Wave propagation responses of porous bi-directional functionally graded magneto-electro-elastic nanoshells via nonlocal strain gradient theory

This study examines the wave propagation characteristics for a bi-directional functional grading of barium titanate(BaTiO_(3)) and cobalt ferrite(CoFe_(2)O_(4)) porous nanoshells,the porosity distribution of which is simulated by the honeycomb-shaped symmetrical and asymmetrical distribution functions.The nonlocal strain gradient theory(NSGT) and first-order shear deformation theory are used to determine the size effect and shear deformation,respectively.Nonlocal governing equations are derived for the nanoshells by Hamilton's principle.The resulting dimensionless differential equations are solved by means of an analytical solution of the combined exponential function after dimensionless treatment.Finally,extensive parametric surveys are conducted to investigate the influence of diverse parameters,such as dimensionless scale parameters,radiusto-thickness ratios,bi-directional functionally graded(FG) indices,porosity coefficients,and dimensionless electromagnetic potentials on the wave propagation characteristics.Based on the analysis results,the effect of the dimensionless scale parameters on the dispersion relationship is found to be related to the ratio of the scale parameters.The wave propagation characteristics of nanoshells in the presence of a magnetoelectric field depend on the bi-directional FG indices.

Vibration analysis of piezoelectric sandwich nanobeam with flexoelectricity based on nonlocal strain gradient theory

A nonlocal strain gradient theory(NSGT) accounts for not only the nongradient nonlocal elastic stress but also the nonlocality of higher-order strain gradients,which makes it benefit from both hardening and softening effects in small-scale structures.In this study, based on the NSGT, an analytical model for the vibration behavior of a piezoelectric sandwich nanobeam is developed with consideration of flexoelectricity. The sandwich nanobeam consists of two piezoelectric sheets and a non-piezoelectric core. The governing equation of vibration of the sandwich beam is obtained by the Hamiltonian principle. The natural vibration frequency of the nanobeam is calculated for the simply supported(SS) boundary, the clamped-clamped(CC) boundary, the clamped-free(CF)boundary, and the clamped-simply supported(CS) boundary. The effects of geometric dimensions, length scale parameters, nonlocal parameters, piezoelectric constants, as well as the flexoelectric constants are discussed. The results demonstrate that both the flexoelectric and piezoelectric constants enhance the vibration frequency of the nanobeam.The nonlocal stress decreases the natural vibration frequency, while the strain gradient increases the natural vibration frequency. The natural vibration frequency based on the NSGT can be increased or decreased, depending on the value of the nonlocal parameter to length scale parameter ratio.

Vibration and wave propagation analysis of twisted micro-beam using strain gradient theory

In this research, vibration and wave propagation analysis of a twisted micro- beam on Pasternak foundation is investigated. The strain-displacement relations （kine-matic equations） are calculated by the displacement fields of the twisted micro-beam. The strain gradient theory （SGT） is used to implement the size dependent effect at micro-scale. Finally, using an energy method and Hamilton＇s principle, the governing equations of motion for the twisted micro-beam are derived. Natural frequencies and the wave prop- agation speed of the twisted micro-beam are calculated with an analytical method. Also, the natural frequency, the phase speed, the cut-off frequency, and the wave number of the twisted micro-beam are obtained by considering three material length scale parameters, the rate of twist angle, the thickness, the length of twisted micro-beam, and the elastic medium. The results of this work indicate that the phase speed in a twisted micro-beam increases with an increase in the rate of twist angle. Moreover, the wave number is in- versely related with the thickness of micro-beam. Meanwhile, it is directly related to the wave propagation frequency. Increasing the rate of twist angle causes the increase in the natural frequency especially with higher thickness. The effect of the twist angle rate on the group velocity is observed at a lower wave propagation frequency.

Variational principles for buckling and vibration of MWCNTs modeled by strain gradient theory

Variational principles for the buckling and vibration of multi-walled carbon nanotubes （MWCNTs） are established with the aid of the semi-inverse method. They are used to derive the natural and geometric boundary conditions coupled by small scale parameters. Hamilton＇s principle and Rayleigh＇s quotient for the buckling and vibration of the MWCNTs are given. The Rayleigh-Ritz method is used to study the buckling and vibration of the single-walled carbon nanotubes （SWCNTs） and double-walled carbon nanotubes （DWCNTs） with three typical boundary conditions. The numerical results reveal that the small scale parameter, aspect ratio, and boundary conditions have a profound effect on the buckling and vibration of the SWCNTs and DWCNTs.

Static and dynamic stability responses of multilayer functionally graded carbon nanotubes reinforced composite nanoplates via quasi 3D nonlocal strain gradient theory

This manuscript presents the comprehensive study of thickness stretching effects on the free vibration,static stability and bending of multilayer functionally graded(FG)carbon nanotubes reinforced composite(CNTRC)nanoplates.The nanoscale and microstructure influences are considered through a modified nonlocal strain gradient continuum model.Based on power-law functions,four different patterns of CNTs distribution are considered in this analysis,a uniform distribution UD,FG-V CNTRC,FG-X CNTRC,and FG-O CNTRC.A 3D kinematic shear deformation theory is proposed to include the stretching influence,which is neglected in classical theories.Hamilton's principle is applied to derive the governing equations of motion and associated boundary conditions.Analytical solutions are developed based on Galerkin method to solve the governing equilibrium equations based on the generalized higher-order shear deformation theory and the nonlocal strain gradient theory and get the static bending,buckling loads,and natural frequencies of nanoplates.Verification with previous works is presented.A detailed parametric analysis is carried out to highlight the impact of thickness stretching,length scale parameter(nonlocal),material scale parameter(gradient),CNTs distribution pattern,geometry of the plate,various boundary conditions and the total number of layers on the stresses,deformation,critical buckling loads and vibration frequencies.Many new results are also reported in the current study,which will serve as a benchmark for future research.

NEW STRAIN GRADIENT THEORY AND ANALYSIS

A new strain gradient theory which is based on energy nonlocal model is proposed in this paper, and the theory is applied to investigate the size effects in thin metallic wire torsion, ultra-thin beam bending and micro-indentation of polycrystalline copper. First, an energy nonlocal model is suggested. Second, based on the model, a new strain gradient theory is derived. Third, the new theory is applied to analyze three representative experiments.

Bending and wave propagation analysis of axially functionally graded beams based on a reformulated strain gradient elasticity theory

A new size-dependent axially functionally graded(AFG) micro-beam model is established with the application of a reformulated strain gradient elasticity theory(RSGET). The new micro-beam model incorporates the strain gradient, velocity gradient,and couple stress effects, and accounts for the material variation along the axial direction of the two-component functionally graded beam. The governing equations and complete boundary conditions of the AFG beam are derived based on Hamilton's principle. The correctness of the current model is verified by comparing the static behavior results of the current model and the finite element model(FEM) at the micro-scale. The influence of material inhomogeneity and size effect on the static and dynamic responses of the AFG beam is studied. The numerical results show that the static and vibration responses predicted by the newly developed model are different from those based on the classical model at the micro-scale. The new model can be applied not only in the optimization of micro acoustic wave devices but also in the design of AFG micro-sensors and micro-actuators.

Small scale effects on buckling and postbuckling behaviors of axially loaded FGM nanoshells based on nonlocal strain gradient elasticity theory

By means of a comprehensive theory of elasticity, namely, a nonlocal strain gradient continuum theory, size-dependent nonlinear axial instability characteristics of cylindrical nanoshells made of functionally graded material(FGM) are examined. To take small scale effects into consideration in a more accurate way, a nonlocal stress field parameter and an internal length scale parameter are incorporated simultaneously into an exponential shear deformation shell theory. The variation of material properties associated with FGM nanoshells is supposed along the shell thickness, and it is modeled based on the Mori-Tanaka homogenization scheme. With a boundary layer theory of shell buckling and a perturbation-based solving process, the nonlocal strain gradient load-deflection and load-shortening stability paths are derived explicitly. It is observed that the strain gradient size effect causes to the increases of both the critical axial buckling load and the width of snap-through phenomenon related to the postbuckling regime, while the nonlocal size dependency leads to the decreases of them. Moreover, the influence of the nonlocal type of small scale effect on the axial instability characteristics of FGM nanoshells is more than that of the strain gradient one.

Bending Analysis of Functionally Graded One-Dimensional Hexagonal Piezoelectric Quasicrystal Multilayered Simply Supported Nanoplates Based on Nonlocal Strain Gradient Theory

In this study,the nonlocal strain gradient theory is adopted to investigate the static bending deformation of a functionally graded(FG)multilayered nanoplate made of onedimensional hexagonal piezoelectric quasicrystal(PQC)materials subjected to mechanical and electrical surface loadings.The FG materials are assumed to be exponential distribution along the thickness direction.Exact closed-form solutions of an FG PQC nanoplate including nonlocality and strain gradient micro-size dependency are derived by utilizing the pseudo-Stroh formalism.The propagator matrix method is further used to solve the multilayered case by assuming that the layer interfaces are perfectly contacted.Numerical examples for two FG sandwich nanoplates made of piezoelectric crystals and PQC are provided to show the influences of nonlocal parameter,strain gradient parameter,exponential factor,length-to-width ratio,loading form,and stacking sequence on the static deformation of two FG sandwich nanoplates,which play an important role in designing new smart composite structures in engineering.

Wave propagation in graphene reinforced piezoelectric sandwich nanoplates via high-order nonlocal strain gradient theory

Αn analytical method is developed to explore the wave propagation characteristics of piezoelectric sandwich nanoplates in the present work.The sandwich nanoplates are composed of a graphene reinforced composite core layer with two piezoelectric surface layers exposed to electric field.The material properties of the nanocomposite layer are given by the Halpin–Tsai model and mixture’s rule.The Euler–Lagrange equation of the nanoplates is obtained by Hamilton's principle and first-order shear deformation theory.Then,combining the high-order nonlocal strain gradient theory with the hygrothermal constitutive relationship of composite nanoplates,the nonlocal governing equations are presented.Finally,numerical studies are conducted to demonstrate the influences of scale parameters,applied external voltage,temperature variation,moisture variation,graphene size,and weight fraction on wave frequency.The results reveal that low-order and high-order nonlocal parameters and length scale parameters have different effects on wave frequency.The wave frequency can be reduced by increasing temperature and the thickness of graphene.This could facilitate the investigation of the dynamic properties of graphene nanocomposite structures.

Wave Propagation in Fluid-Filled Single-Walled Carbon Nanotube Based on the Nonlocal Strain Gradient Theory

A dynamic Timoshenko beam model is established based on the new nonlocal strain gradient theory and slip boundary theory to study the wave propagation behaviors of fluid-filled carbon nanotubes （CNTs） at nanoscale. The nanoscale effects caused by the CNTs and the inner fluid are simulated by the nonlocal strain gradient effect and the slip boundary effect, respectively. The governing equations of motion are derived and resolved to investigate the wave characteristics in detail. The numerical solution shows that the strain gradient effect leads to the stiffness enhancement of CNTs when the nonlocal stress effect causes the decrease in stiffness. The dynamic properties of CNTs are affected by the coupling of these two scale effects. The flow velocity of fluid inside the CNT is increased due to the slip boundary effect, resulting in the promotion of wave propagation in the dynamic system.

Wave propagation analysis of rotating thermoelastically-actuated nanobeams based on nonlocal strain gradient theory

This paper is concerned with the wave propagation behavior of rotating functionally graded（FG）temperature-dependent nanoscale beams subjected to thermal loading based on nonlocal strain gradient stress field.Uniform,linear and nonlinear temperature distributions across the thickness are investigated.Thermo-elastic properties of FG beam change gradually according to the Mori–Tanaka distribution model in the spatial coordinate.The nanobeam is modeled via a higher-order shear deformable refined beam theory which has a trigonometric shear stress function.The governing equations are derived by Hamilton’s principle as a function of axial force due to centrifugal stiffening and displacement.The solution of these equations is provided employing a Galerkin-based approach which has the potential to capture various boundary conditions.By applying an analytical solution and solving an eigenvalue problem,the dispersion relations of rotating FG nanobeam are obtained.Numerical results illustrate that various parameters including temperature change,angular velocity,nonlocality parameter,wave number and gradient index have significant effects on the wave dispersion characteristics of the nanobeam under study.The outcome of this study can provide beneficial information for the next-generation research and the exact design of nano-machines including nanoscale molecular bearings,nanogears,etc.

Vibration analysis of nano-structure multilayered graphene sheets using modified strain gradient theory

In this paper, for the first time, the modified strain gradient theory is used as a new size-dependent Kirchhoff micro-plate model to study the effect of interlayer van der Waals （vdW） force for the vibration analysis of multilayered graphene sheets （MLGSs）. The model contains three material length scale parameters, which may effectively capture the size effect. The model can also degenerate into the modified couple stress plate model or the classical plate model, if two or all of the material length scale parameters are taken to be zero. After obtaining the governing equations based on modified strain gradient theory via principle of minimum potential energy, as only infinitesimal vibration is considered, the net pressure due to the vdW interaction is assumed to be linearly proportional to the deflection between two layers. To solve the goveming equation subjected to the boundary conditions, the Fourier series is assumed for w = w（x, y）. To show the accuracy of the formulations, present results in specific cases are compared with available results in literature and a good agreement can be seen. The results indicate that the present model can predict prominent natural frequency with the reduction of structural size, especially when the plate thickness is on the same order of the material length scale parameter.

On complete and micropolar-based incomplete strain gradient theories for periodic lattice structures

The micropolar(MP) and strain gradient(SG) continua have been generally adopted to investigate the relations between the macroscopic elastic constants and the microstructural geometric parameters. Owing to the fact that the microrotation in the MP theory can be expressed in terms of the displacement gradient components, we may regard the MP theory as a particular incomplete SG theory called the MPSG theory,compared with the existing SG theories which are deemed complete since all the SGs are included. Taking the triangular lattice comprising zigzag beams as an example, it is found that as the angle of the zigzag beams increases, the bending of the beams plays a more important role in the total strain energy, and the difference between the results by the two theories gradually decreases. Finally, the models are verified with the pure bending and simple shear of lattices by comparing with the results obtained by the finite element method(FEM)-based structure analyses.

A FIBER-BRIDGING MODEL WITH STRESS GRADIENT EFFECTS

A fiber-bridging model with stress gradient effects is proposed for unidirectional fiber-reinforced composites. The stress gradient terms are introduced by solving a micromechanical model under a non-uniform stress loading. It is shown that the stress gradient effect is significant on both the fiber-bridging stress distribution and the value of the critical load of fiber failure.

New applications of a generalized Hooke's law for second gradient materials

We provide analytical solutions to the problems of a circular bending of a beam in plane strain and the torsion of a non-circular cross-section beam, the beams obeying a second-gradient elasticity law proposed by the author, following a previous suggestion of delrlsola et al. （2009）. The motivation was to find benchmark analytical solutions that can serve to grasp the physical foundations of second gradient elasticity laws for heterogeneous materials. The analytical solution of the circular beam problem presents the additional advantage to establish some nice properties on the unknown second gradient elastic moduli introduced by Enakoutsa （2014） model and the classical elasticity constants for both incompressible and compressible heterogeneous elastic materials. A framework to find the elastic moduli of the new model is also proposed.

Size effects on the mixed modes and defect modes for a nano-scale phononic crystal slab

The size-dependent band structure of an Si phononic crystal(PnC)slab with an air hole is studied by utilizing the non-classic wave equations of the nonlocal strain gradient theory(NSGT).The three-dimensional(3D)non-classic wave equations for the anisotropic material are derived according to the differential form of the NSGT.Based on the the general form of partial differential equation modules in COMSOL,a method is proposed to solve the non-classic wave equations.The bands of the in-plane modes and mixed modes are identified.The in-plane size effect and thickness effect on the band structure of the PnC slab are compared.It is found that the thickness effect only acts on the mixed modes.The relative width of the band gap is widened by the thickness effect.The effects of the geometric parameters on the thickness effect of the mixed modes are further studied,and a defect is introduced to the PnC supercell to reveal the influence of the size effects with stiffness-softening and stiffness-hardening on the defect modes.This study paves the way for studying and designing PnC slabs at nano-scale.

Wave propagation analysis for a second strain gradient rod theory

In this work,an enriched model describing the longitudinal wave propagation is established based on Mindlin’s Second Strain Gradient(SSG)theory,which can describe the heterogeneity caused by the micro-structure interactions in the frame of continuum mechanics.The governing equation and associated boundary conditions are derived based on Hamilton’s principle,then the dispersion relation of non-classical longitudinal wave together with the extra-waves appearing exclusively in SSG theory model are investigated.The investigations are based on the modal density,energy flow,and forced response of the rod.Wave transmission and reflection through planar interfaces based on the proposed model have been calculated.Finally,the results of the enriched model are well interpreted by comparing with the classical theory results,and some useful conclusions are derived on the SSG theory based model in the wave propagation characterization.

Transverse shear and normal deformation effects on vibration behaviors of functionally graded micro-beams

A quasi-three dimensional model is proposed for the vibration analysis of functionally graded(FG)micro-beams with general boundary conditions based on the modified strain gradient theory.To consider the effects of transverse shear and nor-mal deformations,a general displacement field is achieved by relaxing the assumption of the constant transverse displacement through the thickness.The conventional beam theories including the classical beam theory,the first-order beam theory,and the higher-order beam theory are regarded as the special cases of this model.The material proper-ties changing gradually along the thickness direction are calculated by the Mori-Tanaka scheme.The energy-based formulation is derived by a variational method integrated with the penalty function method,where the Chebyshev orthogonal polynomials are used as the basis function of the displacement variables.The formulation is validated by some comparative examples,and then the parametric studies are conducted to investigate the effects of transverse shear and normal deformations on vibration behaviors.

On the plastic buckling of curved carbon nanotubes

This research,for the first time,predicts theoretically static stability response of a curved carbon nanotube(CCNT)under an elastoplastic behavior with several boundary conditions.The CCNT is exposed to axial compressive loads.The equilibrium equations are extracted regarding the Euler–Bernoulli displacement field by means of the principle of minimizing total potential energy.The elastoplastic stress-strain is concerned with Ramberg–Osgood law on the basis of deformation and flow theories of plasticity.To seize the nano-mechanical behavior of the CCNT,the nonlocal strain gradient elasticity theory is taken into account.The obtained differential equations are solved using the Rayleigh–Ritz method based on a new admissible shape function which is able to analyze stability problems.To authorize the solution,some comparisons are illustrated which show a very good agreement with the published works.Conclusively,the best findings confirm that a plastic analysis is crucial in predicting the mechanical strength of CCNTs.