Experimental Study of Plastic Deformation of an Axisymmetric Protrusion in the Form of a Cone

B. P. Mitrofanov, G. A. Rakhvalova, K. N. Tsukublina

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

Because of the waviness and roughness of the surfaces of solid bodies, their contact is always discrete.

Therefore, for the analytical determination of the deformation of the surface layers of the material of two solid bodies in contact, various models are used [1].

Models of the surface layers of a material usually represent axisymmetric protrusions of different heights, located on a rigid or deformable base.

Knowing the laws of deformation of a single protrusion, as well as the law of distribution of protrusions by height, it is possible to construct a theory of deformation of the rough surface of a body.

Let us consider several experimental and theoretical facts obtained by the authors in studying the plastic deformation of an individual protrusion in the form of a cone. This will make it possible to refine somewhat the regularities of deformation of a single protrusion.

It is known that a cone is a good approximation to the real shape of protrusions existing on a surface [2, 3]. We tested cones made of steels 3, 10, and 20Kh and having, mainly, a base angle of \(15^\circ\). This angle is close to the angle of inclination of microgeometric elements of a real profile to the general direction of the surface [4].

In doing so, the following questions were investigated.

Flow of Material and Friction in the Contact Plane

In the theoretical investigation of the axisymmetric state of stress of an individual surface protrusion, modeled in the form of a truncated cone [3], the author neglects the flow of material and friction in the contact plane.

For a quantitative evaluation of the flow of material in the contact plane, we tested in compression cones made of steel 3 with base angles from \(75^\circ\) to \(15^\circ\).

The results of the experiments, presented in Fig. 1a, show that the flow of metal in the contact plane exists for the given cone

only at certain loads. Within the range of variation of the compressive load, most cones showed that, as the load increased, the contact diameter \(d_{\mathrm{f}}\) at first exceeded the diameter of the undeformed cone at the same height, \(d_{\mathrm{k}}\), and then became smaller than it. This circumstance is manifested most clearly for cones with base angles smaller than \(45^\circ\).

Fig. 1.

Within a certain range of loads, the spreading of the metal increases the area of actual contact, in comparison with the geometrical area, by 40%.

The influence of friction on the deformation of the protrusion was investigated on cones made of 20Kh steel, with a base angle of \(30^\circ\).

In the experiments, friction was reduced by means of lead gaskets and a lubricant consisting of colloidal graphite and machine oil.

The experiments showed that, in the load range up to 50 tons, friction for the cones under investigation changed the contact diameter only very slightly. Compression of two cones joined at the apex and forming a single whole confirmed this result (it is known that in this case contact friction is absent).

Strain Hardening of the Cone Material under Compression

The principal feature of plastic deformation of metals is its nonuniformity, accompanied by localization [5] at certain individual places in the deformation zone. The geometrical shape of the cone increases the localization effect. The latter is inevitably associated with localization of strain hardening.

For a qualitative determination of the nature of the manifestation of localization of strain hardening of the material during plastic deformation of the cone, we studied the change in hardness along planes passing through diametrically opposite generators. The cones were compressed by various forces and then cut along the indicated planes. The planes were carefully polished. Hardness was measured on an instrument

TK-2 with a specially selected cone made of hard-alloy material.

A typical distribution of hardness over the plane of the section is shown in Fig. 2. Thus, the locality of the plastic deformation of the cone was specifically expressed in the formation of the zone of maximum hardening \(AB\).

Fig. 2.

A similar change in hardening was observed by us when measuring microhardness along the section of a deformed specimen modeling a body with a rough surface. The latter was produced by milling a steel surface with a special cutter. The multitude of pyramids obtained in this way imitated the micro-roughnesses of the surface.

Let us determine analytically the dimensions of the zone of plastic deformation of the cone. We shall characterize it by its size along the axis of the cone and call it the depth of the plastic-deformation zone.

It is known that, with increasing loads, regions of flow arise in the body; these are separated from the elastic part of the body by surfaces

\[ \psi = \mathrm{const} = \frac{1}{2G}, \]

where \(\psi\) is the coefficient of proportionality of the stress and strain deviators. Using the continuity of stresses and strains on the surface separating the elastic and plastic regions, we shall find for the cone the depth of the plastic-deformation zone.

For this we shall use A. I. Lur’e’s solution for the elastic deformation of a cone [6] and the Mises plasticity condition.

Consider a cone with aperture angle \(2\alpha\), loaded by a force \(P\) applied to its vertex. We take the vertex of the cone as the origin of a spherical coordinate system \((z,\vartheta,\varphi)\). For the case when the force acts along the axis of the cone—along the \(z\)-axis (for \(P>0\) the cone is compressed)—we have, according to [6], the following stresses at the point with coordinates \((z=R,\ \vartheta=0,\ \varphi)\):

\[ \sigma_r = \frac{Q(m-2)}{8\pi(m-1)R^2} \cdot \left( 1+\cos\alpha-\frac{4m-2}{m-2} \right); \]

\[ \sigma_\vartheta = \frac{Q(m-2)}{8\pi(m-1)R^2} \cdot \left( \frac{1}{2}-\frac{\cos\alpha}{2} \right); \tag{1} \]

\[ \sigma_\varphi = \frac{Q(m-2)}{8\pi(m-1)R^2} \cdot \left( \frac{1}{2}-\frac{\cos\alpha}{2} \right), \]

where

\[ Q=\frac{4P(m-1)}{m(1-\cos^2\alpha)-(m-2)\cos z(1-\cos\alpha)}; \]

\(m\) is Poisson’s ratio.

The Mises plasticity condition for the same point is written in the form

\[ \sigma_r-\sigma_\theta=\sigma_m . \tag{2} \]

From equations (1) and (2) we obtain the following expression for the quantity \(R\):

\[ R=\sqrt{\frac{1}{\sigma_m}\, \frac{Q(m-2)}{8n(m-1)} \left( \frac{1}{2}-\frac{4m-2}{m-2}+\frac{3}{2}\cos z \right)} . \tag{3} \]

A comparison of the depths of the plastic-deformation zones of cones \((2\alpha=140^\circ)\), determined by studying hardness, with the theoretical depth calculated by formula (3), is shown in Fig. 3. The good agreement of the theoretical results with the experimental ones makes it possible to apply formula (3) for the analytical determination of the depth of the plastic-deformation zone of a cone.

In the figure: \(R\), cm; \(P\), t; solid line—“theory”; dashed line—“experiment.”

Fig. 3.

Analysis of formula (3) shows that the depth of the plastic-deformation zone decreases as the angle increases.

Influence of the Base of the Cone on Its Deformation

To determine the influence of the base on the deformation of the protrusion, cones with cylindrical bases of various sizes were tested. Graphs of the relationship between the contact area, the shortening of the cone, and the axial load are presented in Fig. 4.

These relationships show that the influence of the base appears beginning at a certain load and then increases with further growth of the latter.

It is interesting to note that the load at which the influence of the base begins to appear is determined by the moment when the plastic-deformation zone reaches the upper level of the base. Consequently, knowing

the height of the cone, it is possible, using equation (3), to determine the magnitude of this load, taking the value of \(R\) to be equal to the height of the protrusion. Let us note that in the experiments of S. B. Ainbinder the influence of the base on the deformation of the cone was not detected. This can be explained by the fact that in the experiments of S. B. Ainbinder [2] specimens with a large height of the conical part were tested under comparatively small forces.

Fig. 4.

Characteristic of cones with massive bases is the constancy of the mean contact stress, which in our experiments was equal to 3.3 times the tensile yield strength.

In addition, the influence of the base is manifested in an increase in the range of loads at which spreading of the metal along the contact takes place. Let us note that, in the process of deformation, the spreading of the material for specimens with massive bases becomes almost constant (Fig. 1, b) after the plastic-deformation zone reaches the upper level of the base.

A manifestation of the influence of the base will also be the considerable increase in the resistance to deformation of a single protrusion that is observed if several protrusions are compressed simultaneously on a common base.

For a quantitative assessment of this phenomenon, specimens representing cones on cylindrical bases were tested. Around the cone a triangular annular protrusion was turned. It turned out that, with equality of the forces compressing the cones, the presence of the triangular annular protrusion, deformed simultaneously with the cone, reduced the contact area of the latter by 50–70% in comparison with the contact area of a single cone.

In addition to this, indentation of 3 hardened balls with \(d = 8\) mm into the flat surface of a steel specimen was carried out. The reduc-

the distance between the balls reduced the magnitude of their indentation. In experiments with balls touching one another and located at distances of \(3d\), the difference in indentations amounted to 30%.

The experimental facts listed above make it possible to draw the following conclusions:

1. The spreading of the metal along the contact, when compressed along the axis of the cone, first increases with increasing load, and then decreases to zero. With a further increase in load, the contact area becomes smaller than the cross-section at the same level.

2. The influence of frictional forces along the contact on the deformation of cones with an included angle of \(120^\circ\) is insignificant.

3. The base of a specimen with a conical protrusion substantially affects its deformation after the zone of plastic deformation reaches the upper level of the base. This influence increases the resistance of the cone to deformation.

4. When several protrusions that are located next to one another and have a common base are compressed, the resistance to deformation of each of them increases significantly.

REFERENCES

1. Demkin N. B. Actual area of contact of solid surfaces. Publishing House of the Academy of Sciences of the USSR, 1962.

2. Ainbinder S. B. Cold welding of metals. Publishing House of the Academy of Sciences of the Latvian SSR, 1957.

3. Marochkin V. N. Axisymmetric stress state of the zone of actual contact of rough surfaces under conditions of complete plasticity. Author’s abstract of Candidate’s dissertation, Institute of Mechanical Engineering, Academy of Sciences of the USSR, 1958.

4. Levin B. M. Contact method for measuring surface microgeometry. Mashgiz, 1950.

5. Gubkin S. I. Plastic deformation of metals, vol. 1. Metallurgizdat, 1961.

6. Lur’e A. I. Three-dimensional problems of the theory of elasticity. Gostekhizdat, Moscow, 1955.