Estimation of Heat Flux and Thermal Stresses in Functionally Graded Hollow Circular Cylinders

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat flux at the inner surface of a functionally graded hollow circular cylinder from the knowledge of temperature measurements taken within the cylinder. Subsequently, the distributions of temperature and thermal stresses in the cylinder can be determined as well. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. The temperature data obtained from the direct problem are used to simulate the temperature measurements, and the effect of the errors in these measurements upon the precision of the estimated results is also considered. Results show that an excellent estimation on the time-dependent heat flux, temperature distributions, and thermal stresses can be obtained for the test case considered in this study.     

Transient Thermo-Mechanical Analysis of Functionally Graded Hollow Circular Cylinders

Thermo-mechanical analysis of functionally graded hollow circular cylinders subjected to axisymmetric mechanical and transient thermal loads is carried out in this study. Thermo-mechanical properties of functionally graded materials (FGM) are assumed to be temperature independent and vary continuously in the radial direction of the cylinder. Employing the Laplace transform, the Galerkin method and series method for ordinary differential equations, solutions for the time-dependent temperature and transient thermo-mechanical stresses are obtained. As an example, a molybdenum/mullite FGM with material properties obeying the exponential law is considered. Effects of heat transfer coefficients and gradient parameters of FGM on the time-dependent temperature and transient thermal stresses are discussed in detail.     

Thermo-Mechanical Bending of Functionally Graded Plates

In this work the deformations of a simply supported, functionally graded, rectangular plate subjected to thermo-mechanical loadings are analysed, extending Unified Formulation by Carrera. The governing equations are derived from the Principle of Virtual Displacements accounting for the temperature as an external load only. The required temperature field is not assumed a priori, but determined separately by solving Fourier's equation. Numerical results for temperature, displacement and stress distributions are provided for different volume fractions of the metallic and ceramic constituent as well as for different plate thickness ratios. They correlate very well with three-dimensional solutions given in the literature.     

Transient Thermoelastic Analysis for a Functionally Graded Hollow Cylinder

This paper is concerned with the theoretical treatment of transient thermoelastic problem involving a functionally graded hollow cylinder due to uniform heat supply. The transient one-dimensional temperature is analyzed by the method of Laplace transformation. The thermal and thermoelastic constants of the hollow cylinder are expressed as power functions of the radial coordinate. We obtain the one-dimensional solution for the temperature change in a transient state, and thermoelastic response of a functionally graded hollow cylinder. Some numerical results for the temperature change, the displacement and the stress distributions are shown. Furthermore, the influence of the nonhomogeneity of the material upon the temperature change, displacement and stresses is investigated.     

Transient Piezothermoelastic Analysis for a Functionally Graded Thermopiezoelectric Hollow Cylinder

This paper is concerned with the theoretical treatment of transient piezothermoelastic problem involving a functionally graded thermopiezoelectric hollow cylinder due to uniform heat supply. The transient one-dimensional temperature is analyzed by the method of Laplace transformation. The thermal, thermoelastic and piezoelectric constants of the hollow cylinder are expressed as power functions of the radial coordinate. We obtain the one-dimensional solution for the temperature change in a transient state, and piezothermoelastic response of a functionally graded thermopiezoelectric hollow cylinder. Some numerical results for the temperature change, the displacement, the stress and electric potential distributions are shown. Furthermore, the influence of the nonhomogeneity of the material upon the temperature change, displacement, stresses and electric potential is investigated.     

Thermal Postbuckling Analysis of Functionally Graded Beams

Thermal buckling and postbuckling analysis of functionally graded (FG) beams is presented. The governing equations are based on the first-order shear deformation beam theory (FSDT) and the geometrical nonlinearity is modeled using Green's strain tensor in conjunction with the von Karman assumptions. For discretizing the governing equations and the related boundary conditions differential quadrature method (DQM) as a simple and computationally efficient numerical tool is used. Based on displacement control method, a direct iterative method is employed to obtain thermal postbuckling behavior of FG beams with different boundary conditions and geometrical parameters.     

Thermal Buckling Analysis of Perforated Functionally Graded Plates

In this research, the buckling behavior of functionally graded (FG) plates under thermal loading is investigated based on finite element analysis. It is assumed the plate is subjected to a uniform temperature rise across plate thickness. First-order shear deformation theory (FSDT) is utilized for developing the solution method. By using an appropriately designed mesh structure for a perforated plate, the critical thermal buckling temperature is obtained by numerical solution of the problem based on finite element method (FEM). The FG plate is perforated by multiple cutouts. The number of cutouts is assumed one, two, four, or six. Also different geometrical shapes of cutouts including triangle, square, rhombus, pentagon, hexagon, and circle are considered. The influence of the number of cutouts and their geometrical shapes on thermal buckling response is investigated. The effects of the number of sides of cutouts from three (triangle) to infinity (circle) are discussed. Two different boundary conditions are taken into account. Also the influences of the distance between the cutouts and the orientation of cutouts on critical buckling temperature are studied. In addition, the effects of the orientation of ellipse cutouts are studied. Some remarkable conclusions are gained that can be useful in practical applications.     

Three-Dimensional Pyroelectric Analysis of a Functionally Graded Piezoelectric Hollow Sphere

A three-dimensional analytical piezothermoelastic solution is presented for a functionally graded piezoelectric spherical shell subjected to various thermal boundary conditions applied on the inner and outer surfaces. Material properties are assumed to vary along the radius, r, obeying a power law. Both the thermal field and the pyroelectric responses are resolved using the state space method. On introducing three displacement and two stress functions, two independent kinds of state equations are derived from the basic equations of piezoelectricity. It is interesting that the first kind is a homogeneous equation only related to the purely elastic behavior of the sphere, yet the second has an inhomogeneous term associated with the thermal effect to determine the pyroelectric responses. The state equations are solved by expanding the field variables into series of spherical harmonic functions. Numerical examples are performed to investigate the influence of material inhomogeneity on the pyroelectric responses of the spherical shell.     

Transient Thermoelastic Analysis for a Laminated Composite Strip with an Interlayer of Functionally Graded Material

This paper is concerned with the theoretical treatment of transient thermal stress problem involving a laminated composite thick strip with an interlayer of functionally graded material due to nonuniform heat supply in the width direction. The thermal and thermoelastic constants of the interlayer of functionally graded material are assumed to vary exponentially in the thickness direction. We obtain the exact solution for the two-dimensional temperature change in a transient state, and thermal stresses of a simple supported strip under the state of plane strain. Some numerical results for the temperature change, the displacement and the stress distributions are shown in figures. Furthermore, the influence of the thickness and position of the interlayer is investigated.     

Analysis of Thermal Stress in a Functionally Graded Coating on a Parabolic Substrate

The thermal stress field in a functionally graded coating on a parabolic substrate, where the material properties vary along the thickness direction, is considered. The closed-form solutions of thermal stresses related to compositional gradient, coating thickness and substrate curvature were obtained based on force and moment balances, and then numerical results are presented for several special examples. It is found that the magnitude and distribution of thermal stress in the functionally graded coating system with general geometrical shape can be designed properly by controlling the compositional gradient, coating thickness and substrate curvature.     

Plane Thermal Stresses in a Functionally Graded Plate Subjected to a Partial Heating

This paper presents plane thermal stresses in a functionally graded plate (FGP) subjected to a partial heating. The heat conductivity, Young's modulus and the coefficient of the linear thermal expansion are expressed by exponential functions of the position. The analytical solution for the FGP with two-dimensional temperature distribution is obtained by use of the stress function method. General solution of the governing equation of the stress function is derived in the functionally graded materials (FGMs). The numerical calculations are carried out for ZrO₂/Ti-6 Al-4 V and ZrO₂{/} stainless (SUS304) functionally graded plates, when the ceramic surface is partially heated. The numerical results are shown in figures for two cases. Even though the FGM, suitable selection of the compositional materials does not produce thermal stresses in the FGP.     

DQM-Based Thermal Stresses Analysis of a Functionally Graded Cylindrical Shell Under Thermal Shock

The transient thermal stresses of a functionally graded (FG) cylindrical shell subjected to a thermal shock are investigated. The dynamic temperature fields of FG shells are obtained by using the Laplace transform and power series method. The differential quadrature method is developed to obtain the transient thermal stresses by solving dynamic governing equations in terms of displacements. The effects of the material constitutions on the transient temperature and the thermal stresses are analyzed in the cases of obverse thermal shock and reverse thermal shock. It turns out that the thermal stresses could be alleviated by means of changing the volume fractions of the constituents.     

Thermoelastic Analysis of Functionally Graded Rotating Disks with Variable Thickness Involving Non-Uniform Heat Source

The research aims to investigate thermoelastic behavior of functionally graded rotating disks with variable thickness involving a non-uniform heat source. We assume material properties and thickness of rotating disks to vary in the radial direction. Axisymmetric thermal loads including non-uniform heat source, heat flux, and temperature boundary conditions are considered. To conduct corresponding simulations, two user subroutines are edited and incorporated into the commercial finite-element code ABAQUS. For verification, analytical formulations are derived and solved uniquely by symbolic calculations using the computing software Mathematica. The developed finite-element technique is then verified with very good agreement between results by ABAQUS and Mathematica.     

Inverse problem of estimating time-dependent heat generation in a frictional heated strip and foundation

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent frictional heat generation for the tribosystem consisting of a semi-infinite foundation and a plane-parallel strip sliding over its surface, from the knowledge of temperature measurements taken within the foundation. It is assumed that no prior information is available on the functional form of the unknown heat generation; hence the procedure is classified as the function estimation in inverse calculation. Results show that an excellent estimation on the time-dependent heat generation can be obtained for the test case considered in this study. The current methodology can be applied to the prediction of heat generation in engineering problems involving sliding-contact elements.     

Estimation of heat flux and temperature distributions in a composite strip and homogeneous foundation

In this study, an inverse algorithm based on the conjugate gradient method and the discrepancy principle is applied to estimate the unknown time-dependent heat flux and temperature distributions for the system composed of a multi-layer composite strip and semi-infinite foundation, from the knowledge of temperature measurements taken within the strip. It is assumed that no prior information is available on the functional form of the unknown heat flux; hence the procedure is classified as the function estimation in inverse calculation. Results show that an excellent estimation on the time-dependent heat flux can be obtained for the test case considered in this study.     

Inverse estimation of boundary conditions on radiant enclosures by temperature measurement on a solid object

In this paper, we present an inverse analysis to estimate the thermal boundary conditions over a two-dimensional radiant enclosure from the knowledge of the measured temperatures for some points on a solid object within the enclosure. The conduction heat transfer in the solid object and the radiative heat transfer between the surface elements of the enclosure are formulated by the finite volume method and the net radiative method, respectively. The resultant set of nonlinear equations is solved by the Newton's method. The inverse problem for estimation of boundary conditions over the radiant enclosure is solved by the conjugate gradient method.     

Thermoelastic Buckling Analysis of Functionally Graded Circular Plates Integrated with Piezoelectric Layers

Thermal buckling of circular plates made of functionally graded materials with surface-bounded piezoelectric layers are studied. The material properties of the FG plates are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituent materials. The general thermoelastic nonlinear equilibrium and linear stability equations for the piezoelectric FG plate are derived using the variational formulations. Buckling temperatures are derived for solid circular plates under uniform temperature rise, nonlinear and linear temperature variation through the thickness for immovable clamped edge of boundary conditions. The effects of piezo-to-host thickness ratio, applied actuator voltage, boundary condition, and power law index of functionally graded plates on the buckling temperature of plate are investigated. The results are verified with the data in literature.     

Nonaxisymmetric Thermomechanical Analysis of Functionally Graded Hollow Cylinders

Analytical solutions for nonaxisymmetric, thermomechanical response of functionally graded hollow cylinders are obtained in this article. The hollow cylinders are assumed to be subjected to nonaxisymmetric mechanical and transient thermal loads. Properties of functionally graded material are considered as temperature-independent and continuously varying in radial direction. Employing complex Fourier series and Laplace transform techniques, analytical solutions of time-dependent temperature and thermomechanical stresses are obtained. Numerical values of temperature and stresses of a FGM hollow cylinder under assumed thermomechanical loads are presented in graphical form.     

Coupled Thermoelasticity of Functionally Graded Beams

This paper presents the finite element solution of an Euler–Bernoulli beam with functionally graded material (FGM) subjected to lateral thermal shock loads. The FGM beam is assumed to be graded across the thickness. The material properties across the thickness direction follow the volume fraction of the constitutive materials in power law form. The solution is obtained under coupled thermoelastic assumption. The equation of motion and the conventional coupled energy equation are simultaneously solved to obtain the transverse deflection and temperature distribution in the beam. The governing partial differential equations of the problem are solved simultaneously using the Galerkin finite element method with the C ¹-continuous shape function leading to fast convergence of the solution. Results are presented for different power law indexes and coupling coefficients for simply supported boundary conditions. The results are verified with those reported in the literature.