The Complex Variable Function Method Research On The Problem Of Complicated Defects Of Classical Elasticity And Quasicrystals

As one branch of Fracture Mechanics, the development of linear elastic fracture mechanics(LEFM)is the earliest and perfect. The study of fracture mechanics under loading conditions is determining the stress fields and the stress intensity factors (SIFs) of all kinds of complicated defects at the crack tip.Westergaard stress function method and Muskhelishvili method are two significant methods for solving the problems of linear elastic fracture mechanics. The elastic problem about a plane of a circular hole with of three unequal straight cracks and an elliptical hole with four edge cracks are considered by using the Muskhelishvili method. Muskhelishvili method reduced the problems into solving two complex variable functions ? ( z),ψ( z)satisfying the boundary condition. The complex variable function method (Muskhelishvili method and extension) for solving the problems of hole or crack in the plane is common and practical.Muskhelishvili method is mapping many defects in physics plane onto the interior(or exterior)of unit disk or upside(or underside)of plane in phase plane using the technique of conformal mapping. Together with the Cauchy integral theory of complex variable function, exact analytic solutions of the SIFs for mode I, mode II and mode III crack problems are determined. Under the limiting conditions, the present results have shown a good agreement. With the changes of hole's semi-major axis, semi-minor axis and the crack length, these results not only stimulate more practical models, such as T crack, cross crack and so on, but also obtain a new crack, L crack, the stress intensity factors can be gained at the mode L crack tip. These results may have the potential applied value and significant meaning in engineering. Finding proper conformal mapping ,which is difficult and creative, is the key to the using of the complex variable function method.Quasicrystals (QCs) are both a novel structure of solids and a kind of new materials discovered in recent two decades. Quasicrystals are unique structures with long-range order but no periodicity, whose atomic arrangements have symmetries that are forbidden for periodic crystals, including configurations with five-fold, eight-fold, ten-fold symmetry and so on. A theoretical description of the deformed state of QCs requires a combined consideration of interrelated phonon and phason field. The phonon field describes the quasi-periodic arrangement of atoms in the complementary orthogonal space, which interact with one another. Quasicrystals (QCs) research in various aspects has made a number of important achievements since the Quasicrystals (QCs) has been found.This paper devotes to develop the complex variable function method for solving complicated defects including a circular hole with of three unequal straight cracks and an elliptical hole with four edge cracks in a strip of 1D hexagonal QCs. Moreover, the exact solutions of the SIFs for mode III problem of phonon and phason field are obtained. Under the condition of limitation, not only all the known results can be obtained but also With the changes of the crack length, hole's semi-major axis, semi-minor axis, not only these results can yield more practical models. For instance, the cross crack and T crack and so on, but also the solutions of the SIFs at the crack tip at the mode L crack which is found firstly in the paper are got. As far as phonon field is concerned, these results are identical to the classical results.