Nonlocal Strain Gradient Theory Research Articles

Temperature-dependent thermal buckling and free vibration behavior of smart sandwich nanoplates with auxetic core and magneto-electro-elastic face layers

AbstractThis 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.

Nonlinear statics of magneto-electro-elastic nanoplates considering flexomagnetoelectric effect based on nonlocal strain gradient theory

The nonlinear static behavior of hygrothermal magnetoelectroelastic(MEE) nanoplates considering the flexomagnetoelectric (FME) effect is researched by engaging the first-order shear deformation theory (FSDT). The constitutive equations of MEE nanoplates take into account the FME effect and hygrothermal effect. Leveraging the FSDT, von Karman's nonlinear equation and Hamilton's principle, the nonlinear control equation for hygrothermal MEE nanoplates can be derived by using the variational approach. The nonlocal strain gradient theory (NSGT) has application in the size effect of MEE nanoplates. The nonlocal nonlinear term in the control equation is handled by employing the Airy stress function, and then the non-linear mechanical model is solved by the Galerkin method. Thus, the nonlinear load-deflection curves (NLDC) of the MEE nanoplate are obtained via the introduction of material parameters, enabling an examination of the impact of various factors such as the FME effect, two small size parameters of NSGT and other parameters on the nonlinear bending behavior of MEE nanoplates. In conclusion, this study offers a theoretical foundation for taking into account the FME effect in the development of nanodevices, thereby contributing to advancements in this field.

Electroelastic wave dispersion in the rotary piezoelectric NEMS sensors/actuators via nonlocal strain gradient theory

This article introduces a computational means for investigating the electroelastic nonlinear wave dispersion traits of the nano-dimension sandwich pipe, which is composed of a core formed of a bi-directional functionally graded (Bi-FG) material, together with a piezoelectric sensor/actuator. A combination of Hamilton’s principle, first-order shear deformation, along with Von-Karman nonlinearity, is used for modeling and obtaining the nonlinear governing equations of the nano-sized sandwich pipe connected to a piezoelectric part. The nonlinear governing equations to obtain the nonlinear phase velocity of the current system are determined using a combination of analytical and multiple scales approaches. Due to some computational cost for choosing the precise values of both the nonlocality factor and length scale of the nanopipe in the laboratory, for the first time in this research, with the aid of COMSOL multi-physics finite element simulation, the results are verified, and new outcomes for obtaining the exact functions for nonlocal and length scale factors is presented. In addition, an artificial neural network (ANN) is utilized in this study for the prediction of the results. The mathematical and finite element results were used to train the ANN. A newly presented optimization algorithm is exploited for the first time for optimizing the ANN parameters concluding higher accuracy of the ANN predictions. Consequently, to explore the influences of the location of the piezoelectric patch, nonlocality and length scale factors, and applied voltage parameter on the phase velocity characteristics of the nano-dimension sandwich pipe made of a Bi-FG core and electrically patch, an effort is performed. As an applicable result that can be useful in the related nano-industries, the current work presents exact nonlocal and length scale functions for different conditions.

Dynamic stability of the euler nanobeam subjected to inertial moving nanoparticles based on the nonlocal strain gradient theory

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.

Examining the critical speed and electro-mechanical vibration response of a spinning smart single-walled nanotube via nonlocal strain gradient theory

In 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.

Nonlocal strain gradient-based dynamic stability of functionally graded piezoelectric nanoshells considering thermo-electro-mechanical coupling effects

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.

The Nonlinear Bending of Sector Nanoplate via Higher-Order Shear Deformation Theory and Nonlocal Strain Gradient Theory

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.

Nonlinear free vibration of pre-buckled PFG micro/nanotubes via nonlocal strain and velocity gradient theory

Nonlinear free vibration of pre-buckled PFG micro/nanotubes via nonlocal strain and velocity gradient theory

Size effect on vibration properties of axially moving nanoplates under different boundary conditions

The nonlocal strain gradient theory is employed to investigate the transverse free vibration characteristics of two-dimensional nano-plates with axial velocities. A generalized Hamiltonian principle has been used to establish the vibration governing equations for the system as well as the corresponding boundary conditions. By applying complex modal analysis to three boundary conditions, the plate’s natural frequency is determined, including four-end simply supported, four-end clamped, and opposite-edge simply supported and clamped, and comparing the effect of the size parameters on the natural frequency in relation to the boundary conditions; based on different theories, the effects of changing boundary conditions on natural frequencies are systematically compared. In the numerical study, it is demonstrated that the size effect significantly influences only the self-oscillation frequency at the nanoscale, whereas the nonlocal parameter as well as the material characteristic parameter have “softening” and “hardening” effects on the equivalent stiffness of the nanoplates, respectively, which are directly related to their natural frequencies. Compared to simple supports, clamped boundary conditions are more significantly affected by size parameters. In addition, higher order frequencies exhibit greater sensitivity and are susceptible to changes in boundary conditions and size parameters.

Nonlinear guided waves in the sandwich nanostructure coupled with piezoelectric actuator: Introducing machine learning approach to measure nonlinear phase velocities

In today's rapidly changing corporate environment, business executives are turning to cutting-edge technology to solve complex problems efficiently. A machine learning approach allows for a more thorough understanding and quantification of decision-making outcomes, leading to increased confidence in business decisions. In this work for the first time, machine learning approach and mathematical simulation are used to model nonlinear guided waves in the sandwich nanostructure coupled with piezoelectric actuator. Using mathematical modeling, the current work studies the nonlinear phase velocity in a sandwich nanoshell composed of a bi-directional functionally graded (BD-FG) core and piezoelectric patch. The modeling uses first-order shear deformation theory with the fairly thick hypothesis to describe the displacement fields of piezoelectric layers. For core layer, a higher order shear deformation theory with terms of higher order in the Taylor series expansion is presented. The nanosystem is accurately modeled using nonlocal strain gradient theory, incorporating nonlocal and length scale characteristics. The solution approach section outlines the multiple scale method for the time domain and the harmonic solution methodology for the displacement domain to address nonlinear equations. To reduce the computational expenses, a machine-learning approach is introduced to address the nonlinear vibration issue in the current study. Validating the conclusions of the current study by comparing them with machine-learning solutions and another published paper can confirm their accuracy. The results section presents the impacts of factors like area of piezoelectric patch, length scale, applied voltage, nonlocality, geometry condition and FG power index on the nonlinear phase velocity of the BD-FG nanoshell coupled with piezoelectric patch.

Dynamic analysis of short-fiber reinforced composite nanobeams based on nonlocal strain gradient theory

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.

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

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

Vibration analysis of functionally graded carbon nanotubes reinforced composite nanoplates

This work presents the analytical analysis for free linear vibration behavior of functionally graded-carbon nanotubes reinforced composite (FG-CNTRC) nanoplates in the framework of nonlocal strain gradient theory (NSGT) and the first-order shear deformation plate theory (FSDPT). The nanoplate is considered made of a mixture of an isotropic polymer matrix and reinforced carbon nanotubes (CNTs). Four different distributions of CNTs are examined including uniformly distributed and FG reinforcements (FG-O, FG-X, and FG-V). The governing equations of motion are established based on the Hamilton’s principle. The closed-form analytical solution for the natural frequency of FG-CNTRC nanoplates with simply supported all edges is carried out by using the Navier-type solution. The impact of some key parameters on the natural frequencies of FG-CNTRC nanoplates is also studied and discussed. The result shows that FG-CNTRC nanoplates reveal the softening- or hardening-stiffness effects depending on the relationship between the nonlocal parameter and the material length scale parameter. The aspect ratios of FG-CNTRC nanoplates, the volume fraction, and the distribution pattern of CNTs have also an important impact on the vibration behavior of FG-CNTRC nanoplates.

Torsional vibration behavior of a restrained non-circular nanowire in an elastic matrix

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.

Size parameter calibration of nonlocal strain gradient theory based on molecular dynamics simulation of guided wave propagation in aluminum plates

The nonlocal strain gradient theory (NSGT) describes long-range interatomic interactions and higher-order strain gradients by introducing size parameters. However, the NSGT is constrained in studying the elastic wave propagation problems due to the difficulty in obtaining the precise size parameters. In this paper, the molecular dynamics (MD) simulations of the elastic wave propagation in the aluminum nano-plate are conducted, and the size parameters in the NSGT are calibrated based on the MD simulation results. The MD simulation process is divided into three stages: relaxation, excitation, and free vibration analysis. The embedded atom method is used to describe the interactions between the metal atoms. During the MD simulations, appropriate boundary conditions and excitation methods are proposed to obtain the group velocities of the elastic bulk, Lamb and SH waves. In addition, based on the three-dimensional (3D) elasticity theory and analytical integration Legendre polynomial method, the theoretical dispersion curves of the elastic wave propagation in the NSGT model are obtained. It is found that both the nonlocal theory and the NSGT can predict the guided elastic wave group velocity, and the NSGT prediction and MD simulation can be well matched when the size parameters are restricted to a certain calibrated regional band. The vibration of atoms in the Lamb wave propagation process is more complex than that in the SH wave propagation, leading to a more substantial size effect on the Lamb wave dispersions.

Size-dependent thermal-electro-mechanical behaviors of a piezo-flexoelectric micro-beam based on nonlocal strain gradient theory and dual-phase-lagging heat model

Size-dependent thermal-electro-mechanical behaviors of a piezo-flexoelectric micro-beam based on nonlocal strain gradient theory and dual-phase-lagging heat model

Scale-dependent torsional frequency analysis of right triangle anisotropic advanced composite microscale rods

Despite 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.

Wave propagation analysis in transversely isotropic piezoelastic medium based on nonlocal strain gradient theory

The purpose of this research is to study the propagation of surface waves in transversely isotropic piezoelastic medium based on nonlocal strain gradient theory. A characteristics equation for the existence of surface waves is discussed. This equation could be easily reduced to the ones of the gradient strain theory, nonlocal theory, and classical theory. It has also been concluded that there exist cut-off frequency for the wave propagating in size-dependent materials based on the nonlocal strain gradient theory. The dispersion equation which surface wave speed satisfies is derived from the free traction condition on the surface of half-space with consideration of electrically open circuit conditions. The effect of the nonlocal parameter, the strain gradient parameter on the existence of surface waves as well as the Rayleigh wave propagation is illustrated through some numerical examples.

Hygrothermal Effects on Vibration Response of Porous FG Nanobeams Using Nonlocal Strain Gradient Theory Considering Thickness Effect

In this work, the thickness-dependent size effect on nonlinear free vibration response of a Porous Functionally Graded (PFG) nanobeam in a hygrothermal environment is examined via nonlocal strain gradient theory (NSGT). Through the depth of the beam, hygro-thermal loadings are uniformly distributed. The mechanical properties of the beam are assumed to vary continuously in the thickness direction according to the power law expression. The equations of motion are derived using Hamilton’s principle with consideration of von-Kármán nonlinearity. The nanobeam vibration responses are obtained with the aid of the Galerkin technique and the Hamiltonian approach. A vital point of this work is to explore the effects of power law index, slenderness ratio, porosity volume fraction, and hygrothermal changes on the vibration behavior of PFG nanobeams. The results indicate how the thickness effect and porosity volume fraction significantly change the size-dependent vibration response of the FG nanobeam under hygrothermal conditions. The paper findings can be applied to improve the mechanical performance of PFG nanobeams.

Dynamic stability and frequency responses of the tilted curved nanopipes in a supersonic airflow via 2D hybrid nonlocal strain gradient theory

In the presented work, vibrational behavior related to a curved nanopipe which is under supersonic airflow and conveying fluid flow, is examined. To model the size-dependent nanopipe, the Quasi-2D hybrid type of nonlocal strain gradient theory (QHNSGT) is presented. Formulations are obtained by means of Hamilton’s principle for bi-directional functionally graded (Bi-FG) nanopipe. Also, formulations are solved by means of the generalized differential quadrature method (GDQM). By taking into account that the fluid flow is infinite, incompressible, uniform flow, Newtonian, laminar, as well, as viscous, and with the help of the Navier-Stokes equation, the fluid-structure interaction is obtained. A quasi-2D hybrid type of higher-order shear deformation theory is employed to introduce the displacement fields. The verification section shows that the results of this paper are very near to the results of the published article in the literature. One of the important findings of the current research is critical values of the Mach number could have increased with the aid of increasing the rigidity of edges and decreasing fluid flow velocity. Another marvelous output is that opening angle and airflow stability have an indirect relation, and increasing the opening angle provides an expansion in the unstable area.