| This module calculates stresses caused by contact between two isotropic elastic bodies such as a sphere on a plane, a sphere on a sphere, a cylinder on a cylinder, a cylinder on a plane, and similar curved surfaces.
Application of a load over a small area of contact results in unusually high stresses. Situations of this nature are found on a microscopic scale whenever force is transmitted through bodies in contact. There are important practical cases when the geometry of the contacting bodies results in large stresses, disregarding the stresses associated with the asperities found on any nominally smooth surface. The Hertz problem relates to the stresses owing to the contact of a sphere on a plane, a sphere on a sphere, a cylinder on a cylinder, a cylinder on a plane, and similar curved surfaces. The practical implications with respect to ball and roller bearings, locomotive wheels, valve tappets, and numerous machine components are apparent.
Consider the contact without deformation of two bodies having curved surfaces in the vicinity of the point of contact. If now a collinear pair of forces are applied to press the bodies together, deformation will occur, and the point of contact will be replaced by a small area of contact. The first steps taken toward the solution of this problem are the determination of the size and shape of the contact area as well as the distribution of normal pressure acting on the area. The stresses and deformations resulting from the interfacial pressure are then evaluated.
The following assumptions are generally made in the solution of the contact problem:
1) The contacting bodies are isotropic and elastic.
2) The contact areas are essentially flat and small relative to the radii of curvature of the undeformed bodies in the vicinity of the interface.
3) The contacting bodies are perfectly smooth, and therefore only normal pressures need to be taken into account.
The foregoing set of assumptions enables an elastic analysis to be conducted. It is important to note that, in all the solutions presented below, the contact pressure varies from zero at the side of the contact area to a maximum value, P, at its center. |
| The contact problem is defined by first selecting the type of surfaces to be analyzed from the pull down menu. There are currently nine various surfaces to select from. When a selection is made some of the text boxes will turn gray in color and text can not be entered into those fields (uneditable). This is because the various solutions require various input parameters.
1) Two Spherical Surfaces
The contact area is a small circle of radius a. The compressive force, F, causing the contact pressure acts in the direction of the normal axis, perpendicular to the tangent plane passing through the contact area. The data required for this problem is r1, r2, E1, E2, Nu1, Nu2, and F. NOTE: The 1 and 2 subscripts correspond to the two surfaces.
2) Spherical Surface and a Plane
This is a special case of the two spherical surfaces with r2 (the plane) approximately infinite. Therefore, the data required for this problem is r1, E1, E2, Nu1, Nu2, and F. E2 and Nu2 correspond to the plane.
3) Sphere in a Socket
This is a special case of a sphere, r1, in a spherical socket, r2. The data required for this problem is r1, r2, E1, E2, Nu1, Nu2, and F.
4) Two Cylinders
This corresponds to two cylindrical rollers in parallel. If the cylinders are not parallel then the Two Curved Surface or Perpendicular Cylinders solutions must be used. The contact area is a narrow rectangle with a width of 2b and length L, where b is the contact width and L is the length the cylinders are in contact. The data required for this problem is r1, r2, L, E1, E2, Nu1, Nu2, and F.
5) Cylinder and a Plane
This is a special case of the two cylindrical rollers in parallel with r2 (the plane) approximately infinite. Therefore, the data required for this problem is r1, L, E1, E2, Nu1, Nu2, and F. E2 and Nu2 correspond to the plane.
6) Cylinder in a Socket
This is a special case of a cylinder, r1, in a cylindrical socket, r2. The data required for this problem is r1, r2, L, E1, E2, Nu1, Nu2, and F.
7) Perpendicular Cylinders
This is a special case of the two cylinders with the axes of the members being mutually perpendicular. The data required for this problem is r1, r2, E1, E2, Nu1, Nu2, and F. The contact area is a small ellipse with a major semiaxis a1 and a minor semiaxis a2.
8) Two Curved Surfaces of Different Radii
Consider two rigid bodies compressed by a force F. The load lies along the axis passing through the centers of the bodies and through the point of contact and is perpendicular to the plane tangent to both bodies at the point of contact. The minimum and maximum radii of curvature of the surface of the upper body are r1 and r1'; those of the lower body are r2 and r2' at the point of contact. Thus, 1/r1, 1/r1', 1/r2, and 1/r2' are the principal curvatures. The sign convention of the curvature is such that it is positive if the corresponding center of curvature is inside the body. If the center of curvature is outside the body, the curvature is negative. The required theta is the angle between the normal planes in which radii r1 and r2 lie. The contact area is a small ellipse with a major semiaxis a1 and a minor semiaxis a2. Many problems of practical importance can be treated with this solution such as the contact stresses in ball bearings or cam and push-rod mechanisms. The data required for this problem is r1, r1', r2, r2', E1, E2, Nu1, Nu2, theta, and F.
9) Curved Surface and a Plane
This is a special case of the two curved surfaces with r2 and r2' (the plane) approximately infinite. Therefore, the data required for this problem is r1, r1', E1, E2, Nu1, Nu2, and F.
The moduli of elasticity, E1 and E2, and the Poisson's ratios, Nu1 and Nu2, of the materials can be downloaded from the material database of the ETB. To open the materials database and view the properties of reference materials select the MATERIAL PROPERTIES button. Select the appropriate material and the desired units followed selecting the STORE button. A new window will open that displays the relevant material properties, select which material these values are for (Material 1 corresponds to E1 and Nu1). Select the INSERT button and the values are inserted into the appropriate corresponding text fields of the contact stress and area module.
Must be consistent with the units to get correct answers. For example, if the radii and length are entered in m, the force in N, the moment of inertia entered in m4, and the modulus of elasticity in Pa, then the the maximum contact stress will be displayed in Pa, the contact area in m2, and all the other dimensions in m.
Figure 1. Module input form |
| Variables:
E1 = Elastic Modulus, Modulus of Elasticity, Young's Modulus for Surface 1
E1 = Elastic Modulus, Modulus of Elasticity, Young's Modulus for Surface 1
Nu1 = Poisson's Ratio for Surface 1
Nu2 = Poisson's Ratio for Surface 2
L = Cylinder Length
r1 = Minimum Radius for Surface 1
r1' = Maximum Radius for Surface 1
r2 = Minimum Radius for Surface 2
r2' = Maximum Radius for Surface 2
theta = angle between then normal planes in which radii r1 and r2 lie
a = Contact Radius
d = Relative Displacement of the Centers
b = Contact Width
a1 = Contact Major Axis
a2 = Contact Minor Axis
A = Contact Cross-Sectional Area
P = Maximum Contact Stress
PI = Constant ~ 3.141592654 (ratio of a circle's circumference to its diameter)
Ce, Kd, alpha, beta, lambda = Constants
Ce = (1 - Nu1 ^ 2) / E1 + (1 - Nu2 ^ 2) / E2
Sphere on a Sphere:
Kd = (2 * r1 * 2 * r2) / ( (2 * r1) + (2 * r2) )
a = 0.721 * ( F * Kd * Ce ) ^ (1 / 3)
P = (1.5 * F) / (PI * a^2)
A = PI * a ^ 2
d = 1.040 * ( (F ^ 2 * Ce ^ 2) / Kd ) ^ (1 / 3)
Sphere on a Plane:
Kd = 2 * r1
Sphere in a Cylindrical Socket:
Kd = (2 * r1 * 2 * r2) / ( (2 * r2) - (2 * r1) )
Two Parallel Cylindrical Rollers:
Kd = (2 * r1 * 2 * r2) / ( (2 * r1) + (2 * r2) )
b = 0.80 * ( (F / L) * Kd * Ce) ^ 0.5
P = 0.798 * ( (F / L) / (Kd * Ce) ) ^ 0.5
A = 2.0 * b * L
Cylinder Surface on a Plane:
Kd = 2 * r1
Cylinder in a Cylindrical Socket:
Kd = (2 * r1 * 2 * r2) / ( (2 * r2) - (2 * r1) )
Perpendicular Cylinders:
Kd = (2 * r1 * 2 * r2) / ( (2 * r2) + (2 * r1) )
a1 = alpha * ( F * Kd * Ce ) ^ (1 / 3)
a2 = beta * ( F * Kd * Ce ) ^ (1 / 3)
P = (1.5 * F) / (PI * a1 * a2)
A = PI * a1 * a2
d = lambda * ( (F ^ 2 * Ce ^ 2) / Kd ) ^ (1 / 3)
alpha, beta, and lambda depend on r1 / r2
Two General Curved Surfaces of Different Radii:
Kd = 1.5 / ( 1.0/r1 + 1.0/r1' + 1.0/r2 + 1.0/r2')
a1 = alpha * ( F * Kd * Ce ) ^ (1 / 3)
a2 = beta * ( F * Kd * Ce ) ^ (1 / 3)
P = (1.5 * F) / (PI * a1 * a2)
A = PI * a1 * a2
d = lambda * ( (F ^ 2 * Ce ^ 2) / Kd ) ^ (1 / 3)
alpha, beta, lambda depend on r1, r1', r2, r2', and theta
General Curved Surface and a Plane:
Kd = 1.5 / ( 1.0/r1 + 1.0/r1' )
a1 = alpha * ( F * Kd * Ce ) ^ (1 / 3)
a2 = beta * ( F * Kd * Ce ) ^ (1 / 3)
P = (1.5 * F) / (PI * a1 * a2)
A = PI * a1 * a2
d = lambda * ( (F ^ 2 * Ce ^ 2) / Kd ) ^ (1 / 3) |
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| Avallone, E.A., and Baumeister III, T., editors (1987) Mark's Standard Handbook for Mechanical Engineers - 9th Edition, McGraw-Hill (New York). |
| Boresi, A.P., Schmidt, R.J., and Sidebottom, O.M. (1993) Advanced Mechanics of Materials - 5th Edition, Wiley (New York). |
| Faupel, J.H., and Fisher, F.E. (1981) Engineering Design - 2nd Edition, Wiley (New York). |
| Flugge, W. editor (1968) Handbook of Engineering Mechanics, McGraw-Hill (New York). |
| Timoshenko, S.P., and Goodier, J.N. (1970) Theory of Elasticity - 3rd Edition, McGraw-Hill (New York). |
| Ugural, A., Fenster, S. (1995) Advanced Strength and Applied Elasticity - 3rd Edition, Prentice Hall (New Jersey). |
| Young, W.C. (1989) Roark's Formulas for Stress and Strain - 6th Edition, McGraw-Hill (New York). |
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