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.     

Temperature-dependent discrete layer-differential quadrature bending analysis of the multi-layered functionally graded annular plates rested on a two-parameter elastic foundation

A discrete layer approach coupled with the differential quadrature method (DQM) is employed to temperature dependent analyze the laminated functionally graded (FG) annular plates under mechanical loading in a thermal environment. The formulations are derived based on the elasticity theory, which includes the effects of the initial thermal stresses and two-parameter elastic foundation. The material properties are assumed to be temperature-dependent and graded in the thickness direction. In order to accurately evaluate the effect of the thermal environment, the initial thermal stresses are obtained by solving the thermoelastic equilibrium equation. Comparison studies with the available solutions in the literature for FG plates are performed. Then, as an application, three common types of FG sandwich plates, namely, the sandwich with homogeneous face sheets and FG core and the sandwich with FG face sheets and homogeneous metal (soft) and ceramic (hard) core are analyzed. The influences of temperature rise, temperature-dependence of material properties, layers lay-out, foundation stiffness parameters, material graded index, and geometrical parameters on the solution are carried out. The new results can be used as benchmark solutions for future researches.     

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 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.     

The elasto-plastic impact analysis of functionally graded circular plates under low-velocities

In this study, three-dimensional elasto-plastic analysis of functionally graded (FG) circular plates under low-velocity impact loads is presented. The FG circular plate is composed of ceramic (SiC) and metal (Al) phases, varying in a predetermined fashion through the plate thickness. The elasto-plastic behaviour of the FG circular plate is described by the TTO model. In the analyses, the ceramic phase is taken to be an isotropic elastic material whereas the metal phase is taken to be elasto-plastic material in accordance with the TTO model. The locally effective material properties were evaluated using homogenization method which is based on the Mori–Tanaka scheme. The effects of compositional gradient exponent and impactor velocity on the elasto-plastic impact response of the FG circular plates are investigated, and results are presented graphical form. The compositional gradient exponent and impactor velocity have significant effects on the elasto-plastic impact response of the FG circular plates. The elasto-plastic impact response of FG circular plates is similar to those of homogeneous plates. Therefore, the TTO model can be used to describe the mechanical behaviour of FG plates beyond the elastic range, assuming that the material response is essentially governed by the spreading of plasticity in the metal phase.     

An electromechanical higher order model for piezoelectric functionally graded plates

Bidirectional flexure analysis of functionally graded (FG) plate integrated with piezoelectric fiber reinforced composites (PFRC) is presented in this paper. A higher order shear and normal deformation theory (HOSNT12) is used to analyze such hybrid or smart FG plate subjected to electromechanical loading. The displacement function of the present model is approximated as Taylor’s series in the thickness coordinate, while the electro-static potential is approximated as layer wise linear through the thickness of the PFRC layer. The equations of equilibrium are obtained using principle of minimum potential energy and solution is by Navier’s technique. Elastic constants are varying exponentially along thickness (z axis) for FG material while Poisson’s ratio is kept constant. PFRC actuator attached either at top or bottom of FG plate and analyzed under mechanical and coupled mechanical and electrical loading. Comparison of present HOSNT12 is made with exact and finite element method (FEM).     

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.     

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.     

A higher-order shear and normal deformation functionally graded plate model: some recent results

A higher-order shear and normal deformations plate theory is employed for stress analysis and free vibration of functionally graded (FG) elastic, rectangular, and simply (diaphragm) supported plates. Although functionally graded materials (FGMs) are highly heterogeneous in nature, they are generally idealized as continua with their mechanical properties changing smoothly with respect to the spatial coordinates. This idealization is required in order to obtain the closed-form solutions of some fundamental solid mechanics problems and also simplify the evaluation and development of numerical models of the structures made of FGMs. The material properties of FG plates such as Young’s moduli and material density are considered in this case to vary continuously in the thickness direction according to the volume fraction of constituents and mathematically modeled as exponential and power law functions. Poisson’s ratio is assumed to be constant. The effect of variation of material properties in terms of material grading index on the deformations, stresses, and natural frequency of FG plates is studied. The accuracy of the presented numerical solutions has been established with the solutions available of other models and the exact three-dimensional (3D) elasticity solutions.     

Surface effects on nonlinear vibration of embedded functionally graded nanoplates via higher order shear deformation plate theory

In this paper, free vibration behavior of functionally nanoplate resting on a Pasternak linear elastic foundation is investigated. The study is based on third-order shear deformation plate theory with small scale effects and von Karman nonlinearity, in conjunction with Gurtin–Murdoch surface continuum theory. It is assumed that functionally graded (FG) material distribution varies continuously in the thickness direction as a power law function and the effective material properties are calculated by the use of Mori–Tanaka homogenization scheme. The governing and boundary equations, derived using Hamilton's principle are solved through extending the generalized differential quadrature method. Finally, the effects of power-law distribution, nonlocal parameter, nondimensional thickness, aspect of the plate, and surface parameters on the natural frequencies of FG rectangular nanoplates for different boundary conditions are investigated.     

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.     

Meshless local Petrov–Galerkin method for coupled thermoelasticity analysis of a functionally graded thick hollow cylinder

In this article, coupled thermoelasticity (without energy dissipation) based on Green–Naghdi model is applied to functionally graded (FG) thick hollow cylinder. The meshless local Petrov–Galerkin method is developed to solve the boundary value problem. The Newmark finite difference method is used to treat the time dependence of the variables for transient problems. The FG cylinder is considered to be under axisymmetric and plane strain conditions and bounding surfaces of cylinder to be under thermal shock loading. The mechanical properties of FG cylinder are assumed to vary across thickness of cylinder in terms of volume fraction as nonlinear function. A weak formulation for the set of governing equations is transformed into local integral equations on local subdomains by using a Heaviside test function. Nodal points are regularly distributed along the radius of the cylinder and each node is surrounded by a uni-directional subdomain to which a local integral equation is applied. The Green–Naghdi coupled thermoelasticity equations are valid in each isotropic subdomain. The temperature and radial displacement distributions are obtained for some grading patterns of FGM at various time instants. The propagation of thermal and elastic waves is discussed in details. The presented method shows high capability and efficiency for coupled thermoelasticity problems.     

Bending analysis of functionally graded sectorial plates using Levinson plate theory

Based on the Levinson plate theory (LPT) and the first-order shear deformation plate theory (FST), the bending analysis of functionally graded (FG) thick circular sector plates is presented. The LPT solutions of FG sectorial plates are first expressed in terms of the solutions of the classical plate theory (CPT) for homogeneous sectorial plates and then presented using a direct method. It is assumed that the non-homogeneous mechanical properties of plate, graded through the thickness, are described by a power function of the thickness coordinate. The results are given in closed-form solutions and verified with the known data in the literature.     

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.     

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.     

Thermo elasticity solution of functionally graded,solid, circular,and annular plates integrated with piezoelectric layers using the differential quadrature method

Static analysis of functionally graded (FG) solid circular/annular plates imbedded in piezoelectric layers under thermo-electro mechanical load is investigated using the differential quadrature method. The plate has various edge boundary conditions and its material properties are assumed to vary in an exponential law with the Poisson ratio to be constant. The method is validated by comparing numerical results with the results obtained in the literature. The effects of the gradient index, thickness to radius ratio, and edges boundary conditions on the thermoelastic behavior of FG solid circular and annular plates are investigated.     

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.     

Three-Dimensional Solutions of Functionally Graded Piezo-Thermo-Elastic Shells and Plates Using a Modified Pagano Method

A modified Pagano method is developed for the three-dimensional (3D) coupled analysis of simply-supported, doubly curved functionally graded (FG) piezo-thermo-elastic shells under thermal loads. Four different loading conditions, applied on the lateral surfaces of the shells, are considered. The material properties of FG shells are regarded as heterogeneous through the thickness coordinate, and then specified to obey an exponent-law dependent on this. The Pagano method, conventionally used for the analysis of multilayered composite elastic plates/shells, is modified to be feasible for the present analysis of FG piezo-thermo-elastic plates/shells. The modifications include that a displacement-based formulation is replaced by a mixed formulation, a set of the complex-valued solutions of the system equations is transferred to the corresponding set of real-valued solutions, a successive approximation (SA) method is adopted and introduced in the present analysis, and the propagator matrix method is developed for the heat conduction analysis and the coupled piezo-thermo-elastic analysis of the FG shells. The influence of the material-property gradient index on the field variables, induced in the FG piezo-thermo-elastic shells and plates under the thermal load, is studied.     

Supersonic flutter prediction of functionally graded cylindrical shells

The supersonic flutter analysis of simply supported FG cylindrical shell for different sets of in-plane boundary conditions is performed. The aeroelastic equations of motion are constructed using Love’s shell theory and von Karman–Donnell-type of kinematic nonlinearity coupled with linearized first-order potential (piston) theory. The material properties are assumed to be temperature-dependant and graded across the thickness of the shell according to a simple power law. The temperature distribution is assumed to vary in the thickness direction and is obtained by solving the steady-state heat conduction equation. The pre-stresses due to the thermal and mechanical loadings are obtained by exact solution of the equilibrium equations. The Galerkin method is used to solve the aeroelastic equations of motion employing appropriate displacement functions. The effects of internal pressure and temperature rise on the flutter boundaries of the simply supported FG cylinder with different values of power-law index are investigated.     

Stochastic dynamic analysis of a functionally graded thick hollow cylinder with uncertain material properties subjected to shock loading

This paper presents an investigation of the stochastic dynamic response of a functionally graded (FG) thick hollow cylinder with uncertain material properties subjected to mechanical shock loading. The mechanical properties are considered to vary across thickness of FG cylinder as a non-linear power function of radius. To obtain the radial displacement in each point, the Navier equation in displacement form is derived using linear functionally graded elements. The Galerkin finite element and Newmark finite difference methods along with the Monte Carlo simulation are employed to deal with the statistical response of the FG cylinder. The mean and variance of radial displacements are calculated in various points across thickness for different values of volume fraction exponents. The results are used to quantify the effects of variations in the mechanical properties on the dynamic response and safety within the FG cylinder.