Thermal Mechanical Bending Response of Symmetrical Functionally Graded Material Plates

This paper presents an analytical solution for the thermomechanical buckling of functionally graded material (FGM) sandwich plates. The solution is obtained using a four-variable equivalent-single-layer (ESL) plate theory. Two types of sandwich plates are included: one with FGM facesheets and homogeneous core, and vice versa for the other. The governing equations are derived based on the principle of minimum total potential energy. For simply supported boundary conditions, these equations are solved via the Navier method. The results on critical buckling load and temperature increment of simply supported FGM sandwich plates are compared with the available solutions in the literature. Several results are presented considering various material and geometrical parameters as well as their effect on the thermomechanical buckling response of FGM sandwich plates. The relationship between the mechanical load and the temperature increment for uniform/linear temperature rise of FGM sandwich plates under combined mechanical and thermal loads is studied.

A new fluid–solid interface algorithm for simulating fluid structure problems in FGM plates

Functionally Graded Material (FGM) is a heterogeneous composite material that consists of a gradient compositional variation of the constituent materials from one surface of the material to the other. These continuous changes result in gradient material properties. Since ceramic has good heat resistance and metal has high strength, FGM made by ceramic and metal can work at super high temperatures or under a high-temperature-difference field. It is a primary to reduce thermal stress by selection of different effective material properties for the intermediate composition of the EGM and to prevent destruction by thermal stress. FGM is manufactured at a high temperature and then residual thermal stresses are produced during cooling to room temperature. In this paper, the elastic-plastic thermal stresses induced in a ceramic-metal FGM plate (FGP) taking the fabrication process into consideration are discussed. The region near the heat resistant surface is produced by metal particle reinforced ceramic while the region near the cooling surface is vice versa. As the metal and the ceramic near the middle region of the FGM are perfectly mixed, it is impossible to consider the particle-reinforced material. In this study, the FGP is divided into three regions. First, the region near the cooling surface is metal rich and then the metal is considered as a matrix while the ceramic is considered as particles. Second, the region near the heat resistant surface is ceramic rich so that the ceramic is considered as a matrix while the metal is considered as particles. Third, in the middle part between the previous two regions the metal and ceramic are perfectly mixed. In the third region macroscopic analysis is considered because the difference between the volume fractions of the ceramic and the metal is small and it is difficult to consider one of them as a matrix or particles. The effects of the distribution parameter of the composition and the fabrication temperature on the thermal stress variations are discussed.

An electromechanically coupled zigzag theory is presented for static and free vibration responses in functionally graded material (FGM) plates bonded with piezoelectric layers. It has the capability to consider continuity as well as discontinuity in material properties at all interfaces. The displacement field approximation, which is initially layer based, has been made layer independent by using continuity conditions on displacements and transverse shear stresses as well as zero traction conditions on top and bottom surfaces. Material properties in FGM layers vary continuously in the thickness direction. Piezoelectric layers have constant property across their thickness. Extended Hamilton’s principle for piezoelectric continuum has been used in deriving governing equations of motion and boundary conditions. An analytical solution is obtained for elastic as well as piezoelectric layer bonded FGM plates. Static and free vibration responses have been obtained in a number of FGM plates, with each plate considered to contain a number of FGM layers with discontinuity or continuity in material properties at the interfaces. Wherever available, present responses have been compared with three-dimensional exact solutions and other shear deformation theories. The effects of the aspect ratio, length-to-width ratio, and different schemes for effective material property calculation have been illustrated.

This work presents a finite element formulation for the vibration analysis of functionally graded material (FGM) plates containing microstructural defects operating in thermal environments. The material properties are assumed to be temperature-dependent and are modeled as higher-order functions of temperature. The volume fractions of the constituent materials are varied through the plate thickness according to a power-law distribution. The effective material properties of the FGM are evaluated using both the Mori–Tanaka homogenization scheme and the rule of mixtures. The kinematics of the FGM plates are described using an efficient layerwise theory. The governing equations of motion, constitutive relations, and boundary conditions are derived using on Hamilton’s principle. The stiffness coefficients of the plates are computed using a sub-laminate mechanics approach, wherein the plate thickness is discretized into several thin sublayers, each assumed to possess uniform material properties. This approach is particularly effective for micromechanical models where exact analytical integration of stiffness coefficients is difficult due to the complex polynomial variation of properties across the thickness. A four-noded quadrilateral finite element, previously developed for laminated and FGM shells, is extended in this study to analyze the dynamic behavior of FGM plates with microstructural defects and temperature dependent material properties. The natural frequencies are obtained using a subspace iteration technique, while the transient responses under step and impulse excitations are computed using the state-space form of the governing dynamic equations. The proposed finite element model is validated against three-dimensional elasticity and Ritz-based semi-analytical solutions, demonstrating excellent accuracy and reliability across varying thermal environments. A detailed parametric study is also carried out to investigate the effects of temperature, inhomogeneity index, porosity concentration, boundary conditions, and geometric parameters on the natural frequencies and transient responses of porous FGM plates.

Analytical formulations and solutions for natural frequency analysis of functionally graded material (FGM) plates based on two higher-order refined shear deformation theories with 9 and 12 degrees-of-freedom are presented. The displacement model with 12 degrees-of-freedom considers the effect of both transverse shear and normal strain/stress while the other considers only the effect of transverse shear deformation. In addition another higher-order model and the first-order model developed by other investigators and available in the literature are also presented for the evaluation purpose. For mathematical modeling purposes, the Poissons ratio of the material is considered as constant whereas Youngs modulus is assumed to vary through the thickness according to the power law function. The equations of motion are derived using Hamiltons principle. Solutions are obtained in closed-form using Naviers technique and solving the eigenvalue equation. The accuracy of the theoretical formulations and the solution method using the present two higher-order refined models is first established by comparing the results generated in the present investigation with the 3D elasticity solutions already reported in the literature. After establishing the accuracy of predictions, benchmark results for the natural frequencies using all the four models are presented for single layer FGM plate and multi layered FGM sandwich plate with varying edge ratios and side-to-thickness ratios.

This paper presents the convergence behavior of Discrete Kirchhoff Mindlin Triangular (DKMT) element in buckling analysis under uniaxial compression of square plate problems. The DKMT element has a good result for a thin plate and a thick plate. For the Functionally Graded Material (FGM) problem, the DKMT element is reformulated. FGM is a graded composite material that has high-temperature and structural flexibility resistance. The numerical results of mechanical buckling of square FGM plate under uniaxial compression using the DKMT element are reported. The critical buckling of the square FGM plate is compared to the reference existing solutions. The effects of parametric variation, such as the type of meshing, boundary conditions, power-law index, and ratio L/h are presented. The results show that the DKMT element gives good results on the buckling analysis of square FGM plate problems.

The dynamic response of the functionally graded material (FGM) plate under moving load considering the effect of temperature is investigated. The FGM plate properties are graded in the thickness direction based on the power-law distribution in terms. The Mindlin plate theory is employed to calculate the shear strain of the plate. The properties of the FGM plate such as the elastic modulus and coefficient of expansion due to effect of temperature are modeled. It is assumed that temperature variation is only in the direction of the plate thickness and that temperature field is assumed to be constant in the plane. Temperature distribution function along the plate thickness can be obtained by solving the governing equation of temperature transfer. Both mechanical strain and temperature-induced strain are employed to calculate the plate strain. Comparison study with previous results were performed to verify the effectiveness of the proposed model. Effects of temperature, moving load, plate thickness, and foundation parameters on the FGM plate are examined. It is found that they have a significant effect on the displacement of the plate. As to be expected, the plate displacement depends on the temperature, volume fraction index as well as velocity, and magnitude of moving load, being larger when they are larger.KeywordsFGMMEMTemperatureMoving load

摘要: 研究矩形截面功能梯度材料(Functionally graded materials, FGM)梁在不同剪切变形理论下的静力弯曲问题。假设FGM梁由金属和陶瓷两种材料构成,其等效物性参数沿厚度方向连续变化,且遵从简单幂率变化规律。基于最小势能原理,建立以轴向位移、横向位移及转角为未知函数的FGM梁的运动微分方程组。对简支FGM梁,采用Fourier级数法获得5种剪切变形理论下FGM梁的挠度、轴向位移及转角曲线,分析梁的长高比、梯度指标对弯曲变形的影响,分析不同剪切变形理论下FGM梁的切应力和正应力的分布特性,并与均质材料梁的静力弯曲特性进行比较。给出FGM梁的中性轴位置随梯度指标的变化曲线并进行分析。

Functionally graded materials (FGMs) are complicated composites created using the concept of continuous variation of material property in one or more predetermined directions. FGMs have been used in the manufacturing of structural parts that are subjected to non-uniform functioning requirements in recent years. In a thermal protection system, for example, FGMs combine the benefits of traditional ceramics, such as corrosion and heat resistance, with those of metal, such as mechanical strength and rigidity. They are common in engineering practice, so static analysis and dynamic analysis for FGM plate structures are necessary. The radial point interpolation method (RPIM) meshless method has been used based on the point interpolation method (PIM) by including the radial basis function (RPF) in the interpolation formulation and has shown good performance in computational engineering. One of the advantages of this method is that it satisfies the Kronecker’s delta function, which overcomes the limitations of critical boundary conditions for the traditional meshless method. Furthermore, RPIM shape function satisfies the high-order continuity constraint, which the low-order finite element methods (FEM) method does not. This paper presents a meshless approach for the static and dynamic analysis of FGM plates whose material properties vary through the thickness. Numerical examples are solved and the results are compared with reference solutions or the results of FEM given by SOMSOL program to confirm the accuracy of the proposed method.

The effects of advanced nonlinear varied shear coefficient and third-order shear deformation theory (TSDT) on the dynamic responses of thick functionally graded material (FGM) plates under thermal vibration are investigated. The nonlinear coefficient of the displacement field of TSDT is used to obtain the expression of advanced varied shear coefficient for the thick FGM plates. The dynamic displacements, shear rotations and stresses in numerical results under sinusoidal applied heat loads are obtained and investigated. Two parametric effects of environment temperature and FGM power law index on the dynamic responses of thermal stress and center deflection of thick FGM plates are also investigated. The transient responses of center deflection are found for the cases of simply homogeneous equation and fully homogeneous equation. Also, the transient responses of center deflection are found for cases of nonlinear and linear varied-modified coefficient of shear correction.

A partial mixed finite element (FE)–state space method (SSM) semi-analytical approach is presented for the static analysis of piezoelectric smart laminate composite and functionally graded material (FGM) plates. Hence, using the Hamiltonian formalism, the three-dimensional piezoelectricity equations are first worked so that a partial mixed variational formulation, which retains the translational displacements, electric potential, transverse stresses, and transverse electric displacement as primary variables, is obtained; this allows, in particular, straightforward fulfillment of the electromechanical continuity constraints at the laminate interfaces. After an in-plane FE discretization only, the problem is first reduced, for a single layer, to a Hamiltonian eigenvalue problem that is solved using the symplectic approach; then, the multilayer solution is reached via the SSM propagator matrix. The proposed methodology is finally applied to the static analysis of piezoelectric-cross-ply hybrid laminated composite and FGM plates. In a comparison with open literature, available tabulated results show good agreements, thus validating the proposed approach.

Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection

Nonlinear dynamic and vibration analysis of piezoelectric eccentrically stiffened FGM plates in thermal environment

Nonlocal steady-state thermoelastic analysis of functionally graded materials by using peridynamic differential operator