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A state of triaxial stress seldom arises in design. An exception to this occurs when two bodies having curved surfaces are pressed against one another. When this happens, point or line contact changes to area contact, and the stress developed in both bodies is three-dimensional. Contact-stress problems arise in the contact of a wheel and a rail, a cam and its follower, mating gear teeth, and in the action of rolling bearings. To guard against the possibility of surface failure, in such cases, it is necessary to have means of computing the stress states which result from loading one body against another.
When two solid spheres of diameters d1 and d2 are pressed together with a force F, a circular area of contact of radius a is obtained. Specifying E1, mu1 and E2, mu2, as the respective elastic constants of the two spheres, the radius a is given by the equation
a = (3*F/8*(((1-mu1^2)/E1)+((1-mu2^2)/E2))/
(1/d1+1/d2))^(1/3) [2-82]
The pressure within each sphere has a semi-elliptical
distribution, as shown in Figure [2-33]. The maximum
pressure occurs at the center of the contact area and is
p_max = - 3 * F / (2*pi*a^2) [2-83]
Equations [2-82] and [2-83] are perfectly general and also apply to the contact of a sphere and a plane surface or to a sphere and an internal spherical surface. For a plane surface use d = infinity. For an internal surface the diameter is expressed as a negative quantity.
Thomas and Hoersch (H.R. Thomas and V.A. Hoersch, Stresses Due to the Pressure of One Elastic Solid upon Another, Univ. Ill. Eng. Expt. Sta. Bull. 212, 1930. See also Harold A Rothbart (ed.), "Mechanical Design and Systems Handbook," McGraw-Hill Book Company, New York, 1964, and M.F. Spotts, "Mechanical Design Analysis," pp. 166-171, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964) have calculated the stresses for varying depths below the contact surface. Referring to Figure [2-33], we define a three-dimensional right-handed coordinate system with the origin in the center of the contact area and the z coordinate in the direction of the contact force and defining the depth of any stress element below the contact surface. Then, according to Thomas and Hoersch, the stresses in the x, y, and z directions are principal stresses. We first define a factor C by the equation
C = 4*a/pi * (1/d1+1/d2)/
((1-mu1^2)/E1+(1-mu2^2)/E2) [2-84]
Then, for any stress element on the z axis, the stresses are
sig_x = sig_y = C*((1+mu)*(z/a*acot(z/a-1)+
(1/2*a^2/(a^2+z^2))) [2-85]
sig_z = -C*(a^2/(a^2+z^2)) [2-86]
The equations are valid for either sphere, but in equation
[2-85], the value of Poisson's ratio used must correspond
with the sphere under consideration. Note that the
equations would be even more complicated if the stress state at
points off the z axis were to be determined because the x and
y coordinates would also have to be included.
Mohr's circles for the stress state described by equations [2-85] and [2-86] are a point and two coincident circles. Since sig_x = sig_y, tau_xy = 0, and
tau_xz = tau_yz = (sig_x - sig_z)/2
= (sig_y - sig_z)/2 [2-87]
Figure [2-34] is a plot of equations [2-85] to [2-87] for
a distance of 3*a below the surface. Note that tau reaches a
maximum value slightly below the surface. It is the opinion
of many authorities that this maximum shear stress is
responsible for the surface fatigue failure of contacting
elements. The explanation is that a crack originates at
the point of maximum shear stress below the surface and
progresses to the surface and that the pressure of the
lubricant flowing into the crack wedges the chip loose.
When the contacting surfaces are cylindrical, the area of contact is a narrow rectangle of half-width b given by the equation
b = SQRT(2*F/(pi*l)*((1-mu1^2)/E1+
(1-mu2^2)/E2)/(1/d1+1/d2)) [2-88]
where l is the length of the contact area, and the remaining
quantities have the same meaning as before. The pressure
has an elliptical distribution across the width 2*b,
and the maximum pressure is
p_max = - 2*F/(pi*b*l) [2-89]
Equations [2-88] and [2-89] apply to a cylinder and a plane
surface by making d = infinity for the plane surface. They
also apply to the contact of a cylinder and an internal
cylindrical surface; in this case d is made negative.
To picture the stress state, select the origin of a reference system at the center of the contact area with x parallel to the cylindrical axes, y perpendicular to the plane formed by the two cylinder axes, and z in the plane of the contact force. Then, for stress elements on the z axis, three principal stresses, sig_x, sig_y, and sig_z exist, all different. Figure [2-35] is a plot of these stresses for depths to 3*b below the contact surface.
Three different shear stresses exist; the largest of these is
tau_yz = (sig_y - sig_z) / 2 [2-90]
This component, labeled tau_max, is also plotted in
Figure [2-35], and it is seen that it reaches a maximum
value slightly below the surface just as in the case of
contacting spheres.