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The current investigation aims to account for the influence of porosity modeling while studying the dynamic response of functionally graded material (FGM) plates within the framework of a new, higher-order shear deformation theory for the first time. In the newly developed porosity modeling, the reciprocal effect of mass density and Young's modulus is considered. Furthermore, silicon carbide in the ceramic phase and nickel in the metallic phase are selected as the constituent materials of the functionally graded plate. Based upon the new higher-order theory, the kinetic and kinematic relations are derived. This theory does not need any shear correction factor. The dynamic form of the principle of virtual work is employed to achieve governing equations of structure. Afterward, Galerkin's method is implemented to solve obtained governing equations of porous, functionally graded plates. In order to verify the accuracy of the method, the obtained results were compared with the results reported in the literature, and good agreement was found. At last, the influences of some parameters such as porosity coefficient, length to thickness ratio, and aspect ratio of the porous, FGM plate on the dimensionless frequency are studied and the results are illustrated in detail. [ABSTRACT FROM AUTHOR]

Based on modified strain gradient theory, nonlinear vibration analysis of a functionally graded piezoelectric doubly curved microshell in thermal environment has been performed in this research. Three scale parameters have been included in the modeling of thin doubly curved microshell in order to capture micro-size effects. Graded material properties between the top and bottom surfaces of functionally graded piezoelectric doubly curved microshell have been considered via incorporating power-law model. It is also assumed that the microshell is exposed to a temperature field of uniform type and the material properties are temperature-dependent. By analytically solving the governing equations based on the harmonic balance method, the closed form of nonlinear vibration frequency has been achieved. Obtained results indicate the relevance of calculated frequencies to three scale parameters, material gradation, electrical voltage, curvature radius, and temperature changes. [ABSTRACT FROM AUTHOR]

This study presents an investigation of the wave propagation of a rectangular sandwich composite plate with tunable magneto-rheological fluid core. The constituent parts of this rectangular sandwich composite plate are the base layer, magneto-rheological fluid core, and limiter layer. Magneto-rheological fluid core is embedded within the base and limiter layers. Also, the upper and lower layers are made of elastic materials. For obtaining the governing equations of motion, Hamilton's principle and classical plate theory are used. After that, applying an analytical solution, the wave frequency and phase velocity of the propagated waves can be gained by solving eigenvalue problem. By investigating the effect of the magnetic field, the results emphasize that the magnetic field intensity is the most important factor for changing the value of the wave frequency and phase velocity. Besides, results show by enhancing the core-to-top layer thickness ratio, the wave frequencies reduce because magneto-rheological fluid core is softer in comparison to elastic layers. Therefore, magneto-rheological fluid layer operates similar to a damper. At the end, the influence of the thickness of magneto-rheological fluid core on the phase velocity is also discussed. [ABSTRACT FROM AUTHOR]

The objective of the present paper is to comprehensively study the nonlinear frequency response of viscoelastic piezoelectric nanoplates exposed to dual harmonic external excitation and thermo-electro-mechanical loads. To achieve this goal, firstly, a piezoelectric nanoplate resting on a viscoelastic foundation is modeled. Secondly, using the nonlocal piezoelectricity theory, Kelvin-Voigt model, von Karman nonlinear relations and Hamilton's principle, the nonlinear governing differential equation of motion is derived. In the next step, employing the Galerkin technique and multiple time scales method, the partial differential equation is transformed to an ordinary one and solved. Finally, the modulation equation of viscoelastic piezoelectric nanoplates for combinational excitation is obtained. Emphasizing the effect of dual harmonic excitation and thermo-electro-mechanical loads on nonlinear frequency response of the system, jump and resonance phenomena are discussed. A detailed parametric study is conducted to examine the effect of nonlinearity, damping coefficient, nonlocal parameter, combinational excitation, electric voltage, initial stress and thermal environment. [ABSTRACT FROM AUTHOR]

Investigation of flexoelectric effect on nonlinear forced vibration of piezoelectric/functionally graded porous nanocomposite is the objective of this study. The nanocomposite is exposed to electric voltage and external parametric excitation. First, a functionally graded porous core nanoplate is modeled and then two piezoelectric layers are glued with core. It is also rested on a visco-Pasternak foundation. Second, to derive governing equation of motion, two theories including Mindlin and Kirchhoff plate theories and Hamilton's principle are utilized. In the next step, to obtain and solve ordinary differential equation, Galerkin technique and multiple time scales method are used, respectively. At the end, modulation equation of piezoelectric/functionally graded porous nanocomposite for both primary and secondary resonances is obtained and discussed. Emphasizing the effect of piezoelectric and flexoelectric, von Karman nonlinear deformation and parametric external excitation are simultaneously taken into account. It is found that electric voltage has no effect on the performance of piezoelectricity and flexoelectricity of the material on vibration behavior. The results of this study can be useful as benchmark for the next investigations in field of energy harvesting systems and piezoelectric structures. [ABSTRACT FROM AUTHOR]

The present article is majorly arranged to survey the vibration problem of a nanocomposite beam reinforced by graphene oxide powder (GOP) whenever the structure is subjected to a nonuniform magnetic field. A higher-order trigonometric refined beam model is extended to achieve the governing equation of the problem according to Hamilton's principle. The effect of a nonuniform magnetic field is applied to the governing equations by combining the magnetic induction relations of Maxwell with the displacement field equations. Then, the achieved equations are solved by implementing Galerkin's method to consider the influence of different boundary conditions on the vibrational responses of the beam. On the other hand, the accuracy and efficiency of the presented model is verified by comparing the results from this work with that of published researches. Finally, effects of different variables on the dimensionless frequency of GOP reinforced composite beams are highlighted in the framework of tables and diagrams. [ABSTRACT FROM AUTHOR]

In this research, nonlinear free and forced vibration behaviors of the multiscale doubly curved composite shells have been investigated. By using Reddy's third-order shear deformation theory (TSDT) and von-Karman type nonlinearity the strains and stresses are obtained. Three-phase composites shells with polymer/carbon nanotube/fiber and polymer/graphene platelet/fiber and shape-memory alloy (SMA)/matrix according to the Halpin–Tsai model have been analyzed. The governing equations of composite shell have been derived by implementing Hamilton's principle, and according to the multiple scales perturbation method have been solved. The obtained results are validated through the comparison with the available results. Finally, the effects of different parameters such as volume fraction of SMA, temperature rise, various distributions pattern, aspect, and curvature ratio considered on nonlinear frequency have been studied. The results indicate that the changes of the fundamental vibrational mode shape have a significant influence on the nonlinear frequency of the composite shell. [ABSTRACT FROM AUTHOR]

This research studied large amplitude vibration behaviors of multiscale composites doubly curved shells with piezoelectric layer. By using Reddy's third-order shear deformation theory, the strains and stresses are obtained. According to the Halpin–Tsai model three-phase composites layers are considered. The governing equations of the multiscale doubly curved shell are derived by implementing the Hamilton's principle and are solved via homotopy perturbation method. For investigating correctness and accuracy, this article is validated by other previous studies. Finally, the influence of different parameters such as temperature rise, various distributions pattern, applied voltage, magnetic potential, and aspect and curvature ratio are observed in this article. It is found that these parameters have significant influence on nonlinear frequencies. [ABSTRACT FROM AUTHOR]

Herein, the agglomeration effect of nanoparticles on the wave dispersion of multi-scale hybrid nanocomposite beams is investigated. The constituent material consists of both macro- and nano-reinforcements which are dispersed in the polymer matrix. Homogenization is conducted according to the well-known micromechanical methods. Herein, the combination of the Eshelby–Mori–Tanaka model and the rule of the mixture is implemented in order to estimate the equivalent material properties of the nanocomposite beam. Also, a refined higher-order beam theory is used in order to calculate the kinetic relations free from utilizing an additional factor to account for the shear deformation. Furthermore, the governing equations are achieved by applying Hamilton's principle. Then, the governing equations are solved analytically to enrich the wave frequency. The effects of various parameters on the variation in wave frequency and phase velocity of the multi-scale hybrid nanocomposite beam are studied. The results of this study reveal that the mechanical responses of the system decrease whenever the nanotubes are inside the clusters. [ABSTRACT FROM AUTHOR]

In this paper, buckling behavior of a higher order functionally graded microbeam with porosities is investigated based on the modified couple stress theory and the exact position of the neutral axis. Porosities are evenly and unevenly distributed inside the functionally graded microbeam. Material properties of the functionally graded microbeam are assumed to vary in the thickness direction through a modified form of power-law distribution in which the volume fraction of porosities is considered. The governing equations are derived by using Hamilton's principle and an analytical method is employed to solve these equations for various boundary conditions. The present formulation and numerical results demonstrate a good agreement with some available cases in the literature. Influences of several important parameters such as power-law exponent, porosity distributions, porosity volume fraction, slenderness ratio, and various boundary conditions on buckling loads of porous functionally graded microbeams are investigated and discussed in detail. [ABSTRACT FROM AUTHOR]

In this article, the vibration problem of an embedded cylindrical shell consisted of porous metal foam is solved via an analytical method with respect to the influences of various porosity distributions. Three types of porosity distribution across the thickness are covered, namely, uniform, symmetric, and asymmetric. The strain–displacement relations of the shell are assumed to be derived on the basis of the first-order shear deformation shell theory. Then, the achieved relations will be incorporated with the Hamilton's principle in order to reach the Navier equations of the cylindrical shell. Next, the well-known Galerkin's method is utilized to calculate the natural frequencies of the system. The influences of both simply supported and clamped boundary conditions are included. In order to show the accuracy of the presented method, the results of the present research are compared with those reported by former published papers. The reported results show that an increase in the porosity coefficient can decrease the frequency of the shell. Also, the stiffness of the system can be lesser decreased while symmetric porosity distribution is chosen. [ABSTRACT FROM AUTHOR]

This article is primarily organized to analyze the thermo-elastic vibrational characteristics of multi-scale hybrid composite beams according to a refined beam model. In this novel type of composites, multi-scale reinforcing elements, carbon fiber (CF) and carbon nanotube (CNT) in particular, are presumed to be dispersed in an initial resin. The homogenization process is carried out employing a mixture of the Halpin–Tsai model and the rule of mixture. The effect of temperature and its gradient on the mechanical properties of CNTs and epoxy resin is rendered to present a more reliable thermal analysis. On the other hand, a refined trigonometric shear deformable beam theory is extended to derive the kinematic relations of the beam needless of any external shear correction coefficient. On the basis of Hamilton's principle, the partial differential equations of motion are developed. Thereafter, the natural frequencies are achieved by the means of Galerkin's method for both simply supported and fully clamped edge conditions. Then, the validity of the presented model is shown by comparing these results with those of previously published researches. Finally, effects of different parameters on the natural frequency of composite beams are rendered in the framework of some numerical case studies. It can be found that multi-scale hybrid composite beams can satisfy higher frequencies once compared with each of the CF- or CNT-reinforced composite beams. [ABSTRACT FROM AUTHOR]

In this study, the survey of the wave dispersion behaviors of sandwich composite nanoplates is carried out by considering the magnetostriction phenomenon. The nanoplate is assumed to be made up of a central magnetostrictive core in addition to two composite face sheets. The scale influences are covered based on the nonlocal strain gradient theory. Moreover, the equations of plate motion are derived according to the classical plate theory. Afterward, the magnetization effects are considered by introducing a feedback control system. Then, Hamilton’s principle is introduced to obtain the Euler–Lagrange equations of the magnetostrictive sandwich composite nanoplate. Also, by relating the achieved equations with those of the nonlocal strain gradient theory, the nonlocal governing equations of magnetostrictive sandwich composite nanoplate are developed. The wave frequency and phase velocity values are computed by the application of an analytical solution. Finally, the effects of participant coefficients are illustrated separately through certain figures. [ABSTRACT FROM AUTHOR]

This paper deals with the combination of surface and couple stress theories to examine free vibrational behavior of rotating functionally graded (FG) nanobeams under axial magnetic field. The theory introduces a scale parameter based on modified couple stress theory to capture the size effects. The modified couple stress theory is a weakly nonlocal theory and introduces the stiffness-hardening effect. A power-law distribution is used to describe the graded material properties. The governing equations and the related boundary conditions are derived by Hamilton’s principle and they are solved by applying an analytical solution which satisfies clamped-free boundary conditions. A comparison study is performed to verify the present formulation with the provided data in the literature and a good agreement is observed. The parametric study covered in this paper includes several parameters such as length scale parameter, surface energy, angular velocity, magnetic field and power-law exponent on natural frequencies of rotary FG nanobeams in detail. [ABSTRACT FROM AUTHOR]

This paper develops a nonlocal strain gradient plate model for vibration analysis of graphene sheets under nonuniform in-plane mechanical loads. For more accurate analysis of graphene sheets, the proposed theory contains two scale parameters related to the nonlocal and strain gradient effects. Graphene sheet is modeled via a two-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on elastic substrate are derived via Hamilton’s principle. Galerkin’s method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as in-plane loading, load factor, nonlocal parameter, length scale parameter, elastic foundation, and boundary conditions on vibration characteristics of graphene sheets are examined. [ABSTRACT FROM AUTHOR]

The present research deals with the wave dispersion behavior of a rotating functionally graded material (FGMs) nanobeam applying nonlocal elasticity theory of Eringen. Material properties of rotating FG nanobeam are spatially graded according to a power-law model. The governing equations as functions of axial force due to centrifugal stiffening and displacements are obtained by employing Hamilton’s principle based on the Euler–Bernoulli beam theory. By using an analytical model, the dispersion relations of the FG nanobeam are derived by solving an eigenvalue problem. Numerical results clearly show that various parameters, such as angular velocity, gradient index, wave number and nonlocal parameter, are significantly effective to characteristics of wave propagations of rotating FG nanobeams. The results can be useful for next generation study and design of nanomachines, such as nanoturbines, nanoscale molecular bearings and nanogears, etc. [ABSTRACT FROM AUTHOR]

Based on exact position of neutral surface, vibrational behavior of smart nanoplates made of magneto-electro-elastic functionally graded materials is examined by implementing a Galerkin method for the first time. Magneto-electro-elastic properties of nanoplate vary in transverse direction via power-law model. The nonlocal governing equations of functionally graded plate under magneto-electrical field are formulated through Hamilton's principle and nonlocal elasticity theory based on a four-variable refined plate theory, which avoids the use of shear correction factors by capturing shear deformation influences. Importance of various parameters including magnetic potential, electric voltage, various boundary conditions, nonlocality, material distribution, and plate thickness on natural frequencies of the magneto-electro-elastic functionally graded nanoplate are explored. It is elucidated that these parameters play significant roles on the dynamic behavior of magneto-electro-elastic functionally graded nanoplates. [ABSTRACT FROM AUTHOR]

In this research, a free vibration study of a biaxially compressed magneto-electro-elastic functionally graded (MEE-FG) nanoplate resting on an elastic substrate is carried out according to a trigonometric plate formulation. Spatially graded MEE properties of the nanoplate are described according to a power-law distribution. Eringen’s nonlocal elasticity model is utilized to incorporate the small-scale influences. The distributions of magneto-electrical potentials in the thickness direction are considered as a combination of cosine and linear variations. The governing equations of the present plate model are obtained employing Hamilton’s principle. Some admissible functions are introduced to satisfy different boundary conditions and solving these equations. It is indicated that vibration frequencies of the MEE-FG nanoplate are dramatically affected by biaxial compression, magnetic potential, electric voltage, elastic substrate, small-scale parameter, and material gradation. [ABSTRACT FROM AUTHOR]

This paper develops a nonlocal strain gradient plate model for vibration analysis of the graphene sheets under in-plane magnetic field and hygro-thermal environments. For more accurate analysis of the graphene sheets, the proposed theory contains two-scale parameters related to the nonlocal and strain gradient effects. The graphene sheet is modeled via a two-variable shear deformation plate theory needless of shear correction factors. Governing equations of a nonlocal strain gradient graphene sheet on elastic substrate are derived via Hamilton's principle. Galerkin's method is implemented to solve the governing equations for different boundary conditions. Effects of different factors such as moisture concentration rise, temperature rise, nonlocal parameter, length scale parameter, elastic foundation, and magnetic field on vibration characteristics of the graphene sheets are examined. [ABSTRACT FROM AUTHOR]

This paper provides the first examination of buckling behavior of axially functionally graded nanobeams. A nonlocal strain gradient theory consisting of two scale parameter is employed for modeling of size-dependent behavior axially functionally graded nanobeam much accurately. This theory takes into account both nonlocal stress field and strain gradient effects on the response of nanostructures. A power-law model is used to describe the distribution of material properties along the axial direction. The axially functionally graded nanobeam is in contact with a non-uniform elastic medium which consists of a Winkler layer with variable stiffness and also a Pasternak layer with constant stiffness. Linear, parabolic and sinusoidal variations of Winkler foundation in longitudinal direction are considered. A Galerkin-based solution technique is implemented to solve the governing equation obtained from Hamilton’s principle. Buckling loads of functionally graded nanobeam are verified with those of previous papers. It is shown that buckling loads of axially functionally graded nanobeams are significantly influenced by power-law index, nonlocal parameter, length scale parameter, type of elastic foundation and boundary conditions. [ABSTRACT FROM AUTHOR]

In this article, free vibration of functionally graded (FG) viscoelastic nanobeams embedded in viscoelastic foundation exposed to hygro-thermal loading is investigated based on nonlocal strain gradient elasticity theory and a higher order refined beam theory which captures shear deformation influences without the need for any shear correction factor. Also, the exact position of neutral axis is determined. The visco-Pasternak foundation is consists of parallel springs and dashpots as well as a shear layer. Temperature-dependent material properties of FGM beam are graded across the thickness based on the power-law form. Hamilton's principle is used to obtain nonlocal governing equations of embedded strain gradient viscoelastic nanobeam which are solved analytically for various boundary conditions. The results are validated with those available in the literature. The impacts of visco-Pasternak foundation parameters, structural damping coefficient, hygrothermal loading, nonlocal stress parameter, nonlocal gradient parameter, power-law exponent, mode number, boundary conditions and slenderness ratio on the damping frequency of nanoscale viscoelastic FG beams are evaluated. [ABSTRACT FROM AUTHOR]

In this paper, vibration characteristics of magneto-electro-thermo-elastic functionally graded (METE-FG) nanobeams is investigated in the framework of third order shear deformation theory. Magneto-electro-thermo-elastic properties of FG nanobeam are supposed to vary smoothly and continuously along the thickness based on power-law form. To capture the small size effects, Eringen’s nonlocal elasticity theory is adopted. By using the Hamilton’s principle, the nonlocal governing equations are derived and then solved analytically to obtain the natural frequencies of METE-FG nanobeams. The reliability of proposed model and analytical method in predicting natural frequencies of METE-FG nanobeam is evaluated with comparison to some cases in the literature. Numerical results are provided indicating the influences of several parameters including magnetic potential, external electric voltage, temperature fields, power-law exponent, nonlocal parameter and slenderness ratio on the frequencies of METE-FG nanobeams. It is found that the vibrational behavior of METE-FG nanobeams is significantly impressed by these effects. [ABSTRACT FROM AUTHOR]

This article examines the application of nonlocal strain gradient elasticity theory to wave dispersion behavior of a size-dependent functionally graded nanoplate in thermal environments. The theory contains two scale parameters corresponding to nonlocal and strain gradient effects. A quasi-3D plate theory considering shear and normal deformations is employed to present the formulation. Mori–Tanaka micromechanical model is used to describe functionally graded material properties. Hamilton’s principle is employed to obtain the governing equations of nanoplate accounting for thickness stretching effect. These equations are solved analytically to find wave frequencies and phase velocities of functionally graded nanoplate. It is indicated that wave dispersion behavior of functionally graded nanoplates is significantly affected by temperature rise, nonlocality, length scale parameter, and material composition. [ABSTRACT FROM AUTHOR]

In this paper, size-dependent free vibration analysis of curved functionally graded nanobeams embedded in Winkler–Pasternak elastic medium is carried out via an analytical solution method. Three kinds of boundary condition namely, simply supported-simply supported, simply supported-clamped and clamped-clamped are investigated. Material properties of curved functionally graded beam change in thickness direction according to the Mori–Tanaka model. Nonlocal strain gradient elasticity theory is adopted to capture the size effects in which the stress is considered for not only the nonlocal stress field but also the strain gradients stress field. Nonlocal governing equations of curved functionally graded nanobeam are obtained from Hamilton’s principle based on Euler–Bernoulli beam model. Finally, the influences of length scale parameter, nonlocal parameter, opening angle, elastic medium, material composition, slenderness ratio and boundary conditions on the vibrational characteristics of nanosize curved functionally graded beams are explored. [ABSTRACT FROM AUTHOR]

This article investigates vibration behavior of magneto-electro-elastic functionally graded nanobeams embedded in two-parameter elastic foundation using a third-order parabolic shear deformation beam theory. Material properties of magneto-electro-elastic functionally graded nanobeam are supposed to be variable throughout the thickness based on power-law model. Based on Eringen’s nonlocal elasticity theory which captures the small size effects and using Hamilton’s principle, the nonlocal governing equations of motions are derived and then solved analytically. Then, the influences of elastic foundation, magnetic potential, external electric voltage, nonlocal parameter, power-law index, and slenderness ratio on the frequencies of the embedded magneto-electro-elastic functionally graded nanobeams are studied. [ABSTRACT FROM AUTHOR]

Static stability analysis of a size-dependent magneto-electro-elastic functionally graded nanoplate has an immense contribution in identification and improvement of the performance of nano-electro-mechanical systems. A refined trigonometric plate theory is employed to formulate the magneto-electro-elastic functionally graded nanoplate for the first time. Magneto-electro-elastic properties of nanoplate change in spatial coordinate based on power-law form. Regarding the small-scale effects at nanoscales, the size-dependent nonlocal continuum theory is employed to derive governing equations of the nonclassical magneto-electro-elastic functionally graded nanoplate. Analytical solution possessing functions which satisfy different boundary conditions is adopted to solve the equations. The results illustrate the size-dependent buckling behavior of magneto-electro-elastic functionally graded nanoplate affected by magnetic potential, electric voltage, various boundary conditions, small-scale parameter, material composition, plate side-to-thickness ratio, and aspect ratio. [ABSTRACT FROM AUTHOR]

This paper investigates free vibration characteristics of a rotating double-tapered functionally graded (FG) beam made of porous material. Material properties of the FG beam vary continuously through thickness direction according to the power-law which modified to approximate material properties for even and uneven distributions of porosities. The governing differential equations of motion are derived based on Euler–Bernoulli beam theory and using Hamilton's principle and then solved utilizing a semi-analytical technique called the differential transform method (DTM). In order to verify the competency and accuracy of the current analysis, a comparative study with previous researches is performed and good agreement is observed. Several important parameters such as power-law exponent, porosity volume fraction, taper ratios, rotational speed and slenderness ratio which have impacts on natural frequencies of such beams are investigated and discussed in detail. It is concluded that these effects play significant role on dynamic behavior of rotating double tapered FG beam. Numerical results are tabulated in several tables and figures that can serve as benchmarks for future analyses of FGM beams with porosity phases. [ABSTRACT FROM AUTHOR]

In this paper, free vibration analysis of rotating functionally graded thick beams within the context of Timoshenko beam theory is presented. Material properties of functionally graded beam are supposed to vary exponentially along the thickness direction of the constituents. Governing equations are derived through Hamilton’s principle and they are solved applying differential transform method. The good agreement between the results of this article and those available in literature validated the presented approach. The detailed mathematical derivations are presented and numerical investigations are performed while the emphasis is placed on investigating the effect of several beam parameters such as constituent volume fractions, slenderness ratios, rotational speed and hub radius on vibration characteristics of the rotating thick functionally graded beam. It is concluded that these effects play significant role on dynamic behavior of rotating FG beam. Numerical results are presented to serve as benchmarks for future analyses of such beams. [ABSTRACT FROM AUTHOR]