Date of Award

8-2005

Level of Access

Campus-Only Thesis

Degree Name

Master of Science (MS)

Department

Mechanical Engineering

Advisor

Senthil S. Vel

Second Committee Member

Donald A. Grant

Third Committee Member

Justin Poland

Abstract

Functionally graded materials (FGMs) are heterogeneous composites with tailored microstructures. By varying the relative volume fraction and orientation of two or more constituent materials, a shell or plate may exhibit macroscopic material properties that have a continuous spatial variation. FGMs show promise in applications where the operating conditions are severe, including spacecraft heat shields, heat exchanger tubes, plasma facings for fusion reactors, engine components, and high power electrical contacts or even magnets. Two important areas in the development of such novel materials are their analysis and optimization. To this end, we present an analytical solution to the thermoelastic deformations of functionally graded orthotropic cylindrical shells, and we also propose a methodology for FGM optimization.

We first analyze the steady-state response of a functionally graded thick cylindrical shell subjected to thermal and mechanical loads. The functionally graded shell is simply supported at the edges and it is assumed to have an arbitrary variation of material properties in the radial direction. The three dimensional steady-state heat conduction and thermoelasticity equations, simplified to the case of generalized plane strain deformations in the axial direction, are solved analytically. Suitable temperature and displacement functions that identically satisfy the boundary conditions at the simply supported ends are used to reduce the thermoelastic equilibrium equations to a set of coupled ordinary differential equations with variable coefficients, which are then solved by the power series method. Results are presented for two constituent isotropic and fiber-reinforced functionally graded shells that have a smooth variation of material volume fractions, and/or in-plane fiber orientations, through the radial direction. The cylindrical shells are also analyzed using the Flugge and the Donnell shell theories. Displacements and stresses from the shell theories are compared with the three-dimensional exact solution to delineate the effects of transverse shear deformation, shell thickness and angular span.

Secondly, this thesis presents a methodology for the multi-objective optimization of laminated composite materials that is based on an integer-coded genetic algorithm. The fiber orientations and fiber volume fractions of the laminae are chosen as the primary optimization variables. Simplified micromechanics equations are used to estimate the stiffness and strength of each lamina using the fiber volume fraction and material properties of the matrix and fibers. The lamina stresses for thin composite coupons subjected to force and/or moment resultants are determined using the classical lamination theory and the first-ply failure strength is obtained using the Tsai-Wu failure criterion. A multi-objective genetic algorithm is used to obtain Pareto-optimal designs for three model problems having multiple, conflicting, objectives.

Lastly, the proposed methodology for multi-objective optimization of composite laminates is extended to the design of functionally graded plates and cylindrical shells. This is accomplished through the redefinition of the genetic coding used for laminates, such that it may be used to account for a continuous transverse distribution of the constituent volume fractions in an isotropic FGM shell. By representing the constituent volume fractions via range-restricted Hermite polynomials, continuous distributions in material properties may be governed by distinct control points that are spaced evenly through the thickness of the FGM shell. Different shell designs are evaluated using the three-dimensional analytical solution presented earlier. The proposed methodology is used to investigate two model problems. Pareto-optimal solutions are provided for an isotropic Zirconia/Titanium functionally graded shell subjected to both thermal and mechanical loads. We also investigate an isotropic Tungsten/Copper functionally graded shell subjected to a prescribed heat flux on its upper surface while the remaining edges are held at reference temperature. In both cases, the examples serve to demonstrate that the proposed methodology is robust and well suited to the design of functionally graded materials.

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