Higher order shear deformation effects on analysis of laminates with piezoelectric fibre reinforced composite actuators

A complete analytical solution for cross-ply composite laminates integrated with piezoelectric fiber-reinforced composite (PFRC) actuators under bi-directional bending is presented in this paper. A higher order shear and normal deformation theory (HOSNT12) is used to analyze such hybrid or smart laminates subjected to electromechanical loading. The displacement function of the present model is approximated by employing Taylor’s series in the thickness coordinate, while the electro-static potential is assumed to be layer wise (LW) linear through the thickness of PFRC. The equations of equilibrium are obtained using principle of minimum potential energy and solution is by Navier’s technique. Transverse shear stresses are presented at the interface of PFRC actuator and laminate under the action of electrostatic potentials. Results are compared with first order shear deformation theory (FOST) and exact solution.     

Nonlinear analysis of smart functionally graded annular sector plates using cylindrically orthotropic piezoelectric fiber reinforced composite

This work deals with the geometrically nonlinear thermo-electro-elastic analysis of functionally graded (FG) annular sector plates integrated with the annular patches of cylindrically orthotropic piezoelectric fiber reinforced composite (PFRC). The annular patches with an external voltage across their thickness act as the distributed actuators and their performance in controlling the nonlinear flexural deformations of the host FG plates is investigated. The temperature field is assumed to be spatially uniform over the plate surfaces and varied through the thickness of the substrate FG plates. The temperature-dependent material properties of the FG plates are assumed to be graded in the thickness direction of the plates according to a power-law distribution while the Poisson’s ratio is assumed to be a constant over the domain of the substrate plate. A finite element model of the overall smart FG annular sector plate is developed based on the first order shear deformation theory and the Von Karman nonlinear strain–displacement relations. The governing nonlinear finite element equations are derived employing the principle of minimum potential energy and solved using direct iteration method. The numerical results illustrate significant control authority of the cylindrically orthotropic PFRC annular patches for active control of nonlinear deformations of the substrate FG annular sector plates. The numerical results also reveal the best radial and circumferential locations of the annular PFRC patches for effective control. For a specified circumferential stretch of the annular PFRC patches, their minimum radial length is numerically estimated in such a way that the performance of the overall smart FG plate is not affected significantly. The effects of the material properties and the temperature of the host FG plate on the performance of the annular PFRC patches are also discussed.     

Hybrid damping of smart, functionally graded plates using piezoelectric, fiber-reinforced composites

This paper deals with the investigation of active, constrained layer damping (ACLD) of smart, functionally graded (FG) plates. The constraining layer of the ACLD treatment is considered to be made of a piezoelectric, fiber-reinforced composite (PFRC) material with enhanced effective piezoelectric coefficient that quantifies the in-plane actuating force due to the electric field applied across the thickness of the layer. The Young's modulus and the mass density of the FG plates are assumed to vary exponentially along the thickness of the plate, and the Poisson's ratio is assumed to be constant over the domain of the plate. A finite-element model has been developed to model the open-loop and closed-loop dynamics of the FG plates integrated with two patches of ACLD treatment. The frequency response of the plates revealed that the active patches of ACLD treatment significantly improve the damping characteristics of the FG plates over the passive damping. Emphasis has been placed on investigating the effect of variation of piezoelectric fiber angle in the constraining layer of the ACLD treatment on the attenuating capability of the patches. The analysis also revealed that the activated patches of the ACLD treatment are more effective in controlling the vibrations of FG plates when the patches are attached to the surface of the FG plates with minimum stiffness than when they are attached to the surface of the same with maximum stiffness.     

Hygrothermo-mechanical buckling of FGM plates resting on elastic foundations using a quasi-3D model

Hygrothermal and mechanical buckling responses of functionally graded (FG) plates resting on Winkler–Pasternak’s foundations are presented in this paper using a refined quasi-3D model. The effects due to transverse normal strain and shear deformation are both included. The present model exactly satisfies stress boundary conditions on the upper and lower surfaces of the FG plate without using shear correction factors. It is assumed that the material properties vary according to a power law of the thickness coordinate variable. The hygrothermal buckling equilibrium equations are derived from the principle of virtual work for FG plates resting on Winkler–Pasternak’s foundations with simply-supported boundary conditions. Two types of thermal and hygrothermal loading, uniform thermal and hygrothermal rise, linear thermal and hygrothermal distribution through the thickness are considered. Numerical results are presented to verify the accuracy of the present study. The effects played by Winkler–Pasternak’s parameters, plate aspect ratio, side-to-thickness ratio, gradient index, and loading type on the critical buckling of the FG plates are all investigated.     

Static solutions for functionally graded simply supported plates

In this article mixed semi-analytical and analytical solutions are presented for a rectangular plate made of functionally graded (FG) material. All edges of a plate are under simply supported (diaphragm) end conditions and general stress boundary conditions can be applied on both top and bottom surface of a plate during solution. A mixed semi-analytical model consists in defining a two-point boundary value problem governed by a set of first-order ordinary differential equations in the plate thickness direction. Analytical solutions based on shear-normal deformation theories are also established to show the accuracy, simplicity and effectiveness of mixed semi-analytical model. The FG material is assumed to be exponential in the thickness direction and Poisson’s ratio is assumed to be constant.     

Elastic response of functionally graded circular plates under a drop-weight

The three-dimensional impact behaviours of functionally graded (FG) circular plates were studied under a drop-weight. The functionally graded circular plate was composed of ceramic (SiC) and metal (Al) phases and the through-thickness mechanical properties through the region between the metal and ceramic layers vary continuously according to a power-law distribution of the volume fraction of the ceramic. The through-thickness material properties of the FG circular plate were determined using the Mori–Tanaka scheme. The effects of layer number and compositional gradient exponent as well as impactor velocity and plate radius on the elastic impact response of the FG circular plates were investigated. The compositional gradient exponent, impactor velocity and plate radius played an important role on the impact response of the FG circular plates, whereas the layer number through the plate thickness had a minor effect. In addition, the failure strains in all layers were determined using Tamura–Tomota–Ozowa (TTO) model in order to predict the damage regions in each layer through the thickness of FG circular plates.     

A comparison of buckling and postbuckling behavior of FGM plates with piezoelectric fiber reinforced composite actuators

Compressive postbuckling under thermal environments and thermal postbuckling due to a uniform temperature rise are presented for a simply supported, shear deformable functionally graded plate with piezoelectric fiber reinforced composite (PFRC) actuators. The material properties of functionally graded materials (FGMs) 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, and the material properties of both FGM and PFRC layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The initial geometric imperfection of the plate is taken into account. A two step perturbation technique is employed to determine buckling loads (temperature) and postbuckling equilibrium paths. The numerical illustrations concern the compressive and thermal postbuckling behaviors of perfect and imperfect, geometrically mid-plane symmetric FGM plates with fully covered or embedded PFRC actuators under different sets of thermal and electric loading conditions. The results for monolithic piezoelectric actuator, which is a special case in the present study, are compared with those of PFRC actuators. The results reveal that, in the compressive buckling case, the applied voltage usually has a small effect on the postbuckling load–deflection curves of the plate with PFRC actuators, whereas in the thermal buckling case, 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.     

A four-variable refined shear-deformation beam theory for thermo-mechanical vibration analysis of temperature-dependent FGM beams with porosities

This article proposes a four-variable shear deformation refined beam theory for thermo-mechanical vibration characteristics of porous, functionally graded (FG) beams exposed to various kinds of thermal loadings by using an analytical method. Thermo-mechanical properties of functionally graded material (FGM) beams are supposed to vary through the thickness direction, and are estimated through the modified power-law rule in which the porosities with even and uneven types are approximated. The material properties of FGM beams are supposed to be temperature dependent. Porosities possibly occur inside FGMs during fabrication because of technical problems that lead to the creation of microvoids in these materials. The variation of pores along the thickness direction influences the mechanical properties. Thus, it is incumbent to predict the effect of porosities on the thermo-mechanical vibration behavior of FG beam in the present study. Four types of thermal loading, namely, uniform, linear, nonlinear, and sinusoidal temperature rises through the z-axis direction are discussed. The governing differential equations and boundary conditions of FG porous beams subjected to thermal loadings are formulated through Hamilton's principle, based on a four-variable refined theory that considers a constant transverse displacement and higher order variation of axial displacement through the depth of the beam without the need of any shear correction factors. An analytical solution procedure is used to achieve the natural frequencies of porous FG beams subjected to various temperature fields. The impact of several specific parameters such as power-law exponent, porosity volume fraction, different porosity distribution, and thermal effect on the vibration of the porous FG beams is perused and discussed in detail. It is deduced that these parameters play a notable role on the thermo-dynamic behavior of porous FG beams. Presented numerical results can serve as benchmarks for the future analyses of FG beams with porosity phases.     

Optimal Analysis for Shakedown of Functionally Graded (FG) Bree Plate with Genetic Algorithm

The Shakedown of a functionally graded (FG) Bree plate subjected to coupled constant mechanical loading and cyclically varying temperature is analyzed with more accurate approaches and optimized with the genetic algorithm method. The shakedown theorem takes into account material hardening. The variation of the material properties in the thickness of a FG Bree plate is characterized with a piecewise exponential distribution, which can replicate the actual distribution with sufficient accuracy. In order to obtain the best distribution of the mechanical properties in the FG plate, the distribution of the reinforcement particle volume fraction is optimized with the genetic algorithm (GA). Two numerical examples are presented, which demonstrate the validity of the developed method in the analysis of the shakedown of the FG Bree plate.     

Free vibration analysis of functionally graded plates resting on Winkler–Pasternak elastic foundations using a new shear deformation theory

Free vibration analysis of simply supported functionally graded plates (FGP) resting on a Winkler–Pasternak elastic foundation are examined by a new higher shear deformation theory in this paper. Present theory exactly satisfies stress boundary conditions on the top and the bottom of the plate. The material properties change continuously through the thickness of the plate, which can vary according to power law, exponentially or any other formulations in this direction. The equation of motion for FG rectangular plates resting on elastic foundation is obtained through Hamilton’s principle. The closed form solutions are obtained by using Navier technique, and then fundamental frequencies are found by solving the results of eigenvalue problems. The numerical results obtained through the present analysis for free vibration of functionally graded plates on elastic foundation are presented, and compared with the ones available in the literature.     

Exact three-dimensional free vibration analysis of thick homogeneous plates coated by a functionally graded layer

Exact closed-form solutions are carried out for both in-plane and out-of-plane free vibration of thick homogeneous simply supported rectangular plates coated by a functionally graded (FG) layer, based on three-dimensional elasticity theory. The elasticity modulus and mass density of the FG coating are assumed to vary exponentially through the thickness of the coating layer, whereas Poisson’s ratio is remaining constant. The equations of motion are solved using two proposed displacement fields for the in-plane and out-of-plane vibration modes. By inserting the displacement fields in the 3-D elasto-dynamic equations, some independent ordinary equations are obtained and solved analytically. Natural frequencies are extracted by satisfying boundary conditions of interface and surfaces of the structure. The solution procedure is validated by comparing the obtained results with corresponding results of a 3-D finite element analysis. Finally, the influence of the FG coating layer on the natural frequencies of the structure is investigated and discussed. Clearly, the present closed-form solutions can exactly predict both in-plane and out-of-plane vibration modes of thick FG coated plates.     

Interfacial fracture of piezoelectric multilayer actuators under mechanical and electrical loading

The elastic stress and strain fields in a plate of finite thickness containing an elliptical hole are systematically investigated using the 3D finite element method. It is found that the stress and strain concentrations are different in the plate of finite thickness even if the plate is in an elastic state. The relation between the stress and the strain concentration factors depends on Poisson’s ratio, the hole’s geometric configuration and the plate thickness. The stress concentration factor is equal to the strain concentration factor only at the notch root of the plate surface. The stress (or strain) concentration factor at the notch root of the plate surface decreases rapidly with increasing thickness and becomes lower than the stress and strain concentration factors corresponding to the plane stress state or at the notch root of the mid plane. It is too low to reflect the overall stress concentration as the thickness increases or as the b/a ratio decreases. The maximum stress concentration factor occurs on the mid plane only when the plate is thinner than the transition thickness of the stress concentration factor. When the plate is thicker than the transition thickness of the stress concentration factor, the distance between the location of the maximum stress concentration factor and plate surface tends to be constant with increasing thickness for the plate with a given  b/a ratio. The differences between the maximum value and the surface value of the stress and strain concentration factors increase rapidly and tend to their respective constant values with increasing plate thickness. The smaller the b/a ratio, the larger these differences. The difference of the stress concentration factor is larger than that of the strain concentration factor in the same plate.     

On deformation of functionally graded narrow beams under transverse loads

Solution is obtained for functionally graded (FG) narrow beams under plane stress condition of elasticity by using the mixed semi analytical model developed by Kant et al. (Int J Comput Methods Eng Sci Mech 8(3): 165–177, 2007a). The mathematical model consists in defining a two-point boundary value problem (BVP) governed by a set of coupled first-order ordinary differential equations (ODEs) in the beam thickness direction. Analytical solutions based on two dimensional (2D) elasticity, one dimensional (1D) first order shear deformation theory (FOST) and a new 1D higher order shear-normal deformation theory (HOSNT) are also established to show the accuracy, simplicity and effectiveness of the developed mixed semi analytical model. It is observed from the numerical investigation that the present mixed semi analytical model predicts structural response as good as the one given by the elasticity analytical solution which in turn proves the robustness of the present development.     

Nonlinear behavior of functionally graded circular plates with various boundary supports under asymmetric thermo-mechanical loading

The equilibrium equations of the first-order nonlinear von Karman theory for FG circular plates under asymmetric transverse loading and heat conduction through the plate thickness are reformulated into those describing the interior and edge-zone problems of the plate. A two parameter perturbation technique, in conjunction with Fourier series method is used to obtain analytical solutions for nonlinear behavior of functionally graded circular plates with various clamped and simply-supported boundary conditions. The material properties are graded through the plate thickness according to a power-law distribution of the volume fraction of the constituents. The results are verified with known results in the literature. The load–deflection curves for different loadings, boundary conditions, and material constant in a solid circular plate are studied and discussed. It is shown that the behavior of FG plates with clamped or simply-supported boundary conditions are completely different. Under thermo-mechanical loading, snap-through buckling behavior is observed in simply-supported FG plates which are immovable in radial direction. Moreover, it is found that linear theory is inadequate for analyzing FG and also homogenous plates with immovable boundary supports in radial direction and subjected to thermal loading, even for deflections that are normally considered small.     

Bending response of functionally graded plates by using a new higher order shear deformation theory

This paper presents an analytical solution to the static analysis of functionally graded plates, using a recently developed higher order shear deformation theory (HSDT) and provides detailed comparisons with other HSDT’s available in the literature. These theories account for adequate distribution of the transverse shear strains through the plate thickness and tangential stress-free boundary conditions on the plate boundary surfaces, thus a shear correction factor is not required. The mechanical properties of the plates are assumed to vary in the thickness direction according to a power-law distribution in terms of the volume fractions of the constituents. The governing equations of a functionally graded (FG) plate and boundary conditions are derived by employing the principle of virtual work. Navier-type analytical solution is obtained for FG plates subjected to transverse bi-sinusoidal and distributed loads for simply supported boundary conditions. Results are provided for thick to thin FG plates and for different volume fraction distributions. The accuracy of the present code is verified by comparing it with known results in the literature.     

A 3-D Ritz solution for free vibration of circular/annular functionally graded plates integrated with piezoelectric layers

This paper addresses three-dimensional (3-D) free vibration characteristic of thick circular/annular functionally graded (FG) plates with surface-bonded piezoelectric layers on the basis of 3-D Ritz solution. Three displacement components along with electrical potential field of the plate are expressed by a set of Chebyshev polynomials multiplied by geometry boundary functions. Both open-circuit and closed-circuit surface conditions are taken into account. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey either exponent or power law distribution of the volume fraction of the constituents. The effect of thickness-to-radius ratio, inner-to-outer radius ratio, piezo-to-host thickness ratio and gradient index on the natural frequencies of coupled piezoelectric FG circular/annular plates is investigated for different electrical and mechanical boundary conditions. It is observed that, unlike isotropic homogeneous circular/annular plates, frequency parameters of their piezoelectric coupled FG counterparts significantly increase with an enhancement in the host plate thickness to radius ratio. Results also show that the frequency parameters for open-circuit condition are higher than those for closed-circuit condition.     

Thermo-Electro-Mechanical Buckling of Shear Deformable Hybrid Circular Functionally Graded Material Plates

Buckling analysis of perfect circular functionally graded plates with surface-bounded piezoelectric layers based on the first-order shear deformation theory is presented in this article. The material properties of the functionally graded (FG) layer are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituents. The plate is assumed to be under constant electrical field and two types of thermal loadings, namely, the uniform temperature rise and nonlinear temperature gradient through the thickness. Also, the stability of a plate under radial mechanical compressive force is examined. The equilibrium and stability equations are derived based on the first-order shear deformation plate theory using a variational approach. The boundary condition of the plate as an immovable type of the clamped edge is considered. Resulting equations are employed to obtain the closed-form solution for the critical buckling temperature for each loading case. The effects of electric field, piezo-to-host thickness ratio, and power law index of functionally graded plates subjected to thermo-mechanical-electrical loads are investigated. The results are compared with the classical plate theory and verified with the available data in the open literature.     

Thermo-electrical buckling of piezoelectric functionally graded material Timoshenko beams

In this article, buckling analysis of functionally graded material (FGM) beams with or without surface-bonded piezoelectric layers subjected to both thermal loading and constant voltage is studied. Thermal and mechanical properties of FGM layer is assumed to follow the power law distribution in thickness direction, except Poisson’s ratio which is considered constant. The Timoshenko beam theory and nonlinear strain-displacement relations are used to obtain the governing equations of piezoelectric FGM beam. Beam is assumed under three types of thermal loading and various types of boundary conditions. For each case of boundary conditions, existence of bifurcation-type buckling is examined and for each case of thermal loading and boundary conditions, closed-form solutions are obtained which are easily usable for engineers and designers. The effects of the applied actuator voltage, beam geometry, boundary conditions, and power law index of FGM beam on critical buckling temperature difference are examined.     

Nonlinear vibration of smart circular functionally graded plates coupled with piezoelectric layers

Nonlinear vibration analysis of thin circular pre-stressed functionally graded (FG) plate integrated with two uniformly distributed piezoelectric actuator layers with an initial nonlinear large deformation are presented in this paper. Nonlinear governing equations of motion are derived based on classical plate theory (CPT) with von-Karman type geometrical large nonlinear deformations. A nonlinear static problem is solved first to determine the initial stress state and pre-vibration deformations of the plate that is subjected to in-plane forces and applied actuator voltage. By adding an incremental dynamic state to the pre-vibration state, the differential equations that govern the nonlinear vibration behavior of pre-stressed piezoelectrically actuated circular FG plate are derived. An exact series expansion method is used to model the nonlinear electro-mechanical vibration behavior of the structure. Control of the FG plate’s nonlinear deflections and natural frequencies using high control voltages are studied and their nonlinear effects are evaluated. In a parametric study the emphasis is placed on investigating the effect of varying the applied actuator voltage as well as gradient index of FG plate on the dynamic characteristics of the structure.     

A comprehensive analysis for the shakedown of a Bree plate made of functionally graded materials

The shakedown of a functionally graded (FG) plate subjected to coupled constant mechanical load and cyclically varying temperature is analyzed comprehensively. The material of the plate is composed of an elastoplastic matrix and elastic particles, and the particle volume fraction varies through the thickness. The distributions of the effective mechanical and thermal properties of the composites through the thickness are evaluated with mean-field approaches and described with an exponential law. The temperature dependence of the material properties is taken into account. The distribution of temperature change and the shakedown of a typical FG Bree plate are analyzed. The comparison with the results of its homogeneous counterpart and that without considering the temperature dependence of the material properties exhibits marked qualitative and quantitative difference. The effect of the temperature dependence of the elastic properties of materials is also investigated. Since FG structures are usually subject to severe coupled thermal-mechanical loadings, the approach developed and the results obtained are significant for the analysis and design of such kind of structures.