Stochastic analysis of a thermoelastic problem in functionally graded plates with uncertain material properties

The statistics (i.e., mean and variance) of temperature and thermal stress are analytically obtained in functionally graded material (FGM) plates with uncertainties in the thermal conductivity and coefficient of linear thermal expansion. These FGM plates are assumed to have arbitrary nonhomogeneous thermal and mechanical properties through the entire thickness of plate and are subjected to deterministic convective heating. The stochastic temperature and thermal stress fields are analysed by assuming the FGM plate is multilayered with distinct, random thermal conductivity and coefficient of linear thermal expansion in each layer. Vodicka’s method, which is a type of integral transform method, and a perturbation method are employed to obtain the analytical solutions for the statistics. The autocorrelation coefficients of each random property and cross-correlation coefficients between different random properties are expressed in exponential function forms as a non-homogeneous Markov random field of discrete space. Numerical calculations are performed for FGM plates composed of partially stabilised zirconia (PSZ) and austenitic stainless steel (SUS304), which have the largest dispersion of the random properties at the place where the volume fractions of the two constituent materials are both 0.5. The effects of the spatial change in material composition, thermal boundary condition and correlation coefficients on the standard deviations of the temperature and thermal stress are discussed.     

Anti-plane problem of periodic cracks in a functionally graded coating–substrate structure

The static and dynamic anti-plane problem for a functionally graded coating–substrate structure containing a periodic array of parallel cracks, which are perpendicular to the boundary, is considered. Integral-transform techniques are employed to reduce the problem to the solution of an integral equation with hypersingular kernels. Numerical results are presented to show the influence of geometry, material properties and material gradient parameter on the fracture behavior.     

Micromechanics-based thermoelastic model for functionally graded particulate materials with particle interactions

Thermoelastic behavior of functionally graded particulate materials is investigated with a micromechanical approach. Based on a special representative volume element constructed to represent the graded microstructure of a macroscopic material point, the relation between the averaged strains of the particle and matrix phases is derived with pair-wise particle interactions, and a set of governing equations for the thermoelastic behavior of functionally graded materials is presented. The effective coefficient of thermal expansion at a material point is solved through the overall averaged strain of two phases induced by temperature change under the stress-free condition, and is shown to exhibit a weak anisotropy due to the particle interactions within the graded microstructure. When the material gradient is eliminated, the proposed model predicts the effective coefficient of thermal expansion for uniform composites as expected. If the particle interactions are disregarded, the proposed model recovers the Kerner model. The proposed semi-analytical scheme is consistent and general, and can handle any thermal loading variation. As examples, the thermal stress distributions of graded thermal barrier coatings are solved for two types of thermal loading: uniform temperature change and steady-state heat conduction in the gradation direction.     

Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory

In this research, thermal buckling of circular plates compose of functionally graded material (FGM) is considered. Equilibrium and stability equations of a FGM circular plate under thermal loads are derived, based on the higher order shear deformation plate theory (3rd order 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. A buckling analysis of a functionally graded circular plate (FGCP) under various types of thermal loads is carried out and the result are given in closed-form solutions. The results are compared with the critical buckling temperature obtained for FGCP based on first order (1st order plate theory) and classical plate theory (0 order plate theory) given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of FGCP, whereas the first order and classical plate theory overestimates buckling temperature.     

Free vibration of functionally graded sandwich plates using four-variable refined plate theory

This paper uses the four-variable refined plate theory (RPT) for the free vibration analysis of functionally graded material (FGM) sandwich rectangular plates.Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. Two common types of FGM sandwich plates are considered, namely, the sandwich with the FGM facesheet and the homogeneous core and the sandwich with the homogeneous facesheet and the FGM core. The equation of motion for the FGM sandwich plates is obtained based on Hamilton's principle. The closed form solutions are obtained by using the Navier technique. The fundamental frequencies are found by solving the eigenvalue problems. The validity of the theory is shown by comparing the present results with those of the classical, the first-order, and the other higher-ordex theories. The proposed theory is accurate and simple in solving the free vibration behavior of the FGM sandwich plates.     

热冲击载荷作用下的含球形空腔的广义热弹性功能梯度球形各向同性体

This paper is concerned with the determination of thermoelastic displacement, stress and temperature in a functionally graded spherically isotropic infinite elastic medium having a spherical cavity, in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters （Green and Lindsay theory）. The surface of cavity is stress-free and is subjected to a time-dependent thermal shock. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain, which is then solved by an eigenvalue approach. Numerical inversion of the transforms is carried out using the Bellman method. Displacement, stress and temperature are computed and presented graphically. It is found that variation in the thermo-physical properties of a material strongly influences the response to loading. A comparative study with a corresponding homogeneous material is also made.     

Several three-dimensional solutions for transversely isotropic functionally graded material plate welded with circular inclusion

The problem of a transversely isotropic functionally graded material (FGM) plate welded with a circular inclusion is considered. The analysis starts with the generalized England-Spencer plate theory for transversely isotropic FGM plates, which expresses a three-dimensional (3D) general solution in terms of four analytic functions. Several analytical solutions are then obtained for an infinite FGM plate welded with a circular inclusion and subjected to the loads at infinity. Three different cases are considered, i.e., a rigid circular inclusion fixed in the space, a rigid circular inclusion rotating about the x-, y-, and z-axes, and an elastic circular inclusion with different material constants from the plate itself. The static responses of the plate and/or the inclusion are investigated through numerical examples.     

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magneto-electro-elastic (FGMEE) cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.     

Anti-plane shear of an arbitrary oriented crack in a functionally graded strip bonded with two dissimilar half-planes

An internal crack located within a functionally graded material (FGM) strip bonded with two dissimilar half-planes and under an anti-plane load is considered. The crack is oriented in an arbitrary direction. The material properties of strip are assumed to vary exponentially in the thickness direction and two half-planes are assumed to be isotropic. Governing differential equations are derived and to reduce the difficulty of the problem dealing with solution of a system of singular integral equations Fourier integral transform is employed. Semi closed form solution for the stress distribution in the medium is obtained and mode III stress intensity factor (SIF), at the crack tip is calculated and its validity was verified. Finally, the effects of nonhomogeneous material parameter and crack orientation on the stress intensity factor are studied.     

Higher-order crack tip fields for functionally graded material plate with transverse shear deformation

The crack tip fields are investigated for a cracked functionally graded material (FGM) plate by Reissner’s linear plate theory with the consideration of the transverse shear deformation generated by bending. The elastic modulus and Poisson’s ratio of the functionally graded plates are assumed to vary continuously through the coordinate y, according to a linear law and a constant, respectively. The governing equations, i.e., the 6th-order partial differential equations with variable coefficients, are derived in the polar coordinate system based on Reissner’s plate theory. Furthermore, the generalized displacements are treated in a separation-of-variable form, and the higher-order crack tip fields of the cracked FGM plate are obtained by the eigen-expansion method. It is found that the analytic solutions degenerate to the corresponding fields of the isotropic homogeneous plate with Reissner’s effect when the in-homogeneity parameter approaches zero.     

Non-axisymmetric thermal stress of a functionally graded coated circular inclusion in an infinite matrix

This paper is to study the non-axisymmetric two-dimensional problem of thermal stresses in an infinite matrix with a functionally graded coated circular inclusion based on complex variable method. With using the method of piece-wise homogeneous layers, the general solution for the functionally graded coating having radial arbitrary elastic properties is derived when the matrix is subjected to uniform heat flux at infinity, and then numerical results are presented for several special examples. It is found that the existence of the functionally graded coating can change interfacial thermal stresses, and choosing proper change ways of the radial elastic properties in the coating can obviously reduce the thermal stresses.     

Coupled thermoelasticity of functionally graded cylindrical shells

The coupled thermoelastic response of a functionally graded circular cylindrical shell is studied. The coupled thermoelastic and the energy equations are simultaneously solved for a functionally graded axisymmetric cylindrical shell subjected to thermal shock load. A second-order shear deformation shell theory that accounts for the transverse shear strains and rotations is considered. Including the thermo-mechanical coupling and rotary inertia, a Galerkin finite element formulation in space domain and the Laplace transform in time domain are used to formulate the problem. The inverse Laplace transform is obtained using a numerical algorithm. The shell is graded through the thickness assuming a volume fraction of metal and ceramic, using a power law distribution. The results are validated with the known data in the literature.     

Contact vibration analysis of the functionally graded material coated half-space under a rigid spherical punch

This paper studies the contact vibration problem of an elastic half-space coated with functionally graded materials (FGMs) subject to a rigid spherical punch. A static force superimposing a dynamic time-harmonic force acts on the rigid spherical punch. Firstly, we give the static contact problem of FGMs by a least-square fitting approach. Next, the dynamic contact pressure is solved by employing the perturbation method. Lastly, the dynamic contact stiffness with different dynamic contact displacement conditions is derived for the FGM coated half-space. The effects of the gradient index, coating thickness, internal friction, and punch radius on the dynamic contact stiffness factor are discussed in detail.     

Effects of SH waves in a functionally graded plate

A computational method is presented to investigate SH waves in functionally graded material (FGM) plates. The FGM plate is first divided into quadratic layer elements (QLEs), in which the material properties are assumed as a quadratic function in the thickness direction. A general solution for the equation of motion governing the QLE has been derived. The general solution is then used together with the boundary and continuity conditions to obtain the displacement and stress in the wave number domain for an arbitrary FGM plate. The displacements and stresses in the frequency domain and time domain are obtained using inverse Fourier integration. Furthermore, a simple integral technique is also proposed for evaluating modified Bessel functions with complex valued order. Numerical examples are presented to demonstrate this numerical technique for SH waves propagating in FGM plates.     

Two-dimensional thermoelastic analysis of a functionally graded cylindrical panel due to nonuniform heat supply

This paper is concerned with the theoretical treatment of the steady-state thermoelastic problem of a functionally graded cylindrical panel due to nonuniform heat supply in the circumferential direction. The thermal and thermoelastic constants of the cylindrical panel are expressed as power functions of the radial coordinate. We obtain the exact solution for the two-dimensional temperature change in a steady state, and thermal stresses of a simple supported cylindrical panel under the state of plane strain. Some numerical results are shown in figures and tables. Furthermore, the influence of the nonhomogeneity of the material, the radius ratio and the span angle upon the temperature change, displacements and stresses is investigated.     

Reissner-Sagoci problem for a homogeneous coating on a functionally graded half-space

This paper is concerned with the axisymmetric elastostatic problem related to the rotation of a rigid punch which is bonded to the surface of a nonhomogeneous half-space. The half-space is composed of an isotropic homogeneous coating in the form of layer, which is attached to the functionally graded half-space. The shear modulus of the FGM is assumed to vary in the direction of axis Oz normal to the boundary as μ₁(z) = μ₀(1 + αz)^β, where μ₀, α, β are positive constants. The punch undergoes rotation due to the action of the internal loads. By using Hankel's integral transforms, the mixed boundary value problem is reduced to dual integral equations, and next, to a Fredholm's integral equation of the second kind, which is solved numerically for the case of β = 2. The final results show the effect of non-homogeneity on the shear stresses and an unknown moment of punch rotation.     

Impact response of a functionally graded piezoelectric plate with a vertical crack

The dynamic fracture problem for a functionally graded piezoelectric plate containing a crack perpendicular to the free boundaries is considered in this study. It is assumed that the electroelastic properties of the medium vary continuously in the thickness direction. Integral transform techniques and dislocation density function are employed to reduce the problem to the solution of a singular integral equation. Mode I dynamic energy density factors are presented for an internal crack as well as an edge crack for various values of dimensionless parameters representing the size and location of the crack and the material nonhomogeneity.     

Neural network optimization of material composition of a functionally graded material plate at arbitrary temperature range and temperature rise

Summary A neural network model is applied to optimization problems of material compositions for a functionally graded material plate with arbitrarily distributed and continuously varied material properties in the thickness direction. Unsteady temperature distribution is evaluated by taking into account the bounds of the number of the layers. Thermal stress components for an infinite functionally graded material plate are formulated under traction-free mechanical conditions. As a numerical example, a plate composed of zirconium oxide and titanium alloy is considered. In the optimization problem of minimizing the thermal stress distribution, the numerical calculations are carried out making use of the neural network. The optimum material composition is determined by taking into account the effect of temperature-dependence of material properties. The results obtained by neural network and ordinary nonlinear programming method are compared. Received 3 March 1998; accepted for publication 22 May 1998     

Two-dimensional complete rational analysis of functionally graded beams within symplectic framework

Exact solutions for generally supported functionally graded plane beams are given within the framework of symplectic elasticity. The Young’s modulus is assumed to exponentially vary along the longitudinal direction while the Poisson’s ratio remains constant. The state equation with a shift-Hamiltonian operator matrix has been established in the previous work, which is limited to the Saint-Venant solution. Here, a complete rational analysis of the displacement and stress distributions in the beam is presented by exploring the eigensolutions that are usually covered up by the Saint-Venant principle. These solutions play a significant role in the local behavior of materials that is usually ignored in the conventional elasticity methods but possibly crucial to the material/structure failures. The analysis makes full use of the symplectic orthogonality of the eigensolutions. Two illustrative examples are presented to compare the displacement and stress results with those for homogenous materials, demonstrating the effects of material inhomogeneity.     

Non-linear analysis of functionally graded plates under transverse and in-plane loads

Large deflection and postbuckling responses of functionally graded rectangular plates under transverse and in-plane loads are investigated by using a semi-analytical approach. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The plate is assumed to be clamped on two opposite edges and the remaining two edges may be simply supported or clamped or may have elastic rotational edge constraints. The formulations are based on the classical plate theory, accounting for the plate-foundation interaction effects by a two-parameter model (Pasternak-type), from which Winkler elastic foundation can be treated as a limiting case. A perturbation technique in conjunction with one-dimensional differential quadrature approximation and Galerkin procedure are employed in the present analysis. The numerical illustrations concern the large deflection and postbuckling behavior of functional graded plates with two pairs of constituent materials. Effects played by volume fraction, the character of boundary conditions, plate aspect ratio, foundation stiffness, initial compressive stress as well as initial transverse pressure are studied.