Small Length Scale Effect Research Articles

3D nonlocal solution with layerwise approach and DQM for multilayered functionally graded shallow nanoshells

A nonlocal layerwise 3D bending solution for multilayered functionally graded shallow nanoshells is presented. Eringen’s nonlocal elasticity theory is used for approaching the small length scale effect. Curvilinear orthogonal coordinate system is employed for writing the constitutive and equilibrium equations. The middle-plane of displacements are assumed as a Navier solution which are only valid for simply supported spherical, cylindrical panels and rectangular plates. The thickness domain is discretized by the well-known Chebyshev–Gauss–Lobatto, and its derivatives are calculated numerically by the Differential Quadrature method (DQM). Lagrange interpolation polynomials are utilized as the basis functions for DQM. The traction conditions at the top and the bottom of the shell panel and continuity conditions of the interlaminar adjacent layers for transverse stresses are considered, so this method presents layerwise capabilities. The results are compared with higher order theories proposed in the literature, and due to the 3D nature of the presented formulation, benchmark solutions are provided.

Application of the Green's function method for static analysis of nonlocal stress-driven and strain gradient elastic nanobeams

This paper focuses on the static analysis of a generally restrained Euler-Bernoulli nanobeam (GRNB), which is subjected to arbitrary distributed or concentrated loads. The small length scale effect is introduced through a strongly nonlocal integral elastic law called the stress-driven nonlocal Euler-Bernoulli beam model, and a weakly nonlocal elastic law called the strain-gradient Euler-Bernoulli beam model. The stress-driven nonlocal theory and the strain gradient elasticity theory are both employed, to formulate the differential equation governing the Euler-Bernoulli nanobeams. Both theories are ruled by a similar 6th order differential equation, with distinct higher-order boundary conditions. The paper demonstrates that the strain gradient theory can be also reformulated as a stress-driven nonlocal theory with unsymmetrical kernel, in contrast to the initial stress-driven nonlocal theory which has symmetrical exponential kernel. Additionally, some theoretical backgrounds about both nonlocal elasticity theorems are provided. Exact solutions for both nanobeam models are presented, using Green's function method. The parameters of the Green's functions are derived for each beam theory, and then utilized to establish the static displacement function of both nanobeam theories. Exact solutions are provided for the bending of both nanobeams with various end boundary conditions, that account for the small length scale effects. The paper also compares the responses of the stress-driven Euler-Bernoulli beam and the strain gradient Euler-Bernoulli beam, showing close deflections and similar effects of increasing the nonlocal parameter on beam stiffness. A major difference highlighted between the two theories is that the strain gradient theory predicts a homogeneous strain state for homogeneous stress state, as opposed to the stress-driven theory.

A size-dependent 3D solution of functionally graded shallow nanoshells

Abstract An unavailable semi-analytical non-local 3D solution for functionally graded nanoshells with constant radii of curvature is presented. The small length scale effect is included in Eringen’s nonlocal elasticity theory. The constitutive and equilibrium equations are written in terms of curvilinear orthogonal coordinates systems which are only valid for spherical and cylindrical shells, and rectangular plates. The stresses and displacements are assumed in terms of the Navier method which is applicable for simply supported structures. The derivatives in terms of thickness are approximated by the differential quadrature method (DQM). The thickness domain is discretized by the Chebyshev–Gauss–Lobatto grid distribution. Lagrange interpolation polynomials are considered as the basis function for DQM. The correct free surface boundary condition for out-of-plane stresses is considered. Several problems of isotropic and functionally graded shells subjected to different types of loads are analyzed. The results are compared with other three-dimensional solutions and higher-order theories. It is important to emphasize that the radii of curvature are crucial at nanoscale, so it should be considered in the design of nanodevices.

Size Dependent Buckling Analysis of a FG-CNTRC Microplate of Variable Thickness under Non-Uniform Biaxial Compression

This paper combines third-order shear deformation theory (TSDT) and modified couple stress theory (MCST) with the principle of total potential energy to analyze the size-dependent buckling behavior of a functionally graded carbon nanotube-reinforced composite (FG-CNTRC) rectangular microplate of variable thickness subject to non-uniform biaxial compression when resting on an elastic medium. To determine the thickness qualities of the material, the extended rule of mixture was applied. In the context of microplate buckling in the presence of small length scale effects, the three kinds of Carbon Nanotube (CNT) distribution—(a) UD, (b) FG-O, and (c) FG-X—were used and compared. The equations governing various combinations of simply supported or clamped boundary conditions have been solved using the differential quadrature method (DQM). The correctness and precision of the solutions have been compared to another study. A numerical study was conducted to examine the dependence of buckling load on several parameters, including percentage change of thickness, length scale parameter, nonuniform edge loads, boundary conditions, volume percentage of the CNTs, CNT distribution, and elastic medium parameter. The results of their effects are presented in this paper.

Scale effects on the torsional surface waves propagation in an initially stressed dissipative nanoplate

The effects of small length scale on the torsional surface waves propagating in a pre-stressed dissipative nanoplate were examined in this paper. Biot’s incremental deformation principle as well as nonlocal elasticity theory were implemented to form the governing and frequency equations of torsional surface waves in nanoplate under considered conditions. In this work, we proposed a new approach to determine the phase velocity of torsional disturbances. Different categories are used to investigate the dispersion relations such as: wave number, scale coefficient, initial stress parameter, and dissipation coefficient. The results show that small length scale influences causes torsional dispersion of the wave propagation, which disappear in local elasticity theory. In addition, it is recognized that the torsional surface waves propagating in the pre-stressed dissipative nanoplates can be differentiated in a clear manner from those propagated in the absence of small length scale effects. The initial stress parameter can also be utilized to change the characteristics of the torsional surface wave's phase velocity in nanoplates. It is also observed that, the dissipation has a great effect on the phase velocity of torsional surface waves in propagating in the nanoplate. Therefore, this research is theoretically useful to convey information on nonlocal and dissipation properties of the nanomaterials: for example, through a precise measurement of the phase velocity of torsional surface wave propagates in a nanocomposite material by presence of the medium’s dissipation. This work could be useful in understanding the way of designing the Nano devices and Nano resonators.

Exact and approximate analytical solutions for nonlocal nanoplates of arbitrary shapes in bending using the line element-less method

In this study, an innovative procedure is presented for the analysis of the static behavior of plates at the micro and nano scale, with arbitrary shape and various boundary conditions. In this regard, the well-known Eringen’s nonlocal elasticity theory is used to appropriately model small length scale effects. The proposed mesh-free procedure, namely the Line Element-Less Method (LEM), only requires the evaluation of simple line integrals along the plate boundary parametric equation. Further, variations of appropriately introduced functionals eventually lead to a linear system of algebraic equations in terms of the expansion coefficients of the deflection function. Notably, the proposed procedure yields approximate analytical solutions for general shapes and boundary conditions, and even exact solutions for some plate geometries. In addition, several applications are discussed to show the simplicity and applicability of the procedure, and comparison with pertinent data in the literature assesses the accuracy of the proposed approach.

A real-time TEM study of the deformation mechanisms in β-Ti reinforced bulk metallic glass composites

The deformation mechanisms in a β-Ti reinforced Zr-based bulk metallic glass composite (BMGC) are studied by extracting submicron sized tensile coupons of the crystalline and amorphous phase in their monolithic and bilaminate composite forms and tensile testing them inside a TEM. Results show that the monolithic BCC crystalline phase and amorphous phase have high yield strains, owing to small length scale effects, but undergo negligible post-yield elongation before failure. However, the ductility of the bilaminate composite is significantly higher (~12.4%), provided the thickness of its amorphous portion is < 100 nm. Real time videos, which are supplemented by molecular dynamics simulations, show that the negligible post yield elongation of the monolithic BCC crystalline phase is caused by planar slip on one of the {110} planes. However, the bilaminate composite form exhibits strain hardening, despite the occurrence of planar slip, as dislocations pile up at the impervious amorphous/crystalline interface, which in turn, activates slip on other {110} planes. Strain hardening in the crystalline phase ceases when a shear band nucleated in the amorphous phase penetrates the dendrite along one of the slip planes. These mechanisms are extended to explain the flow behavior of β-Ti reinforced BMGCs and strategies to improve the ductility are discussed.

Applying nonlocal strain gradient theory to size-dependent analysis of functionally graded carbon nanotube-reinforced composite nanoplates

In this research paper, as initial endeavors, the vibrational responses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates taking into account the effect of nonlocal parameter and strain gradient coefficient are investigated. The study aims at developing mathematical modeling via an analytical solution to FG-CNTRC nanoplate structure with allowance for the nonlocal strain gradient effect. The four types of CNT distribution are used and compared in the context of the vibration of nanoplate in the presence of the small length scale effects, namely the (a) UD, (b) FG-V, (c) FG-O, and (d) FG-X. Some theoretical equations based on the first-order shear deformation plate theory (FSDT) are presented to provide a lucid understanding of how the small length-scale influences the FG-CNTRC nanoplate. For the vibrational analysis of a nanoplate, which is simply supported boundary condition, Navier solutions are obtained. Also, in contrast to earlier studies, an analytical approach is used to establish the governing equations of the FG-CNTRC nanoplate. Some specific numerical examples are given and compared with the results presented in the literature. In the section of numerical results, the influence of the nonlocal parameter, strain gradient coefficient, geometric parameters and vibrational modes on the non-dimensional natural frequency are investigated and discussed in detail. These could be useful to analysts and designers to estimate the fundamental natural frequencies in each of the four CNT distributions that the FG-CNTRC nanoplate possesses.

A three‐dimensional static analysis of embedded single‐walled carbon nanotubes using the perturbation method

A three‐dimensional (3D) asymptotic local elasticity theory is reformulated for the static analysis of a simply supported, single‐walled carbon nanotube (SWCNT) embedded in an elastic medium under a transverse normal load at its inner and outer surfaces. Eringen's nonlocal constitutive relations are used to account for the small length scale effects in the formulation examining the static behavior of the SWCNT. The interaction between the SWCNT and its surrounding foundation is modeled as either a Winkler‐type or a Pasternak‐type model. The two‐dimensional (2D) nonlocal classical shell theory (CST) is derived as a first‐order approximation of the 3D nonlocal elasticity theory. The 2D nonlocal CST solutions can be modified order‐by‐order to asymptotically approach the exact 3D nonlocal elasticity solutions in a hierarchic and consistent manner. A parametric study related to some effects on the static behavior of the embedded SWCNT is undertaken, including the aspect ratio, the stiffness and shear modulus of the surrounding medium of the SWCNT, and the nonlocal parameter. The current 3D asymptotic solutions can provide a standard to assess the accuracy and convergence rate of various 2D nonlocal shell theories.

A nonlocal continuum mechanics-based asymptotic theory for the buckling analysis of SWCNTs embedded in an elastic medium subjected to combined hydrostatic pressure and axial compression

Based on nonlocal continuum mechanics, the authors develop an asymptotic theory by which to examine the buckling behavior of simply supported, single-walled carbon nanotubes (SWCNTs) embedded in an elastic medium under combined hydrostatic pressure and axial compression using the perturbation method. In the formulation, the Eringen nonlocal constitutive relations are used to account for the small length scale effect; the interactions between the SWCNT and its surrounding medium are modelled as a Winkler foundation model, and the half-thickness- to- the mid-surface radius ratio is selected as the perturbation parameter. After performing some mathematical processes, including non-dimensionalization, asymptotic expansion, and successive integration, among others, the authors separate the three-dimensional (3D) nonlocal elasticity equations into recursive sets of governing equations (GEs) based on the Donnell-type nonlocal classical shell theory (CST) for various order problems. The Donnell-type nonlocal CST is derived as a first-order approximation of the 3D nonlocal elasticity problem, where the GEs for higher-order problems retain the same differential operators as those of the Donnell-type nonlocal CST although with different nonhomogeneous terms. The current asymptotic solutions for the critical load parameters of the embedded SWCNT are obtained in order to assess the accuracy of various nonlocal CSTs and nonlocal beam theories available in the literature.

Three-dimensional asymptotic nonlocal elasticity theory for the free vibration analysis of embedded single-walled carbon nanotubes

Within the framework of three-dimensional (3D) nonlocal elasticity theory, the authors develop an asymptotic theory to investigate the free vibration characteristics of simply supported, single-walled carbon nanotubes (SWCNTs) non-embedded or embedded in an elastic medium using the multiple time scale method. Eringen’s nonlocal constitutive relations are adopted to account for the small length scale effect in the formulation. The interactions between the SWCNT and its surrounding medium are modeled as a two-parameter Pasternak foundation model. After performing a series of mathematical processes, including nondimensionalization, asymptotic expansion, and successive integration, etc., the authors obtain recurrent sets of motion equations for various order problems. The nonlocal classical shell theory (CST) is derived as a first-order approximation of the current 3D nonlocal elasticity problem, and the equations of motion for higher-order problems retain the same differential operators as those of the nonlocal CST, although with different nonhomogeneous terms. The current asymptotic solutions for the natural frequency parameters of non-embedded or embedded SWCNTs and their corresponding through-thickness modal stress and displacement component distributions are obtained to assess the accuracy of various nonlocal shell and beam theories available in the literature.

An efficient approach for prediction of the nonlocal critical buckling load of double-walled carbon nanotubes using the nonlocal Donnell shell theory

This paper aims to apply the nonlocal Donnell shell theory to study the buckling of double-walled carbon nanotubes (DWCNTs) under axial compression taking into account the effects of internal small length scale and the Van der Waals interactions between layers. The DWCNTs is modeled as nonlocal double circular cylindrical elastic shells. On the basis of nonlocal elasticity theory, governing equations for buckling of Donnell shells are obtained taking into account the Van der Waals force. The nonlocal buckling load of DWCNT is derived without any assumption on radius tubes. However, it is a very difficult task to obtain the analytical solution of nonlocal critical buckling load. In this paper, we develop an approach for prediction of the nonlocal critical buckling load of DWCNTs.

Numerical analysis of single-layered graphene sheets by a mesh-free approach

The paper presents a numerical analysis of the behavior of single-layered graphene sheet (SLGS) using a mesh-free approach based on a nonlocal continuum plate model (NCPM). The adopted NCPM constructed by incorporating the nonlocal elasticity theory into the first order shear deformation elastic plate theory is able to capture small length scale effects. Through the NCPM, the SLGS is modeled as a continuous orthotropic nanoplate. the obtained nonlocal nonlinear partial differential equations completed by boundary conditions are solved numerically by a mesh-free approach associating the asymptotic numerical method with the mesh-free collocation method based on moving least square approximation. The effects of the small-scale parameter and aspect ratio on the nonlinear bending and post-buckling behaviors of SLGS are considered. Good agreement has been established between the obtained results and those of the literature.

Vibration control and stability analysis of a nanobeam with boundary prescribed performance

ABSTRACT In this paper, we present dynamics analysis and vibration control for a nanoscale beam. The nanobeam is modelled based on the modified couple stress theory which incorporates small length scale effects. The governing equation and boundary conditions of the nanobeam are obtained by Hamilton's principle. We design model-based control and adaptive control with output constraints for the control part. Employing a time-varying function to the constraint part, the displacement of the nanobeam can be well limited within a trajectory. By choosing designed parameters appropriately, signals in the nanobeam system converge to a small neighbourhood of zero and the vibration of the nanobeam system is suppressed with prescribed performance. Correspondingly, the simulation results of the proposed control are verified suitable.

Cylindrical Bending Vibration of Multiple Graphene Sheet Systems Embedded in an Elastic Medium

Based on the Eringen nonlocal elasticity theory and multiple time scale method, an asymptotic nonlocal elasticity theory is developed for cylindrical bending vibration analysis of simply-supported, [Formula: see text]-layered, and uniformly or nonuniformly-spaced, graphene sheet (GS) systems embedded in an elastic medium. Both the interactions between the top and bottom GSs and their surrounding medium and the interactions between each pair of adjacent GSs are modeled as one-parameter Winkler models with different stiffness coefficients. In the formulation, the small length scale effect is introduced to the nonlocal constitutive equations by using a nonlocal parameter. The nondimensionalization, asymptotic expansion, and successive integration mathematical processes are performed for a typical GS. After assembling the motion equations for each individual GS to form those of the multiple GS system, recurrent sets of motion equations can be obtained for various order problems. Nonlocal multiple classical plate theory (CPT) is derived as a first-order approximation of the current nonlocal plane strain problem, and the motion equations for higher-order problems retain the same differential operators as those of nonlocal multiple CPT, although with different nonhomogeneous terms. Some nonlocal plane strain solutions for the natural frequency parameters of the multiple GS system with and without being embedded in the elastic medium and their corresponding mode shapes are presented to demonstrate the performance of the asymptotic nonlocal elasticity theory.

Settling velocity of fine heavy particles in turbulent open channel flow

The settling velocity of heavy particles with sub-millimetre diameters falling in a turbulent open channel flow is investigated using a two-camera imaging technique of simultaneous particle image velocimetry and particle tracking velocimetry. Flow images of heavy particles are separated from those of the fluid flow field based on different wavelengths of light emitted by the fluorescent heavy particles and flow-following seeding particles. Some flow configurations of weak turbulence are generated in the open channel flow with a turbulence grid. The effect of turbulence intensity, vorticity, and small length scale on the settling velocity of fine solid particles in these cases of relatively weak turbulence is studied. Experimental results reveal that the settling velocity of heavy particles in most cases is increased from its still-water value by the weak turbulence. The increase becomes larger when the turbulent Reynolds number is increased for similar flow configurations. The interaction between particle movements and small turbulent scales appears to be responsible for the enhancement of particle settling velocity. The ratio of particle diameter to the local Kolmogorov length scale is found to correlate well with the increase in particle settling velocity. A reduction in settling velocity is only observed when this ratio is less than 0.5. Even in this case, the enhancement phenomenon of settling velocity in low vorticity regions can also be observed. Possible effects of the nonlinear drag and loitering effect are also investigated.

Lattice and continualized models for the buckling study of nonlocal rectangular thick plates including shear effects

The present study investigates the exact buckling behavior of a new lattice plate theory, called microstructured thick plate model, that accounts for the shear effect. Three different kinds of continuous nonlocal elasticity plate theories for capturing the small length scale effect of this bending-shear lattice plate have been compared: a fourth and a sixth order continualized models based on the derivation of continuous equations from discrete ones and a phenomenological nonlocal Uflyand–Mindlin plate model. The phenomenological nonlocal model is the stress gradient model of Eringen (1983) applied at the plate scale for the bending part and eventually the shear part of the constitutive law. Each nonlocal plate model is calibrated with respect to the lattice plate model. By definition, the length scale coefficients are constant for the two proposed continualized models. For the phenomenological model, these coefficients are structural dependent and vary with different parameters, such as the aspect ratio or the buckling mode. More especially, it is shown that the calibrated small length scale coefficient is influenced by the shear effect. The comparison of the nondimensional buckling loads calculated with the optimal value of the small length scale coefficient, which differ for each nonlocal model, shows that the continualized models constitute a much better, more consistent and more accurate approximation than the phenomenological nonlocal approach.

Nonlocal Approaches for the Vibration of Lattice Plates Including Both Shear and Bending Interactions

The present study investigates the dynamical behavior of lattice plates, including both bending and shear interactions. The exact natural frequencies of this lattice plate are calculated for simply supported boundary conditions. These exact solutions are compared with some continuous nonlocal plate solutions that account for some scale effects due to the lattice spacing. Two continualized and one phenomenological nonlocal UflyandMindlin plate models that take into account both the rotary inertia and the shear effects are developed for capturing the small length scale effect of microstructured (or lattice) thick plates by associating the small length scale coefficient introduced in the nonlocal approach to some length scale coefficients given in a Taylor or a rational series expansion. The nonlocal phenomenological model constitutes the stress gradient Eringen’s model applied at the plate scale. The continualization process constructs continuous equation from the one of the discrete lattice models. The governing partial differential equations are solved in displacement for each nonlocal plate model. An exact analytical vibration solution is obtained for the natural frequencies of the simply supported rectangular nonlocal plate. As expected, it is found that the continualized models lead to a constant small length scale coefficient, whereas for the phenomenological nonlocal approaches, the coefficient, calibrated with respect to the element size of the microstructured plate, is structure-dependent. Moreover, comparing the natural frequencies of the continuous models with the exact discrete one, it is concluded that the continualized models provide much more accurate results than the nonlocal Uflyand–Mindlin plate models.

Bending analysis of embedded nanoplates based on the integral formulation of Eringen's nonlocal theory using the finite element method

Due to the capability of Eringen's nonlocal elasticity theory to capture the small length scale effect, it is widely used to study the mechanical behaviors of nanostructures. Previous studies have indicated that in some cases, the differential form of this theory cannot correctly predict the behavior of structure, and the integral form should be employed to avoid obtaining inconsistent results. The present study deals with the bending analysis of nanoplates resting on elastic foundation based on the integral formulation of Eringen's nonlocal theory. Since the formulation is presented in a general form, arbitrary kernel functions can be used. The first order shear deformation plate theory is considered to model the nanoplates, and the governing equations for both integral and differential forms are presented. Finally, the finite element method is applied to solve the problem. Selected results are given to investigate the effects of elastic foundation and to compare the predictions of integral nonlocal model with those of its differential nonlocal and local counterparts. It is found that by the use of proposed integral formulation of Eringen's nonlocal model, the paradox observed for the cantilever nanoplate is resolved.

Vibration of Initially Stressed Nonlocal Euler-Bernoulli Nano-Beams

This paper is pertinent to the analytical solutions for vibration analysis of initially stressed Nonlocal Euler-Bernoulli nano-beams. In order to take into account of small length scale effect, this vibration problem formulation is depending upon both nonlocal Euler-Bernoulli and also Eringen’s nonlocal elasticity theory. The boundary conditions and governing equation are obtained by use of Hamiltonian’s principle. These equations are solved analytically with different initial stresses (both compressive and tensile) and boundary conditions. The effect of small length scale and the initial stress on the fundamental frequency are investigated. The solutions obtained are compared with the ones depending upon both classical Euler-Bernoulli and Timoshenko beam theory to comprehend the responses of nano-beams under the effect of initial stress and small scale in terms of frequencies for both theories. The results supply a better declaration for vibration analysis of nano-beams which are short and stubby with initial stress.