On modeling of wave propagation in a thermally affected GNP-reinforced imperfect nanocomposite shell

Due to rapid development of process manufacturing, composite materials with porosity have attracted commercial attention in promoting engineering applications. For this regard, in this research wave propagation-thermal characteristics of a size-dependent graphene nanoplatelet-reinforced composite (GNPRC) porous cylindrical nanoshell in thermal environment are investigated. The effects of small scale are analyzed based on nonlocal strain gradient theory (NSGT). The governing equations of the laminated composite cylindrical nanoshell in thermal environment have been evolved using Hamilton’s principle and solved with the assistance of the analytical method. For the first time, wave propagation-thermal behavior of a GNPRC porous cylindrical nanoshell in thermal environment based on NSGT is examined. The results show that by increasing the thickness, the effect of porosity on the phase velocity decreases. Another important result is that by increasing the value of the radius, the difference between the minimum and maximum values of the phase velocity increases. Finally, influence of temperature change, wave number, angular velocity and different types of porosity distribution on phase velocity are investigated using the mentioned continuum mechanics theory. As a useful suggestion, for designing of a GPLRC nanostructure should be attention to the GNP weight function and radius, simultaneously.

     

Application of nonlocal strain–stress gradient theory and GDQEM for thermo-vibration responses of a laminated composite nanoshell

In this article, thermal buckling and frequency analysis of a size-dependent laminated composite cylindrical nanoshell in thermal environment using nonlocal strain–stress gradient theory are presented. The thermodynamic equations of the laminated cylindrical nanoshell are based on first-order shear deformation theory, and generalized differential quadrature element method is implemented to solve these equations and obtain natural frequency and critical temperature of the presented model. The results show that by considering C–F boundary conditions and every even layers’ number, in lower value of length scale parameter, by increasing the length scale parameter, the frequency of the structure decreases but in higher value of length scale parameter this matter is inverse. Finally, influences of temperature difference, ply angle, length scale and nonlocal parameters on the critical temperature and frequency of the laminated composite nanostructure are investigated.

     

Nonlinear resonance responses of size-dependent functionally graded cylindrical microshells with thermal effect and elastic medium

The nonlinear resonance responses of functionally graded (FG) cylindrical microshells with the elastic medium is investigated by considering thermal and scale effects. First, using the modified couple stress theory, the nonlinear dynamics model for FG microshell are established. Then the reduced nonlinear differential equations are derived by Galerkin’s method and static condensation. Finally, subharmonic, superharmonic and primary resonances of FG cylindrical microshells are analyzed by a perturbation method. In addition, the bifurcation characteristics of the nonlinear dynamic responses are investigated by some numerical examples. The effects of key parameters (modal damping, excitation frequency, foundation medium, scale parameter and thermal effect) on the nonlinear resonance responses are also discussed by numerical simulation.

     

A geometrically nonlinear size-dependent hypothesis for porous functionally graded micro-plate

The static bending behavior of porous functionally graded (PFG) micro-plate under the geometrically nonlinear analysis is studied in this article. A small-scale nonlinear solution is established using the Von-Kármán hypothesis and the modified couple stress theory (MCST). To obtain the deflection of the plate, the Reddy higher-order plate theory coupled with isogeometric analysis (IGA) is utilized. The distribution of porosities is assumed to be even and uneven across the plate’s thickness and the effective material properties of porous functionally graded micro-plate are calculated using the refined rule-of-mixture hypothesis. The influence of power index, porosity parameter and material length scale parameter on the nonlinear behaviors of static bending of porous FGM micro-plates are also investigated using several numerical examples.

     

Nonlinear torsional buckling and postbuckling analysis of cylindrical silicon nanoshells incorporating surface free energy effects

In the present study, a size-dependent shell model is developed which can afford to describe the nonlinear torsional buckling and postbuckling characteristics of cylindrical nanoshells in the presence of surface stress effects. To accomplish this purpose, the Gurtin–Murdoch theory of elasticity together with the von Karman geometric nonlinearity is implemented into the first-order shear deformation shell theory. A linear variation through the thickness is considered for the normal stress component of the bulk to satisfy the balance conditions on the free surfaces of the nanoshell. By means of the virtual work principle, the non-classical governing differential equations are constructed in which the transverse displacement and Airy stress function are considered as independent variables. Thereafter, a boundary layer theory is employed including the effect of surface stress in conjunction with the nonlinear prebuckling deformations and the large postbuckling deflections. Subsequently, an efficient solution methodology based on an improved perturbation technique is put to use to obtain the size-dependent critical torsional buckling loads and the associated postbuckling equilibrium paths. It is observed that the torsional load exhibits a significant increase after reaching the minimum postbuckling load. Also, it is revealed that the effect of surface stress becomes negligible at high values of the deflection.

     

Buckling and post-buckling of magneto-electro-thermo-elastic functionally graded porous nanobeams

The problem of the nonlinear thermal buckling and post-buckling of magneto-electro-thermo-elastic functionally graded porous nanobeams is analyzed based on Eringen’s nonlocal elasticity theory and by using a refined beam model. The beams with immovable clamped ends are exposed to the external electric voltages, magnetic potentials, a uniform transverse load and uniform temperature change. For the first time, the four types of porosity distribution in the nanobeam are considered and compared in complex electric–magnetic fields. Besides, the new formula of the effective material properties is proposed in this paper to simultaneously estimate the material distribution and porosity distribution in the thickness direction. The generalized variation principle is used to induce the governing equations, then the approximate analytical solution of the METE-FG nanobeams based on physical neutral surface is obtained by using a two-step perturbation technique. Finally, detailed parametric analyses are performed to get an insight into the effects of different physical parameters, including the slenderness ratio, small-scale parameter, volume fraction index, external electric voltages, magnetic potentials, porosity coefficient and different porosity distributions, for providing an effective way to improve post-buckling strength of porous beams.

     

Structural topology optimization on dynamic compliance at resonance frequency in thermal environments

This paper [r] carries out topology optimization to minimize structural dynamic compliance at resonance frequencies in thermal environments. The resonance response is the main dynamic component, minimization of which could possibly change structural dynamic characteristics significantly. A bi-material square plate subjected to uniform temperature rise and driven by harmonic load is investigated in pre-buckling state. The compressive stress induced by thermal environment is considered as pre-stress in dynamic analysis, which could reduce stiffness of the structure and alter the optimal topology. Sensitivity analysis is carried out through adjoint method efficiently. As natural frequencies are constantly changing during the optimization, the associated sensitivity should be calculated in which multiple-frequency case is briefly discussed. Mode switching may occur during the optimization, and mode tracking technique is adopted. Numerical results show that the topology is mainly determined by the excited modes, and could be altered by the location of the applied load if different modes are excited. The natural frequencies become larger in optimal design and the dynamic compliance decreases in nearby frequency band. The critical buckling temperature increases as optimization proceeds, indicating the structure is always in pre-buckling state.     

Numerical solution of finite modified Reynolds equation for couple stress squeeze film lubrication of porous journal bearings

In this paper, the rheological effect of couple stress fluids on the static and dynamic characteristics of squeeze film lubrication in finite porous journal bearings is studied. The finite modified Reynolds equation is derived from the Stokes constitutive equations for couple stress fluids and is solved numerically by using the finite difference technique. The applied load is considered as a sinusoidal function of time to simulate the bearings operating under cyclic loads. Under a cyclic load, the effect of couple stress is to reduce the velocity of the journal centre and to increase the minimum permissible height of the squeeze film.     

Forced vibrations of fluid-conveyed double piezoelectric functionally graded micropipes subjected to moving load

This article is aimed to study the forced vibrations of double piezoelectric functionally graded material micropipes conveying fluid carrying a moving load based on the flexoelectricity theory and modified couple stress theory. The two micropipes are identical and are connected with each other continuously by visco-Pasternak medium. The top micropipe is traversed by a load and is labeled as the primary micropipe. The bottom micropipe is labeled as the secondary micropipe. Non-classical governing equations together with corresponding boundary conditions based on the Hamilton’s principle are obtained and then solved by differential quadrature and Runge–Kutta methods. The influences of the material gradient index, length scale parameter, potential electrical, visco-Pasternak foundation, fluid velocity, various boundary conditions, velocity of the moving load and flexoelectric effect on the dimensionless dynamic deflections of system are discussed. The results show that the flexoelectric and piezoelectric effects tend to increase/decrease the dimensionless dynamic deflections of the primary/secondary pipe, respectively. In addition, the effects of spring constant, shear constant and damping coefficient of the visco-Pasternak medium on the dynamic displacement of the primary and secondary micropipes are vice versa.     

Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory

In this article, the free vibration response of sandwich plates with porous electro-magneto-elastic functionally graded (MEE-FG) materials as face sheets and functionally graded carbon nanotube-reinforced composites (FG-CNTRC) as core is investigated. To this end, four-variable shear deformation refined plate theory is exploited. The properties of functionally graded material plate are assumed to vary along the thickness direction of face sheets according to modified power-law expression. Furthermore, properties of FG-CNTRC layer are proposed via a mixture rule. Hamilton’s principle with a four-variable tangential–exponential refined theory is used to obtain the governing equations and boundary conditions of plate. An analytical solution approach is utilized to get the natural frequencies of embedded porous FG plate with FG-CNTRC core subjected to magneto-electrical field. A parametric study is led to fulfill the effects of porosity parameter, external magnetic potential, external electric voltage, types of FG-CNTRC, and different boundary conditions on dimensionless frequencies of porous MEE-FG sandwich plate. It is noteworthy that the numerical consequences can serve as benchmarks for future investigations for this type of structures with porous mediums.

     

Size-dependent vibration analysis of multi-span functionally graded material micropipes conveying fluid using a hybrid method

Microscale fluid-conveying pipes and functionally graded materials (FGMs) have many potential applications in engineering fields. In this paper, the free vibration and stability of multi-span FGM micropipes conveying fluid are investigated. The material properties of FGM micropipes are assumed to change continuously through thickness direction according to a power law. Based on modified couple stress theory, the governing equation and boundary conditions are derived by applying Hamilton’s principle. Subsequently, a hybrid method which combines reverberation-ray matrix method and wave propagation method is developed to determine the natural frequencies, and the results determined by present method are compared with those in the existing literature. Then, the effects of material length scale parameter, volume fraction exponent, location and number of supports on dynamic characteristics of multi-span FGM micropipes conveying fluid are discussed. The results show that the size effect is significant when the diameter of micropipe is comparable to the length scale parameter, and the natural frequencies determined by modified couple stress theory are larger than those obtained by classical beam theory. It is also found that natural frequencies and critical velocities increase rapidly with the increase of volume fraction exponent when it is less than 10, and the intermediate supports could improve the stability of pipes conveying fluid significantly.     

Experimental investigation on size-dependent higher-mode vibration of cantilever microbeams

A new perspective is presented to study size-dependent elasticity experimentally within micron scale utilizing dynamic approach. The size-dependent vibration of cantilever microbeams made of nickel is studied for the first three transverse modes. The normalized natural frequency of the first mode manifests strong size effect as reported. Remarkably, the normalized natural frequencies of the second and third mode also increase to 1.9 times as the thickness of microbeams decreases from 15 to 2.1 μm. Similarly the normalized bending rigidity increases to about 3.5 times. It is the first time that elastic size effect is observed in vibration of higher modes. Moreover, the size-dependent vibration of the first mode is interpreted in light of the modified couple stress theory, and the theoretical prediction also fit the experimental results of high modes very well. Hence it is confirmed that modified couple stress theory is valid for vibration of higher modes too.

     

Nonlinear analysis of size-dependent frequencies in porous FG curved nanotubes based on nonlocal strain gradient theory

A nonlocal strain gradient model is developed in this research to analyse the nonlinear frequencies of functionally graded porous curved nanotubes. It is assumed that the curved nanotube is in contact with a two-parameter nonlinear elastic foundation and is also subjected to the uniform temperature rise. The non-classical theory presented for curved nanotubes contains a nonlocal parameter and a material length scale parameter which can capture the size effect. A power law distribution function is used to describe the graded properties through the thickness direction of curved nanotubes. The even dispersion pattern is used to model the porosities distribution. The high-order shear deformation theory and the von Kármán type of geometric non-linearity are utilized to obtain the nonlinear governing equations of the structure. The size-dependent equations of motion for the large amplitude vibrations of curved nanotubes are obtained by employing Hamilton’s principle. The analytical solutions are extracted for the curved nanotube with immovable hinged-hinged boundary conditions. Size-dependent frequencies of the curved nanotube exposed to thermal field are obtained using the two-step perturbation technique and Galerkin procedure. The effects of important parameters such as nonlocal and length scale parameters, temperature field, elastic foundation, porosity, power law index and geometrical parameters are studied in detail.

     

An efficient size-dependent shear deformable shell model and molecular dynamics simulation for axial instability analysis of silicon nanoshells

Understanding the size-dependent behavior of structures at nanoscale is essential in order to have an effective design of nanosystems. In the current investigation, the surface elasticity theory is extended to study the nonlinear buckling and postbuckling response of axially loaded silicon cylindrical naoshells. Thereby, an efficient size-dependent shear deformable shell model is developed including the size effect of surface free energy. A boundary layer theory of shell buckling in conjunction with a perturbation-based solution methodology is employed to predict the size dependency in the buckling loads and postbuckling behavior of silicon nanoshells having various thicknesses. After that, on the basis of the Tersoff empirical potential, a molecular dynamics (MD) simulation is performed for a silicon cylindrical nanoshell with thickness of four times of silicon lattice constant, the critical buckling load and critical shortening of which are extracted and compared with those of the developed non-classical shell model. It is demonstrated that by taking the effects of surface free energy into account, a very good agreement is achieved between the results of the developed size-dependent continuum shell model and those of MD simulation.     

The impact of side walls on the MHD flow of a second-grade fluid through a porous medium

The magnetohydrodynamic flow through a porous medium of a second-grade fluid between two side walls induced by an infinite plate that exerts an accelerated shear stress to the fluid over an infinite plate is examined. Expressions for velocity and shear stress are determined with the help of integral transforms. In the absence of side walls, all the solutions that have been obtained are reduced to those corresponding to the motion over an infinite flat plate. The Newtonian solutions are also obtained as limiting case of the general solution. Finally, influence of magnetic and porosity parameter is graphically highlighted.

     

Nonlocal strain gradient finite element analysis of nanobeams using two-variable trigonometric shear deformation theory

In the present paper, a new trigonometric two-variable shear deformation beam nonlocal strain gradient theory is developed and applied to investigate the combined effects of nonlocal stress and strain gradient on the bending, buckling and free vibration analysis of nanobeams. The model introduces a nonlocal stress field parameter and a length scale parameter to capture the size effect. The governing equations derived are solved employing finite element method using a 3-nodes beam element, developed for this purpose. The predictive capability of the proposed model is shown through illustrative examples for bending, buckling and free vibration of nanobeams. Comparisons with other higher-order shear deformation beam theory are also performed to validate its numerical implementation and assess its accuracy within the nonlocal context.

     

圆柱壳在热载荷及微扰外压作用下的分岔

在考虑温度对圆柱壳材料性能影响的基础上,建立了圆柱壳在扰动外压作用下的几何非线性动力控制方程.并采用伽辽金原理及 Melnikov 法研究了圆柱壳在热载荷及微扰外压作用下的分岔,进一步讨论分析了温度、Batdorf 参数等因素对圆柱壳发生混沌运动区域的影响,得出了随温度、Batdorf 参数的增大,混沌运动区域将越来越大的结论.     

A mathematical model of MHD nanofluid flow having gyrotactic microorganisms with thermal radiation and chemical reaction effects

In this article, we have examined three-dimensional unsteady MHD boundary layer flow of viscous nanofluid having gyrotactic microorganisms through a stretching porous cylinder. Simultaneous effects of nonlinear thermal radiation and chemical reaction are taken into account. Moreover, the effects of velocity slip and thermal slip are also considered. The governing flow problem is modelled by means of similarity transformation variables with their relevant boundary conditions. The obtained reduced highly nonlinear coupled ordinary differential equations are solved numerically by means of nonlinear shooting technique. The effects of all the governing parameters are discussed for velocity profile, temperature profile, nanoparticle concentration profile and motile microorganisms’ density function presented with the help of tables and graphs. The numerical comparison is also presented with the existing published results as a special case of our study. It is found that velocity of the fluid diminishes for large values of magnetic parameter and porosity parameter. Radiation effects show an increment in the temperature profile, whereas thermal slip parameter shows converse effect. Furthermore, it is also observed that chemical reaction parameter significantly enhances the nanoparticle concentration profile. The present study is also applicable in bio-nano-polymer process and in different industrial process.

     

Effect of Porosity on free and forced vibration characteristics of the GPL reinforcement composite nanostructures

This article investigates the influence of porosity on free and forced vibration characteristics of a nanoshell reinforced by graphene platelets (GPL). The material properties of piece-wise graphene-reinforced composites (GPLRCs) are assumed to be graded in the thickness direction of a cylindrical nanoshell and estimated using a nanomechanical model. In addition, because of imperfection of the current structure, three kinds of porosity distributions are considered. The nanostructure is modeled using modified strain gradient theory (MSGT) which is a size-dependent theory with three length scale parameters. The novelty of the current study is to consider the effects of porosity, GPLRC and MSGT on dynamic and static behaviors of the nanostructure. Considering three length scale parameters ( $l_0=5h$, $l_1=3h$, $l_2=5h$ ) in MSGT leads to a better agreement with MD simulation in comparison by other theories. Finally, effects of different factors on static and dynamic behaviors of the porous nanostructure are examined in detail.