ДЕЯКІ ЗАДАЧІ ДЛЯ ТРАНСВЕРСАЛЬНО-ІЗОТРОПНОГО ПРОСТОРУ З ПЕРІОДИЧНИМИ ПАКЕТАМИ АНТИТРІЩИН

In this paper, the stress state of a transversely isotropic space with periodic systems (packets) of plane circular anti-cracks, the centers of which are located on the anisotropy axis, and the planes are perpendicular to it, is investigated for the first time. It is assumed that the space is under a constant bi- axial compressive stress applied at infinity. Each periodic system (packet) is determined by a representative layer whose planes are perpendicular to the anisotropy axis, containing a finite system of anti-cracks of different sizes. Such a system forms a certain configuration. Any odd number of anti- cracks of arbitrary size can be included in a specific configuration, but with certain restrictions: the anti-cracks are symmetrical relative to the middle plane of the layer, their sizes satisfy a certain convergence condition. The given restrictions provide practically uniform conditions with respect to tan- gential stresses and normal displacements on the boundaries of the representative layer (the order of values of these quantities is in the range 10-10 ÷10-14 ), which can be considered as infinity conditions. All problems were solved by the generalized Fourier method, which allowed them to be reduced to infinite systems of linear algebraic equations with Fredholm operators. The results of the study were also based on an extensive computer experiment, within the framework of which stress distributions were calculated not only in periodic problems, but also in non-periodic problems formed by several representative layers. Practical verification of the convergence of the reduction method showed high efficiency of the generalized Fourier method. Thus, doubling the reduction parameter from 10 to 20 led to stabilization of 8 – 14 significant digits in the obtained results. Compari- son of stress intensity factors for different configurations showed that for anti-cracks of the same size they depend little on a specific configuration. A qualitative conclusion that follows from the calculation results is that normal stresses on the surface of a smaller anti-crack outside its boundary in a packet increase with an increase in the size of larger neighboring anti-cracks.