Solutions of Periodic and Doubly Periodic Bending Problems of a Thin Piezo Plate with Holes or Cracks

The solutions of periodic and doubly periodic problems of bending of a piezo plate with elliptical holes or cracks are given with an analysis of the results of numerical studies. In this case, complex potentials of the theory of bending of thin electro-magneto-elastic plates are used, holomorphic functions outside the holes are represented by Laurent series in negative powers of variables from the corresponding conformal mappings and, based on the periodicity or doubly periodicity of the electro-magneto-elastic state of the plate, the coefficients of the series from all the holes are expressed through the coefficients of the series from one, the so-called main hole. The determination of the last coefficients is carried out from the boundary conditions on the contour of the main hole using the generalized least squares method. The results of numerical studies for a plate with circular holes or cracks with full or partial consideration of piezo properties, without taking them into account, are described. The patterns of influence on the values of bending moments and their concentration of the geometric characteristics of the discussed plates and the physico-mechanical properties of their materials are established.