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.

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.

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.

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 in Graphene Platelet-Reinforced Piezoelectric Sandwich Composite Nanoplates with Nonlocal Strain Gradient Effects

This research develops an analytical approach to explore the wave propagation problem of piezoelectric sandwich nanoplates.The core of the sandwich nanoplates is a nanocomposite layer reinforced with graphene platelets,which is integrated by two piezoelectric layers exposed to electric field.The material properties of nanocomposite layer are obtained by the Halpin–Tsai model and the rule of mixtures.The Euler–Lagrange equations of nanoplates are derived from Hamilton’s principle.By using the nonlocal strain gradient theory,the nonlocal governing equations are presented.Finally,numerical studies are conducted to demonstrate the influences of propagation angle,small-scale and external loads on wave frequency.The results reveal that the frequency changes periodically with the propagation angle and can be reduced by increasing voltage,temperature and the thickness of graphene platelets.

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.

传输高速纳米流的石墨烯增强复合纳米管的接触动力响应:尺寸依赖性对局部/全局瞬态响应的影响

流体输送纳米管在纳米机电系统中起着关键作用.本文研究了输送高速纳米流体石墨烯增强复合材料(GRC)纳米管在横向低速冲击下的接触动态响应和应力场,建立了考虑滑移流、非局部应力和应变梯度效应的尺寸相关模型.基于精细梁理论,推导了流致后屈曲和接触振动的控制方程,其中后屈曲平衡路径为冲击振动分析提供了初始构型.采用两步摄动-高阶Galerkin截断-Runge-Kutta(R-K)方法的计算模式研究接触动力响应.通过收敛性分析,得到了保证精度所需的截断项.获得了接触力曲线、跨中位移时程曲线和应力场,以指导强度设计.此外,还揭示了流致后屈曲纳米管在冲击作用下的动态跳跃失稳行为.结果揭示了非局部应力和应变梯度对接触动态响应和应力场的双重影响,并提供了对尺寸效应敏感的流速范围.