Investigation of the electro-magneto-elastic state of a finite multiply connected thin plate

The problem of bending a finite plate with arbitrary holes and cracks is solved with the use of complex potentials of the theory of bending of thin electro-magneto-elastic plates. Moreover, with the help of conformal mappings, expansion of holomorphic functions into the Laurent series or Faber polynomials owing to satisfaction of boundary conditions on the contours of the plate by the generalized least squares method, the problem is reduced to solving an overdetermined system of linear algebraic equations by the method of singular value decompositions. Results of numerical investigations for a circular plate with a circular hole, for a circular plate with an internal or edge crack, for a plate with a two circular internal holes or external recesses are reported. We study how physical and mechanical properties of the plate material and geometric characteristics of holes, cracks and recesses influence the values of the bending moments and moments intensity factors for the crack ends. It is important to consider the piezoproperties of the material on the values of bending moments in the plate. They cannot be neglected in the study of the stress-strain state, that is, it is necessary to solve the problem of electro-magneto-elasticity, and not the problem of the classical theory of bending of an anisotropic plate. Moreover under the electromagnetic field in the piezoelectric plate there are sufficiently large bending moments (hence stresses and deformations), and they can be found only by solving the problem of electro-magneto-elasticity. It is determined that a crack in a plate can be considered as an elliptical hole, in which the ratio of the semiaxes is less than 10-3, and in these cases it is possible to calculate the intensity factors of mechanical and electromagnetic moments. We also outline the distances between the contours, which have an insignificant influence of one of them on the stress-strain state around the other and can be neglected.