Double-layered Nanoplates Research Articles

Shear vibration and buckling of double-layer orthotropic nanoplates based on RPT resting on elastic foundations by DQM including surface effects

In this article, shear vibration and buckling of double-layer orthotropic nanoplates resting on elastic foundations are analyzed subjected to in-plane loadings including surface and nonlocal effects. These effects are considered by Gurtin---Murdoch's theory. Using the principle of virtual work, the governing equations for bulk and surface of double-layer orthotropic nanoplate are derived using refined plate theory. Differential quadrature method (DQM) is implemented. DQM solutions are validated by Navier's method and journal references. The influences of nonlocal parameter, van der Waals, Winkler, shear modulus, orthotropic material properties, boundary conditions, and in-plane biaxial, uniaxial, and shear loadings, are investigated on the surface effects of buckling and vibration modes of out-of-phase and in-phase. Results demonstrate that by augmenting nonlocal parameter, the surface effects on the vibration and buckling modes of both out-of-phase and in-phase increase. This result is in contrast with the works of other researchers in the field. Moreover, by enhancing in-plane loadings, the degree of surface effects on the vibration increase. On the other hand, the effects of nonlocal parameter on the vibration and buckling under in-plane shear load are more influential than those of biaxial and uniaxial, while the surface effects on the biaxial vibration and buckling are more remarkable than those of shear vibration and buckling.

Dispersion Analysis of Thermo Magnetic Effect on Double Layered Nanoplate Embedded in an Elastic Medium

Dispersion Analysis of Thermo Magnetic Effect on Double Layered Nanoplate Embedded in an Elastic Medium

Nonlinear vibration analysis of double-layered nanoplates with different boundary conditions

By the nonlocal theory, the nonlinear equations for double-layered nanoplates (DLNP) with different boundary conditions are established. The relation between aspect ratio and nonlinear frequencies with fixed mode amplitude is discussed. This relation for two vibration modes presents completely distinct trends. The novel fact is observed that there exists a point P where nonlinearity is weakest for the fundamental mode. Furthermore, we notice that P only appears for clamped movable edge when neglecting the nonlocal effect, which is substantially different from the other boundary conditions. It should distinguish whether the edge is movable or immovable for studying nonlinear dynamics of DLNP.

Nonlinear vibration of double layered viscoelastic nanoplates based on nonlocal theory

The nonlinear flexural vibration properties of double layered viscoelastic nanoplates are investigated based on nonlocal continuum theory. The von Kámán strain–displacement relation is employed to model the geometrical nonlinearity. Based on the classical plate theory, the formulations are derived by the Hamilton's principle in conjunction with Eringen's nonlocal elasticity theory, and are further discretized by the Galerkin's method. The coordinate transformation is adopted to obtain the nonlinear governing equations of motion in the modal coordinate system. On the basis of these equations, the frequency responses of double layered nanoplates with simply supported and clamped boundary conditions are derived by the method of multiple scales. The influences of small scale and other structural parameters (e.g. the aspect ratio of the plate, van der Walls (vdW) interaction and the viscidity of the plate) on the nonlinear vibration characteristics are discussed. From the result, the vdW interaction has obvious effects on the nonlinear frequency corresponding to the second nonlinear normal mode (NNM). The nonexistence of the internal resonance is also induced from the vdW forces between the plates. The influence of the elastic matrix is also discussed. The hardening nonlinearity is observed for the primary resonance. Additionally, some interesting phenomena different from the linear vibration are observed.

Axisymmetric free and forced vibrations of initially stressed circular nanoplates embedded in an elastic medium

As a first endeavor, the axisymmetric free and forced vibrations of circular single- and double-layered nanoplates under initial in-plane radial stresses and embedded in an elastic medium are investigated. The governing equations are derived by decoupling the nonlocal constitutive equations of the Eringen theory in polar coordinates in conjunction with the classical plate theory. The elastic medium is modeled as a two-parameter elastic foundation (Pasternak type). Galerkin’s method is employed to solve the resulting equation for vibration frequencies and dynamic response. The effects of small scale together with the other parameters such as initial in-plane load, Winkler and shear elastic foundation coefficients and the radius of the nanoplate are investigated. It is shown that the corresponding natural frequencies obtained by nonlocal elasticity theory are very different from those predicted by classical elasticity theory when the radius of the nanoplate is less than an approximate limit value.

Lamb waves in double-layered nanoplates

This paper is concerned with the propagation of Lamb waves in double-layered nanoplates and a nonlocal continuum model is presented. Each layer of the double-layered nanoplates is coupled through the van der Waals interaction. The equations of wave motion are derived and dispersion relations are obtained. The present model predicts some notable features of the dispersion relations in double-layered nanoplates, in comparison with Lamb waves in classic plates in vacuo. By considering the van der Waals interaction, some modes appear in pairs at the same frequency, their phase velocities are very close at a given frequency. Some modes only propagate in a very narrow frequency band and may disappear above certain frequencies. The small-scale has significant influence on the propagation characteristics of Lamb waves. Larger values of the scale coefficient result in lower escape frequencies and phase velocities of Lamb waves.

The origin of ZnO twin crystals in bio-inspired synthesis

A fabrication of uniform nacre-like hierarchical nanostructures of faceted ZnO twin-crystals was established by a hydrothermal route using gelatin as the structure-directing agent, zinc nitrate hexahydrate as the Zn source, and hexamethylenetetramine to control alkalinity. Early stage growth of ZnO twin-crystals was investigated by powder X-ray diffraction, scanning electron microscopy and transmission electron microscopy. A new formation mechanism is proposed. In the bio-inspired synthesis, Zn5(NO3)2(OH)8·2H2O nanoplatelets (10 to 20 nm in size) undergo orientated aggregation with gelatin to form Zn5(NO3)2(OH)8·2H2O/gelatin mesocrystalline nanoplates (150 to 400 nm in diameter and 20 to 50 nm in thickness). Surface re-crystallization of these nanoplates leads to two thin layers of ZnO separated by gelatin molecules. These double-layer nanoplates, negatively charged on both outer surfaces, are the cores of the twin-crystals. The dipolar Zn5(NO3)2(OH)8·2H2O nanoplates then stack on both sides of the double-layer nanoplates, followed by a phase transformation to ZnO. Eventually, twin-crystals are constructed in a manner reminiscent to that of an hourglass. The hexagonal morphology of the twin-crystals resulted from a late re-crystallization. The microstructure of the ZnO twin-crystals is very similar to the brick and mortar arrangement found in nacre. The present study is expected to shed light on the formation mechanism of many naturally occurring biominerals, as well as many other synthetic twin-crystals.

Thermal effects on vibration properties of double-layered nanoplates at small scales

In this paper, the thermal effects on the vibration properties of the double-layered nanoplates are studied. Based on the nonlocal continuum theory, the governing equations are derived and the expressions of the natural frequencies are presented. The axial stress caused by the thermal effects is considered. The influences of the small scale coefficient, the room or low temperature, the high temperature, the half wave numbers, the temperature change and the ratio of plate widths are discussed. Numerical results show that the small scale effects are significant for larger half wave numbers. The vibration properties can be obviously tuned by the thermal effects, the ratio of the nanoplate widths and the half wave numbers. The influences on the vibration behaviors are usually different for mode I and mode II.

Flexural wave propagation in double-layered nanoplates with small scale effects

In this work, the flexural wave propagation in doubled-layered nanoplates is studied. Based on the nonlocal continuum theory, the equation of wave motion is derived. The frequency, phase velocity, group velocity, and their ratio with different scale coefficients and wave numbers are presented. From the results, it can be observed that the small scale effects should be considered for higher frequencies. The dispersion properties for mode I and mode II are different. The van der Walls (vdW) interaction has significant influence on the wave characteristics for the higher mode, which is similar to the vibration properties of nanoplates. However, not all of the characteristics for mode II can be dominated by the vdW interaction, they can be affected by the wave number and the scale coefficients.

Nonlocal elasticity theory for vibration of nanoplates

Classical plate theory (CLPT) and first-order shear deformation theory (FSDT) of plates are reformulated using the nonlocal differential constitutive relations of Eringen. The equations of motion of the nonlocal theories are derived. Navier's approach has been used to solve the governing equations for simply supported boundary conditions. Analytical solutions for vibration of the nanoplates such as graphene sheets are presented. Nonlocal theories are employed to bring out the effect of the nonlocal parameter on natural frequencies of the nanoplates. The developed theory has been extended to the analysis of double layered nanoplates. Effect of (i) nonlocal parameter, (ii) length, (iii) height, (iv) elastic modulus and (v) stiffness of Winkler foundation of the plate on nondimensional vibration frequencies are investigated. The theoretical development as well as numerical solutions presented herein should serve as reference for nonlocal theories of nanoplates and nanoshells.