Nonlocal Elasticity Research Articles

Numerical investigation on the thermal-nanofluidic flow induced transverse and longitudinal vibrations of single and multi-walled branched nanotubes resting on nonlinear elastic foundations in a magnetic environment

Carbon nanotubes (CNTs) have attracted great attention due to their tremendous mechanical, thermal, structural and electrical properties leading to many promising applications. The unique and interesting properties of CNTs, such as their mechanical and electrical features, have fascinated industries and researchers to implement CNTs for production of different electromechanical devices. However, it is established that the phenomenon of buckling often occurs when CNTs are subjected to compressive loads and their frequencies under vibration are in order of Tera-Hz and Giga-Hz. Moreover, the profound dynamic behaviours of nanotube under the influences of slip when conveying fluid and magnetic field require a serious and meticulous study. Therefore, this study focuses on the vibration analysis of nanofluid-conveying embedded multi-walled branched nanotubes resting on Winkler-Pasternak foundation in a thermal-magnetic environment. Using Euler-Bernoulli theory, Hamilton's principle and nonlocal elasticity theory, fully coupled equations of motion governing the transverse and longitudinal vibrations of the nanotube are developed. Also, the equation of the deformation of the nanotubes as well as the pressure variation in the tubes are developed. Additionally, Navier-Stoke's equation and energy equation for the fluid and nanotube are coupled with the vibration models. The dynamics of the multi-walled carbon nanotubes when coupled with Navier-Stokes and energy equations require meticulous study as it is different from the usual assumption of plug flow. Therefore, in this present work, the developed coupled systems of nonlinear partial differential equations are solved using multi-dimensional numerical PDEs solvers coupled with PDE-tools in MATLAB. With the aids of the solutions, parametric studies were performed. The results indicate that increasing the downstream angle decreases the stability of the system. Also, the results obtained from the dynamic behaviour of the system indicate that the magnetic effect has an attenuating or damping effect of about 20 %. Furthermore, the plug flow assumption deviates from actual working processes by over 11 %. The analytical solutions verified and validated with existing analytical, numerical and experimental results. It is envisaged that the present study will give better insight into nanotubes design and serve as an index for subsequent works in the research area.

Vibrational analysis of size-dependent thermo-piezo-photo-electric semiconductor medium under memory-dependent Moore–Gibson–Thompson photo-thermoelasticity theory

This study presents a comprehensive approach to vibrational analysis in a size-dependent, orthotropic thermo-piezo-photo-electric semiconductor half-space under Moore–Gibson–Thompson (MGT) photo-thermoelasticity theory with memory-dependent derivatives. Following Eringen’s nonlocal elasticity theory, the governing equations for the considered material are derived. The normal mode technique is applied to investigate the photothermal excitation process. Focusing on the interface adjacent to the vacuum, the study considers plasma, thermal, and mechanical stress boundary conditions to determine fundamental physical quantities. Through a detailed exploration, this research examines the impact of various thermo-piezo-photo-electric models, time, nonlocal parameters, and spatial coordinates on different physical quantities. The effects are meticulously analyzed and visually represented through graphical illustrations. While the literature survey reveals numerous studies on vibrational analysis in thermo-piezo-photo-electric semiconductor mediums under different thermoelasticity theories, to the best of the authors’ knowledge, no research emphasizing vibrational analysis in a size-dependent thermo-piezo-photo-electric semiconductor medium has been conducted. Moreover, the effect of memory-dependent MGT photo-thermoelasticity theory on a size-dependent thermo-piezo-photo-electric semiconductor half-space has not been illuminated until now, significantly defining the novelty of the conducted research.

On shear-dependent vibration of nano frames

In this study, the effect of shear deformation on the atomic size-dependent free vibration of nanoframes is investigated. The equations of motion of axial and bending vibrations are obtained via the variational algebra according to nonlocal elasticity theory. Axial and bending behaviors are expressed by using simple rod theory and first-order shear deformation beam theory, respectively. By combining the nonlocal finite element solutions for the equations of motion, an eigenvalue formulation for the free vibration of nanoframe is developed. In the presentation of numerical results, firstly, some comparison studies are carried out to show the accuracy of the present analysis. Then, the variations of nondimensional frequencies of three different nanoframe structures are investigated with respect to various parameters, and the results are discussed in detail. The focus of this study is to better understand the dynamic behavior of nanostructures that are within the scope of nano-electro-mechanical systems (NEMS) technology and can be modelled with discrete members.

A size-dependent 3D solution of functionally graded shallow nanoshells

Abstract An unavailable semi-analytical non-local 3D solution for functionally graded nanoshells with constant radii of curvature is presented. The small length scale effect is included in Eringen’s nonlocal elasticity theory. The constitutive and equilibrium equations are written in terms of curvilinear orthogonal coordinates systems which are only valid for spherical and cylindrical shells, and rectangular plates. The stresses and displacements are assumed in terms of the Navier method which is applicable for simply supported structures. The derivatives in terms of thickness are approximated by the differential quadrature method (DQM). The thickness domain is discretized by the Chebyshev–Gauss–Lobatto grid distribution. Lagrange interpolation polynomials are considered as the basis function for DQM. The correct free surface boundary condition for out-of-plane stresses is considered. Several problems of isotropic and functionally graded shells subjected to different types of loads are analyzed. The results are compared with other three-dimensional solutions and higher-order theories. It is important to emphasize that the radii of curvature are crucial at nanoscale, so it should be considered in the design of nanodevices.

Nonlocal strain gradient-based isogeometric analysis of graphene platelets-reinforced functionally graded triply periodic minimal surface nanoplates

Recent developments in additive manufacturing (AM) technologies have empowered the design and fabrication of intricate bioinspired engineering structures at the nano/micro scale. However, mathematical modeling and computation of these structures are still challenging. The main target of this study is to address an efficient computational approach for predicting the mechanical behavior of graphene platelets (GPLs)-reinforced functionally graded triply periodic minimal surface (FG-TPMS) nanoplates. The computational model integrates both nonlocal elasticity and strain gradient effects into the NURBS-based isogeometric analysis of these small-scale structures. We establish advanced nanoplate models by combining three sheet-based TPMS architectures with two new porosity distribution patterns and three distribution patterns of GPLs across the thickness direction. Moreover, the present work makes a pioneering attempt to elucidate how the stiffness-hardening and stiffness-softening mechanisms influence FG-TPMS nanoplates reinforced with GPLs. Compared with two common cellular solids, the superiority of TPMS architectures' mechanical performance is demonstrated. Among all, P and IWP TPMS types along with symmetric porosity and GPLs distributions exhibit outstanding behaviors under static bending, free vibration, and dynamic instability. Furthermore, we conducted a performance analysis for the first time, showcasing the superior capabilities of TPMS architectures under dynamic in-plane compressive loads, especially when compared to isotropic plates of equal weight. The findings of this study greatly enhance our understanding of the intricate mechanical responses of GPLs-reinforced TPMS architectures at the small-scale level, contributing to future interdisciplinary applications.

Energy distribution of plane waves in functionally graded nonlocal integral thermo-electro-elastic nanoplates in liquid

Based on the nonlocal integral elasticity, this paper focuses on the plane wave propagation in functionally gradient piezoelectric material (FGPM) nanoplates immersed in liquid, especially on the dissipation characteristics of the waves. A mathematical model is established by using the presented Legendre polynomial series approach. Comparisons with the published works not only demonstrate the correctness of the presented method, but also indicate the significant differences between the nonlocal integral and differential elasticities at high frequencies or at large incident angles. Besides, the energy dissipation and the crucial incident angle both increase as the nonlocal parameter increases.

Investigation of vibration of the nano rotating blade coupled with viscous fluid medium by considering the nonlocal elastic theory

In view of the great importance of dynamical behavior prediction of nanostructures in contact with fluid and their vast range of applications in biomedical engineering, aerospace, etc., in this research, the free vibration of a Nanoscale Euler-Bernoulli rotating beam coupled with incompressible viscous fluid is studied. Small-scale effects are applied by using nonlocal elasticity theory. Using the Navier-Stokes relation, the interaction forces between the fluid and nanobeam are obtained. Governing differential equations have been solved by Galerkin method and the system vibrations frequency response has been obtained for clamped-free boundary condition. Based on the results of this research, nonlocal elasticity has a different effect on different vibration modes. The frequency of the nanobeam coupled with the fluid quickly increases when applying this theory, and the presence of fluid reduces the natural frequencies.

A layer-wise theory for the dynamic analysis of functionally graded composite nanobeams under thermal environment using nonlocal boundary conditions and Chebyshev collocation technique

In this study, a layer-wise theory satisfying displacement continuity at each layer’s interface and the Chebyshev collocation technique are adopted to analyze the vibration behavior of functionally graded composite nanobeams under non-uniform temperature variation in the thickness direction. The temperature-dependent mechanical properties vary according to the rule of mixture. Boundary conditions are modified by including the effect of size parameter and named as nonlocal boundary conditions. These nonlocal boundary conditions along with Eringen’s nonlocal elasticity theory are developed to capture the effect of size-dependency for such a nanobeam based on the first-order shear deformation theory (FSDT) and physical neutral plane. The energy principle and the variational approach are used to obtain force-moment equilibrium equations, which lead to governing differential equations and local boundary conditions. Two types of sandwich constructions are considered, namely, Type 1 (functionally graded-homogeneous-functionally graded) and Type 2 (homogeneous-functionally graded-homogeneous), by maintaining the material continuity at the interfaces. Natural frequencies are analyzed in the fundamental mode for various values of volume fraction index, thickness of the layers, size scale parameter, and thermal environment. The convergence rate of the Chebyshev collocation technique is also shown over the differential quadrature method.

Dynamic stability of the sandwich nano-beam system

This article explores the stochastic stability of a three nanobeam system connected by Kerr type elastic layer, subjected to the magnetic effects and compressive axial forces. While considering nano beams rotary inertia is taken into account and differential equations of nanobeams are given according to Eringen's nonlocal elasticity theory. The impact of a longitudinal magnetic field on the behavior of the nanobeam system is described using Lorentz magnetic forces. The objective of this research paper is to investigate the moment and almost sure stability of the system. It analyzes the system's stability under white noise excitation by employing the contact transformation method and perturbation approach. Notably, we consider the case where the system exhibits low damping. The study focuses on examining the system's dynamic behavior and stability characteristics under these conditions. The stochastic stability of this system is investigated through moment Lyapunov exponents and Lyapunov exponents for the first time and interesting conclusions regarding the stability region of this system were obtained by varying various system parameters. These conclusions are presented in a visual manner and discussed.

Torsional Dynamics of Axially Graded Viscoelastic Carbon Nanotubes

Torsional vibration analysis of the axially functionally graded carbon nanotubes has been carried out. Nonlocal stress gradient elasticity theory has been used in continuum mechanics model of the carbon nanotube. Variation of the material properties of the axially graded nanostructure has been assumed in exponential form. Differently from the majority of literature works, viscous damping and nonlocal parameters have been assumed in grading form. Energy functional for the carbon nanotube has been achieved with minimum potential energy principle and weak form solution has been obtained with the Ritz Method. Effects of material grading, nonlocality and viscoelasticity to the torsional dynamics of axially graded carbon nanotube have been investigated. Results of the present work could be useful in modeling and production of axially functionally graded nanostructures.

Dynamic behavior of nanoplate on viscoelastic foundation based on spatial-temporal fractional order viscoelasticity and thermoelasticity

The nanodevices often work in a thermal environment and lead to the viscoelastic and thermoelastic coupling which becomes more complicated when further coupled with the size effects. A spatial-temporal fractional order model is proposed to study the dynamic behavior of thermoelastic nanoplate in the present work. The integer order differential is extended to the fractional order differential with the integer order differential as a special case. This extension makes the modeling of small-scale effects of mechanical behavior and generalized heat conduction more flexible. Non-local strain gradient elasticity and non-local heat conduction are used to describe the spatial non-local effects while the fractional order differential to describe the complicated history-dependent effects or the temporal nonlocal effects. The standard solid viscoelastic model and generalized heat conduction model with the fractional order differentials lead to the fractional-order differential governing equations. The fractional order differential equation is solved by the Laplace transform method, and the analytical solution of the response is expressed in terms of the Mittag-Leffler function. In order to verify the reliability of the analytical solution, the numerical solution is also provided for comparison. Based on the numerical results of dynamic response under the step load, the effects of thermoelastic fractional order parameters, viscoelastic fractional order parameters, and small-scale parameters are discussed. The new spatial-temporal fractional order model is a natural extension of the traditional integer order model which can be recovered from the present model.

Prediction of nonlocal elasticity parameters using high-throughput molecular dynamics simulations and machine learning

Higher-order elasticity theories are able to model the mechanics at nanoscale. However, these theories have length-scale parameters that need to be evaluated either through experiments or molecular dynamics (MD) simulations. In the current literature, these length-scale parameters are assumed to be constant for a chosen material. However, this is not true, as shown in this paper through a set of high-throughput MD simulations on a variety of boundary value problems (BVP). For carbon and boron nitride nanotubes, the length-scale parameter in the modified strain gradient theory is found to vary with the absolute dimensions, type of the BVP, and the deformation level. This complex dependence makes prediction of the length-scale parameter l0 challenging. To address this issue, a supervised machine learning (ML) based framework is developed here. In this new framework, MD, continuum formulation, and ML are used in tandem to predict the length-scale parameter for a given material, dimension, and boundary condition. It is a three-step process. First, a number of MD simulations are performed for varying chirality, length, and boundary conditions. Then, the values of l0 are found using these MD output in the continuum formulation. Finally, an ML-based regression model is used to capture the variability of l0 and subsequently predict its value for any chirality, length, and boundary condition. Here three regression models – Gaussian process regression, support vector machine, and neural network – are used to make the predictions.This predictive tool eliminates need for expensive MD simulations for further continuum scale analysis. In a broader context, this novel three-step framework opens an accurate yet inexpensive door for applying non-classical continuum theories to nanoscale mechanics problems. The accuracy is imposed by the MD simulations, while the computational cost reduction is achieved by machine learning.

Effect of non‐local modified couple stress theory on the responses of axially moving thermoelastic nano‐beams

AbstractThe present study examines the lateral thermal vibrations of a nanobeam subjected to an axial motion while being influenced by a sinusoidal thermal load. The partial differential equation describing the system was obtained using the extended Hamiltonian principle. Also considered were the Euler‐Bernoulli (EB) beam, the nonlocal couple stress theory, and Eringen's nonlocal elasticity model. This type of axially moving beam has multiple applications in design, including industrial, civil, structural, chemical, and computer engineering. The Laplace transform technique is utilized to translate partial differential equations into a thermoelastic differential equation of the sixth order. This study investigates the impact of nanobeam size and velocity on thermo‐mechanical characteristics. To explore the impacts of axial velocity, pulse width, nonlocal index, material length scale coefficient, and phase lag coefficients on the examined studied fields, such as lateral vibration and temperature change for the moving nanobeam are calculated. The specified factors were discovered to impact the flexibility and dynamic response of the nanobeam substantially.

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams

Exact closed-form solutions for the free vibration analysis of multiple cracked FGM nanobeams with arbitrary boundary conditions based on the Nonlocal Elasticity Theory (NET) and the Timoshenko beam theory are presented. The NET considering the size effect of nanostructures is applied. FGM properties vary nonlinearly along the height of the beam. A crack model using two springs with stiffness depending on the crack depth is applied. The proposed solutions are provided explicitly as functions of integration constants determined from the standard boundary conditions. Frequency equations, that established in the form of the determinant of the matrix 3 × 3 order for arbitrary boundary conditions, are applied to analyse free vibration. Expressions for the mode shapes are given explicitly. The effects of geometry, material, nonlocal and crack parameters on the free vibration of the multiple cracked nanobeam are then analysed in detail.

Winkler‐Pasternak foundation effect on the buckling loads of arbitrarily rigid or restrained supported nonlocal beams made of different FGM and porosity distributions

AbstractThe present research investigates lateral stability of a functionally graded nanobeam using Eringen's differential nonlocal elasticity model under rigid (clamped, pinned, free) and deformable (lateral, rotational restraints) boundary conditions. Sigmoid and power law have been employed as grading laws to study the influence of the material distribution on the snap‐buckling analysis of a nanobeam with arbitrary boundary conditions. Moreover, Fourier sine series with Stokes’ transformation are employed to investigate the effects of boundary conditions on the stability response of nanobeams embedded in a Pasternak foundation. A parametric study has been performed to investigate the effect of deformable boundaries, Pasternak foundation and small‐scale parameters on the stability response of the nanobeam and the results have been presented in a series of tables and figures. It has been observed that consideration of the small‐scale parameter, Pasternak foundation, deformable boundaries and functionally grading index (of sigmoid and power‐law) are essential while analyzing the static stability response. The obtained analytical results may be used as benchmarks in future researches of functionally graded nanobeams embedded in an elastic medium.

Dynamic Analysis of a Timoshenko–Ehrenfest Single-Walled Carbon Nanotube in the Presence of Surface Effects: The Truncated Theory

The main objective of this paper is to study the free vibration of a Timoshenko–Ehrenfest single-walled carbon nanotube based on the nonlocal theory and taking surface effects into account. To model these effects on frequency response of nanotubes, we use Eringen’s nonlocal elastic theory and surface elastic theory proposed by Gurtin and Murdoch to modify the governing equation. A modified version of Timoshenko nonlocal elasticity theory—known as the nonlocal truncated Timoshenko beam theory—is put forth to investigate the free vibration behavior of single-walled carbon nanotubes (SWCNTs). Using Hamilton’s principle, the governing equations and the corresponding boundary conditions are derived. Finally, to check the accuracy and validity of the proposed method, some numerical examples are carried out. The impacts of the nonlocal coefficient, surface effects, and nanotube length on the free vibration of single-walled carbon nanotubes (SWCNTs) are evaluated, and the results are compared with those found in the literature. The findings indicate that the length of the nanotube, the nonlocal parameter, and the surface effect all play important roles and should not be disregarded in the vibrational analysis of nanotubes. Finally, the results show how effective and successful the current formulation is at explaining the behavior of nanobeams.

Thermoelastic damping analysis of functionally graded sandwich microbeam resonators incorporating nonlocal and surface effects

Functionally graded (FG) sandwich microstructures/nanostructures stand as one of the most promising composite structures due to the capability to resist thermal shock and high noise in extreme thermal environments. Meanwhile, accurately prediction quality-factor based on thermoelastic damping (TED) is of great significance for the design of high-performance microresonators /nanoresonators. However, the classical TED models fail to explain the size-dependent thermo-mechanical behavior. The nonlocal elasticity and surface elasticity are responsible for the size-dependent phenomenon. In this article, a modified TED model is proposed to estimate the impact of the size-dependent effect on the TED of FG sandwich microbeam resonators in unsteady heat transfer processes by combining the nonlocal elasticity model, surface elasticity model, and dual-phase-lag (DPL) heat conduction model. It is assumed that the FG sandwich microbeam resonators consist of a ceramic core and FG surfaces. The energy equation and the transverse motion equation are derived. The analytical expression of TED is obtained by the complex frequency method. The influences of the factors including the nonlocal parameter, the surface effect, the power-law index, and the vibration modes on the TED are analyzed. These results provide a reasonable theoretical analysis to predicate TED in the design of FG sandwich microresonators/nanoresonators with high performance.

Vibration analysis of higher-order nonlocal strain gradient plate via meshfree moving Kriging interpolation method

In this paper, a higher-order nonlocal strain gradient plate model combining nonlocal elasticity theory with strain gradient theory in the form of adding signs is constructed. The governing equation for the higher-order nonlocal strain gradient plate model is derived by Hamilton’s principle. Two higher-order parameters with nonlocal stress and strain gradient are introduced here to explain the size effect and dispersive behaviour. The meshfree moving Kriging interpolation method is utilized to address the frequency of higher-order nonlocal strain gradient plate under complicated boundary conditions. Nonlocal elasticity theory, strain gradient theory in the form of add signs and minus signs on the effect of frequency are compared by degenerating the higher-order nonlocal strain gradient plate model. The other goal of this paper is to compare the present model with higher-order nonlocal strain gradient theory in the form of minus signs on the effect of frequency. It is shown that these two items can describe the size effect well. The present method can also produce lower natural frequencies corresponding to ultrahigh-order mode shapes and applied in a discrete medium.

Refined quasi-3D shear deformation theory for buckling analysis of functionally graded curved nanobeam rested on Winkler/Pasternak/Kerr foundation

In the present work, a novel refined three-variable quasi-3D shear deformation theory incorporates a correction factor is developed to analyze the buckling behavior of multi-directional functionally graded (FG) curved beams. The proposed displacement field is formulated in accordance with the Euler-Bernoulli beam theory. The research investigated two types of coated Functionally Graded (FG) nanobeams: Hardcore (HC) FG curved nanobeams and Softcore (SC) FG curved nanobeams. Three different material distributions are taken into consideration: a bidirectional material distribution referred to as “2D-FG,” a unidirectional transverse material distribution known as “T-FG,” and a unidirectional axial material distribution called “A-FG.” Eringen’s nonlocal elasticity theory is employed to capture small-scale effects. The total potential energy principle is utilized to derive the equilibrium equations of curved nanobeams. A novel solution, utilizing Galerkin’s method, has been developed to effectively address a range of boundary conditions. The curved FG beam is supported by an elastic foundation following the Winkler/Pasternak/Kerr model. A comprehensive analysis has been conducted to examine the impacts of various FG schemes, curved beam geometry, nonlocal parameter, elastic foundations, and various boundary conditions on the dimensionless critical buckling loads. This analysis aims to provide a comprehensive understanding of how each of these factors influences the critical buckling loads of the nanobeams.

Analysis of nonlocal effects on plane waves in a transversely isotropic visco-thermoelastic medium with variable thermal conductivity

PurposeThis study aims to analyse the behaviour of plane waves within a nonlocal transversely isotropic visco-thermoelastic medium having variable thermal conductivity.Design/methodology/approachThe concept of enunciation is used in the generalized theory of thermoelasticity in accordance with the Green–Lindsay and Eringen’s nonlocal elasticity models. The linear viscoelasticity model developed by Kelvin–Voigt is used to characterize the viscoelastic properties of transversely isotropic materials.FindingsIt has been noticed that three plane waves, which are coupled together, travel through the medium at three different speeds. The derivation of reflection coefficients and energy ratios for reflected waves is carried out by incorporating suitable boundary conditions. Numerical computations are performed for the amplitude ratios, phase speeds and energy partition and displayed in graphical form.Originality/valueThe outcomes of the numerical simulation demonstrate that the amplitude ratios are significantly influenced by variable thermal conductivity, nonlocal parameters and viscosity. It is further observed from the plots that the phase speeds in a transversely isotropic medium depend on the angle of incidence. In addition, it has been established that the energy is preserved during the reflection phenomenon.