Porous Nanobeams Research Articles

On dynamic instability of magnetically embedded viscoelastic porous FG nanobeam

Dynamic instability of viscoelastic porous functionally graded (FG) nanobeam embedded on visco-Pasternak medium subjected to an axially oscillating loading as well as magnetic field is presented in this research. Porosity-dependent material properties of the porous nanobeam are described via a modified power-law function. Viscoelasticity of the nanostructure is considered based on the Kelvin–Voigt model. Employing Eringen's differential law in conjunction with Timoshenko beam theory (TBT), the motion equations are derived via Hamilton's variational principle. Navier's solution as well as Bolotin's approach are utilized to obtain the dynamic instability region of viscoelastic porous FG nanobeam. Some benchmark results related to the effects of structural damping, length to thickness ratio, foundation type, nonlocal parameter (NP), static load factor, power-law index, porosity volume index and magnetic field on the instability region of porous FG nanobeam are evaluated. The results reveal that with increasing power-law index and structural damping, the pulsation frequency decreases and so, instability region shifts to left side while as magnetic field magnifies, the dynamic instability moves to right side. Also, it is represented that the porosity effect on the dynamic stability of FG nanobeam depends significantly on the values of power-low index and magnetic field.

Surface Effect on Static Bending of Functionally Graded Porous Nanobeams Based on Reddy’s Beam Theory

In this paper, the surface effect on the static bending behavior of functionally graded porous (FGP) nanobeams subjected to a concentrated transverse load is studied by using Reddy’s higher-order beam theory. Three types of porosity distributions are considered for the nanobeam, i.e. uniform porosity distribution, symmetric and asymmetric non-uniform porosity distributions. With the consideration of the surface effect, the nanobeams can be abstracted as a composite beam composed of a surface layer and a bulk volume. According to the generalized Young–Laplace equation, the normal stress discontinuity across a surface due to the effect of surface stress is taken into consideration. The analytical solutions of the static bending problem of FGP nanobeams are obtained for the beams with hinged-hinged, clamped-clamped and clamped-free boundary conditions. The effects of the residual surface stress, porosity distribution type, porosity coefficient and length-to-thickness ratio on the transverse displacement of the FGP beams are discussed.

A finite element model for the thermo-elastic analysis of functionally graded porous nanobeams

In this study, for the first time, a nonlocal finite element model is proposed to analyse thermo-elastic behaviour of imperfect functionally graded porous nanobeams (P-FG) on the basis of nonlocal elasticity theory and employing a double-parameter elastic foundation. Temperature-dependent material properties are considered for the P-FG nanobeam, which are assumed to change continuously through the thickness based on the power-law form. The size effects are incorporated in the framework of the nonlocal elasticity theory of Eringen. The equations of motion are achieved based on first-order shear deformation beam theory through Hamilton's principle. Based on the obtained numerical results, it is observed that the proposed beam element can provide accurate buckling and frequency results for the P-FG nanobeams as compared with some benchmark results in the literature. The detailed variational and finite element procedure are presented and numerical examinations are performed. A parametric study is performed to investigate the influence of several parameters such as porosity volume fraction, porosity distribution, thermal loading, material graduation, nonlocal parameter, slenderness ratio and elastic foundation stiffness on the critical buckling temperature and the nondimensional fundamental frequencies of the P-FG nanobeams. Based on the results of this study, a porous FG nanobeam has higher thermal buckling resistance and natural frequencies compared to a perfect FG nanobeam. Also, the format of the porosity distribution is important, that uniform distributions of porosity result in greater critical buckling temperatures and vibration frequencies, in comparison with functional distributions of porosities.

Effects of porosity models on static behavior of size dependent functionally graded beam

In this study, the mechanical bending behaviors of functionally graded porous nanobeams are investigated. Four types of porosity which are, the classical power porosity function, the symmetric with mid-plane cosine function, bottom surface distribution and top surface distribution are proposed in analysis of nanobeam for the first time. A comparison between four types of porosity are illustrated. The effect of nano-scale is described by the differential nonlocal continuum theory of Eringen by adding the length scale into the constitutive equations as a material parameter comprising information about nanoscopic forces and its interactions. The graded material is designated by a power function through the thickness of nanobeam. The beam is simply-supported and is assumed to be thin, and hence, the kinematic assumptions of Euler-Bernoulli beam theory are held. The mathematical model is solved numerically using the finite element method. Numerical results show effects of porosity type, material graduation, and nanoscale parameters on the static deflection of nanobeam.

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.

Wave propagation of functionally graded porous nanobeams based on non-local strain gradient theory

In this paper, attention is paid to the prediction of wave propagation behaviors of functionally graded materials (FG) porous nanobeams based on Reddy’s higher-order shear deformation beam theory in conjunction with the non-local strain gradient theory. The governing equations of the porous nanobeams are derived with the help of the Hamilton principle. By solving an eigenvalue problem, the analytic dispersion relation is obtained. The results of Euler-Bernoulli beam and Timoshenko beam models are also presented. The influences of non-local parameter, strain gradient parameter, power law index and porosity volume fraction on the wave propagation are discussed in detail.

Vibration of FG magneto-electro-viscoelastic porous nanobeams on visco-Pasternak foundation

In this paper, the size-dependent vibrational behaviors of functionally graded (FG) magneto-electro-viscoelastic nanobeams in the presence of porosities and embedded in the viscoelastic medium are studied based on the nonlocal Timoshenko beam theory in conjunction with the Kelvin-Voigt viscoelastic model. The viscoelastic medium is modeled as a visco-Pasternak foundation with consideration of both shear modulus and medium damping coefficient. The FG material properties are supposed to vary along the thickness direction in a power-law exponent form, and two types of porosity distributions are considered. The present model is validated by comparison with several existing theories and a good agreement is achieved. Parametric studies are carried out to investigate the coupling influences of the nonlocal parameter, FG power-law index, porosity volume fraction, porosity distribution, boundary condition, structural damping coefficient and viscoelastic foundation parameters, as well as electric voltage and magnetic potential on the vibrational performance of FG magneto-electro-viscoelastic porous nanobeams. The results are helpful for the design and applications of nano-electro-mechanical systems (NEMS).

Nonlinear vibration and buckling of functionally graded porous nanoscaled beams

Although many researchers have studied the vibration and buckling behavior of porous materials, the behavior of porous nanobeams is still a needed issue to be studied. This paper is focused on the buckling and nonlinear vibration of functionally graded (FG) porous nanobeam for the first time. Nonlinear Von Karman strains are put into consideration to study the nonlinear behavior of nanobeam based on the Euler–Bernoulli beam theory. The nonlocal Eringen’s theory is used to study the size effects. The mechanical properties of ceramic and metal are used to model the functionally graded material through thickness, and the boundary conditions are considered as clamped–clamped (CC) and simply supported–simply supported (SS). The generalized differential quadrature method (GDQM) is used in conjunction with the iterative method to solve the equations. The parametric study is done to examine the effects of nonlinearity, porosity, sized effect, FG index, etc., on the vibration and buckling of porous nanobeam.

Modified porosity model in analysis of functionally graded porous nanobeams

This work studies the mechanical bending and vibration of functionally graded nanobeams using finite elements according to Euler beam theory. We implement a modified porosity model that represents porosity and Young’s modulus in an implicit form, where the density is assumed as a function of the porosity parameter, while Young’s modulus is assumed as a ratio of the mass density with porosity to that without porosity. The effect of nano-scale is described by the nonlocal continuum theory by adding the length scale into the constitutive equations as a material parameter comprising information about nanoscopic forces and its interactions. The material gradation of constituents is described by a power function through the thickness of nanobeam. The beam is simply supported and is assumed to be thin, and hence, the kinematic assumptions of Euler–Bernoulli beam theory are held. The mathematical model is solved numerically using the finite-element method. Numerical results show that the increase of porosity, material graduation, and nano-scale parameters tend to decrease the bending resistance as well as the fundamental frequencies of the nanobeam.

Nonlocal strain gradient-based vibration analysis of embedded curved porous piezoelectric nano-beams in thermal environment

This disquisition proposes a nonlocal strain gradient beam theory for thermo-mechanical dynamic characteristics of embedded smart shear deformable curved piezoelectric nanobeams made of porous electro-elastic functionally graded materials by using an analytical method. Electro-elastic properties of embedded curved porous FG nanobeam are assumed to be temperature-dependent and vary through the thickness direction of beam according to the power-law which is modified to approximate material properties for even distributions of porosities. It is perceived that during manufacturing of functionally graded materials (FGMs) porosities and micro-voids can be occurred inside the material. Since variation of pores along the thickness direction influences the mechanical and physical properties, so in this study thermo-mechanical vibration analysis of curve FG piezoelectric nanobeam by considering the effect of these imperfections is performed. Nonlocal strain gradient elasticity theory is utilized to consider the size effects in which the stress for not only the nonlocal stress field but also the strain gradients stress field. The governing equations and related boundary condition of embedded smart curved porous FG nanobeam subjected to thermal and electric field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is utilized to achieve the natural frequencies of porous FG curved piezoelectric nanobeam resting on Winkler and Pasternak foundation. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality parameter, electric voltage, coefficient of porosity, elastic foundation parameters, thermal effect, gradient index, strain gradient, elastic opening angle and slenderness ratio on the natural frequency of embedded curved FG porous piezoelectric nanobeam are successfully discussed. It is concluded that these parameters play important roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

Influence of porosity and axial preload on vibration behavior of rotating FG nanobeam

In this paper, a nanobeam connected to a rotating molecular hub is considered. The vibration behavior of rotating functionally graded nanobeam based on Eringen's nonlocal theory and Euler–Bernoulli beam model is investigated. Furthermore, axial preload and porosity effect is studied. It is supposed that the material attributes of the functionally graded porous nanobeam, varies continuously in the thickness direction according to the power law model considering the even distribution of porosities. Porosity at the nanoscopic length scale can affect on the rotating functionally graded nanobeams dynamics. The equations of motion and the associated boundary conditions are derived through the Hamilton's principle and generalized differential quadrature method (GDQM) is utilized to solve the equations. In this paper, the influences of some parameters such as functionally graded power (FG-index), porosity parameter, axial preload, nonlocal parameter and angular velocity on natural frequencies of rotating nanobeams with pure ceramic, pure metal and functionally graded materials are examined and some comparisons about the influence of various parameters on the natural frequencies corresponding to the simply-simply, simply-clamped, clamped-clamped boundary conditions are carried out.

Wave propagation analysis of smart rotating porous heterogeneous piezo-electric nanobeams

The present work is mainly focused on studying the influences of angular velocity on the wave propagation responses of functionally graded (FG) piezo-electric nanobeams. Moreover, the effects of porosity are also regarded in the wave propagation analysis of size-dependent FG beams. The distribution of electro-mechanical properties of a piezo-electric beam are precisely described employing power-law formulation. The nonlocal elasticity theory is utilized to account for the influences of small scale. Herein, a classical beam theory is expounded to derive the nonlocal governing equations of the nanobeam. Once the governing equations are completely derived, an analytical solution method is applied to obtain the dispersion relations of propagating waves. A comparison of this model with previous studies is then made to show the validity of the obtained results. Finally, the influences of various variants, such as wave number, nonlocal parameter, gradient index, electric voltage, volume fraction of porosity and angular velocity, are studied in detail to show how these parameters can affect the wave frequency, phase velocity and escape frequency of FG smart rotary porous nanobeams.

Thermal vibration of two-dimensional functionally graded (2D-FG) porous Timoshenko nanobeams

The thermo-mechanical vibration behavior of two dimensional functionally graded (2D-FG) porous nanobeam is reported in this paper. The material properties of the nanobeam are variable along thickness and length of the nanobeam according to the power law function. The nanobeam is modeled within the framework of Timoshenko beam theory. Eringen\'s s nonlocal elasticity theory is used to develop the governing equations. Using the generalized differential quadrature method (GDQM) the governing equations are solved. The effect of porosity, temperature distribution, nonlocal value, L/h, FG power indexes along thickness and length and are investigated using parametric studies.

Dynamic characteristics of curved inhomogeneous nonlocalporous beams in thermal environment

This paper proposes an analytical solution method for free vibration of curved functionally graded (FG) nonlocal beam supposed to different thermal loadings, by considering porosity distribution via nonlocal elasticity theory for the first time. Material properties of curved FG beam are assumed to be temperature-dependent. Thermo-mechanical properties of porous FG curved beam are supposed to vary through the thickness direction of beam and are assumed to be temperature-dependent. Since variation of pores along the thickness direction influences the mechanical and physical properties, porosity play a key role in the mechanical response of curved FG structures. The rule of power-law is modified to consider influence of porosity according to even distribution. The governing equations of curved FG porous nanobeam under temperature field are derived via the energy method based on Timoshenko beam theory. An analytical Navier solution procedure is used to achieve the natural frequencies of porous FG curved nanobeam supposed to thermal loadings with simply supported boundary condition. The results for simpler states are confirmed with known data in the literature. The effects of various parameters such as nonlocality, porosity volume fractions, type of temperature rising, gradient index, opening angle and aspect ratio of curved FG porous nanobeam on the natural frequency are successfully discussed. It is concluded that these parameters play key roles on the dynamic behavior of porous FG curved nanobeam. Presented numerical results can serve as benchmarks for future analyses of curve FG nanobeam with porosity phases.

Phonon Conduction in Periodically Porous Silicon Nanobridges

Thermal conduction in periodically porous nanostructures is strongly influenced by phonon boundary scattering, although the precise magnitude of this effect remains open to investigation. This work attempts to clarify the impact of phonon-boundary scattering at room temperature using electrothermal measurements and modeling. Silicon nanobeams, prepared using electron beam lithography, were coated with a thin palladium overlayer, which serves as both a heater and thermometer for the measurement. The thermal conductivity along the length of the silicon nanobeams was measured using a steady-state Joule heating technique. The thermal conductivities of the porous nanobeams were reduced to as low as 3% of the value for bulk silicon. A Callaway-Holland model for the thermal conductivity was adapted to investigate the relative impact of boundary scattering, pore scattering, and phonon bandgap effects. Both the experimental data and the modeling showed a reduction in thermal conductivity with increasing pore diameter, although the experimentally measured value was up to an order of magnitude lower than that predicted by the model.