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The paper presents a size-dependent high-order shear deformation theory (HSDT) model for static and free vibration and analyses of laminated composite and sandwich nanoplates based on the nonlocal strain gradient theory. To consider the size effect of nanostructures, two scale parameters having relationship with the nonlocal and strain gradient effects are introduced into the classical HSDT model. Due to these parameters, the increase and decrease in the stiffness of nanostructures are confirmed by adjusting these two ones. The virtual work principle is used in order to perform the weak forms, and the size-dependent bending and free vibration isogeometric analysis model are developed using the weak form. As observed numerical results, bending and free vibration characteristics of laminated composite and sandwich nanoplates are changed by the geometry, boundary condition, length-to-thickness ratio, strain gradient parameter and nonlocal parameter. In addition, the pure nonlocal, strain gradient and classical HSDT models can be retrieved from the present model when the strain gradient parameter, nonlocal parameter and these two parameters are taken equal to zero.

For the first time, buckling behavior of functionally graded (FG) nanoplates made of anisotropic material (beryllium crystal as a hexagonal material) is investigated. Also, it is the first time that the size-dependent behavior of nanostructured systems is studied for buckling response of the graded anisotropic material. The properties of graded material are assumed vary exponentially through the z-direction. Nonlocal strain gradient theory is utilized to predicate the size-dependent buckling behavior of the nanoplate. The nanoplate is modeled by a higher order shear deformation refined plate theory in which any shear correction factor not used. Governing equations and boundary conditions are obtained using a virtual work of variational approach. To solve the buckling problem for different boundary conditions, Galerkin’s approach is utilized. Finally, the influences of different boundary conditions, small-scale parameters, geometry parameters and exponential factor are studied and discussed in detail. It is hoped that the present numerical results can help the engineers and designers to understand and predict the buckling response of FG anisotropic materials.

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 (CoFe 2 O 4 ) and barium titanate (BaTiO 3 ). 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.

In this paper, a nonlocal strain gradient isogeometric model based on the higher order shear deformation theory for free vibration analysis of functionally graded graphene platelet-reinforced composites (FG GPLRC) plates is performed. Various distributed patterns of graphene platelets (GPLs) in the polymer matrix including uniform and non-uniform are considered. To capture size dependence of nanostructures, the nonlocal strain gradient theory including both nonlocal and strain gradient effects is used. Based on the modified Halpin–Tsai model, the effective Young’s modulus of the nanocomposites is expressed, while the Poisson’s ratio and density are established using the rule of mixtures. Natural frequencies of FG GPLRC nanoplates is determined using isogeometric analysis. The effects played by strain gradient parameter, distributions of GPLs, thickness-to-length ratio, and nonlocal parameter are examined, and results illustrate the interesting dynamic phenomenon. Several results are investigated and considered as benchmark results for further studies on the FG GPLRC nanoplates.

Many investigators have become interested in nanostructures due to their outstanding mechanical, chemical, and electrical properties. Two-dimensional nanoplates with higher mechanical properties compared with traditional structural applications are a common structure of nanosystems. Nanoplates have a wide range of uses in various sectors due to their unique properties. This paper focused on the static analysis of functionally graded (FG) nanoplates with porosities. The nonlocal strain gradient theory is combined with four-variable shear deformation theory to model the nanoplate. The proposed model captures both nonlocal and strain gradient impacts on FG nanoplate structures by incorporating the nonlocal and strain gradient factors into the FG plate’s elastic constants. Two different templates of porosity distributions are taken into account. The FG porous nanoplate solutions are compared with previously published ones. The impact of nonlocal and strain gradient parameters, side-to-thickness ratio, aspect ratio, and porosity parameter, are analyzed in detail numerically. This paper presents benchmark solutions for the bending analysis of FG porous nanoplates. Moreover, the current combination of the nonlocal strain gradient theory and the four-variable shear deformation theory can be adapted for various nanostructured materials such as anisotropic, laminated composites, FG carbon nanotube reinforced composites, and so on.

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 Al 2 O 3 /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.

Nonlocal strain gradient continuum mechanics is a methodology widely employed in the literature to assess size effects in nano-structures. Notwithstanding this, improper higher-order boundary conditions (HOBC) are prescribed to close the corresponding elastostatic problems. In the present study, it is proven that HOBC have to be replaced with univocally determined boundary conditions of constitutive type, established by a consistent variational formulation. The treatment, developed in the framework of torsion of elastic beams, provides an effective approach to evaluate scale phenomena in smaller and smaller devices of engineering interest. Both elastostatic torsional responses and torsional-free vibrations of nano-beams are investigated by applying a simple analytical method. It is also underlined that the nonlocal strain gradient model, if equipped with the inappropriate HOBC, can lead to torsional structural responses which unacceptably do not exhibit nonlocality. The presented variational strategy is instead able to characterize significantly peculiar softening and stiffening behaviors of structures involved in modern nano-electro-mechanical systems.

In this study, critical temperature and frequency characteristics of a doubly curved panel are reinforced by graphene nanoplatelets (GPLs) with the aid of a two-dimensional generalized differential quadrature method (2D-GDQM) are investigated. The size effects are included using nonlocal strain gradient theory (NSGT) that has two length scale parameters, and the panel is modeled as a panel using high order shear deformation theory (HSDT). The mechanical properties of GPLs are calculated based on the rule of mixtures and the modified Halpin–Tsai model. The novelty of the current study is in considering the effects of the thermal environment, various boundary conditions, and size effects on the frequency and critical temperature of the GPLRC panel. The validation is performed through the comparison of the numerical results for the frequency of the GPLRC panel and the literature. For more verification, a finite element model is presented using the finite element package to simulate the response of the current structure. The results created from a finite element simulation illustrate a close agreement with the numerical method results. The results demonstrate that GPLs’ weight function, the ratio of panel curvature (R1/R2), GPLs’ pattern, and size-dependent parameters have noticeable effects on the frequency and critical temperature characteristics of the GPLs-reinforced composite (GPLRC) curved panel. The favorable suggestion of this survey is that when designing the GPLRC structure, special attention should be paid to size-dependent parameters because the nonlocal and length scale parameters have an essential role in the static and dynamic behaviors of the GPLRC panel.

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.

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 f 14 , and therefore, the maximum deflection decreases as f 14 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.