Nonlocal Continuum Elasticity Research Articles

A fractional nonlocal elastic model for lattice wave analysis

This paper investigates the link between lattice elasticity and nonlocal continuum mechanics from one-dimensional and multi-dimensional wave propagation problems. Eringen (1983) closed the bridge between one-dimensional linear lattice elasticity (called Lagrange lattice) and nonlocal elastic continuum. The Born-Kármán wave dispersive properties of the one-dimensional lattice are accurately fitted by Eringen's nonlocal differential elastic law using asymptotic or phenomenological arguments. A fractional version of Eringen's nonlocal model based on the introduction of a fractional Laplacian for the nonlocal differential operators can improve the accuracy in matching the wave dispersive curve of lattice dynamics. A multi-dimensional generalization of Lagrange lattice for cubic crystals is the lattice of Gazis et al. composed of central and angular interactions. We show in this paper that the differential Eringen's nonlocal elasticity differs from the lattice-based nonlocal elasticity continuum. In particular, Eringen's nonlocal elasticity preserves the isotropic property, whereas the cubic lattice is only isotropic in the low frequency regimes. The fractional generalization of Eringen's model accurately fitted the wave dispersive properties of the cubic crystal in the first Brillouin zone. The fractional generalization of Eringen's model highlights an isotropic response in the low frequency regimes and a cubic symmetry in the high frequency regime.

Influences of defects on the propagation of transverse waves in periodic piezoelectric laminate structure with nanoscaled layers

By the stiffness matrix method based on the nonlocal elastic continuum theory, the transmission spectra of the transverse elastic waves propagating normally in nanoscaled periodic piezoelectric laminates with defects are investigated in detail in this paper. Defect is introduced by replacing a constituent layer or inserting another layer in a unit-cell. The thicknesses of all sub-layers and defect layer are at the nanoscale. Simultaneously, complex band structures and localization factors are calculated and employed to verify the band structures described by the transmission spectra. The influences of the defect type, the defect material, the volume fraction of the defect layer in the unit-cell, the piezoelectric constants of the defect layer, and the nanoscale size-effect on the transmission spectra are analyzed and discussed in detail. The present study provides a theoretical guide for the design and applications of the nanoscaled acoustic filters or transducers with defects.

Band Structure Analysis of SH Wave Propagating in Nanoscale Layered Metamaterial Structures

This study is devoted to the analysis of the band structures of the anti-plane transverse wave (SH wave) in nanoscale layered metamaterial structures. Attention is restricted to normal incidence of waves. The localization factor is introduced to characterize the band structures. The general transfer matrix method based on the nonlocal elastic continuum theory is employed to calculate the localization factor. Based on the analysis of band structures, the influences of random disorder of the internal characteristic length and the external thickness of each sub-layer, the aperiodic arrangements, the location of different material components, the ratio of mass density, the ratio of the transverse wave velocity, the ratio of the internal characteristic length or the external thickness of each sub-layer on the band structures, the cut-off frequency, the peak points and the dense band zones are investigated and discussed in detail, which can provide some new thoughts for the designs and applications of the nanoscale wave devices.

Size-effect on the band structures of the transverse elastic wave propagating in nanoscale periodic laminates

Based on the nonlocal elastic continuum theory, the band structures of the anti-plane transverse elastic wave propagating normally in nanoscale periodic laminates is investigated in detail in this paper. The localization factor computed by using the transfer matrix method is employed to describe the band structures. The variations of the displacement are calculated and analyzed. The influences of the size-effect of each sub-layer, the material and structural parameters on the cut-off frequency and the dense band zones are discussed in detail. The study presented in this paper provides a theoretical guide for the design and applications of the novel nanoscale wave devices.

Waves in nonlocal thermoelastic solids of type III

AbstractIn this paper, the propagation of time‐harmonic thermoelastic plane waves is studied in an infinite nonlocal elastic continuum. The type III Green‐Naghdi model (with energy dissipation) of generalized thermoelasticity and the Eringen's nonlocal elasticity model are adopted to address this problem. We found two sets of the coupled longitudinal waves which are dispersive in nature and experience attenuation. In addition to the coupled waves, there also exists one independent vertically shear‐type wave which is dispersive but experiences no attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear‐type wave is found to face a critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. Reflection phenomenon of an incident coupled longitudinal waves from a rigid and thermally insulated boundary surface of a homogeneous and isotropic nonlocal thermoelastic half‐space is investigated. Using these boundary conditions, the formulae for various reflection coefficients and their respective energy ratios are presented. For a particular model, various graphs are plotted to analyze the behavior of the phase speeds, reflection coefficients and their respective energy ratios. The amplitude ratios of the reflected waves and their respective energy ratios are determined analytically. For a particular model, the effect of elastic nonlocality parameter on the variations of phase speeds, attenuation coefficients, amplitude ratios and corresponding energy ratios of the reflected waves are presented graphically. Finally, analysis of the various results have been interpreted.

On the vibration and buckling analysis of quadrilateral and triangular nanoplates using nonlocal spline finite strip method

In this paper, stability and free vibration characteristics of quadrilateral and triangular orthotropic single-layered graphene sheets are investigated based on Eringen’s nonlocal continuum elasticity theory. The principle of virtual work is used to derive the governing nanoplate equations. The geometry of arbitrary-shaped quadrilateral and triangular nanoplates in the global Cartesian coordinate system is transformed into the natural coordinate system. B3-spline finite strip method is then employed to extract the buckling and vibration equations. The obtained results are first validated against those reported elsewhere. A comprehensive parametric study is performed while the effect of different parameters such as nonlocal parameter, side length, aspect ratio, shape of the nanoplate, mode number, and boundary conditions is investigated. It is demonstrated that a small-scale effect takes a considerable role in the buckling and vibration behavior of quadrilateral and triangular nanoplates.

Application of local elasticity continuum damping models in nonlinear dynamic analysis

In order to have a more generic representation of the damping phenomenon in seismic analysis of structures, in this paper, the already existing nonlocal elasticity continuum damping models have been adapted and extended to the inelastic domain. Adaptations of two nonlocal elasticity based damping models, Russell’s spatial hysteresis model and the extended Sorrentino model, into the nonlinear dynamic analysis is presented. Galerkin based finite element schemes are developed and the numerical implementation of the models are outlined. The performances of the nonlocal elasticity-based damping models are illustrated by studying the nonlinear dynamic responses of a four-story Reinforced Concrete frame designed to the Eurocodes. The incremental dynamic analysis study presented illustrates the fact that the proposed models are devoid of the large spurious damping actions commonly exhibited by the popular classical Rayleigh damping model. It has also been shown that in a nonlinear dynamic analysis, the proposed adaptation of the nonlocal elasticity continuum damping models might be a more realistic alternative to the popular classical Rayleigh damping model. The proposed models may be easily implemented in an existing software framework capable of solving both ordinary and integro-differential equations without adding markedly to the computational effort.

Application of nonlocal elasticity continuum damping models in nonlinear dynamic analysis

Application of nonlocal elasticity continuum damping models in nonlinear dynamic analysis

Exact and Nonlocal Solutions for Vibration of Axial Lattice with Direct and Indirect Neighboring Interactions

This paper investigates the effect of long-range interaction forces in the longitudinal vibration of an axial linear elastic lattice. The eigenfrequencies of this axial lattice with direct and indirect linear elastic interactions are expressed in closed-form solutions for fixed-fixed and fixed-free boundary conditions. The paper starts with the two-neighbor interaction problem, and then generalizes the result to the more general N-neighbor interaction problem. Starting from a continualization of the higher-order difference equations, a nonlocal elastic continuum is constructed to approximate the behavior of the axial lattice with generalized short and long-range interactions. It is shown, while preserving the definite positiveness of the generalized lattice strain energy, that the associated nonlocal continuum is equivalent to a stress gradient type of Eringen's nonlocal model, which is able to capture the main scale phenomena of the generalized lattice. The length scale factor of the nonlocal model is calibrated to the size of the influence domain for the long-range interaction. The paper ends with a discussion about the effect of the discrete interaction kernel on the sensitivity of the eigenfrequencies with respect to scale effects. Exact analytical natural frequencies are calculated for discrete power law or exponential-based kernels. The stress gradient nonlocal theory offers an efficient engineering continuum framework, which accurately fits the response of generalized lattice with N-neighbor interaction.

Free Vibrational Response of Single-Layered Graphene Sheets Embedded in an Elastic Matrix using Different Nonlocal Plate Models

In this paper, the small scale effects are incorporated into the free vibration analysis of single-layered graphene sheets (SLGSs) embedded in an elastic medium. To this end, Eringen’s nonlocal elasticity continuum are applied to the different types of plate theory namely as the classical plate theory (CLPT), first order shear deformation theory (FSDT), and higher order shear deformation theory (HSDT). Winkler and Pasternak foundation models used to simulate the surrounding elastic medium are compared with each other. Explicit expressions are derived to calculate the natural frequencies of square SLGSs corresponding to each type of nonlocal plate model. Selected numerical results are given to indicate the influence of the nonlocal parameter, Winkler and Pasternak elastic moduli, mode number, and the side length of SLGSs in detail. Also, comparison is made between the vibrational responses of SLGSs obtained through different nonlocal plate theories. It is found that the elastic foundation and value of nonlocal parameter have quite significant effects on the natural frequencies of SLGSs and these effects are influenced by mode number as well as side length.DOI: http://dx.doi.org/10.5755/j01.mech.23.5.14883

Thermo-electro-radial coupling nonlinear instability of piezoelectric shear deformable nanoshells via nonlocal elasticity theory

In the present article, the size-dependent nonlinear thermo-electro-mechanical instability of piezoelectric nanoshells under a combined action of hydrostatic pressure, lateral electric field and uniform changes in temperature is examined in the presence of the geometric imperfection sensitivity. The non-classical formulations given herein are built upon the nonlocal continuum elasticity within the framework of the shear deformation shell theory incorporating von Karman nonlinearity due to large deflection. The nonlocal-based governing equations are constructed based on the minimum potential energy of the system and then they are deduced using boundary layer theory of shell buckling. By applying an efficient perturbation-based solution methodology, closed-form solutions including explicit expressions are extracted for nonlocal postbuckling equilibrium paths of thermo-electro-mechanical excited piezoelectric nanoshells. It is displayed that for the both local and nonlocal shell models, the lateral applied electric fields with positive and negative signs induce, respectively, initial contraction and expansion in the piezoelectric nanoshell which causes in order to increase and decrease the critical shortening relevant to the electro-radial instability of piezoelectric nanoshell.

Development of efficient size-dependent plate models for axial buckling of single-layered graphene nanosheets using molecular dynamics simulation

In the current paper, axial buckling characteristics of nanoscaled single-layered graphene sheets (SLGSs) are investigated on the basis of Eringen’s nonlocal elasticity continuum and different plate theories namely as classical plate theory and first-order shear deformation theory. Through implementing of the nonlocal equations into the different types of plate theory, nonlocal plate models are developed to consider the small-scale effects in the axial buckling analysis of SLGSs. Generalized differential quadrature method is utilized to discretize the governing differential equations of the nonlocal elastic plate models along simply-supported and clamped boundary conditions. Afterward, molecular dynamics (MD) simulations are performed for a series of SLGS with various values of side-length and chiralities, the results of which are matched with those of nonlocal plate models to extract the appropriate values of nonlocal parameter. It is found that among the type of boundary conditions, chirality, and nonlocal plate theory, boundary conditions have the most significant influence on the recommended values of nonlocal parameter to predict the axial buckling behavior of SLGSs.

Calibration of developed nonlocal anisotropic shear deformable plate model for uniaxial instability of 3D metallic carbon nanosheets using MD simulations

In comparison with semiconducting carbon structures, the metallic carbon exhibits more novel properties due to their high density of states as well as superconductivity. As a result, prediction of mechanical properties associated with three-dimensional (3D) carbon structures is very attractive. In this paper, the size-dependent nonlinear uniaxial instability of 3D metallic carbon nanosheets containing interlocking hexagonal carbon bonds is investigated based on a novel calibrated nonlocal anisotropic plate model. To accomplish this purpose, Eringen’s nonlocal continuum elasticity is incorporated to an exponential shear deformation plate theory. To extract the proper value of nonlocal parameter, the critical buckling loads and associated postbuckling equilibrium curves obtained by the established size-dependent plate model are compared with those of molecular dynamics (MD) simulations corresponding to various nanosheet side lengths and different boundary conditions. It is displayed that the calibrated nonlocal plate model with the proposed proper value of nonlocal parameter equal to 1.12 nm for fully simply supported edge supports and 1.26 nm for fully clamped edge supports has a very good capability to anticipate accurately both of the buckling and postbuckling characteristics of the uniaxially loaded 3D metallic carbon nanosheets.

Nonlocal size dependency in nonlinear instability of axially loaded exponential shear deformable FG-CNT reinforced nanoshells under heat conduction

The present study deals with size-dependent nonlinear instability characteristics of functionally graded carbon nanotube (FG-CNT) reinforced composite shells at nanoscale subjected to axial compression combined with through-thickness heat conduction. To take size dependency into account, Eringen’s nonlocal continuum elasticity is incorporated to a novel shear deformation shell theory including a refined exponential distribution for transverse shear strain. In addition to the uniform distribution (UD) of CNT reinforcements, three FG patterns are also considered, namely FG-A, FG-V and FG-X. Also, on the basis of polynomial series, the temperature variation due to the through-thickness heat conduction is estimated. Via a perturbation-based boundary layer-type solving procedure, explicit expressions for nonlocal equilibrium curves are proposed relevant to the prebuckling and postbuckling regimes of FG-CNT exponential shear deformable nanoshells with temperature-dependent and temperature-independent material properties. It is observed that by taking the nonlocality size effect into consideration, the influence of the through-thickness heat conduction on the nonlinear axial instability response of FG-CNT reinforced nanoshells becomes more significant.

Development an efficient calibrated nonlocal plate model for nonlinear axial instability of zirconia nanosheets using molecular dynamics simulation

New ceramic materials containing nanoscaled crystalline phases create a main object of scientific interest due to their attractive advantages such as biocompatibility. Zirconia as a transparent glass ceramic is one of the most useful binary oxides in a wide range of applications. In the present study, a new size-dependent plate model is constructed to predict the nonlinear axial instability characteristics of zirconia nanosheets under axial compressive load. To accomplish this end, the nonlocal continuum elasticity of Eringen is incorporated to a refined exponential shear deformation plate theory. A perturbation-based solving process is put to use to derive explicit expressions for nonlocal equilibrium paths of axial-loaded nanosheets. After that, some molecular dynamics (MD) simulations are performed for axial instability response of square zirconia nanosheets with different side lengths, the results of which are matched with those of the developed nonlocal plate model to capture the proper value of nonlocal parameter. It is demonstrated that the calibrated nonlocal plate model with nonlocal parameter equal to 0.37nm has a very good capability to predict the axial instability characteristics of zirconia nanosheets, the accuracy of which is comparable with that of MD simulation.

Nonlocal temperature-dependent postbuckling behavior of FG-CNT reinforced nanoshells under hydrostatic pressure combined with heat conduction

The objective of this paper is to anticipate the size-dependent nonlinear instability response of functionally graded carbon nanotube (FG-CNT) reinforced nanocomposite nanoshells with temperature-dependent material properties under combination of hydrostatic pressure and through-thickness heat conduction. To this purpose, a new size-dependent shell model is constructed based upon implementation of the Eringen’s nonlocal continuum elasticity theory in conjunction with von Karman kinematics nonlinearity into a refined shell theory with exponential function to distribute the shear deformation. In addition to the uniform distribution (UD) of CNT reinforcements, three FG patterns are also considered namely FG-A, FG-V and FG-X. Moreover, the temperature variation due to the through-thickness heat conduction is evaluated using polynomial series. Subsequently, by transforming the problem to a boundary layer-type one and with the aid of a two-stepped perturbation solving process, explicit expressions for nonlocal temperature-dependent load–deflection and load-shortening curves of exponential shear deformable FG-CNT reinforced nanoshells with and without initial geometric imperfection are presented. It is indicated that the reduction in the value of critical buckling pressure due to the nonlocality size effect is more prominent for FG-A distribution pattern in comparison with other ones, especially for thinner FG-CNT reinforced nanoshells.

Band structure properties of elastic waves propagating in the nanoscaled nearly periodic layered phononic crystals

The localization factor is used to describe the band structures for P wave propagating normally in the nanoscaled nearly periodic layered phononic crystals. The localization factor is calculated by the transfer matrix method based on the nonlocal elastic continuum theory. Three kinds of nearly periodic arrangements are concerned, i.e., random disorder, quasi-periodicity and defects. The influences of randomly disordered degree of the sub-layer's thickness and mass density, the arrangement of quasi-periodicity and the location of defect on the band structures and cut-off frequency are analyzed in detail.

Nonlinear instability of hydrostatic pressurized hybrid FGM exponential shear deformable nanoshells based on nonlocal continuum elasticity

Abstract The size-dependent radial buckling and postbuckling behavior of functionally graded cylindrical shells at nanoscale integrated with piezoelectric nanolayers is studied in this paper on the basis of nonlocal elasticity theory within the framework of exponential shear deformation shell theory to capture the influence of transverse shear deformation in a refined form. By introducing a new reference surface, the stretching-bending coupling terms due to unsymmetrical material characteristics related to the functionally graded substrate of nanoshell are removed. After that, with the aid of boundary layer of shell buckling, the non-classical partial nonlinear differential equations are constructed to describe the nonlocal instability of nanoscaled shells. Finally, a perturbation-based solution methodology is utilized to propose explicit expressions for the nonlocal equilibrium paths associated with the both prebuckling and postbuckling domains of hybrid functionally graded nanoshells subjected to the combination of hydrostatic pressure and lateral electric field. It is seen that the nonlocality size effect causes to reduce the critical hydrostatic pressure, but it leads to increase the associated shortening of the movable ends of hybrid functionally graded nanoshells.

Size dependency in axial postbuckling behavior of hybrid FGM exponential shear deformable nanoshells based on the nonlocal elasticity theory

The objective of this study is to examine the nonlocal nonlinear instability of functionally graded cylindrical shells at nanoscale integrated with piezoelectric nanolayers under combination of axial compressive load and lateral electric field. Eringen's nonlocal continuum elasticity is incorporated within the framework of the exponential shear deformation shell theory to consider the influence of transverse shear deformation in a refined form. Additionally, in order to eliminate the stretching-bending coupling terms, the change in the position of physical neutral plane corresponding to different volume fractions is taken into account. With the aid of the boundary layer theory of shell buckling and employing a perturbation-based solution methodology, explicit expressions for the size-dependent equilibrium paths before and after buckling point are proposed for functionally graded hybrid nanoshells with various nonlocal parameters, material property gradient indexes and subjected to different values of lateral electric field. It is indicated that the both width and depth of the snap-through phenomenon related to the axial postbuckling behavior of hybrid FGM nanoshells decrease due to the nonlocality influence.

Band structures of transverse waves in nanoscale multilayered phononic crystals with nonlocal interface imperfections by using the radial basis function method

A radial basis function collocation method based on the nonlocal elastic continuum theory is developed to compute the band structures of nanoscale multilayered phononic crystals. The effects of nonlocal imperfect interfaces on band structures of transverse waves propagating obliquely or vertically in the system are studied. The correctness of the present method is verified by comparing the numerical results with those obtained by applying the transfer matrix method in the case of nonlocal perfect interface. Furthermore, the influences of the nanoscale size, the impedance ratio and the incident angle on the cut-off frequency and band structures are investigated and discussed in detail. Numerical results show that the nonlocal interface imperfections have significant effects on the band structures in the macroscopic and microscopic scale.