From Functionally Graded Materials Research Articles

Investigation of FGM membrane wrinkling using reduced models based on a multi-scale method

In this article, we show that the reduced model proposed in our recent article to study the wrinkling of homogeneous elastic membranes does not reproduce the wrinkling observed in the case of an FGM-type composite membrane. New reduced models are then proposed to investigate the wrinkling of membranes made from Functionally Graded Materials (FGM). We examine, by this way, the influence of material functional parameters on determining the critical load and post-critical behavior under different loads. These reduced models, created using a multi-scale method based on Fourier series with slowly varying coefficients, enrich the reference model. In addition, these reduced models also allow us to significantly decrease the computation time and the number of finite elements compared to the complete model. The robustness of these reduced models is demonstrated in the case of wrinkling of membranes modeled by very thin shells, which are discretized using DKT18-type finite elements as described in literature. The resulting nonlinear problems are solved using a High-Order Continuation Method (HOCM).

Thermal Vibration Analysis of Sandwich Cylindrical Shells with Porous FGM Surface Layers

This study aims at investigating the thermal vibration characteristics of sandwich cylindrical shells consisting of two surface layers crafted from functionally graded materials (FGMs) and a central metal core layer. The sandwich cylindrical shells with FGMs surface layers, with and without porosity, are modelled by using the Kirchhoff–Love shell theory. A porosity function composed of three distinct parts is introduced, including the core-to-thickness ratio, porosity volume fraction, and porosity distribution function. Through the function, the significant effects of porosity that varies with the mixing degree of constituent materials can be analyzed. The material properties are assumed to be temperature-dependent and they show continuous graded variation along the thickness direction. A theoretical approach for analyzing thermal strain energy in the cylindrical shells subjected to thermal environments is established by incorporating Green’s nonlinear strains. The governing equations are derived by applying Hamilton’s principle. Subsequently, analytical solutions for the system’s natural frequencies are determined. Further, to validate the analytical results, a comparative analysis is conducted, drawing upon numerical simulations and other data available in the open literature. Additionally, the thermal vibration characteristics of the composite shell structures are examined in a comprehensive study with respect to various parameters such as porosity type, porosity volume fraction, core-to-thickness ratio, power-law exponent, and temperature changes.

Analytic solutions for static bending and free vibration analysis of FG nanobeams in thermal environment

Beam, plate, and shell structures manufactured from functionally graded materials (FGMs) are becoming increasingly prevalent in practice. The manufacture of these structures is not free from flaws, one of which is that the thickness distribution of the material does not change continuously. This is the first study to apply two distinct analytical solutions to examine the static and free vibration responses of imperfect FG nanobeams in terms of material distribution and temperature influence. The calculation formulas are based on the novel shear strain theory, which, despite being simple, accounts for both bending and shear strain, rendering the calculation procedure highly convenient. Either of the two precise solutions is computed using direct integrals and can be computed for various boundary conditions, which is not possible with Navier’s solutions. The results of the verification indicate that the approaches given in the article guarantee the needed precision. This study also analyzes the effect of several geometrical, material, nonlocal, and imperfection characteristics on the static bending and free oscillation response of imperfect FG nanobeams.

Quasi-3D large deflection nonlinear analysis of isogeometric FGM microplates with variable thickness via nonlocal stress–strain gradient elasticity

Via the nonlocal stress–strain gradient continuum mechanics, the microscale-dependent linear and nonlinear large deflections of transversely loaded composite sector microplates with different thickness variation schemes are investigated. Microplates are assumed to be prepared from functionally graded materials (FGMs) the characteristics of which are changed along the thickness direction. A quasi-3D plate theory with a sinusoidal transverse shear function in conjunction with a trigonometric normal function was employed for the establishment of size-dependent modelling of FGM microplates with different thickness variation schemes. Then, to solve the nonlocal stress–strain gradient flexural problem, the non-uniform rational B-spline type of isogeometric solution methodology was applied for an accurate integration of geometric discerptions. It was found that the gap between load–deflection curves drawn for linear, concave and convex thickness variation patterns became greater by changing FGM composite microplate boundary conditions from clamped to simply supported. In addition, it was found that by considering only the nonlocal size effect, the plate deflection obtained by the nonlocal strain gradient quasi-3D plate model was greater than that extracted by the classical continuum elasticity because of the softening character of nonlocal size effect, while the strain gradient microstructural size dependency acted in opposite way and represented a stiffening character.

Free vibrations of functionally graded material cylindrical shell closed with two spherical caps

ABSTRACT Free vibration response of a cylindrical shell closed with two hemispherical caps at the ends (hermit capsule) is analysed in this research. It is assumed that the system of joined shell is made from functionally graded materials (FGM). Properties of the shells are assumed to be graded through the thickness. Cylindrical and hemispherical shells are unified in thickness. To capture the effects of through-the-thickness shear deformations and rotary inertias, first order theory of shells is used. Donnell type of kinematic assumptions are adopted to establish the general equations of motion and the associated boundary and continuity conditions with the aid of Hamilton's principle. The resulting system of equations are discretised using the semi-analytical generalised differential quadrature (GDQ) method. After proving the efficiency and validity of the present method for the case of isotropic homogeneous joined shells, some parametric studies are carried out for the system of combined FGM hermetic capsule.

Bending analysis of five-layer curved functionally graded sandwich panel in magnetic field: closed-form solution

In this paper, an exact closed-form solution for a curved sandwich panel with two piezoelectric layers as actuator and sensor that are inserted in the top and bottom facings is presented. The core is made from functionally graded (FG) material that has heterogeneous power-law distribution through the radial coordinate. It is assumed that the core is subjected to a magnetic field whereas the core is covered by two insulated composite layers. To determine the exact solution, first characteristic equations are derived for different material types in a polar coordinate system, namely, magneto-elastic, elastic, and electro-elastic for the FG, orthotropic, and piezoelectric materials, respectively. The displacement-based method is used instead of the stress-based method to derive a set of closed-form real-valued solutions for both real and complex roots. Based on the elasticity theory, exact solutions for the governing equations are determined layer-by-layer that are considerably more accurate than typical simplified theories. The accuracy of the presented method is compared and validated with the available literature and the finite element simulation. The effects of geometrical and material parameters such as FG index, angular span along with external conditions such as magnetic field, mechanical pressure, and electrical difference are investigated in detail through numerical examples.

An analytical solution for heat conduction of FGM cylinders with varying thickness subjected to non-uniform heat flux using a first-order temperature theory and perturbation technique

In this research, thermal analysis in cylinders with varying thickness constructed from functionally graded materials (FGMs) which are under non-uniform heat fluxes in their inner and outer layers is presented. The heterogeneous distribution of properties is a modelled power function along the thickness and the length of the cylinder. Using first-order temperature theory (FTT) and the energy method, the governing equations are extracted. The system of governing differential equations is a system of ordinary differential equations with variable coefficients that is performed by using the analytical method of matched asymptotic expansion of the perturbations technique for solving it. Results relating to temperature distribution, heat flux, and thermal strain for different heat fluxes and heterogeneous constants are discussed. The results show that the heterogeneity has a magnificent effect on the temperature field and thermal strain inside FG cylinders. Moreover, it is observed that the heterogeneity has no effect on the direction of heat flux vector inside the body but any changes in the heterogeneity would change the magnitude of heat flux. The results achieved from the analytical method are compared with those predicted by finite element method (FEM). A good agreement was found between the predicted results of both methods.

Analysis of the Functionally Step-Variable Graded Plate Under In-Plane Compression.

A study of the pre- and post-buckling state of square plates built from functionally graded materials (FGMs) and pure ceramics is presented. In contrast to the theoretical approach, the structure under consideration contains a finite number of layers with a step-variable change in mechanical properties across the thickness. An influence of ceramics content on a wall and a number of finite layers of the step-variable FGM on the buckling and post-critical state was scrutinized. The problem was solved using the finite element method and the asymptotic nonlinear Koiter’s theory. The investigations were conducted for several boundary conditions and material distributions to assess the behavior of the plate and to compare critical forces and post-critical equilibrium paths.

Thermal buckling of functionally graded cylindrical shells with temperature-dependent elastoplastic properties

Thermal buckling problem of cylindrical shells made from functionally graded materials (FGM) is investigated under the framework of Donnell shell theory. The elastoplastic properties of FGM are described by Tamura–Tomota–Ozawa model, with a power law hardening model of metallic constituents, and the material-graded characteristic through the thickness is taken into account. $$J_{2}$$ flow theory helps to formulate the power law hardening constitutive relation of FGM, and Mises yield criterion is used to determine the position of elastoplastic material interface. The buckling critical temperature is obtained by employing a semi-analytical method programmed to converge both the prebuckling and the buckling internal forces iteratively. Three thermal cases, i.e., uniform, linear or nonlinear distribution of temperature rises through the thickness, are considered. Results reveal the different behaviors of elastoplastic buckling from those of elastic buckling and highlight the significance of considering the temperature-dependent material properties in the elastoplastic thermal buckling analysis of FGM cylindrical shells.

Nonlinear bending and postbuckling analysis of FG nanoscale beams using the two-phase fractional nonlocal continuum mechanics

The geometrically nonlinear bending and postbuckling of nanoscale beams are investigated herein according to the two-phase fractional nonlocal continuum model. It is considered that the beams have been made from functionally graded materials (FGMs), and the Bernoulli-Euler beam model is employed for their modeling. The variational form of the governing fractional equation is obtained first by means of an energy approach. Thereafter, a novel numerical solution method is proposed named as fractional variational differential quadrature method (FVDQM). In FVDQM, which is applied to the variational statement of the problem in a direct way, a combination of the differential quadrature method and matrix operators is utilized. The efficiency of the proposed fractional nonlocal model is evaluated by molecular dynamics (MD) simulations. Selected numerical results are given to explore the influences of fractional order, nonlocality, length-to-thickness ratio and FG index on the nonlinear bending and postbuckling responses of FG nanobeams with various types of boundary conditions.

Nonlinear Pull-In Instability of Strain Gradient Microplates Made of Functionally Graded Materials

In this paper, the size-dependent nonlinear pull-in behavior of rectangular microplates made from functionally graded materials (FGMs) subjected to electrostatic actuation is numerically studied using a novel approach. The small scale effects are taken into account according to Mindlin’s first-order strain gradient theory (SGT). The plate model is formulated based on the first-order shear deformation theory (FSDT) using the virtual work principle. The size-dependent relations are derived in general form, which can be reduced to those based on different elasticity theories, including the modified strain gradient, modified couple stress and classical theories (MSGT, MCST and CT). The solution of the problem is arrived at by employing an efficient matrix-based method called the variational differential quadrature (VDQ). First, the quadratic form of the energy functional including the size effects is obtained. Then, it is discretized by the VDQ method using a set of matrix differential and integral operators. Finally, the achieved discretized nonlinear equations are solved by the pseudo arc-length continuation method. In the numerical results, the effects of material length scale parameters, side length-to-thickness ratio and FGM’s material gradient index on the nonlinear pull-in instability of microplates with different boundary conditions are investigated. A comparison is also made between the predictions by the MSGT, MCST and CT.

Vibration of sandwich plates considering elastic foundation,temperature change and FGM faces

This study presents a comprehensive nonlinear dynamic approach to investigate the linear and nonlinear vibration of sandwich plates fabricated from functionally graded materials (FGMs) resting on an elastic foundation. Higher-order shear deformation theory and Hamilton's principle are employed to obtain governing equations. The Runge–Kutta method is employed together with the commercially available mathematical software MAPLE 14 to solve the set of nonlinear dynamic governing equations. Method validity is evaluated by comparing the results of this study and those of previous research. Good agreement is achieved. The effects of temperature change on frequencies are investigated considering various temperatures and various volume fraction index values, N. As the temperature increased, the plate frequency decreased, whereas with increasing N, the plate frequency increased. The effects of the side-to-thickness ratio, c/h, on natural frequencies were investigated. With increasing c/h, the frequencies increased nonlinearly. The effects of foundation stiffness on nonlinear vibration of the sandwich plate were also studied. Backbone curves presenting the variation of maximum displacement with respect to plate frequency are presented to provide insight into the nonlinear vibration and dynamic behavior of FGM sandwich plates.

Nonlinear bending and vibration analysis of functionally graded porous tubes via a nonlocal strain gradient theory

In this paper, the nonlinear bending and vibrational characteristics of porous tubes are analyzed for the first time. Within the framework of the nonlocal strain gradient theory, a size-dependent model for the tubes with radial inhomogeneity is formulated. It is assumed that the tube is made from functionally graded materials (FGM). Employed a new model for tubes which takes into account of the shear deformation effects, the motion equations are derived with the help of Hamilton variational principle and determined by the two-step perturbation technique. The validity and feasibility of the method are verified by actual examples. The effects of different parameters such as scaling parameters, porosity volume fraction, power law index and inner-to-outer radius ratio on the nonlinear bending and vibration behaviors of the porous tubes are particularly discussed.

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.

Buckling and postbuckling of elastoplastic FGM plates under inplane loads

Elastoplastic buckling behaviors of rectangular plates made from functionally graded materials (FGMs) are investigated in this paper. The elastoplastic material properties are assumed to vary smoothly through the thickness of the plates. The three dimensional material constitutive relation of FGMs is found by introducing the material homogenization method, named Tamura-Tomota-Ozawa(TTO) model, into J2 deformation theory or J2 flow theory. The uniform strain hypothesis helps to simplify the prebuckling state and derive the analytical expression of the position of the material elastoplastic interface. The buckling governing equations and the buckling critical condition of the structures are formulated under the framework of the classical plate theory. An iterative algorithm is designed to obtain the elastoplastic buckling critical load, a converging result between the prebuckling and the buckling critical internal forces. ABAQUS simulation well verifies the present theoretical predictions from J2 flow theory, and is resorted to investigate the postbuckling behaviors of FGM plates. Discussions are addressed for the effects of the constituent distribution, the material plastic flow, the preloaded states of the plates, and the regions of buckling types are plotted as well.

Vibration and post-buckling of a functionally graded beam subjected to non-conservative forces

Vibration and post-buckling of beams made from functionally graded materials (FGM) subjected to uniformly and tangentially compressing follower forces are studied in this paper. Based on the accurately and geometrically nonlinear theory for extensible beams, the dynamic governing equations for FGM beams under non-conservative load are formulated. By using a shooting method to solve the non-linearly differential equations numerically, the responses of post-buckling and free vibration in the vicinity of post-buckling configuration are obtained, in which the hinged-fixed boundary conditions of beam are considered. Effects of material gradient parameter on the critical buckling, post-buckling and lower frequencies of the FGM beam are discussed in details.

A mathematical boundary integral equation for analysis of the heterogeneous media, using the functionally graded elements

In the present research, a functionally graded (FG) boundary integral equation method capable of modeling quasistatic behavior of heterogeneous media fabricated from functionally graded materials (FGMs) whose distributions of the material properties obey either power or exponential laws is developed. Two heterogeneous material gradation models were employed to present the numerical formulations and solution algorithm. Somigliana's identity in 2D displacement fields of the isotropic heterogeneous domains is numerically implemented, employing FG elements. Based on the constitutive and governing equations and the weighted residual technique, the proposed boundary integral equation formulations are implemented for behavior analysis of the elastic heterogeneous isotropic solid structures. Results are verified and the proposed boundary element (BE) formulation is employed for behavior analysis of the plates and cylinders to demonstrate the proposed procedure more adequately.

Modified continuum model for stability analysis of asymmetric FGM double-sided NEMS: Corrections due to finite conductivity, surface energy and nonlocal effect

Finite conductivity, surface energy and nonlocal effect can influence the electromechanical performance of micro/nano-electromechanical systems (MEMS/NEMS). However, these factors are yet ignored on stability analysis of MEMS/NEMS fabricated from functionally graded materials (FGM). In this paper, dynamic stability of double-sided NEMS fabricated from non-symmetric FGM is investigated incorporating finite conductivity, surface energy and nonlocal effect. The Gurtin–Murdoch model and Eringen's elasticity are employed to consider the surface energy and nonlocal effect, respectively. Effect of finite conductivity of FGM on electrostatic and Casimir attractions is incorporated via relative permittivity and plasma frequency of the material. The stability analysis of the nanostructure is conducted by plotting time history and phase portraits. Moreover, bifurcation analysis is conducted to investigate the stability of the fixed points of the nano-structure. The validity of the proposed model is examined by comparing the results of the present study with those reported in the literature. The impact of various parameters i.e. finite conductivity, nonlocal parameter, surface stresses and material characteristics on the dynamic instability of the NEMS are addressed.

Modified model for instability analysis of symmetric FGM double-sided nano-bridge: Corrections due to surface layer, finite conductivity and size effect

Finite conductivity, size dependency and surface layer are known as crucial factors that affect the electromechanical response and pull-in instability of micro/nano-electromechanical systems (MEMS/NEMS). However, these effects are yet ignored on modeling the instability of MEMS/NEMS fabricated from functionally graded materials (FGMs). Herein, a modified continuum model is developed to incorporate these effects on the dynamic behavior and electromechanical instability of double-sided FGM NEMS bridge. Using Gurtin–Murdoch model in conjunction with nonlocal Eringen elasticity, the governing equations of the nano-bridges is derived considering the surface layer and size dependency. The Coulomb and Casimir forces are incorporated in the governing equation considering the corrections due to the finite conductivity of FGM (relative permittivity and plasma frequency). The validity of the proposed method is elucidated by comparing the results obtained from the present study with the results reported in the literature. The stability analysis of the nanostructure is conducted by plotting time history and phase portraits. The effects of various parameters including finite conductivity, nonlocal parameter, surface stresses and material characteristics on the dynamic pull-in behavior of the nano-bridges are studied. The obtained results imply a significant impact of the abovementioned corrections on the instability threshold and pull-in parameters of the FGM nanobridge.

Vibration characteristics of stepped beams made of FGM using differential transformation method

The present paper is given to investigate free vibration analysis of stepped beams produced from functionally graded materials (FGMs). The differential transformation method is employed to solve the governing differential equations of the beams to obtain their natural frequencies and mode shapes. The power law distribution is used and modified for describing material compositions across the thickness of the stepped beams made of FGM. Two main types of the stepped FGM beams in which their material compositions can be described by the modified power law distribution are selected to investigate the free vibration behaviour. The significant parametric studies such as step ratio, step location, boundary conditions and material volume fraction are also covered in this paper.