There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Theoretical formulation, Navier's solutions of rectangular plates, and finite element models based on the third-order shear deformation plate theory are presented for the analysis of through-thickness functionally graded plates. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The formulation accounts for the thermomechanical coupling, time dependency, and the von Karman-type geometric non-linearity. Numerical results of the linear third-order theory and non-linear first-order theory are presented to show the effect of the material distribution on the deflections and stresses. Copyright © 2000 John Wiley & Sons, Ltd.

Analysis of functionally graded plates

The dynamic thermoelastic response of functionally graded cylinders and plates is studied. Thermomechanical coupling is included in the formulation, and a finite element model of the formulation is developed. The heat conduction and the thermoelastic equations are solved for a functionally graded axisymmetric cylinder subjected to thermal loading. In addition, a thermoelastic boundary value problem using the first-order shear deformation plate theory (FSDT) that accounts for the transverse shear strains and the rotations, coupled with a three-dimensional heat conduction equation, is formulated for a functionally graded plate. Both problems are studied by varying the volume fraction of a ceramic and a metal using a power law distribution.

Thermomechanical analysis of functionally graded cylinders and plates

This paper presents a review of the principal developments in functionally graded materials (FGMs) with an emphasis on the recent work published since 2000. Diverse areas relevant to various aspects of theory and applications of FGM are reflected in this paper. They include homogenization of particulate FGM, heat transfer issues, stress, stability and dynamic analyses, testing, manufacturing and design, applications, and fracture. The critical areas where further research is needed for a successful implementation of FGM in design are outlined in the conclusions. DOI: 10.1115/1.2777164

https://web.mst.edu/~vbirman/papers/Modeling%20and%20Analysis%20of%20FGM_Review_2007.pdf

Modeling and Analysis of Functionally Graded Materials and Structures

Free vibration and stability of functionally graded plates according to a 2-D higher-order deformation theory

The thermal stress problems of functionally graded materials (FGMs), as one of the advanced high-temperature materials capable of withstanding the extreme temperature environments, are discussed. The FGMs consist of the continuously changing composi tion of two different materials. For example, one is an engineering ceramic to resist the severe thermal loading from the high-temperature environment, and the other is a light metal to maintain the structural rigidity. When the FGMs are subjected to extremely severe thermal loading, large thermal stresses are produced in the FGMs. Therefore, one of the most important problems of FGMs is how to decrease thermal stresses and how to increase heat resistance. The optimal composition profile problems of the FGMs in decreasing thermal stresses are discussed in detail. When FGMs are subjected to extremely severe thermal loading, the FGMs are damaged. The crack initiates on the ceramic surface and propagates in the FGMs. It is important to discuss the thermal stresse...

Thermal stresses in functionally graded materials

This paper considers the steady thermal stresses in a hollow circular cylinder and a hollow sphere made of a functionally gradient material (FGM). The aim of this research is to understand the effect of the composition on stresses and to design the optimum FGM hollow circular cylinder and hollow sphere. We discuss the influence of inside radius size on stresses and the available temperature regions. We also compare these results with those of a FGM plate.

Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally gradient material

Crack-Tip Singular Fields in Nonhomogeneous Materials

the current state of thermoelastic problems in materials with temperature dependent properties is described for analytical treatments. We discuss a perturbation method to analyze the thermelastic problems, and an new treatment for heat conduction problem in the non-homogeneous body. Thermoelastic problems in Functional Gradient Plates(FGP) are treated for a steady state and an unsteady state. Thermoelastic problems in the FGP with a crack are also considered.

Naotake Noda

Papers

Thermal stresses in functionally graded materials

Steady thermal stresses in a hollow circular cylinder and a hollow sphere of a functionally gradient material

Crack-Tip Singular Fields in Nonhomogeneous Materials

Thermal Stresses in Materials with Temperature-Dependent Properties

Thermal stress intensity factors for a crack in a strip of a functionally gradient material