On the Shift of Elastic Ellipsoidal Protrusions

V. I. Maksak

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

In works [1, 2], problems were solved on the shift of compressed elastic protrusions having an elliptical contact-zone shape. The distribution of tangential stresses in the plane of contact was found on the basis of the assumption that, as the shifting force increases, the state of limiting equilibrium develops homothetically from the boundary of the contact toward its center. Thus, at any moment of loading the contact area is divided into two parts: a “core,” representing an area bounded by an ellipse, and a ring between this ellipse and the contour ellipse. On the surface of the ring (the slip zone) a limiting state is realized, characterized by tangential stresses \(\tau=f\sigma\), where \(f\) is the coefficient of static friction, and \(\sigma\) is the normal stress (calculated according to Hertz); over the area of the “core” (the adhesion zone), the magnitude of the tangential stress is determined from the condition of absence of slip.

Below, the shift of compressed ellipsoidal protrusions is considered under conditions of more complex loading. Let two elastic ellipsoidal protrusions, compressed by a force \(N\) and initially loaded by a shifting force \(P_y\) along one of the principal axes of the contact ellipse, be shifted by a force \(P_x\) along the second principal axis of the ellipse. It is assumed that at any moment of loading the resultant of the shifting forces is no greater than the static-friction force equal to \(fN\).

To find the magnitude of the displacement under loading by the shifting force \(P_y\), one may use the solution of R. Mindlin [1] or A. I. Lurie [2]:

\[ \Delta = \begin{cases} \dfrac{3fN}{4Ga}\left[1-\left(1-\dfrac{P_y}{fN}\right)^{2/3}\right]\chi(e), & a>b,\\[1.2em] \dfrac{3fN}{4Gb}\left[1-\left(1-\dfrac{P_y}{fN}\right)^{2/3}\right]\chi(e), & b>a, \end{cases} \tag{1} \]

where \(G\) is the shear modulus,

\(a, b\) are the principal semiaxes of the elliptical contact zone,

\(\chi(e)\) is a coefficient depending on the ratio of the semiaxes \(a\) and \(b\).

The first of these formulas is used for a shift along the larger semiaxis, the second along the smaller.

To obtain the solution under loading by the second shearing force \((P_x)\), we use the following assumptions:

1. The coefficient of friction is constant and does not depend on the nature of the loading.

2. The vectors of tangential stresses in the plane of contact are parallel to the vector of the shearing force.

Then the coefficient of friction can be represented in the form

\[ f=\sqrt{f_y^2+f_x^2}, \]

where \(f_y\) and \(f_x\) are the coefficients of friction corresponding to the components of the shearing force \(P_y\) and \(P_x\). Taking into account an analogous dependence between the shearing force \(P\) and its components, we obtain

\[ \left. \begin{aligned} f_y&=f\sqrt{1-\left(\frac{P_x}{N}\right)^2},\\ f_x&=f\sqrt{1-\left(\frac{P_y}{N}\right)^2}. \end{aligned} \right\} \tag{2} \]

Thus, in the general case of loading, when the components of the shearing force do not decrease, the following relations may be used to compute the displacement:

\[ \left. \begin{aligned} \Delta_y&=\frac{3f_yN}{4Ga(b)} \left[1-\left(1-\frac{P_y}{f_yN}\right)^{2/3}\right]\chi_y,\quad a>b,\ (b>a),\\ \Delta_x&=\frac{3f_xN}{4Gb(a)} \left[1-\left(1-\frac{P_x}{f_xN}\right)^{2/3}\right]\chi_x,\quad b>a,\ (a>b). \end{aligned} \right\} \tag{3} \]

It is obvious that

\[ \Delta=\sqrt{\Delta_x^2+\Delta_y^2}, \]

and the vector of the shearing force \(P=\sqrt{P_x^2+P_y^2}\) for the contact under consideration does not coincide with the displacement vector \(\Delta\), while the shearing force \(P_x\) (in the second direction) causes an additional increment of shear in the first direction \((P_y)\).

Introducing the notation \(P_y=P\sin\alpha\), we obtain the direction of the displacement (angle \(\beta\)) as a function of the direction of the shearing force \(P\) (angle \(\alpha\)):

\[ \operatorname{tg}\beta=\frac{\chi_y}{\chi_x}\operatorname{tg}\alpha. \]

The magnitude of the displacement in this direction is determined from the formula

\[ \Delta= \frac{0.75\sqrt{f^2N^2-P^2\cos^2\alpha}}{Ga\sin\alpha} \left[ 1- \left( 1-\frac{P\sin\alpha}{\sqrt{f^2N^2-P^2\cos^2\alpha}} \right)^{2/3} \sqrt{\chi_x^2\cos^2\alpha+\chi_y^2\sin^2\alpha} \right]. \tag{4} \]

From formula (4) the magnitude of the divergence of the vectors of the shearing force and displacement is readily determined:

\[ \operatorname{tg}(\beta-\alpha)= \frac{(\chi_y-\chi_x)\operatorname{tg}\alpha} {\chi_x+\chi_y\operatorname{tg}^2\alpha}, \tag{5} \]

as well as the value of the angle \(\alpha\) at which the divergence of the vectors is minimal,

\[ \operatorname{tg}\alpha=-\sqrt{\frac{\chi_x}{\chi_y}} \tag{6} \]

The conditions used to obtain the dependences (3) were verified experimentally under conditions of elastic preliminary displacement, in which, as is known, the shift of many protrusions occurs in the discrete contact of two contacting solid bodies [3].

REFERENCES

1. R. D. Mindlin. Compliance of Elastic Bodies in Contact. J. of Applied Mechanics, 1949, 16, No. 3.

2. A. I. Lur’e. Spatial Problems of the Theory of Elasticity. State Publishing House of Technical-Theoretical Literature, Moscow, 1955.

3. B. P. Mitrofanov. Elastic preliminary displacement. Izv. vuzov, “Mechanical Engineering,” No. 5, 1965.