Study On Inhomogeneities With Uniform Internal Stress/Strain In 2D Deformations

’Inhomogeneities’ usually refer to the particles or fibers in composite materials.The study of inhomogeneities greatly stimulates the mechanical analysis and design of particle-or fiber-reinforced composite materials.In recent years,some researchers presented a kind of composite materials containing inhomogeneities of special shapes which guarantee uniform stress/strain fields achieved inside the inhomogeneities when a uniform external loading is imposed on the material.It is shown that such a kind of composite materials can achieve optimal elastic properties in comparison with all the other kinds of composite materials for a given volume fraction of inhomogeneities.On the other hand,for such a kind of composite materials,the uniformity of the stress fields inside the inhomogeneities efficiently relieves the interfacial stress concentration and therefore reduces the possibility of interfacial failure such as interfacial sliding or debonding.Consequently,the investigations of inhomogeneities with uniform stress/strain fields are of great value for the design of particle-or fiber-reinforced composite materials.However,since ordinary methods(analytical or numerical)used in the stress analysis of composite materials work only when the shapes of corresponding inhomogeneities are given,it is usually rather difficult to find out the special shapes of the inhomogeneities which ensure the uniformity of the internal stress/strain fields.In the past a few decades,to author?s knowledge,there have been only very few methods that can be used to address the determination of the shapes of inhomogeneities with prescribed stress/strain fields.Therefore,in the respect of rounding out the theory of applied mathemcatics and mechanics,it shall be also significant to study the establishment of inhomogeneities with uniform stress/strain fields.We develop in this thesis an innovative and efficient method to construct multiple inhomogeneities with prescribed uniform stress/strain fields in anti-plane shear or plane deformations.In this method,the interface between each inhomogeneity and matrix can be either treated as ?perfecly bonded? or described by an elastic membrane based on the Gurtin-Murdoch model.We organize the main content of the thesis as follows.In chapeters 2 and 3,we construct multiple inhomogeneities with prescribed uniform stress/strain fields in an infinite elastic plane under either a uniform remote anti-plane shear loading or a uniform remote in-plane loading,and study the admissible range of the uniform stress/strain fields inside such inhomogeneities,the admissible range of the uniform remote loadings(for in-plane case only),the(rotational)symmetry of such inhomogeneities and the dependence of the shape of such inhomogeneities on the remote loading.In chapeter 4,we examinethe existence and construction of multiple inhomogeneities with given uniform anti-plane shear eigenstrains which achieve prescribed uniform(actual)strain fields in an elastic half-plane with a traction-free surface,while discuss the symmetry of such inhomogeneities and the shapes of such inhomogeneities relative to the surface-inhomogeneity distance and the given eigenstrains.In chapters 5 and 6 we check in anti-plane shear elasticity the uniqueness of a nanosized inhomogeneity(incorporating interface effect characterized by the Gurtin-Murdoch model)with a uniform strain field and the existence of a periodic array of nanosized inhomogeneities enclosing prescribed uniform strain fields.The main innovative features of the thesis are listed below:(1)We first showed that the admissible range of the uniform stress/strain fields inside multiple inhomogeneitites in an elastic infinite plane or half-plane can be moderately beyond that of the uniform stress/strain field inside a single elliptical inhomogeneity in an elastic infinite plane.(2)We first derived the condition imposed on the material constants(of inhomogeneities and matrix)for the existence of(rotationally)symmetrical inhomogenieites with uniform stress/strain fields in an elastic infinite plane or half-plane.(3)We first revealed the existence of multiple inhomogeneities with uniform actual strain fields in an elastic half-plane or quarter-plane with traction-free surface(s)when the inhomogeneities are subjected to uniform anti-plane shear eigenstrains.(4)In the context of the Gurtin-Murdoch model,we first verified the existence of single or multiple non-circular nanosized inhomogeneities which achieve(s)uniform strain field(s)in an elastic infinite plane under a uniform remote anti-plane shear loading.This is in sharp contrast to the conclusion established earlier in the literature that only circular nano-inhomogeneity admits a uniform strain field in an elastic infinite plane under a uniform remote loading.