| Summary: | 博士 === 國立成功大學 === 航空太空工程學系碩博士班 === 101 === The problems concerning with singularity and how to obtain accurate results from them are the topics which greatly interest and challenge to the researchers all the while. In solid mechanics, such singularities are resulted from discontinuities of geometry or material heterogeneity, such as a plate with defects (holes, cracks, inclusions) and interfaces. Although a lot of exact solutions for those critical problems have been derived mathematically, the numerical analyses are still needed for some problems which involve the complex geometry and loading conditions. That is why we endeavor to solve such problems engaged with the boundary element method (BEM). The main advantages of BEM are the reduction of the dimension of the problem by one and the exact satisfaction of certain boundary conditions in some particular problems if their associated fundamental solutions are embedded in boundary element formulation. Because the fundamental solutions satisfy the boundary conditions of the defects and interface, say, no meshes are needed along the boundaries of holes, cracks, inclusions or interfaces in BEM. For two-dimensional anisotropic elasticity, many fundamental solutions, the so called Green’s functions of BEM, have been derived by using the complex variable Stroh formalism, such as an infinite space with holes, cracks, inclusions and interfaces, etc. Several researches in this field have proved that by using such fundamental solutions for BEM it makes the task more efficient and the results more accurate when coping with these problems.
Since the mathematical formulation of piezoelectric elasticity can be organized into the same form as that of anisotropic elasticity by just expanding the dimension of the corresponding matrix to include the piezoelectric effects, the solutions to the problems of piezoelectric materials can be obtained immediately through the extension of the solutions to the associated anisotropic elastic materials. Similarly, with the aid of the correspondence between elastic and viscoelastic materials, solutions for the linear anisotropic elastic solids can be applied directly to the linear anisotropic viscoelastic solids in the Laplace domain. With this understanding, one can acquire the fundamental solutions and boundary integral equations for piezoelectric and viscoelastic problems. Moreover, by using the dual reciprocal theorem for BEM, the static fundamental solutions can be used for dynamic boundary element method if the inertia term of the latter is treated as a general body force, which is the source term remained in the domain integral.
Based on the discussion above, in addition to the dynamic analysis of two-dimensional plates containing holes, cracks, and interfaces, several examples are also demonstrated in this dissertation to show that the static fundamental solutions of holes, cracks, inclusions and interfaces in anisotropic elasticity, derived in the sense of Stroh formalism, and the associated boundary integral equations have been extended to the applications for the piezoelectric and viscoelastic problems successfully with the prominence of accuracy and efficiency of the present BEM.
|
|---|
