A review is given on the basic concepts and fundamental framework of the theory of elasticity for quasi-crystalline materials, including some 1D, 2D, and 3D quasi-crystals. The elasticity of quasi-crystals embodies some new concepts, field variables, and equations. It is much more complicated and beyond the scope of classical elasticity which holds only for conventional structural materials, including crystalline materials. Hence, some well-developed methods in classical elasticity cannot be directly applied to solve the problems of elasticity of quasi-crystalline materials. But the ideas of the classical theory of elasticity provide beneficial insight to treat this new subject. A decomposition and superposition procedure is suggested to simplify the elasticity problems of 1D and 2D quasi-crystals. Application of displacement and stress potentials further simplifies the problems. The large number of complicated equations involving elasticity is reduced to a single or a few partial differential equations of higher order by this technique. Also, efforts have been made to simplify the equations for 3D cubic quasi-crystals to a single partial differential equation of higher order. Simplification of the basic equations provides the possibility to solve boundary value or initial-boundary value problems of elasticity. For this purpose, some direct and systematic methods of mathematical physics and function theory are developed, and a series of analytic (classical) solutions, mainly for dislocations and cracks in materials, are derived. In addition, attention is drawn to those variational problems and generalized solutions (weak solutions) of boundary value problems and numerical implementation by the finite element method. The above may be seen as a development of the theory and methodology akin to those of classical elasticity. Based on the exact solutions of crack problems with different configurations under different motion states for different quasi-crystal systems, we put forward a framework of fracture mechanics of quasi-crystalline materials. This may be seen as an extension of the development of fracture mechanics for conventional structural materials. Also, some elastodynamic problems for some 1D and 2D quasi-crystals are studied, related results for dislocation and crack dynamics are found, and possible connections with certain thermal properties of quasi-crystalline materials, eg, specific heat and other thermo-dynamic functions, are discussed. There are 75 references cited in this review article.
Explicit Solutions of the Boundary Value Problems for an Ellipse with Double Porosity
