Functionally Graded Nanoplates Research Articles

A size-dependent nonlinear isogeometric approach of bidirectional functionally graded porous plates

Recent progress in materials science and fabrication technologies, such as additive manufacturing (AM) or 3D printing, has facilitated the design and production of intricate multi-directional functionally graded (FG) materials which are playing a pivotal role in interdisciplinary engineering applications. Developing mathematical modeling for such nanostructures, however, remains challenging. The primary objective of this research, therefore, is to present a mathematical model for studying the static bending characteristics considering the geometric nonlinearity of bidirectional FG nanoplates with internal pores. The mathematical formulations are established by utilizing the first-order shear deformation theory and the von-Kármán assumption within the framework of an isogeometric approach based on NURBS basis functions. To account for the influence of both strain gradient and nonlocal elasticity, we adopt the nonlocal strain gradient theory including two independent material length scale parameters to predict the geometric nonlinearity performance of structures at small-scale levels. The mechanical properties of multi-directional FG material models are assumed to be continuously graded in two directions based on power law, while internal pores can be dispersed into two typical distribution patterns. Several numerical investigations are performed to evaluate the substantial impact of certain parameters on the nonlinear deflections of bidirectional FG nanoplates with pores under various conditions. Compared to existing studies, the key contribution is to present a robust numerical model aimed at elucidating both the stiffness-softening and stiffness-hardening mechanisms of multi-directional FG nanostructures under static bending conditions with geometric nonlinearity. These findings enhance our insights into nonlinear bending performance and contribute valuable design guidelines for future small-scale engineering structures. These innovative designs have become popular and are now crucial in state-of-the-art engineering applications, such as aerospace, nuclear systems, or thermal resistance devices.

A comprehensive investigation of the bending and vibration behavior of size-dependent functionally graded nanoplates via an enhanced nonlocal finite element shear model

This research paper conducts a comprehensive investigation into the linear bending and free vibration of size-dependent functionally graded (FG) nanoplates using an improved first-order shear deformation theory (IFSDT). The present IFSDT offers an enhanced representation and precise computation of normal and shear stresses across the thickness of the nanoplates. Notably, it not only ensures compliance with free conditions on both the upper and lower surfaces but also eliminates the need for a conventional correction factor commonly employed in the first-order shear deformation theory (FSDT). To transcend the assumptions of classical continuum mechanics and address the impacts of small-scale effects in discrete nanoplates, Eringen’s nonlocal elasticity theory is applied. The formulation of the governing equation for the bending and free vibration analyses of the FG nanoplates is achieved through the application of Hamilton’s principle. The proposed IFSDT is implemented with an efficient C0-continuous quadrilateral element which is suitable for analysis of structures with arbitrary shapes and complex forces. The model’s performance is showcased through a comparative evaluation against literature predictions, highlighting its high accuracy and rapid convergence. Additionally, the research scrutinizes the effects of various parameters, such as plate thickness, boundary conditions, aspect ratio, nonlocal parameter, different material compositions, and power-law index on the deflections, stresses, and naturel frequencies of the FG nanoplates. The results underscore the significant impact of size-dependent effects on the linear bending and vibration behaviors of the nanoplates.

The role of nonlocality on low and high frequency behaviors of functionally graded sandwich nanoplates

AbstractThe role of nonlocality on low and high frequency behaviors and modes of the functionally graded (FG) sandwich nanoplates is investigated in this study for the first time using simple nonlocal higher‐order shear deformation theory (HSDT). The simple HSDT consists of only four unknowns in its equation of the displacement field. The nonlocal elasticity theory is used to consider the small‐scale effects on the behaviors of the nanoplates. The governing equations of motion are established by applying Hamilton's principle, then Navier's solution technique is employed to solve these equations to achieve the free vibration behaviors of the FG sandwich nanoplates. The vibrations of the nanoplates under both low and high frequency conditions are investigated, but the high frequency vibration of the nanoplates is discussed extensively. The influences of the geometrical dimensions, material gradient index, and nonlocal parameter on the high frequency vibration of the FG nanoplates are also considered and discussed comprehensively.

A novel dynamic stiffness matrix for the nonlocal vibration characteristics of porous functionally graded nanoplates on elastic foundation with small-scale effects

The present paper deals with the development of a dynamic stiffness matrix to evaluate the free vibration response of functionally graded nanoplate (FG-nP) resting on the Winkler-Pasternak elastic foundation. The complete mathematical modeling of the dynamic stiffness matrix for nanostructures is given for the first time. The equation of motion for rectangular FG-nP plates supported on an elastic foundation is derived using Hamilton’s principle in conjunction with nonlocal elasticity theory. The non-local theory is incorporated to account for the size effect in the small-scale plate. The effective material property of the porous FG-nP has been calculated using three recently developed models of porosity. The developed dynamic stiffness matrix is solved using the Wittrick-Williams algorithm to extract the natural frequencies of the FG-nP. The variation of natural frequencies with the change of numerical values, such as nonlocal parameter, aspect ratio, elastic foundation parameters, and porosity volume fraction is analyzed. The validity and accuracy of the results are confirmed through comparison with the available literature. The use of non-local theory in dynamic stiffness analysis is shown to be effective in predicting the natural frequency of the FG-nP on a Winkler-Pasternak elastic foundation, providing new insights into the dynamic behavior of small-scale structures.

Porosity-dependent wave propagation in multi-directional functionally graded nanoplate with nonlinear temperature-dependent characteristics on Kerr-type substrate

The present study investigates the wave propagation characteristics of porous, multi-directional, functionally graded (FG) nanoplates embedded in a Kerr-elastic substrate. A first-order shear deformation model (FSDM) and the nonlocal strain gradient approach (NSGA) are utilized to address this issue. The NSGA is enhanced by considering softening and hardening material effects, leading to improved accuracy in the obtained results. In accordance with the law of mixing, the material properties of the nanoplates exhibit nonlinear temperature-dependent variations along the structural axes. Both even and uneven porosity profiles are considered in this study. The nonclassical governing equations for the proposed structure are derived using Hamilton's approach. The phase velocity and wave frequency are determined through the analytical solution of an eigenvalue problem, which depends on the wave number. The dispersion properties of waves in porous FG nanoplates are examined with respect to various factors, including porosity distributions, wave numbers, nonlocal and strain gradient effects, elastic foundation coefficients, and volume fraction index. The main findings of the study indicate that increasing the volume fraction index in the thickness direction leads to a decrease in the phase velocity and frequency of wave propagation in a porous FG nanoplate for a given wave number. Additionally, regardless of the porosity pattern, an increase in the porosity coefficient results in a decrease in wave frequency and phase velocity.

Nonlocal Strain Gradient Theory for the Bending of Functionally Graded Porous 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.

Predicting elemental stiffness matrix of FG nanoplates using Gaussian Process Regression based surrogate model in framework of layerwise model

The accuracy of predicting the behaviour of structure using finite element (FE) depends widely on the precision of the evaluation of the stiffness matrix. In the present article, an attempt has been made to evaluate the stiffness matrix of functionally graded (FG) nanoplate using Gaussian process regression (GPR) based surrogate model in the framework of the layerwise theory. The stiffness matrix comprises various matrix terms corresponding to the membrane, membrane-bending, bending-membrane, and bending and shear. Following two different methodologies are adopted for predicting the stiffness matrix at the elemental level, one in which the final elemental stiffness matrix is evaluated, and the second one in which all the matrix terms as stated are evaluated separately using the GPR surrogate model and then are added to get the final stiffness matrix at the elemental level. The effectiveness of both approaches has been worked out by comparing the present results with those available in the literature. Both the proposed methodologies can predict the behaviour of FG nanoplates with good accuracy. However, the second one is found to be outstanding.

Nonlocal vibration of functionally graded nanoplates using a layerwise theory

This work studies the size-dependent free vibration response of functionally graded (FG) nanoplates using a layerwise theory. The proposed model supposes not only a higher-order displacement field for the core but also the first-order displacement field for the two face sheets, thereby maintaining an interlaminar displacement continuity among layers. Unlike the conventional layerwise models, the number of variables is kept fixed and does not increase for an increased number of layers. This is a very important feature compared to conventional layerwise models and facilitates significantly the engineering analyses. The material properties of the FG nanoplate are graded continuously through the thickness direction in accordance with a power-law function. The Eringen’s nonlocal elasticity theory is here adopted to relax the continuum axiom required in classical continuum mechanics and hence hopeful to capture the small size effects of naturally discrete nanoplates. The equations of motion of the problem are obtained via a classical Hamilton’s principle. The present layerwise model is implemented with a computationally efficient C0- continuous isoparametric serendipity elements and applied to solve a large-scale discrete numerical problem. The robustness and reliability of the developed finite element model are demonstrated by a comparative evaluation of results against predictions from literature. The comparative studies show that the proposed finite element model is: (a) free of shear locking, (b) accurate with a fast rate convergence for both thin and thick FG nanoplates, and (c) excellent in terms of numerical stability. Moreover, a detailed parametric analysis checks for the sensitivity of the vibration response of FG nanoplates to the aspect ratio, length-to-thickness ratio, nonlocal parameter, boundary conditions, power-law index, and modes shapes. Referential results are also reported, for the first time, for natural frequencies of FGM nanoplates which will serve as benchmarks for further computational investigations.

Wave propagation analysis of functionally graded nanoplates using nonlocal higher-order shear deformation theory with spatial variation of the nonlocal parameters

In the present paper, an efficient higher-order shear deformation theory is proposed and applied in conjunction with nonlocal elasticity theory to analyze the wave propagation of infinite functionally graded (FG) nanoplates. The novelty of this work is that the effect of the spatial variation of the nonlocal parameters in the wave propagation of the FG nanoplates is considered for the first time. The nonlocal parameters and other material properties are assumed to be material-dependent and vary smoothly across the thickness of the nanoplates. Hamilton’s principle is applied to establish the governing equations of motion. The Navier closed-form solution is used to solve the propagating wave equations. The accuracy and robustness of the proposed algorithm are demonstrated by comparing the present results with those available in the literature. Moreover, a detailed parametric study is carried out to highlight the effect of the material graded index, the spatial variation of the nonlocal parameters on the wave propagation of the functionally graded nanoplates. The outcome of this study highlights the significant effects of spatial variation of the nonlocal parameters on the wave propagation of FG nanoplates.

Flexoelectricity and surface effects on coupled electromechanical responses of graphene reinforced functionally graded nanocomposites: A unified size-dependent semi-analytical framework

Owing to inhomogeneous strain and high surface-to-volume ratio in nanostructures, it is imperative to account for the flexoelectricity as well as surface effect while analyzing the size-dependent electromechanical responses of nano-scale piezoelectric materials. In this article, a semi-analytical ‘single-term extended Kantorovich method (EKM)’ and ‘Ritz method’ based powerful framework is developed for investigating the static and dynamic electromechanical responses of graphene reinforced piezoelectric functionally graded (FG) nanocomposite plates, respectively. The residual surface stresses, elastic and piezoelectric surface modulus, and direct flexoelectric effects are taken into account while developing the unified governing equations and boundary conditions. The modified Halpin Tsai model and rules of mixture are implemented to predict the effective bulk properties. Our results reveal that the static deflection and resonance frequency of the proposed FG nanoplates are significantly influenced due to the consideration of flexoelectricity and surface effects. While such outcomes emphasize the fact that such effects cannot be ignored, these also open up the notion of on-demand property modulation and active control. The effects are more apparent for nanoplates of lesser thickness, but they diminish as plate thickness increases, leading to the realization and quantification of a size-dependent behavior. Based on the developed unified formulation, a comprehensive numerical investigation is further carried out to characterize the electromechanical responses of nanoplates considering different critical parameters such as plate thicknesses, aspect ratios, flexoelectric coefficients, piezoelectric multiples, distribution, and weight fraction of graphene platelets along with different boundary conditions. With the recent advances in nano-scale manufacturing, the current work will provide the necessary physical insights in modeling size-dependent multifunctional systems for active control of mechanical properties and harvesting electromechanical energy.

A Quasi-3D Higher-Order Theory for Bending of FG Nanoplates Embedded in an Elastic Medium in a Thermal Environment

This paper presents the effects of temperature and the nonlocal coefficient on the bending response of functionally graded (FG) nanoplates embedded in an elastic foundation in a thermal environment. The effects of transverse normal strain, as well as transverse shear strains, are considered where the variation of the material properties of the FG nanoplate are considered only in thickness direction. Unlike other shear and deformations theories in which the number of unknown functions is five and more, the present work uses shear and deformations theory with only four unknown functions. The four-unknown normal and shear deformations model, associated with Eringen nonlocal elasticity theory, is used to derive the equations of equilibrium utilizing the principle of virtual displacements. The effects due to nonlocal coefficient, side-to-thickness ratio, aspect ratio, normal and shear deformations, thermal load and elastic foundation parameters, as well as the gradation in FG nanoplate bending, are investigated. In addition, for validation, the results obtained from the present work are compared to ones available in the literature.

On the buckling and bending analysis of FG straight-sided quadrilateral nanoplates using a continuum mechanics-based surface elastic model

In this research, an elastic model based on the continuum mechanics is developed to study the static behaviors of functionally graded (FG) arbitrary straight-sided quadrilateral nanoplates. The model is constructed in the framework of Gurtin-Murdoch’s surface and Mindlin’s plate theories to account for the surface energy and shear deformation effects, simultaneously. The variational differential quadrature (VDQ) method is used along with a mapping technique to do the discretization process in a variational framework by means of differential and integral operators. Consequently, a weak form of governing equations is obtained from the energy quadratic representation of the problem. The solution method is of a distinguished feature as it involves just the first-order derivative of the field components in the mapping and discretization. After assuring the effectiveness of presented model by doing comparative studies, the critical buckling load and static deflection of the FG nanoplates with different shapes in geometry are investigated considering the surface effects. It is found that the surface energies effect on the static behavior of the rectangular nanoplates is more significant as compared to the non-rectangular nanoplates.

Analysis of Axisymmetric Vibration of Functionally-Graded Circular Nano-Plate Based on the Integral Form of the Strain Gradient Model

In this paper, it is aimed to analyze the linear vibrational behavior of functionally-graded (FG) size-dependent circular nano-plates using the integral form of the non-local strain gradient (NSG) model. The linear axisymmetric vibration of the circular FG nano-plates based on the non-local strain gradient (NSG) model is the focal point of this study. In this regard, the non-local elasticity theory (NET) and strain gradient (SG) models are used in conjunction with Hamilton's principle to obtain the governing equations. Discretization of the obtained governing equations is performed with the help of generalized differential quadrature rule (GDQR) and Galerkin weighted residual method (GWRM). The analysis is focused on the effect of non-local and material parameters, as well as the aspect ratio, heterogeneity index of FG material, different boundary conditions, and frequency number on the overall behavior of nano-plate. On using the Galerkin method, a system of linear differential equations is obtained and solved to determine the natural linear frequencies and mode shapes. The obtained results are then compared with the existing results in the literature. On using the proposed procedure in this paper, the dynamic behavior of nano-plate under different boundary conditions can be well described. In addition, the existing deficiencies in other non-local theories can be eliminated. The results of this investigation can be considered as a turning point in the improvement of theoretical results for achieving a better prediction of vibrational behavior in nanostructures.

Size-dependent vibration response of porous graded nanostructure with FEM and nonlocal continuum model

In the present paper, a refined trigonometric higher-order shear deformation theory has been presented with the conjunction of nonlocal theory for the vibrational response of functionally graded (FG) porous nanoplate. The displacement field is chosen based on assumptions that the out of the plane displacement consists of bending and shear components whereas the transverse shear-strain has nonlinear variation along the thickness direction. The number of unknown variables is four, as against five in other renowned shear deformation theories. The governing equations have been derived using Hamilton's principle. A generalized porosity model has also been developed to accommodate both even and uneven type of distribution of porosity in the FG nanoplates. The closed-form solution of simply-supported FG porous nanoplates is obtained and the results are compared with the available reported results. In finite element solution, a C0 continuous isoparametric quadrilateral element has been used with various conventional and unconventional boundary conditions. The effects of various parameters like small-scale effect, aspect ratio, volume fraction index, porosity volume fraction, and thickness ratio have been investigated. The significant influence of small-scale effects and porosity inclusions have been observed in the reported results. It has been reported that both closed-form and finite element solutions with the present theory can make accurate predictions of the free vibration response.

A size dependent meshfree model for functionally graded plates based on the nonlocal strain gradient theory

A nonlocal strain gradient meshfree model based on the nonlocal strain gradient theory (NSGT) and higher-order shear deformation theory (HSDT) is presented and developed to examine the bending and free vibration behaviors of functionally graded (FG) nanoplates. Regarding two scale parameters, nonlocal and strain gradient effects in nanostructures are appeared in the current model. The power index and Mori-Tanaka schemes are utilized to compute material properties. A weak form of governing equations of motion for NSGT can be obtained by using the principle of virtual displacements. Later, the deflections and natural frequencies of the FG nanoplates are resolved by using the moving Kriging (MK) meshfree method. A detailed study to assess effects of exponential factor, length-to-thickness ratio, geometry, boundary condition, strain gradient parameter and nonlocal parameter on the deflections and natural frequencies of the FG nanoplates is revealed by numerical results. It is shown that large different results of deflections and natural frequencies between NSGT and classical theory are indicated through numerical examples.

Influence of initial geometric imperfection on static and free vibration analyses of porous FG nanoplate using an isogeometric approach

In this paper, the size-dependent analysis of geometrically imperfect porous functionally graded (FG) nanoplates is studied by an isogeometric approach based on a new refined plate theory. Simultaneous effects of initial geometric imperfection and porosity on the natural frequency and deflection of the FG nanoplates are investigated. The initial geometric imperfection is considered as an initial curvature and modeled by an analytical function in the governing equations of the nanoplate. Material properties of porous FG nanoplate are defined by a modified power-law function, and two types of distribution for porosity are used. A four-variable refined plate theory with a new polynomial shape function is proposed. Based on Hamilton’s principle, a weak form of static and free vibration problem for nonlocal plate is derived. The discrete system of equations is solved by an isogeometric approach based on NURBS basis functions, and the accuracy of the present study is verified by comparing the results with solutions given in published papers. Present results indicate the importance of porosity parameter, porosity distributions, nonlocal parameter, material index, plate geometrical parameters, and specially imperfection amplitude on the static and free vibration behavior of FG nanoplates.

The effects of multi-directional functionally graded materials on the natural frequency of the doubly-curved nanoshells

Up to now, no studies have been yet reported to study the mechanical behaviors of three dimensionally functionally graded (FG) shell structures. In this paper, the free vibration of three dimensionally FG nanoplates and nanoshells are investigated for the first time. All of the mechanical properties expect passion’s ratio are assumed to be changed along the length, width and thickness directions, which can vary according to an arbitrary function. The small scale effect due to nanostructures is considered based on nonlocal Eringen’s theory where Hamilton’s principle is adopted to derive the equations of motion. Galerkin solution method is adopted to obtain the natural frequencies of the FG nanostructures. Dynamic results are reported for nanoplates, spherical nanoshells, and cylindrical nanoshells for simply supported boundary conditions. The influences of several parameters, such as nonlocal parameter and functionally graded indexes are investigated on the natural frequency of the nanostructures. The results of the presented study can be served as benchmarks for future mechanical analysis of three-dimensionally FG shell structures.

A Finite Element Formulation and Nonlocal Theory for the Static and Free Vibration Analysis of the Sandwich Functionally Graded Nanoplates Resting on Elastic Foundation

This article presents a finite element method (FEM) integrated with the nonlocal theory for analysis of the static bending and free vibration of the sandwich functionally graded (FG) nanoplates resting on the elastic foundation (EF). Material properties of nanoplates are assumed to vary through thickness following two types (Type A with homogeneous core and FG material for upper and lower layers and Type B with FG material core and homogeneous materials for upper and lower layers). In this study, the formulation of the four-node quadrilateral element based on the mixed interpolation of tensorial components (MITC4) is used to avoid “the shear-locking” problem. On the basis of Hamilton’s principle and the nonlocal theory, the governing equations for the sandwich FG nanoplates are derived. The results of the proposed model are compared with published works to verify the accuracy and reliability. Furthermore, the effects of geometric parameters and material properties on the static and free vibration behaviors of nanoplates are investigated in detail.

A finite element formulation using four-unknown incorporating nonlocal theory for bending and free vibration analysis of functionally graded nanoplates resting on elastic medium foundations

A finite element model using four-unknown shear deformation theory integrated with the nonlocal theory is proposed for the bending and free vibration analysis of functionally graded (FG) nanoplates resting on elastic foundations. The present study developed the four-node quadrilateral element using Lagrangian and Hermitian interpolation functions for analysis of the membrane and bending displacement fields of FG nanoplates. Such a finite element formulation is suitable to investigate for the FG nanoplates resting on the elastic medium foundation with the stiffness matrices, the mass matrices and the load vectors using the second derivatives. The material properties of FG nanoplates are assumed to vary through the thickness direction by a power rule distribution of volume-fractions of the constituents. The equation of motion for FG nanoplates resting on the elastic foundation is obtained through Hamilton’s principle. Several numerical results are presented to demonstrate the accuracy and reliability of the present approach in comparison with other existing methods. In addition, the effects of geometrical parameters, material parameters, nonlocal parameters on the static bending and the free vibration responses of the nanoplates is also investigated in detail.

Dynamic stability of viscoelastic porous FG nanoplate under longitudinal magnetic field via a nonlocal strain gradient quasi-3D theory

This research is devoted to analyze the dynamic instability of viscoelastic porous functionally graded (FG) nanoplates under biaxially oscillating loading and longitudinal magnetic field using quasi-3D sinusoidal shear deformation plate theory as well as nonlocal strain gradient theory (NSGT). Modified power-law function is developed to show the effective material properties of the porous FG nanoplate that change uniformly from one surface to another. The motion equations are obtained by employing Hamilton's variational principle. In order to calculate the unstable region of the porous FG nanoplate, Navier as well as Bolotin's methods are utilized. The current theories and formulations are verified by comparing the obtained results with those available in literature. Then, the effects of several remarkable factors like magnetic field, nonlocal parameter (NP), internal damping, power-law index, static load factor, aspect ratio, porosity volume index and length scale parameter (LSP) on the unstable region of viscoelastic porous FG nanoplates are investigated via a comprehensive parametric study which could be utilized as reference results in future research. The numerical results indicated that increasing the porosity volume index, power-law index, internal damping parameter and NP leads to a reduction in the pulsation frequency and thus, unstable region moves to left, whereas magnetic field and LSP have a reverse effect.