Higher-order Nonlocal Strain Gradient Research Articles

The effect of the viscoelastic support and GRPL-reinforced foam material on the thermomechanical vibration response of piezomagnetic sandwich nanosensor plates

This paper investigates the vibration characteristics of a sandwich nanosensor plate composed of piezoelectric materials, specifically barium and cobalt, in the upper and lower layers, and a core material consisting of either ceramic (silicon nitride) or metal (stainless steel) foams reinforced with graphene (GPRL). The study utilized the novel sinosoidal higher-order deformation theory and nonlocal strain gradient elasticity theory. The equations of motion for nanosensor sandwich graphene were derived using Hamilton's principle, considering the thermal, electroelastic, and magnetostrictive characteristics of the piezomagnetic surface plates. These equations were then solved using the Navier method. The core element of the sandwich nanosensor plate can be represented using three distinct foam variants: a uniform foam model, as well as two symmetric foam models. The investigation focused on analyzing the dimensionless fundamental natural frequencies of the sandwich nanosensor plate. This analysis considered the influence of three distinct foam types, the volumetric graphene ratio, temperature variation, nonlocal parameters, porosity ratio, electric and magnetic potential, as well as spring and shear viscoelastic support. Furthermore, an analysis was conducted on the impact of the metal and ceramic composition of the central section of the sandwich nanosensor plate on its dimensionless fundamental natural frequencies. In this context, the use of ceramic as the central material results in a mean enhancement of 33% in the fundamental natural frequencies. In contrast, the incorporation of graphene into the core material results in an average enhancement of 27%. The thermomechanical vibration behavior of the nanosensor plate reveals that the presence of graphene-supported foam and a viscoelastic support structure in the core layer leads to an increase in thermal resistance. This increase is dependent on factors such as the ratio of graphene, porosity ratio of the foam, and parameters of the viscoelastic support. Metal foam or ceramic foam has been found to enhance thermal resistance when compared to solid metal or ceramic core materials. The analysis results showed that it is important to take into account the temperature-dependent thermal properties of barium and cobalt, which are piezo-electromagnetic materials, and the core layer materials ceramics and metal, as well as the graphene used to strengthen the core. The research is anticipated to generate valuable findings regarding the advancement and utilization of nanosensors, transducers, and nano-electromechanical systems engineered for operation in high-temperature environments.

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

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

On Vibration Responses of Advanced Functionally Graded Carbon Nanotubes Reinforced Composite Nanobeams

This article presents an analytical approach to explore the free vibration behaviour of new functionally graded carbon nanotube-reinforced composite beams (FG-CNTRC) based on a two-variable higher-order shear deformation theory and nonlocal strain gradient theory. The beams resting on the Pasternak elastic foundation, including a shear layer and Winkler spring, are considered. The kinematic relations of the shaft are proposed according to novel trigonometric functions. The vibrated nanobeam’s motion equations are obtained via the classical Hamilton’s principle and solved using Navier’s steps. A comparative evaluation of results against predictions from literature demonstrates the accuracy of the proposed analytical model. Moreover, a detailed parametric analysis checks for the sensitivity of the vibration response of FG nanobeams to nonlocal length scale, strain gradient microstructure scale, material distribution, constant spring factors, and geometry. The current work presents the free vibration problem of supported (FG-CNTRC) beams reinforced by different patterns of carbon nanotube (CNT) distributions in the polymeric matrix.

Bending Responses of Bi-Directional Advanced Composite Nanobeams Using Higher Order Nonlocal Strain Gradient Theory

The bending response of two-dimensional (2D) functionally graded (FG) nonlocal strain gradient nanobeams is explored analytically in this work. The longitudinal and transverse orientations vary in material gradation and material characteristics. Kinematic relations of nanobeams are proposed according to hybrid hyperbolic-parabolic functions. The virtual work principle obtains the equilibrium equations, which are then solved using Navier's method. The accuracy and dependability of the suggested analytical model are demonstrated by comparing the results to predictions made in the literature. A thorough parametric study also determines how sensitive the material distribution, the nonlocal length-scale parameter, the strain gradient microstructure-scale parameter, and the geometry are to how the bending response and stresses of 2D FG nanobeams. The results obtained provide benchmark results, which can be used in the design of composite structures.

On the dynamics and stability of size-dependent symmetric FGM plates with electro-elastic coupling using meshless local Petrov-Galerkin method

In present study, the equations of motion of sandwich symmetric functionally graded size-dependent plates integrated with piezoelectric layers are derived using Hamilton’s variational principle (with modified Lagrangian) based on a higher-order nonlocal strain gradient theory in conjunction with Reddy’s third-order shear deformation plate kinematics. For the first time, a set of sixth-order partial differential equations are derived and solved by meshless local Petrov-Galerkin method to investigate the bifurcation buckling and free vibration of plates under electromechanical in-plane forces with considered nanostructure and various boundary conditions. The shifted and scaled moving least-squares approach is utilized for approximation of unknown variables, and the Gauss weight function is used as test function for deriving local weak form of governing equations. The Gauss-Legendre quadrature is employed for numerical evaluation of weak forms. A power-law distribution and a half cosine variation are applied to model the variation of materials properties and electric potentials, respectively. The obtained results are verified with well-known solutions in the literature. The effect of nonlocal parameter, material length scale parameter, power-law index, predefined electric field, axial compressive and tensile forces, volume ratio of FG core and piezoelectric layers, and stiffness of Pasternak foundation on the critical buckling loads and natural frequencies are presented and discussed comprehensively. The results obtained can be used in the analysis, design and control of composite nanostructures that are widely used in MEMS and NEMS devices.

On the Nonlocal Interaction Range for Stability of Nanobeams with Nonlinear Distribution of Material Properties

Abstract The present study analyses the range of nonlocal parameters’ interaction on the buckling behaviour of nanobeam. The intelligent nonhomogeneous nanobeam is modelled as a symmetric functionally graded (FG) core with porosity cause nonlinear distribution of material parameters. The orthotropic face-sheets are made of piezoelectric materials. These kinds of structures are widely used in nanoelectromechanical systems (NEMS). The nanostructure model satisfies the assumptions of Reddy third-order beam theory and higher-order nonlocal elasticity and strain gradient theory. This approach allows to predict appropriate mechanical response of the nanobeam regardless of thin or thick structure, in addition to including nano-sized effects as hardening and softening. The analysis provided in the present study focuses on differences in results for nanobeam stability obtained based on classical and nonlocal theories. The study includes the effect of diverse size-dependent parameters, nanobeams’ length-to-thickness ratio and distributions of porosity and material properties through the core thickness as well as external electro-mechanical loading. The results show a dependence of nonlocal interaction range on geometrical and material parameters of nanobeam. The investigation undertaken in the present study provides an interpretation for this phenomenon, and thus aids in increasing awareness of nanoscale structures’ mechanical behaviour.

Free Vibration of FG-CNTRCs Nano-Plates/Shells with Temperature-Dependent Properties

This article presents a mathematical continuum model to analyze the free vibration response of cross-ply carbon-nanotube-reinforced composite laminated nanoplates and nanoshells, including microstructure and length scale effects. Different shell geometries, such as plate (infinite radii), spherical, cylindrical, hyperbolic-paraboloid and elliptical-paraboloid are considered in the analysis. By employing Hamilton’s variational principle, the equations of motion are derived based on hyperbolic sine function shear deformation theory. Then, the derived equations are solved analytically using the Galerkin approach. Two types of material distribution are proposed. Higher-order nonlocal strain gradient theory is employed to capture influences of shear deformation, length scale parameter (nonlocal) and material/microstructurescale parameter (gradient). Temperature-dependent material properties are considered. The validation of the proposed mathematical model is presented. Detailed parametric analyses are carried out to highlight the effects of the carbon nanotubes (CNT) distribution pattern, the thickness stretching, the geometry of the plate/shell, the boundary conditions, the total number of layers, the length scale and the material scale parameters, on the vibrational frequencies of CNTRC laminated nanoplates and nanoshells.

Mechanical models and numerical simulations in nanomechanics: A review across the scales

This work gives an overview of the theoretical background and of the numerical modelling framework used to describe the mechanical properties and the response of materials on scales ranging from the atomistic, through the microstructure and all the way up to the macroscale. In order to describe the dual nature of the structure of matter, which is continuous when viewed at large length scales and discrete when viewed at the atomic scale, plus the interdependence of these scales, multiscale modelling is required to complement the continuum and the atomistic models. More specifically, what we aim for in this review is to present and discuss the following basic conceptual models, as well as the methodologies that accompany them: (a) discrete models such as ab initio, atomistic / molecular, mesoscopic; (b) continuum mechanics models (CMM) comprising pure CMM, non-local elasticity CMM, higher-order strain gradient and higher-order nonlocal strain gradient elasticity CMM, and surface elasticity CMM; (c) multiscale material models (MMM). Since the field of nanomechanics is currently a rapidly expanding research area, the presented state-of-the art is by no means exhaustive. It simply outlines the research efforts that go behind formulating numerical models for the solution of problems in nanomechanics. Despite the advantages that boundary element methods (BEM) have in solving problems at the physical scale, either as stand-alone or in combination with finite element methods (FEM), their application to multiscale modelling is still limited, despite the promise they seem to hold.

On the piezoelectric effect on stability of symmetric FGM porous nanobeams

In present paper, the piezoelectric effect on bifurcation buckling of symmetric FGM porous nanobeam is presented based on a higher-order nonlocal elasticity and strain gradient theory in conjunction with Reddy third-order shear deformation beam theory. The strain energy density and electric enthalpy density functions are used to derive constitutive relations for porous functionally graded, as well as piezoelectric materials. Equations of motion for elastically supported layered FGM nanobeam with diverse distribution of porosity and electro-elastic coupling are derived based on the modified Hamilton’s variational principle. The formulated boundary value problem is analytically solved. For the first time, the effects of porosity distribution, volume of voids, distance between porosity and FGM core surfaces as well as material gradation, layers size, aspect ratio, stiffness of Kerr foundation, and external electric voltage on critical in-plane force, critical porosity and critical voltage were comprehensively presented and discussed. The investigation enables analysis and precise multi-dimensional control of static response of smart beam-like nanostructures widely used in MEMS and NEMS devices.

Higher order nonlocal viscoelastic strain gradient theory for dynamic buckling analysis of carbon nanocones

The dynamic buckling in viscoelastic carbon nanocones (CNCs) under the magnetic and thermal loads is considered in this article. This class of nano-structures has great application in nano-electro-mechanical system (NEMS) such as scanning probe microscopy. Structural damping influences are studied according to approach of Kelvin–Voigt. The viscoelastic medium around the CNC is assumed applying the model of visco-Pasternak. Utilizing the first order shear deformation theory (FSDT), Hamilton's principle, method of energy, the motion equations are determined where the influences of size are captured by higher order nonlocal strain gradient approach. Method of Bolotin and differential quadrature method (DQM) are utilized to solve the equations of motion in order to access the formation of dynamic instability region (DIR). The influences of different factors like, boundary conditions, nonlocal parameters, structural damping, magnetic field, viscoelastic medium, temperature changes along with CNCs semi vertex angle on the DIR are investigated. The outcomes show that with rising strain gradient parameter, the DIR will be occurred at high frequencies. It is furthermore found that the magnetic load has a positive influence on the DIR of the nano-structure.

A numerical method for magneto-hygro-thermal dynamic stability analysis of defective quadrilateral graphene sheets using higher order nonlocal strain gradient theory with different movable boundary conditions

The objective of this novel research is assessment of dynamic stability of a viscoelastic defective single layered graphene sheet (SLGS). Further, various kinds of defects including single vacancy (SV), double vacancy (DV) and Stone-Wales (SW) are regarded and the effects of vacancy defect reconstruction is studied as well. So as to assume this nanostructure much more realistic, Kelvin model is utilized and it is rested upon elastic foundation. This SLGS is under hygrothermal loads and in-plane magnetic forces. In addition to sinusoidal higher order shear deformation theory for modeling this nanostructure mathematically, Hamilton's principle is utilized to obtain the motion equations and furthermore nonlocal strain gradient theory (SGT) is used which considers size effects accurately. Differential quadrature method (DQM) as well as transformed weighing (TW) will be handled to solve the governing equations and determine the dynamic stability region for this nanostructure. Moreover, various moving boundary edges are studied because of hygrothermal loads. Various parameters focusing on hygrothermal loads, magnetic forces, defect kinds, defect degree, various SLGS, boundary edges and coefficients of elastic foundation and their effects upon the dynamic stability behavior of nanostructure will be investigated. In order to assess the validity, the results are compared with other papers. Based on the results, rise of defect degree causes shift of stability region to the less excitation frequencies.

On the bifurcation buckling and vibration of porous nanobeams

In the present paper, effects of porous material on bifurcation buckling and natural vibrations of nanobeams are investigated based on the higher-order nonlocal strain gradient theory. The displacement field of the nanobeam satisfies assumptions of Reddy higher-order shear deformation beam theory. The displacements gradients are assumed to be small so that the components of the Green-Lagrange strain tensor are linear and infinitesimal. The constitutive relations for functionally graded porous material are expressed by nonlocal and length scale parameters and power-law variation of material parameters in conjunction with cosine functions to create a possibility to investigate the effect of diverse distributions of porosity on mechanics of nanostructures. Effect of Winkler-Pasternak foundation on mechanics of nanobeam is also considered. The Hamilton’s variational principle is utilized to derive governing equations of motion of the composite nanobeam. For the first time, the critical porosity is defined and examined for bifurcation buckling analysis of elastically supported nanobeams with symmetric distribution of porosity. Influence of axial forces and types of porosity distributions on eigenfrequencies of functionally graded nanobeams is studied. Classical theory without nonlocal effects is obtained as a special case and valid for all considered distributions of functionally graded material and volume fractions of voids.

Bending of Functionally Graded Sandwich Nanoplates Resting on Pasternak Foundation under Different Boundary Conditions

This article proposes a refined higher order nonlocal strain gradient theory for stresses and deflections of new model of functionally graded (FG) sandwich nanoplates resting on Pasternak elastic foundation. Material properties of the FG layers are supposed to vary continuously through-the-thickness according to a power function or a sigmoid function in terms of the volume fractions of the constituents. The face layers are made of FG material while the core layer is homogeneous and made of ceramic. In this study, an analytical approach is proposed using the higher-order shear deformation plate theory and nonlocal strain gradient theory with combination of various boundary conditions. Numerical outcomes are reported to display the impact of the material distribution, boundary conditions, elastic foundation parameters and the sandwich nanoplate geometry on the deflections and stresses of FG sandwich nanoplates. The exactness of this theory is determined by comparing it to other published outcomes.

Thermal buckling and free vibration of Euler–Bernoulli FG nanobeams based on the higher-order nonlocal strain gradient theory

A size-dependent Euler–Bernoulli beam model is derived within the framework of the higher-order nonlocal strain gradient theory. Nonlocal equations of motion are derived by applying Hamilton’s principle and solved with an analytical solution. The solution is obtained using the Navier solution procedure. In the case of simply supported boundary conditions, the analytical solutions of natural frequencies and critical buckling temperature for free vibration problems are obtained. The paper investigates the thermal effects on buckling and free vibrational characteristics of functionally graded size-dependent nanobeams subjected to various types of thermal loading. The influence of higher-order and lower-order nonlocal parameters and strain gradient scale on buckling and vibration are investigated for various thermal conditions. The obtained results are compared with previous research.

Thermal buckling and free vibration of Timoshenko FG nanobeams based on the higher-order nonlocal strain gradient theory

Thermal buckling and free vibration of Timoshenko FG nanobeams based on the higher-order nonlocal strain gradient theory

Buckling analysis of porous FGM sandwich nanoplates due to heat conduction via nonlocal strain gradient theory

This article deals with the analysis of buckling behavior of simply supported porous functionally graded (FG) sandwich nanoplates resting on a Kerr foundation. The material properties of the FG sandwich nanoplate considered to be dependent on temperature and graded continuously along the thickness direction. The analytical equations are obtained using Hamilton’s variational principle, by considering the strain energies due to the thermal loads, and higher order nonlocal strain gradient plate theory which captures the shear deformation influences needless of any shear correction factor. Power-law model is adopted to describe continuous variation of material properties of FG sandwich nanoplate. An accurate solution for nonlinear temperature variation across the sandwich nanoplate thickness of is employed taking into account the thermal conductivity, the inhomogeneity parameter and the sandwich schemes. The numerical results computed indicate the influence of volume fraction index, nonlocal parameter, length scale parameter, porosity coefficient, Kerr foundation and temperature difference on the buckling response.

Hygrothermal free vibration of multiple magneto-electro-elastic nanoplate system via higher-order nonlocal strain gradient theory

The free vibration behavior of the coupled multiple magneto-electro-elastic nanoplate system embedded in Kelvin–Voigt visco-Pasternak medium incorporating hygrothermal effects is analyzed using the higher-order nonlocal strain gradient assumption. It should be noted that to obtain a more precise modeling of small-scale structures when using higher-order nonlocal strain gradient theory, both hardening and softening effects of materials are considered. The governing equilibrium equations are derived based upon the classical Kirchhoff thin plate approach and Hamilton’s variational principle. The complex eigenvalues of the system are presented in the exact closed-form solutions and are validated by numerical results found from the trigonometric technique. Numerical studies are carried out to demonstrate the effects of the number of magneto-electro-elastic nanoplates, material length scale parameters, initial external electric and magnetic fields, visco-Pasternak foundation, and hygrothermal loading on how the system damped frequency curves are varied.

Global perspectives on Higher Education: A social constructivist case for diversification of Complementary Medicine curricula

In the present investigation, a new first-order shear deformation theory (OVFSDT) on the basis of the in-plane stability of the piezo-magnetoelectric composite nanoplate (PMEN) has been developed, and its precision has been evaluated. The OVFSDT has many advantages compared to the conventional first-order shear deformation theory (FSDT) such as needless of shear correction factors, containing less number of unknowns than the existing FSDT and strong similarities with the classical plate theory (CPT). The composite nanoplate consisted of BaTiO3-CoFe2O4, a kind of material by which coupling between piezoelectric and piezomagnetic in nanosize was established. The plate is surrounded by a motionless and stationary matrix that is embedded in a hygrothermal surround in order to keep it more stable, and to take into consideration the influences of the moisture and temperature on the plate's mechanical behavior. The governing equilibrium equations for the smart composite plate have been formulated using the higher-order nonlocal strain gradient theory within which both stress nonlocality and second strain gradient size-dependent terms are taken into account by using three independent length scale parameters. The extracted equations are solved by utilizing the analytical approaches by which numerical results are obtained with various boundary conditions. In order to evaluate the proposed theory and methods of solution, the outcomes in terms of critical buckling loads are compared with those from several available well-known references. Finally, after determining the accuracy of the results of the new plate theory, several parameters are investigated to show the influences of material properties of the ceramic composite nanoplate on the critical buckling loads.

A numerical method for magneto-hygro-thermal postbuckling analysis of defective quadrilateral graphene sheets using higher order nonlocal strain gradient theory with different movable boundary conditions

Many researchers work on the mechanical analysis of perfect graphene sheets while these structures have some defects actually. In this paper, postbuckling of the defective quadrilateral single layered graphene sheets (SLGS) is presented for the first time. Three types of defects including single vacancy (SV), double vacancy (DV) and Stone–Wales (SW) in conjunction with the influence of vacancy defect reconstruction are considered. Nonlocal higher order strain gradient theory with three parameters for size effects is used. The quadrilateral graphene sheet is subjected to temperature, moisture and in-plane magnetic loads. The elastic foundation is modelled by Pasternak medium. The equations of motion are obtained imposing higher order shear deformation theory and Hamilton’s principle. The transformed weighing (TW) and differential quadrature (DQ) method are applied for postbuckling load–deflection relation of the structure. Due to the hygrothermal load, different movable boundary conditions are used. The influences of various parameters such as defect types, defect degree, defect reconstruction, hygrothermal load, magnetic field, nonlocal parameters, 7 different quadrilateral SLGS, boundary conditions and elastic medium on the postbuckling behaviour of the quadrilateral graphene sheet are shown. The numerical results are validated with other published works in the literature. Results show that with increasing the defect degree, the buckling load decreases.

Geometrically nonlinear vibration analysis of sandwich nanoplates based on higher-order nonlocal strain gradient theory

In this paper, a new model for studying the effects of small-scale parameters simultaneously, on large amplitude vibrations of sandwich plates is developed using the higher-order nonlocal strain gradient theory. Considering the higher-order theories for capturing the size effects of nanostructures results in a set of nonlinear partial differential (PD) equations, including bi-nonlocal terms. By employing Hamilton's principle, the equations of motion for symmetric and anti-symmetric sandwich plates are derived based on the classical plate theory. The partial nonlinear differential equations of motion are reduced to an ordinary differential equation for transverse vibrations of nanoplates using the Galerkin's method. An analytical solution procedure is employed to obtain the closed-form frequency equation as a function of the vibration amplitude, small-scale parameters and sandwich layers elasticity, density and thickness coefficients. Numerical results are presented in order to investigate the sandwich layers coefficients on nonlinear vibrational behavior of nanoplates as same as small-scale parameters and the amplitude of vibrations. It is found that the vibration amplitude plays the main role in nonlinear vibrational behavior of nanoplates in which, nonlinear frequency and its ratio to linear frequency will be increased by increasing it. Moreover, there are non-uniform behaviors by increasing the sandwich layers coefficients and small-scale parameters. In addition, in the case of large amplitude vibrations, effects of sandwich layers’ coefficients and small-scale parameters on the nonlinear frequency and its ratio to linear frequency will become more noticeable. In order to validate the present solution procedure, the results are compared with those obtained from molecular dynamics simulations, the higher-order nonlocal strain gradient theory and the higher-order shear deformation plate theory.