Longitudinal wave propagation in a piezoelectric nanoplate considering surface effects and nonlocal elasticity theory

The propagation characteristics of the longitudinal wave in a piezoelectric nanoplate were investigated in this study. The nonlocal elasticity theory was used and the surface effects were taken into account. In addition, the group velocity and phase velocity were derived and investigated, respectively. The dispersion relation was analyzed with different scale coefficients, wavenumbers, and voltages. The results showed that the dispersion degree can be strengthened by increasing the wavenumber and scale coefficient.     

Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part II: Parametric studies

The possible usage of nanoplates in transporting of nanovehicles encouraged the author to propose some nonlocal plate models in the companion paper where the nanovehicle (i.e., moving nanoparticle) was modeled by a moving point load by considering its friction with the upper surface of the nanoplate. In this paper, a comprehensive parametric study is carried out to study the effects of length to thickness ratio of the nanoplate, small-scale parameter, and velocity (or angular velocity) of the moving nanoparticle on dynamic response of nonlocal Kirchhoff, Mindlin, and higher-order plates subjected to a moving nanoparticle. Herein, dynamic response of the nanoplate covers both time histories and dynamic amplitude factors of the in- and out-of-plane displacements. The capabilities of various nonlocal plate models in predicting the displacement field caused by friction and mass weight of the moving nanoparticle are then explored through various numerical analyses for two cases: (i) the moving nanoparticle moves along a diagonal of the nanoplate; (ii) the moving nanoparticle orbits on an ellipse path whose center is coincident with the nanoplate's center. The obtained results indicate that due to the incorporation of small-scale effect into shear strain energy of the nanoplate, an appropriate nonlocal plate model should be used. The results show that the choice of the nanoplate model to use relies on the small-scale parameter, geometrical properties of the nanoplate, and velocity of the moving nanoparticle.     

Scale effects on flexural wave propagation in nanoplate embedded in elastic matrix with initial stress

In this paper, the small-scale effects on the flexural wave in the nanoplate are studied. Based on the nonlocal continuum theory, the equation of wave motion is derived and the dispersion relation is presented. Numerical simulations are performed to investigate the influences of the scale coefficient, the surrounding elastic matrix and the initial stress on the wave propagation properties. The results show that the nonlocal model provides an appropriate method to investigate the characteristics of the flexural wave in the nanoplate. Furthermore, the direction and amplitude of the biaxial load, the stiffness of the shearing layer and the Winkler foundation can change the wave properties, significantly.     

Wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes with surface and nonlocal effects

In this paper, the transverse wave propagation in fluid-conveying viscoelastic single-walled carbon nanotubes is investigated based on nonlocal elasticity theory with consideration of surface effect. The governing equation is formulated utilizing nonlocal Euler-Bernoulli beam theory and Kelvin-Voigt model. Explicit wave dispersion relation is developed and wave phase velocities and frequencies are obtained. The effect of the fluid flow velocity, structural damping, surface effect, small scale effects and tube diameter on the wave propagation properties are discussed with different wave numbers. The wave frequency increases with the increase of fluid flow velocity, but decreases with the increases of tube diameter and wave number. The effect of surface elasticity and residual surface tension is more significant for small wave number and tube diameter. For larger values of wave number and nonlocal parameters, the real part of frequency ratio raises.     

Wave propagation in embedded inhomogeneous nanoscale plates incorporating thermal effects

In this article, an analytical approach is developed to study the effects of thermal loading on the wave propagation characteristics of an embedded functionally graded (FG) nanoplate based on refined four-variable plate theory. The heat conduction equation is solved to derive the nonlinear temperature distribution across the thickness. Temperature-dependent material properties of nanoplate are graded using Mori–Tanaka model. The nonlocal elasticity theory of Eringen is introduced to consider small-scale effects. The governing equations are derived by the means of Hamilton’s principle. Obtained frequencies are validated with those of previously published works. Effects of different parameters such as temperature distribution, foundation parameters, nonlocal parameter, and gradient index on the wave propagation response of size-dependent FG nanoplates have been investigated.     

Small-scale effect on the vibration of thin nanoplates subjected to a moving nanoparticle via nonlocal continuum theory

The vibration of elastic thin nanoplates traversed by a moving nanoparticle involving Coulomb friction is investigated using the nonlocal continuum theory of Eringen. The eigen function technique and the Laplace transform method are employed to solve the governing equations of the nanoplate. The explicit expressions of the in-plane and transverse displacements are obtained when the moving nanoparticle traverses the nanoplate on an arbitrary straight line. In a special case, the obtained results are also compared with those of other researchers and a reasonably good agreement is achieved. The effects of small-scale parameters and velocity of the moving nanoparticle on the dynamic response as well as the dynamic amplitude factors (DAFs) of the in-plane and transverse displacements are then explored in some detail. The results indicate that the magnitude of DAF of the transverse displacement of the nanoplate (i.e., DAF_w) increases with the first small-scale effect parameter, irrespective of the values of the second small-scale effect parameter and the velocity of the moving nanoparticle. As the first small-scale effect parameter grows, the maximum values of DAF_w as a function of the moving nanoparticle velocity increase and generally occur in the lower levels of the moving nanoparticle velocity.     

Investigating Bulk Waves in Orthotropic Rectangular Nanoplates Based on Three Dimensional Elasticity Theory and Nonlocal Elasticity Theory

The propagation of bulk waves in rectangular nanoplates is studied on the basis of nonlocal three-dimensional elasticity theory. The nonlocal theory applies to both thin and thick rectangular orthotropic nanoplates. The dispersion relation for the waves is derived analytically. Our results are checked against data for macroplates. The influence of nonlocality and other parameters on the wave frequency and phase velocity is discussed.     

Nonlocal and surface effects on the buckling behavior of functionally graded nanoplates: An isogeometric analysis

The size-dependent static buckling responses of circular, elliptical and skew nanoplates made of functionally graded materials (FGMs) are investigated in this article based on an isogeometric model. The Eringen nonlocal continuum theory is implemented to capture nonlocal effects. According to the Gurtin–Murdoch surface elasticity theory, surface energy influences are also taken into account by the consideration of two thin surface layers at the top and bottom of nanoplate. The material properties vary in the thickness direction and are evaluated using the Mori–Tanaka homogenization scheme. The governing equations of buckled nanoplate are achieved by the minimum total potential energy principle. To perform the isogeometric analysis as a solution methodology, a novel matrix-vector form of formulation is presented. Numerical examples are given to study the effects of surface stress as well as other important parameters on the critical buckling loads of functionally graded nanoplates. It is found that the buckling configuration of nanoplates at small scales is significantly affected by the surface free energy.     

Nonlocal continuum-based modeling of a nanoplate subjected to a moving nanoparticle. Part I: Theoretical formulations

The potential applications of nanoplates in energy storage, chemical and biological sensors, solar cells, field emission, and transporting of nanocars have been attracted the attentions of the nanotechnology community to them during recent years. Herein, the later application of nanoplates from nonlocal elastodynamic point of view is of interest. To this end, dynamic response of a nanoplate subjected to a moving nanoparticle is examined within the context of nonlocal continuum theory of Eringen. The fully simply supported nanoplate is modeled based on the nonlocal Kirchhoff, Mindlin, and higher-order plate theories. The non-dimensional equations of motion of the nonlocal plate models are established. The effects of moving nanoparticle's weight and existing friction between the surfaces of the moving nanoparticle and nanoplate on the in-plane and out-of-plane vibrations of the nanoplate are incorporated into the formulations of the proposed models. The eigen function expansion and the Laplace transform methods are employed for discretization of the governing equations in the spatial and the time domains, respectively. The analytical expressions of the dynamic deformation field associated with each nonlocal plate theory are obtained when the moving nanoparticle traverses the nanoplate on an arbitrary straight path (an opened path) as well as an ellipse path (a closed path). The dynamic in-plane forces and moments of each nonlocal plate model are also derived. Furthermore, the critical velocity and the critical angular velocity of the moving nanoparticle for the proposed models are expressed analytically for the aforementioned paths. Part II of this work consists in a comprehensive parametric study where the effects of influential parameters on dynamic response of the proposed nonlocal plate models are scrutinized in some detail.     

Dispersion of dilatation wave propagation in single-wall carbon nanotubes using nonlocal scale effects

Nonlocal continuum mechanics allows one to account for the small length scale effect that becomes significant when dealing with micro- or nano-structures. This paper investigates a model of wave propagation in single-wall carbon nanotubes (SWCNTs) with small scale effects are studied. The equation of motion of the dilatation wave is obtained using the nonlocal elastic theory. We show that a dispersive wave equation is obtained from a nonlocal elastic constitutive law, based on a mixture of a local and a nonlocal strain. The SWCNTs structures are treated within the multilayer thin shell approximation with the elastic properties taken to be those of the graphene sheet. The SWCNT was the (40,0) zigzag tube with an effective diameter of 3.13 nm. Nonlinear frequency equations of wave propagation in SWCNTs are described through the effect of small scale. The phase velocity and the group velocity are derived, respectively. The nonlinear dispersion relation is analyzed with different wave numbers versus scale coefficient. It can be observed from the results that the dispersion properties of the dilatation wave are induced by the small scale effects, which will disappear in local continuous models. The dispersion degree can be strengthened by increasing the scale coefficient and the wave number. Furthermore, the characteristics for the group velocity of the dilatation wave in carbon nanotubes can also be tuned by these factors.     

Wave propagation in viscoelastic single-walled carbon nanotubes with surface effect under magnetic field based on nonlocal strain gradient theory

The governing equation of wave motion of viscoelastic SWCNTs (single-walled carbon nanotubes) with surface effect under magnetic field is formulated on the basis of the nonlocal strain gradient theory. Based on the formulated equation of wave motion, the closed-form dispersion relation between the wave frequency (or phase velocity) and the wave number is derived. It is found that the size-dependent effects on the phase velocity may be ignored at low wave numbers, however, is significant at high wave numbers. Phase velocity can increase by decreasing damping or increasing the intensity of magnetic field. The damping ratio considering surface effect is larger than that without considering surface effect. Damping ratio can increase by increasing damping, increasing wave number, or decreasing the intensity of magnetic field.     

Propagation of waves in nonlocal porous multi-phase nanocrystalline nanobeams under longitudinal magnetic field

ABSTRACT

This article investigates wave propagation behavior of a multi-phase nanocrystalline nanobeam subjected to a longitudinal magnetic field in the framework of nonlocal couple stress and surface elasticity theories. In this model, the essential measures to describe the real material structure of nanocrystalline nanobeams and the size effects were incorporated. This non-classical nanobeam model contains couple stress effect to capture grains micro-rotations. Moreover, the nonlocal elasticity theory is employed to study the nonlocal and long-range interactions between the particles. The present model can degenerate into the classical model if the nonlocal parameter, couple stress and surface effects are omitted. Hamilton’s principle is employed to derive the governing equations which are solved by applying an analytical method. The frequencies are compared with those of nonlocal and couple stress-based beams. It is showed that wave frequencies and phase velocities of a nanocrystalline nanobeam depend on the grain size, grain rotations, porosities, interface, magnetic field, surface effect and nonlocality.     

Size-dependent dispersion characteristics in piezoelectric nanoplates with surface effects

In this paper, surface effects on the dispersion characteristics of elastic waves propagating in an infinite piezoelectric nanoplate are investigated by using the surface piezoelectricity model. Based on the surface piezoelectric constitutive theory, the presence of surface stresses and surface electric displacements exerting on the boundary conditions of the piezoelectric nanoplate is taken into account in the modified mechanical and electric equilibrium relations. The partial wave technique is employed to obtain the general solutions of governing equations, and the dispersion relations with surface effects are expressed in an explicit closed form. The impacts of surface piezoelectricity, residual surface stress and plate thickness on the propagation properties of elastic waves are analyzed in detail. Numerical results show that the dispersion behaviors in piezoelectric nanoplates are size-dependent, and there exists a critical plate thickness above which the surface effects may vanish.     

Exact analytical solution for free vibration of functionally graded micro/nanoplates via three-dimensional nonlocal elasticity

Using three-dimensional (3-D) nonlocal elasticity theory of Eringen, this paper presents closed-form solutions for in-plane and out-of-plane free vibration of simply supported functionally graded (FG) rectangular micro/nanoplates. Elasticity modulus and mass density of FG material are assumed to vary exponentially through the thickness of micro/nanoplate, whereas Poisson's ratio is considered to be constant. By employing appropriate displacement fields for the in-plane and out-of-plane modes that satisfy boundary conditions of the plate, ordinary differential equations of free vibration are obtained. Boundary conditions on the lateral surfaces are imposed on the analytical solutions of the equations to yield the natural frequencies of FG micro/nanoplate. The natural frequencies of FG micro/nanoplate are obtained for different values of nonlocal parameter and gradient index of material properties. The results of this investigation can be used as a benchmark for the future numerical, semi-analytical and analytical studies on the free vibration of FG micro/nanoplates.     

The effect of surface stress on the propagation of Lamb waves

This work investigates the possibility of the propagation of Lamb waves in thin solid layers with external traction free surfaces, in the presence of surface elasticity, inertia and residual stress. It is demonstrated that such waves do exist and that their characteristics can be quite different from their classical counterparts. The governing equations with non-classical boundary conditions involving the bulk and surface stress are solved exactly in the frequency-wavenumber domain. This solution is utilized to compute the Lamb wave modes for different layer thicknesses. An efficient strategy to capture all the modes of Lamb waves within a given frequency window is outlined. It is shown that the effect of surface elasticity and inertia becomes significant with increasing frequency and decreasing layer thickness, where the number of modes participating within a given frequency window is more than that permitted by the classical theory. Further, it is observed that the nature of the Lamb wave modes (in terms of negative dispersion) in the presence of surface stress is similar to what predicted by the nonlocal theory and microstructure based continuum theory.     

Scale-dependent effects on wave propagation in magnetically affected single/double-layered compositionally graded nanosize beams

This article deals with the wave propagation analysis of single/double layered functionally graded (FG) size-dependent nanobeams in elastic medium and subjected to a longitudinal magnetic field employing nonlocal elasticity theory. Material properties of nanobeam change gradually according to the sigmoid function. Applying an analytical solution, the acoustical and optical dispersion relations are explored for various wave number, nonlocality parameter, material composition, elastic foundation constants, and magnetic field intensity. It is found that frequency and phase velocity of waves propagating in S-FGM nanobeam are significantly affected by these parameters. Also, presence of cut-off and escape frequencies in wave propagation analysis of embedded S-FGM nanobeams is investigated.     

Size-dependent effect on thermo-electro-mechanical responses of heated nano-sized piezoelectric plate

The booming development of nanotechnology motivates the widespread applications of piezoelectric nanomaterials (e.g. ZnO, ZnS, GaN) and their nanostructures (e.g. nanobelts, nanorings nanowires). It is noted that the coupled field analysis of nano-sized piezoelectric structure under non-uniform temperature in-service environment is of great importance for the fabrication and exploitation of nanoelectromechanical devices. In such situation, spatial size effect of heat conduction is necessary to be taken into account due to its important significance in characterizing the nonlocal feature of heat transport in nanosystems. In this study, thermal nonlocal effect is introduced into the thermo-electro-mechanical model based on nonlocal elasticity theory to further shed light on the size-dependent coupling behavior of thermal, electric, and elastic fields. The coupled field equations involving size-dependent parameters are derived. The solutions can be obtained using Laplace transformation methods. Parametric studies are conducted to evaluate the influences of thermal as well as elastic nonlocal parameters on the transient responses. The results indicate that the piezoelectric performance of the nanoplate is greatly improved in the presence of thermal nonlocal effect.     

Axial instability of double-nanobeam-systems

This Letter considers the axial instability of double-nanobeam-systems. Eringen's nonlocal elasticity is utilized for modelling the double-nanobeam-systems. The nonlocal theory accounts for the small-scale effects arising at the nanoscale. The small-scale effects substantially influence the instability (or buckling) of double-nanobeam-systems. Results reveal that the small-scale effects are higher with increasing values of nonlocal parameter for the case of in-phase (synchronous) buckling modes than the out-of-phase (asynchronous) buckling modes. The increase of the stiffness of the coupling elastic medium in double-nanobeam-system reduces the small-scale effects during the out-of-phase (asynchronous) buckling modes. Analysis of the scale effects in higher buckling loads of double-nanobeam-system with synchronous and asynchronous modes is also discussed in this Letter. The theoretical development presented herein may serve as a reference for nonlocal theories as applied to the instability analysis of complex-nanobeam-system such as complex carbon nanotube system.     

Free vibration of nonlocal piezoelectric nanoplates under various boundary conditions

This paper investigates the thermo-electro-mechanical vibration of the rectangular piezoelectric nanoplate under various boundary conditions based on the nonlocal theory and the Mindlin plate theory. It is assumed that the piezoelectric nanoplate is subjected to a biaxial force, an external electric voltage and a uniform temperature rise. The Hamilton's principle is employed to derive the governing equations and boundary conditions, which are then discretized by using the differential quadrature (DQ) method to determine the natural frequencies and mode shapes. The detailed parametric study is conducted to examine the effect of the nonlocal parameter, thermo-electro-mechanical loadings, boundary conditions, aspect ratio and side-to-thickness ratio on the vibration behaviors.