Inverse Problem of Fracture Mechanics for a Perforated Stringer Plate

Abstract

To determine an optimal contour of holes for a perforated stringer plate weakened by a periodic system of cracks, an inverse problem of fracture mechanics is considered. It is assumed that the material of the plate is elastic or elastic-plastic. The stiffeners (stringers) are symmetrically riveted to the plate. The perforated plate is uniformly stretched at infinity along the stringers. It is assumed that rectilinear cracks are located near the contours of the holes and are perpendicular to the riveted stiffeners. The solution of the formulated inverse problem is based on the principle of equal strength. The optimal shape of the holes satisfies two conditions: the condition for the absence of stress concentration on the hole surface and the condition for the zero stress intensity factors in the vicinity of the crack tips. The unknown contour of holes is looked for in the class of contours close to circular. The action of the stiffeners is replaced by unknown equivalent concentrated forces at the points of their connection with the plate. The sought-for functions (the stresses, displacements, concentrated forces and stress intensity factors) are looked for in the form of expansion in small parameter. The solution to the problem is sought using the apparatus of the theory of analytic functions and the theory of singular integral equations, then the conditional extremum problem is solved. As a result, a closed system of algebraic equations is obtained, which allows to minimize the stress state on the contours of holes and stress intensity factors in the vicinity of the crack tips. The obtained system of algebraic equations allows to determine the form of equal strength contour of holes, the stress-strain state of the perforated stringer plate and also the optimal value of the tangential stress.