Porous Nanobeams Research Articles

Nonlinear dynamic buckling and vibration of thermally post-buckled temperature-dependent FG porous nanobeams based on the nonlocal theory

In this paper, nonlinear dynamic snap-through buckling and vibration behavior of the thermally post-buckled functionally graded (FG) porous nanobeams subjected to static and sudden mechanical loads are investigated utilizing the nonlocal elasticity theory. The physical properties of the nanobeam are considered to be functions of temperature based on the Touloukian model. In addition, to describe the FG porous materials, two different patterns of porosity distribution are adopted using trigonometric functions through the thickness of the nanobeam. The equations of motion in conjunction with the von-Kármán nonlinear assumption are established in the framework of Hamilton’s principle. By employing the Chebyshev-Ritz procedure, the nonlinear equations are discretized for three types of edge supports. Following that, the cylindrical arc-length technique is employed to assess the vibrational responses of the post-buckled nanobeam during static snap-through buckling. To evaluate the nonlinear dynamic buckling of the graded nanobeam under a sudden dynamic load, the Newmark time integration scheme together with the Newton-Raphson iterative method are utilized. Next, by means of the Budiansky-Roth criterion and the phase-plane approach, the dynamic snap-through loads are identified. After validating the developed mathematical model, a comprehensive investigation is carried out to determine the role of various physical and geometrical parameters on the dynamic snap-through buckling and vibration characteristics of the post-buckled FG nanobeams.

Stochastic thermo-elastic vibration characteristics of functionally graded porous nano-beams using first-order perturbation-based nonlocal finite element model

This paper emphasized the size-dependent thermo-elastic vibration characteristics of uncertain functionally graded (FG) porous nano-beams by employing the first-order perturbation theory (FOPT) with the finite element method. An isoparametric quadratic beam element having three nodes is used for the finite element analysis of the nano-beams with variation in degree of uncertainty in material properties, geometric configuration, and thermal environment. The power-law model is employed to obtain the effective material properties of the considered beams. The governing equations are presented according to Eringen’s elasticity theory and Reddy’s beam theory using the minimum potential energy. The computed FOPT results for the vibration statistics are assessed with the help of Monte Carlo simulation (MCS) and a well agreement is found. Further, results are presented by analyzing the effects of degree of source uncertainty, size-dependent parameter, volume fraction indices, porosity distribution, porosity index, thermal environment, and aspect ratios on the fundamental frequency of the FGM nano-beams. The numerical results revealed the significant influence of thermal environment and porosity on the vibration statistics of the nano-beams and provided guidance for the precise and reliable design of nano-devices for the application in thermal environment.

Dynamic instability of magnetically embedded functionally graded porous nanobeams using the strain gradient theory

A finite element modeling combined with the strain gradient theory (SGT) and the refined higher-order shear deformation beam theory is developed to study the dynamic instability of magnetically embedded functionally graded porous (FGP) nanobeams. Nanobeams with elastic foundation (EF) subjected to an axially oscillating load are analyzed. Nanobeams are made of functionally graded material (FGM) with an uneven porosity distribution. A three-node beam element with 8 degrees of freedom (DOFs) for two outer nodes and 2 DOFs for the middle node, which has the C1 and C2 continuous Hermite shape functions, is used to simulate nanobeams. Besides, Bolotin’s method is employed to determine the instability region of FGP nanobeams. The accuracy of the proposed method is tested by comparing it with other published works. In addition, the influences of various parameters such as magnetic potential, small-scale parameter, porosity coefficient, stiffness foundation, boundary conditions (BCs) on the dynamic instability of nanobeams are studied in detail.

Mathematical modeling and methods of analysis of generalized functionally gradient porous nanobeams and nanoplates subjected to temperature field

Mathematical modeling and methods of analysis of generalized functionally gradient porous nanobeams and nanoplates subjected to temperature field

Linear and nonlinear free vibration analysis of functionally graded porous nanobeam using stress-driven nonlocal integral model

Linear and nonlinear free vibration analysis of functionally graded porous (FGP) nanobeam with four different porous distribution patterns is performed on the basis of stress-driven two-phase local/nonlocal integral model. The strain is defined according to von Karman nonlinearity, and the differential governing equations as well as standard boundary conditions are derived through the Hamilton’s principle. The integral relation between nonlinear strain and nonlocal stress is transformed equivalently into differential form with constitutive boundary conditions. Neglecting the nonlinear terms, the bending deflection and moment are derived and expressed explicitly through the Laplace transformation with six unknown constants for linear free vibration, and the nonlinear characteristic equation for vibration frequency and linear vibration mode shape can be calculated through standard and constitutive boundary conditions. Two analytic methods including Ritz–Galerkin and perturbation methods based on both local and nonlocal linear vibration mode shape are applied to study the linear and nonlinear free vibration of nonlocal FGP nanobeams under different boundary conditions Meanwhile, direct numerical method based on general differential quadrature method and Newton’s iterative process are utilized to obtain the numerical solutions of nonlinear vibration frequencies. The influence of nonlocal parameters and vibration amplitude on the free vibration frequencies is investigated numerically. The numerical results show that influence of nonlocal parameters on linear vibration mode shape is not significant. Local linear vibration mode shape based Ritz–Galerkin​ and perturbation methods would overestimate and underestimate the linear free vibration of nonlocal simply-supported and clamped–clamped nanobeams. Nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods could provide more accurate prediction on nonlinear vibration frequencies than local mode shape based methods when W(0.5) is small. However, the relative error increases quickly with the increase of W(0.5). Based on the results obtained in this study, local and nonlocal linear vibration mode shape based Ritz–Galerkin and perturbation methods cannot provide accurate prediction on nonlinear vibration frequencies when the vibration magnitude is big, since linear vibration mode shape assumption is not suitable for strong nonlinear vibration.

Analytical and meshless DQM approaches to free vibration analysis of symmetric FGM porous nanobeams with piezoelectric effect

The current study is an first attempt to develop a vibrational-based piezoelectrically actuated smart nanoactuator model based on assumptions of Reddy third-order shear deformation beam model together with nonlocal strain gradient theory. The considered model is a sandwich structure consisting of piezoelectric layers under an external electric field, and symmetric to the midplane functionally graded (FG) porous core. Expressed by displacements equations of motion are derived based on a dynamic version of Hamilton's principle. Eigenvalue problem of free vibration of simply supported nanobeam is solved analytically by using properties of Fourier series, and numerically via meshless Differential Quadrature Method (DQM). The study includes a convergence study for the numerical approach, as well detailed discussion on obtained results involving effects of nonlocal parameters, external mechanical and electrical loads, material gradation coupled with diverse porosity distribution, and influence of external environment on dynamic behavior of the nanostructure. The present paper, in view of included optimization study, may positively affect nanodevices reliability and extend its operation by avoiding resonance phenomenon.

On the wave dispersion in functionally graded porous Timoshenko-Ehrenfest nanobeams based on the higher-order nonlocal gradient elasticity

Dispersion characteristics of flexural waves in a functionally graded (FG) porous nanobeam are analytically examined. Kinematics of the nano-sized beam is assumed to be consistent with the Timoshenko-Ehrenfest beam model. Material behavior of the FG porous beam is considered to vary symmetrically along the beam thickness while appropriate symmetric porosity models are applied to account for two diverse distributions of porosity with variable volume of voids. For the first time, size-dependent response of the symmetric FG porous nanobeam is realized within the framework of the higher-order nonlocal gradient elasticity theory. The integro-differential constitutive laws of the stress resultant fields are established and reinstated with the equivalent differential relations equipped with non-standard boundary conditions. The closed-form solution of the phase velocity of flexural waves is analytically determined. The ensuing numerical results of the flexural wave dispersion detect new benchmarks for numerical analysis and can be effectively exploited in design and optimization of composite nano-structural elements of advanced nano-electro-mechanical systems.

On the nonlinear dynamics of porous composite nanobeams connected with fullerenes

In this study, an attempt is made to investigate the large-amplitude nonlinear vibration response of symmetric porous functionally graded (FG) nanobeams carrying several fullerenes. The influence of various porosity distributions as well as position and mass of fullerenes on the nonlinear vibrational behaviour is particularly studied. To model size effects on the vibration response, a nonlocal elasticity and strain gradient theory in conjunction with the Euler-Bernoulli beam theory is used. Using Hamilton’s variational principles, the strain energy, external work and kinetic energy of the ultrasmall system are formulated. The governing nonlinear equations and the associated boundary conditions that incorporate geometrical nonlinearity are derived. Inhomogeneity in the material parameters of the FG porous nanobeam is modelled according to the power-law mixture of constituent materials with cosine function. A discretization technique based on Galerkin’s approach and a perturbation solution method are employed to estimate the shift in the nonlinear frequencies of the nanobeams due to the attached fullerene. The nonlinear frequency shift of the complex coupled nanosystem is calculated for various boundary conditions. The validity of the estimated results is evaluated by performing several comparison studies with available data in the literature. Several system parameters such as porosity distribution, material index, the number, position and mass of attached fullerenes, as well as size parameters including the length scale and nonlocal coefficients are taken into consideration in the nonlinear vibration analysis.

On the piezoelectric effect on stability of symmetric FGM porous nanobeams

In present paper, the piezoelectric effect on bifurcation buckling of symmetric FGM porous nanobeam is presented based on a higher-order nonlocal elasticity and strain gradient theory in conjunction with Reddy third-order shear deformation beam theory. The strain energy density and electric enthalpy density functions are used to derive constitutive relations for porous functionally graded, as well as piezoelectric materials. Equations of motion for elastically supported layered FGM nanobeam with diverse distribution of porosity and electro-elastic coupling are derived based on the modified Hamilton’s variational principle. The formulated boundary value problem is analytically solved. For the first time, the effects of porosity distribution, volume of voids, distance between porosity and FGM core surfaces as well as material gradation, layers size, aspect ratio, stiffness of Kerr foundation, and external electric voltage on critical in-plane force, critical porosity and critical voltage were comprehensively presented and discussed. The investigation enables analysis and precise multi-dimensional control of static response of smart beam-like nanostructures widely used in MEMS and NEMS devices.

Investigation of thermal preloading and porosity effects on the nonlocal nonlinear instability of FG nanobeams with geometrical imperfection

The aim of the present research is twofold. The first is to present a study on nonlinear thermo-mechanical bending and thermal postbuckling analysis of functionally graded (FG) porous perfect/imperfect nanobeam according to the nonlocal elasticity theory. The second, concurrent aim is to address the snap-through phenomenon in the thermally preloaded graded porous nanobeams due to lateral mechanical load. Two types of thermal loading including, uniform temperature rise and heat conduction through the thickness as well as two cases of porosity distribution, including uniform and uneven, are considered. Geometrically imperfect FG porous nanobeam with four different types of immovable boundary conditions is also taken into account. Thermo-mechanical material properties of the porous nanobeam are assumed to be position-dependent according to the modified rule of mixture. The nonlinear equilibrium equations are constructed using the Euler–Bernoulli beam model and von-Kármán type of geometrical nonlinearity. Chebyshev polynomial based Ritz method is utilized into the principle of virtual displacement to obtain the matrix representation of the nonlocal governing equations. Two different strategies, including the Newton–Raphson technique and direct displacement control scheme, are first proposed to extract the nonlinear bending and postbuckling curves of the graded nanobeam. Afterwards, to capture the snapping behavior and trace the path beyond the limit loads of the thermally preloaded nanobeam, the cylindrical arch-length technique is adopted. Numerical results are provided to explore the effect of different parameters such as the power-law index, nonlocal parameter, boundary conditions, porosity distribution, thermal loading, and imperfection amplitude on the nonlinear thermo-mechanical bending and instability of the FG nanobeam.

Effect of Axial Porosities on Flexomagnetic Response of In-Plane Compressed Piezomagnetic Nanobeams

We investigated the stability of an axially loaded Euler–Bernoulli porous nanobeam considering the flexomagnetic material properties. The flexomagneticity relates to the magnetization with strain gradients. Here we assume both piezomagnetic and flexomagnetic phenomena are coupled simultaneously with elastic relations in an inverse magnetization. Similar to flexoelectricity, the flexomagneticity is a size-dependent property. Therefore, its effect is more pronounced at small scales. We merge the stability equation with a nonlocal model of the strain gradient elasticity. The Navier sinusoidal transverse deflection is employed to attain the critical buckling load. Furthermore, different types of axial symmetric and asymmetric porosity distributions are studied. It was revealed that regardless of the high magnetic field, one can realize the flexomagnetic effect at a small scale. We demonstrate as well that for the larger thicknesses a difference between responses of piezomagnetic and piezo-flexomagnetic nanobeams would not be significant.

RETRACTED: Bending, buckling and vibration analyses of FG porous nanobeams resting on Pasternak foundation incorporating surface effects

AbstractIn the current paper, size‐dependent mechanical analysis of functionally graded porous (FGP) nanobeams resting on Pasternak foundation in thermal environment is presented based on nonlocal strain gradient theory and Gurtin‐Murdoch surface elasticity theory. The nanobeam is modeled based on Euler‐Bernoulli beam theory, Timoshenko beam theory and Reddy's third‐order shear deformation theory and the set of the governing equations are solved for a clamped‐clamped FGP nanobeam using generalized differential quadrature method. Numerical results show that as porosity parameter increases, static deflection increases and critical buckling load decreases, but its effect on the fundamental frequency strongly depends on the porosity distribution pattern. It was revealed that among all studied porosity distribution patterns, the minimum value of the static deflection and maximum values of critical buckling load and fundamental frequency can be achieved using symmetric distribution pattern of pores. It is concluded that tensional residual stress reduces static deflection and increases both critical buckling load and fundamental frequency and temperature rise has reverse effects.

Nonlinear random vibrations of functionally graded porous nanobeams using equivalent linearization method

In the present study, the nonlinear random vibration of a porous functionally graded nanobeam supported by a viscoelastic foundation has been investigated by making use of the statistical linearization method for the first time. Nonlocal Euler-Bernoulli beam theory with von Kármán type nonlinearity is utilized to derive the governing equation of the motion of the FG porous nanobeam. Two types of porosity distribution have been considered for the FG beam. To discretize the governing equation Galerkin's method is applied. The mean square value of the response of the resulted Duffing's equation in the presence of a random input with nonzero mean value has been extracted using statistical linearization method. The results of the present study are compared with the analytical results of the FPK approach and the results of the perturbation method for the classical limit case of the homogeneous beam. It is revealed through detailed numerical analysis that the statistical linearization method (in contrast to the perturbation method) gives reliable results and there is an excellent agreement between its results and the analytical results. The effect of various parameters on the mean value of the response is also investigated numerically and illustrated graphically. It is found that the mean square value of vibrations increases when porosity, power-law index, nonlocal parameter, or the mean value of the input increases.

Effect of Kerr Foundation and in-Plane Forces on Free Vibration of FGM Nanobeams with Diverse Distribution of Porosity

Abstract In the present paper, the effect of diverse distribution of functionally graded porous material and Kerr elastic foundation on natural vibrations of nanobeams subjected to in-plane forces is investigated based on the nonlocal strain gradient theory. The displacement field of the nanobeam satisfies assumptions of Reddy higher-order shear deformation beam theory. All the displacements gradients are assumed to be small, then the components of the Green-Lagrange strain tensor are linear and infinitesimal. The constitutive relations for functionally graded (FG) porous material are expressed by nonlocal and length scale parameters and power-law variation of material parameters in conjunction with cosine functions. It created possibility to investigate an effect of functionally graded materials with diverse distribution of porosity and volume of voids on mechanics of structures in nano scale. The Hamilton’s variational principle is utilized to derive governing equations of motion of the FG porous nanobeam. Analytical solution to formulated boundary value problem is obtained in closed-form by using Navier solution technique. Validation of obtained results and parametric study are presented in tabular and graphical form. Influence of axial tensile/compressive forces and three different types of porosity distribution as well as stiffness of Kerr foundation on natural frequencies of functionally graded nanobeam is comprehensively studied.

A comprehensive study for mechanical behavior of functionally graded porous nanobeams resting on elastic foundation

In this paper, a comprehensive study is presented for mechanical analysis of functionally graded porous (FGP) nanobeams resting on an elastic foundation. The nanobeam is modeled according to different beam theories along with nonlocal strain gradient theory and Gurtin–Murdoch surface elasticity theory. Using Hamilton’s principle, the set of governing equations are derived, and exact solutions are presented using Navier’s method for the static bending, mechanical buckling and free vibration analyses of simply supported FGP nanobeams. The effects of various parameters on static deflection, critical buckling load and fundamental natural frequency of FGP nanobeams are studied. It is shown that for all types of porosity distribution patterns, an increase in porosity parameter leads to an increase in static deflection, and a decrease in critical buckling load and effect of porosity parameter on fundamental natural frequency is dependent on the porosity distribution pattern. Numerical results confirm that tensional residual surface stress decreases static deflection and increases both critical buckling load and fundamental natural frequency, but compressive residual surface stress has opposite effects.

Implementation of Hermite–Ritz method and Navier’s technique for vibration of functionally graded porous nanobeam embedded in Winkler–Pasternak elastic foundation using bi-Helmholtz nonlocal elasticity

Implementation of Hermite–Ritz method and Navier’s technique for vibration of functionally graded porous nanobeam embedded in Winkler–Pasternak elastic foundation using bi-Helmholtz nonlocal elasticity

On the bifurcation buckling and vibration of porous nanobeams

In the present paper, effects of porous material on bifurcation buckling and natural vibrations of nanobeams are investigated based on the higher-order nonlocal strain gradient theory. The displacement field of the nanobeam satisfies assumptions of Reddy higher-order shear deformation beam theory. The displacements gradients are assumed to be small so that the components of the Green-Lagrange strain tensor are linear and infinitesimal. The constitutive relations for functionally graded porous material are expressed by nonlocal and length scale parameters and power-law variation of material parameters in conjunction with cosine functions to create a possibility to investigate the effect of diverse distributions of porosity on mechanics of nanostructures. Effect of Winkler-Pasternak foundation on mechanics of nanobeam is also considered. The Hamilton’s variational principle is utilized to derive governing equations of motion of the composite nanobeam. For the first time, the critical porosity is defined and examined for bifurcation buckling analysis of elastically supported nanobeams with symmetric distribution of porosity. Influence of axial forces and types of porosity distributions on eigenfrequencies of functionally graded nanobeams is studied. Classical theory without nonlocal effects is obtained as a special case and valid for all considered distributions of functionally graded material and volume fractions of voids.

Thermo-mechanical vibration of FG curved nanobeam containing porosities and reinforced by graphene platelets

Recently, the graphene platelets (GPLs) are extensively used as reinforcements to enhance the strength of porous materials. However, most studies are focused on the straight GPLs-reinforced structures under macro-scale. In this paper, the thermo-mechanical vibration of a curved functionally graded (FG) porous nanobeam reinforced with GPLs is studied via the curved refined shear deformation beam theory associated with nonlocal strain gradient size scale effect. Four different porosity distribution patterns are employed to account for the graded distribution of porosity, and the GPLs are assumed to be dispersed evenly over the porous nanobeam. The effective mechanical properties of the curved nanobeam with different FG porosity patterns are evaluated by virtue of the Halpin–Tsai micromechanical-based model. The effects of GPL weight fraction, temperature variation, opening angle, porosity distribution pattern and coefficient, as well as size-scale parameters on the vibrational performance of the GPLs-reinforced curved FG porous nanobeam with different boundary conditions are investigated. It is demonstrated that the fundamental frequency enhances with the increase of the GPL weight fraction, while increasing the porosity coefficient does not always decrease the vibrational frequency, which is related to the porosity distribution pattern and boundary condition. Moreover, the thermal environment and size-scale characteristics also have significant influences on the vibration of GPLs-reinforced porous nanobeam.

Nonlinear thermal behavior of shear deformable FG porous nanobeams with geometrical imperfection: Snap-through and postbuckling analysis

Present paper deals with the nonlinear thermal bending response, thermal postbuckling behavior, and snap-through phenomenon due to lateral mechanical load in a thermally preloaded functionally graded (FG) porous perfect/imperfect nanobeam subjected to two types of thermal loading, including heat conduction across the thickness and uniform temperature rise. Heterogenous material properties of the porous nanobeams are assumed to be position/temperature-dependent, where dependency is obtained according to the modified rule of mixture and Touloukian formulation. Two types of porosity distribution and also geometrical imperfection of the nanobeam are considered. Assuming the Timoshenko beam model and von-Karman nonlinearity, the nonlinear equilibrium equations are extracted on the basis of the nonlocal theory of elasticity. With the establishment of the principle of virtual displacement and Chebyshev polynomial of the first kind as the basic functions, the Ritz method is utilized to obtain the matrix form of the nonlocal governing equations. Two different strategies, including the Newton-Raphson technique and direct displacement control scheme, are first proposed to extract the nonlinear bending and postbuckling trajectories of the graded porous nanobeam. Afterwards, to trace the snapping behavior beyond limit loads of the graded porous nanobeam with thermal preloading, the cylindrical arch-length scheme as a path-following method is adopted. Numerical parametric investigations are given to discuss the influences of the power-law index, two types of porosity distribution and thermal loads, size-dependent nonlocal parameter, imperfection amplitude, and boundary conditions on the snap-through behavior and the nonlinear thermal stability of the FG nanobeam.

Dynamic analysis of functionally graded nonlocal nanobeam with different porosity models

This article presented a nanoscale modified continuum model to investigate the free vibration of functionally graded (FG) porous nanobeam by using finite element method. The main novelty of this manuscript is presenting effects of four different porosity models on vibration behaviors of nonlocal nanobeam structure including size effect, that not be discussed before The proposed porosity models are, uniform porosity distribution, symmetric with mid-plane, bottom surface distribution and top surface distribution. The nano-scale effect is included in modified model by using the differential nonlocal continuum theory of Eringen that adding the length scale into the constitutive equations as a material parameter constant. The graded material is distributed through the beam thickness by a generalized power law function. The beam is simply supported, and it is assumed to be thin. Therefore, the kinematic assumptions of Euler-Bernoulli beam theory are held. The mathematical model is solved numerically using the finite element method. Results demonstrate effects of porosity type, material gradation, and nanoscale parameters on the free vibration of nanobeam. The proposed model is effective in vibration analysis of NEMS structure manufactured by porous functionally graded materials.