The contact problem for the membrane and the impactor in a nonstationary formulation

Аuthors

Okonechnikov A. S.^*, Fedotenkov G. V.^**, Feoktistova E. S.^***

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: leon_lionheart@mail.ru
**e-mail: greghome@mail.ru
***e-mail: feoktistova0206@mail.ru

Abstract

This paper presents an algorithm for solving problems of contact interaction between an indenter and a membrane at various speeds. A distinctive feature of the algorithm proposed for consideration is its applicability to both axisymmetric and flat problems. This approach significantly simplifies solving contact problems by reducing various settings to a generalized solution. The second feature of the algorithm is its applicability to various speed modes of interaction, namely, at speeds above the speed of sound and at subsonic speeds. The paper introduces two hypotheses that take into account the features of the subsonic mode of contact interaction. Based on these hypotheses, the paper applies the thesis that the membrane outside the contact area has the form of a linear function, and in the contact area the shape of the membrane coincides with the shape of the indenter itself. Due to the introduction of the above hypotheses, determining the position of the boundaries of the contact area of the stamp and the membrane at any time, with a known displacement of the frontal point of the stamp, becomes mathematically possible. Determining the position of the boundaries of the contact area is the first step in the proposed algorithm. The next step determines the displacement of membrane points under the action of contact pressure from the indenter, as well as the contact pressure itself under the indenter. In the course of solving the problem, the presence of distributed and concentrated loads was revealed, which is reflected in the obtained expression for the contact pressure. It is worth noting that concentrated loads arise only if the rate of expansion of the contact area coincides with the speed of sound. The article first considers the formulation of the problem in the subsonic interaction mode, and after obtaining the first results, moves on to consider the problem in the supersonic interaction mode. In this case, the problem statement will be similar to the subsonic mode. Taking into account the features of contact interaction in the supersonic mode, it is assumed that the deflection of the membrane when the indenter moves at speeds above the speed of sound will be determined by the shape of the indenter itself and the depth of its penetration. The next step of the algorithm determines the pressure under the stamp. The information on the structure of distributed loads under the stamp in the supersonic contact interaction mode obtained during the solution coincides with the results obtained for the subsonic mode. The paper presents an example of calculating the problem of interaction between a convex indenter and a membrane in a flat formulation, and presents graphs of the obtained results for determining the depth of indenter penetration and the expansion of the contact area depending on time.

Keywords:

Non-stationary contact, impactor, membrane, deflection of the membrane, contact area, influence function, flat setting, axisymmetric setting

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