Elastic and thermoelastic behavior of materials with continuously-varying elastic moduli

Several fundamental problems dealing with the elastic and thermoelastic behavior of radially-inhomogeneous materials are solved. These problems are of importance to many types of polymer composites, metal-matrix composites, concrete, and functionally-gradient materials. Each of the problems studied is governed by ordinary differential equations having regular singular points at the origin or at infinity. The method of Frobenius is used to derive analytical solutions in the form of infinite series.; In Chapter 2 a solution is found for the hydrostatic compression of a sphere whose elastic moduli vary linearly with radius. The stresses are found to concentrate in the region of the sphere where the material is stiffest. The effective bulk modulus is calculated, and is found to lie within both the Voigt-Reuss and Hashin-Shtrikman bounds. In doing so, the Hashin-Shtrikman bounds are extended to materials in which the moduli vary continuously.; The problem of uniform heating of a sphere whose elastic moduli and thermal expansion coefficient each vary linearly with radius is solved in Chapter 3. The radial stress is found to be tensile if the thermal expansion coefficient increases with radius, and compressive if it decreases with radius; the hoop stress changes signs at some intermediate radios. The effective thermal expansion coefficient of the sphere is shown to be very nearly equal to the volumetric average of the local thermal expansion coefficient. In the special case where the moduli are uniform, the results constitute a partial generalization of the Levin-Schapery theorem for n-component materials.; The problem of a spherical inclusion in an infinite matrix, surrounded by a radially-symmetric interphase zone, is solved for both hydrostatic (Chapter 4) and shear (Chapter 6) loading. A weak interphase zone is found to decrease the stress concentrations in and around the inclusion, and vice versa. These solutions are used to estimate the effective bulk and shear moduli of a material that contains a random dispersion of such inclusions. This work is in effect an extension of Eshelby's inclusion problem to the case where the matrix is no longer homogeneous.; In Chapter 5, the inhomogeneous interphase model is used to represent the interfacial transition zone around sand inclusions in concrete. Using experimental measurements of the effective bulk modulus taken from the literature, the model is used in an inverse manner to estimate the elastic moduli at the interface between the sand and cement paste. It is estimated that the elastic moduli at the interface are about 30% less than in the pure cement paste.