A moving punch on an infinite viscoelastic layer

Abstract

When a smooth rigid punch of arbitrary shape is pressed onto an elastic body, an area of contact D is formed within which the contact pressure p is non-negative. Outside of the contact area D, p = 0. The problem is to find the contact pressure p and the shape of the contact region D for a given depth of indentation, or a given total force applied to the punch. In general, the governing equation can be reduced to the following integral equation:

$$w\left( {{x_1},{x_2}} \right) = \iint\limits_D {\left( {{x_1},{x_2},{\xi _1},{\xi _2}} \right)p\left( {{\xi _1},{\xi _2}} \right)d{\xi _1}d{\xi _2}}$$

((1a))

in which K(x ₁, x ₂, ξ₁ ξ₁) is known and w is given by

$$w\left( {{x_1},{x_2}} \right) = \alpha - f\left( {{x_1},x{}_2} \right)$$

((1b))

where a is the depth of indentation and f(x ₁, x ₂) is the function which describes the shape of the punch. Unless K and f are simple in form, the solutions of eq. [1] for p and D in general require a numerical approximation. Since the physical observation demands that p > 0 inside D and p = 0 outside D, the method of linear programming is probably the most effective one in solving this problem numerically. The purpose of this paper is to show that not only the contact problem of a rigid punch on an elastic body yields eq. [1], many contact problems of a rigid punch on a viscoelastic body and the contact problems of a moving punch on a viscoelastic body also reduce to eq. [1]. It should be noticed that depending on the shape of the punch, D may not be a simply-connected region. It should also be mentioned that for a viscoelastic body, D is in general not a constant but varies with time even in the case of a punch which is held stationary on the surface of a viscoelastic body.

The work presented here was supported by the National Science Foundation under Grant GK 35163 with the University of Illinois.