Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations

This paper investigates the nonlinear dynamic response of thick functionally graded materials (FGM) plates using the third-order shear deformation plate theory and stress function. The FGM plate is assumed to rest on elastic foundations and subjected to thermal and damping loads. Numerical results for dynamic response of the FGM plate are obtained by Runge–Kutta method. The results show the influences of geometrical parameters, the material properties, the elastic foundations, and thermal loads on the nonlinear dynamic response of FGM plates.     

Conditions of stress-free deformation of anisotropic FGM interface under constant temperature

Abstract

This article demonstrates conditions of stress-free deformation of anisotropic FGM interface under thermal loading. Existence of such a deformation is theoretically possible in the case of constant temperature, reduction of the covariant differential along direction of functional gradation to the partial differential, and simultaneously the linear gradation of thermal expansion tensor. Nevertheless, two examples of engineering structures, a thick three-layer plate and a thick-walled cylinder, show that either the compatible deformation of three-layer structure or an application of the curvilinear interface results in stress state appearance. Both examples under consideration take the advantage of an alumina-based composite Al-Al₂O₃, the thermomechanical properties of which are subjected to transverse isotropy.     

Thermo-Elasticity Solution of Functionally Graded Plates Integrated with Piezoelectric Sensor and Actuator Layers

Based on theory of piezoelasticity, a functionally graded material (FGM) host plate under applied electric field and thermal or mechanical load is studied. The thermo-elastic constants of the plate vary continuously throughout the thickness direction in the form of an exponential function and the Poisson ratio is held constant. Analytical solutions for the temperature, stress and displacement fields for the plate with simply supported edges are derived by using the Fourier series expansions and state-space method. The theory is assessed by comparison with the previously published results. The effects of surface boundary conditions, gradient index, applied voltage, aspect ratio and length-to-thickness ratio on the behavior of the FGM plate are examined.     

Thermal Post-Buckling of Functionally Graded Material Circular Plates Subjected to Transverse Point-Space Constraints

Axisymmetrical thermal post-buckling of functionally graded material (FGM) circular plates with immovably clamped boundary and a transversely central point-space constraint was studied. The material properties of the plate were assumed to vary as power law functions in the thickness direction and the temperature rise field to change only in the thickness direction. Based on von Karman's non-linear plate theory, governing equations in terms of the displacements of the middle plane were established. Temperature rise field was obtained by solving the one-dimensional heat conduction equation associated with specified boundary conditions at the top and bottom surface of the plate. By using the shooting method, thermal post-buckling deformation of the FGM circular plate was obtained before and after the plate contacting the point-space constraint. The changes in the characteristics of the deformation and the internal forces of FGM plates were discussed. The effects of gradients of material properties and non-uniform temperature rise parameters on the thermal post-buckling behaviors of FGM circular plates were also examined.     

Thermal Buckling of Functionally Graded Plates Resting On Elastic Foundations Using the Trigonometric Theory

In this article, thermal buckling analysis of functionally graded material (FGM) plates resting on two-parameter Pasternak's foundations is investigated. Equilibrium and stability equations of FGM plates are derived based on the trigonometric shear deformation plate theory and includes the plate foundation interaction and thermal effects. The material properties vary according to a power law form through the thickness coordinate. The governing equations are solved analytically for a plate with simply supported boundary conditions and subjected to uniform temperature rise and gradient through the thickness. Resulting equations are employed to obtain the closed-form solution for the critical buckling load for each loading case. The influences of the plate aspect ratio, side-to-thickness ratio, gradient index, and elastic foundation stiffnesses on the buckling temperature difference are discussed.     

Thermal buckling analysis of functionally graded sandwich plates

Thermal buckling of functionally graded sandwich plates are presented in this article. Two common types of FGM sandwich plates, namely, homogeneous face layers with FGM core and FGM face layers with homogeneous core are considered. Material properties and thermal expansion coe?cient of FGM layers are assumed to vary continuously through-the-thickness according to a simple power-law distribution in terms of the volume fractions of the constituents. Equilibrium and stability equations of FGM sandwich plate with simply supported boundary conditions are derived using the higher-order shear deformation plate theory. The influence of the plate aspect ratio, the relative thickness, the gradient index, and the thermal loading conditions on the critical buckling temperature of FGM sandwich plates are investigated. The thermal loads are assumed to be uniform, linear, and nonlinear distribution through-the-thickness. A new simple solution for thermal buckling of FGM sandwich plates under nonlinear temperature rise is presented.     

THERMALLY INDUCED STRESS WAVES IN FUNCTIONALLY GRADED MATERIALS WITH TEMPERATURE-DEPENDENT MATERIAL PROPERTIES

This article is concerned with the dynamic treatment of thermally induced stress waves in an infinite elastic plate subjected to impulsive electromagnetic radiation. The plate is assumed to be a functionally graded material (FGM), meaning that the material is composed of multiconstituents in ceramics and metals, the volume fractions of which distribute continuously inside the material. The mathematical problem is one of wave propagation in a typical nonhomogeneous material The radiation absorption is assumed to occur at a constant rate for the duration of the pulse and to diminish exponentially with distance from the surface of the plate, assuming negligible heat conduction. In treating problems, the nature of the stress-wave buildup in the plate is studied for the case of a temperature-dependent solid, that is, when material properties vary with temperature. The numerical procedure employs the characteristic method based on the integration of the governing equations along the characteristics. Numerical calculations are carried out for ceramic-metal FGM plates showing the influences of the temperature-dependent material properties and the volume fractions of the phases composing the FGM on the magnitude of the dynamic thermal stresses.     

Stress Evolution and Thermal Shock Computation Using the Finite Volume Method

This work presents a novel numerical approach to thermal shock and transient thermal stress distribution based on the Finite Volume method. A unified approach was developed based upon the coupling between a three-dimension thermal solution and a linear thermo-elastic, multi-layered plane stress approximation. Temperature distributions were compared to analytical solutions for a flat plate and to experimental results using a temperature-dependent heat transfer coefficient. The numerical results were assessed against three analytical methods: Timoshenko and Goodier, Lu and Fleck, and Collin and Rowcliffe. This work demonstrates that the FVM provides also accurate results for thermal stress distributions in solids.     

Transient stress analysis of sandwich plate and shell panels with functionally graded material core under thermal shock

A finite element formulation for stress analysis of functionally graded material (FGM) sandwich plates and shell panels under thermal shock is presented in this work. A higher-order layerwise theory in conjunction with Sanders’ approximation for shells is used to develop the finite element formulation for transient stress analysis of FGM sandwich panels. The top and the bottom surfaces of FGM sandwich panels are made of pure ceramic and metal, respectively, and core of the sandwich is assumed to be made of FGM. The temperature profile in the thickness direction of the panels is considered to be varying as per the Fourier’s law of heat conduction equation for unsteady state. The heat conduction equations are solved using the central difference method in conjunction with the Crank–Nicolson approach. Transient thermal displacements of the sandwich panels are obtained using Newmark average acceleration method and the transient thermal stresses are obtained using stress–strain relations, subsequently. Results obtained from the present layerwise finite element formulations are first validated with available solutions in literature. Parametric studies are taken up to study the effects of volume fraction index, temperature dependency of material properties, core thickness, panel configuration, geometric and thermal boundary conditions on transient thermal stresses of FGM sandwich plates and shells.     

THERMAL BUCKLING OF FUNCTIONALLY GRADED PLATES BASED ON HIGHER ORDER THEORY

Equilibrium and stability equations of a rectangular plate made of functionally graded material (FGM) under thermal loads are derived, based on the higher order shear deformation plate theory. Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. The derived equilibrium and stability equations for functionally graded plates (FGPs) are identical to the equations for laminated composite plates. A buckling analysis of a functionally graded plate under four types of thermal loads is carried out and results in closed-form solutions. The critical buckling temperature relations are reduced to the respective relations for functionally graded plates with a linear composition of constituent materials and homogeneous plates. The results are compared with the critical buckling temperatures obtained for functionally graded plates based on classical plate theory given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of functionally graded plates, whereas the classical plate theory overestimates buckling temperatures.     

Generalized Magneto-Thermo-Viscoelastic Problems of Rotating Functionally Graded Anisotropic Plates by the Dual Reciprocity Boundary Element Method

A numerical model based on the dual reciprocity boundary element method (DRBEM) is extended to study the generalized theories of magneto-thermo-viscoelasticity in a rotating plate of functionally graded material (FGM). The material properties of the FGM plate have a gradient in the thickness direction. and are anisotropic in the plane of the plate. In the case of plane deformation, an implicit-implicit staggered scheme was proposed and implemented for use with the DRBEM to obtain the solution for the displacement and temperature fields. A comparison of the results for different theories of magneto-thermo-viscoelasticity is presented graphically. Numerical results that demonstrate the validity of the proposed method are also presented graphically.     

Thermal Buckling Response of Functionally Graded Plates with Clamped Boundary Conditions

In this research work, an exact analytical solution for thermal buckling analysis of functionally graded material (FGM) plates with clamped boundary condition subjected to uniform, linear, and non-linear temperature rises across the thickness direction is developed. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory accounts for parabolic distribution of the transverse shear strains, and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factor. The material properties of FGM 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 governing equations are solved analytically for a plate with simply supported boundary conditions. Resulting equations are employed to obtain the closed-form solution for the thermal force resultant for each loading case. Numerical examples covering the effects of the plate aspect ratio, side-to-thickness ratio and gradient index on thermal force resultant are discussed.     

Thermal stability of eccentrically stiffened FGM plate on elastic foundation based on Reddy's third-order shear deformation plate theory

This article studies the nonlinear thermal buckling and postbuckling of eccentrically stiffened functionally graded plates on elastic foundation subjected to mechanical, thermal, and thermomechanical loads. The noticeable point of this study is using the Reddy's higher order shear deformation plate theory and a general formula for the forces and moments of eccentrically stiffened functionally graded material (FGM) plate, which takes into account the influence of temperature on both the FGM plate and stiffeners. The article used the Galerkin method, stress function, and iterative method to determine the thermal buckling loads and postbuckling response of the eccentrically stiffened FGM plates in three different cases of boundary conditions. The effects of material, temperature-dependent material properties, elastic foundations, boundary conditions, outside stiffeners, and temperature on the buckling and postbuckling loading capacity of the FGM plates in thermal environments are analyzed and discussed. A good agreement is obtained by comparing the present analysis with other available literature.     

Thermal buckling and postbuckling responses of geometrically imperfect FG porous beams based on physical neutral plane

Abstract

Considering the third-order shear deformation and physical neutral plane theories, thermal postbuckling analysis for functionally graded (FG) porous beam are performed in this research. The cases of shear deformable functionally graded materials (FGM) beams with initial deflection and uniformly distributed porosity are considered. Geometrically imperfect FG porous beams with two different types of immovable boundary conditions as clamped–rolling and clamped–clamped are analyzed. Thermomechanical nonhomogeneous material properties of the FG porous beam are assumed to be temperature and position dependent. FG porous beams are subjected to different types of thermal loads as heat conduction and uniform temperature rise. Heat conduction equation is solved analytically using the polynomial series solution for the one-dimensional condition. The governing equilibrium equations are obtained by applying the virtual displacement principle. Assuming von Kármán type of geometrical nonlinearity, equilibrium equations are nonlinear and are solved using an analytical method. A two-step perturbation technique is used to obtain the thermal buckling and postbuckling responses of FG porous beams. The numerical results are compared with the case of perfect FGM Timoshenko beams without porosity distribution based on the midplane formulation. Parametric studies of the perfect/imperfect FG porous beams for two types of thermal loading and boundary conditions are provided.     

Investigation Methods for Thermal Shock Crack Problems of Functionally Graded Materials–Part I: Analytical Method

In most published papers, in order to obtain the analytical solution of the crack problems in functionally graded materials (FGMs), the thermomechanical properties of FGMs are usually assumed to be very particular functions and, hence, may not be physically realizable for many actual material combinations. Very few analytical methods can be used to solve the thermal shock crack problem of an FGM cylindrical shell or plate with general thermomechanical properties. In this article, a set of analytical methods is proposed for the thermal shock crack problem of an FGM plate or cylindrical shell with general thermomechanical properties. The crack problem of a cylindrical shell is modeled by a plate on an elastic foundation. Greatly different from previous studies, a set of analytical methods using both the perturbation method and a piecewise model are developed to obtain the transient temperature field and thermal stress intensity factor (TSIF). The perturbation method is applied to deal with the general thermal properties and the piecewise model is used to deal with the general mechanical properties. In the analytical procedure, integral transform, the residue theorem, and the theory of singular integral equation are used. Several representative examples are considered to check the capability of the present method. The transient thermal shock behavior of a ZrO₂/Ti-6Al-4 V FGM plate with a surface crack and a Rene 41-Zirconia FGM cylindrical shell with a circumferential crack are analyzed.     

PSR波纹板片的热强度分析

提出一种新型紧凑式回热器——一次表面回热器(PSR)的强度设计技术。结合船用ICR燃气轮机一次表面回热器研发,以3．7MW燃气轮机为背景,根据热弹性力学和传热学理论,建立PSR板片热强度分析物理数学模型,并对波纹曲线为椭圆、正弦波和抛物线的3种常见波纹传热板片所受热应力做了对比计算,分析了板片厚度,两侧压差,本身温度及形状对板片热强度的影响。给出基于Von Mise等效应力极值σrmax的板片最小设计厚度δmin,还研究了波纹板片在典型工况下的弹性变形情况。本文工作对PSR的结构设计有重要参考价值：     

Theoretical prediction and experimental verification of temperature distribution in FGM cylindrical plates subjected to thermal shock

This paper focuses on the problem of temperature field and evaluation of the heat transfer coefficient in FGM cylindrical plates subjected to thermal shock. The cylindrical plates are made of five ceramic layers: purely Al₂O₃ layer and composite layers made of Al₂O₃ matrix and 5, 10, 15, 20 wt% content of ZrO₂. The problem is solved in two stages. First, the heat transfer coefficient on the thermal shock surface is estimated by fitting the experimental data with calculated temperatures for a monolithic plate made of alumina. Then, the obtained heat transfer coefficient is used for prediction of the temperature field in the FGM plate subjected to the same boundary conditions. After comparison with the experimental data, the heat conduction characteristics of the alumina/zirconium composite can be estimated.     

Nonlinear thermal stability of eccentrically stiffened FGM double curved shallow shells

This article presents analytical solutions for the nonlinear static and dynamic stability of imperfect eccentrically stiffened functionally graded material (FGM) higher order shear deformable double curved shallow shell on elastic foundations in thermal environments. It is assumed that the shell’s properties depend on temperature and change according to the power functions of the shell thickness. The shell is reinforced by the eccentrically longitudinal and transversal stiffeners made of full metal. Equilibrium, motion, and compatibility equations are derived using Reddy’s higher order shear deformation shell theory and taking into account the effects of initial geometric imperfection and the thermal stress in both the shells and stiffeners. The Galerkin method is applied to determine load–deflection and deflection–time curves. For the dynamical response, motion equations are numerically solved using Runge–Kutta method. The nonlinear dynamic critical buckling loads are found according to the criterion suggested by Budiansky–Roth. The influences of inhomogeneous parameters, dimensional parameters, stiffeners, elastic foundations, initial imperfection, and temperature increment on the nonlinear static and dynamic stability of thick FGM double curved shallow shells are discussed in detail. Results for various problems are included to verify the accuracy and e?ciency of the approach.     

The Refined Theory of Thermoelastic Plane Problems

Based on thermoelastic theory, various two-dimensional equations and solutions for plane problems have been deduced systematically and directly from thick plate theory by using Biot's solution and Lur'e method without ad hoc assumptions. These equations and solutions can be used to construct the refined theory for the plane problems. In the case of homogeneous boundary conditions, the exact governing differential equations and solutions for the plate are derived, which consist of four governing differential equations. It is important note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of non-homogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms. The correctness of the stress assumptions in the classic plane stress problems is revised.