Collision of dissimilar blunt elastic bodies: a plane problem

Abstract

The paper deals with a direct central impact of two infinite cylindrical bodies having differently shaped cross sections and made of different materials. A nonstationary plane problem of elasticity is solved. The contact boundary is moving and determined during the solution. A mixed boundary-value problem is formulated. Its solution has the form of Fourier series. Satisfying mixed boundary conditions gives an infinite system of Volterra equations of the second kind for the unknown coefficients of the series. The basic characteristics of the impact process and their dependence on the physical and mechanical properties of the bodies are determined numerically Introduction. Impact of elastic bodies is a complicated and difficult-to-solve problem in continuum mechanics. The modern formulation of the impact problem leads to a mixed boundary problem with initial conditions and an unknown time-dependent contact boundary determined during the solution. Some recent approaches to and results on the subject can be found in [4,6,[8][9][10]. A variable contact boundary and associates difficulties of its description are also encountered in nonclassical fracture mechanics [7].Here we present a method to solve the central impact problem for elastic cylindrical bodies with different physical and mechanical properties (masses, shapes of frontal surface, materials). This approach was earlier used to study the direct central impact of identical elastic bodies [4,6]. As before, we assume that there is no friction between the bodies, the impact velocities are small, and the frontal surfaces are smooth and have small curvature. At the initial instant of impact, the bodies interact along the common generatrix. In this case, the deformation pattern is the same in all cross sections of the cylinders; therefore, a plane problem can be formulated to study the process. We also assume that the depth of interpenetration of the bodies is small compared with the contact region, which makes it possible to prescribe boundary conditions on the undisturbed surfaces of the bodies. We will examine a period during which the perturbations reflected from the rear and lateral surfaces of the bodies do not yet reach the contact region, which permit formulating somewhat arbitrary boundary conditions on the lateral and rear surfaces.There may be different problem formulations depending on the method used to determine the contact boundary. We will present specific numerical results and analyze the dependence of the stress-strain state of the bodies on their physical and mechanical properties. A central-impact problem is formulated in Sec. 1. The governing system of equations for determining the stress state of and the force of interaction between the bodies is derived in Sec. 2. A simplified governing system of equations for determining the characteristics of the process at its initial stage is derived in Sec. 3. Calculated results are presented and conclusions are drawn in Sec. 4.1. Problem Formulation. Consider two elastic cylindrical bodies having parallel...