Application Of Complex Variable Function Method To Complicated Defects Of Classical Elasticity And Quasicrystals

The fracture phenomenon is always in close relation to defects, such as holes, notches or cracks in materials and structures because of the strong stress concentration there. The defects (holes, cracks, dislocations, etc.) and the stress concentration always cause the destruction of materials and structures. Therefore, determining the stress fields and the stress intensity factors (SIFs) of all kinds of complicated defects at the crack tip is the key to the study of fracture mechanics under loading conditions.The complex variable function method is one of the fundamental methods to solve classical elasticity problems, which is very powerful in solving problems of classical plane elasticity and defects. Using the technique of conformal mapping together with the Cauchy integral theory of complex variable function, many defects in an infinite plane, such as elliptical hole with two symmetry cracks, circular hole with asymmetry collinear cracks, elliptical hole with an edge crack, elliptical hole with asymmetry collinear cracks are considered in the second chapter of this paper. The stress fields of these defects are obtained under uniform unidirectional tension, biaxial tension, shearing stress, anti-plane shear at infinity and under uniform pressure, shearing stress, anti-plane shear of holes with crack. At the same time, 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 numerical solutions as well as they can be reduced to known exact solutions. With the changes of hole's semimajor axis, semiminor axis and the crack length, these results which may have the potential applied value in science and engineering can stimulate more practical models, such as circular hole with an edge crack, circular hole with two symmetry cracks, T crack, cross crack, half-plane with an edge crack and so on.Quasicrystals (QCs) are both a novel structure of solids and a kind of new materials discovered in recent two decades. They exhibit five-, eight-, ten-, or twelve-fold symmetry, which are forbidden in conventional crystals, and long-range order, which is lacking in amorphous solids. Their translational order is not periodic, as in conventional crystals, but quasiperiodic. A theoretical description of the deformed state of QCs requires a combined consideration of interrelated phonon and phason field. The phonon field describes the quasiperiodic arrangement of atoms in the complementary orthogonal space, which interact with one another. Owing to the existence of phason field, the elasticity of QCs is more complex than that of the conventional crystals. As far as the elasticity and defects problem of QCs are concerned, there are some mathematical methods having been proposed such as Green function method, Fourier transform method and perturbation method and so on. But about complex variable function method of the elasticity theory of QCs, it has been done only in the simplest case of one-dimensional (1D) hexagonal QCs with point group 6mm. The third chapter of this paper devotes to develop the complex variable function method for solving complicated defects including elliptical hole with two symmetry cracks, circular hole with asymmetry collinear cracks, elliptical hole with an edge crack, elliptical hole with asymmetry collinear cracks and asymmetrical semi-infinite crack 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. It is shown that the complex variable function method is one of the most effective ways for solving complicated defects and elasticity of QCs. With the changes of hole's semimajor axis, semiminor axis and the crack length, these results can yield more practical models. For instance, the elliptical hole with asymmetry collinear cracks can be reduced to the elliptical hole with an edge crack, circular hole with an edge crack, circular hole with asymmetry collinear cracks, circular hole with two symmetry cracks, elliptical hole with two symmetry cracks, T crack, symmetry cross crack, asymmetry cross crack and half-plane with an edge crack as well as the known Griffith crack. Besides, the asymmetrical semi-infinite crack in a strip can yield the symmetrical semi-infinite crack in a strip and semi-infinite crack in an infinite plane. As far as phonon field is concerned, these results are identical to the classical results.Further, the complex variable function method and the technique of conformal mapping are developed to solve the defects of dynamic mechanics. The forth chapter investigates fast propagating crack with an elliptical hole, and yields the exact analytical solution of the dynamic SIFs at the crack tip. Moreover, the analytical solutions of the fast propagating crack with an elliptical hole and fast propagating T crack are able to be solved out under the limiting conditions. When crack speed V tends to zero, the dynamic solution can be reduced to the static solution. Consequently, the dynamic problems about defects of 1D hexagonal QCs are discussed in the fifth chapter. The analytical solutions of defects including the fast propagating crack with elliptical hole and an asymmetrical fast propagating semi-infinite crack in a strip are given. Under the limiting conditions, the former can be reduced to the fast propagating crack with an elliptical hole, fast propagating T crack and half-plane with an edge fast propagating crack of 1D hexagonal QCs. And the latter can yield a symmetrical fast propagating semi-infinite crack in a strip and fast propagating semi-infinite crack in an infinite plane of 1D hexagonal QCs. These models are more complex and general than motion Griffith crack. The present results make a good way for analyzing dynamic problems of more complicated defects.