Solving the Problem of a Transverse Bending of an Electro-Magneto-Elastic Half-Plane with Holes and Cracks

The paper solves the problem of bending a piezo plate in the form of a half-plane with holes and cracks by using the complex potentials of the theory of electro-magneto-elastic bending of thin plates. In this case, functions that are holomorphic outside the contours of the holes and cracks are decomposed into Laurent series, and functions that are holomorphic in the lower half-planes are expressed using Cauchy-type integrals in terms of functions conjugate to these functions. При таком подходе полученные суммарные функции точно удовлетворяют граничным условиям на прямолинейной границе полуплоскости, а для определения неизвестных коэффициентов рядов Лорана используются граничные условия на контурах отверстий и трещин, которые в работе удовлетворяются обобщенным методом наименьших квадратов, приводящим задачу к переопределенной системе линейных алгебраических уравнений, решаемой методом сингулярного разложения. With this approach, the resulting total functions precisely satisfy the boundary conditions on the rectilinear boundary of the half-plane. To determine the unknown coefficients of the Laurent series we use boundary conditions on the contours of the holes and cracks, which are affected by the generalized least squares method, leading the problem to an overridden system of linear algebraic equations solved by the singular value decomposition method. The numerical results of the electro-magneto-elastic state of a half-plane with a circular hole or crack, with a circular hole and an internal crack in the jumper, with a circular hole having an edge crack in the jumper are described. We establish regularities of changes in the electro-magneto-elastic state of the plate depending on its material and geometric characteristics of holes and cracks, their mutual location. It has been found that as the hole or crack approaches the rectilinear boundary, the values of the moments at the points of the bridge increase sharply and change insignificantly in other zones. A large concentration of moments is also observed at the points of the rectilinear boundary near the jumper. The values of these moments are especially high in the problem for a half-plane with a circular hole having an edge crack in the jumper. The values of bending moments are significantly affected by taking into account the piezo properties of the material, especially in areas of high concentrations of the bending moments, therefore in these cases it is forbidden to limit ourselves to solving the problem of elasticity theory of plate bending, and it is necessary to solve the problem of electro-magneto-elasticity. Keywords: thin piezo plate, half-plane, holes, cracks, complex potentials, Cauchy type integrals, generalized least squares method, concentration of bending moments, moment intensity factors.