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The present article deals with the application of refined plate theory and surface effects to investigate the nonlocal free vibration behavior of a functionally graded nano-plate composed of two piezoelectric face sheets resting on Orthotropic Visco-Pasternak. Surface and nonlocal effects are applied using the surface piezoelectricity and stress-strain gradient theories. Surface effects, refined quasi-3D higher-order theory, neutral surface position and in-plane mechanical preloads are considered to study piezoelectric sandwich nano-plate with a functionally graded core in this study for the first time, simultaneously. The size-dependent governing equations are derived based on 2D and 3D refined theory using Hamilton's principle and different boundary conditions are studied and investigated according to the semi-analytical solution method. Comparing the results of the vibration behavior of 2D and quasi-3D models, local and non-local models, and also, examining different foundations simultaneously can be said as one of the innovations of the current research. Finally, a parametric study is presented to examine the effects of different parameters such as surface effects, small scale parameters, mechanical preloads, thickness stretching effects, neutral surface position, non-homogeneous coefficient, applied voltage, different foundation constants, boundary conditions, geometrical ratios and different theories in details. It is indicated that the inclusion of surface effects and thickness stretching effects leads to a significant influence on the vibration behavior of three-layered nano-scale plates. The presented results in this article may provide useful guidance for the design and development of the next generation of nano-devices. [ABSTRACT FROM AUTHOR]

In this paper, a nonclassical shell model is utilized to investigate the dynamic stability of functionally graded piezoelectric (FGP) nanoshells considering the thermo-electro-mechanical coupling effects. Also, the FGP nanoshells on elastic foundations are subjected to periodic axial loading. The theoretical formulations are derived by implementing the first-order shear deformation (FSD) shell theory into the nonlocal strain gradient theory (NSGT). Hamilton's principle is utilized to obtain the equations of motion and related boundary conditions of the present nanoshell model. Moreover, the Mathieu-Hill equations determining the instability regions are presented and analyzed based on Bolotin's method. Results demonstrate that the external electric voltage, the elastic foundations, the temperature rise, the static load factor, the power-law index and the small-scale parameters have significant influences on the dynamic stability responses of FGP nanoshells. [ABSTRACT FROM AUTHOR]

This study introduces an approach to analyze torsional vibration in non-circular nanowires within magnetic fields, considering various boundary conditions on an elastic foundation. Analytical formulas for natural angular frequencies are obtained using nonlocal strain gradient theory. The analysis covers three non-circular cross-sections, incorporating the warping effect. Elastic springs at the wire ends simulate support conditions, restricting rotation around the wire's axis. The torsion function around the axis is represented by a Fourier series, discretized at the spring points and linked using the Stokes' transform alongside the boundary values. This leads to an eigenvalue problem that includes higher-order material size parameters (strain gradient, nonlocal), spring parameters, and the warping function. The study's novelty lies in effectively solving torsional vibration for non-circular sections, addressing warping function, elastic medium, and size effects under both deformable and non-deformable boundary conditions. The presented solution is capable of determining vibration frequencies for both rigid and deformable boundary conditions. This is accomplished by specifying the torsional spring stiffness values, thereby obviating the necessity for additional recalculations. In order to verify the obtained results and compare them with the existing literature, nanowires with free and clamped boundary conditions are solved numerically by changing the spring parameters in the eigenvalue problem. In the formulation of natural angular frequency; length scale parameters, warping, magnetic field, elastic medium effects are included. In addition, since the support conditions are modeled with elastic springs, the resulting formulas are quite general and can be used to solve different types of torsional vibration problems. [ABSTRACT FROM AUTHOR]

Current research presents the nonlinear static/dynamic response of functionally graded (FG) porous nanotubes in the thermal pre- and post-buckling regimes. The case of temperature-dependent FG nanotubes with even distributed porosities subjected to uniform temperature rise is considered. The stability and vibration characteristics are analyzed for the uniformly heated nanotube surrounded by a nonlinear elastic foundation. The nonlinear equations of motion are derived using the Hamilton principle for the thermally pre/post-buckled nanotube in the framework of nonlocal strain gradient theory. The size-dependent governing equations are established based on the uncoupled thermoelasticity, the von Kármán assumption and the high-order shear deformation theory. The nonlinear governing equations are reduced for the nonlocal strain gradient-based model of nanotubes with immovable simply supported boundary conditions. The system of equations is solved for the nonlinear problems of free vibration and post-buckling of the heated tube using two-step perturbation technique and Galerkin procedure. The novel parametric examples are carried out to investigate the effect of nonlocal and length scale parameters, porosity distribution, functionally graded pattern, elastic foundations and geometrical parameters on the thermoelastic response of tubes. The obtained results highlight the importance of static/dynamic analysis of FG porous nanotubes in the thermal pre-buckling and post-buckling regimes. [ABSTRACT FROM AUTHOR]

The aim of this research is to investigate the impact of a viscoelastic foundation on the vibration of a curved beam structure with clamped and simply-supported boundary conditions. The structure considered is a micro-scale laminate composite beam made up of two piezoelectric face layers and a carbon nanotube-reinforced composite core. The study employs non-classical elasticity theory and nonlocal strain gradient theory to account for both nano-scale and microscale effects. The equations of motion are derived using the energy terms of the beam and a variational principle, with boundary conditions assumed to be clamped at one end and simply supported at the other end. A semi-analytical method of general differential quadrature method is used to solve the equations due to their nonlinear terms. To capture the effects of various parameters on the in-plane vibration of sandwich arches, an artificial neural network is designed and implemented. The study analyzes the impact of nonlocal and gradient parameters, geometrical aspect ratios, and substrate constants of the structure on the natural frequency and amplitude of vibration of the laminate beam. Results show that increasing nonlocal and gradient parameters have different effects on the amplitude and frequency of vibration of the beam. [ABSTRACT FROM AUTHOR]

AbstractIn this investigation, the analysis of the nonlinear vibroacoustic and sound transmission loss behaviors of nanoplates made of functionally graded materials considering the nonlocal strain gradient theory is presented. It is presumed that the properties of the functionally graded nanoplate are in the form of the simple power law scheme and continuous along the thickness, under nonlinear temperature rise and incident oblique acoustic wave as well as the first-order shear deformation theory. For this purpose, applying Hamilton’s principle, the nonlinear partial differential equations of motion are derived by the displacement field function approach and by considering the von Kármán nonlinear strain-displacement relations. To solve the equations, using the Galerkin method, the nonlinear partial differential equations of motion lead to the Duffing equation. Then, applying the homotopy analysis method, the equation of the transverse movement of the nanoplate is solved semi-analytically to obtain the nonlinear frequencies. Lastly, the nonlinear vibration and acoustic response of functionally graded nanoplate are investigated by considering the variation of the significant factors such as material gradient indices, nonlocal parameters, length scale parameters, aspect ratio, dimensionless amplitude, nonlinear temperature changes, and sound characteristics of the functionally graded nanoplate. [ABSTRACT FROM AUTHOR]

This research addresses the three-dimensional thermomechanical wave propagation behavior in sandwich composite nanoplates with a metamaterial honeycomb core layer and double functionally graded (FG) ultra-stiff surface layers. Due to its potential for high-temperature applications, pure nickel (Ni) is preferred for the honeycomb core layer, and an Al2O3/Ni ceramic-metal matrix is preferred for the surface layers. The functional distribution of graphene platelets (GPLs) in three different patterns, Type-U, Type-X, and Type-O, in the metal-ceramic matrix with a power law distribution provides double-FG properties to the surface layers. The mechanical and thermal material characteristics of the core and surface layers, as well as the reinforcing GPLs, are temperature-dependent. The pattern of temperature variation over the plate thickness is considered to be nonlinear. The sandwich nanoplate's motion equations are obtained by combining the sinusoidal higher-order shear deformation theory (SHSDT) with nonlocal integral elasticity and strain gradient elasticity theories. The wave equations are established by using Hamilton's principle. Parametric simulations and graphical representations are performed to analyze the effects of honeycomb size variables, wave number, the power law index, the GPL distribution pattern, the GPL weight ratio, and the temperature rise on three-dimensional wave propagation in an ultra-stiff sandwich plate. The results of the analysis reveal that the 3D wave propagation of the sandwich nanoplate can be significantly modified or tuned depending on the desired parameters and conditions. Thus, the proposed sandwich structure is expected to provide essential contributions to radar/sonar stealth applications in air, space, and submarine vehicles in high or low-temperature environments, protection of microelectromechanical devices from high noise and vibration, soft robotics applications, and wearable health and protective equipment applications. [ABSTRACT FROM AUTHOR]

This article addresses the thermomechanical thermal buckling and free vibration response of a novel smart sandwich nanoplate based on a sinusoidal higher-order shear deformation theory (SHSDT) with a stretching effect. In the proposed sandwich nanoplate, an auxetic core layer with a negative Poisson's ratio made of Ti-6Al-4V is sandwiched between Ti-6Al-4V rim layers and magneto-electro-elastic (MEE) face layers. The MEE face layers are homogenous volumetric mixtures of cobalt ferrite (CoFe2O4) and barium titanate (BaTiO3). The mechanical and thermal material properties of the auxetic core and MEE face layers are temperature-dependent. Using Hamilton's principle, governing equations are constructed. To characterize the size-dependent behavior of the nanoplate, governing equations are adapted with the nonlocal strain gradient theory (NSGT). By applying the principles of Navier's technique, closed-form solutions are obtained. Parametric simulations are carried out to examine the effects of auxetic core parameters, temperature-dependent material properties, nonlocal parameters, electric, magnetic, and thermal loads on the free vibration and thermal buckling behavior of the nanoplate. According to the simulation results, it is determined that the auxetic core parameters, temperature-dependent material properties, and nonlocal factors significantly affect the thermomechanical behavior of the nanoplate. The outcomes of this investigation are expected to contribute to the advancement of smart nano-electromechanical systems, transducers, and nanosensors characterized by lightweight, exceptional structural integrity and temperature sensitivity. Also, the auxetic core with a negative Poisson's ratio provides a metamaterial feature, and thanks to this feature, the proposed model has the potential to be used as an invisibility technology in sonar and radar-hiding applications. [ABSTRACT FROM AUTHOR]

Flexoelectricity refers to the link between electrical polarization and strain gradient fields in piezoelectric materials, particularly at the nano-scale. The present investigation aims to comprehensively focus on the static bending analysis of a piezoelectric sandwich functionally graded porous (FGP) double-curved shallow nanoshell based on the flexoelectric effect and nonlocal strain gradient theory. Two coefficients that reduce or increase the stiffness of the nanoshell, including nonlocal and length-scale parameters, are considered to change along the nanoshell thickness direction, and three different porosity rules are novel points in this study. The nanoshell structure is placed on a Pasternak elastic foundation and is made up of three separate layers of material. The outermost layers consist of piezoelectric smart material with flexoelectric effects, while the core layer is composed of FGP material. Hamilton’s principle was used in conjunction with a unique refined higher-order shear deformation theory to derive general equilibrium equations that provide more precise outcomes. The Navier and Galerkin-Vlasov methodology is used to get the static bending characteristics of nanoshells that have various boundary conditions. The program’s correctness is assessed by comparison with published dependable findings in specific instances of the model described in the article. In addition, the influence of parameters such as flexoelectric effect, nonlocal and length scale parameters, elastic foundation stiffness coefficient, porosity coefficient, and boundary conditions on the static bending response of the nanoshell is detected and comprehensively studied. The findings of this study have practical implications for the efficient design and control of comparable systems, such as micro-electromechanical and nano-electromechanical devices. [ABSTRACT FROM AUTHOR]

AbstractIn this article, the analytical solution for free oscillation of non-uniform thickness functionally graded porous (FGP) rectangular nanosheets resting on a Pasternak foundation with various boundary conditions (BCs) based on the nonlocal strain gradient theory and surface effect are discovered. The nonlocal stress and strain effects are captured on the nonlocal strain gradient hypothesis, while the surface energy effects are based on the surface elasticity hypothesis. The FGP nanosheet has a nonlinear thickness change in both x and y directions and is placed on the Pasternak elastic foundation. The unique point of this study is to investigate the change of nonlocal and length-scale coefficients along the thickness as mechanical properties of the material. Applying classical shear deformation theory (ignore the effect of shear deformation) combined with Hamilton’s principle, the motion balance equation of non-uniform thickness nanosheets is established. The accuracy of the proposed method is verified through reliable publications. A set of results of natural vibration frequencies of variable thickness FGP nanosheets under the influence of factors are discovered and discussed. The results of this study can be used in design calculations of MEM and NEMs systems in mechanics. [ABSTRACT FROM AUTHOR]

The present study explores dispersion characteristics of thermal, viscoelastic and mechanical waves in graphene sheets subjected to uniform thermal loading and supported by the visco-Pasternak foundation. Kinematic relations for graphene sheets are deduced within two-variable refined higher-order plate theory. Damping effects of the viscoelastic medium are modeled using the Kelvin–Voigt model. The research extensively investigates the size-dependent behavior of graphene sheets by incorporating nonlocal strain gradient theory. Nonlocal governing equations are formulated under Hamilton's principle and solved analytically to determine wave frequency values. To validate the results, a comparative analysis is conducted, and the outcomes are tabulated to confirm the effectiveness of the approach. Finally, graphical representations are employed to depict the influence of each parameter on the wave propagation responses of graphene sheets. [ABSTRACT FROM AUTHOR]

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

This work investigates the free vibrations of innovative thermally loaded nanoplates constructed by integrating magneto-electro-elastic (MEE) layers with functionally-graded graphene platelet-reinforced composite cores (FG-GPLRC) and accounting for viscous fluid interactions. An advanced multiphysics model is developed using the Navier–Stokes equations to capture fluid structure coupling effects, Halpin–Tsai, and the rule of mixtures micromechanics to predict the non-uniform effective properties, third-order shear deformation plates theory (TSDPT) to incorporate thickness stretching, and the nonlocal strain gradient theory (NSGT) to characterize size dependencies. The Galerkin technique is used to solve the governing equations, which are derived from the Hamilton’s principle. Parametric analyses quantify the influences of fluid depth, temperature fluctuations, temperature profiles, nonlocal and strain gradient parameters, electric and magnetic potentials, graphene distribution patterns, graphene weight fractions, and boundary conditions on the vibration response. The outcomes of this study provide design guidelines and predictive tools enabling active vibration control systems for next-generation thermally-loaded nanocomposite structures with widespread applications from aerospace vehicles to nanoelectronics. [ABSTRACT FROM AUTHOR]

The free vibration behavior of a new advanced functionally graded (FG) nanobeam is presented in this work using the recently proposed nonlocal higher-order shear deformation theory. In the present theory, the stress tensor can satisfy the parabolic variation of the shear stress distribution throughout the thickness direction and also fulfill the requirement that the shear stress on the top and bottom surfaces of the FG nanobeam is zero. Two common types of FG structures, namely, FG hardcore and FG softcore, are considered here for analysis with three schemes. The material properties of the FG nanobeam are assumed to vary continuously in both the longitudinal and transversal directions according to a combined simple power-law distribution in terms of the volume fractions of the constituents. The governing equations of the FG nanobeam with simply supported boundary conditions are derived using the proposed higher-order shear deformation plate theory. The nonlocal strain gradient theory is employed to capture the microstructure-dependent effect. The influence of the structural geometry, the gradient index, and the nonlocal and length scale parameters on the vibration frequency is investigated. Finally, many new results are also reported in the current study, which will serve as a benchmark for future research. [ABSTRACT FROM AUTHOR]

This paper presents analytical solutions for the bending behavior of bi-directional functionally graded (BDFG) micro and nanobeams, wherein the material properties vary along both the thickness and axial directions, following power-law and exponential-law profiles, respectively. This study employs two size-dependent theories, nonlocal modified couple stress theory (NCST) and nonlocal strain gradient theory (NSGT), to account for size effects inherent in nanoscale structures. The governing differential equations are derived using Hamilton's principle, and the Laplace transform technique is utilized for their solution. The study critically compares the size effects captured by NCST and NSGT and assesses the influence of material gradation parameters in both directions. Additionally, the impacts of nonlocal and length scale parameters are thoroughly investigated. The findings indicate that NSGT tends to overestimate size effects compared to NCST. This research enhances the understanding of the mechanical behavior of BDFG nanobeams, offering valuable insights for the design and analysis of nanoscale structures across diverse applications.

Presented herein is an investigation for the nonlinear vibration and stability analysis of rotating functionally graded (FG) piezoelectric nanobeams based on the nonlocal strain gradient theory. The present model can be regarded as a simplified version for the rotating nanowire of biomechanical nanogenerators. The Hamilton principle is used to derive nonlinear equations of motion and their related boundary conditions, which are then discretized to form a set of algebraic equations. Accordingly, the nonlinear vibration frequencies and buckling loads of the nanobeams can be determined by an iterative method. A parametric study of rotational velocity, nonlocal parameter, material length parameter, power-law index, and electrostatic voltage on the dynamic stability behavior of such nanobeams is also presented. In the cantilever case, increasing the nonlocal parameter and material length parameter can result in a stiffness-hardening effect that is unaffected by rotational velocity and the material length parameter to nonlocal parameter ratio. Yet, this has not been reported previously. More importantly, incorporating the effect of geometric nonlinearity on the dynamic responses and stability results of the nanobeams is indispensable. In particular, new observations for the coupling effect of vibration amplitude and power-law index on the electric potential effect are useful for the design of rotating microelectromechanical devices. [ABSTRACT FROM AUTHOR]

Present study deals with the nonlinear analysis of thermally induced deflection and snap-through buckling in functionally graded porous (FGP) curved nanotubes. It is assumed that the curved nanotube is rested on a two-parameter nonlinear elastic foundation. The curved nanotube is subjected to uniformly distributed transverse pressure loading and also uniform temperature rise. Distribution of thermomechanical properties in the thickness of curved nanotubes is radially graded according to a power law function. The even type of porosity distribution patterns is included into the formulation and all properties are considered as temperature dependent. The nonlocal strain gradient and uncoupled thermoelasticity theories are provided to analyze the geometrically nonlinear behaviors of curved nanotubes. The equilibrium equations are obtained based on the high-order shear deformation theory. To capture the large deformations, the von Kármán type of nonlinear kinematic assumptions is included. Two types of immovable boundary conditions are considered which are simply-supported and clamped-clamped. The equilibrium equations are reformulated and transferred into the dimensionless presentation. The closed-form expressions are provided to trace the thermal/mechanical load-deflection paths of curved nanotubes using the two-step perturbation technique. Numerical results of this study are compared with the available data in the literature for the macro-scale curved tubes. After that novel parametric results are provided to discuss the effects of nonlocal and length scale parameters, boundary conditions, porosity coefficient, foundation stiffness, power law exponent, and geometrical parameters. [ABSTRACT FROM AUTHOR]

The geometrically nonlinear stability analysis of restrained nanobeams under an elastic medium is considered in the presented manuscript. The investigation is based on the nonlocal strain gradient elasticity theory, Euler-Bernoulli beam theory and geometrical nonlinearity which is a significant change in geometry. To the best of the authors' knowledge, the nonlinear stability response of a nanobeam with elastic boundary conditions and on an elastic foundation has not been presented before via the theory of nonlocal strain gradient. The aim of this paper is to fill this gap in the literature by offering a method that can give a solution under general elastic boundary conditions. Using a Fourier sine series, a Fourier coefficient suitable for constructing an eigenvalue problem is computed. The constructed problem has been treated in the transformed region with the help of the Stokes' transform, then the stability analysis is executed for arbitrary boundary conditions (rigid or restrained) based on the nonlocal strain gradient elasticity. The influences of various nonlinear parameters, deformable boundary conditions, small-scale parameters and elastic foundation parameters on the stability of the constrained nanobeam are studied. It is observed that the investigated variables produce significant changes in buckling loads. [ABSTRACT FROM AUTHOR]

The nonlinear free vibration of simply supported porous functionally graded (PFG) size-dependent tubes under compressive axial loading in pre-buckling regime are investigated. The nonlinear Euler–Bernoulli beam hypothesis is employed within the framework of the nonlocal strain and velocity gradient theory to derive the nonlinear partial differential equation of motion. It is assumed that the material properties are gradually graded in the radial direction. Additionally, two different porosity distribution patterns are used in the radial direction. The modified power-law function is employed to estimate the effective properties of PFGM/NTs. The partial differential equation governing the lateral nonlinear vibration of prestressed PFGM/NTs is achieved by employing Hamilton's principle. The method of multiple scales is utilized to solve the nonlinear ordinary differential equation system obtained by applying the Galerkin method to the nonlinear partial differential equation of lateral motion. The nonlinear frequencies of pre-buckled PFGM/NTs, which reflect the effects of the amplitude of different modes involved in vibration response, are formulated. The effects of different parameters, such as length scale parameters, porosity distribution pattern, and material composition, on the nonlinear frequencies of pre-buckled porous functionally graded size-dependent tubes are examined. The pre-buckling behavior stage demonstrates that porosity reduces the static stability of FGM/NTs, but the power-law index value and modulus of elasticity ratio can modify the softening effects of porosity. Research indicates that the presence and distribution pattern of porosity, along with material composition, can enhance the nonlinear frequencies of M/NTs. [ABSTRACT FROM AUTHOR]

AbstractIn this study, the stability and vibration analyses of a spinning smart nanotube under electrical loads are examined for the first time. The smart nanotube structure is made from a single-walled zinc oxide nanotube (SWZnONT) due to its extraordinary piezoelectric and magnetic properties, which have great potential for use as a rotating component in nanoelectromechanical systems (NEMS). To achieve this aim, nonlocal strain gradient theory and Maxwell’s electrostatic equations are used to capture the structure’s small-scale and piezoelectric effects, respectively. In addition, the nanotube’s structural model is based on the Euler–Bernoulli beam theory, deriving governing equations and boundary conditions via the Hamilton principle. A technique employing Galerkin-based closed-form solutions is utilized to solve the equations of motion and get the vibration response of the spinning smart nanotube. Finally, the effects of the material length scale, nonlocal parameter, rotating speed, external voltage, and boundary conditions on natural frequency and critical rotational speed are investigated. The study results indicate that applying an external voltage to spinning SWZnONT effectively adjusts the critical speed and enhances structural stability. [ABSTRACT FROM AUTHOR]

In this paper, a novel analytical approach based on nonlocal strain gradient theory is proposed to investigate small-scale effects on the radial vibration of isotropic spherical nanoparticles interacting with a viscoelastic fluid. The viscoelastic fluid is assumed to be compressible and takes into account both compressional and shear relaxation processes. The frequency equation is derived by imposing the fluid-nanosphere interface continuity conditions. To demonstrate the validity and accuracy of the approach, a comparison is made with the literature results in some particular cases, which shows a good agreement. Numerical examples are finally conducted to reveal the significance of small-scale effects in the radial vibrations, which need to be included in the nonlocal strain gradient model of submerged spherical nanoparticles. It is found that the vibration behavior greatly depends on the nanosphere size, nonlocal parameter, strain gradient parameter and glycerol-water mixture. Particularly, the small-scale effects play a very pronounced role when the spherical gold nanoparticle radius is smaller than 3 nm. Thus, the obtained frequency equation is very useful to interpret the experimental measurements of vibrational characteristics of nanospheres submerged in a viscoelastic fluid. [ABSTRACT FROM AUTHOR]

This study deals with the dynamic behavior of short-fiber reinforced composite nanobeams. It is assumed that short-fibers are aligned or randomly distributed in the composite nanobeams. Nonlocal strain gradient theory is applied to composite nanobeam mechanics including Euler–Bernoulli and Timoshenko beam models. The transverse vibration of these composite nanobeams is investigated for various boundary conditions. Approximate Ritz method is used for obtaining the natural frequencies of short-fiber reinforced composite nanobeams. In addition to vibration analysis, wave propagation in short-fiber reinforced composite nanobeams is investigated and wave dispersion relations are analytically obtained for both Euler-Bernoulli and Timoshenko beam models. The vibration and wave dispersion results of short-fiber reinforced composite nanobeams are obtained for aligned and randomly distributed cases. The results obtained from this paper showed that there is no significant difference between the aligned and randomly oriented short-fiber composite nanobeams. This provides great convenience to designers where it is not possible to orient the reinforcement material in composites. The present study may be useful for the mechanical analysis and design of micro/nano-electromechanical systems (MEMS/NEMS), nanoprobes, nanosensors, nanoactuators, and atomic force microscopes. [ABSTRACT FROM AUTHOR]

Purpose: Flexoelectricity refers to the phenomenon of the link between electrical polarization and strain gradient fields in piezoelectric materials, particularly at the nanoscale. The goal of this study is to look into the bending and vibration properties of piezoelectric sandwich nanoplates in more detail, focusing on the flexoelectric effect and nonlocal strain gradient. Method: The plate consists of three distinct layers, including two outside skin layers formed of piezoelectric smart material exhibiting the flexoelectric effect and an inner core layer consisting of functionally graded porous (FGP) material. The general equations of motion with improved accuracy are derived using Hamilton's principle and the revised higher order shear deformation plate theory (RPT). The use of the Galerkin–Vlasov method allows for the determination of the static bending characteristics and particular vibration frequencies of plates with various boundary conditions. A computational program is implemented using the Matlab software platform. Then, the program's correctness is assessed by doing a comparative analysis using previously published, dependable findings in specific instances of the model described in the article. Moreover, it is crucial to carry out a comprehensive analysis to evaluate the influence of various attributes on a system's static bending response and free vibration. The parameters include the flexoelectric effect, nonlocal and strain gradient parameters, elastic foundation stiffness coefficient, porosity coefficient, and geometric conditions. Results: The results of this research have practical consequences for the effective planning and management of similar systems, such as micro-electro-mechanical and nano-electromechanical devices. The results of this work are an important premise for developing more complex problems shortly, such as buckling analysis and dynamics problems. In addition, this is also a valuable reference for engineers designing models in engineering practice. Conclusion: The article presents a set of numerical experiments that provide significant findings. The stiffness of the plate is increased by the parameter f14, and therefore, the maximum deflection decreases as f14 rises. Nonlocal and strain gradient coefficients have contrary effects on the structure, so modulating these two coefficients is crucial for regulating the structure's stiffness. The ratio between the thicknesses of the piezoelectric layer and the FGP core layer plays a crucial role in the study of increasing or decreasing the overall structure's stiffness. An augmentation in the grading index will lead to a decrease in the overall stiffness of the structure, while an augmentation in the elastic base stiffness would result in an increase in the overall stiffness. Furthermore, the porosity coefficient, contingent upon the characteristics of the object, will induce either an augmentation or diminution in the magnitude of the inherent vibration frequency. However, it is quite probable that the displacement value of the plate will be augmented. Some new points in the article: The plate is composed of three layers with two skin layers made of piezoelectric smart material with flexoelectric effect and a core layer made of functionally graded porous material. The nonlocal and strain gradient coefficients change along the thickness direction and the porosity. Navier and Galerkin–Vlasov analytical methods are applied with many different boundary conditions. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate bandgap formation in an Euler-Bernoulli nanobeam using the nonlocal strain gradient theory. Such a model enables the capture of both softening and stiffening size-dependent behavior of the nanobeams using two length-scale parameters, namely, a nonlocal parameter and a strain gradient parameter. Hamilton's principle is used to develop the normalized equation of motion yielding a sixth order differential equation. The nanobeam is assumed to be infinitely long and the dispersion of an elastic wave in a single representative periodic unit cell of the structure is used to estimate the bandgap edge frequencies. To this end, two unit cell approaches are employed to obtain the dispersion relations, namely, the homogenization and transfer matrix approaches. Both methods yield similar dispersion curves and showed good agreement with classical cases from the literature. A parametric study was carried out to investigate the effect of several parameters on the bandgap formation in the metamaterial nanobeams. These parameters include, the nonlocal parameter, the strain gradient parameter, the length of the unit cell and the resonator mass to unit cell mass ratio. The study conducted using a purely strain gradient model showed a comparable response to that of the classical macro problem with the usual hardening when increasing the value of the strain gradient length scale. Nonlocal and nonlocal strain gradient models introduced coupling between parameters of the study with the occasional introduction of non-monotonic dispersion curves. Such non-monotonic behavior is believed to be characteristic of large values of the nonlocal length-scale. • Bandgap formation in a nonlocal strain gradient theory nanobeam using local resonators. • Nanobeam is governed by a 6th order differential equation with six boundary conditions and two length-scale parameters. • Transfer matrix and homogenization approaches were used to predict bandgaps. • Effect of length scales, in addition to other parameters on bandgap formation were investigated. [ABSTRACT FROM AUTHOR]

• Small scale analysis of functionally graded triply periodic minimal surface (FG-TPMS) nanoplates. • The generalized weak form formulation for the FG-TPMS nanoplate is proposed. • A fitting technique based on a two-phase piece-wise function for porous structures of TPMS. • The length scale parameters can high-efficiently predict size effects. • Novel benchmark numerical results are illustrated and introduced. In recent years, the triply periodic minimal surface (TPMS) has emerged as a remarkable solution for constructing structures, drawing inspiration from natural architectures. TPMS offers several outstanding features, including porous architectures with high interconnectivity, smooth surfaces and the ability to achieve mathematically controllable geometry features. However, it is evident that the current research has not fully harnessed the extensive potential and benefits of TPMS structures. In this paper, a groundbreaking approach for analyzing functionally graded triply periodic minimal surface (FG-TPMS) nanoplate, which is utilized a novel nonlocal strain gradient isogeometric analysis, is provided. Three patterns of the FG-TPMS nanoplate, namely Primitive (P), Gyroid (G) and I-gragh and Wrapped Package-graph (IWP), are utilized in this study. The primary focus is to investigate size dependent problems with two types of density distributions. The proposed model successfully incorporates both nonlocal effects and strain gradient effects into nanoplate structures. The paper demonstrates how the mechanisms responsible for both reducing and enhancing stiffness in the nanoplate can be understood by fine-tuning the nonlocal and strain gradient parameters. The findings of this study offer promising prospects for future design and optimization as they provide a robust approach to address the complex mechanical behavior observed in the FG-TPMS nanoplate. The proposed model not only captures the behavior accurately but also opens up new avenues for exploring the capabilities of FG-TPMS structures. [ABSTRACT FROM AUTHOR]

Harnessing vibrations in multifunctional nanostructured plates is pivotal to next-gen microsystems but necessitates understanding scale effects under multifield loading. Investigated in this work are the forced and unforced vibrations of multilayered piezoelectric nanoplates supported on a variable viscoelastic medium and loaded hygrothermo-electromechanically with functionally graded porous (FGP) cores. The viscoelastic foundation is supposed to demonstrate non-linear changes in both stiffness and damping characteristics in the x-coordinate direction. The goal is to improve the accuracy of the results by using the nonlocal strain gradient theory (NSGT) and the third-order shear deformation assumption (TSDA), which include the effects of hardening and softening materials. The FGP core layer considers four states of porosity distribution patterns. These porosity distributions are expected to change in both the in-plane and thickness directions. The governing partial differential equations resulting from Hamilton's principle can be reduced to a system of algebraic equations by applying the Galerkin method. The parametric studies evaluate the effects of several factors, such as the initial electric voltage, viscoelastic medium parameters, moisture rise, temperature changes, porosity distributions, FG index, nonlocal and strain gradient features, and boundary conditions, on the vibration response. The novelty lies in the incorporation of advanced theories, such as the NSGT and TSDT, which capture size-dependent effects and material hardening/softening phenomena. Additionally, the consideration of four distinct porosity distribution patterns in the FGP core layer provides insights into the influence of porosity gradients on the dynamic response. The proposed model can guide the design and optimization of multifunctional nanostructured plates for applications in next-generation microsystems, energy harvesting devices, and smart structures.

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The waves propagation characteristics in porous functionally graded piezoelectric nanoshells were investigated based on the nonlocal strain gradient theory. The governing equations were developed under Hamilton's principle and the 1st-order shear theory. The scale-dependent characteristic equations were obtained through combination of the nonlocal strain gradient theory and the harmonic solutions. The effects of the scale parameter, the wave number, the gradient index, the thickness, the porosity and the voltage on the wave propagation characteristics were discussed numerically. The results show that, the influences of the nonlocal parameter and the strain gradient parameter on the wave propagation frequency are closely related to the wave number, and the larger the wave number is in a certain range, the greater the influence of scale parameters on the frequency will be. In addition, the porosity and the gradient index have a coupling effect on the frequency. [ABSTRACT FROM AUTHOR]

AbstractDespite several theoretical investigations on the torsional dynamic behavior of non-circular micro/nanorods, the torsional behavior of right triangle cross-section is still unevaluated. Therefore, Timoshenko-Gere theory in conjunction with nonlocal strain gradient theory is developed for modeling the scale-dependent torsional dynamic of right triangle microscale rods. Various anisotropic materials, i.e. hexagonal, trigonal, triclinic, and monoclinic materials are regarded as the constituent material of microscale rods. The shear stress function is developed for right triangle shape and according to the calculated shear stress function, the warping function of right triangle shape is obtained. The nonlocal governing equation of the torsional dynamic of anisotropic microrods is obtained with the help of the energy approach and solved using an analytical method by regarding two different boundary conditions. The accuracy of the proposed model is validated with recent similar investigations. Because of the lack of mechanical research on right triangle microrods, the proposed methodology was validated with equilateral triangle nanowire reported in the literature. Eventually, the various remarkable variants’ effects on change of scale-dependent torsional frequency are assessed comprehensively. It is found that hexagonal and triclinic materials have the largest and smallest values of frequency. Also, nonlocal parameter, b/h ratio play a decreasing role and length scale parameters and mode number plays an increasing role. [ABSTRACT FROM AUTHOR]

The bending analysis due to thermoelastic loading in nanoscale materials is primarily determined by the physical aspect of the material and the modeling of structures. This paper examines the prediction of bending characteristics of the Euler-Bernoulli beam using Moore-Gibson-Thompson (MGT) thermoelasticity theory in conjunction with nonlocal strain gradient theory (NSGT). The coupled equations for dimensionless deflection and temperature change with ramp-type heating boundary conditions are formulated and solved using the Laplace transform method and the wavelet approximation method. The stiffness softening and hardening effects due to nonlocal and length-scale parameters are assessed, and their discrepancies are discussed. Findings further reveal that the impact of the thermal relaxation parameter on deflection and temperature is negligible. Moreover, the different aspect ratios of the beam's structure depict the prominently behavior of structural simulations in bending. [ABSTRACT FROM AUTHOR]

In this work, a nonlocal strain gradient model is developed for the buckling analysis of laminated nanocomposite plates. The nonlocal and strain gradient constitutive equations are invoked to reformulate the governing equations of classical plate theory and third-order shear deformation plate theory. The resulting governing equations are solved using Navier's approach. The potential of the proposed model, to capture the effects of nonlocality and strain gradient is illustrated using a parametric study. A comparison between the critical buckling loads predicted by both the nonlocal strain gradient model and the classical continuum models is shown via different examples. The results obtained using the proposed model agree well with the literature in the limiting sense of no nonlocal and strain gradient effects. [ABSTRACT FROM AUTHOR]

This research studied the dynamic stability of the Euler-Bernoulli nanobeam considering the nonlocal strain gradient theory (NSGT) and surface effects. The nanobeam rests on the Pasternak foundation and a sequence of inertial nanoparticles passes above the nanobeam continuously at a fixed velocity. Surface effects have been utilized using the Gurtin-Murdoch theory. Final governing equations have been gathered implementing the energy method and Hamilton's principle alongside NSGT. Dynamic instability regions (DIRs) are drawn in the plane of mass-velocity coordinates of nanoparticles based on the incremental harmonic balance method (IHBM). A parametric study shows the effects of NSGT parameters and Pasternak foundation constants on the nanobeam's DIRs. In addition, the results exhibit the importance of 2T-period DIRs in comparison to T-period ones. According to the results, the Winkler spring constant is more effective than the Pasternak shear constant on the DIR movement of nanobeam. So, a 4 times increase of Winkler and Pasternak constants results in 102 % and 10 % of DIR movement towards higher velocity regions, respectively. Furthermore, the effect of increasing nonlocal and material length scale parameters on the DIR movement are in the same order regarding the magnitude but opposite considering the motion direction. Unlike nonlocal parameter, an increase in material length scale parameter shifts the DIR to the more stable region.

This article analyzes the vibrational behavior of the double-layered micro-nanosphere to simulate the drug delivery mechanism in the biological structure under the influence of the temperature environment and the viscoelastic substrate for the polar lipid fraction E (PLFE) liposome isotropic material model. In order to obtain the micro-nano structural equations of the double-layer spherical shell, the displacement-strain relations of the shell have been expressed according to the first-order shear deformation theory and the non-local strain gradient theory. The partial differential equations of motion have been obtained by applying Hamilton's principle. The system of linear couple equations have been solved using the Galerkin method. After validating the model with the results of the articles available in the literature to determine the accuracy of the presented model, numerical results are presented to investigate the effects of various parameters such as the radius of curvature ratio to length, damping coefficient, Kelvin-Voight damping coefficient, boundary conditions and temperature on vibration frequency response are provided and discussed. Results of the current research indicates that natural frequency decreases as the damping coefficient related to the viscous effect between the liposome bilayers increase. Results of the current research can be used as a benchmark for drug delivery applications. [ABSTRACT FROM AUTHOR]

A novel stability model is analytically reformulated for the nano-sized beam resting on a one-parameter elastic foundation. The stability solution is based on the nonlocal strain gradient elasticity theory. To corporate the small size effects, two small scale parameters are introduced. The six-order ordinary differential form of the buckling equation, together with two force boundary conditions, are utilized to examine the stability equation in terms of lateral deflection. The infinite terms of linear equations are discretized with the help of the Stokes' transformation and Fourier sine series. The present work can investigate the effects of elastic spring parameters at the ends, nonlocal properties, elastic medium properties, strain gradient parameter, and buckling behavior of the nanobeam. The predictions of the proposed analytical model with deformable boundary conditions are in agreement with those available in the scientific literature for the nanobeam on elastic foundation based on a closed form of solution. The presence of the deformable conditions, elastic foundation, nonlocal, and strain gradient properties change the buckling loads and buckling mode shapes. [ABSTRACT FROM AUTHOR]

• The temperature-dependent material properties of BaTiO 3 and CoFe 2 O 4 were defined in the model for the first time. • Sinusoidal higher-order shear and nonlocal strain gradient theories are employed to model the dynamic behavior. • The free vibration and buckling behavior of a magneto-electro-elastic (MEE) laminated nanoplate were investigated. • The effects of nonlinear thermal field, electric and magnetic potentials are presented. • The faceplate composition and electric/magnetic potentials are helpful to keep the MEE nanoplate in adjusted response. This study investigated the free vibration and buckling behavior of a magneto-electro-elastic (MEE) sandwich porous nanoplate under nonlinear thermal loads and electric and magnetic potentials. The equation of motion of the nanoplate consisting of barium-titanate and cobalt-ferrite components, with a functionally graded (FG) porous core and two face plates, was modeled by a sinusoidal higher-order shear (SHSDT) and the nonlocal strain gradient (NSGT) theories. Due to the significant effect of the temperature in the dynamics of the magneto-electro-elastic nanoplate, the temperature-dependent material properties of barium-titanate (BaTiO 3) and cobalt ferrite (CoFe 2 O 4) were defined and applied to the modeling for the first time in the literature. The laminated plate consists of upper and lower surface auxiliary plates and a main FG core plate. The core plate consists of functionally grading cobalt-ferrite and barium-titanate, while face plates can be in three different material compositions consisting of pure cobalt-ferrite, pure barium-titanate, or a combination of the two. It has been shown in this study that the dynamic behavior of the core plate, exposed to high temperatures, can be controlled through auxiliary surface plates. Especially in nanosensor applications, the disadvantage of thermal load can be reduced by the electric and magnetic field's strength applied to the surface plates. This study will make an essential contribution to the design and application of nanosensor or nanoelectromechanical systems, especially by controlling nanosensor responses in high-temperature applications. [ABSTRACT FROM AUTHOR]

Abstract Artificial intelligence has revolutionized the process of solving engineering problems by facilitating predictions and optimizations with ease. The use of artificial neural networks (ANNs) for predicting the mechanical behavior of small-scale structures, particularly in the instability field, is widely prevalent. The present study aims to use an ANN to predict the frequency response of microscale spinning cylindrical laminate shells. Our ANN design utilizes a novel modified Adam optimizer for dimensional consistency, and a genetic algorithm is used for optimizing the number of nodes in the hidden layers of the ANN. ANNs require a valid dataset for training and testing, thus, a numerical solution of the size-dependent elasticity theory is generated, and a parametric study is conducted to find the dependency of vibrational responses on different parameters of the cylindrical shell resting on an elastic foundation equipped with a piezoelectric layer. These numerical results are utilized to train and test the designed ANN. The unique feature of the current ANN is the utilization of several well-known optimizers to observe their efficacy in solving the vibrational problem. The results of the vibration study from both numerical and ANN are presented using transverse displacement of the structure. The study finds that the AdaDelta optimizer performs satisfactorily in comparison to other optimization algorithms in terms of both total error and convergence rate. [ABSTRACT FROM AUTHOR]

In order to examine the impact of structural and thermal scale parameters on thermoelastic vibrations of Euler-Bernoulli nanobeams, this article intends to provide a size-dependent generalized thermoelasticity model with the help of nonlocal strain gradient theory (NSGT) in conjunction with dual-phase-lag (DPL) heat conduction model. To highlight the role of each scale parameter in size-dependent motion and heat conduction equations, normalized forms of these nonclassical coupled thermoelastic equations are extracted by introducing and exploiting some dimensionless parameters. By exploiting power series expansion as a general solution for arbitrary boundary conditions, system of partial differential equations reduces to system of ordinary differential equations in time domain. The Laplace transform is then utilized to attain the analytical solution of this set of differential equations. By conducting a case study on a hinged-hinged nanobeam, with the aim of illustrating the impact of nonclassical structural and thermal parameters on thermoelastic vibrations, size-dependent numerical results are compared with those estimated by classical elasticity theory and heat conduction model. Meaningful disparity between classical and nonclassical results reveals the pivotal role of scale parameters in realistic assessment of thermoelastic behavior of nanobeams. Outcomes also indicate that corresponding to the relationship between the magnitudes of two scale parameters of NSGT, the nonclassical model of nanobeam can exhibit either softening or hardening behavior compared to the classical one. This affirms the ability of NSGT to justify both stiffness softening and stiffness hardening phenomenon in nanostructures. [ABSTRACT FROM AUTHOR]

Abstract The use of nanoscale arch structures in vibration detection has become popular due to their ability to capture resonance in two independent directions. This study investigates the vibration behavior of a small-scale cantilever curved beam structure made of a functionally graded carbon nanotube reinforced nanocomposite core and piezoelectric surface layers on a viscoelastic substrate. The piezoelectric layers convert motion into electrical signals, allowing for the harvesting of high-frequency motions in a delicate manner. Various theories, including the first-order shear deformation theory, strain gradient theory, and nonlocal theory, are used to capture the mechanical properties of the FG nanocomposite core at both nano- and micro-scales. The equations of motion and natural boundary conditions are derived using Hamilton’s principle and solved using Akbari-Ganji’s method. An artificial intelligence method is also presented to simplify the solution procedure and calculate numerical results based on important parameters in the vibrational behavior analysis. The study demonstrates the significance of viscoelastic properties of the substrate and the volume fraction of CNT in composite core in the frequency analysis of the arch structure, as shown by the results of both the numerical and AI methods. [ABSTRACT FROM AUTHOR]

This article presents the nonlinear investigation of the thermal and mechanical buckling of orthotropic annular/circular single-layer/bilayer nanoplate with the Pasternak and Winkler elastic foundations based on the nonlocal strain gradient theory. The stability equations of the graphene plate are derived using higher-order shear deformation theory (HSDT) and first-order shear deformation theory (FSDT) considering nonlinear von Karman strains. Furthermore, this paper analyses the nonlinear thermal and mechanical buckling of the orthotropic bilayer annular/circular nanoplate. HSDT provides an appropriate distribution for shear stress in the thickness direction, removes the limitation of the FSDT, and provides proper precision without using a shear correction coefficient. To solve the stability equations, the differential quadratic method (DQM) is employed. Additionally, for validation, the results are checked with available papers. The effects of strain gradient coefficient, nonlocal parameter, boundary conditions, elastic foundations, and geometric dimensions are studied on the results of the nondimensional buckling loads. Finally, an equation is proposed in which the thermal buckling results can be obtained from mechanical results (or vice versa). [ABSTRACT FROM AUTHOR]

In this context, the nonlinear bending investigation of a sector nanoplate on the elastic foundation is carried out with the aid of the nonlocal strain gradient theory. The governing relations of the graphene plate are derived based on the higher-order shear deformation theory (HSDT) and considering von Karman nonlinear strains. Contrary to the first shear deformation theory (FSDT), HSDT offers an acceptable distribution for shear stress along the thickness and removes the defects of FSDT by presenting acceptable precision without a shear correction parameter. Since the governing equations are two-dimensional and partial differential, the extended Kantorovich method (EKM) and differential quadrature (DQM) have been used to solve the equations. Furthermore, the numeric outcomes were compared with a reference, which shows good harmony between them. Eventually, the effects of small-scale parameters, load, boundary conditions, geometric dimensions, and elastic foundations are studied on maximum nondimensional deflection. It can be concluded that small-scale parameters influence the deflection of the sector nanoplate significantly.

Due to nonlocal and strain gradient effects with rigid and deformable boundary conditions, the thermal vibration behavior of perforated nanobeams resting on a Winkler elastic foundation (WEF) is examined in this paper. The Stokes transformation and Fourier series have been used to achieve this goal and to determine the thermal vibration behavior under various boundary conditions, including deformable and non-deformable ones. The perforated nanobeams' boundary conditions are considered deformable, and the nonlocal strain gradient theory accounts for the size dependency. The problem is modeled as an eigenvalue problem. The effect of parameters such as the number of holes, elastic foundation, nonlocal and strain gradient, deformable boundaries and size on the solution is considered. The effects of various parameters, such as the length of the perforated beam, number of holes, filling ratio, thermal effect parameter, small-scale parameters and foundation parameter, on the thermal vibration behavior of the perforated nanobeam, are then illustrated using a set of numerical examples. As a result of the analysis, it was determined that the vibration frequency of the nanobeam was most affected by the changes in the dimensionless WEF parameter in the first mode and the changes in the internal length parameter when all modes were considered. [ABSTRACT FROM AUTHOR]

Thermoelastic damping (TED) has been discerned as a definite source of intrinsic energy loss in miniaturized mechanical elements. The size-dependent structural and thermal behavior of these small-sized structures has been proven through experimental observations. As a first attempt, this article exploits nonlocal strain gradient theory (NSGT) and nonlocal dual-phase-lag (NDPL) heat conduction model simultaneously to acquire a mathematical formulation and analytical solution for TED in nanobeams that can accommodate size effect into both structural and heat transfer fields. For this purpose, the coupled equations of motion and heat conduction are firstly extracted via NSGT and NDPL model. Next, by deriving the distribution of temperature from heat conduction equation and substituting it in the motion equation, the unconventional thermoelastic frequency equation is established. By deriving the real and imaginary parts of the frequency from this equation and employing the definition of quality factor, an explicit solution is given for approximating TED value. The veracity of the proposed model is checked by comparing it with the solutions reported in the literature for specific and simpler cases. A diverse set of numerical results is then presented to appraise the influence of some factors like structural and thermal nonlocal parameters, strain gradient length scale parameter, geometrical parameters, mode number and material on the amount of TED. According to the results, use of NDPL model yields a smaller value for TED than DPL model, but prediction of NSGT about the magnitude of TED, in addition to the relative amounts of its two scale parameters, strongly depend on other factors such as aspect ratio, vibration mode and material type. [ABSTRACT FROM AUTHOR]

The vibration and stability of spinning viscoelastic Y-shaped bifurcated nanotubes conveying fluid in complex environments by considering additional concentrated masses and springs are studied based on the nonlocal strain gradient theory (NSGT). A detailed investigation is also performed to clarify the effect of influential parameters such as Knudsen number, magnetic nanoflow, scale parameter ratio, spin speed, fluid velocity, downstream bifurcation angle, viscoelastic coefficient, attached springs, localized masses, boundary conditions, and magneto-hygro-thermal environments on the system dynamics. The size-dependent dynamical equations of the system are derived utilizing Hamilton's principle. The Galerkin discretization scheme is adopted, and the eigenvalue problem is numerically solved. Campbell diagrams, forward and backward frequencies, divergence and flutter instability maps are acquired. Besides, the static instability threshold of the system is determined analytically. Results revealed that although the magnetic nanoflow has a decreasing effect on system vibrational frequencies, it delays the occurrence of the dynamical instability and prevents the buckling phenomenon. It is demonstrated that by considering simultaneous stiffness-softening effects induced by nonlocality and hygro-thermal environments, the flutter instability could occur instead of divergence condition in the system stability evolution. The present modeling and results could be applied as a benchmark for the performance improvement of innovative bi-gyroscopic nanofluidic devices. [ABSTRACT FROM AUTHOR]

This paper examines the free lateral vibration of a cracked nano-beam based on Euler-Bernoulli beam theory and nonlocal strain gradient theory (NSGT). Due to the importance and application of nanostructures, their mechanical and mechanical properties are essential. The governing equations and boundary conditions related to using the Hamilton principle have been extracted. The beam separation with the nano-beams division into two parts attached to the Torsion spring is modeled. The model calls the excess strain energy due to crack and increases the discontinuity in the deflection slope. This study investigated the effects of crack propagation, crack intensity, material length scale parameter, and various nonlocal parameters. A comparison of previous studies has been published, where a good agreement is observed. The results show that the parameters mentioned above play an important role in dynamical behavior. [ABSTRACT FROM AUTHOR]