On the Anisotropy of Preliminary Displacement
Abstract Generated abstract
This paper examines preliminary displacement in contacts whose frictional properties are anisotropic, where the displacement vector need not be opposite to or collinear with the shearing force. It develops a work-based geometrical model using the hodograph of limiting shearing forces, approximated by an ellipse, and derives relations between the directions of shearing force and displacement, including the angle of their divergence and the condition for maximum divergence. The analysis identifies principal displacement directions, zones of increased and reduced stability, and argues that the divergence angle is unchanged during loading under homothetic force hodographs. Experimental comparisons for metal specimens with directional roughness are reported to agree with the calculated dependencies, and the approach is extended qualitatively to more general hodograph shapes and to an anisotropic static-friction cone.
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ON THE ISSUE OF ANISOTROPY OF PRELIMINARY DISPLACEMENT
V. I. Maksak
(Presented by the Department of Strength of Materials)
The shear of contacting surfaces caused by an incomplete friction force is called preliminary displacement.
It was previously assumed that the friction force is always directed opposite to the vector of preliminary displacement. However, in those cases where the contacting surfaces are anisotropic in their geometrical and mechanical characteristics, this statement is not always valid. Thus, in experiments it was found that, in the presence of directional roughness, the displacement vector does not coincide with the vector of the shearing force. Moreover, when the vectors diverge, the acute angle between the displacement vector and the direction of the machining traces decreases. Larger displacements occur in the direction of the smaller coefficient of friction, i.e., along the machining traces [1].
It is quite clear that displacement of a body in some direction by a total shearing force is possible only if the work of the friction force on any elementary displacement in this direction is, in absolute value, less than or equal to the work of the corresponding shearing force on the same displacement, i.e.,
\[ |dA_{\mathrm{fr}}| \le |dA_{\mathrm{p}}|. \]
The work performed by such a force during displacement in a given direction and referred to a unit magnitude of displacement may be called the specific work of the force in that direction. For the case of an anisotropic surface, the magnitude of the specific work of the friction force will be a function of the direction of displacement, which can be represented graphically.
However, for a rough surface it is simpler to determine the hodograph of the limiting shearing forces. Thus, by experimentally determining in different directions the breakaway force, i.e., the force at which preliminary displacement passes into uniformly accelerated sliding, and plotting it in the corresponding directions from the origin of coordinates, we obtain a curve of limiting shearing forces, which passes through the ends of the force vectors. The form of this curve generally depends on the physical and geometrical properties of the contacting surface and is usually close to an ellipse.
The direction in which displacement is possible exists in the case when the circle \(l\), constructed on the vector of the shearing force \(F\) as on a diameter, is tangent to the curve of specific works \(K\) at some point \(M\) (Fig. 1). The line \(OM\) will be the direction of displacement.
Thus, the following states are possible:
- Stable equilibrium in the case where the circle \(l\) is located inside the curve of specific works and does not touch it.
- Critical, limiting equilibrium if the circle \(l\) touches the curve of specific works.
- Motion if the circle \(l\) intersects the curve of specific works.
Let us consider the state of critical equilibrium. To establish the dependence between the directions of the displacement vectors and of the shear force, we shall assume that the hodograph of the limiting shear forces is an ellipse,
\[ \frac{x_1^2}{f_x^2}+\frac{y_1^2}{f_y^2}=1, \tag{1} \]
where \(f_x>f_y\) are the coefficients of static friction in the corresponding directions.
Fig. 1. Diagram of the divergence of the vectors of the shear force and of the preliminary displacement
The equation of the circle \(l\):
\[ x^2-xx_1+y^2-yy_1=0. \tag{2} \]
The simultaneous solution of (1) and (2) gives a family of circles \(l\) for various directions of the shear force,
\[ \Phi(\operatorname{tg}\alpha)=0, \tag{3} \]
or
\[ x^2-x\frac{f_x f_y}{\sqrt{f_y^2+f_x^2\operatorname{tg}^2\alpha}} +y^2-y\frac{f_x f_y\operatorname{tg}\alpha}{\sqrt{f_y^2+f_x^2\operatorname{tg}^2\alpha}}, \]
where \(\alpha\) is the angle between the direction of the shearing force and the major semi-axis of the ellipse. The curve of specific works \(K\) will be the envelope of the family of circles; the tangent to them, determined from (3), will be common,
\[ \frac{d\Phi(x,y,\operatorname{tg}\alpha)}{d\operatorname{tg}\alpha}=0, \]
and after transformation we have
\[ f_x^2 x\operatorname{tg}\alpha+f_y^2 y=0. \tag{4} \]
Solving (3) and (4) simultaneously, we obtain the equation of the points of tangency, or of the envelope of the family of circles,
\[ x^2\left(\sqrt{f_x^2x^2+f_y^2y^2}-f_x^2\right) +y^2\left(\sqrt{f_x^2x^2+f_y^2y^2}-f_y^2\right)=0. \tag{5} \]
Let us recall that the point of tangency \(M\) of the curve of specific works \(K\) and the circle \(l\), with coordinates \(x\) and \(y\), lies on the displacement vector, where
\[ \frac{y}{x}=\operatorname{tg}\beta \]
The simultaneous solution of (5) and (3) determines the dependence between the directions of the displacement and of the shearing force,
\[ \operatorname{tg}\beta=\frac{f_x^2}{f_y^2}\operatorname{tg}\alpha. \tag{6} \]
The magnitude of the divergence of the displacement vectors and the shear force (the angle \(\beta-\alpha\)), as a function of the direction of the latter, is obtained from formula (6).
\[ \tg(\beta-\alpha)=\frac{(f_x^2-f_y^2)\tg\alpha}{f_y^2+f_x^2\tg^2\alpha}. \tag{7} \]
The maximum divergence of the vectors will occur under the condition
\[ \Phi'[\tg(\beta-\alpha)]=0, \]
or
\[ \tg\alpha=\frac{f_y}{f_x}. \tag{8} \]
The dependences obtained satisfy the special cases \(\alpha=0\), \(\alpha=90^\circ\), \(f_x=f_y\), and also agree well with experiment.
In Fig. 2 the direction of displacement is shown by the solid line—experimental, and by the dash-dot line—calculated, as a function of the direction of the shearing force—the dashed line. Specimens: the upper one—steel, hardened, \(HRC=45\), У6; the lower one—lead. The roughness is directed along the \(y\)-axis. Similar results were obtained for steel and copper specimens as well.
The dependences (6, 7, 8) were obtained for the case of limiting critical equilibrium. However, this state is preceded by stable equilibrium, when the displacement is caused not by the full friction force.
Since the hodographs of shearing forces equal to \(\dfrac{P}{n}\), where \(n>1\), are homothetic, the curves of specific work corresponding to these hodographs are also homothetic. Therefore the angle of divergence of the displacement vector and the shearing force will be the same both for a shearing force equal to \(P\) and for \(\dfrac{P}{n}\), i.e., it does not change from the beginning of loading of the contact.
Fig. 2. Directions of displacement: experimental (solid line) and calculated (dash-dot line) for the corresponding directions of the shearing forces (dashed line)
The directions along which the displacement coincides with the line of action of the shearing forces may be called the principal directions of displacement. They are, obviously, the axes of symmetry of the hodograph of shearing forces, since on these axes \(\beta=\alpha\). Symmetrically with respect to the principal directions of displacement there are zones of reduced and increased stability. Along the direction of the minimum coefficient of static friction lies a zone of increased stability, since the displacement vector is distributed closer to this direction than the shearing-force vector. Along the direction of the maximum coefficient of friction, on the other hand, lies a zone of reduced stability.
The validity of the obtained dependences is confirmed experimentally for all three of the above-mentioned states.
In the general case, the hodograph of the limiting shearing force may be a closed analytic curve of a somewhat different form; however, the calcu-
the divergence of the displacement and shear-force vectors may also be obtained from the principle of minimum work. If the hodograph of the limiting shear forces consists of definite segments, then, because of the difficulty of an analytical calculation, the divergence of the vectors can be obtained graphically.
If it is assumed that the modulus of the vector of the limiting shearing force is directly proportional to the normal compressive force, then, by translating the hodograph in the direction of the external normal to the surface by the magnitude of the compressive force, we obtain a cone of static friction, whose vertex lies on the anisotropic surface at the center of the hodograph.
LITERATURE
- V. I. Maksak. On the stiffness of the contact of machine parts. Second Scientific-Technical Conference of the Ural Polytechnic Institute (abstracts of reports). Sverdlovsk, 1968.