Periodic And Quasiperiodic Problems In The Mechanics For Some Innovative Materials

Innovative materials are more and more important in the modern engineer. Much investigation has been done in this area in the world. In the recent years, functionally graded materials and quasicrystal materials have been obtained widely attention.Because of the intrinsic brittleness, the fracture problems of these materials are studied detailedly. Some numerical methods such as the finite element method, the boundary element method, the complex analysis method and the integral equation method are adopted to study the mechanical problems for these materials. Practices show that the integral equation method has the advantage of less computational efforts but high precision computation, so it is an effective method for investigating the fracture mechanics. This thesis studied the periodic cracking problems for the functionally graded materials and the quasicrystal materials, and researched the transmission properties of a one-dimensional periodical multi-barrier structure and a quasi-periodical structure.There are seven chapters in this thesis. The briefly introduction for the functionally graded materials, the quasicrystal materials, and the artificial periodic and quasiperiodic structure are expounded in the first chapter. In the second chapter the periodic interfacial cracks contained in a sandwiched layered composite is considered. The hollow cylinder which consists of an inner functionally graded elastic substrate and an outer functionally graded elastic layer with cyclically symmetric cracks is considered under anti-plane shear load in the third chapter. For the periodic interfacial cracks and cyclically symmetric cracks problem which are discussed in the second and third chapters, the variables separation and dislocation density functions are employed to reduce the mixed-boundary value problem to singular integral equations, which are solved numerically by standard Lobatto-Chebyshev quadrature technique.And the effects of the functionally graded non-homogeneity, the layers thickness, and the crack spacing on the fracture parameter are discussed in detail. The plane strain problem for an infinite one-dimensional hexagonal quasicrystals body containing a doubly-periodic array of cracks in periodical and aperiodical plane are considered in the fourth and fifth chapters. By introducing the periodic and quasi-periodic structure into the theoretical analysis model of superlattice, the penetrability of one-dimensional periodic and quasi-periodic potential barrier units is discussed in the sixth chapter. Finally, the summary of our work and the forecast for relevant work are prospected in the seventh chapter.