Material Property Gradient Index Research Articles

Three-dimensional semi-analytical solutions of arbitrary functionally graded piezoelectric doubly curved shell panel under thermo-electric load

In this article, a methodological approach for the three-dimensional (3D) static behavior of functionally graded (FG) thermo-electro-elastic doubly curved shells subjected to a thermal and electrical load is presented. A coupling procedure based on the state space, a multilayered Runge-Kutta and transfer matrix methods is elaborated for numerical solution of the 3D thermo-electro-elastic governing equations. In the thickness direction, recursive relationships using a propagator matrix are elaborated allowing to handle arbitrary functionally graded effects. A global transfer matrix is formulated permitting to investigate the effects of various types of bottom and top surface temperature loads and heat flux. Some benchmark tests are presented to validate the presented approach and to analyze the influence of the material-property gradient index, the curvature radius ratio, the span-to-thickness ratio, the temperature, normal heat flux, and the electric voltage on the mechanic, electric, and thermal variables of doubly curved shells.

A Hermite-Family C1 Finite Layer Method for the Three-Dimensional Free Vibration Analysis of Exponentially Graded Piezoelectric Microplates Based on the Consistent Couple Stress Theory

Based on the consistent couple stress theory (CCST), we develop a Hermite-family [Formula: see text] finite layer method (FLM) for the three-dimensional (3D) free vibration analysis of a simply-supported, exponentially graded (EG) piezoelectric microplate under open- and closed-circuit surface conditions. In the formulation of the FLM, the microplate is artificially divided into a number of finite microlayers, and Fourier functions and Hermite polynomials are used to interpolate the in-plane and out-of-plane variations of a number of primary variables, respectively, including elastic displacement components and the electric potential variable for each individual layer. The Hermite-family [Formula: see text] FLM for analyzing EG piezoelectric microplates is reduced to the Hermite-family [Formula: see text] FLM for analyzing EG piezoelectric macroscale plates and functionally graded (FG) elastic microplates by assigning a value of zero to the material length scale parameter and by ignoring the piezoelectric and flexoelectric effects in the formulation, respectively. The accuracy and convergence rate of the FLM are assessed by comparing their solutions with the benchmark solutions of both the EG piezoelectric macroplates and the power-law-type FG elastic microplates that are available in the relevant literature. We examine and discuss some key effects on the free vibration characteristics of an EG piezoelectric microplate, including the impact of the material length scale parameter, the material-property gradient index, the length-to-thickness ratio, the piezoelectric effect, and the flexoelectric effect.

A three-dimensional weak formulation for stress, deformation, and free vibration analyses of functionally graded microscale plates based on the consistent couple stress theory

A three-dimensional (3D) weak formulation based on the consistent couple stress theory (CCST) is developed for displacement, stress, and free vibration analyses of simply-supported, power-law-/exponential-type functionally graded (FG) microscale plates. In the formulation, the microscale plate is artificially divided into numerous finite microscale layers, where Fourier functions and Lagrange/Hermite polynomials are used to interpolate the in- and out-of-plane variations in the displacement variables for each individual layer, respectively. Based on the weak formulation, the authors develop layer-wise C0 quadratic and cubic finite element methods (FEMs), as well as layer-wise C1 two- and three-node FEMs to address the current issue. The accuracy and convergence of these layer-wise C0 and C1 FEMs are validated by comparing their solutions with the 3D exact and the quasi-3D results available in the literature. A 3D weak formulation based on the modified couple stress theory (MCST) is also developed for comparison purposes. The similarity and differences between the results obtained using the CCST- and MCST-based weak formulations are examined and discussed. Some key effects on stress, deformation, and free vibration characteristics of the FG microscale plate are examined, including the material length scale, the material-property gradient index, and the length-to-thickness ratio effects, which are shown to be significant for the FG microscale plate.

Size-dependent nonlinear bending behavior of porous FGM quasi-3D microplates with a central cutout based on nonlocal strain gradient isogeometric finite element modelling

With the aid of the non-uniform rational B-spline (NURBS)-based isogeometric technique, for the first time, the size-dependent geometrically nonlinear bending characteristics of microplates made of porous functionally graded materials (FGMs) having a central cutout with different shapes are studied. The nonlocal strain gradient continuum elasticity within the framework a hybrid higher-order quasi-3D plate theory is adopted to describe the kinematic relations via only four unknowns. To capture the effective material properties, a porosity-dependent rule of mixture is employed. The nonlocal strain gradient nonlinear load–deflection responses are obtained corresponding to various geometrical and material parameters as well as different boundary conditions. It is revealed that the significance of both the nonlocal and strain gradient reduces. This prediction is the same for all values of the material property gradient index as well as the porosity index. Also, it is demonstrated that a central cutout leads to change the trend of load–deflection response, and this change occurs at a specific value for the applied distributed load which depends on several parameters such as the cutout geometry and boundary conditions. In addition, it is displayed that corresponding to different maximum deflections, the significance of the strain gradient size effect in the absence of nonlocality on the nonlinear flexural stiffness of a porous FGM microplate is more than that of the nonlocal size effect in the absence of the strain gradient size dependency.

On the NURBS-based isogeometric analysis for couple stress-based nonlinear instability of PFGM microplates

A porosity-dependent nonlinear postbuckling analysis for microplates prepared from a porous functionally graded material (PFGM) is performed based on the modified couple stress theory of elasticity (MCSTE). To accomplish this purpose, the modified couple stress-based nonlinear differential equations are derived by third-order shear deformation plate theory (TSDPT). To extract PFGM microplate effective mechanical properties, a power-law function is utilized which is capable of incorporating porosity dependency and material gradient, simultaneously. Thereafter, the non-uniform rational B-spline (NURBS)-based isogeometric analytical method is employed as an effective discretization method which could satisfy C−1 continuity condition. It is demonstrated that the gap between equilibrium paths relevant to various couple stress length scales reduces by going deeper in postbuckling regime. Also, one can see that for certain porosity and material property gradient index values, by boundary condition transformation from simply supported to clamped, the contribution of the effect of couple stress size in PFGM microplate postbuckling response becomes more significant, especially for a higher maximum deflection values.

Buckling and postbuckling response of nonlocal strain gradient porous functionally graded micro/nano-plates via NURBS-based isogeometric analysis

The prime objective of the present investigation is to analyze the nonlinear buckling and postbuckling characteristics of micro/nano-plates made of a porous functionally graded material (PFGM) in the presence of nonlocality and strain gradient size dependencies. In accordance with this purpose, a modified porosity-dependent power-law function is put to use to estimate the effective mechanical properties of PFGM micro/nano-plates with various porosity distribution patterns. To solve the constructed nonlinear nonlocal strain gradient problem, the non-uniform rational B-spline (NURBS)-based isogeometric analysis is utilized as an efficient discretization technique. It is concluded that by taking the geometrical nonlinearity into account and moving to deeper part of the postbuckling regime, the significance of the both nonlocality and strain gradient size dependencies decreases. Also, by increasing the material property gradient index, an enhancement in the nonlocal and strain gradient size effects is found which is more considerable at deeper part of the postbuckling domain. In addition, it is demonstrated that for a PFGM micro/nano-plate, the value of the porosity index has a negligible influence on the significance of size dependencies, and this observation is the same for the both types of simply supported and clamped boundary conditions.

Strong and Weak Formulations of a Mixed Higher-Order Shear Deformation Theory for the Static Analysis of Functionally Graded Beams under Thermo-Mechanical Loads

The strong and weak formulations of a mixed layer-wise (LW) higher-order shear deformation theory (HSDT) are developed for the static analysis of functionally graded (FG) beams under various boundary conditions subjected to thermo-mechanical loads. The material properties of the FG beam are assumed to obey a power-law distribution of the volume fractions of the constituents through the thickness of the FG beam, for which the effective material properties are estimated using the rule of mixtures, or it is directly assumed that the effective material properties of the FG beam obey an exponential function distribution along the thickness direction of the FG beam. The results shown in the numerical examples indicate that the mixed LW HSDT solutions for elastic and thermal field variables are in excellent agreement with the accurate solutions available in the literature. A parametric study related to various effects on the coupled thermo-mechanical behavior of FG beams is carried out, including the aspect ratio, the material-property gradient index, and different boundary conditions.

Large-amplitude oscillations of composite conical nanoshells with in-plane heterogeneity including surface stress effect

In the current investigation, a new formulation for nonlinear vibration behaviors of functionally graded (FG) composite conical nanoshells are constructed using Gurtin-Murdoch elasticity theory based on higher-order shear deformation shell theory (HSDFST) framework. Both Voigt and Reuss homogenization procedures are considered for the estimation of the mechanical characteristics of FG materials. Using generalized differential quadrature method (GDQM) together with Galerkin technique, the surface elastic-based nonlinear frequency-responses of FG composite conical nanoshell are obtained. It has been illustrated that the decrease of material property gradient index or transformation of boundary condition from full simply supported to full clamped, surface stress effect on the nonlinear frequency of a FG composite conical nanoshell reduces. Also, decreasing semi-vertex angle increases the frequency ratio of ωNL/ωL which reveals higher geometrical nonlinearity. However, it is seen that surface elasticity effect on the nonlinear vibration behavior of FG composite conical nanoshells is not significant.

Isogeometric nonlinear oscillations of nonlocal strain gradient PFGM micro/nano-plates via NURBS-based formulation

The present investigation deals with the size-dependent analysis of the geometrically nonlinear vibration response of micro/nano-plates with and without a central cutout made of a porous functionally graded material (PFGM) in the presence of nonlocality and strain gradient size dependencies. In accordance with this purpose, a modified porosity-dependent power-law function is put to use to estimate the effective mechanical properties of PFGM micro/nano-plates with various porosity distribution patterns. To solve the constructed nonlinear nonlocal strain gradient problem, the non-uniform rational B-spline (NURBS)-based isogeometric analysis is utilized as an efficient discretization technique having the capability to satisfy C−1 continuity. It is seen that for specific values of the material property gradient index, porosity index and the plate deflection, the enhancement in the nonlinear frequency due to the strain gradient size effect is more than the reduction caused by the nonlocality. Furthermore, it is found that there is a specific value of the length to thickness ratio, corresponding to which the nonlocal strain gradient frequency ratio becomes minimum. This minimum value enhances by increasing the value of the porosity index of PFGM micro/nano-plates. Also, by increasing the value of the material property gradient index, the minimum point of the nonlocal strain gradient frequency ratio shifts to a higher ratio of the length to width ratio.

Modified couple stress-based geometrically nonlinear oscillations of porous functionally graded microplates using NURBS-based isogeometric approach

In this work, porosity-dependent nonlinear large-amplitude oscillation responses of rectangular microplates with and without a central square cutout made of a porous functionally graded material (PFGM) is explored using modified couple stress theory of elasticity (MCSTE). The associated nonlinear size-dependent modified couple stress-based differential motion equations are obtained based on third-order shear deformation plate model (TSDPM). A new power-law function incorporating simultaneously the material gradient and porosity dependency is employed for the extraction of the effective mechanical characteristics of PFGM microplates. Afterwards, the non-uniform rational B-spline (NURBS)-based isogeometric technique is put to use as an efficient discretization method taking the C−1 continuity satisfaction into account. It is observed that among various patterns of porosity distribution, the lowest and greatest effects of couple stress size dependency are observed on the nonlinear frequencies of microplates in which the porosity from top and down surfaces to center is increased and decreased, respectively. Also, it was observed that increase of the material property gradient index as well as plate deflection reduces couple stress size effect on the nonlinear oscillations of PFGM microplates. It was shown that there is a specific length to thickness ratio, corresponding to which the modified couple stress-based frequency ratio becomes minimum. This minimum value enhances with the increase of the porosity index of PFGM microplates.

Couple stress-based nonlinear buckling analysis of hydrostatic pressurized functionally graded composite conical microshells

The present study deals with the size-dependent nonlinear buckling characteristics of conical microshells made of functionally graded (FG) composite materials under uniform hydrostatic pressure based upon the modified couple stress theory of elasticity. Accordingly, a modified couple stress-based shell model within the framework of the higher-order shear deformation shell theory and von Karman geometrical nonlinearity is constructed. Using the virtual work's principle in conjunction with the adjacent equilibrium criterion, the non-classical governing differential equations are established. The material properties of FG composite conical microshells are estimated on the basis of different homogenization schemes. To solve the size-dependent nonlinear problem, the generalized differential quadrature discretization pattern together with the Galerkin technique is employed. It is seen that among various types of homogenization scheme, the Voigt and Reuss models represent, respectively, the overestimated and underestimated critical buckling pressures. Also, it is found that for a FG composite conical microshell with higher semi-vertex angle, the influence of the material property gradient index on the nonlinear critical buckling pressure diminishes. In addition, it is observed that the couple stress size dependency plays more important role in the nonlinear buckling behavior of FG composite conical microshells with lower ratio of R1/h. These patterns are the same for all types of boundary conditions.

Size-dependent shear buckling response of FGM skew nanoplates modeled via different homogenization schemes

The size effects on the shear buckling behaviors of skew nanoplates made of functionally graded materials (FGMs) are presented. The material properties are supposed to be changed uniformly from the ceramic phase to the metal one along the plate thickness. To estimate the associated effective material properties, various homogeniza-tion schemes including the Reuss model, the Voigt model, the Mori-Tanaka model, and the Hashin-Shtrikman bound model are used. The nonlocal elasticity theory together with the oblique coordinate system is applied to the higher-order shear deformation plate theory to develop a size-dependent plate model for the shear buckling analysis of FGM skew nanoplates. The Ritz method using Gram-Schmidt shape functions is used to solve the size-dependent problem. It is found that the significance of the nonlocality in the reduction of the shear buckling load of an FGM skew nanoplate increases for a higher value of the material property gradient index. Also, by increasing the skew angle, the critical shear buckling load of an FGM skew nanoplate enhances. This pattern becomes a bit less significant for a higher value of the material property gradient index. Furthermore, among various homogenization models, the Voigt and Reuss models in order estimate the overestimated and underestimated shear buckling loads, and the difference between them reduces by increasing the aspect ratio of the skew nanoplate.

Magneto-electro-elastic vibration of moderately thick FG annular plates subjected to multi physical loads in thermal environment using GDQ method by considering neutral surface

The vibration analysis of an annular plate made up of functionally graded magneto-electro-elastic materials subjected to multi physical loads is presented. The plate is in thermal environment and temperature is distributed non-uniformly in its thickness direction. In addition, the plate is assumed moderately thick, the material properties vary through the thickness, and the exact neutral surface position is determined and took into account. According to Hamilton’s principle and the first-order shear deformation theory, the governing motion equations are extracted. Numerical results for various boundary conditions are obtained via the generalized differential quadrature method and are validated in simpler states with those of the literature. The effects of different parameters such as material property gradient index, multi physical loads, temperature variations, boundary conditions and geometric specifications of the plate on the natural frequencies and mode shapes are investigated. Temperature changes have little effect on the natural frequencies and the effect of electric potential on them is opposite of magnetic one. In other words, by increasing the magnetic potential, the rigidity of the plate increases too, and the frequency increases. The results of this study are useful to design more efficient sensors and actuators used in the smart or intelligent structures.

Non-classical plate model for FGMs

In this work, we developed a non-classical plate model for the analysis of the behavior of microplates prepared from functionally graded materials (FGMs). It was assumed that the material properties of FGM microplates vary along the direction of thickness and could be estimated using Mori–Tanaka homogenization technique. A variational approach based on Hamilton’s principle was employed to achieve motion equations and corresponding boundary conditions simultaneously. Unlike classical plate theory, the proposed plate model contained three material length scale parameters which allowed size effect to be taken into account effectively. Moreover, the proposed non-classical plate model could be divided into classical and modified couple stress plate models when all material length scale parameters were set at zero. To approve the applicability of the proposed size-dependent plate model, analytical solution for free vibration problem of a simply supported FGM microplate was obtained according to the Navier solution, where generalized displacements were stated as multiplication of undetermined functions with known trigonometric functions to identically satisfy simply supported boundary conditions for all edges. The results obtained by the proposed non-classical plate model were compared with those obtained from reduced ones relevant to different values of material property gradient index and dimensionless length scale parameter. It was found that the differences observed between the sets of predicted natural frequencies obtained from different types of plate models were more significant for lower values of dimensionless length scale parameter.

Boundary Layer Modeling of Nonlinear Axial Buckling Behavior of Functionally Graded Cylindrical Nanoshells Based on the Surface Elasticity Theory

The main purpose of the present study is to examine the axial buckling and postbuckling response of cylindrical nanoshells made of functionally graded (FG) materials in the presence of surface free energy effects. To accomplish this purpose, an efficient size-dependent shell model is introduced through combination of the classical shell theory with the well-known Gurtin–Murdoch surface elasticity theory. The volume fraction of FG nanoshells made of the mixture of aluminum and silicon is defined using a power law function. Also, in order to eliminate the stretching–bending coupling terms, the change in the position of physical neutral plane corresponding to different volume fractions is taken into account. Based upon the principle of virtual work, the non-classical governing differential equations and boundary conditions are derived. Subsequently, the governing equations are deduced to the boundary layer-type equations incorporating simultaneously the nonlinear large deflections and size effect. Finally, a perturbation-based solving process is put to use to predict the size-dependent critical buckling loads and related postbuckling equilibrium curves for FG nanoshells with various values of material property gradient index and shell thickness. It is found that the surface free energy effect is more prominent for FG nanoshell with lower shell thickness and higher material property gradient index.

Imperfection sensitivity of the nonlinear axial buckling behavior of FGM nanoshells in thermal environments based on surface elasticity theory

A size-dependent shell model which accounts for geometrical imperfection sensitivity of the axial postbuckling characteristics of a cylindrical nanoshell made of functionally graded material (FGM) is proposed within the framework of the surface elasticity theory. In accordance with a power law, the material properties of the FGM nanoshell are supposed to vary through the shell thickness. In order to eliminate the stretching-bending coupling terms, the change in the position of physical neutral plane corresponding to different volume fractions is taken into account. Based upon the virtual work’s principle, the non-classical governing differential equations are derived and then deduced to boundary layer-type ones. After that, a perturbation-based solution methodology is employed to predict the size dependency in the nonlinear instability of perfect and imperfect axially loaded FGM nanoshells with various values of shell thickness, material property gradient index and different uniform temperature changes. It was seen that for thicker FGM nanoshells in which the surface free energy effects diminish, the influence of the initial geometric imperfection on the critical buckling load is higher than its influence on the minimum load of the postbuckling domain. It is also found that through reduction of the surface free energy effects, the influence of material property gradient index on the critical end-shortening of FGM nanoshell decreases.

Size dependent buckling analysis of functionally graded piezoelectric cylindrical nanoshell

In this paper, the size-dependent formulation of shear deformable functionally graded piezoelectric (FGP) cylindrical nanoshells is developed based on a new modified couple stress theory. After the general formulation, the buckling of FGP cylindrical nanoshells under pressure is investigated by using the first order shear deformable shell model. The material properties are assumed to be varied along thickness direction according to the power law distribution. The equilibrium equations and boundary conditions are obtained by using the minimum potential energy principle. A buckling analysis of simply-supported FGP cylindrical nanoshells under uniform lateral external pressure is carried out and the effects of different parameters on the critical pressure are examined. The effects of geometrical, electrical and material properties, such as material length scale parameter, length, thickness, external electric voltage and material property gradient index, on the critical pressure are illustrated. It is indicated that the critical pressure is significantly size-dependent.

Size-dependent torsional buckling analysis of functionally graded cylindrical shell

Abstract The size-dependent torsional buckling behavior of functionally graded (FG) cylindrical shell is investigated on the basis of modified couple stress theory using the shell model. The material properties of FG nanoshell are considered change through thickness direction according to power law distribution. The modified couple stress shell theory with the von Karman geometrical nonlinearity is utilized to establish theoretical formulations. The governing equations and boundary conditions are derived using the minimum potential energy principle. As a special case, the torsional buckling of simply supported and clamped FG cylindrical shell is examined using the GDQ method. Afterwards, the influences of geometrical parameters, such as length scale parameter, length, and thickness, as well as material property gradient index of the FG cylindrical shell on the critical torsional buckling moment are studied.

Coupled electro-mechanical effects and the dynamic responses of functionally graded piezoelectric film-substrate circular hollow cylinders

Based on Reissner's mixed variational theorem (RMVT), we developed finite cylindrical layer methods (FCLMs) to investigate the quasi-three-dimensional (3D) dynamic responses of simply-supported, two-layered functionally graded piezoelectric material (FGPM) film-substrate circular hollow cylinders with open- and closed-circuit surface conditions. The FGPM film-substrate cylinder considered in this work consists of a thick and soft FGPM substrate with a surface-bonded thin and stiff homogeneous piezoelectric material (HPM) film. The material properties of the FGPM layer are assumed to obey an exponent-law exponentially varying with the thickness coordinate, and the piezoelectric ceramic material PZT-4 is taken to be the reference material. The accuracy and convergence rate of FCLMs with different orders are assessed by comparing their solutions with the exact 3D ones available in the literature. The convergent solutions of FCLMs for the lowest frequency parameters and their corresponding modal electric and elastic variables of the FGPM film-substrate cylinder are presented. The influences of various factors with regard to some of the key dynamic responses of the cylinder are examined, such as the coupled electro-mechanical characteristics, material-property gradient index, surface boundary conditions, aspect ratio, and film-substrate thickness ratio.

Bending analysis of FG microbeams resting on Winkler elastic foundation via strain gradient elasticity

In this study, bending response of non-homogenous microbeams embedded in an elastic medium is investigated based on modified strain gradient elasticity theory in conjunctions with various beam theories. The governing differential equations and related boundary conditions are derived with the aid of minimum total potential energy principle. Elastic medium is modeled by Winkler foundation model. Bending problem of simply supported microbeams made of functionally graded materials is solved by Navier’s solution procedure. A detailed parametric study is performed to investigate the effects of the material length scale parameters-to-thickness ratio l/h, material property gradient index k, slenderness ratio L/h, Winkler modulus kw and shear correction factor on the bending behavior of embedded functionally graded microbeams.