Research On Three Dimensional Fracture Problem Of Several Types Of Materials With Rectangular Crack

With the rapid development of science and technology, the modern technology engineering on the performance for materials and structures were increasingly demanded. Due to kinds of materials(elastic materials, piezoelectric materials and piezoelectric/prezomagnetic materials) own characteristics, there will be a variety of internal defects(cracks or holes, etc.) in the process or it will be different degrees of damage in the service of the environment, which seriously affect the safety of the material structure itself. For the previous works of the fracture problem, the anti-plane and plane fracture problems were largely studied, and the three-dimensional fracture problem was relatively weak investigated. Due to the three-dimensional fracture problem closer to the actual real situation, the increase of the number of material constants and the control equation greatly increases the difficulty of solving the mathematical. At the same time, the rectangular crack stress intensity factor is greater than the same level of the real class elliptical crack stress intensity factor. Therefore, three-dimensional fracture analysis of these materials not only has important academic value, but also has important practical engineering significance. In this paper, from the perspective of classical elasticity theory and nonlocal continuum mechanics, the three-dimensional fundamental fracture mechanics behavior of the materials with rectangular crack under various loading were studied. The main contents are as follows:Firstly, based on the classical elasticity theory, the static fracture behavior of a rectangular crack and two rectangular cracks in three-dimensional orthotropic elastic material were analyzed. The problems were formulated through the two-dimensional Fourier transform technique into three pairs of dual integral equations, and the unknown variables are the jumps of displacements across the crack surfaces. The Schmidt method was used for solving numerical solution. The expressions of the stress intensity factor at the crack edges were given. The results showed that: in the case of a single rectangular crack, the stress intensity factors were relevant with the geometry shape of rectangular crack; in the case of two rectangular cracks, the stress intensity factors were relevant with the geometry shape of rectangular crack and the distance between two rectangular cracks. Meanwhile, the dynamic fracture behavior of two rectangular cracks in three-dimensional transverse isotropic elastic materials was sloved. The expressions of the dynamic stress intensity factor along each rectanglar crack edges were given. Numerical results show that the dynamic stress intensity factors were relevant with the geometry shape of the rectangular crack, the distance between two rectangular cracks and the frequency of incident wave.Secondly, based on classical elasticity theory, the dynamic fracture problems of two rectangular cracks in three-dimensional transversely isotropic piezoelectric materials under electric permeable condition was investigated by means of two-dimensional Fourier transform, Almansi’s theorem and Schmidt method. The mathematical expression of the dynamic stress intensity factor and the dynamic electric displacement intensity factor along each rectangular crack edges were given, and the numerical results show that the effects of the geometric shape of the rectangular crack, the frequency of the incident wave and the distance between two rectangular cracks were summarized on the fracture properties of piezoelectric materials. Meanwhile, the static fracture problem of four rectangular cracks in three-dimensional transversely isotropic piezoelectric/piezomagnetic materials under the electromagnetic limited-permeable conditions was studied. Numerical results show that the effects of the geometric shape of the rectangular crack, the distance between four rectangular cracks, the electric permittivity and the magnetic permeability of the air inside the crack on the stress, the electric displacement and magnetic flux intensity factors were discussed.Again, based on the non-local theory, the static and dynamic fracture behaviors of a single rectangular crack in three-dimensional transversely isotropic elastic materials were solved. The Schmidt method was used to overcome the mathematical difficulties encountered in similar problems in the past, and this problem was transfored the fracture problems into three pairs of dual integral equations. Compared with the classical theory solution and it is an infinite value. The mathematical expression of the stress field along the crack edge was given, and results show that the rules of the effect of the geometric shape of the rectangular crack and the lattice parameters of the material on the stress field. Meanwhile, the static fracture behavior of a single rectangular crack in three-dimensional orthotropic elastic materials was further solved.Finally, the application of non-local theory was extended to study the static fracture problem of a rectangular crack in three-dimensional transversely isotropic piezoelectric materials under the electric permeable boudary condition. By using the two-dimensional Fourier transform, Almansi’s theorem and Schmidt method, the mathematical expressions of the stress field and electric displacement field along the rectangular edges were obtained. Numerical results show that the rules of the influence of the geometric shape of the rectangular crack and the lattice parameters of the material on the stress field and the electric displacement field near the crack edges. The finite values of the stress field and electric displacement field are obtained by non-local theory, so that the maximum stress and the maximum electric displacement can be used to predict the destroy conditions of piezoelectric material.The work of this article may be providing theoretical supports for the analysis and evaluation of three-dimensional fracture behavior of elastic material, piezoelectric materials and piezoelectric/piezomagnetic materials and structures.