickness on the Frederiks transition for different values of anchoring conditions are summarized in the phase diagrams.The numerical results show that by increasing the nanoparticles size and nanoparticle volume fraction in the ferronematic system,the Frederiks threshold field is strongly reduced.

Surface effects on the persistence length of nanowires and nanotubes

Surface effects on the persistence length of quasi-one-dimensional nanomaterials are investigated by using the theory of surface elasticity and the core-shell model of nanobeams. A simple and unified expression is provided to determine the persistence length of nanowires and nanotubes with any regular polygonal cross-sections. It is demonstrated that surface effects have a distinct in- fluence on the persistence length when the characteristic sizes of materials shrink to nanometers. This work is helpful not only for understanding the size-dependent behavior of nanomaterials but also for the design of devices based on nanotubes or nanowires.

Vibration analysis of fluid-conveying nanotubes embedded in an elastic medium considering surface effects

An analytical model is developed to study the surface effects on the vibration behavior including the natural frequency and the critical flow velocity of fluid-conveying nanotubes embedded in an elastic medium.The effects of surface elasticity and residual surface stress are accounted through the surface elasticity model and the Young-Laplace equation.A Winkler-type foundation is employed to model the interaction of nanotubes and the surrounding medium.The results show that the surface effects have more prominent influences on the nature frequency with smaller nanotube thickness,larger aspect ratio and larger elastic medium constants.Both surface layers and the elastic medium enhance the stability of nanotubes.This study might be helpful for designing the fluid-conveying nanotube devices in NEMS and MEMS systems.

Analytical Solutions for an Isotropic Elastic Half-Plane with Complete Surface Effects Subjected to a Concentrated/Uniform Surface Load

Within the context of Gurtin-Murdoch surface elasticity theory,closed-form analytical solutions are derived for an isotropic elastic half-plane subjected to a concentrated/uniform surface load.Both the effects of residual surface stress and surface elasticity are included.Airy stress function method and Fourier integral transform technique are used.The solutions are provided in a compact manner that can easily reduce to special situations that take into account either one surface effect or none at all.Numerical results indicate that surface effects generally lower the stress levels and smooth the deformation profiles in the half-plane.Surface elasticity plays a dominant role in the in-plane elastic fields for a tangentially loaded half-plane,while the effect of residual surface stress is fundamentally crucial for the out-of-plane stress and displacement when the half-plane is normally loaded.In the remaining situations,combined effects of surface elasticity and residual surface stress should be considered.The results for a concentrated surface force serve essentially as fundamental solutions of the Flamant and the half-plane Cerruti problems with surface effects.The solutions presented in this work may be helpful for understanding the contact behaviors between solids at the nanoscale.

Closed-Form Solution for a Circular Nanohole with Surface Effects Under Uniform Heat Flux

This paper investigates the steady-state thermoelastic problem of a circular nanohole embedded in an infinitely large elastic plane subjected to a uniform far-field heat flux.A lowly conductive surface model is used to account for the effects of surface phonon scattering,while the complete Gurtin–Murdoch model is utilized to characterize the effects of surface tension and surface elasticity.The closed-form solution to the temperature and stress field surrounding the hole is derived in the context of complex variable methods.Several numerical examples are presented to analyze the influence of surface effects on thermal stress fields.It is shown that surface effects induce notable increases in normal and shear stresses around the hole.Specifically,all three stress components(hoop,normal,and shear)in the vicinity of the hole exhibit substantial augmentation with increasing surface tension and surface modulus.In particular,it is found that the presence of surface effects amplifies the variation in stress gradients and intensifies stress concentration around the hole.

Nonlinear Vibration of FGM Sandwich Nanoplates with Surface Effects

Anovel functionally graded material(FGM)sandwich nanoplatemodel with surface effects is developed in thiswork.By using the Gurtin–Murdoch theory of surface elasticity,surface effects are taken into account.Governing equations for nonlinear vibrations are obtained though the balance of forces.The Galerkin method is employed to obtain the approximate solutions for nonlinear free and forced vibrations of the FGM laminates.Numerical results show that considering surface effects changes the equivalent Young’s modulus and bending stiffness of FGM sandwich nanoplates.In addition,the influences of surface effects are related to the geometric size of the nanoplate.Numerical examples are proposed to verify the effectiveness of the present results.