Nonlocal Elasticity Theory Research Articles

Investigation of fluid viscosity and density on vibration of nano beam submerged in fluid considering nonlocal elasticity theory

Predicting the dynamical behavior of nanostructures in contact with fluid is of paramount importance owing to their vast range of applications in civil engineering, aerospace, etc. Therefore, the free vibration of a nonlocal Euler-Bernoulli beam in contact with fluid is studied in this research. Using the Navier-Stokes relation, the interaction forces between the fluid and nanobeam are obtained. The equation of motion is solved using the Galerkin method of weighted residuals, and the natural frequency values in different fluids are presented, considering the viscosity effect. Further, the fluid-solid interaction model on a macroscopic scale is compared with the finite element analysis results. By considering the fluid parameters to be zero, the results of the nonlocal beam have been validated with past research. According to the obtained results, when studying the vibration of nanobeams, while density has a remarkable impact on the reduction of nanobeam natural frequencies, the fluid viscosity can be neglected. The fluid has the greatest impact on lower modes, whereas the nonlocal parameter exerts the biggest influence on higher modes. Moreover, in all vibration modes, with the increase of the nonlocal parameter, the effect of the fluid on the reduction of the natural frequencies increases, and in the nanoscale, it has a more significant role in changing the vibrational behavior of the system compared to the classical condition.

Free vibration analysis of graded porous circular micro/nanoplates with various boundary conditions based on the nonlocal elasticity theory

AbstractIn this article, the free vibrations of graded porous circular nanoplates subjected to various boundary conditions have been investigated. It is assumed that the performance of graded porous materials varies continuously in the whole thickness, and two cosine forms of non‐uniform porosity distribution along its thickness are considered. In order to capture the size effect, the Eringen's nonlocal elastic theory is applied. Then, Mindlin plate theory, combined with Hamilton's principle, is utilized to derive the governing equations of motion. Various types of boundary conditions are assumed for the plate with a combination of clamped, simply supported and free edges. Finally, the governing equations of motion are solved numerically using the shooting technique. The effects of various factors such as porosity coefficient, porosity distribution pattern, thickness‐diameter ratio, and the nonlocal scale effect and boundary conditions on the natural frequencies of grade porous circular nanoplates are discussed in detail.

The Sub and Super-Tangential Nonconservative Load in Stability Problem of Nanobeams with Sprung Masses

In this study, the critical load and natural vibration frequency of Euler–Bernoulli single nanobeams based on Eringen’s nonlocal elasticity theory are investigated. Cantilever nanobeams with attached sprung masses were subjected to compressed concentrated and distributed follower forces. The parameter that determines the direction of nonconservative follower forces was given the positive and negative values, therefore, sub-tangential and super-tangential load were analyzed. The stability analysis is based on dynamical stability criterion and was carried out using a numerical algorithm for solving segmental nanobeams with many boundary conditions. The presented algorithm is based on the exact solutions of motion equations which are derived from equilibrium conditions for each separated segment of the nanobeam. Two comparison studies are conducted to ensure the validity and accuracy of the presented algorithm. The excellent agreement of critical load for Beck’s nano-column on Winkler foundation observed was confirmed as reported by other researchers. The effect of different values of the nonlocality parameter, tangency coefficient, spring stiffness coefficient, location of sprung mass and the greater number of attached sprung masses on a critical load of nanobeams compressed by nonconservative load are discussed. One of the presented results shows that significant differences between local and nonlocal theory appear when the beam subjected to follower forces loses its stability by flutter.

The nonlocal elasticity theory for geometrically nonlinear vibrations of double-layer nanoplate systems in magnetic field

The geometrically nonlinear vibrations of simply supported double-layer graphene sheet systems under in-plane magnetic field are considered in the presented manuscript. The interaction between layers is taken into account due to van der Waals forces. The investigation is based on the nonlocal elasticity theory, Kirchhoff plate theory and von Kármán theory. The effect of the magnetic field is due to the Lorentz force based on Maxwell’s equations. The governing equations are used in mixed form by introducing the stress Airy function. The analytical presentation of the nonlinear frequency ratio for in-phase vibration and anti-phase vibration modes is presented. It is shown that the nonlocal parameter in the compatibility equation can significantly change the vibration characteristics.

Analytical determination of non-local parameter value to investigate the axial buckling of nanoshells affected by the passing nanofluids and their velocities considering various modified cylindrical shell theories

We analytically determine the nonlocal parameter value to achieve a more accurate axial-buckling response of carbon nanoshells conveying nanofluids. To this end, the four plates/shells’ classical theories of Love, Flügge, Donnell, and Sanders are generalized using Eringen’s nonlocal elasticity theory. By combining these theories in cylindrical coordinates, a modified motion equation is presented to investigate the buckling behavior of the nanofluid-nanostructure-interaction problem. Herein, in addition to the small-scale effect of the structure and the passing fluid on the critical buckling strain, we discuss the effects of nanoflow velocity, fluid density (nano-liquid/nano-gas), half-wave numbers, aspect ratio, and nanoshell flexural rigidity. The analytical approach is used to discretize and solve the obtained relations to study the mentioned cases.

On uncertainty modeling of thermoelastic vibration for porous nanosandwich beams with gradient core based on nonlocal higher order beam model

In the present paper, the size-dependent random thermoelastic vibration analysis of porous nanosandwich beams with gradient core are demonstrated based on the nonlocal higher order beam model. The nonlocal elasticity theory incorporation with the higher order shear deformation theory is used to describe the kinematic and constitutive relations. The governing equations are obtained from minimum potential energy principle. The finite element method is used to solve the governing equations. Three types of the FG sandwich nanobeams with three types of porous cores are taken into account, and the influences of these structures and porosity types on the deterministic and stochastic behavior of sandwich beams are thoroughly discussed. Initially convergence and validation study is preformed to validate the present theoretical model by comparing the present results with the several previous works and Monte Carlo Simulation (MCS). Further, a comprehensive parametric study is performed to demonstrate the influence of the length to thickness ratio, degree of uncertainty in the material properties, nonlocal parameter, porosity distribution type, porosity volume fraction, temperature change, and gradation of material on the deterministic and stochastic behaviors of the FG nanosandwich beams. The developed model and computed results may be useful in the precise and reliable design of the components of nanodevices and systems.

Thermoelastic characteristics of moving viscoelastic nanobeams based on the nonlocal couple stress theory and dual-phase lag model

Thermal behavior of a moving viscoelastic nanobeam under the influence of periodic thermal load is considered in the framework of Kelvin-Voigt viscoelastic model with fractional operators. The equation of motion for axially moving nanobeam is modeled by employing the Eringen’s nonlocal elastic theory in conjunction with the couple stress hypothesis and the conventional Euler–Bernoulli beam model. The thermoelastic features is then established by employing the generalized dual phase-lag heat conduction model. After utilizing the Laplace transform, the thermomechanical equations are coupled and solved. The current results are validated by presenting numerical examples and comparing with previous solutions obtained by traditional theories in the literature. According to the provided numerical simulations, the deflection of the axially moving nanobeam as well as its temperature change reduce with the axial velocity and the influences of small scale and nonlocal parameters are also revealed and discussed.

Effect of nonlocal parameters and Kerr foundation on nonlinear static and dynamic stability of micro/nano plate with graphene platelet reinforcement

It is a fact that size effects appear when studying the mechanical behavior of nano/microscopic materials. This paper, for the first time, applies an analytical method to determine the size effect dependence and Kerr foundation on nonlinear static and dynamic stability of micro/nano plate with graphene platelet reinforcement. For this purpose, the nonlocal elasticity theory along with classical plate theory is used to derive the basic equations. Both motion and geometric compatibility equations of the plate are deduced by introducing the Von Kármán strain–displacement relationship. Applying the analytical method (Bubnov–Galerkin and the stress function methods), the nonlinearity of buckling and postbuckling, the vibration and the dynamic response of micro/nano plate with graphene platelet reinforcement are analyzed. The fourth-order Runge–Kutta methods are implemented to solve the governing equation of the dynamic system. To prove the reliability of the formulas, the results of the present study in specific cases are compared with those of relevant published papers in the reference and good agreements are observed. In the final part, the effects of nonlocal parameters, Kerr foundation, initial imperfection, GPL weight fraction and geometrical parameters on the overall nonlinear static and dynamic stability of the micro/nano plate with graphene platelet reinforcement are evidently demonstrated.

Size-dependent nonlinear bending analysis of nonlocal magneto-electro-elastic laminated nanobeams resting on elastic foundation

Employing Reddy’s third-order shear deformation theory (RTSDT) and nonlocal elasticity theory, a nonlinear bending model of the nonlocal three-layer magneto-electro-elastic (MEE) laminated nanobeam resting on elastic foundation is established by considering von Karman’s geometrically nonlinear equations. Nonlinear higher order partial differential governing equations of MEE laminated nanobeams can be obtained employing Hamilton variational principle. The three-layered MEE laminated nanobeam is considered to have simply supported boundary condition in the present paper. Meanwhile, the electric and magnetic potential distributions in the laminated nanobeam are determined through Maxwell’s magnetic-electro equations and boundary conditions. The governing equations of laminated nanobeams are re-expressed in the dimensionless form by introducing the non-dimensional terms. Employing Galerkin method, the nonlinear higher order partial differential governing equations are simplified into lower order equations. Several cases are explored to indicate the effects of foundation parameters, nonlocal parameter, stacking sequence, external electric voltage and external magnetic potential on bending behaviors of MEE laminated nanobeams.

Free vibrations of small-scale plates with complex shape based on the nonlocal elasticity theory

Free vibrations of the orthotropic micro/nanoplate with nonclassical shape are investigated. The considered model is based on the nonlocal elasticity theory. The developed method uses the Ritz method as well as R-function theory for the construction of the system of coordinate functions. The linear frequencies are obtained for a rectangular plate with two cutouts on opposite sides, while the boundary conditions are considered of several types, including simply supported and clamped edges. The small-scale effects for various sizes of cutouts are discussed.

Influence of third-order elastic modulus as material nonlinearity on free and forced vibration of a nonlocal nanobeam rested on viscoelastic medium

In this study, the nonlinear free and forced vibrations of a nanobeam resting on a viscoelastic medium have been investigated. The quadratic material nonlinearity, which is neglected in the literature, is included in the stress–strain relation, and then using nonlocal elasticity theory, the nonlinear vibration of the nanobeam considering both material and geometrical nonlinearities is investigated. To this end, the governing equation of motion is extracted using the Euler–Bernoulli beam theory and utilizing Hamilton’s principle. Applying Galerkin’s method, the nonlinear differential equation of the nanobeam is obtained. The cubic and quantic nonlinearities in the governing differential equation are due to geometrical and material nonlinearities, respectively. Using Modified Homotopy Perturbation Method, the nonlinear differential equation is solved, and the nanobeam time response and frequency are obtained. Also, the frequency response of nanobeam in the presence of a viscoelastic medium and harmonic excitation is obtained. The results illustrate that material nonlinearity has an important effect on the free and forced vibration responses of the system. For validation, the obtained results of nanobeam are compared with the results obtained from the fourth-order of Runge Kutta numerical method and previous research.

Size-Dependent Mechanical Behaviors of Defective FGM Nanobeam Subjected to Random Loading

Considering the uncertainties of the materials and loads, the nanobeam made of functionally graded materials were investigated based on the non-local elastic theory. The spline finite point method (SFPM) was established to analyze the bending behavior of the nanobeam-based Timoshenko theory. In comparison with finite element method (FEM), SFPM has higher accuracy. Further, the multi-source uncertainties are considered, material properties are quantified as interval parameters, and loads are taken as random parameters. To deal with the problems with two types of uncertainties coexisting, a hybrid uncertain analysis model was established, and the method of polynomial chaos expansion and dimensional wise (PCE–DW) analysis was proposed to predict the response of nanobeam in the hybrid uncertain system. Numerical examples ultimately illustrate the effectiveness of the model and solution techniques, compared with MCS. The results furtherly verify the efficiency and accuracy.

Nonlocal Free Vibration of Embedded Short-Fiber-Reinforced Nano-/Micro-Rods with Deformable Boundary Conditions.

An efficient eigenvalue algorithm is developed for the axial vibration analysis of embedded short-fiber-reinforced micro-/nano-composite rods under arbitrary boundary conditions. In the formulation, nonlocal elasticity theory is used to capture the size effect, and the deformable boundary conditions at the ends are simulated using two elastic springs in the axial direction. In addition, to determine the reinforcing effect of restrained nano-/micro-rods, a new system of linear equations with the concept of the infinite power series is presented. After performing the mathematical processes known as Fourier sine series, Stokes’ transformation and successive integration, we finally obtain a coefficient matrix in terms of infinite series for various rigid or deformable boundary conditions. Some accurate eigenvalue solutions of the free axial vibration frequencies of the short-fiber-reinforced micro-/nano-composite rods with and without being restrained by the means of elastic springs are given to show the performance of the present method. The presence of the elastic spring boundary conditions changes the axial vibration frequencies and corresponding mode shapes.

A homotopy analysis solution to large deformation of a nanowire based on nonlocal elasticity theory

A homotopy analysis solution to large deformation of a nanowire based on nonlocal elasticity theory

The role of defected/wavy/aggregated CNTs on the wave propagation in size-dependent nanocomposite beams

Lack of attention to the real structure of nanocomposites such as agglomeration of nanotubes, defects in nanotubes, and curvatures of their structures can be the reason for inaccuracies in the modeling of nanocomposites. Here, a few steps have been taken to bring these models closer to reality. Three modifications will be studied in this article. According to this fact that nanotubes, as fibers in nanocomposites, do not necessarily have straight structure and they have curvatures. Their curliness was included in the present model. Obviously in reality, when nanotubes are produced, they have defective structures and this can be a reason for including the defective nanotubes in the present model for nanocomposites. Finally, it is clear that the production of nanocomposites is not perfect and contrary to expectations, there may be problems when manufacturing nanocomposites. One of these problems is the agglomeration of nanotubes, which will be examined in this article. Also, strain gradient theory and nonlocal elasticity theory are used to capture the size effects on these types of nanocomposites which have not been considered yet for nanocomposite beams. From present results, it is concluded that increasing the agglomeration volume of CNTs will increase the circular frequencies.

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.

STATIC BENDING ANALYSIS OF ANNULAR NANOPLATES RESTING ON ELASTIC FOUNDATION USING NONLOCAL ELASTICITY THEORY

This article presents an application of finite element algorithm for static bending analysis of the functionally graded porous (FGP) annular nanoplate resting on the elastic foundation (EF) using nonlocal elasticity theory. The FGP materials with two parameters are the volume fraction index (k) and the porosity volume fraction (ξ) was used in two cases of even and uneven porosity. The EF includes Winkler-stiffness (k1) and Pasternak-stiffness (k2). For the first time, the stress and displacement of the FGP annular nanoplates are established using an eight-node plate element (Q8). Numerical results of the proposed method are compared with those of published works to verify the accuracy and reliability. Furthermore, the impacts of some factors such as the elastic foundation and material on the static bending of FGP nanoplates resting on the EF are studied in detail.

Nonlocal chaotic analysis of double layered viscoelastic nanoplates embedded in Winkler–Pasternak elastic foundation and under in‐plane biaxial loads

AbstractIn this paper, based on the extended Melnikov method the homoclinic analysis for chaotic vibration of a double layered viscoelastic nanoplates (DLNPs) embedded in Winkler–Pasternak elastic foundation and under in‐plane excited biaxial loads using the Eringen's nonlocal elasticity theory has been investigated. The governing nonlinear coupled partial differential equations (PDEs) of motion for DLNP considering viscoelastic Kelvin–Voigt model in the Kirchhoff's thin plate theory in conjunction with the von‐Karman geometrical nonlinear strain–displacement relations have been derived. Then, applying the Galerkin method the governing nonlocal coupled PDEs of motion of DLNP have been converted into the nonlinear coupled ordinary differential equations (ODEs) using mode summation technique and discretized in terms of time. The criteria for the homoclinic transverse intersection for synchronous and asynchronous types of buckling are studied to predict the homoclinic orbits and chaotic motions applying the generalized Melnikov method. Besides, the critical buckling load has been determined for asynchronous and synchronous types of buckling. The obtained results are validated with comparing them with the results presented in the literature for a special case. Then, the effect of changes of different parameters including nonlocal small‐scale parameter, foundation coefficients, biaxial load ratio, geometric aspect ratio on the critical buckling load and ratio of nonlocal to local natural frequency have been studied. The critical threshold surface for the synchronous and asynchronous manifolds to predict transversely condition for the chaotic motion are discussed on the parametric space.

Effects of thermal environment and external mean flow on sound transmission loss of sandwich functionally graded magneto-electro-elastic cylindrical nanoshell

Similar to many new engineering ideas that are actually interdisciplinary subjects, magneto-electro-elastic (MEE) structures also need accurate ways of describing their structural behaviors. Based on nonlocal elasticity theory, the present article investigates the vibroacoustic behavior of a hollow multilayer cylindrical nanoshell where the core layer is made of isotropic functionally graded material (FGM) and other layers are made of MEE materials. The proposed system also is subjected to combined loads which contain a plane sound wave, the initial external electric and magnetic loads, and external mean airflow. The displacement field of structure is described using the third-order shear deformation assumption (TSDA). The derivation of vibroacoustic equations in the form of coupled relations is realized by implementing Hamilton's principle. The material properties of FGM core layer are supposed to vary along the in-plane and thickness directions based on the power-law model. The final objective is to analyze the sound transmission loss (STL) characteristics of the structure and inspect the accuracy of the developed method against existing data followed by comparing the results in terms of geometric and acoustic parameters. The obtained results and the described method can be used advantageously to better optimize such structures by choosing appropriate initial electric and magnetic potentials, flow Mach number, nonlocal parameter, material gradient index, incident angles.

Analytical solutions for bending, buckling and vibration of nanobeams on Winkler–Pasternak foundation

The nonlocal elasticity theory and the Euler–Bernoulli (EB) beam theory are used to present closed-form analytical expressions for static bending, axial buckling, and free vibration of nanosized beams resting on an elastic foundation. The differential constitutive relations of Eringen are utilized to represent the small-scale effects of the nanobeam’s mechanical response. The governing equation of motion is derived by employing Hamilton’s principle. Utilizing the Laplace transform approach, analytical expressions of the bending displacements, the critical buckling force, and the vibration frequency of nanobeams with simply supported (S-S), clamped, cantilevered, and propped cantilevered boundary conditions are produced. In order to confirm the correctness of the offered closed-form equations, their outputs are compared to those of the available numerical method solutions. The effects of the Winkler parameter, the Pasternak parameter and the nonlocal parameter on bending, buckling, and vibration characteristics of nanobeams have been explained. Presented analytical expressions and graphical representations demonstrate how increasing Winkler and Pasternak parameters reduce bending displacements while raising the critical buckling load and the natural frequency of nonlocal nanobeams. Benchmark numerical results are also presented to investigate and discuss the effects of all parameters on bending deflections, buckling loads, and natural frequencies of nanobeams.