STEADY-STATE CREEP OF A LONG NARROW MEMBRANE INSIDE A HIGH RIGID MATRIX AT VARIABLE TRANSVERSE PRESSURE

This paper describes a problem of steady-state creep of a long narrow rectangular membrane in the case of a linear dependence between the transverse pressure and time. The membrane is located inside a high long rigid matrix with a rectangular cross section, the ratio of the height to half the width is greater than unity. A power relationship between the stress intensities and creep strain rates of the membrane is used. Two variants of contact conditions for the membrane and the matrix are considered: ideal sliding and adhesion. The analysis of the problem is carried out until the time at which the membrane is almost completely adjacent to the matrix walls. It is shown that, if the relative height of the membrane is smaller than a certain value, then the creep duration of the membrane until the moment at which it almost completely adjoins the matrix walls is shorter in the case of ideal sliding than in the case of adhesion, and vice versa. An explanation of this effect is provided.


Introduction
Consider the creep of a long narrow rectangular membrane fixed along the long sides and loaded with uniform transverse pressure q , which increases in proportion to time t . The solution to this problem under various physical and geometric conditions is given in the monographs of L.M. Kachanov [1], Odqvist [2], N.N. Malinin [3] and others. Of particular interest is the study of the creep of the considered membrane inside a rigid matrix. In monographs [3,4], a cycle of problems on the creep of such a membrane inside a rigid matrix is considered. In [4], solutions of problems are given taking into account various forms of matrices: wedge-shaped, curvilinear and rectangular under various conditions at the contact between the membrane and the matrix.

Problem Statement
This paper investigates the creep of a long narrow rectangular membrane of thickness 0 H inside a rigid rectangular matrix (Fig. 1). Membrane width 2a and length L satisfy the inequality 2 / 1 a L << . The ratio of the height of the matrix b to half of its width a satisfies the inequality / 1 b a ≥ . To describe the deformation of the membrane at 0 t > , a power-law model of the steady-state creep of the material is used where u σ , u p  -stress intensity and creep strain rates intensity, respectively, 0 σ , 0 t and n -constants of a corresponding dimensions. These values are determined as a result of a series of creep tests of several tensile specimens and the subsequent approximation of the dependence of the steady-state creep rate on stress in the form of a power function. Here, the proportional dependence of the magnitude of the transverse pressure q on time t is considered: It is of interest to consider the features of the almost complete adherence of the membrane to the space inside the matrix at different rates of increase of the transverse pressure at which the ratio of the radius of curvature of the middle surface ρ of the membrane to a equals a given small value / 1 a ρ = ∆ << . When modeling the stress-strain state at 0 t > , the radial rr σ , circumferential θθ σ and axial zz σ principal stresses and the corresponding components of the creep strain tensor rr p , p θθ and zz p are considered. Considering a membrane element; taking the stresses in the element uniformly distributed over the thickness and writing the equilibrium equations in projections onto the normal and tangent, it is obtained: ICMIE 123-3 where H -membrane thickness. Therefore, (3) Comparing equalities (2) and (3), it is concluded that the middle surface of the membrane during its deformation is a part of the surface of a circular cylinder with an opening angle 2α [4]. Consequently, according to equality (2), the circumferential stress along the circumference of the radius ρ does not change.
The membrane creep is studied in three successive stages.

Free Deformation of the Membrane under Creep Conditions (The First Stage)
At this stage, the membrane is deformed under conditions of steady creep until it touches the side walls of the rigid matrix.
Dimensionless variables are introduced as follows: Further, the dashes over all dimensionless variables are omitted, and the dot denotes the derivative with respect to the dimensionless time everywhere.
As a connection between the components of stress tensors and creep strain rates, the hypothesis of proportionality of the corresponding deviators is accepted (see, for example, [4]): In the considered plane deformed state, the rate of axial creep strain zz p  is taken to be zero: Accept, as usual for thin-walled cylindrical shells, the equality: In this case, from the creep hypothesis (4), taking into account (5) -(6), it follows: Considering two close deformed states of the membrane, the increment of the circumferential creep strain is determined: Consequently, the circumferential creep strain rate is From the incompressibility condition in the case of a plane deformed state, it follows: As a result of a series of transformations, a connection between α and 1 τ is obtained, where 1 τ is the time of the first stage: ( 0 x , 0 y -coordinates of the points of contact between the membrane and the matrix), further on, the dashes above these dimensionless variables are also omitted.
In this paper, the creep of the membrane inside a relatively high matrix is considered ( 1 b ≥ ). Due to the axial symmetry of the membrane and the matrix, the creep of the right half of the membrane in coordinates

Third stage ( )
After a series of transformations, the dependence of the coordinate 0 y on time 2 τ is obtained:  ( ) ( ) ( ) Introduce the expressions for u σ and u p  into (1). After a series of transformations, the dependence of the coordinate 0 y on time 3 τ is obtained:

Conclusion
The results of a study of the steady-state creep of a long narrow membrane inside a high rigid matrix with a proportional dependence of the transverse pressure on time are presented. Two variants of the conditions of contact between the membrane and the matrix are considered -ideal sliding and adhesion. The analysis is carried out until the time of almost complete adherence of the membrane to the matrix. These times are compared for ideal slip and adhesion for different values of the exponent in the constituting equation and the relative height of the matrix.