The Analytic Research On The Problem Of A Circular Hole With Several Periodic Cracks About Fracture Mechanics

As one branch of Fracture Mechanics, the development of linear elastic fracture mechanics(LEFM)is the earliest and perfect. The main job is to find the stress intensity factors for crack configurations. Westergaard method and Muskhelishvili method are two significant methods for solving the problems of linear elastic fracture mechanics. 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 easiest and strictest. Muskhelishvili method and extension base on conformal mapping function, map the area in physics plane onto the interior(or exterior)of unit disk or upside(or underside)of plane in phase plane. We can get the solutions of hole and crack by using the conformal mapping function, although the computerizing is complex, this method is called conformal mapping method. Several cracks and the hole with several cracks is investigated in this paper. The cracks are familiar and important model I, model II and mixed model of I-II. The main job in this paper includes the following three parts.First, by means of complex variable function method, this paper deals with the elastic problem about a plane of a circular hole with four straight cracks(a pair of asymmetry collinear cracks and a pair of symmetry collinear cracks) by using the technique of conformal mapping, the case of functionsφ(ζ),ψ(ζ)and analytic solutions of the stress intensity factors at the crack tip are obtained.The results can also simulate the circular hole with three straight cracks, the circular hole with four symmetry straight cracks, the circular hole with two asymmetry collinear cracks, the circular hole with two symmetry collinear cracks, the asymmetry cross crack,the symmetry cross crack and the mode T crack.Second, by using the new conformal mapping, this paper deals with the elastic problem about a plane of star crack, the elastic problem about a plane of a circular hole with 4k periodic radial straight cracks and the elastic problem about a plane of a circular hole with 2k periodic radial straight cracks. The analytic solutions of the stress intensity factors at the crack tip are obtained. Under the condition of limitation, some known results and some new results can be obtained. For example: the three straight cracks with common point and uniform distribution, the eight symmetry cracks, the circular hole with four symmetry straight cracks, the circular hole with eight symmetry straight cracks and so on. The analytic solutions of the stress intensity factors at the crack tip also can be obtained.Third, use above-mentioned method in new materials quasicrystals,the anti-plane shear problem about a circular hole with four straight cracks(a pair of asymmetry collinear cracks and a pair of symmetry collinear cracks)in one-dimensional hexagonal quasicrystals was investigated, and the analytic solutions of the stress intensity factors at the crack tip are obtained. Under the condition of limitation, the present results can also simulate the circular hole with three straight cracks, the circular hole with four symmetry straight cracks, the circular hole with two asymmetry collinear cracks, the circular hole with two symmetry collinear cracks, the asymmetry cross crack, the symmetry cross crack and the mode T crack in one-dimensional hexagonal quasicrystals.