ON THE QUESTION OF MODELING A ROUGH SURFACE

I. R. Konyakhin

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

Introduction

The question of modeling a rough surface arises in connection with the need to calculate the strength of the material of mating bodies at the places of their true contact, both in movable and in immovable fits of various machine parts.

The calculation of the strength and reliability in operation of the indicated parts includes establishing relationships between the magnitudes of loads, on the one hand, and the magnitudes of the deformations of those microprotrusions which always cover a rough surface, on the other. These microirregularities have a highly indefinite geometric form, as a result of which calculation of their deformation is practically impossible.

However, if the real microprotrusions are conditionally replaced by certain others having a simple geometric form, then such a calculation becomes quite feasible.

It is true that a surface modeled by artificial roughness will not be fully equivalent to the real surface, but with a successful choice of the form of the modeling body the indicated discrepancy can be reduced to a minimum. The minimum discrepancy will occur in the case when the selected body begins to deform according to the same law as the microprotrusions of the real rough surface.

Modeling of a Rough Surface Carried Out by Certain Researchers

I. V. Kragel’skii [5] modeled the surface in the form of a plane covered with rods of different lengths. When such a surface is pressed, the highest protrusions enter into contact first; then, as the deforming load increases, ever new rods come into operation, and therefore the resistance to deformation gradually increases. It is assumed that in this case approximately the same thing should occur as when pressing on a real rough surface.

V. A. Zhuravlev [2], for modeling a surface, used de-

protrusions in the form of hemispheres. It was assumed in this case that their deformation under compression occurs elastically.

O. T. Il'chenko [3], as the model body, took a truncated pyramid ending in a convexity. Such a body shape was supposed to create a model of elastoplastic deformation.

V. N. Marochkin [6], in order to create a model of a rough surface, used a truncated cone. A body of this shape imitates a surface capable of elastoplastic deformation.

G. Ernst and M. Merchant [7] believe that a rough surface is deformed only plastically, while the contact stress remains constant during the process of deformation.

According to our experiments, such deformation conditions are satisfied by a sharp, i.e., untruncated, protrusion.

Method of testing a rough surface in compression

To establish the most rational shape of the modeling body, it is necessary to compare the law of its deformation with the law of deformation of a real rough surface. Conducting experiments to determine these laws for a body of simple geometric shape presents no particular difficulties, but such experiments with a rough surface are associated with great difficulties because of the small magnitude of the deformations. These difficulties were the reason why different authors chose the shape of the modeling body in different ways, relying only on conjecture.

However, for some time experiments aimed at determining the law of deformation of a rough surface have become possible to a certain extent.

The first such experiment was described by P. E. Dyachenko [1] and co-workers. The authors carried out experiments on compression of a steel surface machined to cleanliness class 4. The instrument on which the subsequent experiments on surface compression were performed gave a 9800-fold magnification of the displacements of the bodies in contact with one another.

As a result of the experiment, the authors obtained curves both for loading the surface and for its unloading. The experimental data indicate that the initial deformation of the surface occurs plastically, while unloading is accompanied by elastic recovery of the microprotrusions.

Unfortunately, the authors did not determine the law of plastic and elastic deformation of roughness.

In our experiments on compression of a surface, the magnitude of the deformation was determined from the approach of the specimens toward one another, occurring as a result of the action of a load applied to the upper specimen standing on the lower one.

The magnitude of such approach is very small; it is measured in microns. Therefore, to carry out the experiments, a device is required that would make it possible to measure microdeformations on the order of several fractions of a micron.

The diagram of our apparatus, on which the experiments with microdeformation of a rough surface were carried out, is shown in Fig. 1.

The distance from the mirror to the scale is 1500 mm.

As a result of pressing the upper specimen 1 with force N, the surface of the lower specimen 2 undergoes compressive deformation. Specimen 2 is pressed to the bed of the apparatus by means of bolts, owing to which the plastic deformation of its lower surface proves to have been produced in advance and therefore does not appear in the experiment.

The compliance of the instrument itself is subsequently taken into account during the processing of the experimental results.

Deformer 1 has a very hard working surface (pobedit alloy); therefore its displacement relative to the lower body 2 occurs only through compression of the microprotrusions of the surface of the latter.

Deformer 1, by means of its tailpiece 3, transmits displacement to rod 4, which turns the bell-crank lever 5. The latter, by its upper arm, pushes driver 7, which, in turn, turns needle 8, resting on prisms. Together with the needle, the mirror 9 fastened to it turns through a certain angle \(\alpha\). The ray from scale 11, reflected in the mirror, enters the viewing tube 10. Since the angle of incidence is equal to the angle of reflection, the ray turns through an angle \(2\alpha\), i.e., twice as large as the mirror.

Fig. 1. Diagram for measuring the amount of approach of specimens occurring as a result of deformation of microprotrusions:
1—deforming specimen (deformer); 2—test specimen with annular working surface; 3—tailpiece of the deformer; 4—transmitting rod; 5—bell-crank lever with arm ratio \(\frac{b}{a}=6\); 6—driver spring; 7—driver with weight; 8—needle of diameter \(0.7\) mm; 9—mirror reflecting the scale; 10—viewing tube for observing displacement in the contacts.

The magnification of the instrument is obtained from the ratio of the arms of the bell-crank lever, the ratio of the distance from the scale to the mirror to the diameter of the needle, and the doubling due to the equality of the angle of incidence and the angle of reflection.

Thus,

\[ \mathrm{Magn.}=\frac{b}{a}\cdot\frac{l}{d}\,2\cdot6\,\frac{1500}{0.7}\,2 \simeq 25000. \]

Consequently, when the upper specimen is displaced by \(1\ \mu\), the reflection of the scale in the mirror is displaced by \(25\) mm.

Readings on the free scale can be made with an accuracy of up to 1 mm, which corresponds to a displacement of the upper specimen of 0.04 micron.

Before the experiment, the surface under test was treated with emery paper of one or another grit size and then washed with alcohol to remove from the surface contaminating substances that might introduce distortion into the results of the experiment.

The load on the upper specimen, placed on the lower one, was applied in portions of 5 kg. After each additional 5 kg of load to the already applied force, the displacement was read from the scale. Immediately after unloading the specimens, loading was repeated and the corresponding readings were taken of the magnitudes of what was now the elastic deformation of the asperities hardened by the initial deformation.

In processing the experimental results, the displacement due to the elastic compliance of the instrument itself was subtracted from the scale readings; this displacement had been determined beforehand by pressing with a force on the lower specimen, bypassing the upper one, through a bushing.

The character of the deformation of the rough surface is shown in the graph in Fig. 2, constructed from the data of one of the experiments carried out with a copper specimen.

Fig. 2. Curves of plastic (1) and repeated elastic (2) deformation of a rough surface

About one hundred experiments carried out with surfaces on various materials and at all possible roughness intensities showed that, in the overwhelming majority of cases, both the initial deformation—plastic—and the repeated deformation—elastic—proceed according to the law of a parabola of the second degree; moreover, the two parabolas have different parameters.

The ratio of these parameters depends on the material of the surface being tested, but does not depend on the magnitude of the roughness.

Experiments previously carried out by us on compression of a sharp metallic cone [4] showed that its deformation likewise proceeds according to the law of a quadratic parabola.

Consequently, as a modeling body for a rough surface, in most cases the most applicable is a sharp, untruncated cone.

On the Form of Roughness

However, in the course of carrying out numerous experiments on the compression of surface microprotrusions, there were cases in which the deformation proceeded according to the law of a parabola with a degree different from two. On the basis of this fact it may be considered that, in general, the law of deformation of a rough surface can be expressed by the equation

\[ h^n = BN, \]

where \(h\) is the magnitude of deformation, \(N\) is the load. \(B\) is a constant quantity, which is a parameter of the parabola.

In all cases in which experiments were conducted, the exponent of the parabola \(n\) did not go outside the limits \(n=1 \div 4\). There were also cases when the primary deformation occurred according to the law of a parabola of one degree, and the repeated deformation according to a parabola of another, usually higher, degree. Thus, for example, when compressing the surface of ruby, there were cases of deformation according to the law of a parabola of the 3rd degree, after which the repeated deformation occurred according to a parabola of the 4th degree. When testing the surface of the mineral talc, the primary compressive deformation occurred at \(n=1.3\), and the repeated deformation at \(n=2.3\).

In our opinion, a degree of the parabola less than 2 pertains to roughness having a strong difference in the height of the irregularities covering the surface. In this case, the highest microprotrusions begin to deform first, and then, already in the process of compression, more and more of them come into action. In such a process the mean-statistical irregularity should be a protrusion in the form of a solid of revolution with a concave generatrix.

A degree of the parabola greater than 2 means that compression of the rough surface proceeds in the same way as the deformation of microprotrusions that are solids of revolution with a convex generatrix.

The degree of the parabola of the graph in the coordinates “load—deformation” should characterize the roughness of the surface by its mean-statistical form.

Before testing the surface of ruby, we treated it with emery paper. When the surface was then examined under a microscope at 20-fold magnification, no slightest scratches were found; the surface of the specimen was covered with rounded convex tubercles in the form of hillocks of identical height. This roughness gave deformation curves in the form of a parabola with degrees greater than two. Thus, the results of the experiments indicate that, in general, for modeling a rough surface, the most suitable body is a sharp, untruncated protrusion in the form of a solid of revolution with a parabolic generatrix.

Then the law of deformation of the surface will be represented in the form of the expression

\[ h^n = BN. \]

Here the coefficient \(B\) characterizes the magnitude of the microprotrusions for the given material of the surface, and \(n\) characterizes the mean-statistical form of the roughness.

In the process of deformation of the microprotrusions, their form may change. In this case it is necessary to keep in mind two coefficients of form: \(n\) before deformation and \(n_y\) after it.

At present, roughness is usually judged by its profilogram, but this characteristic is insufficient. The profilogram still does not

gives quantitative indices characterizing the surface from the standpoint of its service properties.

For the same height of irregularities, the more stable surface will be the one that has the larger so-called bearing surface. But the stress at the contact during compression of a projection depends, among other things, also on the slope of the side of the projection at the given point; therefore knowledge of the magnitude of the bearing surface is likewise insufficient.

The nature of the deformation of the body’s microprojections will depend on the magnitude of the bearing surface, and also on the slope of the generatrix of the mean statistical microprojection; all this taken together is reflected in the coefficient $n$, which characterizes roughness by form.

Conclusions

1. As a modeling body for a rough surface, the most acceptable is a sharp projection in the form of a body of revolution with a convex or concave generatrix of parabolic form.

2. In most cases, a sharp cone is suitable for modeling.

3. When calculating surface deformations, it is necessary to take into account not only the magnitude of the roughness, but also its form.

REFERENCES

1. P. E. Dyachenko, N. V. Tolkacheva, K. N. Goryunov. Determination of the contact area of surfaces. Study of wear of machine parts by means of radioactive isotopes. Academy of Sciences of the USSR, 1957.

2. V. A. Zhuravlev. On the question of a theoretical justification of the Amontons–Coulomb law for the friction of unlubricated surfaces. Journal of Technical Physics, issue 16, 1940.

3. O. G. Ilyenko. Determination of the area of actual contact of two plane bodies. Vestnik Mashinostroeniya, No. 10, 1958.

4. I. R. Konyakhin, L. M. Seldkov, M. S. Gorbenko. Determination of acting forces by means of conical marks. Zavodskaya Laboratoriya, No. 5, 1958.

5. I. V. Kragelskii. Static friction of two identical rough surfaces. Friction and Wear in Machines. Academy of Sciences of the USSR, Collection IV, 1949.

6. V. I. Marochkin. The stressed state in the contact zone of conjugate surfaces. Kiev, 1958.

7. G. Ernst and M. Merchant. Friction and surface quality in cutting. Applied Mechanics and Mechanical Engineering, No. 2, 1952.