On equilibrium problem for a plate having a crack under the creep condition

摘要

We consider a boundary problem for equations describing an equilibrium of a plate being under the creep law. The plate is assumed to have a vertical crack. The main peculiarity of the problem is determined by a presence of an inequality imposed on a solution which represents a mutual nonpenetration condition of the crack faces [W] ν ≥ ∣[(partial deriv w)/(partial deriv ν)]∣, where W = (w~1,w~2),w are horizontal and vertical displacements of mid-surface points of the plate, v is the normal to the crack shape, [•] is the jump of a function at crack faces. The presence of a crack alone implies a domain wherein the solution is determined to have a nonsmooth boundary, and boundary conditions given at crack faces are of the inequality type. An existence theorem for the equilibrium problem of the plate is proved. A complete system of equations and inequalities fulfilled at the crack faces is found. Solvability of the optimal control problem with a cost functional characterizing an opening of the crack is established. The solution is shown to belong to the space C~∞ near crack points provided the crack opening is equal to zero.