Nonstationary plane contact problem in theory of elasticity for conformal cylindrical surfaces

doi:10.1016/j.camss.2017.03.002

Acta Mechanica Solida Sinica

Volume 30, Issue 2, April 2017, Pages 190-197

https://doi.org/10.1016/j.camss.2017.03.002 Get rights and content

Abstract

A numerical–analytical approach is described to investigate the process of impact interaction between a long smooth rigid body and the surface of a circular cylindrical cavity in elastic space. A non-stationary mixed initial boundary value problem is formulated with a priori unknown boundaries moving with variable velocity. The problem is solved using the methods of the theory of integral transforms, expansion of desired variables into a Fourier series, and the quadrature method to reduce the problem to solving a system of linear algebraic equations at each time step. Some concrete numerical computations are presented. The cylindrical body mass and radius impact on the profile of the transient process of contact interaction has been analysed.

Introduction

In mechanics of contact interaction, which embraces the problems of elastic impact of bodies, investigators distinguish between cases of interaction of bodies whose shapes are conformal and non-conformal [1]. The interacting bodies are commonly considered non-conformal if their profiles, at least in the place where they contact another body, do not match. For such bodies, as a rule, the contact zone is significantly smaller than their characteristic sizes. When simulating the impact problem for such bodies, this zone can be approximated by a plane surface [2], [3], [4], [5]. A conformal contact, as a rule, is inherent to "inner" contact problems. It occurs when the surfaces of both bodies in the contact zone have sufficiently close or conformal profiles at the initial moment. In engineering practice, in particular, such a contact occurs between the shaft (axle) neck and the bearing part surface, between the teeth of Novikov's mesh gears, or in other cases [6], [7]. The inapplicability of Hertz's theory to solving the above application problems due to commensurability of the sizes of the contact zone with the radii of curvature of mating surfaces, in particular, of cylindrical form, has sparked interest in developing appropriate approaches to such investigations. Monographs [1], [6], [8] and papers [9], [10], [11], [12], [13], [14], [15] can be mentioned among a big variety of publications dealing with developing effective methods to solve the problems of contact interaction between conformal cylindrical bodies. Thereat recent publications have been focused on building approximate solutions to the problems in the static statement with the least number of independent parameters. At the same time, the impact contact interaction between conformal bodies has not been investigated adequately to date [16], [17].

This paper offers a numerical–analytical method in solving the non-stationary problem of impact "inner" contact between cylindrically shaped conformal bodies. The case is considered when a rigid body with a cross-section defined by a curve sufficiently smooth in the frontal part collides at a given initial velocity with the surface of a round cylindrical cavity in elastic space. The stress–strain state of elastic medium and the characteristics of body motion upon penetration into the medium are to be defined. The essence of the approach offered consists in applying the Laplace integral transform in the time domain and expanding desired variables into Fourier series for the corresponding angular coordinates. The formulated non-stationary mixed boundary problem is reduced to solving dual integral equations whose kernels are determined using the Volterra integral equation. The solution of the latter is computed using a regularisation algorithm, and the system of dual equations is solved by time discretisation and solving a system of algebraic equations at each time step. Numerical results are obtained for two cases: 1) the radii of the cavity and the body are equal (for the simulation of the impact of a crankshaft against a journal bearing); 2) the radius of the body is half of the cavity radius. For bodies with different masses, the following basic characteristics of the impact process are defined: stress in the contact zone, medium resistance force, and body motion velocity.

Section snippets

Generalities

Impact interaction of a body with an elastic medium causes, on the one hand, formation of elastic waves in the medium, which dissipate part of the impact energy, and on the other hand, a change in the body velocity up to a reversal of its sign and motion in the opposite direction. Adequate simulation of the process considered requires solving a coupled system of equations comprising the elasticity theory equations describing the motion of an elastic medium and the body motion equations.

Problem statement

In general, the above considerations allow for a statement of the mixed initial boundary value problem with an a priori unknown boundary varying with time and space.

A long rigid cylindrical body is placed in a cylindrical cavity and, at t = 0, velocity V(t) is instantaneously imparted perpendicular to the body axis. Gravity is not considered in this model. Let velocity V(t) be significantly less than the velocity of sound in the medium, and the depth of body penetration into the medium be

Form of solution

To solve the stated problem, the integral Laplace transform in time t domain with parameter s [19] shall be applied to equations (2). Respective transforms shall be denoted by superscript L. Then Eq. (2), with the zero initial conditions (8) taken into account, shall take the form $\begin{matrix} \frac{\partial^{2} Φ^{L}}{\partial t^{2}} + \frac{1}{r} \frac{\partial Φ^{L}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} Φ^{L}}{\partial θ^{2}} - s^{2} Φ^{L} = 0 \\ \frac{\partial^{2} Ψ^{L}}{\partial t^{2}} \frac{1}{r} \frac{\partial Ψ^{L}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} Ψ^{L}}{\partial θ^{2}} - s^{2} ξ^{2} Ψ^{L} = 0 \end{matrix}$ and their general solution decaying at infinity (see (7)) can be written as $\begin{matrix} Φ^{L} = \sum_{n = 0}^{\infty} A_{n} (s) K_{n} (s r) \cos (n θ); Ψ^{L} = \sum_{n = 1}^{\infty} B_{n} (s) K_{n} (s ξ r) \sin (n θ) \end{matrix}$

Here, K_n(x) is the

Solution technique

To obtain a solution of the mixed boundary problem formulated in section III, first we shall consider the solution of an auxiliary problem for the system of equation (2) under the following boundary conditions: the tangential stress on the cavity surface (r = 1) is absent, but its strain rate is specified as a certain function W(t, θ), i.e. $\begin{matrix} {\frac{\partial u}{\partial t} |}_{r = 1} = W (t, θ); {σ_{r θ} |}_{r = 1} = 0 \end{matrix}$

Function W(t, θ) should be presented as a Fourier series $\begin{matrix} W (t, θ) = \sum_{n = 0}^{\infty} W_{n} (t) \cos (n θ) \end{matrix}$

With Laplace transforms and solutions (14), the

Inverse laplace transform for $P_{n}^{L} (s)$

To solve the coupled system of Eqs (22), (9) and (12), it is necessary to have an effective algorithm of computing integrands P_n(t) whose transforms are given by formula (20). Here we shall give the technique of computing these functions. Function $P_{n}^{L} (s)$ is given as: $\begin{matrix} P_{n}^{L} (s) = \frac{S_{n}^{L} (s)}{R_{n}^{L} (s)} \end{matrix}$ where the numerator and denominator respectively have the form $\begin{matrix} R_{n}^{L} (s) & = & \frac{1}{s^{4}} [T_{1 n} T_{4 n} - T_{2 n} T_{3 n}] \\ = & - \frac{ξ^{2}}{2} n \frac{1}{s} K_{n} (s) K_{n} (s ξ) + (\frac{ξ^{2}}{2} - n \frac{1}{s^{2}}) K_{n + 1} (s) K_{n} (s ξ) \\ - ξ n \frac{1}{s^{2}} K_{n} (s) K_{n + 1} (s ξ) + ξ \frac{1}{s} K_{n + 1} (s) K_{n + 1} (s ξ) \\ S_{n}^{L} (s) & = & \frac{1}{s^{4}} [T_{4 n} T_{5 n} - T_{3 n} T_{6 n} + \frac{ξ^{2}}{2} (T_{1 n} T_{4 n} - T_{2 n} T_{3 n})] \end{matrix}$

Numerical results

Two cases are considered as examples of implementing the proposed algorithm. The first one deals with the most typical case in the applied sense when the body and the cavity have a circular cylinder shape and their radii differ so little that they can be considered equal (R_b = R). In this case, it is evident that the value of polar angle θ* that defines the surface contact boundary is equal to π/2. Coefficient ξ during computations is taken to be 1.87, which corresponds to Poisson's ratio of the

Conclusions

Within the framework of the plane problem in the linear theory of elasticity the method of numerical–analytical solving of the problem in impact interaction (unsteady indentation) between a solid smooth body and the surface of a cylindrical cavity in elastic medium is presented. This method allows to compute all the characteristics of the process of interaction in case when penetration velocity much less than the velocity of elastic waves in the medium. The calculated velocity and displacement

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Nonstationary plane contact problem in theory of elasticity for conformal cylindrical surfaces

Abstract

Introduction

Section snippets

Generalities

Problem statement

Form of solution

Solution technique

Inverse laplace transform for PnL(s)

Numerical results

Conclusions

Int. J. Solids Struct.

Compos. Struct.

Mech. Mach. Theory

Contact Mechanics

Supersonic and subsonic stages of dynamic contact between bodies

Proc. R. Soc.

Impact of blunted bodies on a liquid or elastic medium

Int. Appl. Mech.

Methods of the Classical Theory of Elastodynamics

An exact solution for the superseismic stage of dynamic contact between a punch and an elastic body

J. Appl. Mech.

To the theory of analysis of a two-layer cylindrical bearing

Mech. Solids

The state of stress induced by the plane frictionless cylindrical contact. II. The general case (elastic dissimilarity)

Int. J.Solids Struct.

Inverse laplace transform for $P_{n}^{L} (s)$