PLANE CONTACT PROBLEM FOR AN ELASTIC BODY TAKING INTO ACCOUNT THE INFLUENCE OF THE SURFACE LAYER

B. P. MITROFANOV

(Presented by the scientific seminar of the Department of Strength of Materials)

The rough surface of a solid body possesses mechanical properties that differ from those of the underlying layers. Therefore, under the force interaction of bodies, the deformation of the rough surface to some extent distorts the deformation of the main volume of the bodies [1].

The geometric structure of the rough layer determines its increased compliance, which is manifested, in particular, in a change in the distribution of contact pressure.

Let us consider the solution of the contact problem for an elastic half-plane, on whose boundary a thin elastic layer of increased compliance is placed. It is known [2] that if a normal pressure \(p(x)\) acts on the boundary of an elastic half-plane in the region \(-a \leq x \leq a\), then a point lying on the boundary of the half-plane and having abscissa \(x\) undergoes the normal displacement

\[ \vartheta \int_{-a}^{a} p(t)\ln \frac{1}{|t-x|}\,dt+\mathrm{const}, \]

where \(\vartheta=\dfrac{2}{\pi E}(1-\mu^2)\), \(E\) is the modulus of elasticity, and \(\mu\) is Poisson’s ratio.

Following I. Ya. Shtaerman [3], we shall assume that the additional displacement caused by deformation of the surface layer is proportional to the normal pressure. Then the normal pressure \(p(x)\), acting in the region \(-a \leq x \leq a\), will cause for the points of this region an additional displacement \(\kappa p(x)\), where \(\kappa\) is a coefficient depending on the deformational properties of the rough surface.

The total normal displacement of a point located on the boundary of the half-space, caused by the normal pressure \(p(x)\), will be equal to the sum of the displacements indicated above:

\[ \kappa p(x)+\vartheta \int_{-a}^{a} p(t)\ln \frac{1}{|t-x|}\,dt+\mathrm{const}. \tag{1} \]

As an example, let us consider the problem of the pressure of a plane elastic body with a configuration determined by the function \(f(x)\), on a plane

rigid punch. The usual geometric relations and condition (1) lead to the following integral equation

\[ \kappa p(x)+\vartheta \int_{-a}^{a} p(t)\ln \frac{1}{|t-x|}\,dt=\alpha-f(x), \tag{2} \]

where \(\alpha\) is a constant.

Fig. 1. Distribution of the normal pressure \(p(x)\) over the contact:
1 — smooth surface, compressive force \(P=0.86\cdot 10^{5}\) kg/cm;
2 — \(\kappa=10^{-7}\), \(P=0.8\cdot 10^{5}\) kg/cm;
3 — \(\kappa=10^{-6}\), \(P=0.7\cdot 10^{5}\) kg/cm;
4 — \(\kappa=10^{-5}\), \(P=0.7\cdot 10^{5}\) kg/cm, \(E=10^{6}\) kg/cm\(^2\), \(\mu=0.3\). Radius of the circle bounding the elastic body, 10 cm.

Equation (2) was reduced to Prandtl’s equation, which was approximately replaced by a finite system of linear algebraic equations [4]. Figure 1 gives graphs of the pressure \(p(x)\) over the contact of an elastic body bounded by a circle. From the nature of these graphs one may conclude that the presence of a surface layer with increased compliance leads to a smoothing of the pressure diagram and to a certain increase in the dimensions of the contact area.

REFERENCES

1. A. N. Dinnik. Impact and Compression of Elastic Bodies. Kiev, 1909.
2. S. P. Timoshenko. Theory of Elasticity. ONTI, 1937.
3. I. Ya. Shtaerman. Contact Problem of the Theory of Elasticity. Gostekhizdat, 1949.
4. L. V. Kantorovich, V. I. Krylov. Approximate Methods of Higher Analysis. Fizmatgiz, 1962.