On Two Features of the Deformation of the Surface of a Solid Body
Abstract Generated abstract
This paper examines two experimentally observed features of deformation in discrete contacts between solid bodies under compression. The authors analyze the near constancy of the ratio of elastic approach under repeated loading to total elastoplastic approach under first loading, using a model of roughness as spherical asperities and indentation relations, and compare the resulting expressions with experiments on several metals. They also consider how mutual influence among contact spots and boundary compliance make contact deformation dependent on the size and shape of the contour contact area, contradicting a simple additive asperity model. Experimental results and estimates indicate that these effects can be comparable to the local displacement of individual contact spots and are important for refining laws of surface deformation.
Full Text
ON TWO FEATURES OF DEFORMATION OF THE SURFACE OF A SOLID BODY
B. P. MITROFANOV, G. A. RAKHVALOVA, K. N. TSUKUBLINA
(Presented to the scientific seminar of the Department of Strength of Materials)
In [1] certain features of the deformation of the discrete contact of solid bodies under compression were considered. Additional results obtained by the authors are given below.
Constancy of the Ratio of the Magnitude of Elastic Approach to the Total Approach
An experimental study of the approach of two bodies caused by deformation of the contact made it possible to discover the constancy of the ratio of the magnitude of the elastic approach to the total approach [2]. Elastic approach is observed under repeated loadings of the contact, while the total elastoplastic approach is observed under the first loading.
The constancy of the ratio of the magnitude of the elastic approach to the total approach was observed by us on contacts made of various metals: steel, cast iron, copper, brass, zinc, and others, with surface finish classes 4–7 (Fig. 1).
Fig. 1. 1 — contact: rough surface of a specimen of steel 10 (surface finish class 4) with the polished surface of a specimen of hard alloy; 2 — contact: rough surface of a copper specimen (surface finish class 4) with the polished surface of a specimen of hard alloy.
To explain the mentioned fact, let us consider some regularities of deformation of a discrete contact, representing the latter
as an aggregate of spherical protrusions of the same radius, arranged with constant density on a plane and distributed in a definite manner by height [4].
For the contact of a rough surface, modeled by a set of spheres, with an absolutely rigid smooth surface, we use two known experimental relations [3, 5]:
\[ \delta_{\mathrm{p}}-\delta_{\mathrm{y}}=\alpha N, \tag{1} \]
\[ N=ad^n, \tag{2} \]
where \(\delta_{\mathrm{p}}\) is the approach of the ball and the plane during the first indentation;
\(\delta_{\mathrm{y}}\) is the same for repeated indentations;
\(d\) is the diameter of the impression;
\(N\) is the compressive force;
\(\alpha, a, n\) are experimental constants.
For small approaches the following is valid:
\[ \delta=\frac{\left(\dfrac{d}{2}\right)^2}{2R}, \tag{3} \]
where \(R\) is the radius of the ball.
Substituting (3) into (2), we obtain
\[ \delta_{\mathrm{p}}=\frac{1}{8R\,a^{2/n}}\,N^{2/n}. \tag{4} \]
From (1), taking (4) into account, we obtain
\[ \frac{\delta_{\mathrm{y}}}{\delta_{\mathrm{p}}} = 1-\alpha 8R a^{2/n} N^{1-2/n}. \tag{5} \]
An experimental verification of condition (5) during indentation of a ball with \(D=10\ \mathrm{mm}\) into the flat surface of a specimen made of steel 10 showed good agreement of the theoretical values with the experiments (Fig. 2a).
Fig. 2a. Dependence of the indentation of a hardened ball 10 mm in diameter into a lapped surface of a steel specimen.
1 — first loading, 2 — second loading.
The surface layers of metals are considerably strengthened during treatment; therefore \(n \to 2\), and relation (5) can be written in the form:
\[ \frac{\delta_{\mathrm{y}}}{\delta_{\mathrm{p}}} = 1-8\alpha Ra=\mathrm{const}. \tag{6} \]
The experiments we carried out confirm the results obtained (Fig. 1). An analogous conclusion is obtained in processing the experiments of N. B. Demkin [4] and P. E. D’yachenko [6].
If the compression of several individual asperities is begun simultaneously, then, as follows from formulas (1) and (4), the dependences between the approaches and the load will be linear for \(n \to 2\). Such a condition is usually fulfilled when a hardened ball is pressed into a rough surface. In Fig. 2b graphs are shown of the dependence of the approach on the load for the first and repeated penetration of a ball into a rough surface.
Fig. 2b.
The results of the experiments of I. R. Konyakhin [2], N. B. Demkin [4], and those carried out by us allow the magnitude of the approach for a discrete metallic contact to be represented in the form
\[ \hat{\delta}_{y}=A_{y}\cdot N^{m}, \]
\[ \hat{\delta}_{n}=A_{n}N^{m}, \tag{7} \]
where \(A_{y}\), \(A_{n}\), \(m\) are certain constants for the given contact, and \(N\) is the compressive force.
It follows from formulas (2) and (6) that the parameter \(m\) in expressions (7) depends on the distribution of asperities by height, which, as is known [4], is determined by the parameter \(\nu\) of the bearing-surface curve. Consequently, taking [6] into account, one should expect \(\nu\) to be constant in a certain range of loads.
On the influence of the magnitude and shape of the contour contact area on its deformation
For the theoretical description of the deformation of a discrete contact, the following model is currently used. The real surface of a body is represented as a set of axisymmetric asperities. This set is endowed with the property of additivity, which considerably simplifies the derivation of the laws of contact deformation.
In [1] the existence of mutual influence of individual surface asperities on one another during their deformation is shown, which rules out the additivity property mentioned above.
The mutual influence of asperities leads to an increase in the approach, the magnitude of which, at constant contour pressure, will depend on the shape and magnitude of the contour area.
We shall analyze the significance of such an influence using a model of the contour area of a discrete contact consisting of a set of circular contact spots with radius \(R\) and with distance \(L\) between the centers of neighboring circles. To simplify the further calculations, we shall assume that the pressure on these contact spots \(p\) is uniform. Then the additional displacement \(\delta_{1}\) of the center of a circle, caused by the pressure \(p\) distributed over the neighboring spot, will be [8]
\[ \delta_{1}= \frac{4(1-\mu^{2})pL}{\pi E} \left[ A-\left(1-\frac{R^{2}}{L^{2}}\right)B \right], \]
where \(\mu\), \(E\) are Poisson’s ratio and the modulus of elasticity,
\(A, B\) are complete elliptic integrals determined by the modulus
\[ k=\frac{R}{L}. \]
The mean value of the displacement of the points of a circle loaded by a uniform pressure \(p\) is equal to
\[ \delta_{\mathrm{av}}=0.54\,\frac{\pi R p(1-\mu^2)}{E}. \]
Then the ratio
\[ \frac{\delta_1}{\delta_{\mathrm{av}}} =0.753\,\frac{1}{k}\,[A-(1-k^2)B] \]
over the range of variation of \(k\) may, with sufficient accuracy, be represented by the linear function
\[ \frac{\delta_1}{\delta_{\mathrm{av}}}=0.617k. \]
Knowing the mutual influence of two contact spots, one can determine it for other cases by using the principle of independence. As an example, let us calculate the increase in displacement \(\delta\) for a contact spot located at the center of a circular contour area. Let the radii of the circles on which the neighboring spots are located be \(L, 2L, 3L,\ldots nL\); let \(n\) spots be located on the first circle, then \(2n\) on the second, \(3n\) on the third, and so on.
It is easy to show that the additional displacements of the central contact spot caused by forces applied along different circles will be equal to one another:
\[ \delta_1=\delta_2=\ldots=\delta_k =n\cdot 0.617\,\frac{R}{L}\,\delta_{\mathrm{av}}. \]
Then the contact spots located on \(k\) circles will cause a total additional displacement equal to
\[ \delta_{\mathrm{add}}=kn\,0.617\,\frac{R}{L}\,\delta_{\mathrm{av}}. \]
The coefficient \(\delta_{\mathrm{av}}\) determines the value of the additional displacement. In [9] experimental data are given for the mean values of \(R\) and \(L\). Thus, for example, for specimens of steel 20 with a contour area of \(100\ \mathrm{mm}^2\) and compressive forces of \(100\ \mathrm{kgf}\), the value of \(k\) varies in the range \(0.063\)–\(0.75\) for surface-cleanliness classes 4–10, respectively. Thus, \(\delta_{\mathrm{add}}\) will be a quantity of the same order as, or greater than, \(\delta_{\mathrm{av}}\), which determines its significance.
On the other hand, asperities located near the boundaries of the nominal area possess increased compliance, which increases approach for small contour areas (at constant contour pressure).
Thus, under conditions of constant contour pressure and shape of the contour area, the dependence of approach on the magnitude of the contour area can be represented by a function whose graph is shown in Fig. 3a.
Using the instrument of I. R. Konyakhin [2], we carried out experiments to investigate the dependence of approach on the magnitude of the contour area. The results obtained (Fig. 3b) confirm the increase in approach for small contour areas. Similar results were obtained in [7]. In addition, compression of specimens with equal contour-
...with equal areas but different perimeters showed that, for example, for organic glass an increase in the perimeter by a factor of 1.5 leads to an increase in approach by 25–30%.
The features of contact deformation considered here are essential for refining the laws governing deformation of the surface of a solid body and are of direct practical significance.
Fig. 3a.
(Vertical axis: $\delta$; horizontal axis: contour area.)
Fig. 3b. Dependence of approach on the magnitude of the contour area. Contact: rough (class 4) and smooth surfaces. Material: steel 3; contour pressure $0.3\ \mathrm{kg/mm^2}$.
(Vertical axis: $\delta$, $\mu$; horizontal axis: $\mathrm{mm^2}$.)
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