Nonstationary indentation of a blunt rigid body into an elastic layer: An axisymmetric problem

Abstract

The nonstationary indentation of a blunt rigid body into an elastic layer is studied. The general formulation of the problem includes different boundary conditions in the contact area and on the free surface of the layer. The simplified nonmixed problem that arises at the early stage of interaction and allows obtaining approximate results for later times is solved exactly. The solution obtained is compared with that for the plane case Keywords: nonstationary indentation, blunt body, elastic layer Introduction. The current state of the art in the research on nonstationary contact interaction and contact problem of elasticity is elucidated in, e.g., [4,6,9]. In the general case, the modern problem of impact of a body against an elastic medium or structural member is formulated as a nonstationary mixed initial-boundary-value problem of elasticity with an unknown time-dependent boundary, which is determined during problem solving. The problem statement includes the equations of elastic deformation of the impacted body, the equation of motion of the impactor, the relationship between the interaction force and the unknown contact boundary, the relationship between the contact area and the displacement of the impactor, and the boundary and initial conditions. In the general case, the problem is coupled one with fuzzy input data, which predetermines the difficulties of its solution.Note the following physical factor that has a direct influence on the ways of solving the problem of impact of a body against an elastic medium. During an impact, the contact boundary between the bodies moves over their surfaces with variable velocity dependent on their shapes. If the bodies are blunted, the velocity of the contact boundary can be very high during the initial period (at least, higher than the velocity of elastic waves (supersonic)). During this period, waves in the elastic body do not interact with its free surface and, hence, the boundary conditions on it can be chosen as one sees fit. This would allow us, at least for the early stage of interaction, to formulate a nonmixed boundary-value problem and, thus, to simplify the solution procedure. Eventually, the velocity of the contact boundary reduces to transonic and then subsonic magnitudes because of the geometry of the bodies and the deceleration of the impactor, which causes the waves to reach (and interact with) the surface of the body outside the contact region. However, as demonstrated in a number of studies, the supersonic solution is in some cases enough to make assessments.Such an approach was used in [11,12] to solve the plane problem of nonstationary indentation of a smooth blunt body into a layer of perfect fluid and into a layer of elastic material. The solution found demonstrates the behavior and features of the elastic waves in the layer, taking into account their repeated reflections from the layer boundaries. The problem of impact of an elastic cylinder against a rigid barrier was solved in [14], this problem being similar to that treated with asymptotic appro...

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