The steady-state creep of long membrane in a rigid matrix at a variable transverse pressure

The problem of the steady-state creep of a long rectangular membrane in constrained conditions inside a rigid matrix is investigated with a piecewise constant dependence of the transverse pressure $q$ on time $t$. The problem considers a long matrix of rectangular cross section, in which the ratio of its height to width is not less than 0.5. As an example, the creep of the membrane is investigated with a single change in the magnitude of the transverse pressure over time. Three variants of the contact conditions of the membrane and the matrix are considered: perfect sliding, adhesion and sliding taking friction into account. In this paper, four stages of membrane deformation were investigated. At the first stage (elastic deformation), the membrane, flat in the initial state, under the action of pressure, instantaneously is deformed elastically, acquiring the form of an open circular cylindrical shell with a central angle $2\alpha _1 $. At the second stage, the membrane is deformed under steady-state creep conditions up to the moment when the side walls of the matrix touch. The third stage ends when the membrane touches the transverse wall of the matrix. In the fourth stage, the membrane is in contact with the matrix on the transverse and lateral sides. The analysis is carried out until the time of almost complete adherence of the membrane to the matrix, at which the ratio of the radius of the membrane near the corners of the matrix to the initial width of the membrane is 0.005. For the third and fourth stages, the friction force of the membrane on the matrix walls is additionally taken into account. The dependences of the thickness of various parts of the membrane on time and on the intensity of stresses in the membrane on time are obtained. In relation to this formulation of the problem, deviations from the rule of summing the partial times of filling the matrix are considered.