Nonlinear finite element analysis of functionally graded plates integrated with patches of piezoelectric fiber reinforced composite

In this paper, a nonlinear static finite element analysis of simply supported smart functionally graded (FG) plates in the presence/absence of the thermal environment has been presented. The substrate FG plate is integrated with the patches of piezoelectric fiber reinforced composite (PFRC) material which act as the distributed actuators of the plate. The material properties of the FG substrate plate are assumed to be temperature dependent and graded along the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The derivation of this nonlinear thermo-electro-mechanical coupled finite element model is based on the first order shear deformation theory and the Von Karman type geometric nonlinearity. The numerical solutions of the nonlinear equations of the finite element model are obtained by employing the direct iteration method. The numerical illustrations suggest the potential use of the distributed actuator made of the PFRC material for active control of nonlinear deformations of smart FG structures. The effects of volume fraction index of the FG material of the substrate plates and the locations of the PFRC patches on the control authority of the patches are investigated. Emphasis has also been placed on investigating the effect of variation of piezoelectric fiber orientation angle in the PFRC patches on their actuation capability for counteracting the large deflections of FG plates.     

3D thermo-mechanical bending solution of functionally graded graphene reinforced circular and annular plates

Within the framework of three-dimensional elasticity theory, this paper investigates the axisymmetric bending of novel functionally graded polymer nanocomposite circular and annular plates reinforced with graphene nanoplatelets (GPLs) whose weight fraction varies continuously and smoothly along the thickness direction. The generalized Mian and Spencer method is utilized to obtain the analytical solutions of nanocomposite circular and annular plates under a combined action of a uniformly distributed transverse load and a through-thickness steady temperature field. Three different distribution patterns of GPLs within the polymer matrix are considered. The present analytical solutions are validated through comparisons against those available in open literature for the reduced cases. A parametric study is conducted to examine the effects of GPL weight fraction, distribution pattern, plate thickness to radius ratio, and boundary conditions on the stress and deformation fields of the plate. The results show that GPL nanofillers with a low content can have a significant reinforcing effect on the bending behavior of the thermo-mechanically loaded plate.     

The propagation of elastic waves in composite materials reinforced with piezoelectric fibres

The propagation of longitudinal elastic waves in composite materials, consisting of a polymer matrix reinforced by continuous fibres in one direction, is considered. The reinforcing fibres have piezoelectric properties and have a thin current-conducting coating (“shunted fibres”). The scattering of electric energy in such materials leads to dispersion of the velocity of the elastic waves and to their attenuation. The effective-field method is used to determine the macroscopic electroelastic constants of such composites. These constants enable one to obtain, in explicit form, the frequency dependence of the real and imaginary parts of the wave number of a longitudinal wave, propagating along the reinforcement direction, and also their dependenc on the physical and geometrical characteristics of the components.     

Nonlinear in-plane buckling of fixed shallow functionally graded graphene reinforced composite arches subjected to mechanical and thermal loading

The nonlinear in-plane buckling analysis for fixed shallow functionally graded (FG) graphene reinforced composite arches which are subjected to uniform radial load and temperature field is presented in this paper. The arch is composed of multiple graphene platelet reinforced composite (GPLRC) layers with gradient changes of concentration of graphene platelets (GPLs) in each layer. The principle of virtual work, combined with the effective materials properties estimated by the Halpin-Tsai micromechanics model for GPLRC layer, is used to derive the nonlinear buckling equilibrium equations of the FG-GPLRC arch, and then the analytical solutions for the limit point and bifurcation buckling loads are obtained. Comprehensive parametric studies are conducted to explore the effects of various distribution patterns and geometries of GPL, temperature field and arch geometry on the nonlinear equilibrium path and buckling behavior of the composite arch. The influence of temperature on the geometric parameters which are defined as switches between limit point buckling, bifurcation buckling and no buckling are also discussed. It is found that a higher temperature field can increase the buckling loads of the FG-GPLRC arch but reduce the value of the minimum geometric parameters that switching the buckling modes. The results also show that even a small amount of GPLs filler content can increase the buckling loads of the FG-GPLRC arch considerably, and distributing more GPLs near the surface layers is the best pattern to enhance the buckling performances of FG-GPLRC arches.     

Approximation of functionally graded plates with non-conforming finite elements

In this paper rectangular plates made of functionally graded materials (FGMs) are studied. A two-constituent material distribution through the thickness is considered, varying with a simple power rule of mixture. The equations governing the FGM plates are determined using a variational formulation arising from the Reissner–Mindlin theory. To approximate the problem a simple locking-free Discontinuous Galerkin finite element of non-conforming type is used, choosing a piecewise linear non-conforming approximation for both rotations and transversal displacement. Several numerical simulations are carried out in order to show the capability of the proposed element to capture the properties of plates of various gradings, subjected to thermo-mechanical loads.     

Compressive and thermal postbuckling behaviors of laminated plates with piezoelectric fiber reinforced composite actuators

This paper studied compressive postbuckling under thermal environments and thermal postbuckling due to a uniform temperature rise for a shear deformable laminated plate with piezoelectric fiber reinforced composite (PFRC) actuators based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The material properties are assumed to be temperature-dependent and the initial geometric imperfection of the plate is considered. The compressive and thermal postbuckling behaviors of perfect, imperfect, symmetric cross-ply and antisymmetric angle-ply laminated plates with fully covered or embedded PFRC actuators are conducted under different sets of thermal and electric loading conditions. The results reveal that, the applied voltage usually has a small effect on the postbuckling load–deflection relationship of the plate with PFRC actuators in the compressive buckling case, whereas the effect of applied voltage is more pronounced for the plate with PFRC actuators, compared to the results of the same plate with monolithic piezoelectric actuators.     

Active control of dynamic behaviors of graded graphene reinforced cylindrical shells with piezoelectric actuator/sensor layers

This work investigates the active vibration control and vibration characteristics of a sandwich thin cylindrical shell whose intermediate layer is made of the graphene reinforced composite that is bonded with integrated piezoelectric actuator and sensor layers at its outer and inner surfaces. The volume fraction of graphene platelets in the intermediate layer varies continuously in the shell's thickness direction, which generates position-dependent effective material properties. The constitutive relations of the graphene reinforced composite and piezoelectric materials are given by taking one-dimensional steady thermal field into account. Considering Donnell's shell theory, a final equation of motion in terms of the generalized radial displacement is derived by using Hamilton's principle and Galerkin method. Shell's natural frequencies are derived considering influences of the thermo-electro-elastic field. Introducing a constant velocity feedback control algorithm, active vibration control of the sandwich cylindrical shell is presented by employing the Runge-Kutta method. The feedback control gain has a pronounced effect on the damping, as well as the inertia of the system. Comparisons between the present results and those in other papers are done to validate the present solutions. Influences of weight fractions, distribution patterns and geometrical sizes of graphene platelets, temperature variations, thicknesses of layers and the feedback control gain on the vibration characteristics and active vibration control behaviors of the novel sandwich cylindrical shell are discussed.     

Analysis of functionally graded plates using higher order shear deformation theory

This work addresses a static analysis of functionally graded material (FGM) plates using higher order shear deformation theory. In the theory the transverse shear stresses are represented as quadratic through the thickness and hence it requires no shear correction factor. The material property gradient is assumed to vary in the thickness direction. Mori and Tanaka theory (1973) [1] is used to represent the material property of FGM plate at any point. The thermal gradient across the plate thickness is represented accurately by utilizing the thermal properties of the constituent materials. Results have been obtained by employing a C° continuous isoparametric Lagrangian finite element with seven degrees of freedom for each node. The convergence and comparison studies are presented and effects of the different material composition and the plate geometry (side-thickness, side–side) on deflection and temperature are investigated. Effect of skew angle on deflection and axial stress of the plate is also studied. Effects of material constant n on deflection and the temperature distribution are also discussed in detail.     

Dynamic analysis of multi-directional functionally graded annular plates

Dynamic analysis of multi-directional functionally graded annular plates is achieved in this paper using a semi-analytical numerical method entitled the state space-based differential quadrature method. Based on the three-dimensional elastic theory and assuming the material properties having an exponent-law variation along the thickness, radial direction or both directions, the frequency equations of free vibration of multi-directional functionally graded annular plates are derived under various boundary conditions. Numerical examples are presented to validate the approach and the superiority of this method is also demonstrated. Then free vibration of functionally graded annular plates is studied for different variations of material properties along the thickness, radial direction and both directions, respectively. And the influences of the material property graded variations on the dynamic behavior are also investigated. The multi-directional graded material can likely be designed according to the actual requirement and it is a potential alternative to the unidirectional functionally graded material.     

Complex guided waves in functionally graded piezoelectric cylindrical structures with sectorial cross-section

For the purpose of the design and optimization of piezoelectric transducers, the modified double orthogonal polynomial series method is proposed to investigate guided waves in functionally graded piezoelectric(FGP) cylindrical structures with sectorial cross-section. The real, imaginary and complex solutions are obtained simultaneously without iterative process. The real solutions represent propagative waves; the imaginary and complex solutions are evanescent waves. The boundary conditions are incorporated into the constitutive equations by virtue of the Heaviside function. Subsequently, the amplitudes are expanded into the double orthogonal polynomial series, and the motion equations are converted into a matrix eigenvalue problem about complex wavenumber. Numerical comparison with available reference result confirms the validity of the present method. Dispersion curves and the Poynting vector distributions are illustrated. The influences of angular measure, radius-thickness ratio and graded index on dispersion curves are analyzed. Results show that there exist some evanescent guided wave modes that have higher velocities than that of the propagative wave modes and simultaneously have low attenuation at high frequencies. These results can be utilized to improve the performance of transducers.     

Influence of nonlinear terms on dynamical behavior of graphene reinforced laminated composite plates

This research is focused on the effects of nonlinear terms on the dynamical behavior of graphene reinforced laminated composite plates. Firstly, the governing equations of the graphene reinforced composite thin plate subjected to transverse excitations are derived by using the Hamilton's principle and the von Karman deformation theory. Then numerical method is applied to investigate the nonlinear behaviors of graphene reinforced composite plates. Bifurcation diagram, waveform and phase portrait are demonstrated to analyze the nonlinear dynamics of the graphene reinforced laminated composite plates. Furthermore, the effects of nonlinear terms on the dynamical behavior are discussed in detail, where both the stronger and weaker nonlinear characteristics of lower modes of the plate are presented. Moreover, some interesting phenomena are obtained in numerical simulation.     

On wave propagation in two-dimensional functionally graded porous rotating nano-beams using a general nonlocal higher-order beam model

This paper studies the wave propagation of two-dimensional functionally graded (2D-FG) porous rotating nano-beams for the first time. The rotating nano-beams are made of two different materials, and the material properties of the nano-beams alter both in the thickness and length directions. The general nonlocal theory (GNT) in conjunction with Reddy's beam model are employed to formulate the size-dependent model. The GNT efficiently models the dispersions of acoustic waves when two independent nonlocal fields are modelled for the longitudinal and transverse acoustic waves. The governing equations of motion for the 2D-FG porous rotating nano-beams are established using Hamilton's principle as a function of the axial force due to centrifugal stiffening and displacement. The analytic solution is applied to obtain the results and solve the governing equations. The effect of the features of different parameters such as functionally graded power indexes, porosity, angular velocity, and material variation on the wave propagation characteristics of the rotating nano-beams are discussed in detail.     

Free vibration of a functionally graded annular sector plate integrated with piezoelectric layers

Based on the first order shear deformation theory, free vibration behavior of functionally graded (FG) annular sector plates integrated with piezoelectric layers is investigated. The distribution of electric potential along the thickness direction of piezoelectric layers which is assumed to be a combination of linear and sinusoidal functions, satisfies both open and closed circuit electrical boundary conditions. Through a reformulation of governing equations and harmonic motion assumption, a novel decoupling method is suggested to transform the six second order coupled partial differential equations of motion into two eighth order and fourth order equations. A Fourier series method is then employed to present analytical solutions for free vibration of smart FG annular sector plates with simply supported radial edges and arbitrarily supported circular edges. The results, which can be used as a benchmark and suitable for design purposes, are verified with those reported in the literature. Finally, by presenting extensive ranges of frequencies, the effects of geometric parameters, power law index, FG and piezoelectric materials, electrical and mechanical boundary conditions as well as the piezoelectric layer thickness on vibration response of smart annular sector plates are discussed in detail.     

Longitudinal modeling and properties tailoring of functionally graded carbon nanotube reinforced composite beams: A novel approach

Carbon Nanotubes (CNTs) are considered a promising reinforcement for engineering advanced structures, such as aerospace applications. It became inevitable to tailor structures not only by conventional geometric but also by mechanical properties tailoring, to acquire the maximum load-carrying capacity. Therefore, this research presented a novel Finite Element (FE) modeling of Axially Functionally Graded CNT Reinforced Composite (AFG-CNTRC) beams, based on Timoshenko beam theory. However, an obstacle is raised due to properties discontinuity produced between consecutive elements in axial gradation. A solution is proposed based on the ladder-pattern concept. It defined a locally specific value at each element, satisfying the overall required distribution. It followed by a convergence test, static, and dynamic analysis with comprehensive parametric studies. The obtained results showed that the model is convergent, accurate, free of shear-locking, and suitable for axial gradation. Moreover, this study introduced an advanced technique for axial properties tailoring, utilizing privileges of CNT and aggregation. The longitudinal tailoring depends on both the CNT orientation angle and gradation index, which played a crucial role in response control and adjusting beam stiffness together with strength. This approach could solve enormous engineering problems, especially at critical cross-sections design and constraint response, without affecting the total weight.     

Two-dimensional solution for electro-mechanical behavior of functionally graded piezoelectric hollow cylinder

In this paper, the general theoretical analysis for a hollow cylinder made of functionally graded piezoelectric material subjected to two-dimensional electromechanical load, is developed. The material properties, except the Poisson’s ratio, are assumed to vary with the power law function through the thickness of the cylinder. The mechanical and electrical displacements are assumed to be a function of radial and circumferential directions. By using the separation of variables method and complex Fourier series, the Navier equations in terms of displacements are derived and solved.     

Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells

This paper presents the report of an investigation into thermoelastic vibration and buckling characteristics of the functionally graded piezoelectric cylindrical, where the functionally graded piezoelectric cylindrical shell is made from a piezoelectric material having gradient change along the thickness, such as piezoelectricity and dielectric coefficient et al. Here, utilizing Hamilton’s principle and the Maxwell equation with a quadratic variation of the electric potential along the thickness direction of the cylindrical shells and the first-order shear deformation theory, and taking into account both the direct piezoelectric effect and the converse piezoelectric effect, the thermoelastic vibration and buckling characteristics of functionally graded piezoelectric cylindrical shells composed of BaTiO₃/PZT − 4, BaTiO₃/PZT − 5A and BaTiO₃/PVDF are, respectively, calculated. The effects of material composition (volume fraction exponent), thermal loading, external voltage applied and shell geometry parameters on the free vibration characteristics are described, and the axial critical load, critical temperature and critical control voltage are obtained.     

Characteristics of wave propagation in piezoelectric bent rods with arbitrary curvature

The wave propagation in the piezoelectric bend rods with arbitrary curvature is studied in this paper. Basic three-dimensional equations in an orthogonal curvilinear coordinate system (r, θ, s) are established. The Bessel functions in radial co-ordinate and triangle series in the angular co-ordinate are used to describe the displacements and electrical potential. Characteristics of dispersion, distributions of displacements and electrical potential over the cross section are calculated, respectively. In the numerical examples, the effects of the ratio of the two ellipse axes on the dispersion relations of the first three modes are observed. The characteristics of the distribution of displacements and electric potential in the cross section, along the radial and s direction are investigated.     

Elasticity solutions for functionally graded rectangular plates with two opposite edges simply supported

England (2006) [13] proposed a novel method to study the bending of isotropic functionally graded plates subject to transverse biharmonic loads. His method is extended here to functionally graded plates with materials characterizing transverse isotropy. Using the complex variable method, the governing equations of three plate displacements appearing in the expansions of displacement field are formulated based on the three-dimensional theory of elasticity for a transverse load satisfying the biharmonic equation. The solution may be expressed in terms of four analytic functions of the complex variable, in which the unknown constants can be determined from the boundary conditions similar to that in the classical plate theory. The elasticity solutions of an FGM rectangular plate with opposite edges simply supported under 12 types of biharmonic polynomial loads are derived as appropriate sums of the general and particular solutions of the governing equations. A comparison of the present results for a uniform load with existing solutions is made and good agreement is observed. The influence of boundary conditions, material inhomogeneity, and thickness to length ratio on the plate deflection and stresses for the load x²yq are studied numerically.     

Generalized shear deformation theory for bending analysis of functionally graded plates

In this study, the static response is presented for a simply supported functionally graded rectangular plate subjected to a transverse uniform load. The generalized shear deformation theory obtained by the author in other recent papers is used. This theory is simplified by enforcing traction-free boundary conditions at the plate faces. No transversal shear correction factors are needed because a correct representation of the transversal shearing strain is given. Material properties of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of the volume fractions of the constituents. The equilibrium equations of a functionally graded plate are given based on a generalized shear deformation plate theory. The numerical illustrations concern bending response of functionally graded rectangular plates with two constituent materials. The influences played by transversal shear deformation, plate aspect ratio, side-to-thickness ratio, and volume fraction distributions are studied. The results are verified with the known results in the literature.     

Effects of initial stress on guided waves in orthotropic functionally graded plates

Because of the limitation of the manufacturing technology, initial stress in functionally graded materials (FGM) and structures is inevitable. Based on the theory of “Mechanics of Incremental Deformations”, the guided wave propagation in FGM plates under gravity, homogeneous initial stress in the thickness direction and inhomogeneous initial stress in the wave propagation direction is investigated. The Legendre polynomial series method is used to solve the coupled wave equations with variable coefficients. The convergence of the polynomial series method is discussed through the numerical examples. The effects of the initial stress on the Lamb-like wave and on the SH wave are investigated respectively and the numerical results show they are quite distinct. The effect of the gravity on the wave propagation can be ignored. The effects of the initial stress in the thickness direction are very different from those of the initial stress in the wave propagation direction, both on the dispersion curves and on the displacement and stress distributions.