Porous Nanobeams Research Articles

Application of machine learning methodology for investigating the vibration behavior of functionally graded porous nanobeams

This study presents a semi-analytical solution that can calculate the free vibration frequencies of functionally graded nanobeams with three distinct pore distributions under both deformable and rigid boundary conditions, based on nonlocal elasticity and Levinson beam theories. The novelty lies in the incorporation of transverse springs at both ends of porous functionally graded nanobeams and introducing a general eigenvalue problem dependent on the stiffness of these springs. This solution provides vibrational frequencies considering Levinson beam theory, non-local elasticity theory, spring stiffnesses, porosity coefficients, and temperature change. Additionally, the vibrational behavior of these porous nanobeams is explored through machine learning (ML) techniques. Four ML models namely artificial neural network (ANN), support vector regression (SVR), decision tree (DT), and extreme gradient boosting (XGB) are trained to predict the natural frequencies of nanobeams with varying pore distributions. The Sobol quasi-random space-filling method is employed to generate samples by altering input feature combinations for different porous nanobeam distributions. Model performance is evaluated using different performance indicators and visualization tools, with optimal hyperparameters determined via a Bayesian optimization algorithm. Results underscore the efficacy of ML models in predicting natural frequencies, with SVR and ANN demonstrating superior performance compared to XGB and DT. Notably, SVR and ANN exhibit exceptional R2 values of 0.999, along with the lowest MAE, MAPE, and RMSE values among the models assessed. At the end of the study, the effects of various parameters on porous gold (Au) nanobeams using the solution of the presented eigenvalue problem are discussed.

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

AbstractRETRACTION: S. Enayat, M. Hashemian, D. Toghraie, E. Jaberzadeh, “Bending, buckling and vibration analyses of FG porous nanobeams resting on Pasternak foundation incorporating surface effects,” Zeitschrift für Angewandte Mathematik und Mechanik (Early View): https://doi.org/10.1002/zamm.202000231.The above article, published online on 27 September 2020 in Wiley Online Library (wileyonlinelibrary.com), has been retracted by agreement between the journal Editor‐in‐Chief, Holm Altenbach, and Wiley‐VCH GmbH. The retraction has been agreed as there is unambiguous evidence that this article was accepted on the basis of a compromised peer review process. Therefore, the review of this article is deemed to be insufficient. The authors disagree with the retraction.

Thermomechanical Response of Smart Magneto-Electro-Elastic FGM Nanosensor Beams with Intended Porosity

AbstractThis study investigates the behavior of free vibrations in a variety of porous functionally graded nanobeams composed of ferroelectric barium-titanate (BaTiO3) and magnetostrictive cobalt-ferrite (CoFe2O4). There are four different models of porous nanobeams: the uniform porosity model (UPM), the symmetric porosity model (SPM), the porosity concentrated in the bottom region model (BPM), and the porosity concentrated in the top region model (TPM). The nanobeam constitutive equation calculates strains based on various factors, including classical mechanical stress, thermal expansion, magnetostrictive and electroelastic properties, and nonlocal elasticity. The study investigated the effects of various factors on the free vibration of nanobeams, including thermal stress, thermo-magneto-electroelastic coupling, electric and magnetic field potential, nonlocal features, porosity models, and changes in porosity volume. The temperature-dependent mechanical properties of BaTiO3 and CoFe2O4 have been recently explored in the literature for the first time. The dynamics of nanosensor beams are greatly influenced by temperature-dependent characteristics. As the ratios of CoFe2O4 and BaTiO3 in the nanobeam decrease, the dimensionless frequencies decrease and increase, respectively, based on the material grading index. The dimensionless frequencies were influenced by the nonlocal parameter, external electric potential, and temperature, causing them to rise. On the other hand, the slenderness ratio and external magnetic potential caused the frequencies to drop. The porosity volume ratio has different effects on frequencies depending on the porosity model.

Hygro-Thermo-Magneto-Elastic Vibration of Multidirectional Graded Porous Nanobeams with Axial Motion by Considering Rotary Inertia and Thickness Effects

Scale-dependent vibration and stability of three-dimensional functionally graded (FG) porous Rayleigh nanobeams with axial motion in varying environmental conditions are investigated by considering scale effects in the thickness orientation. It is supposed that the effective material characteristics of the nanobeam in the longitudinal, width, and thickness orientations are graded according to exponential and power-law functions. Different patterns involving uniform, non-uniform, and logarithmic distributions are considered to model the porosity impacts. Based on the nonlocal strain gradient Rayleigh beam model and incorporating thickness effects, the dynamical equation of the system is derived by considering linear, parabolic, and sinusoidal elastic foundations. The Galerkin discretization technique and Laplace transform are adopted to solve the eigenvalue problem and accomplish the stability analysis. Several comparative studies are performed with the available data in the open literature to validate the solution approach and results. Also, detailed parametric investigations are accomplished to interpret the importance of scale factor, rotary inertia factor, porosity characteristics, magneto-hygro-thermal loads, foundation parameters, gradient indices, and boundary conditions on the dynamical behavior, critical divergence, and flutter axial speeds. The results revealed that, contrary to the effect of the material gradient in thickness orientation, the longitudinal and width gradation of materials can lead to the increment of vibrational frequencies. The outcomes declared that the instability threshold improves significantly by considering the scale effects along the thickness orientation. Also, it is demonstrated that when the middle surface of the cross-section has the highest porosity, the stability regions expand by increasing the porosity coefficient. The present research outcomes can offer valuable insights into the optimum design of high-speed moving multi-directional graded porous nanosystems.

Hygrothermal-Magnetic Dynamics of Functionally Graded Porous Nanobeams on Viscoelastic Foundation

Hygrothermal-Magnetic Dynamics of Functionally Graded Porous Nanobeams on Viscoelastic Foundation

Vibration analysis of embedded porous nanobeams under thermal effects using boosting machine learning algorithms and semi-analytical approach

This study presents a thermal vibration analysis of functionally graded porous nanobeams using boosting machine learning models and a semi-analytical approach. Nonlocal strain gradient theory is employed to explore vibration behavior, accounting for thermal and size effects. A semi-analytical approach solution utilizing Fourier series and Stokes’ transform to establish an eigenvalue problem capable of examining vibrational frequencies of porous nanobeams in both rigid and deformable boundary conditions is presented. Four boosting models including gradient boosting (GBoost), light gradient boosting (LGBoost), extreme gradient boosting (LGBoost), and adaptive boosting (AdaBoost) are employed to study the impact of seven crucial parameters on natural frequencies of a nanobeam. Sobol quasi-random space-filling method is used to generate the samples by varying input feature combinations for different porous nanobeam distributions. The model performance is assessed using statistical metrics, visualization tools, 5-fold cross-validation, and SHAP analysis for feature importance. The results highlight the effectiveness of boosting ML models in predicting natural frequencies, particularly XGBoost, which achieved an exceptional R2 value of 0.999, accompanied by the lowest MAE, MAPE, and RMSE values among the models assessed. LGBoost and AdaBoost follow XGBoost in performance, while GBoost exhibits relatively lower effectiveness, as highlighted by radar plots. The SHAP analysis revealed the significant impact of the foundation parameter on frequency prediction, with porosity coefficients notably influencing higher vibration modes.

Examination of the high-frequency behavior of functionally graded porous nanobeams using nonlocal simple higher-order shear deformation theory

Examination of the high-frequency behavior of functionally graded porous nanobeams using nonlocal simple higher-order shear deformation theory

Rotary inertia effect on dynamic analysis of embedded FG porous nanobeams under deformable boundary conditions with the effect of neutral axis

Rotary inertia effect on dynamic analysis of embedded FG porous nanobeams under deformable boundary conditions with the effect of neutral axis

The random thermo-elastic nonlinear vibration analysis of porous FGM nano-beams using the first order perturbation theory

The objective of present work is to study the nonlocal nonlinear thermo-elastic vibrations of porous nanobeams made up of functionally graded material (FGM) with random material properties. The mechanical properties of the nano-beam are calculated using a porosity based power law model by assuming the cosine type porosity distributions. The randomness present in the material properties cause the mechanical characteristics of the nano-beams to be random, hence first order perturbation theory has been employed to take care of this randomness. The Reddy’s beam theory in association with the Eringen’s non classical elasticity has been used to account for nonlocal effects, and Von-karman nonlinearity has been employed to take care of geometric nonlinearity. The governing equations are derived from the minimum potential energy principle and solved them using the finite element method. To validate the present model, a convergence and comparison study has been performed. Further, the effects of amplitude ratio, aspect ratio, uncertain material properties, size parameter, porosity, thermal environment, and gradation of material on the stochastic behavior of the porous FGM nano-beams are demonstrated. As a result of the present research, the significance of considering thermal environment, porosity, and random material properties has been revealed for the design of nanodevices and systems to be more reliable.

Dynamic analysis of 2DFGM porous nanobeams under moving load with surface stress and microstructure effects using Ritz method

This paper investigates the dynamic behavior of micro/nanobeams made of two-dimensional functionally graded porous material (2DFGPM) under accelerated, decelerated, and uniform moving harmonic load, using surface elasticity and modified couple stress theories. The key feature of this formulation is that it deals with a higher order shear deformation beam theory. The non-classical equilibrium equations are developed using Lagrange's equation and the concept of physical neutral surface. The equations of motion are derived using the same approach, accounting for the porosity effect and the modified power-law distribution of material properties. The trigonometric Ritz method is used with sinusoidal trial functions for the displacement field, and the Newmark method is applied to obtain the dynamical response of 2DFGPM nanobeams. The results are compared with previous studies, and the impact of critical parameters such as gradation indices, volume fraction ratio, pattern of porosity, velocity, frequency, and motion type of the applied force are explored. This study highlights the importance of considering the porosity effect, as neglecting it can lead to significant errors in the predicted results. Additionally, the study found that the accelerated and decelerated motions of the applied load have a greater impact on the dynamical deflection of 2DFGPM nanobeams than the uniform motion. The findings of this study can provide guidance for the optimal design of micro/nanobeams subjected to a moving force with multifunctional properties.

On nonlinear stability analysis of saturated embedded porous nanobeams

In this work, the nonlinear stability behaviors of the saturated porous nanobeams embedded in an elastic foundation are investigated. The restrained nanobeam is modeled using geometrical nonlinear equations in conjunction with the constitutive law of saturation. Three patterns for saturation along the thickness of the nanobeam are considered as porous/monotonous, porous/nonlinear symmetric, and porous/nonlinear non-symmetric distributions. The infinite terms of the power series are discretized thanks to Stokes' transformation and trigonometric series. Then, the effects of saturation and nonlinearity on buckling loads are studied by considering the restrained boundary conditions. Moreover, the nonlinear results are validated with the saturated nanobeam with the rigid supporting conditions.

Influence of surface effect on post-buckling behavior of graded porous nanobeam subjected to follower force

Influence of surface effect on post-buckling behavior of graded porous nanobeam subjected to follower force

Statistical analysis and Neural Network Modeling of functionally graded porous nanobeams vibration in an elastic medium by considering the surface effects

The natural frequency of a clamped–clamped functionally graded porous (FGP) nanobeam is predicted in this study. Material distribution is considered based on monotonous, symmetric, and non-symmetric patterns in the thickness direction. This paper deals with governing equations of nanobeams based on third-order shear deformation beam theory in conjunction with nonlocal strain gradient theory (NSGT) and surface effects. Artificial neural network (ANN) is utilized to predict the effect of eight parameters including temperature gradient, residual surface stress, porosity distribution pattern, porosity parameter, nonlocal and material length scale parameters, and elastic and shear coefficients of Pasternak foundation on the fundamental frequency of FGP nanobeam. Different training methods are selected to simulate input and output dependency. Results show that the dependency of the natural frequency is inverse to the temperature gradient and nonlocal parameter in the sense that increasing these factors will decrease the natural frequency. Also, increasing the material length scale parameter grows the effect of the nonlocal parameter. Residual surface stress, material length scale, and Pasternak foundation parameters have a direct effect on the output and among them; the material length scale parameter has a more noticeable effect. Finally, it was found that by increasing the porosity parameter value, the diversity of natural frequency levels up drastically

Vibration analysis of functionally graded porous nano-beams: A comparison study

Porous nano-beams has been widely used in macro as well as nano and micro-sized systems and devices such as biosensors, nano actuators, and nano-electro-mechanical systems. Presence of porosity in the nano-beams significantly affects the material properties and consequently the structural behavior of the nano-beams. Hence it is very important to take care of porosity effects during the modelling of nanostructures. In the present article, vibration characteristics of porous nano beams has been studied along with a comparative study between the two porosity models by taking the cosine type porosity distribution. One porosity model is based on the incorporation of porosity by reducing the average material properties of the constituents in the same amount. While second model is based on the open cell type porosity theory. The effective material properties of the nanobeams are obtained by modifying the power law model according to the different methods of porosity incorporation. The Reddy’s beam theory in conjunction with Eringen’s nonlocal elasticity theory is used to define the displacement field by taking care of size effects. Further, governing equations are derived using the Lagrange’s equation. Finite element method by considering the three noded beam element with four degrees of freedom at each node is used to solve the governing equation. In addition, the influence of porosity co-efficient, the porosity incorporation techniques, and power law index on the effective material properties and vibration characteristics of the functionally graded nano-beams has been studied. A significant influence of porosity incorporation technique on the effective material properties and non-dimensional fundamental frequency of nanobeams has been found for higher values of porosity fraction. For 20% volume of porosity, maximum 6.68% difference between the frequency is found for the considered power law index by using different porosity incorporation methods. While this difference is found 35.74% for 80% volume of porosity with the considered power law index.

Nonlocal strain gradient finite element procedure for hygro-thermal vibration analysis of bidirectional functionally graded porous nanobeams

ABSTRACT This paper presents the hygro-thermal vibration analysis of bidirectional functionally graded (BDFG) porous nanobeams resting on elastic foundation based on a nonlocal strain gradient theory and the finite element method. The effects of temperature and moisture on structures are assumed to cause tension load in the plane and do not change the material's mechanical properties. Nanobeams are made from functionally graded (FG) materials which the mechanical properties change along with the thickness (z-axis) and length (x-axis) of nanobeams according to the power-law model. The porosity distribution is considered to be even and uneven. For the first time, a two-node beam element with nine degrees of freedom at each node is developed to analyze the hygro-thermal vibration behavior in the nanoscale. The elastic foundation is used in this work to be the Winkler-Pasternak type. Applying Hamilton's principle based on the refined higher-order shear deformation beam theory (RBT) and the~nonlocal strain gradient model, governing equations of nanobeams are derived. The accuracy of the proposed method is verified by comparing the obtained numerical results with those of the published works in the literature. The nonlocal coefficient makes the nanobeam softer, while the strain gradient coefficient makes it stiffer. In addition, the effects of the nonlocal coefficient, porosity coefficient, foundation stiffness, boundary conditions on the natural frequency of nanobeams are investigated in detail.

Size-dependent vibrations of bi-directional functionally graded porous beams under moving loads incorporating thickness effect

This article analyzes the forced and free vibrations of bi-directional functionally graded (BDFG) porous nanobeams under moving loads by considering the thickness effect based on the nonlocal strain gradient theory (NSGT). It is assumed that the material properties of the system are graded across the transverse and axial directions according to power and exponential laws, respectively. Various porosity models, including even, uneven, and asymmetric porosity distribution patterns, are considered to model the porosity effects. The dynamic equation of the system is derived by implementing the physical neutral surface concept. Analytical and numerical approaches are adopted to acquire the vibration response of the system. Mathematical closed-form expressions are provided for various dynamic phenomena, such as two types of cancelation, two types of resonance, maximum resonance, cancelation disappearance, and resonance disappearance. In addition, for the first time in this article, it is demonstrated that BDFG porous nanobeams under moving loads are prone to experience the double cancelation phenomenon. Several comparison studies with published data are conducted to validate the results. Also, comprehensive parametric studies are carried out to elucidate the impacts of system parameters such as thickness power index, axial gradient parameter, scale parameter ratio, axial force, porosity coefficient, and porosity distribution pattern on the dynamic magnification factor, critical load speed, and maximum free-response. The outcomes established that by considering the thickness effect, the stiffening behavior of the system is amplified depending on the slenderness ratio. Also, it is concluded that, among the considered porous models, the critical load speed for the system with an even porosity distribution is more dependent on the thickness power index.

Nonlinear Eigen frequencies of a functionally graded porous nano-beam with respect to the coulomb and Casimir forces

A new mathematical model of a porous functionally graded micro/nano-beam under the action of the Casimir force and the Coulomb force is constructed in this paper. The construction of the mathematical model takes into account the Euler-Bernoulli kinematic model. The dimensional-dependent parameter is taken into account by modified moment theory of elasticity. The variational, differential equations, boundary and initial conditions are derived from Ostrogradsky-Hamiltonian variational principle. The problem of non-linear natural oscillations of a beam under the action of force Casimir and Coulomb force is solved. The influence of the Casimir and Coulomb forces on the nonlinear eigenfrequencies of the micro/nano-beam is shown.

DETECTION OF NONLOCAL CALIBRATION PARAMETERS AND RANGE INTERACTION FOR DYNAMICS OF FGM POROUS NANOBEAMS UNDER ELECTRO-MECHANICAL LOADS

The present investigation examines the range of effect of nonlocal parameters on dynamic behavior of a smart beam-like nanostructure modeled as sandwich functionally graded porous nanobeam with piezoelectric layers. Therefore, the study is concentrated on determining length of the structure for which nonlocal effects are observed for vibration of nanobeam under in-plane electro-mechanical forces. The nanobeam-based NEMS device model is obtained based on assumptions of the nonlocal strain gradient theory in conjunction with Reddy higher-order shear deformation theory. The investigation present differences in obtained results for nanostructure’s free vibration based on classical and nonlocal assumptions. To study range of application of nonlocal parameters for different length of simply supported nanobeam, defined eigenvalue problem is solved in view of variation of length to thickness ratio, distribution of material properties, as well as electro-mechanical loads. What is more, the study attempts to determine and calibrate values of size-dependent coefficients based on expected natural frequencies, material properties, and applied loads. The results are completed with extensive discussion on the dependence of nonlocal parameters on nanobeam’s dynamic response, thus may be an important step forward to extend understanding of ultra-small structure’s behavior.

Thermomechanical vibration and buckling response of nonlocal strain gradient porous FG nanobeams subjected to magnetic and thermal fields

This study investigates the free vibration and thermal buckling behavior of functionally graded porous nanobeams in magnetic and thermal fields using high-order trigonometric shear stress and nonlocal strain gradient elasticity theories. The results demonstrated the effects of nonlocal differential and strain gradient elasticities on softening and stiffness enhancements, respectively. Additionally, the Lorentz force induced by the magnetic field makes nanobeam’s vibratory motion difficult, causing the natural frequencies to increase. This situation can contribute to the dynamic stability of nanobeams exposed to the nonlinear temperature distribution. This study’s results will assist in designing and implementing micro/nanoelectromechanical systems.

A new model for non-linear vibration of functionally graded porous nano-Beam based on non-local curvature and strain gradient tensors

In this paper, non-linear free vibration analysis of nano-beam has been studied. The non-local strain gradient theory and curvature tensor are used to show the size effect. The length scale parameter expresses the effect of strain gradient tensor in the non-local strain gradient theory. However, the aim of this article is to show the simultaneous effect of curvature and strain gradient tensors in non-linear vibration of functionally graded porous nano-beams. The effect of curvature tensor is demonstrated with the curvature tensor dependent parameter. Considering non-linear Von Kármán strains and Euler–Bernoulli beam theory, the governing vibrational equation of FG porous nano-beams are derived using Hamilton’s principle in the presence of strain gradient and curvature tensors simultaneously. The non-linear differential equation is extracted by using Galerkin’s method and the non-linear natural frequency of nano-beam is obtained according to Hamiltonian approach. Results represent the simultaneous effects of the length scale and curvature tensor dependent parameters on dimensionless non-linear natural frequencies. Also effects of different parameters such as non-local parameter, length scale parameter, porosity volume index, and power-law index are discussed in the presence and absence of the curvature tensor dependent parameter. Also, the beginning points of stiffness-hardening and stiffness-softening of nano-beam are always constant values in the non-local strain gradient theory, whereas considering the curvature tensor changes the beginning points of stiffness-hardening and stiffness-softening. The results are also compared with previous researches for validation.