Nonlocal Scale Parameter Research Articles

Nonlocal wave propagation analysis of a rotating nanobeam on a Pasternak foundation

In this paper, the wave propagation analysis of a cantilever rotating nanobeam modeled as a thin beam based on the Euler–Bernoulli theory on a Pasternak foundation has been investigated using the nonlocal theory of elasticity. The governing partial differential equation of motion for a uniform rotating nanobeam is derived using the Hamilton principle considering the nonlocal parameter of small scale effect. The Spectrum and dispersion relations of non-dimensional wave number are obtained analytically. The effect of changes of different parameters including nonlocal scale parameter, non-dimensional rotational speed, non-dimensional rotational wave frequency ratio, and shear stiffness of the Pasternak foundation on the wave dispersion behavior of the non-dimensional wave number and phase and group speeds dispersions for the rotating nanotube have been studied. It is observed that the wave dispersion characteristics of the rotating nanobeam are extremely influenced by rotational speed, wave number, nonlocal length scale parameter, and shear stiffness of the Pasternak foundation. Moreover, the propagated flexural wave has been shown to exhibit non-dispersive behavior at very high rotational speeds.

Dynamic analysis of double-walled carbon nanotubes embedded in elastic medium under a nanoparticle delivery

Dynamic behaviors of double-walled carbon nanotubes (DWCNTs) embedded to elastic foundation under a moving nanoparticle are studied based on the nonlocal elasticity theory. The DWCNTs embedded an elastic medium are considered as two-scale shell model coupled with the van der Waals (vdW) interaction between two adjacent nanotubes. The elastic medium surrounding DWCNTs is formulated as Winkler spring model. Using the proposed models and solutions, we investigate the influences of various parameters on the mechanical properties, such as the nanoparticle delivery, the elastic foundation, and nonlocal coefficient. The dynamic response of DWCNTs depends largely by the moving velocity of nanoparticle, the stiffness of the elastic medium, and the nonlocal coefficient. The results show that increasing constant of elastic medium causes deceases in the dynamic deflection of DWCNTs due to the improved stiffness property. The influence of the nonlocal scaling parameter on the dynamic response of the flexural displacement is enhanced with increasing nanoparticle velocity.

Transverse free vibration of nanobeams with intermediate support using nonlocal strain gradient theory

In the present study, the transverse free vibration of intermediately supported nanobeams is investigated in the framework of size-dependent nonlocal strain gradient theory. Unlike the classical elasticity theory, nonlocal strain gradient theory considers the micro/nano scale effects in mechanical analysis of size-dependent structures. The potential and kinetic energies have been derived for the nanobeam model and the Ritz method has been used to determine the natural frequencies of intermediately supported nanobeams. A cantilever nanobeam model with intermediate support is considered in the analysis. By changing the position of the intermediate support, dimensionless vibration frequencies of the nanobeam model have been obtained. Obtained results from the present nonlocal strain gradient theory showed that hardening or softening material responses have been observed according to the classical elasticity theory depending on the magnitude of the material length scale parameter and nonlocal scale parameter. This mechanical behavior can provide an advantage to designers while modeling the micro/nanostructures. Present results can be used for the design of nano-sensors, nanotube-based resonators, and oscillators.

A gradient-enhanced formulation for thermoviscoplastic metals accounting for ductile damage

In this work, we extend a gradient-enhancement formulation for ductile damage analysis to thermoviscoplastic metals. The original contribution of the study is the nonlocal extension of our previously developed thermoviscoplastic Gurson's model accounting for plastic work heat generation, thermal diffusion and void shearing mechanism. The gradient enrichment is applied to the porosity evolution equation by adding a nonlocal length scale parameter together with the Laplacian of the void volume fraction, following a void diffusion law. A mixed finite element formulation is used to discretize the system with an implicit time integration and a monolithic solution of the linearized system at every time step. Three ductile fracture problems involving plates under plane-strain conditions and tensile loading are simulated, in order to compare the local and nonlocal formulations. Results show that the nonlocal term delays the damage evolution and somewhat alleviates mesh sensitivity in terms of the stress-strain response and the porosity distribution along specific lines. Nevertheless, the most notable effect of the nonlocal formulation is the regularization (or smoothing) of the porosity field, reducing its gradients and mesh dependence, and leading to more consistent porosity contours. The choice of the nonlocal parameter is found to be dependent of the strain rate employed. • A gradient-enhanced thermoviscoplastic ductile damage formulation is proposed. • Coupling of Gurson's porous plasticity, void shearing mechanism, thermoviscoplasticity and thermal diffusion. • Three structural problems involving plates under plain strain assumption are numerically analyzed in detail. • The model is suitable to alleviate mesh dependence and to smooth the porosity field.

The propagation of plane waves in nonlocal visco-thermoelastic porous medium based on nonlocal strain gradient theory

The present paper deals with propagation of plane harmonic waves in nonlocal visco-thermoelastic porous medium based on the nonlocal strain gradient theory. For the first time, the assumptions of Green-Naghdi model type III of hyperbolic thermoelasticity with voids in conjunction with nonlocal strain gradient theory are taken into account in formulation of the problem. The constitutive relations and field equations for nonlocal thermoelastic materials with voids are presented. In addition to transverse waves, three longitudinal waves (elastic, thermal, volume fraction) can be observed. The fundamental solution to equations for steady oscillations is derived. The effects of nonlocal length scale parameter, voids, viscosity on phase velocities, attenuation coefficients and penetration depths of waves with diverse frequencies are comprehensively studied and graphically presented.

Electromechanical coupling characteristics of double-layer piezoelectric quasicrystal actuators

Electromechanical coupling characteristics of double-layer piezoelectric quasicrystal micro/nano-actuator are investigated by considering the effects of nanoscale interactions and nonlocal scale. The analytical models in the phonon, phason, and electric fields are derived via state vector formulation and propagator matrix method based on quasicrystal nonlocal elasticity theory. Governing equations of the models with geometrical nonlinearity are solved using the secant method. All nonlinear effects of the electrostatic force, van der Waals force, Casimir force, and nonlocal parameters on the mechanical behaviors of quasicrystal actuators are studied. Our results show that the increment of van der Waals force and Casimir force enlarges phonon stress and displacement. When the initial gap is larger than 3.72 nm, the effect of van der Waals force is more significant than that of Casimir force. The phonon-phason coupling elastic coefficient and the nonlocal scale parameter respectively have positive and negative effects on the static responses. Besides, the influence of the Casimir force on mechanical behavior depends on the volume ratio and scale of the quasicrystal nano-actuator.

Hygro-thermal buckling of porous FG nanobeams considering surface effects

Hygro-thermal buckling of the porous FG nanobeam incorporating the surface effect is investigated. The even distribution of porosities is assumed in this paper. Various porous FG nanobeam models including classical beam theory (CBT), Timoshenko beam theory (TBT), Reddy beam theory (RBT), sinusoidal beam theory (SBT), hyperbolic beam theory (HBT) and exponential beam theory (EBT) are developed in this paper. The nonlocal strain gradient theory with material length scale and nonlocal parameters is adopted to examine the buckling behavior. The governing equations of the porous FG nanobeam are derived from principle of minimum potential energy. In the numerical examples, the effect of the nonlocal parameter, material length scale parameter, the temperature rise, the moisture concentration, surface effect, material gradient index, and porosity volume fraction on the buckling temperature and moisture are analyzed and discussed in detail. The results show that the stiffness of the beam depends on the relation of size between nonlocal parameter and length scale parameter. The paper will be helpful for the design and manufacture of the FG nanobeam under complex environments.

A Nonlocal Strain Gradient Approach for Out‐of‐Plane Vibration of Axially Moving Functionally Graded Nanoplates in a Hygrothermal Environment

Vibration analyses on axially moving functionally graded nanoplates exposed to hygrothermal environments are presented. The theoretical model of the nanoplate is described via the Kirchhoff plate theory in conjunction with the concept of the physical neutral layer. By employing the nonlocal strain gradient theory, the governing equation of motion is derived based on Hamilton’s principle. The composite beam function method, as well as the complex modal approach, is utilized to obtain the vibration frequencies of axially moving functionally graded nanoplates. Some benchmark results related to the effects of temperature changing, moisture concentration, axial speed, aspect ratio, nonlocal parameter, and the material characteristic scale parameter on the stiffness of axially moving functionally graded nanoplates are obtained. The results reveal that with increasing the nonlocal parameter, gradient index, temperature changing, moisture concentration, and axial speed, the vibration frequencies decrease. The frequencies increase while increasing the material characteristic scale parameter and aspect ratio. Moreover, there is an interaction between the nonlocal parameter and material characteristic scale parameter, influencing and restricting each other.

Closed form solution for dynamic analysis of rectangular nanorod based on nonlocal strain gradient

ABSTRACT In this study, the free torsional vibration of the nanorod with noncircular cross-section based on the nonlocal strain gradient theory is investigated. The rectangular cross-section has been selected. The governing partial equation of the torsional vibration of the noncircular nanorod is derived using Hamilton’s principle. Using a Galerkin method, and by consideration of the rectangular shape function in series, the natural frequency of noncircular nanorod is obtained. The obtained results indicate the convergence of the nondimensional natural frequency of the nanorod by enhancement of the number of the rectangular shape function. Furthermore, the nondimensional natural frequency of the noncircular nanorod for different nondimensional nonlocal parameter and nondimensional material length scale parameter is studied. Eventually, the effects of both nonlocal parameter and material length scale parameter on the nondimensional natural frequency, which causes torsional stiffness softening and torsional stiffness hardening of the nanorod, is also illustrated.

Wave dispersion characteristics of high-speed-rotating laminated nanocomposite cylindrical shells based on four continuum mechanics theories

This paper investigates the wave propagation behavior of a high-speed rotating laminated nanocomposite cylindrical shell. The small-scale effects are analyzed based on nonlocal strain gradient theory (NSGT). The governing equations of the cylindrical laminated composite nanoshell in a thermal environment were obtained using Hamilton’s principle and solved by the analytical method. For the first time in this study, the wave propagation behavior of a high-speed rotating nanocomposite cylindrical shell is studied based on classic, strain gradient, nonlocal and nonlocal strain gradient theories (4 continuum theories) with considering the calibrated values of the nonlocal constant and material length scale parameter. The results show that wave number, angular velocity, and different types of laminated composites have an important role in the phase velocity of the nanocomposite structure using mentioned continuum mechanics theories. Another significant result is that in the higher values of angular velocity, three layers of laminated composite has the highest phase velocity in comparison with the other layers. The outputs of the present work can be used in structural health monitoring and ultrasonic inspection techniques.

Vibration Analysis of Size-Dependent Piezoelectric Nanobeam Under Magneto-Electrical Field

The damping vibration characteristics of magneto-electro-viscoelastic (MEV) nanobeam resting on viscoelastic foundation based on nonlocal strain gradient elasticity theory (NSGT) is studied in this article. For this purpose, by considering the effects of Winkler-Pasternak, the viscoelastic medium consists of linear and viscous layers. with respect to the displacement field in accordance with the refined shear deformable beam theory (RSDT) and the Kelvin-Voigt viscoelastic damping model, the governing equations of motion are obtained using Hamilton’s principle based on nonlocal strain gradient theory (NSGT). Using Fourier Series Expansion, The Galerkin’s method adopted to solving differential equations of nanobeam with both of simply supported and clamped boundary conditions. Numerical results are obtained to show the influences of nonlocal parameter, the length scale parameter, slenderness ratio and magneto-electro-mechanical loadings on the vibration behavior of nanobeam for both types of boundary conditions. It is found that by increasing the magnetic potential, the dimensionless frequency can be increased for any value of the damping coefficient and vice versa. Moreover, negative/positive magnetic potential decreases/increases the vibration frequencies of thinner nanobeam. Also, the vibrating frequency decreases and increases with increasing nonlocal parameter and length scale parameter respectively.

Nonlinear free vibrations analysis of geometrically imperfect FG nano-beams based on stress-driven nonlocal elasticity with initial pretension force

Size-dependent flexural nonlinear free vibrations of geometrically imperfect straight Bernoulli-Euler functionally graded nano-beams are investigated by the stress-driven nonlocal integral model (SDM). By using the Galerkin method, the governing equations is reduced to a nonlinear ordinary differential equation. The closed form analytical solution of the nonlinear natural flexural frequency for Simply-Supported, Clamped-Simply Supported and Clamped-Clamped nano-beams is then established using the Hamiltonian approach to nonlinear oscillators. Effects of nonlocal scale parameter and an initial axial tension force on fundamental frequencies are examined and compared with those obtained by Eringen’s nonlocal model. It is shown that the nonlinear approach based on nonlocal stress model, with the appropriate constitutive boundary conditions, is capable of capturing the dynamical responses of the nano-beams and provides an advantageous method for the design and the optimization of a wide range of nano-scaled beam-like components of Nano-Electro-Mechanical-Systems (NEMS).

Nonlinear vibration of piezoelectric laminated nanobeams at higher modes based on nonlocal piezoelectric theory

The presented paper investigates the nonlinear vibration of a nanobeam with a piezoelectric layer bounded to its top surface considering the nonlocal piezoelectricity theory. To do this, Hamilton’s principle is implemented to derive the governing nonlinear vibration equations of the nanobeam by assumption of nonlocal piezoelectricity and a nonlinear strain–displacement relation. Then, the Galerkin separation method is applied to transform and simplify the partial differential equation of the nonlinear oscillation to an ordinary one with quadratic and cubic nonlinearities in the time domain. By implementing the multiple-scale perturbation method, an analytical relation for the nonlinear natural frequencies is obtained as a function of the oscillation amplitude and the nonlocal size scale parameter. Then, the nonlinear vibration characteristics of the nanobeam are investigated at higher modes of vibration and the size scale effects are reviewed comprehensively. It is observed that the nonlocal parameter decreases the nonlinear natural frequencies and becomes noticeable at higher modes of vibration. Moreover, by increasing the amplitude ratio, the nonlocal effects are decreased and the nonlocal nonlinear frequency approaches the local one. Also, the amplitude ratio has increasing effects on the nonlinear frequencies.

Stability of Vibrating Functionally Graded Nanoplates with Axial Motion Based on the Nonlocal Strain Gradient Theory

Considering both the nonlocal scale and material length scale effects, we investigate vibration and stability behaviors of functionally graded nanoplates with axial motion in order to model the two-dimensional nanobelt in nanoengineering. The nonlocal strain gradient theory is applied and the differential nonlocal strain gradient constitutive relation is adopted. Using the physical neutral plane of a functionally graded thin plate, we derive the governing equation of motion for the functionally graded nanoplate with axial motion via Hamilton’s principle, where the kinematic characteristics are introduced into the dynamic behaviors. The governing equation is numerically solved using the Galerkin method. Effects of the nonlocal scale and material length scale parameters, axial velocity, gradient index, biaxial pre-tensions and aspect ratio are discussed. The results demonstrate that complex frequencies of the functionally graded nanoplate with axial motion decrease with an increase of axial velocity in the subcritical region, while the moving nanoplate experiences a divergent instability or flutter instability in the supercritical region. Natural frequency and critical speed decrease with the increase of the nonlocal scale parameter while increase with the increase of the material length scale parameter, reflecting the nonlocal softening and strain gradient hardening mechanisms, respectively. Besides, natural frequency and critical speed increase with the increase of the biaxial pre-tensions and aspect ratio, but decrease with the increase of the gradient index. In particular, the influences of the gradient index and size or weight of functionally graded nanoplates on the critical speed are explored.

Dynamic and pull-in instability analyses of functionally graded nanoplates via nonlocal strain gradient theory

The main subject of this study is the dynamic and static pull-in analyses of rectangular nanoplates made of functionally graded materials based on the nonlocal strain gradient theory compared with the strain gradient theory, Eringen’s differential method, and classical theory considering disparate boundary conditions (BCs). To this end, the governing equation is derived based on Kirchhoff’s plate theory and then, the Galerkin method is implemented to arrive at an ordinary differential equation of second-order in the time domain. The homotopy analysis method is used as an analytical solution methodology to solve the nonlinear ordinary differential equation which is under different nonlinear forces such as intermolecular, electrostatic and hydrostatic one. Likewise, an efficient technique is exerted for omitting secular terms. Accordingly, the dynamic pull-in voltage is obtained for different non-classical continuum theories, material gradient indices, nonlocal parameters and internal length scale parameters. The dynamic and static pull-in instabilities are also compared in the same situation. Additionally, the effects of the nonlocal parameter and internal length scale parameter on the non-dimensional frequency against the initial amplitude, electrostatic force and hydrostatic actuation are explored in disparate boundary conditions. Communicated by Eleonora Tubaldi.

Nonlinear vibration and stability analysis of piezo-harmo-electrostatic nanoresonator based on surface/interface and nonlocal strain gradient effects

In the current study, nonlinear vibration and stability analysis of piezo-harmo-electrostatic nanoresonator (PHENR) based on cylindrical nanoshell is investigated to compare nonlocal (NLT), nonlocal strain gradient (NSGT) and Gurtin–Murdoch surface/interface (GMSIT) theories with classical theory (CT). PHENR is simultaneously subjected to direct electrostatic voltage DC and harmonic excitations, structural damping and two piezoelectric layers. For this analysis, Hamilton’s principle and Galerkin technique, respectively, are used to obtain the governing equations and boundary conditions and also to solve the equation of motion. Complex averaging method combined with arc-length continuation is used to investigate dimensionless natural frequency (DNF), pull-in voltage, nonlinear frequency response and stability analysis of the piezo-harmo-electrostatic nanoresonator considering the nonlocal, material length scale, surface/interface effects, electrostatic and harmonic excitation. For this work, three nonclassical theories of NLT, NSGT and GMSIT are compared with classical theory (CT). It is concluded that ignoring surface/interface and small-scale effects leads to inaccurate results in vibrational response of the PHENR. It is found that in all boundary conditions, nonlocal scale parameter $$\bar{\mu }$$ and material length scale parameter $$\bar{\eta }$$, respectively, lead to increase and decrease in PHENR stiffness and in the nonclassical theories of NLT and NSGT, the oscillation amplitude, the range of instability and all nonlinear behavior is greater than that of the classical one. Also an increase in surface/interface parameters leads to increase or decrease in the dimensionless natural frequency, resonance amplitude, resonant frequency, the system’s instability and nonlinear behavior of PHENR.

Nonlinear magneto-thermo-elastic vibration of mass sensor armchair carbon nanotube resting on an elastic substrate

Abstract The present paper aims at studying the nonlinear ultrasonic waves in a magneto-thermo-elastic armchair single-walled (SW) carbon nanotube (CNT) with mass sensors resting on a polymer substrate. The analytical formulation accounts for small scale effects based on the Eringen’s nonlocal elasticity theory. The mathematical model and its differential equations are solved theoretically in terms of dimensionless frequencies while assuming a nonlinear Winkler-Pasternak-type foundation. The solution is obtained by means of ultrasonic wave dispersion relations. A parametric work is carried out to check for the effect of the nonlocal scaling parameter, together with the magneto-mechanical loadings, the foundation parameters, the attached mass, boundary conditions and geometries, on the dimensionless frequency of nanotubes. The sensitivity of the mechanical response of nanotubes investigated herein, could be of great interest for design purposes in nano-engineering systems and devices.

Nonlocal bending and buckling of agglomerated CNT-Reinforced composite nanoplates

The study presents a nonlocal bending and buckling analysis of agglomerated carbon nanotube-reinforced composite nanoplates resting on a Pasternak foundation. A two-parameter micromechanics model incorporating agglomeration is used to obtain the effective mechanical properties of the nanoplates. Using Hamilton's principle, the governing differential equations are derived based on the Eringen's nonlocal elasticity theory and the sinusoidal shear deformation theory. The deflection and critical buckling load of the nanoplates are obtained by Navier's analytical solution. To verify the approach, the results are compared with experimental, analytical, and numerical findings in the literature. Detailed parametric studies are then performed to discuss the influences of the following parameters on the static bending and buckling response of the nanoplates with agglomerated CNTs: degree of agglomeration, nonlocal material scale parameter, temperature, foundation properties, volume fraction of CNTs, and length-to-thickness aspect ratio for the plate.

Small scale and spin effects on free transverse vibration of size-dependent nano-scale beams

This study aims to investigate the transverse vibration of size-dependent nano-beam which spins about its longitudinal axis. To this end, nano-beams are modeled based on the Euler–Bernoulli and Timoshenko theories. To consider the size dependency of the nano-beams, the nonlocal elasticity theory is used. By using an exact analytical solution, the governing equations of motions are solved. Comprehensive results are presented to show the effects of various boundary conditions, nonlocal scale parameter, rotational speed, and geometrical parameters on natural frequencies and critical speeds of spinning nano-beam. As spinning nano-beams are the main part of any rotary nano-machines, conclusions of present analysis promote researcher’s information about the vibrational behavior of spinning nano-beams.

Nonlinear transverse vibration of nano-strings based on the differential type of nonlocal theory

The nonlinear vibration responses of nano-strings are studied based on the theory of Eringen’s nonlocal elasticity. Firstly, the nonlocal differential constitutive model in one-dimensional form which is suitable for a string structure is used, and then the governing equation of motion for the nonlinear vibration of nano-strings is derived by considering the expression of classical Lagrangian strain. In order to solve the non-dimensional nonlinear governing equation of motion, the Galerkin method or Rayleigh-Ritz method is applied, and the nonlinear partial differential equation is approximately transformed into the a set of ordinary differential equations. The ordinary differential equations are then solved by a numerical method, and the nonlinear vibration responses under different time histories are thus obtained. Subsequently, the approximate numerical solution of the nonlinear displacement is solved by the second-order multi-scale method. The nonlinear phenomena in the transverse displacement and the influences of nonlocal scale parameter on the nonlinear vibration characteristics of nano-strings are analyzed accordingly. The results will provide a basis for understanding and controlling the nonlinear dynamics of nano-strings which may act as key components in the booming intelligent nano-systems.