Some Analytical Solutions For The Problem Of An Elliptical Inhomogeneity Embedded In Orthotropic Media

The existence of various microstructures in composite materials can greatly alter their overall behavior. Some main aspects for tackling inhomogeneity problems rest in determining the eigenstrain included stress fields in microstructures (inclusions and inhomogeneities) as well as in matrix materials, which play a considerable influence in controlling fracture, fatigue strength and failure behavior of the materials and structures.Complex analysis of anisotropic elasticity indicates that the degree of anisotropy (orthotropy) is characterized by two basic complex parameters. The two parameters and their conjugates being four roots of a characteristic equation emanating from the compatibility condition could be purely imaginary numbers or complex numbers for any ideal orthotropic elastic material. This paper deals with an elastic orthotropic inhomogeneity problem due to eigenstrains. A complete procedure for seeking for analytical solution for such orthotropic inhomogeneity problem is given and some closed-form solutions for the induced stress field are derived based on the polynomial conservation theorem, the complex function method and conformal transformation. The specific form of the distribution of eigenstrains is assumed to be some polynomial functions in Cartesian coordinates of the points of the inhomogeneity. The elastic energy of the inhomogeneity/matrix system is expressed in terms of some unknown real coefficients, which are analytically evaluated by means of the principle of minimum potential energy, and the corresponding strain and stress fields in the inhomogeneity are obtained finally. The resulting stress field in the inhomogeneity is verified by checking the continuity conditions for the normal and shear stresses at the interface between the inhomogeneity and matrix. In addition, the present analytical solutions can reduce to the known results for some special cases of uniform eigenstrains and isotropic materials.The dissertation consists of eight chapters. Chapter one reviews some developments in the inhomogeneity problems in the filed of composite materials. Chapter two presents some fundamental governing equations for orthotropic elasticity due to complex analysis. Chapter 3 and 4 are concerned with analytical solutions for the problems of an elliptical inhomogeneity embedded in orthotropic media with complex and purely imaginary roots, respectively. The specific form of the distribution of eigenstrains is assumed to be a linear function in Cartesian coordinates of the points of the inhomogeneity. Further, some numerical examples are given, and distribution of normal and shear stresses at the interface are illustrated in Chapter 5. Chapter 6 focuses on the case of the eigenstrains having the form of a quadratic polynomial function. The resulting solution reflects the coupling effect of the quadratic terms and zero terms in the polynomial expression on the elastic fields. Chapter 7 presents an analytical solution for the elastic fields induced by uniform eigenstrains in an elliptical inhomogeneity embedded in the orthotropic matrix under tension at infinity and inclined at any angle. Especially, explicit expression is obtained for the stress in the inhomogeneity under the remote tension. Finally, Chapter 8 draws some conclusions.