## What is/are Functionally Graded?

Functionally Graded - In this work, a novel asymmetric micro-cylinder composed of functionally graded (FG) materials is designed to evaluate computationally the role of flexoelectricity in the electromechanical response.^{[1]}In the present work we study the static response of functionally graded (FG) porous nanocomposite beams, with a uniform or non-uniform layer-wise distribution of the internal pores and graphene pla.

^{[2]}This paper presents the damping characteristics of nonlinear vibrations of functionally graded (FG) plates utilizing piezoelectric composite material.

^{[3]}The in-plane vibration problem of functionally graded (FG) sandwich circular arch made up of two layers of power law FGM face sheet and one layer of homogeneous core is investigated.

^{[4]}This paper constitutes an attempt to explore the influence of porosity on free vibration analysis of functionally graded (FG) beams with different boundary conditions using different efficient analytical and numerical approaches.

^{[5]}8-noded hexahedral elements are functionally graded to appropriately simulate the stress concentration developed in each layer for a particular load.

^{[6]}GRMMC layers having different graphene volume fractions can facilitate the formation of functionally graded (FG) patterns of the beams.

^{[7]}A recent proof on non-existence of Stoneley waves in functionally graded (FG) plates with continuous transverse inhomogeneity may indicate that in real situations, when there is no abrupt change of material properties, the Stoneley waves arise less frequently than it can be supposed.

^{[8]}For the exact solution of the stress in the functionally graded (FG) cylindrical/spherical pressure vessel, this paper presents a unified form of the basic equations of the FG hollow cylinder/spherical shell by introducing parameter, and then, the axial/spherical symmetry mechanical problems of FG hollow shells are studied under three different boundary conditions, respectively.

^{[9]}ABSTRACT This paper develops a new solution approach to solve nonlinear forced vibrations of functionally graded (FG) piezoelectric shells in multi-physics fields.

^{[10]}

## carbon nanotube reinforced

In this paper, a first known study on the large deformation dynamic behavior of functionally graded carbon nanotube-reinforced (CNTR) nanocomposite cylinder considering the particle/matrix interphase property is presented.^{[1]}On the basis of that, an accurate and analytical method for investigation the dynamic instability of laminated functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell surrounded by the elastic foundations is presented in this work based on the first-order shear deformation theory.

^{[2]}The dynamical analysis of two directional functionally graded carbon nanotube reinforced composite (2DFG-CNTRC) plates are studied for various boundary conditions.

^{[3]}The effective properties of nanocomposite materials Functionally Graded Carbon Nanotube-Reinforced Composite (FG-CNTRC) were estimated using the rule of mixtures.

^{[4]}This paper investigates the thermal vibration and buckling of the functionally graded carbon nanotube reinforced composite (FG-CNTRC) quadrilateral plate using the meshless method.

^{[5]}This article presents a numerical investigation on the free vibration characteristics of rotating pretwisted sandwich conical shell panels with two functionally graded carbon nanotube-reinforced composite (FG-CNTRC) face sheets and a homogeneous core in uniform thermal environments.

^{[6]}These shells possess a viscoelastic core, and face sheets made of functionally graded carbon nanotube-reinforced magneto-electro-elastic (FG-CNTMEE) materials.

^{[7]}In this work, the nonlinear primary resonance of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) circular cylindrical panels is numerically studied.

^{[8]}The paper is dedicated to the vibration and flutter analysis of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) regular polygon built-up (RPB) structures with general boundary conditions.

^{[9]}This study presents the multi-stepped functionally graded carbon nanotube reinforced composite (FG-CNTRC) plate model for the first time, and its free and forced vibration is analyzed by employing the domain decomposition method.

^{[10]}A full understanding of the mechanical behaviors of a three-dimensional (3D) functionally graded carbon nanotube reinforced composite (FG-CNTRC) cylindrical panel is important for structural design of engineering composite components.

^{[11]}Dynamic instability behavior of functionally graded carbon nanotube reinforced hybrid composite plates subjected to periodic loadings is studied.

^{[12]}In the present work, a geometrically nonlinear finite shell element is first presented to predict nonlinear dynamic behavior of piezolaminated functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shell, to enrich the existing research results on FG-CNTRC structures.

^{[13]}In the present paper, the geometrical nonlinear behaviour of the sandwich shallow spherical shells made of the functionally graded carbon nanotube-reinforced composites is assessed.

^{[14]}In this research paper, as initial endeavors, the vibrational responses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoplates taking into account the effect of nonlocal parameter and strain gradient coefficient are investigated.

^{[15]}Three distributed types of functionally graded carbon nanotube-reinforced coatings and the honeycomb lattice form of the auxetic core are investigated.

^{[16]}This paper presents the analytical solution on vibro-acoustic characteristics of three-dimensional graphene foam core and functionally graded carbon nanotube-reinforced polymer face sheet (3DGrFC-CNTRPFS) sandwich panel.

^{[17]}In this regard, we conduct a comparative study of six typical dynamic condensation techniques utilized for addressing damage identification problems of composite plates made of functionally graded materials (FGM) and functionally graded carbon nanotube-reinforced composite (FG-CNTRC) materials.

^{[18]}There is no literature on the dynamic property of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates with a viscoelastic core.

^{[19]}The current research is concerned with the small- and large-amplitude free vibrations of a composite beam made of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) media.

^{[20]}Mechanical properties of functionally graded carbon nanotube reinforced composite (FG-CNTRC) structures have been extensively investigated in the published literature.

^{[21]}The free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) conical shell is carried out by using element-free kp-Ritz method.

^{[22]}In this study, an effective analytical solution for the stability problem of functionally graded (FG) carbon nanotube reinforced composite (CNTRC) conical shells (CSs) exposed to external lateral and hydrostatic pressures is presented.

^{[23]}The main objective of this paper is to analyze the free vibration of arbitrary shaped thick functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates based on the higher-order shear deformation theory (HSDT) using a variational differential quadrature approach.

^{[24]}The present approach captures the small scale effects on the geometrically nonlinear behaviors of functionally graded carbon nanotube reinforced composite (FG-CNTRC) micro-plate with four patterns distribution.

^{[25]}The main object of this research is to formulate the linear free vibration as well as static stability of embedded functionally graded carbon nanotube-reinforced composite microbeams in thermal environment.

^{[26]}

## finite element method

OBJECTIVE: The aim of this work is to present the methodology for grading the Functionally Graded Material (FGM) using Level Set (LS) sign distance function around the multiple holes and parametrically analyse the maximum stresses for a pair of inclined unequal circular holes surrounded by the FGM layer in an infinite plate subjected to uniaxial tensile load using the Extended Finite Element Method (XFEM).^{[1]}In this paper, the free vibration of the functionally graded porous (FGP) non-uniform annular-nanoplates lying on Winkler foundation (WF) is studied by using the smoothed finite element method based on the first-order shear deformation theory (FSDT).

^{[2]}The three-dimensional free flexural vibrations of functionally graded circular cylindrical shell with curvilinear radius variation fully or partially filled with incompressible and inviscid fluid are investigated for the first time using a coupled axisymmetric solid-fluid hierarchical finite element method based on curved elements.

^{[3]}The paper deals with the development of the finite element method (FEM) model of piezoelectric beam elements, where the piezoelectric layers are located on the outer surfaces of the beam core, which is made of functionally graded material.

^{[4]}There is not enough mixed finite element method (MFEM) model developed for static and dynamic analysis of functionally graded material (FGM) beams in the literature.

^{[5]}Stress intensity factors (SIFs) and propagation paths of cracks in functionally graded materials (FGMs) are simulated by a novel numerical method called the field-enriched finite element method (FE-FEM).

^{[6]}OBJECTIVES The aim of this work was to evaluate the biomechanical behavior of one-piece zirconia implants with a functionally graded bioglass (BG) layer as compared to monolithic zirconia and BG-coated implants, using the finite element method (FEM).

^{[7]}This article proposes a finite element method (FEM) based on a quasi-3D nonlocal theory to study the free vibration of functionally graded material (FGM) nanoplates lying on the elastic foundation (EF) in the thermal environment.

^{[8]}In this study, nonlinear transient thermo-elastic analysis for a thick hollow 1D-FGM (functionally graded material) axisymmetric cylinder with finite length is investigated using higher-order graded finite element method.

^{[9]}In this paper, the free vibration results of a functionally graded conical shell under nonlinear thermal loading are presented employing finite element method wherein the shell kinematics is based on first-order shear deformation theory.

^{[10]}In this work, an improved control volume finite element method (ICVFEM) is proposed and implemented for thermoelastic analysis in functionally graded materials (FGMs) at steady state.

^{[11]}The aim of the present work is to provide a parametric analysis of stress intensity factor (SIF) in different functionally graded material (FGM) plates using the extended finite element method (XFEM).

^{[12]}In this study, continuous contact problem in the functionally graded (FG) layer loaded with a FG flat punch resting on the elastic semi-infinite plane was analyzed by the finite element method (FEM).

^{[13]}This paper proposes a finite element method (FEM) based on a nonlocal theory for analyzing the free vibration of the functionally graded porous (FGP) nano-plate with different shapes lying on the elastic foundation (EF).

^{[14]}A cell-based smoothed finite element method with discrete shear gap technique is used to study the stochastic free vibration behavior of functionally graded plates with material uncertainty.

^{[15]}This paper applies the stochastic finite element method (SFEM) to perform the natural frequency analysis of functionally graded material (FGM).

^{[16]}Vibration characteristics of cracked functionally graded structures have been investigated using extended finite element method (XFEM).

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## higher order shear

In this paper, a higher order shear deformation theory for bending analysis of functionally graded plates resting on Pasternak foundation and under various boundary conditions is exposed.^{[1]}This paper introduces a comprehensive investigation of bi-directional functionally graded sandwich plates using higher-order shear deformation theory and finite element method for the first time.

^{[2]}A novel unified framework of higher-order shear deformation theories is developed for modeling and analysis of laminated and functionally graded plates.

^{[3]}A mixed layer-wise (LW) higher-order shear deformation theory (HSDT) is developed for the thermal buckling analysis of simply-supported, functionally graded (FG) beams subjected to a uniform temperature change.

^{[4]}In this article, the vibrational behavior of conical panels in the nonlinear regime made of functionally graded graphene platelet–reinforced composite having a hole with various shapes is investigated in the context of higher-order shear deformation theory.

^{[5]}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.

^{[6]}Within the framework of a variational mixed formation and higher-order shear deformation theory (HSDT), a numerical approach is developed in this research to investigate the buckling and post buckling behaviors of variously-shaped plates made of functionally graded graphene platelet-reinforced composites (FG-GPLRCs) taking the effect of porosity into account.

^{[7]}In this paper, the bending behavior of functionally graded single-layered, symmetric and non-symmetric sandwich beams is investigated according to a new higher order shear deformation theory.

^{[8]}In this work, the dynamic response of functionally graded beams on variable elastic foundations is studied using a novel higher-order shear deformation theory (HSDT).

^{[9]}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.

^{[10]}Time-dependent analysis based on layerwise theory (LT) along with higher-order shear deformation theory (HSDT) is studied to determine the stress distribution in a simply supported sector of spherical sandwich shell with piezoelectric face sheets and functionally graded carbon nanotube (FG-CNT) core subjected to the low-velocity impact.

^{[11]}This paper extends the higher-order shear deformable mixed beam element model with rational shear stress distribution to vibration analysis of functionally graded (FG) beams.

^{[12]}Buckling analysis of functionally graded piezoelectric nanoshell is studied in this paper based on the higher-order shear and normal deformation theory and accounting thickness stretching effect.

^{[13]}

## first order shear

Based on the geometrically exact beam theory, a first-order shear deformable curved beam element is developed for geometrically nonlinear analysis of functionally graded (FG) curved beams.^{[1]}This article proposes a general method to investigate the vibration characteristics of functionally graded corrugated plate structure (FGCPS) according to first-order shear deformation theory (FSDT) by adopting differential quadrature finite element method (DQFEM).

^{[2]}In this paper, the isogeometric analysis (IGA) integrated with a simple first-order shear deformation theory (S-FSDT) with only four variables per node is extended to study geometrically nonlinear natural frequency, nonlinear static bending and transient dynamic response of two types of smart piezoelectric functionally graded plates (PFGPs) in thermal environment.

^{[3]}Nonlinear behavior of functionally graded (FG) sandwich circular sector plates with simply supported radial edges under transverse loading is investigated using the first-order shear deformation theory with von Karman geometric nonlinearity.

^{[4]}In this paper, the first order shear deformation theory is used to derive an analytical formulation for shrink-fitted thick-walled functionally graded cylinders.

^{[5]}The present paper investigates size-dependent buckling analysis of a functionally graded cylindrical micro/nano shell integrated with a regular pattern of piezoelectric segments based on first-order shear deformation theory and virtual work principle.

^{[6]}This paper is concerned to investigate the static bending and buckling response of Functionally Graded (FG) nanobeams by employing a new refined first order shear deformation beam theory.

^{[7]}Based on the first order shear deformation theory and meshless method, the governing equation of functionally graded graphene reinforced composite (FG-GRC) quadrilateral plates is derived and discretized to analyze the vibration behavior.

^{[8]}In this article, a comprehensive study is conducted on the free vibration analysis of rotating truncated conical shells reinforced with functionally graded agglomerated carbon nanotubes The shell is modeled based on the first-order shear deformation theory, and effective mechanical properties are calculated based on the Eshelby–Mori–Tanaka scheme along with the rule of mixture.

^{[9]}A mathematical model based on the first-order shear deformation theory and the von Karman’s nonlinear kinematics for buckling, postbuckling and failure analysis of elastic–plastic Functionally Graded Material (FGM) plate under thermomechanical is presented.

^{[10]}This article deals with thermal free vibration analysis of functionally graded material (FGM) plates and panels using an improved first-order shear deformable (I-FSDT) shell model.

^{[11]}In this study, we present a numerical solution for geometrically nonlinear dynamic analysis of functionally graded material rectangular plates excited to a moving load based on first-order shear deformation theory (FSDT) for the first time.

^{[12]}In this study, the first-order