Study On Anti-plane Fracture Problems Of Multiple Cracks Emanating From A Hole In One-dimensional Hexagonal Quasicrystals With Piezoelectric Effects

Compared with the traditional crystalline, quasicrystalline material has wide application prospects in many areas of science and technology due to its unique quasi-periodic structure, which shows some unique performance. Because of the existence of phason fields, piezoelectric properties of quasicrystals (QCs) are much more complex than traditional crystalline. At present, the study on the fracture problem of defects in QCs has achieved some results. The research on the piezoelectric properties of the QCs is inadequate, especially for the research on the fracture problems of defects in QCs with piezoelectric effects. In addition, multiple cracks emanating from a hole model widely exist in various kinds of materials, which has brought great threat to the engineering practice and production development of human. For this case, based on the mechanics theory and applied mathematics model, the fracture problems of multiple cracks emanating from a hole in point group for the 6mm of in one-dimensional (ID) hexagonal QCs with piezoelectric effects will be considered using the complex variable function method and the analytic function theory, and some practical conclusions are given.In the first chapter, the briefly introduction are expounded, including the research background and significance, the development trend and fracture problem research of QCs, the main research contents of this article. In the second chapter, the anti-plane fracture problems about a circular hole with a crack, two symmetry collinear cracks, two asymmetry collinear cracks and four straight cracks (a pair of asymmetry collinear cracks and a pair of symmetry collinear cracks) were solved in 1D hexagonal QCs with piezoelectric effects, and the analytical expressions of the stress intensity factors (SIFs) for the phonon field and the phason field, and the electric displacement intensity factors (EDIFs) were presented. With the variation of the hole-size and the crack length, the present results can yield more practical models, for example, three straight cracks and four symmetry straight cracks emanating from a circular hole, the asymmetry cross crack, the symmetry cross crack, the mode T crack and Griffith crack. In engineering practice, cracks usually occur groups. A single crack is rare, and some groups of cracks can be idealized as periodic cracks. Therefore, anti-plane fracture problems about 4k periodic radial straight cracks and 2k periodic radial straight cracks originating from a circular hole in 1D hexagonal QCs with piezoelectric effects were studied in the third chapter. Meanwhile the analytic expressions of the SIFs for the phonon field and the phason field and the EDIFs are obtained. With the variation of the radius or the number of cracks, the present results can be reduced to the cases of two symmetrical edge cracks and a single edge crack emanating from a circular hole were given in the second chapter. Moreover, new models are obtained, such as 4k periodic radial straight cracks and 2k periodic radial straight cracks with common point. In the fourth chapter, anti-plane fracture problems of k periodic radial straight cracks emanating from a circular hole in 1D hexagonal QCs with piezoelectric effect was considered, and the analytic expressions of the SIFs for the phonon field and the phason field and EDIFs were given. With the variation of the radius, or a different positive integer k is given, results were given in the second chapter, and some new results can be derived as special cases from the general solutions, such as star crack, a circular hole with the three straight cracks and uniform distribution. Two kinds of electrical boundary conditions in the first three chapters are considered, that is, impermeable and permeable. Considering the partially electrical boundary conditions, based on Stroh formula, anti-plane fracture problems of two asymmetry collinear cracks emanating from an elliptic hole in 1D hexagonal QCs with piezoelectric effect was investigated in the fifth chapter, and the analytic expressions of the field intensity factors and energy release rate were obtained. With the changes of hole’s semimajor axis, semiminor axis and the crack length, analytic solutions of the field intensity factors can yield the cases of two symmetry collinear cracks and two asymmetry collinear cracks emanating from a circular hole, the mode T crack and the asymmetry cross crack were given in the second chapter. And corresponding results of Griffith crack, a crack and two symmetry collinear cracks of emanating from an elliptic hole were given. As far as phonon field is concerned, the obtainable results in this thesis are identical to the classical results.In this thesis, it can be found that the stress intensity factor is independent of the permeability of the electric field, however, the electric displacement intensity factor is a function of the permeability of the electric field. The electric displacement intensity factor is determined by the stress field for a permeable crack. Based on the analytic expressions of the dimensionless field intensity factor, the influences of the geometric parameters and the number of cracks on the field intensity factors are examined. And it can be concluded that the geometric parameters and the number of crack have obvious effects on the fracture characteristics of material. In the absence of the electric field, the obtainable results in this thesis can be degraded into anti-plane shear problem of multiple cracks emanating from a circular hole in ID hexagonal elastic QCs, which are the same with the existing results in the literature. And the analytic expressions for energy release rate of the crack tip is given in 1D elastic QCs, and it can be found that energy release rate is dependent on the phonon field as well as the phason field. Therefore, the energy release rate can be used as the fracture criterion of the 1D elastic QCs materials. According to experimental analysis and theoretical prediction of quasicrystals parameters in the existing literature, and the numerical results reveal the effects of material parameters, the number of cracks, the geometric parameters, the stress for the phonon field and the phason field on the fracture characteristics of QCs. In addition, when the missing phason field, the fracture problems of multiple cracks emanating from a hole can be used to discuss in isotropic piezoelectric materials, which is consistent with the existing literature. And the analytic expressions for energy release rate of the crack tip is given in isotropic piezoelectric materials. Numerical analysis is conducted to discuss the influences of crack length and applied mechanical/electric loads on energy release rate in isotropic piezoelectric materials, respectively.Research on such issues will provide reliable theoretical value for materials preparation and application of QCs in the engineering, which has practical application value. In addition, some analytic results can be used to test the correctness of the numerical solution.